Weyl and the Problem of Space: from science to philosophy 9783030115265, 3030115267

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Studies in History and Philosophy of Science 49

Julien Bernard Carlos Lobo Editors

Weyl and the Problem of Space From Science to Philosophy

Studies in History and Philosophy of Science Volume 49

Series Editor Stephen Gaukroger, University of Sydney, Australia Advisory Board Rachel Ankeny, University of Adelaide, Australia Peter Anstey, University of Sydney, Sydney, Australia Steven French, University of Leeds, UK Ofer Gal, University of Sydney, Australia Clemency Montelle, University of Canterbury, New Zealand Nicholas Rasmussen, University of New South Wales, Australia John Schuster, University of Sydney/Campion College, Australia Koen Vermeir, Centre National de la Recherche Scientifique, Paris, France Richard Yeo, Griffith University, Australia

More information about this series at http://www.springer.com/series/5671

Julien Bernard • Carlos Lobo Editors

Weyl and the Problem of Space From Science to Philosophy

123

Editors Julien Bernard Assistant Professor in Philosophy Centre Gilles Gaston Granger (CGGG) UMR 7304, Aix-Marseille-University Marseille, France

Carlos Lobo Collège International de Philosophie Paris, France Centro de Filosofia das Ciencias Universidade de Lisboa Lisboa, Portugal

ISSN 0929-6425 ISSN 2215-1958 (electronic) Studies in History and Philosophy of Science ISBN 978-3-030-11526-5 ISBN 978-3-030-11527-2 (eBook) https://doi.org/10.1007/978-3-030-11527-2 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Introduction: Structure and Philosophical Foundations of Hermann Weyl’s Work on Space

History of This Joint Scientific Venture This collection gathers various specialists of Hermann Weyl to evaluate the contribution of the German mathematician and philosopher to the reflection on space in the twentieth century. But following the interpenetration of philosophy and mathematics constantly advocated by Weyl, the title should not be interpreted in a narrow technical sense. The expression “problem of space” in Weyl’s works sometimes means, according to a tradition opened by Riemann, the search for arguments to justify that the infinitesimal metric to be adopted, in mathematics as in physics, is a quadratic differential form. Weyl dealt with the “problem of space” (Raumproblem) in this narrow sense in his series of lectures in Barcelona and Madrid1 and in related articles. This problem concerns us, but the scope of our work far exceeds it. Indeed, as Weyl himself explains in the introduction to his Mathematische Analyse des Raumproblems, the problem of space in this technical sense is only one part of a much broader reflection. To understand properly the status of space, and the way we can know it, one must synthetize all its faces: mathematical, physical, phenomenological, and psychophysiological. Space does not refer only to a set of structures (affine, projective, conformal, or metrical) but also, just as much, to the simple idea of a homogeneous field of points, which presupposes, in order to become clearer, that we take care of the problem of the continuum, in its intuitive as much as mathematical aspects. Therefore, in order to carry out a work on Weyl’s problem of space in this broader sense, we had to gather specialists from numerous disciplines: historians of

1 H.

Weyl, Mathematische Analyse des Raumproblems, Berlin: Julius Springer, 1923. Julien Bernard presented during the Constance workshop the new French-German critical edition of the text: H. Weyl, L’Analyse mathématique du problème de l’espace, introductions, notes, and translation by E. Audureau and J. Bernard, Presses Universitaires de Provence, Nov. 2015, with the help of the French typescripts from Barcelona, ISBN:979-1-03200-010-6, two volumes. The pagination is the same as in the original German edition. v

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science, physicists, philosophers, phenomenologists, and mathematicians of course. After all, did not Weyl himself wear these different hats, often within the same monograph, to embrace the problem of space in all its extent? In fact, the few works that manage to capture Weyl’s thought faithfully adopt this same interdisciplinary perspective.2 In order to gather a research group with common language, problems, and references, we organized a workshop “Weyl and the Problem of Space” at the University of Constance, from the 27th to the 29th of May 2015.3 After three days of debate, we were surprised to create such fruitful scientific exchanges, despite the disciplinary barriers. Undoubtedly, Weyl’s personality and his resistance to any confinement of thought were a sure guide for establishing dialogue. The chapters in this book come in part from the reflections and exchanges made during these three intense days. However, we have expanded the group of contributors by receiving chapters by Pierre Kerszberg, Luciano Boi, and Antoni Roca-Rosell. Our colleagues, Christophe Eckes, Philippe Nabonnand, and Thomas Ryckman, present during the workshop, were not able to participate in the collective work, for reasons independent of the project. We thank them with all the other contributors for their active participation in the debates that made this book possible.

Weyl and the Problem of Space The notion of space has never been far from the concerns of Hermann Weyl, as a mathematician. However, it is the stimulation brought by the emergence of general relativity, which led him to put the concept of space at the heart of his scientific and philosophical concerns, from 1916 onward. Thanks to Weyl’s texts that are indissolubly historical and philosophical,4 and thanks to his autobiographical remarks,5 we can roughly reconstruct the tradition of the problems in which his thought of space fits. The philosophical problem of space gathers a set of questions of Kantian tradition, themselves inherited from the debates that accompanied the

2 We

are thinking particularly of Hermann Weyl’s Raum-Zeit-Materie and a General Introduction to His Scientific Work, edited by Erhard Scholz, Birkhäuser: Basel, 2001; but also of Hermann Weyl and the Philosophy of the New Physics, special issue of Studies in History and Philosophy of Modern Physics, S. de Bianchi and G. Catren Eds, 2017. 3 Our partners were the Zukunftskolleg (University of Constance), the Centre G.G. Granger UMR 7304 (Aix-en-Provence), the CFCUL (Center for Philosophy of Science, University of Lisbon), l’Institut Français de Münich, the CiPh (Collège International de Philosophie), the Ecole Normale Supérieure (Paris), and the department of philosophy of the University of Constance. 4 Without attempting to be exhaustive, one can refer to the introduction of Space-Time-Matter, to the first conference of Mathematische Analyse des Raumproblems, and to Philosophy of Mathematics and Natural Science. 5 “Erkenntnis und Besinnung (ein Lebensrückblick)”, Studia Philosophica, Jahrbuch der schweizerischen philosophischen Gesellschaft, 13, 1954.

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constitution of mathematical physics (Galileo, Descartes, Newton, Leibniz, etc.). According to this tradition, space is an entity with a complex and ambiguous status, being at once a matter of intuition and of conceptual construction. As a form of intuition, space makes possible measurements and construction of the concepts of physics and consequently is at the basis of the very possibility of mathematical physics, at least for “external phenomena.” For, by contrast, although the psychical phenomena are submitted to the general nomological and mechanical form of nature, Kant considered that the constitution of a scientific psychology was impossible, because of the nonspatial nature of psychic facts.6 With this Kantian background, Weyl was receptive to phenomenology and recognized from the start that Husserl’s own transcendental aesthetics was much richer than Kant’s. Like Husserl he attributes an absolute character to consciousness as a form of experience and hence to subjectivity. Yet, while maintaining a continuity of method in the position of the philosophical problem, Weyl does not deprive himself, as he says, of “gathering nectar and pollen” from different philosophical flowers (Fichte, Leibniz, Bergson, Hobbes, Plato, Hume, Galileo, etc.), on the occasion of his readings. Thus, even if he seems to espouse for a time the genetic preoccupations of Husserlian phenomenology, running through the constitutive layers leading from the most subjective forms of spatial experience to the most exact objective forms of “real” space(-time), Weyl constantly mixes this kind of preoccupation with some “glimpses,” striking by their depth and their philosophical elegance, always based on mathematical and physico-mathematical constructions. If he does so, it is not that Weyl would take philosophy and his attempt at systematicity not seriously enough. Rather, as he explains, he must take charge of urgent problems that the science of his time has put forward, at a time when the theory of knowledge has not yet reached the stage of maturity sufficient to precede de facto, as it should de jure, science in its theoretical venture. In this delicate balance between science and philosophy, Weyl does not submit to any preconstructed system. For Weyl, space when properly analyzed reveals a succession of layers. The first important distinction is between a nonstructural layer of space and a structural layer, which is superimposed on the first and asks philosophical questions that are specific to it. In between7 we find a topological level, which consists in characterizing space as a continuous and homogeneous manifold of points. This stratification holds mutatis mutandis for the time continuum.

6 See the preface of Kant, Immanuel (1786): Metaphysische Anfangsgründe der Naturwissenschaft.

In: Immanuel Kant: Gesammelte Schriften, Bd. IV. Berlin: Walter de Gruyter 1968, pp. 465–566. 7 In several of his texts, Weyl seems to consider a “topological level” of space (or space-time) that is

not already structural. Cf. l’Analyse mathématique du problème de l’espace (see detailed reference within footnote 1), p. 2, and note 12 of the editors; and Philosophy of Mathematics and Natural Science, chap. 16, Dover edition, p. 130. Weyl was however very conscious about the possibility to axiomatize topology since, as early as 1913, in his monograph on Riemann surfaces, he proposed his own axiomatic of the notion of neighborhood, inspired by Brouwer’s one. Probably, when he speaks about topology as nonstructural, Weyl simply means that the shapes of topology are amorphous or undetermined structures.

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The nonstructural characterization of space is subjected to deep philosophical investigations. Weyl adopts a rather Kantian starting point, by differentiating between, on one hand, matter and its sense qualities, and on the other hand, space conceived as an a priori form of intuition. Here lies the “essence of space,” an outer object being able to exist for a consciousness, only by the conforming of sensation (the “matter” of intuition) to this form of intuition.8 When considered at this prestructural level, as a pure form, space can already fulfill in the purest way its function of principle of individuation.9 The mathematization of space begins with its characterization as a continuum, by shifting from the intuitive continuum, which is a phenomenological datum irreducible to a point-by-point construction, to the mathematical atomistic continuum, which serves as a support for mathematical analysis and hence physics. These are the questions that led Weyl to move away, at least temporarily, from his master Hilbert and to propose his own predicativist views on the continuum, before getting closer for a time to Brouwer’s position. In Das Kontinuum, by a typical interpenetration of philosophical and mathematical reflection, Weyl moves away from the superficial and confuse blending of empiricism and formalism that is typical of the naive philosophy of the common mathematicians,10 and he investigates the layering of mathematical constructions of the continuum, especially in their logical part. Weyl explicitly places these investigations under the patronage of Husserl phenomenological theory of knowledge, which displaces the traditional cut between intuition and concept.11 Weyl consequently presents his own essay from 1918 as a contribution: to a critical epistemological investigation into the relations between what is immediately (intuitively) given and the formal (mathematical) concepts through which we seek to construct the given in geometry and physics.12

8 See Weyl’s characterization of space, in Raum-Zeit-Materie, as a “form of our intuition” (eine Form unserer Anschauung) or “form of the appearances“ (Form der Erscheinungen). This Kantian (in a broad sense) starting point is repeated, for example, in the first page of Mathematische Analyse des Raumproblems and in Philosophy of Mathematics and Natural Science, p. 130–131 “B. The essence of space”, Dover edition, 1949: “Kant contradistinguishes it as form of intuition from ‘the matter of phenomena’, i.e. that which corresponds to sensation.” 9 See below, in particular the footnote 39. 10 See p. 2 of The Continuum, Dover edition, Stephen Pollard and Thomas Bole eds., Dover, New York, 1987. 11 By a kind of epistemological normalization, Weyl’s explicit reference to his Husserlian inspiration was skipped by some commentators and translators of Das Kontinuum. See S. Feferman “The Significance of Weyl’s Das Kontinuum,” in: Hendricks V.F., Pedersen S.A., Jørgensen K.F. (eds) Proof Theory: History and Philosophical Significance, Dordrecht, p. 174 passim) or Jean Largeault’s French tranlation, Le continu et autres écrits, Vrin, p. 49. While Weyl says indeed that “regarding the epistemology of logic, he subscribes to the views at the basis of the Logical Investigations” and the “Ideas for a pure phenomenology,” Largeault weakens obviously this central reference (op. p. 46). 12 Ibid., preface, p. 2.

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What about the structural layer of space, which contains its projective, conformal, and metric properties? Weyl asserts that, even if space is an a priori form, this does not disengage us from the need to describe and analyze the precise structures that are attached to it. According to Weyl, neither the structures of the space of perception nor the structures of the space of mathematical sciences can be legitimately considered as unanalyzable and only given through an immediate intuition. The analysis of the foundations of space begins at the aprioristic level. The structure of space must indeed be based on a priori judgments. Nevertheless, we are led to: a split of Kant’s judgments a priori into two directions. On the one hand, there are the non-empirical laws (Wesensgesetze), which express the manner in which data and strata of consciousness are founded upon each other, but do not claim to involve statements of fact; this line of pursuit culminated in Husserl’s phenomenology, in which the a priori is much richer than in the Kantian system. On the other hand, principles of theoretical construction are formulated, which according to the most extreme point of view (Poincare) rest on pure convention.13

Once the aprioristic (phenomenological, rational, conventional, or whatever their statuses) aspects of the structures of space have been clarified, it remains the complex task of articulating space, as an aprioristic form, with the empirical reality. The whole history of nineteenth-century geometry testifies that the field of possible geometrical structures is vast. It is therefore necessary for Weyl to investigate why, among the infinite multiplicity of conceivable abstract metrics, only one stands out as the structure of “real” space. This reference to a “reality” does not have the same meaning according to the scale (respectively, infinitesimal and finite).14 Weyl, however, opposes in each case to conventionalism, seeking discriminating reasons to justify the metric structure adopted. For Weyl, from 1916 onward, the epistemological emergency consists in univocally characterizing and epistemologically justifying the structure that is given by the dynamic metric of general relativity. The reading of Einstein’s memoir convinced Weyl that a new, deeper stage in our understanding of space had been achieved. Weyl then felt a certain urgency to criticize the ancient philosophical assumptions on space, to provide a better understanding of the foundations of Einstein’s work. Nevertheless, Weyl’s epistemological investigation is not directly characterized as a simple search for the principles of general relativity but rather as a more general philosophical and scientific inquiry on space, only partly guided by Einstein’s theory. For comparison, during the elaboration of his critical philosophy, Kant first admitted Euclidean geometry and the Newtonian physics as

13 Philosophy

of Mathematics and Natural Science, Dover edition, p. 134. reality of finite structures is physical and accessible to experience, whereas the infinitesimal structures have a transcendental status. See J. Bernard, l’idéalisme dans l’infinitésimal. Weyl et l’espace à l’époque de la relativité, Presses Universitaires de Provence, 2013. 14 The

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technically valid15 and then wondered what made them possible. Analogically, Weyl first accepted that Riemann’s mathematical developments and Einstein’s general relativity provided a new stage in the understanding of the foundations of the notion of space; only afterward did he develop a conceptual and epistemological theory in order to legitimate these new theses. However, neither Kant nor Weyl considered that their respective epistemologies were derived from or were based on the scientific theories they had to account for. This would have been a vicious circle, since the sought-after epistemological justifications are supposed to hold a priori. That is why Weyl and Kant thought that the respective scientific theories had been simple opportunities to reveal certain a priori epistemological elements. Weyl is peculiarly lucid and subtle when he thinks about the relationships between theory of knowledge, as aiming at a priori claims, and the factual development of positive science.16 Therefore, Weyl wonders: What did the new theory of Einstein teach us (or confirm us) as to the nature of space and to our way of knowing it scientifically? How can we justify epistemologically that the correct notion of metric is the one that was finally used by Einstein (after having been announced by Riemann)? The fact that it is not Einstein’s theory alone that interests Weyl, but more generally any physical theory with a dynamic metric field, is attested by two facts. Firstly, Weyl prefers to name Riemann rather than Einstein as a representative of the idea of dynamic metric. Secondly, Weyl tried to generalize Einstein’s theory by proposing his own “gauge” theories, including that of 1918, which unifies the gravitational and electromagnetic fields. Weyl’s generalization of the Riemannian metric consists in replacing the Levi-Civita connection by a connection that does not act only on the direction but also on the lengths of vectors. The invariance is then obtained through the introduction of a gauge function. Focusing on the dynamic metric, the justification of the structural layer of space will itself be subdivided into two major problems, with radically different epistemological statuses: 1. The first problem, according to the logical order, is to justify that the metric adopted for space-time is of “Pythagorean type,” as Weyl says, i.e., that it has the same properties, in the infinitesimal realm, as the (pseudo-)Euclidean metric. It is precisely the “problem of space,” in the technical sense given by Weyl.

15 Kant showed that Newton’s physics was partially derivable from the first metaphysical principles

of natural science although the position of an absolute space and time remained suspended by the very position of the “axiom” of relativity. This “first principle” (Grundsatz) states that “every movement, as an object of a possible experience, can arbitrarily be considered either as the movement of a body in a space at rest, or conversely, the body being at rest, as the movement of space in the opposite direction with an equal speed” (First Metaphysical Principles of the Science of Nature, see the reference to the German text in our footnote 6). 16 See Weyl’s texts quoted in Alain Michel, “La fonction de l’histoire dans la pensée mathématique et physique d’Hermann Weyl,” Kairos, Presses universitaires du Mirail, 2006, n◦ 27 “Monde de la vie et histoire”, p. 209–235.

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Nevertheless, according to the “split” of the a priori,17 the Euclideanity or Pythagoreanity of space(-time) can make sense at two different levels. Firstly, we can attempt to show that the “intuitive space” of our perception has a Euclidean structure, which can be analyzed either by phenomenological methods, as Oskar Becker tried to do,18 or by psychophysiological methods. Secondly, we can try to justify that the spatiotemporal metric that we should use to build physics on is of the Pythagorean type. This is precisely the problem tackled by Weyl in his Barcelona and Madrid lectures, which seems to be independent of an analysis of perception. 2. The second problem is to justify that the metric adopted for space-time, beyond the infinitesimal realm, is a metric whose curvature is everywhere intrinsically indeterminate; the determination is taking place only a posteriori, when geometry is articulated with physics (more precisely when the curvature is articulated to the distribution of physical matter). Weyl and Einstein, by acknowledging that the metric, at the finite scale, is dependent of empirical facts, vindicated a tradition in the epistemology of the metric, which started with Gauss and Riemann and was partly shared by Helmholtz.19 The metrical structure, at finite distance, is empirically emerging. Weyl remarks here that the aposterioristic foundations of the spatiotemporal metric are in turn split in two: Within the a posteriori one has thus to make yet another distinction, between what is necessitated by natural law and what even under their rule remains free and thus appears as contingent.20

For example, Einstein’s equations express a law according to which the curvature of the metric evolves in interaction with the matter distributed in space-time. Then the particular values of the Einsteinian metric, at a specific event of space-time, depend contingently of the distribution of the mass around that event and of the contingent past state of the metric field.

17 Cf.

Weyl’s text quoted above. Becker, the structure of space emerges on the basis of eidetic intuitions. See O. Becker, “Beiträge zur phänomenologischen Begründung der Geometrie und ihrer physikalischen Anwendung,” in Jahrbuch für Philosophie und phänomenologische Forschung (Vol. IV), 1923, and Weyl’s report on it in Philosophy of Mathematics and Natural Science, Dover edition, p. 137. 19 Concerning the interpretation of Helmholtz as an empiricist, cf. Weyl, Philosophy of Mathematics and Natural Sciences, Princeton University Press, 1949, p. 119; Philosophie des mathématiques et des sciences de la nature, fr. Tr. C. Lobo, p. 209). In contrast with Riemann, Helmholtz tried to legitimate the infinitesimal properties of the metric with the same epistemological means as for the finite properties. Moreover, it is not absolutely correct to consider Helmholtz as a mere empiricist. He wavers between empiricism and a kind of conventionalism. See Julien Bernard, “Riemann’s and Helmholtz-Lie’s Problems of Space, from Weyl’s relativistic perspective.”, section 3.6, in Studies in History and Philosophy of Modern Physics, special issue “Hermann Weyl and the Philosophy of the ‘New Physics”, S. Bianchi et G. Catren Eds. 20 Philosophy of Mathematics and Natural Science, Dover edition, p. 135. 18 For

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Overall View of Weyl’s Positions on the Two Structural Moments of the Problem of Space Concerning the first problem, Weyl relies on a philosophical a priori analysis to justify the selected essence (Wesen) of the metric structure.21 The selected essence, that of a “Pythagorean” metric, stands out among other choices, which can be expressed either in the context of Finsler’s metrics or in the context of transformation groups acting on the infinitesimal neighborhood of a point. The selection of the essence of the metric is made in two moments, which Weyl calls “analytic” and “synthetic a priori.” The two synthetic axioms posited by Weyl, namely, (1) the axiom of maximal freedom for the orientation of the metric and (2) the axiom of unique determination of the affine connection, seem to be requirements that the rational subject imposes on the metric, without any link with perceptive experiences. This is difficult to understand how Weyl intend to conciliate the nature of these two axioms with his Kantian starting point, according to which space would be a “form of appearances.” Apparently, the fact that Weyl’s two synthetic axioms are “synthetic a priori” means that (1) they are independent of any empirical knowledge, being posited prior to any empirical measurement and (2) they make empirical measurements possible. However, the “synthetic a priori,” as it appears in Weyl’s text, has apparently nothing to do with a form of sensibility, as in Kant. It seems indeed difficult to pretend that these axioms, being expressed thanks to highly elaborated mathematical notions (connection, gauge, properties of Lie groups, and algebras), could emerge from space as a form of our intuition. Whatever be the precise status of the synthetic axioms, they make sense in the framework of an epistemological analysis of the role of the embodied subject in the constitution of the objectivity of measurements. The objectivity is conquered 21 For

a list of Weyl’s works on the problem of space in its technical meaning, and an historical discussion, see the bibliography of Julien Bernard, “les tapuscrits barcelonais sur le problème de l’espace de Weyl,” in Revue d’Histoire des Mathématiques, 21 (2015), pp. 147–167. Secondary reading on this subject is abundant; see in particular: – Robert Coleman and Herbert Korté. “Hermann Weyl: Mathematician, Physicist, Philosopher, 4.11 The Laws of Motion and Mach’s Principle”, in: Hermann Weyl’s Raum-Zeit-Materie and a General Introduction to His Scientific Work (2001), E. Scholz Ed., II. 4.5–4.7, pp. 262–270 – Erhard Scholz, “Hermann Weyl’s Analysis of the ‘Problem of Space’ and the Origin of Gauge,” in: Structures. Science in Context (2004), pp. 165–197 – Detlef Laugwitz, “Über eine Vermutung von Hermann Weyl zum Raumproblem,” in: Archiv der Mathematik, 9 (1958), pp. 128–133 – Julien Bernard, “Becker-Blaschke Problem of Space,” in: Studies in History and Philosophy of Modern Physics 52, Part B (2015), pp. 251–266 – Julien Bernard, “Riemann’s and Helmholtz-Lie’s Problems of space.” Complete reference in footnote 18 – Hermann Weyl. L’analyse mathématique du problème de l’espace. Complete reference in footnote 1

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by the subject by eliminating progressively what makes the contingent singularity of his position. At the end of this “de-subjectivization,” a non-eliminable residue remains within the physical theory itself: the coordinate system. The “coordination” of those coordinate systems through groups of transformation is consequently the mathematical expression of the way the schematic and crystallized subjects (or point-like observers) reach the level of an intersubjective constitution. Concerning the second problem, Weyl’s thought is part of what we can call the third great period of the problem of space. In the first period, that of Kant, it was a question of giving its philosophical justification to a geometrical structure, that of Euclid, supposed to be the only one able to impose apodictically on empirical reality.22 In the second period, which opens with the discovery of Lobachevski geometry and elliptic geometry, it was from now on necessary to give selective arguments to determine the metric structure of space, among a plurality of possible structures.23 It is the period of the Riemann-Helmholtz-Lie’s problem of space.24 Finally, in the third period, which opens notably with certain remarks in Riemann’s memoir,25 it is no longer a matter of setting and justifying a single metric structure.

22 We

cannot enter here in the famous question, among Kantian scholars, whether Kant was aware or not of the possibility to have other geometries than Euclid’s. Weyl seems to have always supported the idea that Kantian transcendental aesthetics would be inseparable from Euclidean geometry. This is why we retain this position here. 23 To be exhaustive here, we should also consider the other pathway, that followed by Poincaré, which consists not of selecting a single metric but of giving its philosophical basis to a plurality of possible metrics, between which we are free to choose by convention. 24 In Part III of his habilitation text (1854), after having characterized the infinite family of metrics now called “Riemannian,” Riemann wonders what is the appropriate metric for “space.” According to Riemann, the empirical measurements, carried out up to his time, confirm that space is Euclidean, at least in first approximation at our scale. Riemann gives several possible characterizations of the Euclidean metric, among the Riemann spaces. In particular, Euclidean space shares with the two other spaces of constant curvature (elliptic and hyperbolic spaces) the property that any “rigid” figure can move and reach any point and can orient itself toward any direction. Helmholtz uses the same idea speaking about “free mobility.” Unlike Riemann, Helmholtz applies free mobility also to justify the infinitesimal structure of the metric. As is well known, Lie filled some holes in Helmholtz’s argumentation and demonstrations. See J. Merker’s edition of Lie’s texts and Julien Bernard “Riemann’s and Helmholtz-Lie’s problems of space from Weyl’s relativistic perspective,” complete reference in footnote 18. In Mind and Nature, Weyl proposes an interesting philosophical interpretation of the idea of “free mobility” (in a sense however different from Helmholtz’ precise notion). It is interpreted as the requirement for the transition from an “ego-centered” space to the homogeneous space, where the bodily ego takes on a position on equal terms with other bodies [ . . . ] This is accomplished by the possibility of our own body in space and by the intention of our will directed toward such motions. Not before this last step do I become capable of imagining myself as being in the position of another person, only this space may he thought as the same for different subjects, it is a medium necessary for constructing an intersubjective world. (p. 101) 25 In

a few sentences of his memoir, which fascinated Weyl (see the beginning of Space-TimeMatter §12, and Weyl’s endnotes in his own edition of Riemann’s text), Riemann conjectures that,

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Rather, it is a matter of determining a family of metrics with freely variable curvature and then of justifying that the selection of an individual metric, within this family, is only made a posteriori by a dynamic law that connects the metric to the physical content of space. However, the adoption of a metric with variable and dynamic curvature poses problems insofar as it comes into conflict with the idea of the homogeneity of space. Here’s how Weyl poses the problem: Space is a form of phenomena, and, by being so, is necessarily homogeneous. It would appear from this that out of the rich abundance of possible geometries included in Riemann’s conception only the three special cases mentioned come into consideration from the outset, and that all the others must be rejected without further examination as being of no account: parturiunt montes, nascetur ridiculus mus! Riemann held a different opinion, as is evidenced by the concluding remarks of his essay. Only now that Einstein has removed the scales from our eyes by the magic light of his theory of gravitation do we see what these words actually mean.26

The problem is thus the following. Riemann discovered an infinitely rich universe of metrics with variable curvatures (today, “Riemannian metrics”). However, it seems that “the mountain labours and brings forth a mouse,” because, when the challenge is to characterize space, only three among the Riemannian metrics are admissible, as they are the only ones to respect the requisite of homogeneity. More precisely, Weyl refers to the well-known result of Riemann (taken up by Helmholtz and Lie), according to which there are only three geometries (those with constant curvature) which respect a requirement of “free mobility.”27 Therefore we have a tension between, on the one hand, the infinitely rich universe of the Riemannian metrics that we would like to apply to geometry – in the strict sense of a science of the metrical properties of space – and, on the other hand, the requisite of homogeneity that violently constrains the acceptable metrics. As we are going to develop now, the requisite of homogeneity comes historically from both a mathematical tradition and a philosophical tradition. Concerning the mathematical tradition of homogeneous spaces, we think of the so-called “synthetic” geometries and the characterization of a geometry by a group of transformations, as in the works of Helmholtz and Lie and in Klein’s Erlangen Program. In this tradition, the use of a group of transformations that acts transitively is justified from within mathematics, by the fact that it unifies a large part of the geometric practices of the nineteenth century, including projective,

in a hypothetical future, physics would have reached a new stage in which the Euclidean metric observed on our scale would be considered as an approximation of a more complex physical reality, in which the metric would be in fact dynamic and of variable curvature, being correlated with the forces at play in matter. See p. 3 of J. Bernard, “Riemann’s and Helmholtz-Lie’s problems of space from Weyl’s relativistic perspective,” complete reference in footnote 18. 26 Space-Time-Matter, Henry Brose’s edition, p. 96–97. The issue of the acceptance of nonhomogeneous Riemannian metric in the physic of space-time will be dealt with more extensively in Scholz’ and Bernard’s chapters in the current volume. 27 See our footnote 23 and Scholz’ chapter in the current volume.

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affine, Euclidean, spherical, and Lobachevski geometries. This unification also has a heuristic value. The conflict between this geometrical tradition of homogeneous spaces and Riemann’s analytical metrics was already at the heart of Klein’s preoccupations. This also led Poincaré to express a suspicion on the general Riemannian metric when it is used to ground the notion of space.28 Similarly, at Weyl’s time, Elie Cartan considered the conflict between Riemann’s freely variable metrics and the Erlangen program viewpoint as the central problem of the modern mathematical notion of space. Indeed, from 1924,29 Cartan considers that we cannot properly ground differential geometry as long as we cannot reconcile the two great traditions coming, respectively, from Riemann and Klein. It is a question of giving back its central place to the notion of group, even in differential geometry where it apparently loses its legitimacy. So, Cartan writes: Which role does the notion of group play, or rather: should play, in this new domain of geometry? Can all the new geometries and an infinity of others fit into the framework of the Erlangen Program in a broad sense?30

For Cartan, the notion of group is absolutely primitive in differential geometry as in Klein geometry. Space, in differential geometry, consists of an infinite multiplicity of infinitesimal spaces, provided with the same Klein geometry. The problem is then to glue together these infinitesimal spaces to form a unified whole. In such a “gluing problem,” which allows global space not to be Klein-homogeneous, the notion of group can still play a preponderant role. Cartan’s solution uses the notion of holonomy group, which emerges in his works on “connection spaces.”31

28 In

la Science et l’Hypothèse, Chap. III, p. 66–68, Poincaré presents this problem through the tension between his paragraph “le théorème de Lie” and his paragraph “les géométries de Riemann”: [Riemann] constructs an infinite number of geometries [ . . . ]. All depends, he says, on the way the length of a curve is defined. Now, there is an infinite number of ways of defining this length, and each of them may be the starting-point of a new geometry. That is perfectly true, but most of these definitions are incompatible with the movement of a variable figure such as we assume to be possible in Lie’s theorem. These geometries of Riemann, so interesting on various grounds, can never be, therefore, purely analytical, and they cannot be subjects of proofs analogous to those of Euclid.

29 Concerning

this date and additional information on Cartan’s program to reconcile the Riemannian and Erlangen traditions, see also Renaud Chorlay, “Passer au global: le cas d’Elie Cartan, 1922-1930,” Revue d’histoire des mathématiques, 15 (2, 2009), p. 251-sq. See also Scholz, Erhard. 2012. H. Weyl’s and E. Cartan’s proposals for infinitesimal geometry in the early 1920s. Newsletter European Mathematical Society 84:22–30. 30 E. Cartan, “La théorie des groupes et les recherches récentes en géométrie différentielle,” Ens. Math., 24 (1925), p. 1–18 31 See notably P. Nabonnand, “La notion d’holonomie chez Élie Cartan,” Revue d’histoire des sciences, 62 (1, 2009), p. 221-245, and P. Nabonnand, “L’apparition de la notion d’espace généralisé dans les travaux d’Elie Cartan en 1922,” in Éléments pour une biographie de l’espace géométrique, Lise Bioesmat-Martagon (ed.), pp; 313–336, 2016. Nancy: PUN-Édulor.

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In many of his general texts of differential geometry in the 1920s, Weyl does not bother to reconstruct differential geometry from the recollection of such infinitesimal spaces. He often prefers to give directly the gμν of Riemannian geometry or the (gμν, ϕμ ) of Weylian geometry. But in the texts in which he deals with the foundations of differential geometry,32 Weyl’s point of view is very similar to Cartan’s. The groups of rotations (Drehungsgruppe), attached to the various points of the manifold,33 become the primitive elements of differential geometry. The quadratic differential form, in the “Pythagorean” case, is then just derived from the nature of the group involved. Even the role that the notion of group can play in thinking about the recollection of infinitesimal spaces is implicitly present in Weyl’s texts. In the §18 of the fourth edition of Space-Time-Matter, Weyl highlights the fact that the mappings of “transport” (in particular, Levi-Civita’s parallel transport), which make possible to connect the different infinitesimal spaces together, have a structure of pseudo-group. More precisely, the word “group” does not appear in Weyl’s text (a fortiori “pseudo-group”), but the structure itself is well highlighted.34 Even if Weyl’s and Cartan’s mathematical heritages differ, and even if the reference to the Erlangen program is important only to the second,35 these differences manifest themselves only from a common background, that is, the legacy of the two greatest geometrical movements since the middle of the nineteenth century: on the one hand, the development of differential geometry (from Gauss to Levi-Civita, including of course Riemann, Christoffel, and Ricci) and, on the other hand, the foundational role given to the notion of group in geometry (Helmholtz, Klein and Lie in particular). To sum up, Weyl, Klein, Cartan, and Poincaré considered the tension between the notion of group and differential geometry as one or even the central epistemological question on space opened by the nineteenth-century geometry. Weyl’s and Cartan’s solutions to resolve this tension do not consist in abandoning one of the two elements of the tension (homogeneity/differential geometry). Rather, they try each

32 See

Space-Time-Matter, 4th ed. §18, and above all Weyl’s articles about the problem of space (Raumproblem) in its technical sense. See our footnote 14. 33 Nabonnand, however, insists on a difference between Weyl and Cartan concerning the status of the manifold. For Weyl, the manifold is a primitive datum that serves as a support for infinitesimal structures, the latter being defined on the tangent spaces. According to Nabonnand, Cartan thinks, contrary to Weyl, that “the geometric space that circulates (through the connection) and the continuum are not necessarily geometrically linked.” See the article by Nabonnand quoted above and E. Scholz, “The problem of space in the light of relativity: the views of H. Weyl and E. Cartan,” Éléments pour une biographie de l’espace géométrique, L. Bioesmat-Martagon Ed., Nancy: PUNÉdulor, 2016, p. 255–312. 34 Cf. Space-Time-Matter, H. Brose’s edition, p. 138–142. In Mathematische Analyse des Raumproblems, p. 47, Weyl speaks explicitly of a structure of group (Gruppeneigenschaft). 35 In The Classical Groups: Their Invariants and Representations (Princeton mathematical series. Princeton University Press, 1939, chap I, §4), Weyl adopts explicitly the Erlangen Program perspective. Nevertheless, as Christophe Eckes remarked during a discussion, Weyl is (mainly) focussed on Klein-homogeneous geometries in this book. Therefore, the reference is not conclusive.

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in their own way to metamorphose the notion of group – and the correlative notion of homogeneity – to make them compatible with the point of view of differential geometry. The main difference between both authors concerns the way they legitimize the adoption of the requisite of homogeneity of space. Cartan refers mostly to the inner mathematical practice and to the Erlangen Program as program of unification of geometry. Weyl rather refers to a different and more philosophical tradition. Concerning the philosophical tradition of homogeneous space, Weyl relies on both Kant and Leibniz. As announced, it is rather by these philosophical references, than by the mathematical importance of the notion of group, that Weyl justifies the necessary homogeneity of the space. Nevertheless, quickly after these philosophical references, Weyl moves naturally from the adoption of a homogeneous space to the possibility of applying the techniques of group theory.36 The reference to the Kantian conception of space appears very early in Weyl’s writings, through the characterization of space as “form of appearances” (Form der Erscheinungen) or more rarely as “form of our intuition” (Form unserer Anschauung). The second conception, that of Leibniz, is put forward by Weyl from 1927, in relation with the critical reference to space as principium individuationis. In both cases, it is a matter of attributing to space a sort of ideality. Maybe, Weyl is not careful enough, about the philosophical difference between these two forms of ideality.37 But at any rate, the status of form of appearances, as well as that of principium individuationis, involves the homogeneity of space: [Space] is a form of phenomena. Precisely the same content, identically the same thing, still remaining what it is, can equally well be at some place in space other than that at which it is actually. The new portion of Space S then occupied by it is equal to that portion S which it actually occupied. S and S are said to be congruent. To every point P of S there corresponds one definite homologous point P of S which, after the above displacement to a new position, would be surrounded by exactly the same part of the given content as that which surrounds P originally[ . . . ].38

36 See

how Weyl shifts from the notion of homogeneity to that of congruence and congruence group, in Space-Time-Matter, p. 5-6, 11-15, or in Mathematische Analyse des Raumproblems, p. 44-49. See Scholz’ article in the present volume, for a survey of the different ways Weyl expressed the idea of homogeneity in the context of differential geometry. 37 In the paragraph quoted below, Weyl passes without further comment from Kant to Leibniz. The ideality that Leibniz claims for space, in the name of the principle of identity of indiscernibles, is not of the same kind as the transcendental ideality of space in Kant and represents an infinitesimal, in Leibniz’s sense, and metaphysical version of it, infinitesimal since any two narrow states of two monades delineate an infinite continuum of intermediary states, following the principle of continuity. The mathematical notion of infinitesimal proposed by Leibniz (PMNS, p. 45) is transferred to the metaphysical level (PMNS, p. 131). See also Weyl about the ideality of space in Leibniz, which is explicitly connected to the problem of the intersubjective underpinnings of relativity theory, in Mind and Nature, p. 116 and Philosophy of Mathematics and Natural Science, p. 119, and F. Balibar and C. Lobo’s introduction “La philosophie impliquée dans la science” to Weyl, Philosophie des mathematiques et des sciences de la nature, Métis Presses, p 56–59. 38 Space-Time-Matter, p. 11, and Raum-Zeit-Materie, 3rd ed., p. 10. See also the first page of the conferences in Barcelona and Madrid, where the opposition between form and matter takes a more

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When he later assumes that space is a principle of individuation, Weyl continues to conclude that it is homogeneous. He even proposes a connection between these two conceptions: Since the mere Here is nothing by itself that might differ from any other Here, space is the principium individuationis. It makes the existence of numerically different things possible which are equal in every respect. That is why Kant contradistinguishes it as the form of intuition from “the matter of phenomena, i.e. that which corresponds to sensation.” Here lies the root of the concepts of similarity and congruence. Leibniz infers from this the ideality of space and time; for they violate the principle of the identity of indiscernibles, which – along with Spinoza– he postulates as necessary in the domain of substances (namely as a consequence of the principle of sufficient reason).39

Now, since Weyl wants to think about space in the context of differential geometry, he must adapt the notion of homogeneity and that of group to the infinitesimal level. As we said, he accomplishes this program thanks to the groups of rotations (Drehungsgruppe), all isomorphic, which act within each tangent space of the manifold, and with the pseudo-group of connections (whether affine connections, as in Levi-Civita, or length connections, as in Weyl’s gauge theory of 1918). But Weyl preserves the ideality of space, so much so that it is subjected to a priori determinations. In this sense, Weyl, like the Neokantians or Husserl, wishes to safeguard a place for transcendental idealism. Husserl (see Ding und Raum) would doubtless not adhere to Weyl’s conservatism with regard to the notion of space as “forms of appearances.” It is rather for him a “form of the thinghood” (Dinglichkeit), and he seeks after 1909 to account for the homogeneity and tridimensionality of space through a transcendental monadology. But Husserl agrees with Weyl and neoKantians that space involves some a priori foundations. Weyl finds the correct place of the a priori determinations in the realm of infinitesimal metric relations. One has to move the boundaries between a priori and a posteriori and to overcome the Kantian dual schema space/matter, in favor of a ternary schema space/metric/matter. Space is a form based on an a priori essence. Matter (sensible qualities or physical properties) refers to its content, which is knowable only a posteriori. Finally, the metric is an in-between which includes, on the one hand, an a priori essence (the nature of the metric), linked to infinitesimal structures, and, on the other hand, a contingent variation of its orientation, from point to point, linked to finite structures, and determined a posteriori by its relationships with matter. On all this, the first pages of Mathematische Analyse des Raumproblems are decisive.

psychological turn, in the sense that matter designates the sensible content of perception. The opposition is then related to Kant. 39 Philosophy of Mathematics and Natural Science, Dover edition, p. 131. See also J. Bernard, Becker-Blaschke Problem of Space, p. 256 for the idea of principium individuationis in Weyl.

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Late Developments of Weyl’s Thought on Space This whole philosophical program on the notion of space, from the pre-structural to the structural level, from the a priori to the a posteriori, and from the subjective to the objective, is carried out by Weyl simultaneously and in dialogue with his scientific works on pure infinitesimal geometry and on the physical applications of his gauge theories. The subsequent works of Weyl, after 1927, are dedicated to the classification of Lie groups and the application of a group theoretical approach to quantum mechanics. New (hidden) dimension of the problem of space emerges in that perspective, another and deeper version of the principium individuationis related to Pauli’s exclusion principle and other symmetries related to the behavior of electron (right and left, past and future, positive and negative) requiring a mathematical and philosophical clarification. This new setting of the problem of space (and time), tackled here by some contributions, would motivate another full volume.

Plan of This Collection Weyl’s problem of space, in the broad sense we have just defined, is constituted by a system of issues on space, which would suffer from a division into separated parts. The chapters you will find in this book were not conceived according to a division into sub-themes or disciplines, imposed a priori. In accordance with Weyl’s unity of thought, each article ties together different moments of the problem of space, intertwining philosophical, historical, mathematical, and physical perspectives. Despite this, we have decided not to simply juxtapose the chapters in an arbitrary order (e.g. alphabetically) but to group them by thematic affinities. This a posteriori grouping is partly arbitrary, but it stimulates the important resonances between the chapters.

Weyl’s Intellectual Neighbourhoods and the Theory of Subjectivity The first articles aimed at contextualizing Weyl’s work on space within the history of ideas and the history of scientific institutions. Antoni Roca-Rosell’s chapter opens this series by studying Weyl’s stay in Barcelona and Madrid in the spring of 1922, which was so decisive for the history of the problem of space. This chapter gives a lively picture of the development of one of Weyl’s major texts, in the complex context of scientific relations between European countries, in the first half of the twentieth century.

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In Charles Alunni’s chapter, we go from history of institutions to an internal history of epistemological thought. Applying Gaston Bachelard’s epistemological categories, he analyses the “spectral” influence of Hermann Weyl on a “constellation” of “super-rational” thinkers, of which Bachelard is the herald. Weyl’s thought is used here as a model for thinking about the influence on the twentieth century of a certain shift initiated by Riemann in the relationship between mathematics and physics. The importance of contact actions Nahwirkungen is admitted both in pure geometry and in physics, since Riemann. How did this acknowledgement induce a displacement of the epistemological boundaries between the a priori and the a posteriori and those between the axiomatic construction and the teaching of experience? This decisive moment in Weyl’s thought, located in his texts on infinitesimal geometry and on the programme of geometrical unification of field physics, haunts the thought of super-rationalists. It is an important topic to understand the articulation between the two moments of Weyl’s thought that we have distinguished above (pp. x–xi). Carlos Lobo proposes a modelling of the parallel paths of Husserl and Weyl and cross-references as an epistemological crossover. They both share a common philosophical and mathematical heritage, among other things, Riemannian and Kantian. For both, substantial parts of the problem of space found a solution with the recent development of mathematics and physics. For both, however, a philosophical residuum remains, that of the relation of space as a form of intuition to the real world as it appears and is posited through ordinary experience and intersubjective communication. But, starting from a Kantian conception of space as an a priori subjective form, Weyl repeatedly adapts this idea to the recent development of geometry and physics until he converted it in a form of radicalization of Riemann’s view, infinitesimal geometry. Conversely the starting point of Husserl’s investigation on space, from 1892 onwards, is motivated by Riemann’s break-up with former conceptions (including Gauss’s), a revolution that Husserl will “thematize”, after his phenomenological breakthrough, as a historical example of formalization, unclosing the larger field of formal conceptions of “abstract spaces” as manifolds provided with additional structures (field, group, etc.), while the a priori form of space (and time) appears as a multilayered and complex formation, clearly noticed and praised by Weyl, as an enrichment of Kant’s transcendental aesthetics. Norman Sieroka’s chapter which tackles the same problem, that of intersubjectivity, has two objectives: (1) to ask about Weyl’s place within a constellation of authors who are either his contemporaries (Medicus, Husserl) or important prior idealistic philosophers (Fichte, Kant) and (2) to study certain structuring analogies, which build bridges between different parts (mathematics, physics, theory of subjectivity) of Weyl’s work. By a fortunate tour de force, Sieroka succeeds in making the two objectives correspond, through a detailed analysis of the different uses by Weyl of the notion of neighbourhood Umgebung. Sieroka uses this notion at two different levels. Firstly, it sheds light on the mutual relationships between Weyl’s positions on the continuous and the discrete, on intersubjectivity and the individual, and on law and freedom. Secondly, Sieroka characterizes the notion of individual as a mere “limit idea”, which is abstracted from a web of

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neighbourhood relations. Then, he applies this to the individual Weyl himself. His intellectual trajectory becomes clearer by taking into account all his intellectual neighbourhoods, both in Zurich and Göttingen. By this double use of the notion of neighbourhood, Sieroka’s chapter provides a natural transition between the first series of chapters, about the institutional and intellectual contextualization, and the second series, which is about Weyl’s analysis of the nonstructural notion of space and of the problem of the continuum.

Weyl’s Theory of the Continuum: Intuitionism and Dimensionality of Space Mark van Atten’s chapter starts with a remarkable observation: Weyl has never completely denied his theoretical attachment to the intuitionist position – whether that of Brouwer or his own – concerning the foundations of mathematical analysis and of the continuum. He departed from it in practice, only because he failed to find in the intuitionistic mathematics of his time sufficient resources for the needs of physics. Therefore, van Atten seeks, in the intuitionist mathematics of the second half of the twentieth century, possible developments that would have done justice to Weyl’s intuitionist position on the continuum. He finds them in the intuitionistic developments proposed by the young Brouwer, Vesley, and Reeb of infinitesimals and non-standard analysis. Dominique Pradelle’s chapter takes as a framework the evolutions of Weyl’s thought concerning the mathematical definition of the continuum and the foundations of analysis. However, it is not the mathematical problem of the continuum that mainly motivates Pradelle’s chapter. This problem is rather considered as the most accurate place where one can pose the methodological questions linked to a search for a phenomenological philosophy of mathematics. Should the construction of the mathematical continuum ultimately rest on the ground of intuition? What is the place of symbolic thought in the foundations of analysis? By positioning himself on these questions, Weyl defines what must be for him a philosophy of mathematics that is phenomenological rather than historical or logico-syntactical. We thus see the resurgence of problems that were already posed by Husserl, when he questioned the place of the intuitive and the symbolic or when he questioned the relationship between the historical and the eidetic. The last chapter in this part, that of Silvia de Bianchi, progressively shifts the problem of space from the prestructural layer of the continuum and mathematical analysis to the structural layer where the metric is defined. Indeed, de Bianchi studies Weyl’s answer to a typically topological problem: that of the dimension of space(-time). However, de Bianchi, just like Weyl in “Why Is the World Four-Dimensional?”, is led to build bridges between the two domains (prestructural/structural) of the problem of space. In the first place, the question of dimensionality brings themes, belonging to Weyl’s philosophy of space, which are

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also present at the metric level: the importance of symbolic thought, the question of the essence Wesen of space, and the criticism of Kant’s transcendental idealism. In the second place, de Bianchi explains that Weyl’s bold justification for the fourdimensionality of space-time takes into account not only the structural layers of the notion of space but even the fundamental physical theories. The argument is based on causality constraints, expressed in the context of Maxwell’s electromagnetic theory and Dirac’s equations for quantum electron theory.

From Aprioristic to Physical Foundations of the Metric The next four chapters focus on the structural layers of the notion of space. This is where we find in particular, but not only, the problem of space in its technical sense. In fact, this part opens with the chapter by Erhard Scholz which considers the history of the problem of space, in the strict sense of a justification for the infinitesimal metrical structures. Scholz develops this history both from the strictly internal point of view of the evolution of Weyl’s thought on the foundations of geometry and from the point of view of its comparison with Elie Cartan’s point of view and the possible physical applications of their two points of view on field theory. Luciano Boi sums up the leading idea of Weyl’s philosophy of space and his mathematical programme under the thesis: that physics is geometry in act (i.e., topology, differential geometry, and algebraic geometry). To substantiate this thesis, he provides an in-depth and updated account of Weyl’s contributions to quantum field theory, by examining two examples: the concept of gauge invariance and the two-component relativistic wave equation of the neutrino. Gauge invariance appeared as an extension of Weyl’s initial programme of infinitesimal geometry, where groups and their associated algebras play a key role by building a bridge between the physical concepts of gauge field theory and those of differential geometry and topology of fibre spaces. The second example, the unification part of the programme, stems from a similar mathematical insight. The concept of connection already central in Cartan’s and Weyl’s account of general relativity, looked at through the lenses of infinitesimal geometry, receives here an essential extension, since not only the direction but also the length of the vectors is dependent on the path. Invariance is obtained through the introduction of a (non-integrable) gauge function, which provides an intrinsic account of the evolution of the electron in space-time. Initially rejected, this idea revealed afterwards pioneering as testified by Yang-Mills gauge field theory for the quark. Alexander Afriat’s chapter also discusses the way in which Weyl constructs and justifies the metric relationships of space. However, as for Boi, the focus is not on the “nature of the metric” and its Pythagorean character but on rather the so-called gauge. It is a matter of thinking about the possibility that the length of vectors on a manifold can undergo infinitesimal transformations, so that length could no longer be transported in an integrable manner. Now, the two gauge theories proposed by Weyl are based on different “logics”, i.e. on justificatory principles

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that have different epistemological statuses. The structures of group which are involved in the freedom of the gauge are essential to reflect the symmetries of the physical objects supported by the fundamental theories (electromagnetism/YangMills theory). Afriat thus invites us to consider not only the history of Weyl’s gauge theories but more deeply the story of their “logics”. The chapter by Julien Bernard deals also with the foundations of the metric. But we are no longer in the infinitesimal realm but in the finite one. The focus is not anymore on the “problem of space” in the technical sense, nor on the gauge symmetries, but on the problem of the determination – complete or not – of the finite metric relationships of space-time by the distribution of matter. Bernard considers the complex history of the relations of Weyl’s thought on space with Mach’s principle and with the principle of homogeneity of space. The reconstruction of this story is possible, thanks to the analysis of a thought experiment of Weyl, which Bernard calls “the Plasticine Ball Argument” and which reappears in always changing forms within Weyl’s corpus, from 1918 to 1949. The modifications of this nomadic argument constitute a useful historical tool for grasping the philosophical and technical evolutions of Weyl’s thought about the relations between matter and metrics. The chapter by Francesca Biagioli allows a natural transition between this part of the collection and the last. Indeed, Biagioli contextualizes Weyl’s thought within the intense debates that animated geometers and philosophers at the turn of the nineteenth and twentieth centuries (Helmholtz, Klein, Schlick). The challenge concerns the relationship between intuition and concept to give foundations to geometry. This historical recontextualization of Weyl’s thought makes possible to highlight the specific role he reserves for the subject in science and to go further into the dialectic between intuition and concept in Weyl’s epistemology.

Weyl’s Methodological Issues: Intuition, Symbolic Thought and Manifolds of Possibilities Benoît Timmermans examines Weyl’s work on the representation of semi-simple complex Lie groups. He then tests the hypothesis that this important episode in Weyl’s mathematical career played a fundamental role in the development of what he calls “constructive knowledge” from 1926 onwards. It consists of “projections of symbols on the background of an ordered manifold of possibilities [ . . . ]”. This leads Timmermans to clarify how this moment in Weyl’s intellectual path led him to redefine classical couples of notions such as algebraic vs. transcendent, abstract vs. concrete, empirical vs. transcendental, a priori vs. a posteriori, and becoming vs. freedom. Jairo da Silva also questions the role of symbolic thought according to Weyl. However, unlike Timmermans, he does not take as a framework a wellcircumscribed mathematical theory of Weyl but instead starts with an overview of

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Weyl’s philosophy of space. Da Silva runs through certain layers of the notion of space, from the concrete egoic phenomenal space, and its qualitative content, to the mathematical and intersubjective space, which is notably the result of an abstraction and an idealization. Through the analysis of these layers, da Silva questions the place given by Weyl to of the symbolic thought. Weyl shares with Husserl the thesis that intuition is the original ground on which science must be built. On the other hand, according to da Silva, the two authors differ fundamentally concerning the manner the symbolic thought can complement the intuition. For Husserl, the symbolic is always auxiliary, in principle dispensable. For Weyl, the movement of science compels us ineluctably to leave the ground of original intuitions, to embark on the adventure of “cognition”, which is a mode of knowledge irreducibly symbolic, language being the vector of symbolic creativity. By this strong emphasis on the symbolic, this chapter complements the intuitionist point of view that we find, for example, in van Atten’s chapter. The methodological problem tackled by Pierre Kerszberg, in the last chapter, is of a great generality. Indeed, it deals with the relations between science and epistemology. On behalf of a scientist like Weyl, one would expect him to defend a dependence of epistemology on science. However, he insists (also) on the converse relation, i.e. on the “scientific implications of epistemology”. Kerszberg leads us to investigate this issue within the framework of Weyl’s critical interpretation of Husserl’s phenomenological idealism. What is the nature of primitive intuition in the world of science, more and more focused on the symbolic? Can intuition admit a mathematical representation? These questions are dealt with through Weyl’s reflection on the temporal flow of consciousness.

Contents

Part I Weyl’s Intellectual Neighborhoods and the Theory of Subjectivity 1

Internationalization of Scientific Activity in Spain in the Interwar Period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Antoni Roca-Rosell

2

Hermann Weyl chez Gaston Bachelard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Charles Alunni

3

Le résidu philosophique du problème de l’espace chez Weyl et Husserl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Carlos Lobo

4

Neighbourhoods and Intersubjectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Norman Sieroka

3 25

35 99

Part II Weyl’s Theory of the Continuum: Intuitionism and Dimensionnality of Space 5

Weyl and Intuitionistic Infinitesimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Mark van Atten

6

Entre phénoménologie et intuitionnisme: la définition du continu. . . . 161 Dominique Pradelle

7

From the Problem of Space to the Epistemology of Science: Hermann Weyl’s Reflection on the Dimensionality of the World . . . . . 189 Silvia De Bianchi

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Contents

Part III From Aprioristic to Physical Foundations of the Metric 8

The Changing Faces of the Problem of Space in the Work of Hermann Weyl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Erhard Scholz

9

H. Weyl’s Deep Insights into the Mathematical and Physical Worlds: His Important Contribution to the Philosophy of Space, Time and Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 Luciano Boi

10

Logic of Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 Alexander Afriat

11

The Plasticine Ball Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 Julien Bernard

12

Intuition and Conceptual Construction in Weyl’s Analysis of the Problem of Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 Francesca Biagioli

Part IV Weyl’s Methodological Issues: Intuition, Symbolic Thought and Manifolds of Possibilities 13

Espace et variété de possibilités chez Hermann Weyl . . . . . . . . . . . . . . . . . . 371 Benoît Timmermans

14

Husserl and Weyl on the Constitution of Space . . . . . . . . . . . . . . . . . . . . . . . . . 389 Jairo José da Silva

15

The Scientific Implications of Epistemology: Weyl and Husserl . . . . . . 403 Pierre Kerszberg

Part I

Weyl’s Intellectual Neighborhoods and the Theory of Subjectivity

Chapter 1

Internationalization of Scientific Activity in Spain in the Interwar Period The 1922 Course of Hermann Weyl in Barcelona and Madrid Antoni Roca-Rosell

In 1922, Hermann Weyl (1885–1955) was invited to teach a course of eight lessons in Barcelona and Madrid. At that time, Weyl was working on his theory of space in order to tackle several problems posed by the Theory of Relativity. The lessons he gave in Barcelona and Madrid constituted a very significant contribution and he published them in a book in 1923.1 We study the impact of the visit of Weyl in the Spanish scientific community. In this paper, we analyse the context in which this invitation to come to Spain was issued, the development of the course and some of its consequences for the emergent Spanish mathematical community. In fact, Weyl’s book contains a dedication full of praise to Esteve Terradas, the person who had invited him to lecture in Spain. Weyl expressed his gratitude to Terradas and his colleagues for the opportunity to develop his proposal on Space through the preparation of the classes he gave in Spain in March and April, 1922. He also stated that he was proud to collaborate in the promotion of science in Spain. Some years ago, the Institut d’Estudis Catalans undertook a project to publish a Catalan translation of Weyl’s 1923 book on the occasion of the celebration of the centenary of Esteve Terradas’ birth. For several reasons, this editorial project never came to fruition.2 For the purpose of this study, I consulted the archives of the Institut d’Estudis Catalans, mainly its “Fons Terradas”. Thanks to Manuel

1 Weyl

(1923); for the first French translation and comments, see Bernard (2015b).

2 We prepared a study on the course of Weyl in Spain: Roca Rosell (1987), accessible on the Internet

since 2014. The present paper forms part of the project HAR2016-75871-R. A. Roca-Rosell () Departament de Matemàtiques, Universitat Politècnica de Catalunya - Barcelona Tech, Barcelona, Spain e-mail: [email protected] © Springer Nature Switzerland AG 2019 J. Bernard, C. Lobo (eds.), Weyl and the Problem of Space, Studies in History and Philosophy of Science 49, https://doi.org/10.1007/978-3-030-11527-2_1

3

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García Doncel and Karl von Meÿenn, I was given access to the correspondence from Terradas belonging to the Weyl Archive at the ETH Library in Zürich. Von Meÿenn also helped me to translate the letters written in German.

1.1 Context: For the Regeneration of Spain In Spain, a serious attempt was made during the first third of the XX century to set up a modern system of scientific research. There are several studies that analyse this attempt, most of which are aimed at recovering the memory of scientific activity prior to the Spanish Civil War (1936–1939).3 In the last third of the nineteenth century, with the loss of the Spanish colonies in America (Cuba and Puerto Rico) and Asia (Philippines) in 1898, there arose a movement in Spain to regenerate the strength of the country. This was aptly called the “regeneration” and it was based on the promotion of science and technology.4 The creation in 1900 of a ministry devoted to state education (Instrucción Pública) is usually interpreted as a consequence of this desire for regeneration. The creation in 1907 of the Junta para Ampliación de Estudios e Investigaciones Científicas (JAE, Board for the Extension of Studies and Scientific Research) was of particular relevance for research. This board offered grants to teachers (from elementary school teachers to university lecturers) and researchers for stays abroad to develop their research or to improve their training. This scheme, which lasted until 1936, had a real impact on the world of education and scientific research.5 In addition, from 1909, the JAE set up institutes of research devoted to biology, chemistry and physics.6 In 1915, a centre of mathematical research was created, the Laboratorio Matemático (Laboratory for Mathematics), on the initiative of Julio Rey Pastor (1888–1962). Thomas Glick has stated that the real growth of science and technology in Spain in the early XX century was based in the creation of a civil discourse for science and technology; that is, an agreement to free them from the constraints of politicization.7 This movement of regeneration in Barcelona was characterized by its links to the official recognition of Catalan identity.8 The Catalanist political strategy consisted in participating in local elections for municipal and regional administrations (the Diputaciones, one for each province), and from 1901 the Catalan Nationalist parties

3 See,

among others, Glick (1986); Sánchez Ron (ed.) (1988a, b). for example, Suárez Cortina; Salavert Fabiani (ed.) (2007). 5 The main reference is Sánchez Ron (ed.) (1988b). There is an update in Sánchez Ron; GarcíaVelasco (ed.) (2010). 6 The first president of JAE was the biomedical researcher Santiago Ramon y Cajal (1852–1934), Nobel Prize winner in 1906. 7 Glick (1986) centered his analysis on the diffusion of Einstein’s idees in Spain within this concept of civil discourse. We have applied these concepts to the Catalan case in Roca-Rosell (2007a). 8 Roca-Rosell (2007a). 4 See,

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went on to win several elections. In 1907, Enric Prat de la Riba (1870–1917) became president of the Diputació of Barcelona and pushed for the unification of the four Catalan provinces with the 1914 constitution known as the Mancomunitat de Catalunya, a regional administration for Catalonia with some degree of autonomy. Also in 1907, Prat de la Riba created the Institut d’Estudis Catalans (IEC, Institute for Catalan Studies) in which a Science Section was set up in 1911.9 This Section was charged with the development of scientific research and technical services, such as the Geological and Geographical Services organised in 1915; later, in 1919, the Catalan Meteorological Service was established, and in 1920 the Institute for Physiology was created as a research centre. All these services or centres were supervised by the IEC Science Section, and in 1914 the Section participated in drawing up a series of courses organised by the Mancomunitat with the aim of providing higher educational studies for Catalans. It should be noticed that the University of Barcelona was not allowed to have doctoral studies.10 Created in 1914, the courses were called Cursos monogràfics d’alts estudis i d’intercanvi (Monographic courses of higher studies and for interchange) and were approved by the Mancomunitat Pedagogical Council. The initial objective was to invite foreign researchers and to facilitate the stay of Catalan researchers abroad, but these plans were seriously affected by the beginning of the World War I. In the first series of the courses, held in the spring of 1915, there were no foreign teachers,11 a situation that changed after 1918. Despite the initial plans, the Mancomunitat did not fund stays of Catalan researchers abroad; this function was fulfilled by the JAE.12 The Cursos monogràfics were proposed by the IEC Science Section and were supervised by several members belonging to the Section. Eugeni d’Ors (1881– 1954) was the organizer of the courses on philosophy, education and social sciences; August Pi i Sunyer (1879–1965) promoted the courses on biology and medicine; Esteve Terradas Illa (1883–1950) organized the courses on exact and physical sciences. Naturally, the course given by Hermann Weyl formed part of the series supervised by Terradas. The list of foreign scientists invited by Terradas during the period 1916–1923 is quite impressive: Béla Szilárd (1916); Tullio Levi-Civita, Jacques Hadamard (1921); Hermann Weyl, Arnold Sommerfeld (1922); Albert Einstein, and Béla Kerékjártó (1923).13 These lectures were complemented by lessons given by Spanish teachers (Rey Pastor, Julio Palacios, and Terradas himself on several occasions). As stated above, the objective of the courses was to provide local audiences (students, teachers, philosophers, engineers, architects, etc.) with the opportunity to learn about the new developments in European research in physical

9 Roca-Rosell;

Camarasa (2008, 2011). have signaled the first doctoral thesis in Barcelona in 1934; see Roca-Rosell (2016a, b). 11 An exception was a course given by Béla Szilard (1884–1926), a refugee who had settled in Barcelona. See Herran (2008). 12 Roca-Rosell (1988b). 13 For informations about these courses, see, for exemple, Roca-Rosell; Sánchez-Ron (1990). 10 I

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and exact sciences. Terradas was trying to set up a research group in Barcelona, composed of University teachers and young students who intended to pursue a career in research.14 Nevertheless, it should be pointed out that Spanish universities offered a very limited opportunity for the research; in fact, the JAE established some research centres that were independent of the university. These centres, most of them in Madrid, initiated professional research in Spain, since outside Madrid the conditions for research were virtually non-existent. As mentioned above, in Barcelona, the Institut d’Estudis Catalans set up some research centres (in general, applied research) that yielded relevant results despite the scarcity of their resources. After the Spanish Civil War (1936–1939), the Franco regime restored some research groups, but one of the impacts of the conflict had been a serious loss of human capital.15 It was not until the 1970’s and 1980’s that genuine science and technology research groups began to emerge in Spain.

1.2 Hermann Weyl in Spain 1.2.1 The Course in Barcelona There is a letter from Terradas to Weyl dated 30th March 192116 in which Terradas invited Weyl to lecture in Barcelona with a course of “8 lessons” on any subject on which Weyl was engaged in research at that time. Terradas outlined the courses on mathematics and physics on which Hadamard and Levi-Civita had already delivered lectures. He also mentioned that they were in the process of trying to invite Einstein. It was worth noticing that the structure of the course –eight lessons- was already defined in the invitation. Terradas asked Weyl to accept the invitation for a fee of 2000 pesetas. In addition, he would also undertake to repeat the course in Madrid for a similar amount of money. To give an idea of the potential audience, Terradas explained that he had lectured on the theory of relativity up to gauge theory. He added that he knew the papers published by Weyl, and his book Raum, Zeit, Materie, which appeared in 1918 (and ran to five editions until 1923). Some copies of the 1921 (fourth) edition can be found in libraries in Barcelona, while a copy of the 1923 edition belongs to Terradas’ own personal collection.17

14 Roca

(2016a, b). Garcia (1996). 16 Terradas to Weyl, Barcelona 30th March 1921, Hs. 91-761, ETH Library, Zürich. 17 Terradas’ personal collection of books is preserved at the Biblioteca de Catalunya, in Barcelona. See Soler Mòdena (1994). 15 López

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At the end of the letter, Terradas stated that Weyl’s lectures would be published, and as a model he sent him the course given by Julio Rey Pastor.18 He finished the letter with the comment that the Spanish audience was not familiar with the German language. Weyl’s reply to Terradas’ letter is unfortunately not available, but there is a letter from Rafael Campalans (1883–1933), the director of Public Instruction of the Mancomunitat de Catalunya Pedagogical Council, dated 26th April, thanking Weyl for his acceptance to participate in the Barcelona Courses.19 Terradas had already explained to Weyl the project of the Courses, to which outstanding lecturers had been invited. In his letter written in French, Campalans explained the technical details: to meet Weyl’s travel expenses and his stay in Barcelona, he confirmed the fee of 2000 pesetas, remarking that while it might seem a modest sum, he trusted Weyl’s commitment to science and his sympathy for Catalonia. Campalans also commented that there was a real possibility that Weyl could repeat the Course in Madrid, probably with the support of the JAE.20 He went on to reaffirm Terradas’ wish that the course should be aimed at a higher level for a select group of students and teachers. Nevertheless, he suggested that Weyl might lecture to a wider audience at the Institut d’Estudis Catalans or at the Barcelona Academy of Sciences and Arts. Weyl replied in a letter dated 13th May in which he sent a detailed programme of his course.21 He made no mention of the open lecture, and suggested that the beginning of March 1922 –“the end of the winter semester”- would be a good date for him. He told Campalans that he would prepare the lectures in French, under the title “Analyse mathématique du problème de l’espace”. He also sent the programme of his lessons, a German copy of which is preserved, which he probably included in the same letter of 13th May22 : Mathematische Analyse des Raumproblems. 8 Vorlesungen in französicher Sprache, von Hermann Weyl, Professor an der Eidgenössichen Technischen Hochschule Zürich. Die auf quadratischen Differentialform beruhende Riemannische Infinitesimalgeometrie. Befreiung von der Voraussetzung der Integrabilität der Streckenübertragung. Fundamentalsatz der InfinetisimalGeometrie. Die Projektive und der konforme Standpunkt. Die homogenen Räume. Physikalische Bedeutung des metrischen Feldes in der vierdimensionalen wirklichen Welt. Das von Helmholtz und Zie behandelte Raumproblem. Das neue, vom Standpunkt der Relativitätstheorie an seine Stelle tretende Raumproblem. Seine Erkenntnis Theoretische Bedeutung. Zurückführung auf einen Satz der Gruppentheorie und Skizzierung des mathematischen Beweises.

18 Rey

Pastor [1916]. to Weyl, Barcelona 26 April 1921, copy, Folder (Lligall) 3372, exp. 2, Arxiu de la Diputació de Barcelona (ADB). 20 Finally, the JAE was not involved. The course was eventually supported by the University of Madrid. 21 Weyl to Campalans, Zürich 13th May 1921, Folder 3372, exp. 2, ADB. 22 Folder 3732, ADB. The programme was not found with the letter, but next to the Catalan and Spanish translations used for the diffusion of the course at the beginning of 1922. 19 Campalans

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It seems that Weyl already had a clear idea of how to develop his theory on space. However, the lectures in Spain led him to make some changes in the eighth lesson.23 Campalans wrote to Weyl on 21st July informing him that he was unable to confirm the course in Madrid, “quoi que j’ai les impressions les plus favorables”, i.e., he thought that he would eventually manage to organise it. He also told Weyl that he found the programme of the course to be of great interest, a remark that was probably made as a gesture of courtesy by Campalans, but one which certainly reflected Terradas’ own opinion, since he was very well informed on these subjects. Weyl wrote to Campalans on 3rd January 1922 to inform him that he had received the printed programme and that everything was in order.24 Weyl was planning to arrive to Barcelona several days before the beginning of his course and hoped they would have the opportunity to meet. On this same sheet we find a manuscript annotation dated 23rd February, 1922, probably made by Terradas, saying that they offered to find Weyl accommodation at a hotel. A note from Campalans to Terradas dated 14th January also exists in which he asks if it had been possible to organise the repetition of the Weyl’s course in Madrid.25 In the same letter, Campalans stated that he was working on the arrangements for Einstein’s visit. In a letter dated the 17th January, Terradas welcomed Weyl to Spain shortly after his arrival in Málaga, in the South of Spain.26 This letter is written in Spanish, since in the meantime Weyl had been learning the language, and Terradas complemented him for “taking the trouble and courtesy to learn it fluently”.27 In fact, Weyl’s wife, the philosopher Helene (“Hella”) Weyl (born Helene Joseph: 1893–1948), became an expert in Spanish literature, and it seems that she took advantage of this 1922 trip to Spain to visit places with literary and historical associations (the Alhambra, Granada; Toledo), as well as several people. She later went on to translate some of the works by the philosopher José Ortega y Gasset (1883–1955) into both German and English.28 In a further letter dated 17th January, 6 weeks before the beginning of the course, Terradas apologized to Weyl for the fact that in Spain he would not be able to introduce him to anyone with a similar status in the field of mathematics. Terradas stated that there were many reasons, difficult to summarize, to explain the

23 Manuel

García Doncel pointed this out in 1987. See the analysis by Bernard (2015a, b). to Campalans, 3 January 1922, Folder 3372, exp. 2, ADB. 25 Campalans to Terradas, Barcelona, 14 January 1922, Folder 3372, exp. 2, ADB. 26 Terradas to Weyl, Barcelona 17 January 1922, Hs. 91-762, ETH Library, Zürich. 27 The quotation is: “con toda fluidez y fineza se ha tomado usted el trabajo de aprender”. 28 Gesine Märtens has published the correspondence between Helene Weyl and José Ortega Gasset. She states that Helene kindled her enthusiasm for Spanish literature during the time she accompanied her husband’s 3-months trip to Spain during the summer of 1923. This is obviously a mistake, since the trip took place in the spring of 1922. See: Märtens (ed) (2008). 24 Weyl

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backwardness of Spanish mathematical culture, and that invitations to experts such as Weyl himself were an attempt to remedy this situation.29 In this sense, Terradas organised a session in preparation for Weyl’s course on 1st March, the day before it was due to begin.30 One year before, Terradas had prepared a complete course on Relativity to coincide with the course to be given by Tullio Levi-Civita, one of whose lessons dealt with Relativity.31 Levi-Civita was well known as one of the authors of tensor calculus (then called Absolute Differential Calculus), a theory that was instrumental for Einstein’s General Relativity. In accordance with the usual plans of the secretary of the Pedagogical Council, Weyl’s course was widely announced. The text included Weyl’s lesson programme and described him as the main follower of Einstein’s theories. As on previous occasions, the announcement was sent to the press and to a selective list of people in official authority and other persons belonging to the small circle of mathematics and mathematical physics. Several Barcelona newspapers published the programme notes prepared by the organisers of the course.32 On 27th February, Campalans sent a note in reply to Terradas, who had asked for a hotel reservation for Weyl and for two invitations for his students.33 Campalans told him that they had booked a room at the Hotel Majestic. He also asked him to prepare an article on Weyl’s course for the press, since the year before he (Campalans) had written a paper on Levi-Civita’s course that had been published in the mainstream press, but on this occasion he feared that the contents of Weyl’s lectures would be “more than I can chew”.34 The course began the Thursday 2nd March and continued on the 4th, 6th, 7th, 9th, 11th, 13th and 15th March, 8 days in all with one lesson each day. We have a direct testimony of the course thanks to a letter by Terradas to Levi-Civita dated in 22nd March 1922.35 They were in active correspondence at that time because the course that had been given by the Italian mathematician in Barcelona was in the process of publication. Terradas stated as follows36 :

29 Terradas

remarks in this regard read as follows: “No hallará Vd. en nuestro país, talento que al suyo iguale, ni en proporción suficiente a su medida. Causas diversas, de difíciles y largas razones cuajadas, nos han llevado a un estado singular de la cultura, en especial de cultura matemática”. 30 Note, 28 February 1922, Folder 3372, exp. 2, f. 102, ADB. 31 Glick (1979); Roca-Rosell; Glick (1982); Roca-Rosell; Sánchez Ron (1990). 32 For example, on 7 December 1921, La Publicidad (p. 4) published the series of courses organised by the Mancomunitat, including the detailed program of Weyl’s course. La Veu de Catalunya announced on 25 February 1922 (p. 8) that Weyl’s course would begin in a few days. 33 Campalans to Terradas, Barcelona 27 February 1922, copy, Folder 3372, exp. 2, ADB. 34 His remark in Catalan was as follows: “En Weyl és massa gruixut per mi, i m’esmorçaria les dents”, “Weyl is more than I can chew; it would break my teeth”. The article on Levi Civita: R. Campalans: Levi-Civita a Barcelona, La Publicidad, 12 January 1921 (edición de noche), p. 1. 35 Reproduced in Roca-Rosell; Glick (1982). The letter is in the Archives of the Accademia Nazionale dei Lincei, Rome. 36 “Nous avons ici Mr. Le Prf. Weyl. Il nous explique sa profonde théorie de la structure de l’espace faisant suite aux recherches de Riemann-Helmholtz-Lie, et se rapportant aux nouveaux points de

10

A. Roca-Rosell Professor Weyl is here with us at the moment. He is explaining to us his profound theory of the structure of space, developing the research work by Riemann-Helmholtz-Lie and incorporating the new points of view introduced by Einstein. He is lecturing in French, because most of the audience do not find German easy to follow. That means of course that his ideas are sometimes difficult to understand, and I am obliged to provide explanations for those attending. I am sending you the first batch because I think you will find them interesting. The original text in German will be sent to me shortly, and I will be able to make a free translation in order to make the explanations clearer and more accessible for my fellow citizens.

First of all, it is necessary to point out a slight discrepancy in the date of the letter. On 22nd March, Weyl was lecturing in Madrid, not in Barcelona. Nevertheless, if Terradas sent to Levi-Civita some of the complementary explanations from the beginning of the course, it is quite possible that there is a mistake in the typescript, the 22nd being written instead of the 2nd of March (or perhaps even the 12th). As he told Levi-Civita, Terradas made short abstracts of Weyl’s lectures, stating that the difficulty of the course obliged him to prepare texts to help those who attended to understand them better. Some of these abstracts are preserved in the Archive of the Institut d’Estudis Catalans and of the Diputació de Barcelona. Weyl himself had also prepared texts in French with the same objective, some of which have been found in the Archive of the Diputació of Barcelona.37 Finally, Terradas announced the publication of his “free” translation of the text to help his colleagues. A list of those who registered for the course can be found in Archive of the Diputació, which includes their names and addresses, and in one case some comment: 1.- P. Puig, Ali Bey 10, 1, 2, Gratuit 2.- C. Meisterhans, Sarrià 3.- J. Orriols, Urgell, 116, 1, 2 4.- J. Manyer 5.- A. Robert, Rosselló, 224, p. 6.- B. Lassaletta, Aragó 222, 2, 1 7.- J. Galí, Vergara, 12 8.- Pòlit 9.- Pere Martínez 10.- Manuel Álvarez Castrillón

vue que Einstein y a introduits. Il s’exprime en langue française, l’allemand n’étant pas facile à suivre par la plupart du public. Cela fait, sans doute, que l’on épreuve quelques difficultés pour bien saisir ses idées et je suis obligé de rédiger des explications pour l’usage des auditeurs. Je me permets de vous envoyer les premières, elles pourront peut être vous intéresser.//L’original allemand me sera envoyé par la suite et je pourrai en faire une traduction que je me propose de faire très librement à fin d’essayer de rendre l’explication claire et facile pour mes concitoyens”. Terradas to Levi-Civita 22 March 1922, reproduced in Roca-Rosell; Glick (1982). 37 The abstract of the first lesson was published in a newspaper, La Publicidad, on 11 March 1922, p. 2. The abstract, in Catalan, is relatively technical, and states that Weyl delivered the first lesson in the “Cervantes” room at the Institut d’Estudis Catalans. It should be remembered that this first lesson was delivered on 1 March, that is, 10 days previously. We have not found any other abstract in the newspaper.

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11.- Josep Gorini 12.- Ramon Jardí 13.- Francesc Planell.

Despite the fact that some of the names are not identified, the list enables us to characterize the small group of people in Barcelona who were interested in advanced mathematics and physics. First of all, the order of registration: according to our research into other courses,38 the secretary recorded the names of the attendees as they arrived, which means that the first name on the list was the first person to register. Secondly, we proceed to identify this group of participants. The first name is P. Puig, which corresponds to the mathematician Pere Puig Adam (1900–1960),39 who had presented his doctoral thesis on some aspects of the theory of relativity. Given that Puig Adam was probably without a post, his registration was free (“gratuit” is written after his address) and could possibly be one of the free invitations that Terradas had requested some days before.40 Puig Adam would have had to occupy a relevant position in Spanish higher education, teaching at the university and at the School of Industrial Engineering in Madrid, and was to become a promoter of reforms in the teaching of mathematics in primary and secondary schools. The second name is C. Meisterhans. We have little information about him, except that in 1926 he was the translator from German to Spanish of a handbook on radiotelephony,41 where Conrado Meisterhans is referred to as an engineer.42 He also translated a handbook on electrical measurements, published in 1927, in a review of which it is said that the standing of the translator lends quality to the publication.43 We have no precise information about the following name, J. Orriols, but it could very well be Josep Orriols Artigas, an industrial engineer who graduated in 1912. Neither is much known about the following name, J. Manyer. In our paper of 1987, we stated that there could be a mistake in the registration and that the name should have been Josep Mañas Bonví (1885–1941), a professor at the School of Industrial Engineering. He was someone who frequented Terradas’ circle and was sometimes referred to as “Manyas”.44 Mañas published several handbooks, perhaps the most noteworthy of which was his book on Applied Optics. Furthermore, it

38 Roca-Rosell

(2016a). Sales (1995); Alsina Català (2001). 40 Terradas to Campalans, Barcelona, 20 February 1922, Folder 3732-2, num. 140, ADB. 41 Schönbauer, Carlos; Zeemann, Antonio (1926). Manual de montajes y ejercicios prácticos para aficionados; traducido del alemán por Conrado Meisterhans, Barcelona, Luis Gili. 42 In our unpublished paper of 1987, we stated that Meisterhans was a German diplomat. This is a mistake. The German consul in Barcelona was Ulrich von Hassell (1881–1944), who still occupied this post in 1923, the year in which Einstein visited Barcelona. 43 “Tratado de medidas eléctricas, por A. Linker. Traducción de la tercera edición alemana por C. Meisterhans, — Luis Gili, Córcega, 415 Barcelona.— Precio, 22,50 pesetas”, Ingeniería y Construcción, Año IV, vol. IV, núm. 48, December 1926, p. 575–576. 44 There is a biography of Mañas in the Vinapèdia: http://www.vinapedia.es/letra-m/manas-bonvijosep/. See also: Glick (1986). 39 Alsina;

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is known that Mañas was among those who attended the celebratory lunch at the end of the Course. The next name on the list is another professor at the Barcelona School of Industrial Engineering, Antoni Robert Rodriguez (1878–1951), who taught Industrial Physics. Bernat Lassaletta Perrin (1882–1948) was also a professor at the same School, where he taught Electrotechnics. The same may be said of the following name, Josep Galí Fabra (1877–1927), who was also a professor at the School of Industrial Engineering, where he taught machine theory. Notice that the last five names refer to people who were connected with the Barcelona School of Industrial Engineering, most of whom taught at the Industrial School of Barcelona, where Terradas was in charge of the Institute of Applied Electricity and Mechanics. The next name on the list is Isidre Pòlit Buxareu (1880– 1958), astronomer and physicist, who was a teacher at the Faculty of Sciences of the University of Barcelona as well as being a professor at the Industrial School of Barcelona. It has been impossible to found anything more about the following name, Pere Martínez. He did not graduate in Industrial Engineering, and it would be impossible to identify him as student or graduated in Sciences45 without the full name.46 Name number 10 on the list is Manuel Álvarez Castrillón (1886–1957), a well-known mathematician and meteorologist who was a teacher at the Faculty of Sciences and member of the Meteorological Service of Catalonia.47 We have no information about Josep Gorini, but name number 12 is the physicist Ramon Jardí Borràs (1881–1972), a teacher at the Faculty of Sciences and the Industrial School of Barcelona and member of the Meteorological Service.48 Finally, the last name is Francesc Planell Riera (1886–1973), an industrial engineer and teacher at the Industrial School of Barcelona, where he was the main collaborator of Terradas.49 Thus, most of the people who registered on the list were engineers, seven in total and most of them industrial engineers, the only speciality available in Barcelona. There are four Science graduates (Mathematics or Physics), while the qualifications of the two remaining people are unknown. Nevertheless, it should be said that the people belonging to this group were not fully active in research.50 In fact, at that time, no centre of research in the field of exact sciences existed in Barcelona. Some of the persons who eventually registered had already been included in a letter sent by Terradas to Campalans on 20th February, 10 days before the beginning of the course.51 This list includes the following names: Josep M. Bartrina, 45 There

was a medical doctor, Pere Martínez Garcia, an expert in Paediatrics, who was professor at the Faculty of Medicine of Barcelona in 1931. It does not appear to be the same person. 46 In Spain, people have two surnames, the first from the father, and the second from the mother. In addition, Pere or Pedro, and Martínez, are very common names and surnames. 47 Roca-Rosell (2007b). 48 Batlló et al. (2015). 49 For the professors belonging to the School of Industrial Engineering before 1942, see Castells (1943); and the professors from the Industrial School of Barcelona, see Roca-Rosell (ed.) (2008). 50 As exceptions, Álvarez Castrillón and Jardí Borràs worked for the Meteorological Service of Catalonia, and Pòlit, at the Fabra Observatory. 51 Terradas to Campalans, Barcelona, 20 February 1922, Folder 3732-2, num. 140, ADB.

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Teodor Sabràs, Ferran Tallada, Paulí Castells, Ramon Jardí, Isidre Pòlit, Pere Puig, Manuel Sastre, Bonaventura Bassegoda, Antoni Torroja, Francesc Planell, Adolf Florença, Antoni Darder, Eduard Fontserè, Manuel Alvarez Castrillón, Josep Mur, “two students from the University”, Antoni Robert, Josep Galí, Ramon Vilamitjana, Georges Dwelshauvers, and “German personnel from the Siemens factory in Cornellà”. Terradas also suggested that the German Consulate might be informed of the event. It is known that Bartrina was teaching at the Instituto de Bachillerato (state secondary school), while the others were teachers at the Faculty of Sciences, either at the School of Industrial Engineers or the Industrial School. As far as I know, the only exceptions were Dwelshauvers, a Belgian experimental psychologist and supervisor of a laboratory of the Diputació, the Siemens engineers, and three architects (Bassegoda, Florença, and Darder).52 As may be seen above, some of them (Jardí, Pòlit, Puig, Planell, Álvarez Castrillón, Robert, Galí) accepted the invitation. On 18th March, shortly after the course ended, a celebratory lunch was organized for Weyl and his wife at the Ritz Hotel. There is a text signed by those who attended the lunch that was presented to the professor himself and is now preserved in the Weyl file at the ETH Archives Zürich.53 It is possible to recognise most of the hand-written signatures, most of them belonging to those who attended the course: Meisterhans, Lassaletta, Terradas, Josep Gorini, Pòlit, Bassegoda, Antonio Torroja, Robert, Campalans, José Mur, Eduard Alcobé, Planell, Adolf Florensa, Jardí, and José Mañas.54 In the text, the thanks expressed by the group for the course was accompanied by statements of great veneration. It is interesting to mention that in the final paragraph the signatories say that: Receive, master, a tribute of our sincere admiration. And do not forget the Land where your name is the object of such esteem. Upon awakening from her lethargy, may she add new elements to universal mathematical wisdom, just as she has been contributing in other disciplines and in Art. The attention we have paid you would then be greatly rewarded and you will have helped effectively in that task.55

52 I

have not been able to identify Teodor Sabrás or Manuel Sastre. 91: 469, ETH Archives, Zürich. 54 This banquet was reviewed in the press. See, for example, El Diluvio, 22 March 1922, p. 15, La Veu de Catalunya, 22 March 1922, p. 9. In these reports it said that Weyl was leaving Barcelona and returning to his country, whereas in fact he was going to Madrid to repeat the course. 55 “Recibid, maestro, tributo de admiración sincera. Y no olvidéis la Tierra donde vuestro nombre es objeto de tamaña estima. Ella tal vez, al despertar de su letargo, añada nuevos elementos a la universal sabiduría matemática, como los viene aportando en otras disciplinas y en el Arte. Nuestra labor de atención se verá entonces con creces recompensada y vos habréis ayudado eficazmente en la tarea.” Text presented to H. Weyl, Barcelona, 18 March 1922, HS 91: 469, ETH Archives, Zürich. 53 HS

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1.2.2 The Lectures in Madrid The course in Madrid was organised by Josep M. Plans Freyre (1878–1934), a professor at the Faculty of Sciences in Madrid and another close friend of Terradas. Plans was perhaps the only Spanish researcher involved in research work related to Weyl’s subjects. In 1919 he published a paper in the journal of the Spanish Mathematical Society devoted to the relevance of Weyl’s theory as a development of Einstein’s theories.56 The reference to Weyl’s publications, however, seems to be somewhat indirect. In 1921, Plans published a handbook on relativistic mechanics, a 1919 prize of the Royal Academy of Exact Sciences of Madrid. Although the book is mainly devoted to special relativity, the two last chapters deal with the theory of gravitation, in which Plans refers briefly to Weyl’s work, commenting on Raum, Zeit, Materie as follows: A highly relevant work that constitutes the most complete mathematical development, and in which, in addition, the Author provides us with his most recent theory. It also contains an extensive bibliography of the most notable original memoirs.57

Plans lectured at the plenary of the Spanish Association for the Advancement of Science congress, held in Porto in 1921. He chose as his subject the “history” of absolute differential calculus, i. e. tensor calculus, the main mathematical language for General Relativity.58 Plans referred to Weyl’s contribution in a short sentence in which he mentioned the 3rd edition of Raum, Zeit, Materie, stating that Weyl was attempting to construct a real geometry, a theory of space itself, and not merely as Euclid’s geometry and as almost everything that is cultivated with the name Geometry, but rather a theory of all possible forms in space.59

When Weyl taught the course in Madrid, Plans was member of the Laboratorio y Seminario Matemático (Mathematical Laboratory or Seminar), created in 1915 by the Junta para Ampliación de Estudios on the initiative of Julio Rey Pastor. After the appointment of Rey Pastor to the University of Buenos Aires in 1921, the Laboratory was directed by Plans and José Álvarez Ude (1878–1958). The April issue of the Revista Matemática Hispano-Americana, published by the Spanish Society of Mathematics and the Laboratory, carried a portrait of Weyl with his signature on the cover, and included an article concerning his career as mathematician who was educated in Göttingen and had been a professor at the ETH in Zürich since

56 Plans

y Freyre (1919). importantísima que constituye el desarrollo matemático más completo y en la que, además, se expone la moderna teoría del autor. Contiene también una copiosa bibliografía de las Memorias originales más notables”. Plans y Freyre, 1921a. 58 Plans y Freyre (1921b). 59 “Geometría real, una teoría del espacio mismo, y no meramente como la Geometria de Euclides y casi todo lo que se cultiva con el nombre de Geometría, una teoría de las formas posibles en el espacio”. 57 “Obra

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1913. In the article, it was stated that despite his youth Weyl was an outstanding mathematician, and concluded with a list of his publications. The article was written before Weyl’s arrival in Spain and constitutes a welcome from the Spanish (and Madrid) community. The piece is unsigned, but it is reasonable to assume that Josep M. Plans wrote the article, probably with some text provided by Weyl himself.60 In a letter to Terradas, Plans explained that, before March, the Junta para Ampliación de Estudios had no budget to cover the expenses of Weyl’s course in Madrid, but he managed to get the Faculty of Sciences to approve the 2000 pesetas required to pay Weyl. Plans made preparations for the course in Madrid by personally inviting several of his own colleagues. He arranged for Patricio Peñalver (1889–1979) from the University of Seville to meet Weyl on his arrival in Spain.61 During his stay in the Spanish capital, Weyl was invited to attend at a meeting of the Spanish Mathematical Society, by which he was made an honorary member.62 At the meeting, José Agustín Pérez del Pulgar (1875–1939) and Emilio Herrera (1879–1967) expounded their somewhat heterdox ideas on the theory of relativity and listened to the Weyl’s response.63 Pérez del Pulgar, who belonged to the Society of Jesus, had set up a school of engineering in Madrid (Instituto Católico de Artes e Industrias), while Emilio Herrera was a military engineer and a pioneer of aviation in Spain. Both possessed a good mathematical knowledge and their objections were the fruit of their experience in engineering; in electrotechnics, in the case of Pérez del Pulgar, and in many other fields, in the case of Herrera, who was a really multifaceted character.64 During his course in Madrid, Weyl wrote two letters, one to Campalans and the other to Terradas. We do not know the contents of his letter to Campalans, only Campalans’ reply.65 However, both Weyl’s letter to Terradas and Terradas’ reply are available. Campalans was very grateful to receive Weyl’s letter and insisted that they were working for the “Renaissance” of Catalonia. According to him, in the field of science they were in the initial stages of this renaissance, what Aristotle refers to as the “astonishment”.66 He concluded by conveying the wishes of the German Consul for a meeting with Weyl before he left Spain. The Weyls planned to finish their trip to Spain with a one-week stay in Sitges, a tourist resort on the coast not far from Barcelona.

60 “El professor Hermann Weyl”, Revista Matemática Hispano-Americana, vol. IV, April 1922, pp. 50–54. 61 Plans to Terradas, 1 January, 2 January, 15 January 1922, Fons Terradas, Arxiu IEC. 62 “Acta de la sesión celebrada por la Sociedad Matemática Española el día 1◦ de abril de 1922”, Revista Matemática Hispano-Americana, vol. IV, num. 6, June 1922, p. 101. 63 On this debate, see Glick (1986). 64 On Herrera, see Glick (ed) (1984). 65 Campalans to Weyl, Barcelona 29 March 1922, folder 3732-2, num. 195, ADB. 66 Campalans wrote in French and said “étonnement”.

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Weyl wrote a letter to Terradas on 21 March67 in which he expressed his peace of mind after having successfully overcome some “difficult” moments, i.e. the course in Barcelona, about which he said that he had been worrying with a certain “anxiety” for three quarters of a year. He hoped he had been able to convey to his audience his thoughts on such fundamental problems of mathematics, despite the fact that he had done so in an unfamiliar language (French). Weyl expressed the good impression he had gained of the scientific life in Barcelona, which he hoped to be renewed in a very short time. He also mentioned that he had met Josep M. Plans, with whom he thought it would be possible to collaborate. However, his relationship with him was hampered because Plans did not speak German. As an aside, Weyl remarked that since his arrival in Madrid he had heard no other language spoken except Spanish. He also told Terradas that the weather in Madrid had been bad, and he and his wife remembered with nostalgia “the blue sky and the beautiful landscape” of Barcelona, and for that reason they were looking forward to their stay in Sitges. Terradas later wrote to Weyl at his address in Sitges.68 First of all, he hoped that Weyl had recovered from the health problem that his wife had mentioned in a letter, and that the fine weather in Sitges would help him to do so. Weyl had asked Terradas previously if it would be possible for them to meet again soon, and Terradas explained that he was planning to spend his holidays in Italy and perhaps also in Germany. He also said that had been invited to lecture in South America in 1923 and needed to prepare the course. He went on to say that Weyl’s lectures had had a deep impact on them all, and that despite their lack of mathematical culture they trusted that the course would arouse a new interest. It was announced in the press on 15 March69 that the course in Madrid would take place from 21st to 31st that month at the Faculty of Sciences, registration being free, and that the lessons would be in French. There was also further mention, such as an article by Juan Usabiaga (1879–1953) that appeared in the newspaper El Sol.70 Usabiaga was a Basque industrial engineer and a professor at the Madrid School of Industrial Engineers. In his review, Usabiaga described Weyl as Einstein’s main collaborator in the Physics revolution, but declined to discuss the details of the course to spare his readers difficulties. The purpose of his article

67 Weyl

to Terradas, Madrid, 21 March 1922, Fons Terradas, Institut d’Estudis Catalans. to Weyl, Barcelona 10 April 1922, Hs 91:763, ETH Library, Zürich. 69 See El Heraldo de Madrid, 15 March 1922, p. 4; El Imparcial, 15 March 1922, p. 4, where there are the titles of the lessons: Día 21, «La axiomática elemental en la Geometría»; día 22, “Geometría riemanniana; la noción de desplazamiento paralelo infinitesimal de vectores”; día 24, “Geometría infinitesimal métrica generalizada: el torbellino segmento”; día 25, “E1 torbellino vectorial: los espacios métricos homogéneos”; día 27, “El problema del espacio, según Helmholtz-Lie”; día 28, “La solución de las investigaciones sobre la teoría de los grupos de Lie”; día 30, “El problema del espacio, con una métrica variable, según el punto de vista de la teoría de la relatividad”; día 31, “La solución; esquema de la demostración”. 70 Ortega y Gasset was the main inspiration behind El Sol. 68 Terradas

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was to show that there had been a witness of Weyl’s activities in Madrid.71 He congratulated the Faculty of Sciences and professor Plans for inviting such a great scientist. A very interesting article on Weyl’s course appeared in the journal Ibérica, published by the Ebro Observatory, which belonged to the Society of Jesus.72 Plans was a frequent contributor to the journal and it may be supposed that he was the author of the article, although it was published unsigned. It gave the programme of the eight lessons in the course and stated that: The object of the lecturer was the presentation of the most modern theories on the philosophical and physical problem of space. From Euclidean geometry to the latest conception of the lecturer himself, passing through Lobatchewski, Riemann, Helmholtz, Lie, Minkowski, Levi-Civita, and Einstein, all was carefully revised. The lecturer distinguished the role played in each theory by the notion of straight line (trajectory of a free point in space and in Minkowski’s universe) and of null element (the trajectory of light). Several axioms were thereby established, but what occurs is that, as the problem becomes more complicated, the a priori and a posteriori elements of any metric are emphasized. That of the lecturer, developed in a brilliant way in the two last lessons, is included in the fourth edition of his book Raum, Zeit, Materie, already translated in French . . . but the whole synthesis was achieved [by this author] less than one year ago, and in these lectures, which we hope to see soon in print, has been set out for the first time in the Castilian language.73

The author of the review was aware of the fact that Weyl’s course in Spain had been a significant event for its presentation of a new development of the theory of space.74 It is worth noting the reference to Raum, Zeit, Materie, a book that had already become a key work worldwide, and was reviewed in Spanish journals such as the Revista Matemática Hispano-Americana and Ibérica.

71 Usabiaga

stated that the course was held in Room 6 of the Faculty of Sciences. This may be interpreted as a nod to his readers that he attended the lessons. 72 “Conferencias de los profesores H. Weyl, A. Sommerfeld, O. Hönigschmid y K. Fajans en la Universidad de Madrid”, Ibérica, vol. XVII, núm. 430, 3 June 1922, p. 340–341. 73 The original text: “el objeto del disertante fue presentar las teorías más modernas sobre el problema filosófico y físico del espacio. Desde la geometría euclídea, hasta la última concepción del propio conferenciante, pasando por Lobatchewski, Riemann, Helmholtz, Lie, Minkowski, LeviCivita y Einstein, todo fue revisado cuidadosamente. Distinguió el conferenciante el papel que en cada teoría desempeña la noción de línea recta (trayectoria de un punto libre, así en el espacio como en el Universo de Minkowski) y de elemento nulo (trayectoria de la luz). Con ellos se establecen las diversas axiomáticas; lo que sucede es que a medida que se complica el problema, se van poniendo más de relieve los elementos a priori y a posteriori de una métrica cualquiera. La del conferenciante, expuesta de manera deslumbradora en las dos últimas conferencias, está esbozada en la cuarta edición de su libro Raum, Zeit, Materie, traducido ya al francés...; pero su síntesis global no cuenta un año todavía, y en estas conferencias, que esperamos verán pronto la luz pública, ha sido expuesta por primera vez en lengua castellana”. 74 We see that the author says that in Madrid Weyl lectured in Spanish. In Barcelona, he lectured in French, probably an easier language for Weyl.

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1.3 Publication of the Lectures Terradas wished to publish the courses given during the Cursos Monogràfics d’Alts Estudis i d’Intercanvi series. Some of the volumes were completed before 1922, those corresponding to the courses by Julio Rey Pastor, Terradas and Julio Palacios (1916), and the books by J. Hadamard and Levi-Civita (1922). He planned to publish the volume devoted to Weyl, and, according to what he told to several people,75 he was working on it from 1922, although the volume was never published. It is clear from Terradas’ biography that at that time he was involved in many technical projects, which probably hindered his preparation of the text. Terradas and Weyl developed a close friendship subsequent to Weyl’s sojourn in Spain in 1922. There is a letter from Terradas dated 1st February, 1923,76 in reply to one from Weyl, in which he proposed a dedication for the book containing the texts of the course he had given in Spain the year before. Terradas was very impressed by the text, and it is clear that the dedication, which was highly favourable, appeared at the beginning of the book that came out that same year 1923. It reads as follows: D. Esteban Terradas Dedication Dear friend! Please accept this book, written as a faithful record of the time you enabled me to spend in Barcelona in March of last year, and as a testimony of recognition, of cordial sympathy and the highest respect for your person, as well as a sign of admiration for the constructive task undertaken by you and your companions in the service of technology, science and education in Catalonia! I believe that I have yet to find such a passionate desire, such a clear vision of what is needed and available, with the corresponding work and energy, so harmoniously gathered. I hope that, thanks to your activity, the fertilizing seeds of a polyvalent education, powerfully developed in complete independence, will spread and grow among the men and things around you. Let us reap the richest harvest!77

Weyl’s words are a reflection of the impression he received during his stay in Spain, mainly in Barcelona, where he touched by the extraordinary personality of Terradas.

75 Already

in January 1923, Terradas told his colleague Bofill that he was engaged on the text decoted to Weyl. Terradas to Bofill, Barcelona 27 January 1923, Arxiu de l’Institut d’Estudis Catalans. 76 Terradas to Weyl, Barcelona 1st February 1923, HS.91-770, ETH Library, Zürich. 77 “D. ESTEBAN TERRADAS//zugeeignet.// Verehrter Freund! Nehmen Sie dieses Buch -das in so enger Beziehung steht zu meinem in erster Linie durch Sie veranfaßten Aufenthalt in Barcelona im März vorigen Jahres -von mir entgegen als ein Zeichen der Dankbarkeit, herzlicher Sympathie und höchster Achtung für Ihre Person, zugleich aber auch als ein Zeichen der Bewunderung für das aufbauende Werk, das Sie mit Ihren Arbeitsgefährten zusammen im Dienste der Technik, der Wissenschaft und des Unterrichts in Katalonien errichtet haben! Niemals und nirgendwo, will mir scheinen, habe ich so harmonisch wie dort miteinander verwachsen gefunden guten, ja begeisterten Willen, klaren Blick für das Erforderliche und Erreichbare, nüchterne Arbeitsenergie. Ein befruchtender Strom vielseitiger, kräftiger und zu freier Selbständigkeit fort-schreitender Bildung hat sich von Ihrer Tätigkeit aus auf die Menschm und Dinge Ihrer Umgebung ergossen. Möge der Blüte eine reiche Ernte folgen!”

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In the same letter dated 1st February, Terradas told Weyl that his plans had been upset by a series of circumstances and he had been obliged to leave aside many things. He was preparing a course of lectures in Buenos Aires the following summer, and informed Weyl that he would be able to finish the Catalan edition of his course in Barcelona at the end of the year. He said that he was preparing Einstein’s forthcoming course (Weyl had mentioned that he knew about Einstein’s visit to Barcelona), and went on to congratulate Weyl for giving the course in Madrid in Spanish, a gesture which he said was remembered there with gratitude. The correspondence between Terradas and Weyl was particularly intense during 1923, from which there are five letters from Terradas in the Weyl archives in Zürich. The letter dated 26th February78 from Terradas was written in a state of shock after the death of his daughter Helena, 2 weeks before.79 Terradas wrote Weyl to send him a copy of a letter of introduction of a young scholar “recommended” (recomendado) by Weyl. Although the name of this person does not figure in the letter, the scholar referred to was the Hungarian mathematician Béla Kerékjártó (1898–1946), who was invited to lecture in Barcelona in 1923 and subsequently spent some time there doing research with Terradas.80 In this same short letter of 26th February, Terradas explained that his wife was pregnant and feeling very weak, the date of the birth being 3 weeks later. It was for this reason that she had not written to Weyl’s wife. Terradas further informed Weyl that Einstein was currently in Barcelona, where he had begun his lectures the day before. Finally, he told Weyl that he had postponed the visit to South America. The following letter is dated 15th May 1923.81 It is a longer text in which Terradas reflected on his own situation. He told Weyl that his son David had been born 25 days before and that both mother and child were doing well. He went on to say that he had received Weyl’s book on Riemann surfaces, the first edition of which he already had, but he would take advantage of the new edition, given that Weyl’s disciple, Kerékjárto, was in Barcelona, and he would discuss the contents with him. He said that he had made arrangements for Kerékjarto’s stay and that he was a good and talented person who had written a demonstration of Poincaré’s Theorem in half a page! Terradas was thinking about preparing a monography based on Kerékjarto’s lessons in order to provide him with some extra income.82 On 1st July, 1923, Terradas wrote to Weyl from Cadaqués, a village on the Northern coast of Catalonia where he spent his holidays.83 In this letter he again 78 Terradas

to Weyl, Barcelona 26 February 1923, HS.91-764, ETH Library, Zürich. find the news of the death of Helena Terradas in La Vanguardia, 13 February 1923, p. 6. 80 See Filipiak, Alicia (2015) Les débuts de carrière de Béla von Kerékjártó vus à travers sa correspondance avec Maurice Fréchet, Paris, Université Pierre et Marie Curie. In this correspondence, Kerékjárto gave the house of Terradas as mailing address. 81 Terradas to Weyl, Barcelona 15 May 1923, HS.91-765, ETH Library, Zürich. 82 In the letters reproduced by Filipiak (2015), we see that Kerékjarto wrote some entries for the Espasa Encyclopaedia and was acclimatizing himself to life in Barcelona: he said that he was studying Catalan. 83 Terradas to Weyl, Cadaqués 1st July 1923, HS.91-766, ETH Library, Zürich. 79 We

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expressed his gratitude to Weyl for the dedication in the book, probably soon after he had received a copy. The letter of 25th December 1923 is a reflection on recent events in Spain.84 This was outlined at the beginning of the letter, and then Terradas stated that Spain was in such a political turmoil (“marejada”) that he feared it would all end badly. This is an interesting reference to General Primo de Rivera’s coup in September 1923. A Dictatorship was imposed to put an end to worker movements and democratic protests against the corruption of the Monarchy. It should be said that sometime after this Terradas was not adverse to cooperating with the regime by accepting a post at the so-called National Assembly, created in 1927 as a means of providing the Dictatorship with a better image.85 It eventually came to an end in 1930 and this opened the way to the proclamation of the Second Spanish Republic in 1931. For his part, Terradas said that he was very busy with work during 1923: he was supervising the project for the Transverse Railway (which would be the second underground line in Barcelona), with 6 km of tunnels and “big stations”. He was in charge of the operation and had 1000 workers under his supervision. He hoped that on completion of the work he would be able to take up his academic projects again, one of which was the Catalan edition of Weyl’s course. Meanwhile, thanks to his tasks on the Transverse Railway, he was assuring the future of his children. Towards the end of this letter, Terradas expressed his deep sadness for the death of his daughter Helena, and concluded it by wishing Weyl and his wife a Happy New Year.

1.4 Opening New Avenues of Research In our previous researches into Weyl’s visit, we discussed Josep M. Plans’ attempts to conduct more in-depth research into areas closely related to Weyl’s own field.86 After his visit to Barcelona in 1922, Weyl’s theories continued to arouse the interest of other researchers. When Einstein visited Spain, he included in his lectures an extensive reference to Weyl’s contributions. For this part, Plans, whose interests were centred on the theory of relativity and its mathematical challenges, tried to make inroads into this field and promoted the research work of one of his disciples, Fernando Peña Serrano. Peña published three papers on the matter between 1926 and 1932, while Plans himself devoted one paper to the same question, but the research was not fully completed. Plans died prematurely in 1934, and Fernando Peña Serrano (1894–1960), who was a forestry engineer, suspended his interest in the subject and eventually became a professor at the School of Forestry Engineers (Ingenieros de Montes) as well as occupying important posts in the corresponding

84 Terradas

to Weyl, Barcelona 25 December 1923, HS.91-767, ETH Library, Zürich. Terradas and his trajectory, see Roca-Rosell (2005). 86 Roca-Rosell (1987). 85 On

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Corps. Peña Serrano was a man of considerable scientific culture; he taught statistics, and after his interest in Weyl’s theories he turned his attention to quantum mechanics.87 Shortly after Weyl’s course, Arnold Sommerfeld (1868–1951) gave a series of lectures in Barcelona as part of the Monographic Courses on March 27th-31st, and first April, which were also supported by the Mancomunitat. As on previous occasions, after Barcelona he also went on to lecture in Madrid.88 Terradas was very satisfied with the outcome of invitating of these two outstanding scientists. On 15th April, he wrote a formal letter to the director of Public instruction of the Mancomunitat, Rafael Campalans.89 We do not know exactly what the purpose of this letter was, but it included an expression of gratitude to the Catalan administration for giving support to the courses by foreign scientists. Terradas seemed to be enthusiastic about the situation. He believed that the courses would be useful for drawing attention to the backwardness of the country, as well as pointing the way forward to creating a genuine scientific culture and overcoming the lack of tradition and the influence of false intellectuals. He declared his commitment to the struggle against the superficiality that was killing the “race”.90 However, Primo de Rivera’s coup in September 1923 changed the situation dramatically. The Mancomunitat was dissolved and the Institut d’Estudis Catalans was stripped of all its resources. On completion of the Barcelona Tranverse Railway, Terradas, who initially seemed to be against the Dictatorship, accepted different posts in the new regime. In fact, in 1927 he moved to Madrid, where he became a professor at the University (1928) and Director of the Spanish Telephonic Company (1929–1930). Subsequently, however, after the proclamation of the Republic, objections were raised against his appointment at the University of Madrid and he returned to Barcelona, where he resumed his tasks at the Institut d’Estudis Catalans. In 1933, he proposed the creation of a Centre of Mathematical Studies, which was inspired by several similar centres as well as his own experiences, such as the course given by Hermann Weyl 10 years before. This constituted the last attempt to institutionalize research in exact sciences in Barcelona before the outbreak of the Spanish Civil war in 1936.91

87 Valentín-Gamazo,

2015, especially p. 99–103. Ron (1982). 89 Terradas to Campalans, 15 April 1922, Folder 3732, exp. 2, ADB, reproduced in Roca Rosell (1988a). 90 At that time, the word “race” meant “people” or national collective. 91 Roca-Rosell (2016a, b). 88 Sánchez

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References Alsina i Català, Claudi. 2001. Pere Puig i Adam: ahir, avui i sempre. Butlletí de la Societat Catalana de Matemàtiques 16 (1): 43–60. Alsina, Claudi, and Josep Sales. 1995. Pere Puig Adam. Barcelona, 1900-Madrid, 1960. La nova didàctica de les matemàtiques. In Ciència i tècnica als Països Catalans. Una aproximació biogràfica, ed. J.M. Camarasa and A. Roca-Rosell, 1367–1399. Barcelona: Fundació Catalana per a la Recerca. Batlló, Josep, Àgata Pedrero, and Joan Arús. 2015. Ramon Jardí i Borràs: semblança biogràfica. Barcelona: Institut d’Estudis Catalans, Secció de Ciències. Bernard, Julien. 2015a. Les tapuscrits barcelonais sur le problème de l’espace de Weyl. Revue d’Histoire des Mathématiques 21: 151–171. ———. 2015b. L’Analyse mathématique du problème de l’espace, with E. Audureau, FrenchGerman commented edition of Weyl’s Mathematische Analyse des Raumproblems, Presses Universitaires de Provence, 2 volumes. Castells, Paulí. 1943. Escuela Especial de Ingenieros Industriales. Establecimiento de Barcelona. Reseña histórica, Barcelona. Facsimile in Lusa Monforte, G. (ed.) (2008) Documentos de la Escuela de Ingenieros Industriales de Barcelona, vol. 18, 131–213. http://upcommons.upc.edu/ handle/2099/8088. Filipiak, Alicia. 2015. Les débuts de carrière de Béla von Kerékjártó vus à travers sa correspondance avec Maurice Fréchet. Paris: Université Pierre et Marie Curie. Glick, Thomas F. 1979. Einstein y los españoles: Aspectos de la recepción de la relatividad. Llull: Boletín de la Sociedad Española de Historia de las Ciencias 2 (4, December): 3–22. ———. (ed.). 1984. Flying: The Memoirs of a Spanish Aeronaut Emilio Herrera. Albuquerque, New Mexico: University of New Mexico. ———. 1986. Einstein y los españoles: ciencia y sociedad en la España de entreguerras. Madrid: Alianza. Reprint, Madrid, CSIC, 2005. English edition, Princeton University Press, 1987. Herran, Néstor. 2008. Aguas, semillas y radiaciones. El Laboratorio de Radiactividad de la Universidad de Madrid, 1904–1929. Madrid: Consejo Superior de Investigaciones Científicas. López García, S. 1996. La investigación científica y técnica antes y después de la guerra civil. In Economía y sociedad en la España Moderna y Contemporánea, coord. A. Gómez Mendoza, 265–276. Madrid: Síntesis. Märtens, Gesine. (ed.). 2008. Correspondencia: José Ortega y Gasset, Helene Weyl. Madrid: Biblioteca Nueva: Fundación José Ortega y Gasset. Plans y Freyre, Josep M. 1919. Weyl.-Una nueva teoría de las relaciones entre el campo electromagnético y el gravitatorio. Revista Matemática Hispano-Americana I: 285–286. ———. 1921a. Nociones fundamentales de mecánica relativista. Real Academia de Ciencias Exactas, Físicas y Naturales, Madrid (Memorias, serie 2ª, tomo 2). ———. 1921b. Proceso histórico del cálculo diferencial absoluto y su importancia actual. In Congreso de Oporto de la Asociación Española para el Progreso de las Ciencias, vol. 8, 23–43. Madrid. Rey Pastor, Julio. 1916. Teoria de la representació conforme. Barcelona: Institut d’Estudis Catalans. Roca Rosell, Antoni, and José Manuel Sánchez Ron. 1990. Esteban Terradas (1883–1950). Ciencia y Técnica en la España contemporánea. Barcelona: INTA/Ed. El Serbal. Roca-Rosell, Antoni. 1987. Hermann Weyl entre nosaltres. El curs de 1922 i algunes de les seves repercussions. Barcelona. Published in 2014 at the website: http://upcommons.upc.edu/handle/ 2117/22995. ———. 1988a. La ciència internacional a Catalunya (1914 1923). In Història de la Física, ed. Luis Navarro Veguillas, 319–332. Barcelona: CIRIT. ———. 1988b. Científicos catalanes pensionados por la Junta. Algunos aspectos de su papel en el desarrollo científico catalán. In ed. Sánchez Ron, vol. II, 349–379.

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———. 2005. Professionalism and Technocracy: Esteve Terradas and Science Policy in the Early Years of the Franco Regime. Minerva 43: 147–162. ———. 2007a. El discurso civil en torno a la ciencia y la técnica. In ed. Suárez Cortina, Salavert Fabiani, 241–159. ———. 2007b. Manuel Álvarez Castrillón (1886–1957), Orígens de la Meteorologia matemàtica. In XIII Jornades de Meteorologia Eduard Fontserè, 69–76. Barcelona: ACAM. ———. (coord.). 2008. L’Escola Industrial de Barcelona. Cent anys d’ensenyament tècnic i d’arquitectura. Barcelona: Diputació de Barcelona, Ajuntament de Barcelona, Consorci de l’Escola Industrial de Barcelona. ———. 2016a. La recerca en ciències exactes i enginyeria a l’IEC: aportacions des de la història de la ciència: el cas del Centre d’Estudis Matemàtics (1933). Barcelona: Institut d’Estudis Catalans, Secció de Ciències i Tecnologia. ———. 2016b. L’IEC i els orígens de la recerca en ciències exactes. SCM/Notícies 39: 50–57. Roca-Rosell, Antoni, and Josep Maria Camarasa. 2008. La promoción de la investigación en Cataluña: el Institut d’Estudis Catalans en el siglo XX. In Cien años de política científica en España, ed. A. Romero De Pablos and M.J. Santesmases, 39–77. Madrid: Fundación BBVA. ———. 2011. The Foundation of the Sciences Section of the Institute for Catalan Studies (1911) and its early years. Contributions to Science 7 (2): 197–205. Roca-Rosell, Antoni, and Thomas F. Glick. 1982. Esteve Terradas (1883–1950) i Tullio Levi Civita (1873–1941): una correspondència. Dynamis 2: 387–402. Sánchez Ron, José Manuel. 1982. Documentos para una Historia de la Física Moderna en España: Arnold Sommerfeld, Miguel Angel Catalán y Blas Cabrera. Llull: Revista de la Sociedad Española de Historia de las Ciencias y de las Técnicas 5 (8–9): 97–110. ———. (ed.). 1988a. Ciencia y sociedad en España. Madrid, ediciones el arquero/CSIC. ———. (ed.). 1988b. 1907–1987. La Junta para Ampliación de Estudios e Investigaciones Científicas 80 años después. Madrid: CSIC, 2 volumes. Sánchez Ron, José Manuel, and José García-Velasco, eds. 2010. 100 JAE. La Junta para Ampliación de Estudios e Investigaciones Científicas en su centenario. Madrid: Fundación Francisco Giner de los Ríos-Institución Libre de Enseñanza, Publicaciones de la Residencia de Estudiantes. Soler i Mòdena, Rosa. 1994. Catàleg del fons bibliogràfic Esteve Terradas. Barcelona: Institut d’Estudis Catalans. Suárez Cortina, M., and V. Salavert Fabiani. (eds.). 2007. El regeneracionismo en España. Valencia: Universitat de València. Usabiaga, Juan. 1922. En la Universidad central. El professor Hermann Weyl. El Sol, Madrid, 4 April 1922, p. 6. Valentín-Gamazo, Gonzalo Gimeno. 2015. La matemática de los quanta en España. El andamiaje de la física teórica en el intervalo (1925,1955) [borroso], Doctoral thesis. Barcelona: Universitat Autònoma de Barcelona. http://www.tesisenred.net/bitstream/handle/10803/313454/ vggg1de1.pdf?sequence=1. Weyl, Hermann. 1923. Mathematische analyse des raumproblems: vorlesungen gehalten in Barcelona und Madrid. Berlin: Verlag Von Julius Springer.

Chapter 2

Hermann Weyl chez Gaston Bachelard Un héritage riemannien Charles Alunni

J’aborderai ici de biais la question de l’ « École de l’ETH » dans l’œuvre de Gaston Bachelard, et plus spécifiquement de la figure spectrale d’Hermann Weyl. À son propos, je traiterai de sa place centrale et permanente dans la constitution bachelardienne d’une philosophie qui se veut à hauteur de la nouvelle « géométrie physique » rigoureusement construite dans un esprit riemannien. Une reconstruction de l’entreprise « surrationaliste » prend place ici. Posons dès le départ le syntagme « École de l’ETH » comme un marqueur spectral d’une constellation de pensées singulières, et donc uniques, mais néanmoins articulées (et non isolées dans un solipsisme) – Hermann Weyl, mais également Wolfgang Pauli, Ferdinand Gonseth ou Gustave Juvet. « Spectral » est ici à prendre d’abord comme opérateur de déclinaisons « discrètes » des singularités philosophiques exprimées dans chaque corpus référé, comme zone d « interférences » de domaines d’explication et de problématisation ; le modèle de cet « opérateur spectral » tient dans cette connexion « magique » du Janus mathématico-physique sous sa forme de physique mathématique. C’est la caractéristique « spectrale » purement mathématique de l’espace abstrait de David Hilbert (élaboré par lui dès 1910) qui permettra plus tard à Werner Heisenberg d’induire de manière géniale qu’il constitue en réalité la lisibilité possible des « formes » apparaissant sur le spectre de fréquences d’un corps (sur la base du principe de combinaison ou « loi de composition » de Walter Ritz et Johannes Robert Rydberg). Je rappelle que Walter Ritz est ce physicien suisse mort à 32 ans (1878–1909), qui enseigna à Zurich et Göttingen, est qui est cité par Bachelard dès 1931, comme un héraut de la « nouvelle physique »:

C. Alunni () Directeur du Laboratoire disciplinaire « Pensée des sciences », Paris Cedex 05, France e-mail: [email protected] © Springer Nature Switzerland AG 2019 J. Bernard, C. Lobo (eds.), Weyl and the Problem of Space, Studies in History and Philosophy of Science 49, https://doi.org/10.1007/978-3-030-11527-2_2

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C. Alunni « Mais voici que la Physique contemporaine nous apporte des messages d’un monde inconnu. Ces messages sont rédigés en “hiéroglyphes”, suivant l’expression de Walter Ritz. »1

Dès lors, tout corps physique trouve sa signature dans son « spectre quantique »2 . Le mathématicien Alain Connes n’a de cesse, dans nombre de ses conférences récentes, d’insister sur la « magie » de ce lien entre caractéristique « spectrale » d’un espace abstrait (espace de Hilbert) et la « spectroscopie » concrète en physique. Il renvoie ce nœud paradigmatique du physico-mathématique contemporain au « fantôme » de Georg Friedrich Bernhard Riemann (à son « spectre » en quelque sorte) et à sa révolution d’une géométrie littéralement « faite » pour la physique. Je note encore au passage que l’ « exemple pratique » et paradigmatique du « mixte » mathématique chez Albert Lautman n’est autre que ce même espace de Hilbert : continu pour la topologie de ses éléments ; discontinu pour ses décompositions structurales. Mais « spectral » renvoie également à ces présences « fantomales » qui hantent le grand œuvre de Gaston Bachelard, sous la forme de cette constellation épochale que Mario Castellana qualifie de « “néo-rationalisme” italo-francophone », et que je qualifie d’ « Internationale du surrationalisme ». Lieu d’instauration d’une immense « tradition » épistémologique, notre présent commence à en prendre toute la mesure. À titre d’exemplification « historique », l’un des référents hautement symbolique de cette constellation fut en son temps représenté par le groupe d’opposition théorique au Wiener Kreis lors du congrès Descartes de 1937 (Bachelard, « l’hôte muet » du congrès, Federigo Enriques l’Italien, Ferdinand Gonseth le Suisse, Jean Cavaillès et Albert Lautman les deux Français ; on pourrait y ajouter les « protagonistes anonymes » tels que Paulette Destouches-Février, Jean-Louis Destouches, André Lalande, etc.)3 . Nous allons voir maintenant comment Bachelard a très précisément pointé ce dispositif chez Riemann et chez Hermann Weyl. C’est dire que la mobilisation par Bachelard de ces différents dispositifs mathématiques, physiques et philosophiques n’y est pas plus occasionnelle que vague. Car il ne s’agit pas, comme c’est la règle chez nombre de ses contemporains philosophes (et bien des nôtres), de les exhiber comme de pures références autojustificatives ou illustratives d’une philosophie déjà refermée sur son propre système de présupposés. L’enjeu est bien plutôt de les

1 Gaston

BACHELARD, « Noumène et microphysique », in Gaston BACHELARD, Études, Paris, Vrin, 1970, p. 12. 2 Sur ce point, je renvoie au dialogue de Marc Schützenberger et Alain Connes dans Alain C ONNES , André LICHNEROWICZ, Marc SCHÜTZENBERGER, Triangle de pensées, Paris, Odile Jacob, 2000. 3 Voir sur ce point les Travaux du IXe Congrès international de philosophie (Congrès Descartes) de 1937. En particulier, Marie-Anne COCHET, Le Congrès Descartes (Paris-Sorbonne, 1937). Réflexions sur quelques-unes des orientations qui s’y sont manifestées, Bruges, Imprimerie Sainte Catherine, 1938 ; Mario CASTELLANA, Alle origini della « nuova epistemologia ». Il Congrès Descartes del 1937, Lecce, Il Protagora, 1990 ; Charles ALUNNI, « Le Congrès Descartes 1937 : l’arène philosophique européenne », in Actes de la Recherche en Sciences Sociales, Paris, Seuil, n◦ 141-142, mars, p. 130-131.

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habiter de manière active et « ouverte », de les accompagner dans l’ascétisme de leurs techniques spécifiques, pour leur faire sécréter in situ et in actu cette puissance spéculative, toujours « disponible », engagée dans leurs gestes de pensée : « [ . . . ] aucun spectre n’est plus étendu que le spectre qui aide à classer les philosophèmes des sciences physiques. Il est d’ailleurs bien entendu que toutes les parties d’une science ne sont pas au même point de maturité philosophique. C’est donc toujours à propos d’expériences et de problèmes bien définis qu’il faut déterminer les valeurs philosophiques de la science. »4

C’est donc en tant que « située » en des lieux et sur des nœuds textuels parfaitement identifiables, qu’il nous faut tenter de retracer cette présence spectrale de Weyl (ses enjeux théoriques, à la fois scientifiques et philosophiques – pour ne pas dire « métaphysiques » –, ses solidarités induites, ses potentialités produites). Bachelard s’est ressourcé en permanence auprès de penseurs comme lui, diffusant et prolongeant par là même son travail réflexif. C’est ce qu’il a fait également avec Albert Einstein, dès 1929, avec un texte absolument remarquable, La Valeur inductive de la relativité5 . Je dirais qu’Hermann Weyl apparaît dès l’origine comme une présence « initiatique » dans l’œuvre bachelardienne. Il est convoqué tant dans son Essai sur la connaissance approchée6 qui constitua sa thèse principale pour le doctorat présentée devant la faculté des lettres de l’université de Paris, le 23 mai 1927, que dans sa thèse complémentaire présentée le 28 mars de la même année, sous le titre Étude sur l’évolution d’un problème de physique. La propagation thermique dans les solides. Weyl ouvre et clôt pratiquement l’Essai en deux occurrences fondamentales. Penchons nous d’abord sur l’ouverture du chapitre concernant « Les formules de dimension ». Déployons, avant tout, les plis contextuels de cette référence inaugurale. Quelle est l’idée fondamentale de Bachelard dans ce chapitre ? C’est l’interrogation philosophique de la « nouvelle métrologie » face à la dualité absolu (de l’unité) et arbitraire (de la mesure). « Par mesure absolue, on ne doit pas entendre une mesure exécutée avec une précision particulière, ni par unité absolue une unité d’une construction parfaite ; en d’autres termes, en faisant usage des mots mesures ou unités absolues, on ne veut pas dire que les mesures faites ou les unités de mesure sont absolument parfaites, mais seulement que ces mesures, au lieu d’être établies par une simple comparaison de la quantité à mesurer avec une quantité de même espèce sont rapportées à des unités fondamentales dont la notion est admise comme axiome. 4 Gaston

BACHELARD, Le Rationalisme appliqué, Paris, Presses universitaires de France, 1949, p. 7. 5 À quoi il faut ajouter, Gaston B ACHELARD , « La dialectique philosophique des notions de la Relativité », in L’Engagement rationaliste, Paris, Presses Universitaires de France, 1972, p. 120136 ; originellement dans Albert EINSTEIN, Philosopher-Scientist, Paul Arthur Schilpp [éd.], Evanston, The Library of living philosophers, 1949, vol. II, p. 563-580, sous le titre « The philosophic dialectic of the concepts of relativity ». 6 Voir sur ce point, qui touche à la mathématique « pure », Charles A LUNNI , « Gaston Bachelard face aux mathématiques », in Revue de synthèse, « Philosophie et mathématique », Paris, Lavoisier, Tome 136, 6e série, N◦ 1-2, 2015, p. 9-32.

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C. Alunni Ainsi la métrologie est précédée, elle aussi, d’une véritable axiomatique puisqu’elle a pour base des éléments d’une pureté parfaite et posés arbitrairement. Ces éléments, comme des axiomes, seront seulement astreints à former un système cohérent, irréductible, et à être indépendants. Enfin, de même que diverses géométries dérivent de différents ensembles de postulats, de même des systèmes fondamentaux différents s’offrent pour soutenir toutes les mesures de la physique. »7

Le décor est planté et, à lui seul, permettrait déjà de lire en filigrane tout un ensemble de problèmes à venir et connexes, ainsi que d’autres faisceaux de citations. Comment intervient ici la première mobilisation de Temps, espace, matière de Weyl dans sa traduction Juvet-Leroy de 1922 ? « Ainsi [à propos de l’arbitraire masqué par des habitudes] croit-on que l’arbitraire soit éliminé de la définition de l’unité de volume dès qu’on lie cette unité à l’unité de longueur en choisissant le cube comme volume type ? La mémoire est évidemment soulagée puisqu’elle suit la pente de la géométrie élémentaire traditionnelle [ . . . ]. Mais il y a des points de vue qui s’éclairciraient peut-être avec un autre choix. Ainsi la sphère présente à certains égards des avantages rationnels indéniables. C’est elle qui est le volume de définition minima, sa symétrie est d’une richesse inépuisable. [ . . . ] De même encore dans un espace physiquement anisotrope, il peut y avoir intérêt à dilater ou à contracter certaines coordonnées suivant des fonctions plus ou moins compliquées. C’est un artifice souvent employé dans les nouveaux espaces généralisés. On peut toujours disposer des unités réunies en complexes pour retrancher des diverses mesures géométriques les coefficients numériques – ou tout au moins réduire tous les coefficients à l’unité précédée du signe + ou du signe –. Dans une forme quadratique, seuls les nombres des signes + et des signes – restent des caractéristiques invariantes (voir Weyl, Temps, espace, matière, p. 20). »8

La référence nous renvoie ici au chapitre I de Temps, espace, matière, « L’espace euclidien ; son expression mathématique et son rôle en physique », § 4 : « Les bases de la géométrie métrique ». Le renvoi de pagination est erroné, et il faudrait lire pages 24-27 ; Weyl y affronte les conditions de l’invariance des transformations linéaires orthogonales en coordonnées cartésiennes. C’est techniquement le lieu de passage d’une « théorie de l’invariance », pour des transformations linéaires avec conditions d’orthogonalité, à une théorie de l’ « invariance généralisée » dite « calcul tensoriel », corps mathématique de la relativité générale si minutieusement et si génialement étudié par Gaston Bachelard. Weyl conclut ce § 4 par son programme : « Nous développerons donc cette théorie de l’invariance [ . . . ] mais de telle manière qu’elle ne rende pas seulement possible l’étude des objets mathématiques, mais encore et surtout l’étude des lois physiques. »9

Bachelard reprendra très précisément le fil de ce programme weylien, deux ans plus tard, en 1929, dans La Valeur inductive de la relativité. Telle est la conclusion

7 Gaston

BACHELARD, Essai sur la connaissance approchée, Paris, Vrin, 1928, p. 85. BACHELARD, ibidem, p. 82. 9 Hermann W EYL , Espace, Temps, Matière, 1922, p. 27. Sur l’œuvre de Weyl et les tentatives contemporaines de sa reconstruction, voir SCHOLZ, Hermann Weyl’s Raum, Zeit, Materie and a general introduction to his scientific work, Bâle, Birkhaüser, 2001. 8 Gaston

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pro domo que tire, quelques pages plus loin, l’auteur de l’Essai après ce premier « passage » à travers Weyl : « Il semble qu’en allant des mesures aux idées, une connaissance se perde rapidement dans le logicisme [peu propre à susciter l’expérience]. C’est par une autre voie, en revenant de l’esprit vers les choses, qu’on pourra mobiliser encore la connaissance et lui donner la souplesse suffisante à toucher le réel. »10

Le point de contact est déjà gros de la trajectoire partagée. C’est la page 282 du chapitre conclusif de l’Essai, intitulé « Rectification et réalité », qui va déployer toute la puissance de connexion et de fibration des approches weyliennes et bachelardiennes. C’est la raison pour laquelle nous allons reconstruire plus patiemment les implicites du contexte. « Une géométrisation de la matière ne peut être un point de départ, c’est un schéma, c’est un but, bref une découverte tardive. En fait, dans la science contemporaine, l’étendue conçue a priori comme une qualité uniforme et générale a fait place à une étendue chargée de caractères et saisie par son côté différentiel. Et c’est maintenant l’élément différentiel qui détermine “l’explication”. C’est peut-être le trait le plus frappant de la nouvelle physique. L’idée de Riemann de définir la fonction mathématique par ses variations infinitésimales vient de pénétrer la physique elle-même. Et par un singulier retournement des principes qui va entraîner un véritable bouleversement de l’épistémologie, c’est la loi intégrale qui, de principe, devient la simple conséquence de la relation différentielle. Les “lois d’action de contact doivent être considérées comme la vraie expression des dépendances entre les actions qui s’exercent dans la nature” (Weyl, Temps, espace, matière, p. 55. La citation est tirée du chapitre I, § 9, “Le champ électromagnétique stationnaire”). »

Et la citation de Weyl continue ainsi : « “L’idée de comprendre l’univers par son aspect dans l’infiniment petit est la raison épistémologique qui anime la physique des actions de contact et la géométrie riemannienne” (Weyl, Temps, espace, matière, p. 79. La citation est tirée du chapitre II, “Le continuum métrique”, § 11 “Géométrie riemannienne”). »11

Dans le premier texte, Weyl compare « la loi de Coulomb comme loi d’action à distance, [qui] exprime que le champ en un point dépend des charges situées en tous les autres points de l’espace, les plus éloignés comme les plus proches », aux lois d’action de contact : « [ . . . ] beaucoup plus simples d’ailleurs [ . . . ], puisque pour la détermination de la dérivée d’une fonction en un point, il suffit de connaître l’allure de cette fonction dans un voisinage arbitrairement petit autour de ce point ; les valeurs de r [densité de charge] et e [le vecteur champ] en un point et dans un voisinage immédiat sont liées par les équations (51) ; [ . . . ] l’équation (49) ne doit être considérée que comme une conséquence mathématique des équations (51) : grâce aux équations (51) dont

10 B ACHELARD , 11 Gaston

Essai, 1928, p. 92. BACHELARD, Essai, 1928, op. cit., ibidem, p. 282.

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C. Alunni la signification intuitive est si simple, nous croyons comprendre d’où vient la loi de Coulomb. Certainement, nous obéissons avant tout à une contrainte d’ordre épistémologique. »12

La recherche de cette « contrainte épistémologique » est évidemment la finalité de Bachelard ; et c’est chez Riemann qu’il va en chercher l’axiomatique originaire. C’est aussi dans le rapport Riemann-Weyl que Bachelard exhibe en quelque sorte le « chiffre » de cette révolution géométrique de la physique véritablement imposée par Weyl, et tout particulièrement dans le cadre de la relativité générale. L’interface géométrico-physique à l’avenir prometteur est ainsi mis à nu, avec son déplacement fondamental du rapport a priori / a posteriori : « Des simples lois différentielles rot e = 0 et div e = ρ qui expriment que le rotationnel du champ électrique e est nul et que sa divergence en tout point est égale à la densité électrique en ce point, on déduit la loi de Coulomb suivant laquelle les corps électrisés s’attirent par une force en raison inverse du carré de la distance. La loi générale cesse donc d’être a priori, en facile accord avec un système de catégories, apparentées aux principes logiques, toute proche de l’intuition intellectuelle. Elle est, dans toute l’acception du terme, la conséquence d’un fait, mieux d’un nombre prodigieux de faits. Mais elle ne les résume pas, car elle s’embarrasse de constantes d’intégration. »13

Il n’y a pas moins de sept occurrences riemanniennes dans l’Essai. Cela n’est pas un hasard, pas plus qu’un accident, mais la conscience lucide du lien à Weyl. Reconstruisons partiellement le spectre épistémologique de ces références riemanniennes : « La définition de la fonction [riemannienne] par simple correspondance a ici encore une tout autre souplesse. “Cette définition, dit Riemann, ne stipule aucune loi entre les valeurs isolées de la fonction, car lorsqu’il a été disposé de cette fonction pour un intervalle déterminé, le mode de son prolongement en dehors de cet intervalle reste tout à fait arbitraire”. Ainsi la connaissance parfaite d’un être analytique dans un domaine déterminé n’implique plus la moindre connaissance en dehors de ce domaine. L’être, en Analyse, nous apparaît donc comme le résultat d’une construction qui, dans son principe, sinon toujours en fait, est une construction libre. « En analyse comme en géométrie, les conditions restrictives qui fixent les règles de la construction ne ruinent pas le caractère hypothétique de l’élément analytique défini. Ainsi, en une analogie curieuse, on retrouve pour définir une transcendante, les mêmes types de relations conditionnelles que dans l’Axiomatique de la géométrie. “Comme principe de base dans l’étude d’une transcendante, écrit Riemann, il est, avant toute chose, nécessaire d’établir un système de conditions indépendantes entre elles suffisant à déterminer cette fonction”. La transcendante n’établit ainsi entre ses éléments que les seules liaisons qui sont spécifiées par le système des conditions. Elle n’a pas de réalité en dehors de ce système qui doit être, comme un système de postulats, complet et fondamental. »14

C’est sur ce socle que s’ouvre toute une problématique à venir concernant les catégories de « réel », de « possible » et de « virtuel », catégories au travail dans toute

12 Hermann W EYL , Espace, Temps, Matière, 1922, p. 79. Il serait sans doute très stimulant de traiter

la question de ce que les théories de Feynman doivent à ce schème de l’« action de contact ». BACHELARD, Essai, op. cit., p. 282-283. 14 Gaston B ACHELARD , Essai, op. cit., p. 184-185. 13 Gaston

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l’œuvre, et particulièrement thématisées dans La Valeur inductive de la relativité de 1929. Et c’est ici un « constructivisme fonctionnel » que Bachelard pointe chez Riemann, qui pourrait se prolonger dans une « ontologie nécessairement projetée », dans « l’existence métaphorique » attribuée à l’être mathématique : une « ontologie constructive [qui] n’est jamais à son terme puisqu’elle correspond plutôt à une action qu’à une trouvaille » portant sur une « réalité de second ordre »15 . Mais ce qui intéresse également Bachelard, c’est ce que relèvera plus tard Albert Lautman dans sa propre thèse de 193716 , le fait que, des mathématiques, Riemann a incontestablement une conception structurale17 : « Certes, au sens de Riemann, qui est le sens profond, la fonction [mathématique] ne traduit que l’idée de correspondance »18 . Revenons à Weyl « géomètre de la matière » et à l’importance des lois d’action de contact riemanniennes qu’y relève Bachelard. On y voit une forme de communauté spectrale se dégager, quant à certaines conséquences épistémologiques. Le lien Riemann-Weyl tient ici au fait que, du point de vue de la théorie de la connaissance, Riemann a su imaginer que l’infiniment petit renfermait beaucoup plus d’informations essentielles sur la nature, que l’infiniment grand. Il pense à une sorte de solidarité profonde, non contingente entre les modèles mathématiques de l’infiniment petit et les lois physiques suivant lesquelles cet infiniment petit s’exprime et se manifeste dans la nature des phénomènes. Weyl a reconnu à la base de la nouvelle géométrie différentielle de Riemann les mêmes principes théoriques qui ont animé la nouvelle physique des actions de contact. D’où la possibilité d’établir un parallélisme entre la géométrie de Riemann et la physique de Maxwell, que relève à son tour Bachelard comme déterminant le tournant de la « nouvelle physique ». Dans un article important (contemporain de la première édition de Raum, Zeit, Materie, 1918) où Weyl entreprend le projet d’élaborer une « pure

15 Gaston B ACHELARD , La Valeur inductive, 1929, p. 186. Sur le statut singulier de la métaphore scientifique (et philosophique) chez Bachelard, voir Charles ALUNNI, « Pour une métaphorologie fractale », Revue de synthèse, Paris, Albin Michel, n◦ spécial : « Objets d’échelles », t. CXXII, 1, janvier-mars, p. 154-171. 2001. 16 « Essai sur les notions de structure et d’existence en mathématiques », in Albert L AUTMAN , Essai sur l’unité des mathématiques et autres écrits, Paris, 10/18, 1977, p. 23-154. 17 Albert L AUTMAN , « Les schémas de structures », in Essai, ibidem, p. 31-86. 18 Gaston B ACHELARD , Essai, op. cit., p. 201. « Le rationalisme est une activité de structuration. Si Bachelard n’a pas consacré d’étude spéciale à l’épistémologie structurale, c’est que toute sa recherche épistémologique est précisément structurale, ce n’est pas faute, on en conviendra, d’ignorer que la mathématique contemporaine est purement – mais non simplement – formelle, opérationnelle, structurale (voir La Philosophie du non, p. 133) », in Georges CANGUILHEM, « Dialectique et philosophie du non chez Gaston Bachelard », Revue internationale de philosophie, 66, fasc. 4, 1963, p. 441-452. Ce texte important, l’un des très rares à interroger sérieusement le concept bachelardien de « dialectique », a été repris dans Georges CANGUILHEM, Études d’histoire et de philosophie des sciences, ici 5e éd. augmentée, Paris, Vrin, 1968 p. 202.

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géométrie infinitésimale » (reine Infinitesimalgeometrie) et où, par là, il poursuit celui de construire une théorie généralisée purement géométrique des phénomènes physiques, il affirme : « La théorie de la relativité générale admet, conformément à l’esprit de la physique moderne, des actions de contact seulement de ce qui a une validité dans l’infiniment petit [c’est-à-dire localement], et en ce qui concerne la métrique de l’Univers (Weltmetrik), elle fait appel au concept général de détermination métrique fondée sur une forme différentielle quadratique, proposé par Riemann dans son Habilitationsvortrag de 1854. Mais l’élément vraiment important de cette nouveauté est la vue selon laquelle la métrique n’est pas une propriété de l’Univers en soi (an sich) ; bien plutôt, comme forme des phénomènes, l’espace-temps est un continuum quadridimensionnel complètement amorphe, au sens de l’Analysis situs, la métrique exprimant toutefois quelque chose de réel qui a une existence dans l’Univers, exerçant des effets sur la matière par le biais de forces centrifuges et gravitationnelles et dont l’état, inversement, est également conditionné, selon les lois de la nature (naturgesetzlich), par la distribution et la constitution de la matière. »

Enfin Weyl conclut : « D’après cette théorie [la “géométrie infinitésimale pure”], tout ce qui est réel, c’est-à-dire tout ce qui existe dans l’Univers, est une manifestation de la métrique de l’Univers ; les concepts de la physique ne sont pas quelque chose d’autre que ceux de la géométrie (die physikalischen Begriffe sind keine andern als die geometrischen). La seule différence entre la géométrie et la physique tient dans ce que la géométrie sonde de manière générale la nature essentielle des concepts métriques, mais c’est la physique qui, de son côté, enquête sur la loi en vertu de laquelle l’Univers réel se distingue de tous les espaces métriques quadridimensionnels possibles, d’après leur géométrie. »19

Au plus près de la pensée de Riemann, de la manière la plus cohérente et la plus profonde, Weyl développe ici l’idée philosophique que la métrique exprime à la fois un élément a priori et a posteriori de l’espace. On voit ainsi que « la conception riemannienne ne néglige pas l’existence d’un élément a priori dans la structure de l’espace ; seulement, la frontière entre l’a priori et l’a posteriori se trouve déplacée ». Nous ne pouvons ici analyser tout ce que cela implique de « déplacement » par rapport au kantisme – ce que Bachelard thématisera sous le concept de non-kantisme20 . Que tire de son côté Bachelard du « bougé » riemannien opéré par la relation différentielle de la loi d’action de contact ? « La loi générale [de Coulomb] cesse donc d’être a priori [ . . . ]. On objectera que le général est tangent au particulier, que les cadres euclidiens sont une première simplification du donné infinitésimal lui-même. Mais un système de référence euclidien qu’on doit transporter de proche en proche d’une manière en somme non-euclidienne pour suivre la pseudo-généralité a-t-il vraiment la valeur euclidienne qu’on lui attribue ? La description sur place pourrait peut-être rentrer dans le cadre euclidien en première approximation. Mais il s’agit d’une description essentiellement relative, c’est-à-dire qui doit servir ailleurs et en un autre temps, qui doit lier par la pensée les états successifs et prochains du réel. Le mouvement descriptif doit donc se plier sous la courbure de l’Univers. Il en résultera une géométrie a posteriori, post-expérimentale, qui n’aura pas la valeur de prévision qu’on

19 Hermann

WEYL, Espace, Temps, Matière, op. cit., p. 384 et 411. Gaston BACHELARD, La Philosophie du non. Essai d’une philosophie du nouvel esprit scientifique, Paris, Presses Universitaires de France, 1940.

20 Voir

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attribuait à une géométrie informatrice a priori, mais qui, en échange, sera apte à enregistrer le discontinu du devenir et de l’être [ . . . ]. La matière nous apparaît donc sous la forme d’une contingence en quelque sorte feuilletée. »21

Cet écho potentialisant à Weyl peut être qualifié comme la marque typique (et topique) d’un grand pouvoir d’« auscultation » philosophique, répondant aux « méthodes d’auscultation utilisées par le mathématicien » comme il le formulera dans sa conclusion. Ce qu’il « ausculte » ici du rapport Riemann-Weyl, « c’est l’allure rectificative d’une pensée. Rien de plus clair et de plus captivant que cette jonction de l’ancien et du nouveau. La rectification est une réalité, mieux, c’est la véritable réalité épistémologique (la “contrainte” de Weyl), puisque c’est la pensée dans son acte, dans son dynamisme profond. »22

Il faudrait trouver le temps d’analyser à nouveau la présence récurrente de Weyl dans La Valeur inductive de la relativité de 1929. Il s’agit ici du Weyl de la constitution d’une « véritable géométrie du caractère électrique [ . . . ], en liaison réciproque avec les caractères purement mécaniques de la Relativité générale » : c’est ce qu’il qualifie de « fusion de Weyl », comme tentative d’assimilation de l’électrique au dynamique. Il en tire à nouveau une leçon pour l’épistémologie : « La méthode de M. Weyl consiste essentiellement dans un élargissement de l’axiomatique. » Puis, solidairement, Bachelard pointe la « géométrie des jauges » : « Avant les travaux de M. Weyl, [ . . . ] l’unité de longueur gardait la même valeur après un cycle fermé de transformations dans l’espace. Qu’on abandonne maintenant le postulat de l’intégrabilité de la longueur, et dans la pangéométrie ainsi constituée (“la géométrie des jauges”) on se rendra compte que le champ électromagnétique est entièrement définissable par les moyens algébriques. »23

Conscient des difficultés soulevées, il n’en conclut pas moins « que le sens de la tentative de M. Weyl doit retenir l’attention de l’épistémologue. Cette tentative est propre, croyons-nous, à préparer cette conclusion : l’unité mathématique qui se constitue dans une axiomatique de la Physique commande entièrement l’unité du phénomène. »24

C’est pour défendre plus loin Hermann Weyl dans un débat avec Stanislaw Zaremba qui porte précisément sur « l’axiomatisation » et sur la définition d’un corps rigide en relativité générale. Il insiste positivement sur la définition « toute en virtualité » de Weyl. Enfin il aborde la dialectique « généralités/spécifications » dans sa théorie unitaire, pour terminer avec la « soudure axiomatique » de Weyl, « axiomatique » dans laquelle Weyl a découvert « la trace des potentiels électriques » (à nouveau sa théorie de jauge). La restitution de cette économie « en partage » est essentielle, mais difficile, faute de place, à exposer ici.

21 Gaston

BACHELARD, Essai, op. cit., p. 283. BACHELARD, ibidem, p. 300. 23 Gaston B ACHELARD , La Valeur inductive de la relativité, op. cit., p. 136-137. 24 Gaston B ACHELARD , ibid., p. 137. 22 Gaston

Chapter 3

Le résidu philosophique du problème de l’espace chez Weyl et Husserl Un crossing-over épistémologique Carlos Lobo

« Morgan has explained linkage by the process of crossing-over. ( . . . ) Crossing-over consists in breaking the joins ab and a*b* and joining instead a with b* and b with a*. ( . . . ). Linkage between two points a, b of a chromosome will be the looser the more ways there exist to separate them by crossing-over. ». Weyl, Philosophy of Mathematics and Natural Science.

La position philosophique et scientifique de Weyl est indéniablement placée sous le signe de la singularité. Sa philosophie des sciences ne relève pas du genre « philosophie spontanée des savants ». Du reste, contrairement à nombre de ses confrères et comme nous le voyons par ses déclarations et ses lectures, la philosophie ne se réduit pas chez lui à l’imposition d’une Weltanschauung « scientifique » (positiviste, scientiste ou technoscientifique). Weyl ne se contente pas d’une mise en scène des gestes et des concepts scientifiques présentés, selon la circonstance et le public, sous tel ou tel habillage philosophique. Pas d’usage intempestif de mots en « –isme », visant à revêtir de l’austère vêtement de la tradition une simple conviction personnelle plus ou moins articulée et réfléchie. Aussi déroutant, instable, voire inconsistant qu’il puisse paraître aux yeux du lecteur pressé de trancher (ce n’est pas toujours le fait du philosophe de métier), son rapport à la philosophie n’a rien d’épisodique, d’irréfléchi et d’incohérent. Weyl est pleinement philosophe et l’insaisissabilité de son profil tient à une position placée sous le titre

C. Lobo () Directeur de programme au Collège International de philosophie, Paris, France Centro de Filosofia das Ciências da Universidade de Lisboa, Faculdade de Ciências da Universidade de Lisboa, Lisbon, Portugal © Springer Nature Switzerland AG 2019 J. Bernard, C. Lobo (eds.), Weyl and the Problem of Space, Studies in History and Philosophy of Science 49, https://doi.org/10.1007/978-3-030-11527-2_3

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d’« interpénétration »1 (entre philosophie, mathématiques et sciences de la nature) qu’il exprime en des formules définitives. Il assume ainsi explicitement un héritage kantien. Pour cette raison, sa philosophie ne tombe pas tout uniment dans le registre d’une « pensée des sciences » en réserve et en attente que la finesse analytique de l’épistémologue ressourçant inlassablement sa puissance spéculative et dialectique à l’école des sciences et s’astreignant, pour ce faire, à une connaissance de première main des théories scientifiques, afin de nous montrer la genèse des concepts théoriques nouveaux tirés de la réserve des gestes ou des expériences de pensée2 . Sous ses dehors de rupture avec la théorie de la connaissance néo-kantienne, il est permis de se demander dans quelle mesure cette attitude n’hérite pas, finalement, d’un certain pessimisme de la raison spéculative plus radical encore, selon lequel tout usage de la raison serait fatalement condamné à l’errance et à l’illusion, dès lors qu’il prétend s’affranchir des conditions de construction des concepts et/ou de celles de l’intuition sensible (intuition géométrique, perception de signes ou perception sensible). La situation posée par la Critique de la raison pure constitue le véritable point de départ de Weyl, ce qui n’est pas sans incidence sur le partage du problème de l’espace qui nous occupe ici3 . Sur ce problème plus qu’aucun autre, il est manifeste que la réflexion spéculative chez Weyl n’est pas à ce point inchoative, qu’il exigerait l’intervention de cet ouvrier de la dernière heure qu’est l’épistémologue, tout à la tâche, noble et indispensable par ailleurs, d’extraire de la technicité de la construction symbolique mathématique, le potentiel de concepts essentiels pour les traduire – avec une déperdition plus ou moins importante – dans la langue philosophique, elle-même rédimée par cette opération. Loin que les gestes de pensée se déploient de façon muette et en-dehors de toutes coordonnées philosophiques, ils sont eux-mêmes intimement institués et instruits de première main, dès le départ, à partir d’une tradition philosophique dont Weyl manifeste à la fois une connaissance intime et une fréquentation assidue, même si l’on peut être dérouté par l’éclectisme apparent. Cela explique la résistance qu’il oppose à une analyse (spectrale) des positions philosophiques comme aux catégorisations d’une histoire de la philosophie ou d’une histoire des sciences prisonnières d’une typologie ou de coupures trop rigides. La position et l’analyse du problème de l’espace sont exemplaires à cet égard. En raison du rôle historique de la géométrie comme modèle d’une connaissance rationnelle, (il le fut pour les mathématiques, la physique et la philosophie), il est particulièrement instructif de suivre la manière dont Weyl pose, analyse et résout le

1 Cf.

Weyl, Philosophy of Mathematics and Natural Science, abrégé ci-dessous en PMNS (1949: v); Weyl, (1954: 31); Scholz (2006); Scholz (2005: 335–339). 2 Cf. Châtelet (1993: 53, 72, 82, 93), Kerszberg (2014: 27–28). 3 Cf. Weyl, 1955. Cf. Julien Bernard, in (Weyl, 2016), Voir sur ce point le magistral parcours de P. Kerszberg (2014): 269.

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problème de l’espace, car elle manifeste en gestes et en formules, le partage et la connexion entre science et philosophie ici à l’œuvre4 . En gestes et en formules, car Weyl ne se contente pas d’illustrer ce qu’est l’interpénétration qui, selon lui, représente l’idéal d’une pratique scientifique sensée et d’une réflexion philosophique instruite. il aura parsemé son œuvre de scientifique et de philosophe de formules fulgurantes, parfaitement ciselées qui représentent autant de « théorèmes épistémologiques », et dont la puissance d’élucidation critique et la valeur heuristique s’éprouvent dans les applications à des contextes scientifiques précis et les plus brûlants, à commencer par la formule qui désigne le problème de l’espace comme le lieu de manifestation exemplaire de l’interpénétration entre philosophie et sciences. En raison de son importance, il n’est pas inutile de la rappeler : « Nulle part les mathématiques, les sciences de la nature, et la philosophie ne s’interpénètrent aussi intimement que dans le problème de l’espace. » Pour mesurer sa portée et son sens, il faut se garder de la couper de ses attendus, à savoir que les « présuppositions pour la discussion de ce problème », loin d’être abandonnées au philosophe, « sont apparues dans le cours de la recherche mathématique » elle-même, et qu’en outre Weyl se propose de les mettre en relief. Ce qui se signale ici à nous sous le titre de « présupposition », c’est la zone de transaction profonde entre mathématiques et philosophie ; mais aussi une position philosophique qui, sous une apparence d’éclectisme, résulte de la composition de plusieurs courants d’un spectre philosophique, entre autres de l’idéalisme moderne (Kant, Husserl, Fichte, Dilthey et Leibniz), et leur assigne aussi une fonction dans une vision historique et dynamique des interactions entre physique, mathématique et philosophie dans la réflexion sur ces présuppositions5 . En raison de leur « contemporanéité »6 , la relation complexe à Husserl mérite une attention particulière. Cette « contemporanéité » est indissociablement épistémologique et spirituelle ; et au risque de paraître énoncer des truismes, ajoutons qu’il s’agit d’une relation commutative et transitive. Commutativité : Husserl est contemporain de la théorie de la relativité dans la mesure exacte où Weyl est contemporain de la phénoménologie husserlienne. Transitivité : la prise en charge et la résolution d’une partie de l’a priori par les mathématiques et la physique mathématique se traduit, de manière complémentaire, par une nouvelle délimitation et une redirection du travail d’élucidation philosophique de l’a priori résiduel. C’est dans ce contexte que la phénoménologie apparaît comme un enrichissement de l’a priori esthétique kantien7 , i.e. desdites formes a priori de l’intuition. Cette contemporanéité reste cependant tributaire de contingences historiques, au sens le plus ordinaire, et n’échappe pas au principe bien compris de la relativité. 4 For

a general and systematic overview, see Boi, 1995a. (1949) « Relativity theory as a stimulus in mathematical research », Proc. Amer. Philos. Soc. 93 , (1949), pp. 535–541 ; Ges. Abh. , 4, pp. 394–400 ; Scholz (2009 : 215 passim) ; Hawkins (1998 : 70) 6 Sur cette notion de contemporanéité épistémologique entre Einstein et Husserl, cf. Châtelet; 2010. Et mon commentaire dans Lobo, 2017b. 7 Weyl, (1949 : 134) 5 Weyl

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La lente et équivoque genèse de la phénoménologie transcendantale, la rigueur et la difficulté de sa méthode, l’état du corpus husserlien, de sa publication, le tumultueux héritage de la phénoménologie, dans le contexte des guerres les plus meurtrières de l’histoire de l’humanité, etc., expliquent qu’il y ait en la circonstance, comme pour la théorie de la relativité générale, des phénomènes de retard ou de préfiguration, comme il en va en science, où selon Weyl, Riemann n’apparaît (que) comme le prophète d’Einstein8 , en raison de circonstances historiques similaires et parallèles. Il reste que le rapport de Husserl à la référence kantienne est l’inverse de celui de Weyl. Ce qui explique sans doute certaines ambiguités ou certains flottements qu’on a pu observer dans le traitement de la référence phénoménologique, explicite ou implicite. D’une position anti-kantienne (ou si l’on préfère a-kantienne), Husserl a peu à peu évolué dans le sens d’une appropriation de plus en plus fine de ce que les néo-kantiens (Cassirer ou Grete Hermann, par exemple)9 avaient tendance à abandonner. Inversement, Weyl partant d’une référence kantienne, qui précède et provoque sa vocation scientifique, n’a cessé de travailler dans l’horizon de la référence kantienne, que ce soit dans sa traversée de la phénoménologie husserlienne ou son appropriation de la référence Fichtéenne. Mais ce n’est pas le point le plus important, cette différence de traitement et de position vis-à-vis des coordonnées kantiennes s’inscrit dans une différence plus profonde. Deux traversées des champs disciplinaires enveloppés dans le problème de l’espace. Dans un cas, une trajectoire (celle de Husserl) qui mène des mathématiques à la philosophie en passant par la psychologie et la phénoménologie. Dans l’autre, un parcours qui a son point de départ dans la philosophie kantienne, et retourne à la philosophie, en passant par les mathématiques et la physique (Weyl). Dans l’un et l’autre cas, la dimension logique du problème se trouve mentionnée en relation avec la question des fondements ou présupposés de la géométrie. C’est le cas chez Weyl, dès l’article de 1910 : Über die Definitionen der mathematischen Grundbegriffe10 , comme chez Husserl, bien avant les textes souvent cités de la Krisis, dans ses leçons de 1890-92 sur la question de l’espace. Comme l’atteste la correspondance, les années 1918–1920 sont des années décisives pour l’un et l’autre. La publication de l’essai de Weyl sur Le continu et la première édition de Raum. Zeit. Materie (1918) semblent annoncer un débouché scientifique pour la phénoménologie, comme en porte témoignage l’enthousiasme avec lequel Husserl l’accueille : « Enfin un mathématicien qui montre qu’il comprend la nécessité, dans toutes les questions relatives à l’élucidation des concepts fondamentaux, de modes de considération phénoménologiques, et qui se retrouve, par conséquent, sur le sol originaire de l’intuition logico-mathématique » 11

8 Weyl,

(1952/2009: 102). Ainsi que Mathematische Analyse des Raumsproblems, Vorlesungen gehalten in Bercelona und Madrid, Berlin, Springer, 1923; traduction française: Weyl (2016: 324). 9 Hermann (2000: 115–119) Cassirer (1972 : 461 sq.) 10 In Mathematisch-naturwissenschaftliche Blätter, 7, 93–95 et 109–113 (1910). Weyl, GA I , p. 299 11 Lettre de Husserl à Weyl, d’avril 1918. Cf. Lobo, 2009.

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Il n’en reste pas moins que la position de Weyl vis-à-vis de Husserl est comparable à celle d’un Leibniz ou d’un Pascal à l’égard de Descartes. Ce que dit la phénoménologie de l’essence de l’espace est « en gros » vrai. Mais son exploration a priori du détail de la constitution génétique de l’espace mathématisée, si elle prétend se dispenser du passage par la construction symbolique, est sinon ridicule, du moins précipitée et « incertaine ». Weyl épingle ainsi les exemples avancés par Husserl pour illustrer l’opposition entre loi contingente de la nature (« tous les corps sont lourds ») et loi d’essence nécessaire (« tous les corps ont une extension spatiale »). « Il a sans doute raison », commente Weyl, « mais on sent, même sur le premier exemple, à quel point deviennent incertaines les distinctions épistémologiques posées dans leur généralité dès qu’on descend de la généralité aux applications concrètes particulières. »12 Ces doutes rejoignent le diagnostic sur l’impatience du métaphysicien, qui vaut également, semble-t-il, pour le phénoménologue, qui croit pouvoir accéder à l’essence de l’espace sur la base d’une seule intuition exemplaire 13 . Le caractère a priori des relations d’essence apparaît lui-même comme douteux : « dans quelle mesure ces aspects manifestes à la conscience sont-ils l’expression d’une structure a priori et dans quelle mesure est-ce une simple affaire de convention ? » 14 . Les trajectoires de Weyl et de Husserl se croisent précisément en ce point, et si une connaissance superficielle de la phénoménologie husserlienne semble indiquer une divergence, une connaissance plus approfondie et débarrassée de certains préjugés nous conduit à une prise de conscience progressive de leur affinité et de leur complémentarité. Le rôle que vient jouer la référence leibnizienne chez l’un et chez l’autre est un marqueur précieux à cet égard15 . Pour nous en tenir à son schéma le plus pauvre, nous dirons que, sur la position du problème de l’espace, se produit un crossing-over ou enjambement. On parle de crossing-over lorsqu’une liaison génétique (un linkage) se produit par disjonction et jonction croisée de deux allèles appartenant à des gènes distincts, qu’on peut considérer ici comme des points, ou des « influences », soit : (a, b) et (a*,b*). Plus il y a de manières de séparer les points et plus la liaison génétique sera faible et inversement. Schématiquement, et en réduisant les éléments génétiques à Kant (K) et Riemann (R) : nous aurions pour Weyl (K, R) et Husserl (R*, K*). L’enjambement épistémologique (R, R*), (K, K*) signale dans ces conditions une liaison génétique

12 Weyl,

(1954/2009 : 212) Cela est imputable à l’impatience des philosophes qui, sur la base d’un seul acte d’intuition exemplaire, croient pouvoir donner une description adéquate de son essence. » 14 Outre la référence, elle-même convenue, à Poincaré (cf. supra.) il est possible que Weyl ait ici en tête l’entreprise néo-kantienne d’un Dingler. « Sa thèse : une fondation complète qui n’est pas exposée à une infinie régression par la question ( . . . ) ne peut être visée qu’en s’appuyant uniquement sur des circonstances qui dépendent uniquement de moi, c’est-à-dire sur des conventions arbitraires (freiwillige Festsetzungen), sur une synthèse pure » . (Weyl 1925a: 872] (Weyl: Compte rendu de : Dingler, H., Physik und Hypothese, Jahrbuch über die Fortschritte der Mathematik, 48 1921–1922, 871.) 15 Scholz, E. 2008. 13 «

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des plus fortes, si l’on veut bien considérer que, selon les termes mêmes de Weyl, « cette ligne trouve son point culminant dans la phénoménologie de Husserl, dans laquelle l’a priori est bien plus riche que dans le système kantien ». Ce schéma, très pauvre, est appelé à se compliquer en y introduisant d’autres « influences », telle que celle de Herbart16 , de Fichte, de Helmholtz (1867, 1883) ou de Grassmann (1844). Pour autant qu’il y a enjambement épistémologique, cela n’abolit pas les exigences historiographiques élémentaires, telle que la lecture des textes (y compris la prise en compte des lectures réciproques entre Weyl et Husserl). Mais la multiplication des références respectives et de leurs usages (Leibniz, Kant, Descartes, Einstein) produit un effet de relâchement de la liaison génétique sans la supprimer. La prise en compte de telles possibilités d’enjambement épistémologique nous invite à aborder avec plus de circonspection encore les catégories généalogiques de l’histoire des sciences. Il ne suffit pas de se méfier des anachronismes ou des réécritures a posteriori (Gauss à la lumière de Riemann, et ce dernier à la lumière des acquis ultérieurs), de proscrire les anticipations (préfigurations et prophéties, Riemann par rapport à Einstein). L’application de ces notions comme celles de « contemporanéité épistémologique » ne peuvent se décider qu’au terme d’une analyse interne de la position et composition du problème et des opérations mises en œuvre pour le résoudre. À cette complication s’ajoute une complication proprement épistémologique qui constitue un enjeu polémique majeur, et que présuppose toute position de problème scientifique : le partage entre l’a priori et l’a posteriori. Par Weyl, nous apprenons, explicitement et en acte, que c’est là une frontière doublement mouvante, et cela de manière remarquable en ce qui concerne le problème de l’espace.

3.1 Position et analyse du problème de l’espace chez Weyl 3.1.1 Un point de départ kantien équivoque L’ébauche d’autobiographie intellectuelle que Weyl propose dans les dernières années de sa vie a valeur testamentaire. « Je n’ai jamais oublié comment, durant mes deux dernières années de scolarité, il m’est arrivé de tomber, dans le grenier de la maison familiale, sur une copie d’un bref commentaire sur la Critique de la raison pure de Kant, datant de 1790, jaunie et roussie à force d’avoir été entreposée là. L’enseignement de Kant sur l’« “idéalité de l’espace et du temps” » exerça un puissant effet sur moi ; je fus soudainement tiré de mon “sommeil dogmatique” et l’esprit du jeune garçon trouva le monde mis en question d’une manière radicale ».17 16 A

titre de complication supplémentaire, il faut tenir compte du fait que Kant n’est pas absent chez Riemann, Merker, (1982 : 15–19) qui souligne l’influence de Herbart qui critique Kant sur le point qui nous occupe. Voir également les études d’E. Scholz, (1982b ; 1985 ; 2001), sans oublier le Riemann philosophe (Scholz, 1982c et Boi, 1995b, 1996). 17 (Weyl, 1954 : 205)

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Même si ce genre de déclarations sont à prendre cum grano salis, et qu’un point de départ personnel ne coïncide pas nécessairement avec un commencement scientifique, dans le cas d’espèce, par sa cohérence même, la thèse posant l’idéalité de l’espace et du temps, telle qu’il la trouve chez Kant, constitue bel et bien le fil conducteur du travail scientifique de Weyl, et l’idée de l’idéalité, le point de fuite de ses exceptionnelles contributions en théorie des groupes, en géométrie différentielle, en topologie, en algèbre, en physique (en théorie de la relativité, et en mécanique quantique). Parmi les multiples passages où Weyl pose explicitement le problème de l’espace en tant que problème philosophique, mentionnons celui qui forme l’ouverture de la première conférence18 de l’Analyse mathématique du problème de l’espace. Comme cela semble être la règle, la présentation du « problème philosophique » (philosophische Raumproblem) est assez succincte. Mais compte tenu de l’enchevêtrement ou, plus précisément, de l’interpénétration déjà mentionnée, il ne faut pas s’étonner d’en voir le fil ressurgir plus loin, au beau milieu d’un développement mathématique, ou encore, comme c’est ici le cas, qu’une partie de ce problème trouve une traduction mathématique. Le point de départ du problème n’est autre que le « classique » problème de l’individuation. La mobilisation de la référence kantienne est équivoque à deux titres au moins. À bien y regarder, elle n’est pas purement kantienne. Weyl pose une distinction, au sein de la réalité, entre « contenu qualitatif » (c’est-à-dire les phénomènes, voire la « matière » au sens phénoménologique) et « forme », entendue comme extension spatio-temporelle ; ce qu’il justifie aussitôt par une variation que ne renierait ni Locke19 ni Descartes, ni plus généralement aucun des philosophes modernes ayant pris acte de la révolution galiléenne. La variabilité et la relativité de la « localisation » spatio-temporelle ne sont nullement la marque d’un défaut d’objectivité, mais la condition de « discernabilité » du « contenu » qualitatif ; de là le glissement de l’espace comme « forme a priori » de l’intuition à l’espace comme « milieu extensif du monde extérieur » (spatial et temporel) pour les corps, parallèle au glissement du « contenu qualitatif » en tant que « divers de la sensation », aux « corps réels » dotés de leurs propriétés physiques, voire à la matière prise substantivement ; le premier assumant le rôle d’individuation du second. « C’est ainsi qu’il est possible, dans le milieu extensif du monde extérieur (dans lequel il faut compter non seulement l’espace mais aussi le temps), de distinguer des choses individuellement, qui, sont semblables par essence, du fait de leurs propriétés. »20

D’autre part, quand bien même nous défendrions une conception intuitionniste (au sens kantien) de la construction mathématique, la possibilité de traduire mathématiquement les « formes a priori » (principes d’individuation spatio-temporelle) 18 Celle-ci

reprend presque terme à terme l’article « Das Raumproblem », Jahresbericht der Deutschen mathematikervereinigung 30, 92–93 (Weyl, 1921), in GA II, Texte Nr 45. Weyl (1968a II : 212–213) 19 Sur cette référence à Locke, cf. Weyl (1952/2009: 121–122) et Weyl (1949 : : 97, 111). 20 Weyl (2016 : 9).

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en termes de groupe de « congruences » qui sous-tendent la structure métrique revient à en contredire la lettre et l’esprit. L’individuation en jeu, comme nous le verrons plus en détail, est sous-tendue par un système d’assignation des « places » (spatio-temporelles), qui présuppose à son tour la possibilité d’une identification des places elles-mêmes (autrement indiscernables) par l’affectation arbitraire mais fixe, d’un système de nombres (doublets, triplets, quadruplets, . . . ) formant la « carte d’identité » des éléments ainsi paramétrés. Ce système est un « espace numérique »21 . Une telle individuation mathématique n’a pas de caractère absolu, mais seulement un sens relatif. Très exactement : elle n’a de sens que pour autant que l’on introduit un système de coordonnées, dont les « axes » et les coefficients restent, à ce stade, indéterminés. La détermination du nombre de ces « axes » (ou dimensions) comme celle des coefficients incombera aux mathématiques. Le recours à un tel supplément de détermination mathématique ne se justifie que par le défaut ou l’insuffisance de déterminations intrinsèques ou de propriétés manifestes. À ce stade, le partage entre propriétés apparentes et propriétés réelles n’est pas encore fondé. C’est pourquoi, d’ailleurs, il est justifié de maintenir un certain flottement quant au statut du « contenu qualitatif » venant « remplir » ces « formes » spatio-temporelles : « contenu de sensation » ou « propriétés d’un corps », en tant que « propriétés sensibles susceptibles d’être éprouvées » ou encore « événements ». Enfin, la possibilité même de substituer à la forme a priori de la sensibilité une structure métrique semble rendre inutile toute forme a priori et contredit l’irréductibilité que Kant assigne aux dites « formes » (formes de l’intuition aussi bien que formes de la pensée) et par suite le caractère primitif et non-dérivable qu’il attribue à l’intuition formelle. Le « formel » dont il est ici question doit être entendu, sans anachronisme, en sens strictement kantien, comme un synonyme de transcendantal, c’est-à-dire comme relevant du « synthétique a priori » non construit, non mathématique. Et pourtant, un tel point de départ reste fidèle à Kant sur un point essentiel : l’a priori de l’intuition est lui-même l’index d’un a priori dont le siège reste la subjectivité. À nous en tenir à cette première formulation du problème philosophique de l’espace, on ne voit guère en quoi consiste ce résidu kantien. À la bipartition en deux sources hétérogènes de connaissances (sensibilité et entendement), chacune dotée d’éléments initiaux purs et a priori (de « formes ») et de contenus dérivés (sensations et concepts empiriques ou construits), Weyl substitue une tripartition originale, qui exprime la nouvelle solution au problème mathématique de l’espace : espace et temps comme médium extensif du monde extérieur ; la structure métrique et le remplissement matériel par des qualités changeantes de lieu en lieu. Les ambiguïtés signalées plus haut sont reprises ici. Elles annoncent à la fois la solution apportée par l’analyse mathématique et l’analogie juridico-politique sur laquelle nous reviendrons ; elle ne dit rien encore à ce stade de la « nature de la métrique », ni de la latitude de variabilité accordée au « contenu matériel ». 21 Cette

identité « administrative » (symbolique et juridique) reste une application (géographique et historique) de la première.

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Mais ce problème est si peu kantien qu’il prend une tournure nettement critique à son égard. Se trouve en particulier contestée, la thèse (a) de l’espace comme « forme de l’intuition » ; (b) sinon, son statut transcendantal de « forme a priori » ; Weyl soupçonne qu’une investigation empirique plus exacte (et plus patiente) inscrite dans la lignée d’un Helmholtz, conduirait à destituer l’espace de ce statut et le ravaler au rang de structure, dont la genèse empirique peut et doit être retracée22 ; (c) enfin, dans cette perspective, son unicité même comme forme « commune » est douteuse, et il est légitime de se demander si l’unité supposée de l’espace « commun » ne se pas à la réduirait coordination d’espaces sensoriels distincts (visuel, tactile), entre autres, et si l’exercice du discernement, de l’appréhension d’un intervalle quelconque au sein de l’un de ces champs ne suppose pas à son tour tout un système de relations et de « transformations », par l’enchaînement desquels se constitue la perception d’une distance ; par exemple, la formation d’un relief avec avant- et arrière-plan, de croissance ou diminution de la plénitude, etc. Ce questionnement critique est si dirimant que l’on est en droit de se demander s’il reste encore un champ de questionnement philosophique et que l’on ne peut que s’étonner de le voir rebondir, par une nouvelle salve de questions. Mais loin qu’il s’agisse d’un simple questionnement rhétorique, nous avons affaire à une question indépendante de la précédente. Que le statut de l’espace soit a priori ou empirique, que sa signification soit, dans ce dernier cas, psychologique ou physiologique, il reste à comprendre pourquoi l’espace entendu en son sens ordinaire a été privilégié comme vecteur du développement et d’élaboration des structures mathématiques ; autrement dit de jugements a priori. La question kantienne se reformule dès lors ainsi : pourquoi les jugements et les constructions aprioriques des mathématiques se sont-ils développés sous la forme de propositions (apparemment) a priori synthétiques, en particulier géométriques? Ce point de départ coïncide très exactement avec celui de la philosophie des mathématiques de Weyl, et se trouve confirmé par sa théorie du jugement mathématique comme par sa théorie de l’a priori. Cela explique sans doute également pourquoi, à la différence de l’intuitionnisme mathématique d’un Brouwer et du formalisme, Weyl refuse de se débarrasser de la question kantienne, pour la raison qu’elle enveloppe une question toujours actuelle : celle de l’étonnante efficacité des mathématiques, ou si l’on préfère, de manière moins emphatique, de leur applicabilité. Or, bien que mise à mal en première approximation, la question kantienne sur la géométrie contient le germe de ce questionnement plus vaste dont les Premiers principes métaphysiques de la science de la nature (1786) avaient tenté de dériver les premières conséquences. Il reste que la question doit se reformuler dans un contexte plus étroit, où les principes transcendantaux dynamiques se sont déployés, comme un corollaire, et suivant la puissance inductive d’une théorie 22 La

référence psychologique est, si l’on veut, dans l’air du temps. Même si elle n’est pas une référence explicite à Husserl, elle correspond à la position initiale du problème dans les leçons de 1892. Et celle-ci explique en grande partie l’anti-kantisme qui prévaut encore ultérieurement, du moins en relation la question de l’espace (cf. infra la critique husserlienne de l’expression même de « forme de l’intuition »).

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hautement mathématique : le calcul tensoriel23 . La question relève ainsi de la « métaphysique de la nature » en son sens kantien, donc d’une « logique transcendantale » exhibant les conditions de possibilité d’une connaissance objective et de l’objectivité scientifique. « Par suite, la philosophie a, en outre, pour tâche de comprendre quelle est la provenance métaphysique et quelle est la signification métaphysique de l’espace ; si l’on place notre but aussi haut que le fait le métaphysicien, le plus impatient des savants, on devrait alors chercher à comprendre la nécessité de l’espace et de ce qui la caractérise en propre, à partir de l’idée de réalité telle qu’elle est donnée à la conscience. C’est dès lors un problème épistémologique que présente à la pensée philosophique la nature de la connaissance géométrique, son caractère a priori apparent ou réel. Comment se fait-il que les propositions de la géométrie présentent une puissance si grande de conviction, même pour celles dont on ne peut établir la justesse par aucune expérience, ou du moins aucune expérience accessible ? Tel est le problème qui fournit à la Critique de la raison pure de Kant son point de départ. »24

Il reste que cette question doit se déployer de manière plus modeste et plus patiente. En d’autres termes, si l’entrée en scène du mathématicien ne congédie pas le philosophe, la solution que le premier apportera aux problèmes qu’il formule n’élimine pas entièrement la question philosophique.

3.1.2 Un terminus ad quem riemannien idéalisé Ce terminus ad quem riemannien permet de donner à la question transcendantale son ouverture et sa profondeur maximales. Dans la mesure où le champ des possibles mathématiques y est plus vaste que chez Kant, la « déduction transcendantal » en est également plus ample et plus radicale. Il s’agit de « déduire », c’est-àdire d’expliquer pourquoi parmi l’infinité des variétés et des géométries possibles, l’espace au sens euclidien s’impose comme « forme a priori des phénomènes ». Riemann fournit donc un nouveau point de départ pour cette déduction transcendantale25 , en ce que la forme même de cet espace est beaucoup plus indéterminée, du moins si on le restreint aux « variétés » continues. Si, pour les variétés discrètes, la métrique est donnée a priori par le nombre des éléments, la métrique d’une variété continue ne peut être « donnée que de l’extérieur » 26 . Cela ne revient cependant pas à céder à l’empirisme, mais à comprendre différemment les liens intimes entre géométrie et physique, mathématique et « réalité physique ». Weyl interprète dans une perspective kantienne la formule de Riemann : si « l’espace n’est rien de plus qu’une variété tridimensionnelle dépourvue de toute forme », et que c’est le « remplissement » qui, par un « contenu matériel », lui confère sa métrique, nous 23 Bachelard,

La valeur inductive de la relativité, (Bachelard 1929, 2016). (2016 : 8) (Je traduis). 25 Weyl (1921: 96). 26 Weyl (1921: 97). 24 Weyl

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n’avons d’autre issue que de poser cet a priori comme une forme informe, la plus indéterminée possible et d’en déduire la métrique à partir du principe transcendantal dynamique qui touche à la possibilité d’une nature unifiée, formaliter spectata. Dans cette ouverture maximale de la question kantienne, il n’est pas même exclu que la métrique, comme la réalité physique, soit en dernière instance discrète, et que donc des métriques continues ne soient valides que localement ou en première approximation, « surtout de nos jours si l’on considère les résultats de la théorie quantique » 27 . Cette inversion de l’architectonique des principes transcendantaux tient à la prise en compte de la signification profondément physique de toute mesure sur laquelle Einstein nous a définitivement ouvert les yeux. « Les relations métriques ne proviennent pas de ce que l’espace est une forme des phénomènes, mais du comportement des règles et des rayons lumineux tels qu’ils sont déterminés par le champ gravitationnel »28 . Mais cette inversion n’est elle-même rendue possible que par la médiation de la théorie des groupes de transformations qui permettent de concilier la « mobilité maximale » (ou liberté maximale) et l’invariance la plus générale, qui se confond, en termes philosophiques, avec la préservation de leur caractère nomologique. C’est elle qui permet de décider parmi les trois possibilités des déplacements de la variété euclidienne celle qui préserve la congruence sous les transformations (déplacements locaux). Pour que Riemann annonce Einstein, il aura fallu que le second élève la physique au niveau de l’intuition mathématique du premier. Il reste que ni l’un ni l’autre ne sont encore à la hauteur de cette idée. Chez l’un et l’autre subsiste encore un « résidu » (remnant) de géométrie à distance (Ferngeometrie) que Weyl se propose d’éliminer afin de proposer une « théorie de l’espace en lui-même », qui en respecte l’exigence essentielle, celle d’une liberté maximale de la métrique, ou philosophiquement parlant, l’indétermination maximale de l’espace en tant que forme des phénomènes, compatible avec une nature commune et la constitution d’un monde commun (d’une expérience communicable). Cela suppose une nouvelle médiation mathématique : celle de la géométrie infinitésimale. Conformément à ce nouveau point de départ, la nécessité (et le caractère a priori) de l’espace revêt une nouvelle et triple signification : logique, arithmétique et analytique, et corrélativement, trois niveaux des structures mathématiques : relationnel, numérique et fonctionnel29 . « Pour le mathématicien, il s’agit donc de ce qui est quantitativement saisissable, des relations qui se fondent dans l’essence de l’espace et de la structure spatiale, pour autant qu’ils peuvent être saisis au moyen de la logique, de l’arithmétique et de l’analyse, et de leurs connexions nomologiques exprimables par leurs moyens ; en outre, d’établir les postulats les plus simples dont on puisse déduire par une inférence logico-arithmétique la nécessité de la théorie de l’espace que l’on obtiendra ainsi. Les résultats d’une telle analyse

27 «

Il we discard the first possibility « that reality which underlies space forms a discrete manifold » 28 Weyl (1921: 102). 29 Cette triade correspond aux trois niveaux de l’édifice mathématique exploré dans Philosophy of Mathematics and Natural Science.

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C. Lobo ne doivent pas être écartés par le philosophe. Car je suis convaincu au moins de ce fait que, sur ce terrain, simplicité mathématique et ‘originarité’ métaphysique se tiennent dans une étroite liaison (Verbindung) »30

Le postulat d’une telle liaison (linkage) ne implique-t-il pas la possibilité d’enjambements (de cross-over)?31 Que la présentation kantienne du problème philosophique apparaisse après-coup comme un « habillage philosophique » ne doit ni dispenser d’examiner la teneur du « corps » du problème, ni conduire à confondre ce « vêtement » avec ce qu’il manifeste de contours et de formes. S’il y a incontestablement un transfert de l’a priori du domaine philosophique au domaine mathématique32 , cet a priori mathématique conserve une dimension synthétique, et c’est à ce titre qu’il est relatif, contextualisé, et par suite non exclusif. C’est à cette condition qu’il peut assumer pleinement et sans perte la fonction d’a priori philosophique, c’est-à-dire, à prendre ce terme de manière rigoureuse, « transcendantal »33 . Notons qu’il faut toutefois le dépouiller de ses oripeaux absolutistes et écarter les caractères d’unicité et de nécessité qui affectaient l’a priori synthétique kantien (mathématique aussi bien que transcendantal). Rétrospectivement, il devient possible de dire que ces caractères découlaient de la confusion entre un élément contingent et accidentel et un élément nécessaire et essentiel. L’élément accidentel (par rapport à l’essence de l’espace, comme à celle du temps) et contingent, sans doute empirique, semble se confondre avec l’élément intuitif, puisque, comme le précise ailleurs Weyl, la mathématisation est fondamentalement une « arithmétisation », laquelle implique une exclusion de tous les termes, ou l’élimination dans ces termes, de tout ce qui renvoie à l’intuition (sensible) (espace, temps et qualités sensibles)34 . Mais si la mathématisation signifie symbolisation et élimination de l’intuition, il semble qu’il n’y ait d’autres voies que celle indiquée par le formalisme, voire le logicisme. Que reste-t-il de synthétique a priori dans ces conditions ? C’est ici que prend

30 «

Für den Mathematiker handelt es sich darum, das quantitativ Erfaßbare, die im Wesen des Raumes und der räumlichen Struktur gründenden Relationen, soweit sie mit den Denkmitteln der Logik, Arithmetik und Analysis erfaßt werden können, und ihre mit diesen Hilfsmitteln ausdrückbaren gesetzmäßigen Zusammenhänge aufzudecken; ferner die einfachsten Postulate zu ermitteln, aus denen sich die Notwendigkeit der so zustande gekommenen Raumtheorie durch logisch-arithmetische Schlüsse ergibt. Die Resultate solcher Analyse darf der Philosoph nicht beiseite schieben. Ich wenigstens bin fest davon überzeugt, daß auf diesem Felde mathematische Einfachheit und metaphysische Ursprünglichkeit in enger Verbindung miteinander stehen“ (Weyl, 1923a, b, 2; Weyl; 2016: 9). 31 Anamnèse involontaire, je retrouve en contrôlant les références cette notion déjà employée par Scholz (2009 : 221–225) à propos de la liaison Einstein-Weyl. 32 Voir dans le présent volume l’article d’E. Scholz. 33 La traduction anglaise de manière symptomatique tend à traduit « Transzendent » par « transcendental », alors que ces deux termes sont entendus par Weyl dans une perspective clairement phénoménologique. Le « transcendant » est le réel existant dans le monde (« out there »). Le transcendantal, sans se confondre avec une intériorité psychologique, est pour la phénoménologie immanent et eidétique. Il se confond avec ce que Husserl nomme l’a priori corrélationnel. 34 (Weyl, 1952/2009: 107).

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tout son sens et son relief, l’expression récurrente chez Weyl de « construction symbolique ». Ce régime constructif requiert toute l’attention, si l’on veut comprendre comment un a priori synthétique continue de se loger au cœur des mathématiques dites formelles, a fortiori au sein d’une théorie physique comme la relativité. L’interrogation des « hypothèses » sur lesquelles se fondait le primat de la géométrie euclidienne, « un nombre infini d’axiomes propres de la géométrie », a levé un coin du voile sur l’étendue des formes de géométries possibles. Le développement des méthodes purement formelles (analyse « supérieure », topologie, algèbre, théorie des groupes, théorie des ensembles, combinatoire, etc.) a contribué à rétrograder ce « modèle » (au sens de représentant paradigmatique) au rang de simple modèle (au sens sémantico-formel), et même de simple cas particulier (imparfaitement formalisé) parmi un ensemble de systèmes équivalents tenus pour isomorphes, du moins du point de vue syntaxique. L’intuition du géomètre postriemannien prédomine chez Weyl, même dans le traitement des questions purement logiques. Bien plus, elle aboutit à reléguer la géométrie euclidienne, sous sa forme intuitive et constructive, au rang de simple discipline empirique, externe à la sphère de la mathématique pure, relevant de ses applications. Il n’en demeure pas moins que les mathématiques n’ont pas pour autant abandonné le problème de l’espace dit intuitif, ou de l’intuition de l’espace, à l’investigation empirique (psychologique et/ou physiologique). Parallèlement, cela ne réduit pas à néant la problématique philosophique, si l’on veut bien entendre par là l’exploration de l’a priori subjectif dont l’espace est tout à la fois le titre et l’index. En 1949, date de la traduction anglaise de PMNS35 (Weyl, 1949), le bilan de l’analyse de cette problématique, du partage entre ce qui est résolu et reste à résoudre, est exposé par Weyl en des termes qui ne deviennent intelligibles qu’à la lumière de l’exposé de l’histoire interne de ce problème, des phases de sa position et de sa solution progressive. Relevons quelques-uns de ces points de contact. Soit, par exemple, la distinction de la droite et de la gauche dont Kant fait un trait déterminant du caractère a priori, formel et intuitif, i.e. indécomposable de l’espace. S’il y a quelque chose d’a priori dans l’espace, ce n’est pas une forme de l’intuition inanalysable, mais une structure symboliquement constructible, relevant de la théorie des groupes de transformations. Alors que Kant croit trouver « la clé de l’énigme de la gauche et de la droite dans l’idéalisme transcendantal », dans un a priori que la mathématique serait condamnée à présupposer, sans jamais pouvoir l’épuiser, le mathématicien en opère une traduction qui ne laisse rien à désirer, comme le dit Weyl (établissant non pas l’identité, mais l’équivalence des versions, y compris de l’original). C’est ainsi qu’il est conduit à voir derrière la forme a priori de la sensibilité « le fait combinatoire de la distinction entre permutations paires et impaires »36 . Poussant plus loin la mathématisation, cette traduction, comme toute traduction, laisse cependant un résidu inéliminable, celui de la subjectivité. Dans leur « quête des

35 L’édition 36 Weyl

allemande enrichie datant de 1927 (Weyl 1927). (1949 :84).

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racines des phénomènes que le monde nous offre », le choc et le renversement des fronts entre la recherche du philosophe et celle du mathématicien sont ici exemplaires. Cela ne contredit pas la thèse d’un a priori subjectif de l’ordre de la sensibilité, non plus du reste que celle de son caractère empirique, ou encore la possibilité d’une « intuition formelle » au sens de Kant, mais réfute, en tout cas, l’affirmation que sa solution relève exclusivement d’une analytique transcendantale, comme le prétend Kant, car, à moins de supposer que Kant ne vise « quelque point plus subtil »37 , Weyl démontre ainsi que « le phénomène au sujet duquel Kant s’étonne, peut [ . . . ] être subsumé de manière plus satisfaisante sous des “concepts” généraux et abstraits » (ibid.). Quant à la différence entre synthétique a priori et analytique, après avoir montré comment on s’élève de la géométrie (intuitive) à la géométrie idéalisée par les définitions créatrices, qui sont une forme particulière de définition abstractive et, de là, à une version totalement ensembliste dans le cadre d’une formalisation achevée38 , Weyl voit dans l’histoire progressive des mathématiques et de la physique un processus historique d’abstraction continue, l’extraction d’un système de propositions analytiques, à partir de jugements synthétiques. Il juge « obscure » la détermination kantienne de l’analytique dans son opposition au synthétique en comparaison de la « définition husserlienne » (que l’on trouve dans la deuxième édition des Recherches logiques). Selon cette dernière, le caractère purement formel de tous les concepts contenus dans une proposition analytique constitue le critère d’analyticité. Husserl dérive la distinction entre analytique et synthétique de celle entre deux modes d’individuation des éléments constituant l’extension de ces concepts. La première est formelle en ce qu’elle présuppose pour les concepts universels, à extension non vide, un champ de pures singularités supposées distinctes et cependant non individuées, puisque l’on a suspendu toute supposition d’existence réelle, tandis que la présence de concepts matériels posant « une existence individuelle » devient caractéristique des jugements synthétiques. Par résolution du problème mathématique – ou résolution mathématique du problème – de l’espace, il faut désormais entendre la solution a priori (par voie mathématique) d’une question d’origine physique : prouver l’homogénéité de l’espace physique (continuité, isotropie, isométrie et homothétie), homogénéité dont dépend la possibilité de mesures « objectives », et par suite de lois de la nature exprimant les rapports constants entre ces mesures. C’est sur ce problème que se manifeste de manière exemplaire et radicale le déplacement de frontière entre l’a priori et l’a posteriori, où les mathématiques sont à la manœuvre, quand bien même l’impulsion première viendrait de la physique ou de la philosophie. C’est aussi en ce lieu que se trouve levée, au prix de quelque subtilité d’un autre ordre, l’ambiguïté concernant le statut de l’espace comme forme a priori (« des phénomènes », de l’intuition, du monde, etc.). Comme le réaffirme Weyl, la partie a priori incombe à une théorie mathématique : la théorie de groupes continus de transformations de Lie. Il est hors de propos d’entrer ici dans le détail de cette « démonstration », qui couvre 37 Weyl 38 Weyl

(1949 :80). (1949 :18).

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une série de contributions décisives (de 1923 à 1926)39 , et touche aux problèmes les plus profonds de la théorie de la relativité générale40 . Rappelons simplement que la difficile « dérivation » transcendantale de la nature et de la réalité de l’espace physique (au cœur des Premiers principes métaphysique d’une science de la nature de Kant) se trouve remplacée par la déduction et la construction du groupe le plus adéquat permettant d’atteindre le même objectif épistémologique. Dans sa formulation restreinte, ce problème se formule ainsi : quel est le groupe de transformations garantissant une certaine homogénéité de l’espace compatible avec la plus grande liberté métrique possible, c’est-à-dire avec le minimum de rigidité géométrique possible ? La réponse initiale, que reprend la première édition de PMNS (§ 14) est : le « groupe euclidien des rotations » ; par quoi il faut entendre non le groupe des « simples » rotations « dans l’espace euclidien » (c’est-à-dire le groupe orthogonal O(n) ou le groupe spécial orthogonal SO(n) que l’on a pu, après coup, associer à l’espace euclidien), mais une notion plus générale, indépendante de toute métrique prédéfinie, prise au sens de la « géométrie infinitésimale » (Nahegeometrie) promue par Weyl, un sous-groupe de transformations continues (sur le corps des réels, R) préservant l’analogue du « volume », c’est-à-dire le « corps de vecteur » associé à un point d’une variété riemannienne. Dans l’édition de 1949, Weyl place ces développements sous l’égide de la notion fondamentale de « congruence » (promue au rang de « seul et unique concept fondamental de la géométrie »)41 . Dans cette perspective, le « groupe des mouvements euclidiens » se définit comme le groupe de toutes les applications congruentes et l’espace correspond au groupe de toutes les transformations continues. Cette réponse appelle explicitation et surtout une mise en garde concernant une assimilation hâtive de la variété riemannienne à un espace. Weyl se situe, comme Einstein, dans la perspective ouverte par Riemann42 . Il tient la géométrie euclidienne pour une discipline mathématique et physique. Les déterminations métriques propres à l’espace euclidien n’ont rien d’une évidence a priori, mais reposent sur des hypothèses qui n’ont pas le caractère d’unicité et d’universalité qu’on leur a accordé durant des siècles. Par suite, son statut d’a priori de la physique disparaît également. Abstraction faite de la question du nombre de dimensions de l’ « espace », nous sommes placés a priori devant un choix illimité de variétés (Mannigfaltigkeiten) continues qui pourraient également prétendre au titre d’espace de notre nature. La technique, usuelle depuis l’avènement des géométries non-euclidiennes, du « plongement » dans l’espace euclidien, ne doit pas nous 39 Mathematische

Analyse des Raumsproblems, Vorlesungen gehalten in Bercelona und Madrid, Berlin, Springer, 1923 et H. Weyl, “Theorie der Darstellung kontinuierlicher halbeinfacher Gruppen durch lineare Transformationen, I, II, III und Nachtrag”, in Mathematische Zeitschrift, Springer, 1925, 23, p. 271–301 ; (Weyl, 1926), 24, p. 328–376, p. 377–395. 40 Cf. J. Bernard (2013, 2016), nous permet d’y renvoyer le lecteur soucieux d’en suivre les péripéties. 41 Weyl (1949 :79). 42 Sur les idées géométriques de Riemann, cf. Weyl (1921; 1925b) et bien entendu Riemann luiméme (1854, 1876a, 1876b, 1990).

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faire oublier que, depuis Riemann, sinon Gauss, une géométrie au sens strict est une caractérisation (intrinsèque) de la métrique d’un espace (du nombre de ses dimensions, de la mesure et du signe de courbure, de sa constance) et non plus l’étude des propriétés et des relations entre « objets » (points, lignes, surfaces ou corps) plongés « dans l’espace », présupposé a priori unique. L’analyse a priori du concept de métrique proposée par Riemann permet ainsi de déterminer de manière « formelle » ce qu’est une métrique en général, sans pouvoir déduire a priori la « véritable métrique » de la nature, c’est-à-dire l’unité, l’universalité et l’unicité de la métrique propre à l’« espace physique ». Nous qualifions cette caractérisation de « formelle », suivant en cela l’usage husserlien, reconnu par Weyl, qui dispose à entendre de manière plus précise ce qu’il entend par « nature » de la métrique, et par suite par « essence mathématique » de l’espace. Chaque métrique possible se définit comme une variété topologique dont les points sont repérés par un système de coordonnées, ce dernier concept se trouvant pris ici dans sa généralité et sa « souplesse » maximales. Si l’on se souvient que tout système de coordonnées est inversement un point de la variété, et qu’il correspond au résidu inéliminable de subjectivité (d’idéalisation de cette subjectivité), il s’ensuit que la conquête d’un espace objectif se fera par « raccordement » (connexion) ou « mise en communication » (transformation) de ces points, dotés d’un degré de liberté maximale, et des éléments linéaires qui en procèdent, ou le touchent. Une variété n’étant pas un espace, il n’y a pas de sens à disposer ces « points » de la variété en présupposant des relations fixées d’avance. Les types de relation et les dimensions doivent être construits pas à pas, à partir et en fonction de ces repères. C’est ainsi que les lignes sont des variations linéaires continues croissantes, et les rapports sont eux-mêmes soumis à des variations continues, la longueur de l’élément linéaire étant supposée invariable lors de déplacements infinitésimaux. Pour établir cette invariance, Riemann recourt à une fonction (que Weyl nommera « fonction de lieu » – Ortsfunktion –, par quoi il faut entendre une fonction de position ou de localisation « au point P »), permettant de déterminer la longueur (ds) d’un élément linéaire (dx) au point P. Cette fonction, que Riemann suppose toujours positive, le conduit à la célèbre expression différentielle du deuxième ordre (ds2 ), et par suite aussi à sa racine carrée ds, qui varie comme l’élément linéaire dx : « ce doit donc être ds2 multiplié par une constante et donc ds est la racine carrée d’une fonction homogène entière toujours positive du second ordre des quantités dx ». C’est ainsi que se trouve déterminée, par Riemann, la « nature » de l’espace (à 3 dimensions) dont la « position » des points se trouve exprimée par les coordonnées rectilignes, ds = (dx)2 , où le nombre de dx sommés correspond au nombre de dimensions. La généralisation aux « variétés » à n dimensions s’ensuit aisément. C’est en cela que consiste la « nature de la métrique », et par suite l’« essence géométrique » de l’espace, ce que Weyl nomme encore sa « pythagoricité »43 . Corollaire, qui rejoint la compréhension de la formalisation riemannienne : en dépit d’un usage reçu, parler d’« espace topologique », pour une variété dépourvue de métrique, ou a

43 Weyl

(1949: 137).

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fortiori dépourvue de toute structure conforme, affine ou projective, est impropre ; une telle variété ne mérite qu’improprement le nom d’espace44 . Pour autant, ce terme perd également de son univocité, car la possession d’une telle «nature» (la « pythagoricité ») est compatible avec l’existence d’une courbure globale constante, voire d’une courbure localement variable. Cette « nature » est donc compatible avec l’existence d’une diversité de métriques (d’orientation et de courbure) indiquée par les coefficients. Mathématiquement, la variabilité sera donnée, en relativité générale, par les coefficients variables qui affectent la forme quadratique : le tenseur g μν doublement covariant. Ce dernier pas, dans la lente émancipation de toute conception rigide de l’espace, consistera à se donner des systèmes de coordonnées dotés du maximum de « liberté » compatible avec la possibilité de mesures (comparaison effective de « longueurs » et invariance de la distance par transformation des coordonnées). En cela consiste aussi la généralisation maximale du principe de relativité, conduisant à un nouveau et radical partage de l’a priori et de l’a posteriori. Ce partage est luimême rendu possible par une compréhension plus profonde du principe de relativité du mouvement – qui est chez Kant, rappelons-le, le seul et unique axiome a priori dont dérivent tous les autres principes métaphysiques de la nature45 –, car « la théorie de la relativité générale ne nie pas en bloc qu’il existe en ce sens quelque chose d’apriorique dans la structure du médium extensif du monde extérieur, sauf que la ligne entre a priori et a posteriori est placée ailleurs » (ibid.). L’a priori (la nature d’une métrique de l’espace) est déduit, ou plus exactement mathématiquement construit, et l’a posteriori se laisse déterminer « empiriquement » (les coefficients déterminant la courbure, l’orientation réciproque des systèmes de coordonnées varient en fonction de la distribution de la « matière »).

3.1.3 Reformulation du problème de l’expérience possible Quant à la démonstration de la nécessité d’un tel partage, elle est de nature ambiguë et renvoie, chez Weyl, au célèbre « argument de la boule de pâte à modeler »,46 qui fonctionne comme une procédure de vérification de consistance. Dans RZM, il permet d’écarter l’objection d’absurdité adressée à Riemann et Einstein47 . Si la métrique de l’espace-temps varie en fonction (et sous l’effet) de la « matière »

44 Voir

à ce sujet la position de Husserl in Logique formelle et logique transcendantale, § 30. principes métaphysiques de la nature, tr. fr. F. de Gandt, partie I, Œuvres philosophiques, tome II, Gallimard, 1985, p. 388, dont la formulation (surprenante de prime abord) est : « Tout mouvement, en tant qu’il est l’objet d’une expérience possible, peut à volonté être considéré comme le mouvement d’un corps dans un espace en repos, soit au contraire, le corps étant au repos, comme le mouvement de l’espace en sens opposé avec une vitesse égale. » 46 Argument qui est le pendant de l’argument du trou que l’on trouve chez Einstein. Voir sur ce point, l’étude de J. Bernard (2016 : 209–214). 47 Weyl (1921: 90–91). 45 Premiers

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(densité du champ gravitationnel), comment expliquer (objecte-t-on) que l’on puisse voir une boule sphérique de pâte à modeler revêtir des formes variables – sousentendu dans un « espace » rigide qui ne subit aucune déformation. L’argument possède lui-même une plasticité étonnante qui n’a pas manqué d’arrêter les commentateurs. Nous le verrons revêtir la valeur d’un paradigme au sens platonicien, y compris au niveau métathéorique dans l’étude des rapports entre la pensée mathématique dans sa dimension irréductiblement synthétique et ses mises en forme axiomatiques. Dans RZM, l’argument se retourne, rétorque Weyl, car l’attribution de cette forme géométrique (sphérique), conformément au principe de relativité, repose en effet sur une pétition de principe et une absolutisation d’une apparence (liée à un référentiel) imposée arbitrairement à tous les référentiels possibles. En d’autres termes, on suppose que, quel que soit le lieu d’observation, la boule apparaîtrait avec la même forme – qu’elle possède donc une forme géométrique indépendamment de tout point d’observation (de tout système de coordonnées). La forme initiale elle-même dépend d’un certain arrangement de la matière, et par une variation de cette distribution, il est possible de donner à n’importe quelle portion de l’espace une forme différente. Dans PMNS, l’argument réapparaît en divers lieux, sous une forme radicalisée, au § 16, dans le contexte d’une étude de la structure physique de l’espace-temps et de la relativité restreinte48 . Mais dès les paragraphes introductifs sur la structure mathématique (sur la structure axiomatique), l’argument fonctionne de manière plus générale, en établissant l’impossibilité de réduire la géométrie riemannienne à un simple jeu formaliste, et permet en retour de déterminer la forme de l’argument initial et sa fonction. De l’une à l’autre formulation, le rapport de la forme (conçue comme jeu symbolique) à l’intuition trouvent leur véritable articulation. Corrélativement, se trouve établi le statut (physique ou purement mathématique) de la géométrie euclidienne et de ses objets, et par suite celui des géométries non-euclidiennes en général et de la géométrie riemannienne en particulier. Supposons que nous effectuions une déformation continue de l’espace (comme s’il était rempli de pâte à modeler), et supposons que nous entendions à présent par lignes, plans et figures congruentes de telles courbes, surfaces et figures résultant par déformation des lignes réelles, des plans réels et des figures réelles congruentes. Alors, à l’évidence, tous les faits de la géométrie valent pour les concepts nouvellement introduits. Il est donc impossible de distinguer conceptuellement entre le système de lignes et le système de courbes qui résultent d’eux par une déformation spatiale49 . L’attribution d’une forme visuelle à une portion d’espace suppose que celle-ci, conformément au principe de relativité générale, soit déterminée par les conditions matérielles de l’observation (comme l’invariant de la variation des coefficients affectant la fonction de localisation rattachée au système de coordonnées, au « point de vue » pour lequel telle ou telle forme spatiale apparaît). Parallèlement, la géométrie euclidienne ne se laisse pas plus

48 Weyl 49 Weyl

(1949: 105). (1949: 25).

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réduire à l’étude des figures données dans l’intuition qu’à un pur jeu formellement réglé de relations définies sur des symboles dépourvus de signification, selon la célèbre provocation formaliste due à Hilbert (1902). Il est possible d’obtenir une géométrie à partir de l’autre selon un système de déformations réglées et d’établir si et dans quelles conditions ces géométries sont logiquement « isomorphes » ou axiomatiquement « équivalentes ». Les termes se référant à l’espace, au temps et plus généralement aux contenus des formes a priori ne sont en effet que des index de “termes” purement symboliques, dont la référence n’est ni un phénomène, ni une chose en soi, mais certains invariants exprimant des rapports constants entre « variables », qui sont toutes des termes spatio-temporels, représentant des aspects et des paramètres de phénomènes. La théorie de la correspondance entre ce système logique de symboles et la réalité est saluée chez Schlick, mais « cum grano salis », car en réalité l’expérience qui doit être en concordance n’est pas seulement l’expérience actuelle comme le dit l’empiriste, mais l’expérience possible, comme le souligne Weyl. « Ainsi la correspondance ne vaut pas entre le monde réel un et les perceptions actuelles d’un observateur, mais il y a d’un côté le monde objectif quantitativement déterminé et de l’autre, les perceptions possibles découlant de tous les états possibles d’un observateur : des choses telles que la position et la vitesse de l’observateur, par exemple, font partie de l’élément variable à l’intérieur de cette correspondance »50

Telle est la corrélation qu’occultent ceux qui, au nom d’une conception réaliste et empiriste et, en tous cas, naïve de l’observateur, évacuent cet arrière-fond d’une totalité d’observation et perdent ainsi la signification subjective constituante du système de coordonnées. Cette corrélation est aussi un contraste entre « fixité » (de l’être) et liberté (de l’observation). Corrélativement au système « rigide » d’assignation des places, nous avons le « système de la liberté » comme système de la « libre mobilité ». Un « je peux » qui est aussi un indicateur de la sphère du possible, de l’ « espace des possibles ». Avec l’introduction de cet arrière-plan de possibles, et l’appariement d’un ensemble de déterminations objectives du monde à l’ensemble de déterminations subjectives possibles, et la dérivation de celles-ci d’un ensemble d’états physiques possibles de ce même sujet observateur, nous saisissons mieux la teneur spéculative de la proposition : L’espace-temps est la mesure de cette liberté. La part de l’a posteriori trouve donc elle-même sa traduction formelle dans la multiplicité et la contingence des orientations de cette structure métrique, i.e. du cours quantitatif du champ métrique. Il est possible d’éclairer cette situation au moyen d’une analogie juridicopolitique et d’éclairer celle-ci en retour: De ce point de vue, il est évident que pour la solution du problème de l’espace, il ne s’agit plus de comprendre rationnellement le champ métrique dans sa configuration quantitative contingente, dépendante de la matière, mais bien uniquement la nature pythagoricienne immuable de cette métrique, dans laquelle se révèle l’essence a priori de l’espace [ . . . ]. À la place de l’homogénéité du champ métrique requise par Helmholtz, est apparue la possibilité d’assujettir le champ métrique à des variations virtuelles quelconques, au sein du cadre fixé

50 Weyl

(1954/2009: 108).

54

C. Lobo par la nature de la métrique. Jusque-là, c’est-à-dire en ayant statué sur une telle possibilité, nous n’avons en vérité encore rien affirmé sur la nature de sa métrique.51

Weyl insiste de nouveau sur la nécessité de remplacer l’étagement binaire par un étagement ternaire où l’a priori se trouve scindé de manière à faire place à des lois naturelles nécessaires, mais mathématiquement contingentes, de sorte que « ce qui, même sous leur législation, reste libre et apparaît donc comme contingent », en relève52 . L’analogie déploie le parallèle sur trois niveaux, qu’il faut soigneusement distinguer : celui de la « constitution de l’État » (en tant que loi obligatoire pour tous les citoyens, système d’obligations et de droits fondamentaux) (= la nature de la métrique) ; celui de la liberté des citoyens (= la possibilité de métriques variables aux différents points de l’espace) ; celui du « bonheur du tout », du « bien général » (Wohl des Ganzes), (= la détermination univoque de la connexion affine, nécessaire si l’on veut que cette variabilité des métriques ne compromette pas la possibilité d’un seul et même tout, d’un monde objectif réel commun)53 . Cette analogie gagnerait à être éclairée par un travail spécifique sur le volet du droit et de la politique, en particulier de la critique parallèle que Husserl donne de la Métaphysique des mœurs kantienne et en particulier de sa doctrine du droit54 . La critique de l’enfermement de l’a priori de la physique mathématique dans le carcan de principes synthétiques a priori (axiomes de l’intuition) et de la rigidité de la métrique aurait pour parallèle la critique de l’a priori juridique par le mécanisme, i.e. le système de contraintes « naturelles » et d’obligations morales qui président à la formation d’une constitution qui redouble physiquement et moralement ce système de contraintes s’imposant aux individus comme une puissance extérieure et supérieure. Plus radicalement, elle se ramène à la distinction entre analytique et synthétique a priori déjà évoquée ci-dessus. Il est remarquable que l’on observe un glissement parallèle dans les distinctions entre analytique et synthétique, a priori et a posteriori dans la sphère de la raison pratique55 . Cherchant à approfondir cette question nous sommes renvoyés par-delà l’analogie entre théorie du droit politique et métrique, à la question de la constitution transcendantale au sens phénoménologique du terme, et de ce que Husserl a luimême nommé durant un temps, la théorie du droit (Rechtslehre)56 .

51 Weyl

(1923a, b: 45–46). Weyl (2016 : 104). (1949: 135). 53 Weyl (1923a, b: 46–47). Weyl (2016 : 104–105). Cette détermination relève précisément, selon la Philosophie de l’histoire de Hegel, de l’esprit objectif et fait face aux critiques unilatérales et abstraite de l’esprit subjectif. 54 Voir la discussion de Kant par Hegel dans les Principes de la philosophie du droit. 55 Sur ces parallélismes et l’enrichissement parallèle de l’a priori synthétique a priori et la conversion de l’impératif catégorique en principe analytique, celle des impératifs hypothétiques (techniques) en impératifs synthétiques a posteriori, cf. Husserl (1988/2009: 132). 56 La Rechtslehre en tant que noétique dont il est question en 1906 (in Husserl (1984b : 115–124) ne doit pas conduire à retomber dans un contresens normativiste (Husserl 1984a, b: 40–42; 50–59) ou à l’interprétation de la logique pure en termes de « lois de la pensée » (des lois naturelles ou empiriques) (Husserl, 1984a : 65–66; 73–75). 52 Weyl

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3.2 Terminus a quo riemannien de Husserl et son terminus ad quem kantien Pour voir de quelle manière la trajectoire husserlienne vient recroiser celle de Weyl, et former avec celle-ci ce que nous nous sommes risqués à nommer un crossing-over ou « enjambement » épistémologique, nous devons d’abord en marquer les jalons essentiels.

3.2.1 Un mathématicien réfléchissant et généalogiste Cherchant à cerner la spécificité du moment riemannien, Husserl est conduit à distinguer deux processus constitutifs à l’œuvre chez Riemann et ses prédécesseurs ou contemporains. Quand bien même Husserl se situe à l’époque aux antipodes d’une approche transcendantale, et même si l’interprétation de la constitution de l’espace géométrique euclidien en termes d’idéalisation de l’espace intuitif l’oriente dans un premier temps vers une enquête de type psychologique ou psychophysiologique57 , dès cette époque, il isole chez Riemann une opération d’une autre nature qui, seule à ses yeux, mérite le titre de « formalisation » et qu’il faut se garder de confondre avec d’autres formes d’extensions ou d’élargissement (dont la simple généralisation). En regard de cette rupture instauratrice dont Riemann sera le prototype, à côté de quelques autres, de la mathématique formelle, la généralisation opérée par Gauss reste encore dans l’orbite de l’idéalisation euclidienne, ce qui est également le cas des autres voies que Husserl qualifie de synthétiques : celles de Bolyai et Lobatchevski, par exemple. Cherchant à déterminer la nature de l’activité constitutive, à l’œuvre dans les différents élargissements de la géométrie euclidienne, Husserl sera conduit à s’intéresser à l’autre volet de la philosophie kantienne, l’esthétique transcendantale, dont il cherchera à élargir le cadre, en l’affranchissant lui aussi de ses présuppositions euclidiennes. C’est alors que s’amorce l’approfondissement de l’a priori esthétique qui nous occupe ici. La première idéalisation euclidienne ou proto-euclidienne sera elle-même envisagée ultérieurement dans le cadre de ce nouvel a priori. Cet

57 Selon

une analogie suggérée par Weyl et reprise par Enriques, il est même tentant de chercher à associer des formes d’espace à des niveaux de constitution, voire à des champs sensoriels: « lois suivant lesquelles notre appareil visuel perçoit les objets extérieurs ressemblent plus précisément à celles qui caractérisent la géométrie projective bidimensionnelle » il reste que « à l’époque en général on s’accordait pour nier que l’espace géométrique puisse être fondé sur l’espace physiologique, lequel ne possède en aucun cas les propriétés d’homogénéité et d’isotropie qui caractérisent le premier » (L. Boi 1997: 20) L’analogie suggérée par Enriques faisant « correspondre aux trois « pré-espaces » kinesthésique, tactile et visuel, les trois groupes de représentations correspondant à la théorie du continu, géométrie métrique et différentielle et à géométrie projective respectivement » appellerait ainsi la même remarque. (Ibid).

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approfondissement est nécessaire si l’on veut comprendre, selon une histoire intrinsèque, que ce qui se joue dans la prétendue « généralisation », opérée par Riemann, de la formule de courbure de Gauss est autre chose qu’une généralisation, mais un passage à une autre pratique des mathématiques : une formalisation. La méthode employée par Gauss dans ses travaux pionniers, pour déterminer la courbure intrinsèque des surfaces (courbure, etc.), passe encore par un plongement dans l’espace euclidien. Pour parvenir à une détermination de la courbure, sans plongement, il faut se libérer de restrictions tacites qu’impose le point de départ dans l’idéalisation euclidienne. L’a priori subsistant se borne à définir la possibilité mathématique d’autres métriques, ou même de métriques discrètes, et récuse les prétentions de la forme a priori exposée par Kant au rang de condition de possibilité des phénomènes (de leur existence et de leur connaissance). Tel est l’a priori subjectif plus profond que Husserl explore dans ses leçons de 1907, valable pour un sujet absolument quelconque, placé dans n’importe quelle condition d’observation, dont le monde n’est pas pré-tracé spatio-temporellement comme l’est celui que nous posons selon notre auto-aperception naturelle, dans ce que Husserl nomme la « thèse-naturelle ». L’exploration du nouvel a priori transcendantal implique donc une neutralisation de cette « thèse du monde », en d’autres termes une épochè transcendantale. Ce n’est qu’alors que l’on découvre quelle était la base pour l’intuition de Riemann, combinée à celle de Grassmann, qui permit pour la première fois une véritable théorie formelle et généralisée des variétés, affranchie des hypothèses restrictives : véritablement intrinsèque, sans plongement, valant pour tout système de coordonnées, non restreinte aux variétés continues, sans dimensionnalité préfixée, etc. Et de dégager aussi la propriété essentielle : la connexité – dont l’a priori transcendantal sera l’ensemble des synthèses ou syntaxes explorées par Husserl jusqu’à la fin de sa vie (en y comprenant entre autres les synthèses passives spatiales, temporelles, kinesthésiques, affectives, etc. et surtout les synthèses de l’intersubjectivité). Sur cette trajectoire, dans le traitement de cette notion de connexité, nous trouverions un autre enjambement Weyl-Husserl, plus lâche comme nous le verrons, car il implique chez l’un et l’autre une série plus importante de références croisées : aux connexions entre variétés riemanniennes (Levi-Civita, Cartan, Leibniz, Dilthey) du côté de Weyl ; aux connexions de multiplicités subjectives constituantes d’association, de motivation, de remplissement, d’aperception analogisante ou « empathique » (Leibniz, Dilthey, Hume, Lipps, etc.) du côté de Husserl. La connexion dite de Levi-Civita en particulier, établissant l’unicité de la connexion (de torsion nulle) sur une variété riemannienne préservant sa métrique s’exprime en termes de groupes comme l’invariant par les transformations des systèmes de coordonnées mobiles quelconques. Cette analogie entre connexion et communication intersubjective est clairement à l’œuvre chez Elie Cartan qui relève à son tour une lacune dans la géométrie infinitésimale de Weyl, là même où ce dernier prétendait combler une lacune de Riemann. Cette nouvelle « généralisation » (formalisante) va conduire Cartan à proposer une notion généralisée de connexion affine, dont la connexion métrique et, en particulier, la connexion euclidienne ne sont que des cas particuliers. La lacune est la suivante :

3 Le résidu philosophique du problème de l’espace chez Weyl et Husserl

57

« Dans la théorie de M. H. Weyl, ce repérage de proche en proche est soumis a priori à une certaine restriction, dont on ne voit pas bien la nécessité logique, et qui consiste dans l’existence, au voisinage de chaque point, de ce qu’il appelle un système de coordonnées géodésiques. » 58

Pour y suppléer, Cartan repart d’une notion plus générale et dissocie variété à connexion affine et espace affine proprement dit, cette différence se traduisant par « un déplacement affine associé à tout contour fermé infiniment petit », qui se laisse décomposer en translation et rotation, exprimant respectivement la torsion et la courbure de la variété. Or dans la théorie de Weyl, « la torsion est constamment nulle ». « Toutes ces notions s’étendent aux variétés à connexion métrique ou euclidienne: la théorie classique des espaces définis par un ds2 (espaces de Riemann) n’est au fond que celle des variétés à connexion euclidienne sans torsion ; c’est sur elles, comme on le sait, que repose la théorie de la relativité généralisée édifiée par M. Einstein » (Ibid.).

La généralisation de la notion de connexion affine se fera elle aussi au fil conducteur d’une réflexion sur les groupes de transformations des coordonnées. Cartan nous invite ainsi à imaginer « un ensemble continu d’observateurs, réduits à des points, et dont chacun adopte un système de coordonnées pour l’étude de l’espace (E), ces systèmes étant naturellement tous équivalents entre eux. La variété formée par ces observateurs-points est, je suppose, à p dimensions, chaque point étant défini d’une manière quelconque par p coordonnées u1 , . . . , up . Si l’on passe d’un point m de la variété à un point infiniment voisin m’, on passera dans l’espace (E) d’un certain système de coordonnées à un autre que nous supposerons infiniment voisin ; autrement dit, on passe de coordonnées x, utilisées par l’observateur m aux coordonnées x’ utilisées par l’observateur m’ en effectuant une certaine transformation infinitésimale du groupe G » (Cartan (1923: 384). Comme nous allons le voir, cette lacune correspond très précisément à celle que Husserl pointe chez Riemann, et le conduit à conclure que ce dernier se tient en deçà de l’idée (formelle) dont il est incontestablement le premier à avoir présenté une ébauche grandiose et précise à la fois.

3.2.2 La position et l’analyse anti-kantiennes du problème de l’espace chez Husserl Dans les leçons de (1892/1893)59 , lors d’une série de questions philosophiques, le problème de l’espace se trouve divisé en trois problèmes clairement distincts, mais coordonnés : logique, psychologique et métaphysique. Mais ces problèmes en recouvrent et couvent, pour ainsi dire, une multitude d’autres. L’amphibologie a

58 «

Sur les variétés à connexion affine et la théorie de la relativité généralisée » Cartan (1923 : 326). Voir aussi Scholz (1999a). 59 Texte N◦ 1 sur l’espace daté de 1892/93 , intitulé « Questions d’une philosophie de l’espace ». Husserl (1983 : 262–266). Ci-après donné sous la forme Hua 21 : 262–266.

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gagné le terrain de l’esthétique transcendantal et s’attaque à la racine même du projet critique. Suivant le pli aristotélicien de son maître Brentano, Husserl est conscient des équivoques grandissantes qui affectent l’usage du terme d’espace, selon qu’on désigne ainsi « l’espace de la vie quotidienne », « l’espace de la géométrie pure » (entendu comme intuition géométrique), « l’espace de la géométrie appliquée, des sciences de la nature » ou l’espace « métaphysique », transcendant, absolu (celui de Bruno ou de Newton, peut-être celui de Pascal). Et suivant le même pli, il ordonne cette diversité d’acceptions en leur assignant le rôle de moment dans une structure, et d’étape dans une genèse : « Nous avons affaire ici manifestement à une succession d’étapes génétiques de formations »60 . Corrélativement, le terme de « représentation spatiale » (Raumvorstellung) se trouve lui-même affecté des mêmes équivoques, « auxquelles il convient de prêter attention », selon qu’on se réfère à la « conscience extrascientifique » (außerwissenschaftlichen Bewußtseins) de l’espace, i. e. l’intuition de l’espace, ou à la pensée scientifique de l’espace, élaboration logique d’une formation purement conceptuelle de la représentation de l’espace, totalement sevrée de l’intuition qui a pu lui servir de point de départ61 . Or cette pensée en tant que purement conceptuelle est symbolique, ce qui ne sous-entend pas, prévient Husserl, que toute pensée symbolique soit conceptuelle62 . Le symbolique non-conceptuel mentionné ici en passant n’est pas à entendre au sens algébriste ou formaliste, mais fait référence à ces « moments de représentation impropre » de la perception (interne ou externe) ; les « moments (de) vide(s) de l’intuition », qui fonctionnent comme signes d’autres représentations intuitives, peuvent donc être comblés, remplis, sans qu’aucune pensée conceptuelle n’intervienne63 . Mais qu’en est-il de la formation de « l’intuition géométrique elle-même », de sa genèse ? Si l’on peut parler d’intuition, c’est en un sens purement analogique, prévient Husserl, dans une remarque où se manifeste la finesse d’analyse psychologique qui fournira le terreau propice au développement d’une phénoménologie pure. En décrivant l’espace intuitif, nous sommes en mesure de comparer les idées morphologiques produites ainsi avec les concepts géométriques proprement dits. « Par conséquent, lorsque nous décrivons l’intuition, il va de soi qu’il faut distinguer la pensée médiate (la description) de l’intuition elle-même (ce qui est décrit), qui n’est pas concernée, par cette élaboration conceptuelle et ne peut l’être. Nous pouvons de cette manière comparer aussi l’espace intuitif avec l’espace géométrique, bien que celui-ci aussi soit de part en part intuitif. Car son concept renvoie symboliquement à une chose à laquelle, si elle était donnée intuitivement, doivent correspondre adéquatement ces déterminations conceptuelles, que nous avons formées par des combinaisons purement symboliques et

60 Hua

21 : 270–271. Ce pli génétique et aristotélisant est aussi celui de Riemann, sauf que la genèse est conceptuelle chez Riemann. Cf. sur ce point Merker (2010 : 27 et 30, notes 57 et 64). 61 « Ein begriffliches Gebilde log y)) The non-standard infinite elements in Robinson’s non-standard model give rise to a non-Archimedean order, and he proceeded in such a way that the larger structure is again a field, R∗ . By definition, a field F contains, for each of its elements except 0, also its multiplicative inverse: ∀x ∈ F (x = 0 → ∃y ∈ F (x · y = 1)) The multiplicative inverses of the infinitely large elements in R∗ are infinitesimals. Robinson’s method enabled him to show R∗ |= φ ⇐⇒ R |= φ for φ limited to first-order formulas. First-order logic is not strong enough to distinguish between non-isomorphic models of the theory of real numbers. The equivalence here is called the Transfer Principle because it states that, for first-order φ, truth in R∗ transfers to truth in R, and vice versa. Robinson observed that the philosophical importance of the Transfer Principle is that, within the limitation to stay with first-order logic, it is a mathematical rendering of Leibniz’s Continuity Principle: G.W. Leibniz argued that the theory of infinitesimals implies the introduction of ideal numbers which might be infinitely small or infinitely large compared with the real numbers but which were to possess the same properties as the latter.3 However, neither he nor his disciples and successors were able to give a rational development leading up to a system of this sort. As a result, the theory of infinitesimals gradually fell into disrepute and was replaced eventually by the classical theory of limits.

3 [Note

MvA: An example of Leibniz’ saying this is found in his well-known letter to Varignon of February 2, 1702 (Leibniz 1859, pp. 93–94).]

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It is shown in this book that Leibniz’ ideas can be fully vindicated and that they lead to a novel and fruitful approach to classical Analysis and to many other branches of mathematics. (Robinson 1966, p. 2, original emphasis)

A largely constructive version of non-standard analysis had been developed by Schmieden and Laugwitz (1958) even before Robinson’s classical work. Their nonstandard objects can be considered to be constructive – they are infinite sequences of arbitrary rational numbers – but classical reasoning is used to reason about their properties. Also their transfer principle was limited; their R∗ was not a field but only a partially ordered ring. Inspired by Schmieden and Laugwitz, a fully constructive counterpart to Robinson’s work has been developed by Erik Palmgren. It turns out that full Transfer would require a change in the logic: the following is the argument presented in Palmgren 1998, p. 234. Theorem 1 (Moerdijk and Palmgren) A full constructivisation of Robinson’s Transfer Principle demands a nonstandard interpretation of the logical symbols. Proof 2 Assume that ∀x P(x) is an as yet undecided formula in arithmetic, where P contains no unbounded quantifiers. (Goldbach’s conjecture is of this form.) We can decide all instances up to any given bound, so N |= ∀m (∀n < mP (n) ∨ ∃n < m¬P (n)) Suppose now we have a Transfer Principle: N∗ |= φ ⇐⇒ N |= φ Let m ∈ N∗ be infinite. By transfer from N to N∗ , either for all n we have N∗  P(n∗ ), where n∗ is the image of n ∈ N in N∗ , or N∗  ∃ n < m ¬ P(n). Transfer back from N∗ to N yields N  ∀ xP(x) ∨ ¬ ∀ xP(x). This contradicts our assumption. Inspired by Robinson’s theory, but wishing to reconstruct it in syntactic rather than model-theoretical terms, Edward Nelson proposed Internal Set Theory (IST) (Nelson 1977).4 Another syntactic approach had been invented just before him by Hrbaˇcek, and closely related work (but incompatible with ZFC) by Vopenka had begun even earlier; for the historical details, I refer to the rich footnote 7 in Kanovei and Reeken 2004, p.vii. I will here look at IST, and in some detail, not so much because it is the best known and most used syntactical nonstandard analysis, but because it was part of Reeb’s Brouwerian approach that we will see in Sect. 5.8. It should be mentioned, however, that further development of both Nelson’s and Hrbaˇcek’s work have led to theories with more attractive metamathematical properties.5

4 For

an introduction to Nelson’s approach, see his more explanatory first chapter of a projected book Nelson 2002 and Robert 1988. 5 The culmination of this is HST in Kanovei and Reeken 2004; but it does not contain full ZFC as a proper part. See also footnote 45, below.

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Instead of enriching the ontology by adding nonstandard objects to the set of classical real numbers, Nelson enriches the language we have to talk about the latter.6 The idea is that, from a formal point of view, a distinction between standard and nonstandard numbers useful for the development of analysis can already be made within the set of classical real numbers; what matters is that this is done in such a way that the right formulas become provable. Nelson adds an undefined predicate ‘standard’ to the language of ZFC and adds three axioms to the theory that regulate its use; just as in ZFC, the relation ∈ is undefined. As the phrase goes, ‘its meaning is implicitly defined by the axioms’; but course that is not a specification of a meaning in the sense of presenting a construction method or a meaning explanation in the sense of, for example, Dummett and Martin-Löf. In the axioms for ‘standard’, the following shorthand is used:7 ∃st x φ(x) for ∃x (x standard ∧ φ(x)) ∀st x φ(x) for ∀x (x standard → φ(x)) ∀st fin x φ(x) for ∀x ((x standard ∧ x finite) → φ(x)) ∀st inf xφ(x) for ∀x ((x standard ∧ x infinite) → φ(x)) Then the new axioms are introduced, formally, and without pausing to motivate them. Formulas not containing the predicate ‘standard’ are said to be internal (namely, to ZFC), those containing it external: The axioms of IST are the axioms of ZFC together with three additional axiom schemes which we call the transfer principle (T), the principle of idealization (I), and the principle of standardization (S). They are as follows. Let A(x, t1 , . . . , tk ) be an internal formula with free variables x, t1 , . . . , tk and no other free variables. Then

6 It

is, in fact, possible to look at Robinson’s nonstandard analysis in an entirely formalistic way, and take it not to introduce new objects, but new ways of deducing theorems. Robinson points this out at the very end of his book: Returning now to the theory of this book, we observe that it is presented, naturally, within the framework of contemporary Mathematics, and thus appears to affirm the existence of all sorts of infinitary entities. However, from a formalist point of view we may look at our theory syntactically and may consider that what we have done is to introduce new deductive procedures rather than new mathematical entities. (Robinson 1966, p. 282)

I have not highlighted this in the main text, so as to be able to show the contrast between the model-theoretical and the syntactical approaches. 7 The predicates ‘finite’ and ‘infinite’ are defined as usual, in terms of the presence or absence of a bijection between x and the set {m | m < n} for some n ∈ N.

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M. van Atten   ∀st t1 . . . ∀st tk ∀st xA (x, t1 , . . . , tk ) → ∀xA (x, t1 , . . . , tk ) .

(T)

Let B(x, y) be an internal formula with free variables x, y and possibly other free variables. Then ∀st fin z∃x∀y ∈ zB (x, y) ↔ ∃x∀st yB (x, y) .

(I)

Finally, let C(z) be a formula, internal or external, with free variable z and possibly other free variables. Then ∀st x∃st y∀st z (z ∈ y ↔ z ∈ x ∧ C(z)) .

(S)

(Nelson 1977, p. 1166)

We may not use external predicates to define subsets, as the axioms of ZFC that would have to be used to prove the existence of these subsets do not know how to interact with the undefined predicate ‘standard’. (It is for this reason that not all of Robinson’s nonstandard analysis can be reconstructed in IST.) It is the role of the standardization axiom (S) to form standard subsets of standard sets. Note that standard sets may well contain nonstandard elements; we will see that N does, and in a sense that is the whole point of IST. As IST is not an ontological enrichment of R, Weyl’s question how the dimensions of the finite and the infinitesimal are related for IST does not point to a problem with an ontological aspect. William Powell proved, by model-theoretical means, that IST is conservative over ZFC and hence consistent relative to ZFC (Nelson 1977, section 8).8 In the following, we will look at a few theorems of IST, in order to demonstrate how IST proves that N contains nonstandard numbers, which are greater than any standard number; this mostly with an eye on the discussion of Reeb’s Brouwerian take on IST in Sect. 5.8.9 Their reciprocals are infinitesimals.10 First, a strengthening of (T) together with its dual form: Theorem 3  s  st   T ∀ t1 . . . ∀st tk ∀st x A (x, t1 , . . . , tk ) ↔ ∀x A (x, t1 , . . . , tk ) Proof 4 From (T) and ∀xφ(x) → ∀st xφ(x).

8 Nelson

himself later provided a purely syntactical proof that proofs in IST can be reduced to proofs in a standard system ZFC[V] which is itself conservative over ZFC (Nelson 1988). Kanovei and Reeken have shown that actually the presentation of IST there is stronger than that in Nelson 1977, and that not all properties of the later version are shared by the earlier one. However, they add that this is the case if only bounded sets are considered, and that in practice these are the ones that matter (Kanovei and Reeken 2004, p. 128). 9 The references for theorems 3–15 are Nelson 1977, 2002. 10 In IST it is also possible to prove of R directly that it contains infinitesimals. But the approach through N fits Reeb’s motivation better.

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Theorem 5     Tsd ∀st t1 . . . ∀st tk ∃st x A (x, t1 , . . . , tk ) ↔ ∃x A (x, t1 , . . . , tk ) Proof 6 Apply Ts to ¬A, negate both sides of the bi-implication, and use ¬ ∀ x ¬ A ↔ ∃ x A. An immediate consequence of Tsd is: Theorem 7 If ∃!x A(x), and A(x) is internal, then x is standard. In particular, N is standard. There is a dual of (I): Theorem 8   Id ∃st fin z∀x∃y ∈ zB (x, y) ↔ ∀x∃st yB (x, y) Proof 9 Apply (I) to ¬B(x, y), negate both sides, push the negations inward, and cancel double negations. The key theorem of IST is this, which entails that nonstandard objects exist formally: Theorem 10 Let X be a set. Then X is a standard finite set ⇐⇒ Every element of X is standard Corollary 11 Every infinite set has a non-standard element. In fact, it has infinitely many non-standard elements, because whenever we remove one non-standard element from it, the theorem applies again. First we prove Lemma 12 X is a subset of a standard finite set ⇐⇒ Every element of X is standard Proof 13 Set B(x, y) = x ∈ X ∧ x = y and apply the dual of idealisation (Id ) to ¬B(x, y): ∃st fin z∀x∃y ∈ z¬ (x ∈ X ∧ x = y) ↔ ∀x∃st y¬ (x ∈ X ∧ x = y) Applying logic to the right hand side, we get ∃st fin z∀x∃y ∈ z¬ (x ∈ X ∧ x = y) ↔ ∀x ∈ X (x standard) and to the left hand side, ∃st fin z (X ⊆ z) ↔ ∀x ∈ X (x standard)

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Proof 14 (Proof of Theorem 10) Left to right: The assumption that X is a standard finite set gives, together with X ⊆ X, ∃st fin z(X ⊆ z), and now apply Lemma 12 from left to right. Right to left: Assume that every element of X is standard. By Lemma 12, from right to left, we have ∃ st fin z(X ⊆ z). By ZFC, the power set of z, P(z), exists: ∃x∀y (y ∈ x ↔ y ⊆ z) This formula is internal, and P(z) is unique, so, by Theorem 7, P(z) is standard. It is also finite, because z is. Applying the proof in the previous paragraph for the direction from left to right to P(z), all elements of P(z) are standard, and as X ∈ P(z), in particular X is. Finally, X is finite because z is. By the corollary to Theorem 10, N contains a nonstandard number; this may be thought of as a proof that the natural numbers we usually work with, 0, 1, 2, . . . do not exhaust N. This idea became important to Reeb; see below, Sect. 5.8. Also by the corollary, there is no set containing exactly those natural numbers that are standard natural numbers; as mentioned, the set-forming principles of ZFC do not have a grip on the predicate ‘standard’. Finally, we have Theorem 15 The nonstandard numbers in N are greater than all its standard elements. First we prove Lemma 16 Two standard sets are equal if they have the same standard elements. Proof 17 Apply (T) to A(x, t1 , t2 ) = x ∈ t1 ↔ x ∈ t2 . Proof 18 (Proof of Theorem 15) Let n ∈ N be nonstandard. By (S), there exists a standard subset of N, notation S {z ∈ N| z < n}, such that it includes all standard elements of N that satisfy z < n; by Lemma 16, that set is unique.11 Obviously, any standard element of S {z ∈ N| z < n} is a standard element of N. On the other hand, if z is a standard element of N, then the set {w ∈ N| w ≤ z} is a standard finite set. Theorem 10 entails that all its elements are standard, hence n ∈ {w ∈ N| w ≤ z}, and z < n. It follows that S {z ∈ N| z < n} and N have the same standard elements. Both are standard sets, so Lemma 16 applies and S {z ∈ N| z < n} = N. From a radically formalist position the axioms could be left unmotivated, once the axioms are shown or at least believed to be consistent. Nelson’s paper includes a (relative) consistency proof; in his later book chapter, there are informal considerations for accepting them. I single out the one for (I), as it is the one that formally implies the existence of nonstandard objects:

11 But

(S) does not guarantee that it does not also contain nonstandard elements.

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The intuition behind (I) is that we can only fix a finite number of objects at a time. To say that there is a y such that for all fixed x we have A [i.e., B(x, y)] is the same as saying that for any fixed finite set of x’s there is a y such that A holds for all of them. (Nelson 2002, p. 5)

Nelson acknowledged of course that there is informal discourse in mathematics, and the term ‘fixed’ belongs to that realm (Nelson 2002, p. 1). But since Nelson’s reflected judgement is that there is no mathematical reality, be it intuitionistic or Platonic, and that strictly speaking mathematics consists in formal systems,12 the strict counterpart of the informal discourse’s notion of being fixed for him must be found in a property of formal proofs. The statement ‘we can only fix a finite number of objects at a time’ is then mapped to the fact that each proof in the formal system at hand is a finite object, which therefore leaves room for only finitely many occasions to define (fix) individual objects and prove or assume their existence. An analogous argument for a simpler case is this (presentation after Palmgren 1993, p. 1195): Theorem 19 Extend Peano Arithmetic with a constant ω and the axiom schema ω > n, to obtain a nonstandard theory PA∗ . Then PA∗ is conservative over PA. Proof 20 Assume we have a formal proof of A(ω). As the formal proof is finite, only finitely many instances of the schema occur in it, ω > n1 , . . . , ω > nk . Define m = max(n1 , . . . , nk ) + 1, and replace, in the original proof, ω by m everywhere. Below, in Sect. 5.8, we will see that according to Reeb, who was not a formalist and who held that there is a mathematical reality which furthermore is constructive, there is in mathematical reality a motivation for introducing the predicate ‘standard’ in ZFC, and from there for accepting the formal theory IST.

5.3 Three Brouwerian Desiderata Turning now to Brouwer’s writings, one may distill three desiderata for constructions of infinitesimals: 1. They should be intuitionistic constructions, i.e., be built up starting from ‘the basic intuition of mathematics’: the substratum of all perception of change, which is divested of all quality, a unity of continuous and discrete, a possibility of the thinking together of several units, connected by a ‘between’, which never exhausts itself by the interpolation of new units. (Brouwer 1907, p. 8, trl. Brouwer 1975, p. 17)

Further on in the dissertation Brouwer specifies that this basic intuition consists in the awareness of time as pure change (Brouwer 1907, pp. 98–99), that it and our

12 See

on this also chapter 32, ‘A modified Hilbert Program’, in Nelson 1986.

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construction acts on it are not of a linguistic nature (Brouwer 1907, p. 169), and that there is not also a spatial continuum that is a priori given to us (Brouwer 1907, p. 121). 2. Logical reasoning about them should be done according to the nature of mental constructions. This respects the essential non-linguistic character of mathematical construction, and the nature of logic, such as Brouwer describes it, as a study of the patterns in descriptions of that activity (Brouwer 1907, pp. 131–132, 1908). Whatever logical principle one has recognised as correct on this conception should be allowed in one’s reasoning. 3. They should be geometrical in nature. Brouwer defines geometry as follows: Geometry is concerned with the properties of spaces of one or more dimensions. In particular it investigates and classifies sets, transformations and transformation groups in these spaces. The spaces under consideration are built up out of one or more Cartesian simplices,13 which can be connected in different ways; consequently a space is not completely defined by its dimension alone. (Brouwer 1909, p. 15, trl. Brouwer 1975, p. 116)

The classical and constructive nonstandard-models of the previous section obviously do not meet these desiderata, and neither does a purely axiomatic approach. But it is of course just as clear that there will be no direct intuitive construction of infinitesimals on the one-dimensional continuum. If in the next section the reason for this is spelled out, that is because it adds relief to Brouwer’s construction in Sect. 5.6 of a real number that is greater than 0, but of which we cannot indicate a positive distance from 0; this is the kind of construction that Vesley took up, as we will see in Sect. 5.7. At the same time, it should also be noted that even in classical nonstandard analysis there is a large constructive element in the following sense: Once non-constructive methods have been employed to obtain infinitesimals, the reasoning often proceeds constructively, employing standardisation at the very end to return to the realm of the standard. That topic has recently been explored in great detail in Sanders 2018.

5.4 The Impossibility of a Direct Construction on the One-Dimensional Continuum The intuitive continuum as given in what Brouwer calls the basic intuition of mathematics has no scale on it.14 Brouwer’s ‘between’ is not intrinsically tied to intervals of any size, because if there is no scale then there are no sizes, in particular

13 In

two dimensions a simplex is a triangle with all its interior points; in three dimensions a pyramid with a triangle as its base. 14 The primary reference for this paragraph and the next is Brouwer’s dissertation, Brouwer 1907, pp. 8–11, but the argument is general.

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no infinitesimal ones. This intuitive ‘between’ precedes the construction of a scale, and the scale is constructed by ‘the interpolation of new units’ on it. Putting a scale on the intuitive continuum is itself a construction process that takes place over time. The human mind is limited in such a way that in a given time interval we can only place finitely many points of a scale on the intuitive continuum, or begin a potentially infinite sequence of such placements. Between any two previously placed points, an intuitive continuum remains, and if we choose to do so, we can place a further point on this ‘between’, and thereby continue the construction of our scale into it. If we iterate this everywhere, in a potentially infinite process, we construct a countable, everywhere dense scale. We can correlate the points on this scale with any number system we have constructed of the order type of Q; we may begin by correlating an arbitrary point on the scale with 0 and another one with 1. We thus obtain a ‘measurable continuum’ (Brouwer 1907, p. 11). Once the construction process of the rational scale has begun, we then construct points or real numbers (including the embedding of the rationals) p by constructing potentially infinite sequences of nested intervals with endpoints on the scale p0 , p1 , p2 , . . . As a point does not exist on the continuum prior to our construction, it is identified with the developing sequence, as opposed to an independently existing limit to which the sequence converges. If two points p and q are not equal, this unequality must consist in the fact that starting from a finite index n, the intervals pn and qn do not overlap. As the intervals are determined by rationals, we can determine a natural number m such that mp > q or, as the case may be, mq > p. So the system is Archimedean, and infinitesimals or intervals of infinitesimal length do not exist. To construct an infinitesimal interval on the intuitive continuum, we would have to be able to construct a point p such that ¬(p = 0) but of which it is contradictory to assume that the unequality to 0 arises at some pn for natural n (i.e., at a rational distance from 0). That is impossible. We will see in Sect. 5.6 that, with the admission into intuitionism of choice sequences, we can construct a real number r that is unequal to 0 and of which we cannot indicate the interval rn at which the unequality to 0 arises until a certain proposition P has been decided. But before P has been decided we can already show that it is contradictory to suppose that the unequality to 0 arises at no finitely indexed interval. The fact that there are, in the basic intuition of mathematics, no direct motivation and no direct construction for infinitesimals (as objects constructed on the onedimensional continuum), the development of a theory would have to proceed, just as in the classical case, either by an embedding of the standard real numbers into a more-dimensional structure, and thereby no longer take propositions in analysis of the one-dimensional continuum at face value, or construe talk about nonstandard objects as talk about certain standard objects.

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5.5 Brouwer’s Non-Archimedean Numbers Early Brouwer’s construction of non-Archimedean numbers discussed in this section was not meant to lead up to a form of infinitesimal analysis. However, that would have been a first step, so the general considerations are of interest to the present discussion. When Brouwer was working on his dissertation, non-Archimedean fields and geometries had been constructed by notably Veronese, Levi-Civita, Pasch, Hilbert, and Vahlen,15 and these he refers to in his notebooks and in his thesis, with an emphasis on Hilbert. Brouwer criticised these approaches: Veronese’s was not constructive in his sense, and those of Pasch, Hilbert and Vahlen are not geometrical in his sense. In Veronese’s Fondamenti di Geometria of 1891, a real number is construed as the ratio of two magnitudes (both of the same species), one of which is designated to be the unit; and the existence of a segment that is infinitesimal with respect to another is postulated. Veronese can do so because, as he states in his introduction, ‘A thing postulated by thought one can consider as given to thought, and inversely’ (Veronese 1891, Introduzione, section 18, trl. mine). For Brouwer, on the other hand, only what has been constructed from the basic intuition qualifies as given. In a notebook that predates his dissertation, he comments: Veronese’s fuss, with his constantly introducing hypotheses, is nothing but forming logical assemblies; if for certain things (I do not know whether they exist) such and such relations hold, then also such and such relations. (Brouwer 1904–1907, Notebook 3, p. 35, trl. mine)

Hilbert’s non-Archimedean geometries are criticised for their non-geometrical nature. In the synopsis for his dissertation, Brouwer writes: Hilbert’s pseudo-geometries are (in contrast to the non-Euclidian) of little importance, because they have been built within a rather ‘far-fetched’ building [i.e., construction] (while the non-Euclidian in the ordinary Cartesian space). (Brouwer 1904–1907, p. 405, trl. mine)

Brouwer is referring to the fact that the coordinates of points in the space are not real numbers but objects of a higher type, namely certain algebraic functions on the real numbers (Hilbert 1899, p. 25). Such algebraic functions may themselves be represented (extensionally) geometrically, but Brouwer’s hesitation here would be that each such representation is not a point in a (n-dimensional) Cartesian space. Hilbert, of course, proposed his non-Archimedean geometry first of all in the service of an independence proof of the Archimedean axiom, and then Brouwer’s considerations are not important. But such geometries soon turned out to be of interest in their own right. In his dissertation, Brouwer presents an alternative non-Archimedean continuum, a(n intended) construction in his specific, non-axiomatic sense of that term, in an

15 An

extensive historical treatment is Ehrlich 2006.

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ordinary Cartesian space (Brouwer 1907, pp. 67–73). It is, he states (Brouwer 1907, p. 140n) a generalization of that in Hilbert 1899, section 34. By Brouwer’s de facto phenomenological consideration, given in the previous section, a non-Archimedean (mathematical) continuum cannot be constructed on the one-dimensional intuitive one. His strategy therefore was to construct a multidimensional mathematical continuum and define a subset on it which he calls ‘the pseudo-continuum’. The pseudo-continuum can be linearly ordered and the onedimensional continuum embedded into it. Brouwer starts with an infinite-dimensional Cartesian space of (ω∗ + ω)n dimensions, where ω∗ is . . . , − 3, − 2, − 1. Each coordinate has (instead of a letter) an ordinal number in between −ωn and ωn , which can be written in the form a1 ωn − 1 + · · · + an − 1 ω + an , with −ω < ai < ω. To the coordinate then is associated the n-tuple of indices a1 , . . . , an . The ‘pseudo-continuum’ now consists of the subset of points in the space with the property that for all their coordinates whose value is not 0 we can indicate lower bounds on the ai : the property, in other words, that non-zero values are not found at arbitrarily low coordinate numbers. This in turn means that the coordinate numbers corresponding to those n-tuples form a well-ordered set (i.e., a set of which each non-empty subset has a first element). So for any two distinct points p, q there will be a smallest coordinate number at which they differ, and therefore the set can be linearly ordered. The one-dimensional Archimedean continuum is embedded into the pseudocontinuum by assigning the point on the former with coordinate x to the point on the latter whose coordinates are all 0 except that its 0-coordinate is x. One may view the pseudo-continuum as a real continuum with infinitely many points inserted to the right and left of each real point, and with infinitely many pseudo points to the left and right of the real continuum as a whole. The operations + and × are understood group-theoretically, that is, as parametrised transformation operations +a and ×a. Theorem 21 (Brouwer 1907) On a measurable continuum, there is only one construction for a two-parameter continuous uniform transformation group x  = c1 × x + c2 namely the one in which + and × are ordinary addition and multiplication (and hence commutative). (Brouwer 1907, pp. 32–33). On the pseudo-continuum he then defines a two-parameter continuous uniform transformation group that preserves + and × on the embedded one-dimensional continuum, but whose multiplication is not commutative on the pseudo-continuum as a whole. The group operation + on the pseudo-continuum is induced by the + operation on the scale of each of the coordinates. It is associative and commutative. The operation × on the pseudo-continuum is defined as an operation that shifts coordinates; 11 × shifts the number of a coordinate to the right by one while mapping

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the 1-points of the scales of each coordinate onto one another. Likewise, 1ω × shifts the number of a coordinate to the right by ω while mapping the 1-points onto one another. Brouwer determines conditions on the the choices of the 1-points on the scales of coordinates 1, ω, ω2 , . . . that will guarantee that is × associative and distributive with +. But × on the pseudo-continuum has been defined so as not to be commutative: for example, 11 × 1ω = 1ω+1 but 1ω × 11 = pω+1 where p is the point on the scale of coordinate 1 chosen to be the 1-point on that scale. In general, 1ω + 1 and pω + 1 are not equal. Theorem 21 then implies that this pseudo-continuum is not a measurable one. Brouwer remarks that this pseudo-continuum is not continuous in Dedekind’s sense (which would have implied it is Archimedean), but it is in Veronese’s (Brouwer 1907, pp. 72–73).16 An objection to the way the pseudo-continuum is constructed is that it presupposes the Principle of the Excluded Middle: In order to obtain the linear ordering of its points, it must be possible to decide whether a sequence that proceeds infinitely to both sides is from a certain element onward constant zero to the left. This is similar to a problem in another part of his thesis, which is flagged and discussed in the corrections that he published in 1917: When moving down along a branch in a tree, one cannot, in general, decide whether each future node will have a unique descendant (Brouwer 1917, p. 440). At the time, Brouwer accepted the Principle of the Excluded Middle because he took P ∨ ¬ P to be equivalent to ¬P → ¬ P (van Dalen 1999, pp. 106–107). Constructively, it is not; Brouwer presented the correct reading, according to which PEM is valid only for decidable propositions, in Brouwer 1908, ‘The unreliability of the logical principles’.17 Moreover, Brouwer’s construction, if successful, would not be a field extension of the real numbers, so it could not have been used to develop a nonstandard analysis. It is not surprising, then, that Brouwer did not develop the theory of this pseudo-continuum any further. The work on non-Archimedean numbers was superseded by Hahn’s paper ‘Über die nichtarchimedischen Größensysteme’ (Hahn 1907), which appeared just too late to be taken into consideration in Brouwer’s thesis, which was defended on February 19 of the same year. But in 1917,

16 See 17 A

on this point Ehrlich 2006, 69–71. recent English translation and introduction is van Atten and Sundholm 2017.

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Brouwer referred to it in his list of additions and corrections to his dissertation specifically for its treatment of commutative principal operations (Brouwer 1917, p. 441). One of the few people who seem actually to have studied Brouwer’s pseudocontinuum is Kurt Gödel. In 1941, by which time he had emigrated to the United States, he asked his brother, who had remained in Vienna, to order a copy of Brouwer’s dissertation for him (van Atten 2015, pp. 190–191). And indeed, in one of Gödel’s notebooks, probably filled in 1942, one finds reading notes on Brouwer’s construction.18 Gödel at the time was interested in the question if there could be non-human beings in whose awareness time is ordered in a non-Archimedean way (Gödel 1906–1978, Max Phil VI (?–July 1942), pp. 431–432). Brouwer made one last remark on non-Archimedean geometry in his second lecture in Vienna 1928 – with Hahn, who had in the meantime become a friend of his, in the audience:19 The initial, negative attitude towards these [non-Euclidean or non-Archimedean] geometries was completely overcome by their arithmetisation due to Riemann, Beltrami, Cayley, and, respectively, Levi-Civita and Hahn. In the process, the peculiar fact came about that the non-Archimedean continuum, which had proved to fulfill the a priori conditions on the continuum just as well as the Archimedean, was brought about in a plausible manner only with the aid of the latter, so that the calling into question of the a priori necessity of the Archimedean continuum had to be founded precisely on the a priori consistency of this continuum. (Brouwer 1930, p. 1, trl. mine)

5.6 Brouwer: A Real Number That Is Greater Than 0, But Not Measurably Greater Around 1916, Brouwer introduced choice sequences into intuitionistic analysis.20 A choice sequence is a sequence of natural or rational numbers that are freely chosen by the Creating Subject, which is moreover free to impose restrictions on its choices. Thus we have sequences without any restriction on the choices (lawless sequences) and sequences determined by an algorithm or law (lawlike sequences). For Brouwer these are the extreme cases, with many other kinds of choice sequence in between, notably also choice sequences for which the Creating Subject lets its choices depend on some of its other mathematical activities. (We will see examples of this latter kind in this section and the next.)

18 Gödel

1906–1978, Arbeitsheft 14, pp. 21–23; see its page 14 for the year.

19 Brouwer’s Vienna lectures were invited by a committee of which Hahn was a member (van Dalen

2005, p. 561). an introduction to choice sequences, with particular attention to philosophical and mathematical differences between Brouwer’s theory and Weyl’s adaptation of it, see van Atten et al. 2002; for their history, Troelstra 1982.

20 For

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Brouwer’s rationale for reconstructing analysis in a theory of choice sequences is that this gives a mathematical, fully constructive model of the intuitive continuum that faithfully mirrors, not only epistemologically but ontologically, the fact that the latter is not a composition out of discrete elements. Weyl, in his intuitionistic period, accepted the theory of choice sequences in a modified form that, however, made it incoherent (van Atten et al. 2002, section 3). Be that as it may, of some interest for our present theme is Weyl’s intention to accept universal quantification over lawless sequences but to insist that instantiations are lawlike, for in this way he is in effect treating lawless sequences as nonstandard objects. For Weyl, only lawlike choice sequences could exist as individuals. Brouwer’s particular choice sequence that is the topic of the present section would not have been acknowledged as an individual mathematical object by Weyl. The introduction of choice sequences did not affect early Brouwer’s observation on the impossibility to construct a non-Archimedean scale. The reason is that the latter observation is made at such a high level of generality that it also subsumes choice sequences. Yet, in the Cambridge Lectures (1946–1951) Brouwer states that the intuitive ‘between’ surely requires as well that the continuum contains further point cores between, for instance, the origin on the one hand and all rational point cores on the other. (Brouwer 1981, p. 50)

A ‘point core’ is an equivalence class (or rather a ‘species’) of choice sequences, the criterion being that they are all co-convergent. Brouwer seems to be saying here that one can construct points that are not 0 yet whose distance from 0 is smaller than any rational number we will ever construct; he seems to be saying that we can construct infinitesimals. It will turn out that this is not quite what is meant.21 To show Brouwer’s argument for this claim, his definitions of some order relations are needed (Brouwer 1949c, p. 1246n). Let β and γ be two real numbers, i.e., two convergent infinite sequences of rational numbers β(n) and γ (n). Define β  γ , ‘β is measurably smaller than γ ’ as 

1 ∃m, n ∈ N∀v ∈ N v ≥ m → γ (v) − β(v) > n 2



Correspondingly, γ  β means that γ is ‘measurably greater’ than β (Brouwer 1951, p. 3). Write β = γ for ¬ (β = γ ) β ≥ γ for ¬ (β  γ ) 21 The

same construction is also in Brouwer 1948, but there Brouwer does not add the comment quoted above.

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β > γ for β ≥ γ ∧ β = γ So being measurably greater, defined as a double existential statement, is a positive property, while being greater, defined as in effect a conjunction of two negations, is a negative property. The apartness relation (Brouwer 1949c, p. 1246n) is defined as follows:   1 β # γ ≡ ∃k ∈ N |β − γ | > k 2 or, equivalently, β #γ ≡β γ ∨β γ The definition of a choice sequence, and so in particular of a point core that it represents, may be made to depend on what goes on in the subject’s other activities in between the choices of the elements in this sequence, notably with respect to attempts to settle a certain problem. For example, in between two choices, the subject may have decided a proposition P, or have tested it. A proposition P is decided by either proving P or proving ¬P; it is tested by either proving ¬P or ¬¬P. Decidability implies testability. If P holds, then so does ¬¬P, and if ¬P, then ¬P; so P ∨ ¬P implies ¬P ∨ ¬¬P. But testability does not imply decidability. For suppose we can prove ¬¬P but not (yet) P; then P has been tested but is still undecided. Weak Counterexample 22 (Brouwer) There is no hope of showing that ∀α(α > 0 → α  0).22 Plausibility Argument 23 Let P be a proposition that we cannot test, in the weak sense that we do not now possess a proof of ¬P ∨ ¬¬P.23 The Creating Subject constructs a real number α in a choice sequence of rational numbers α(n), as follows: • As long as, when making the choice of α(n), the Creating Subject has obtained evidence neither of P nor of ¬P, α(n) is chosen to be 0.

22 In

the next section, we will see that Brouwer also had a proof of the actual negation, ¬ ∀ α(α > 0 → α  0) (Theorem 28). 23 Brouwer could have given this argument in terms of an undecidable proposition instead of an untestable one. The reason he uses an untestable one is that in his paper he exploits almost the same construction to prove that = cannot be defined as a disjunction of < and >, as that would lead to the contradiction that an untestable proposition is testable. For further discussion of Brouwer’s weak and strong counterexamples, see van Atten 2018.

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• If between the choice of α(n − 1) and α(n), the Creating Subject has obtained evidence P, or has obtained evidence of ¬P, α(n) and all α(n + k) are chosen  of n 1 to be 2 . The choice sequence α converges, hence α is a real number.24 We have α = 0 ↔ ¬P ∧ ¬¬P Hence α = 0. We also have ¬(α  0) because, by definition of α, no α(n) is ever smaller than 0; and their conjunction gives α > 0. But we do not have the stronger α  0 because if we had, then   1 ∃m, n ∈ N∀v ∈ N v ≥ m → α(v) > n 2 and this is only possible if P would have been decided, and hence tested; but this contradicts the hypothesis that P cannot be tested yet. This is what Brouwer means when, in the quotation from the Cambridge Lectures above, he says that there are point cores between the origin and all positive rational point cores. If, by developing more mathematics, we do come in a position to test P, that is we can find a proof of ¬P or a proof of ¬¬P, then the number α becomes rational, and, in the sense of these order relations, no longer lies between 0 and all rational point cores. Note that this does not mean that α was irrational before.25 Brouwer does not go on to connect his example of a number between 0 and all the rationals in any way to infinitesimals. That would be done by Vesley.

5.7 Vesley’s α-Infinitesimals Vesley realised that real numbers like the one in Brouwer’s example are, although not infinitesimals in an ontological sense, in an important respect similar to infinitesimals (Vesley 1981).26 The same observation was made independently in van Dalen 1988, p. 191. Vesley appeals to Kripke’s Schema:

 be given, and determine an n such that 2−n < . Construct the sequence α up to α(n), which can be done as each choice is decidable. If α(n) = 0, all further choices will be in the interval [0, 2−(n + 1) ] and hence within  from one another. If α(n) = 0, then the choices in α have already been fixed, and hence within  from one another. 25 Before, it was a growing construction for a real number that had yet acquired neither the property of being rational, nor that of being irrational. 26 I don’t think Vesley knew of that particular passage in Brouwer, which was published only in 1981, but he was of course very familiar with this kind of reasoning, e.g. Kleene and Vesley 1965. 24 Let

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∃α (∃n α(n) = 1 ↔ P )

(KS)

where P is a variable for propositions, and α for choice sequences. Brouwer had demonstrated this before Kripke did but never used it again, and instead reasoned from the general principles from which KS quickly follows.27 These principles were later codified by Kreisel in the so-called ‘Theory of the Creative Subject’ (or ‘Creating Subject’). For discussion of the Creating Subject and KS, see Myhill 1966; Kreisel 1967; Troelstra and van Dalen 1988, ch.4; van Atten 2004, ch.5; and van Atten 2018. I will adapt Vesley’s construction somewhat to Brouwer’s way. Let α be a choice sequence (whether of convergent rationals or not). Define a real number x as follows: • As long as, when making the choice of x(n), the Creating Subject has obtained evidence of neither ∀nα(n) = 0 nor of ¬ ∀ nα(n) = 0, x(n) is chosen to be 0. • If between the choice of x(n − 1) and x(n), the Creating Subject   has obtained n

evidence of ∀nα(n) = 0, x(n) and all x(n + k) are chosen to be 12 . • If between the choice of x(n − 1) and x(n), the Creating Subject has  obtained n evidence of ¬ ∀ nα(n) = 0, x(n) and all x(n + k) are chosen to be − 12 . Then we have ∃x ∈ R [(x  0 ↔ ∀nα(n) = 0) ∧ (x  0 ↔ ¬∀nα(n) = 0)] and, since α was an arbitrary choice sequence, ∀α∃x ∈ R [(x  0 ↔ ∀nα(n) = 0) ∧ (x  0 ↔ ¬∀nα(n) = 0)] Define for every choice sequence the species of real numbers L(α) and M(α): x ∈ L (α) ≡ [x # 0 ↔ ∀nα(n) = 0 ∨ ¬∀nα(n) = 0] x ∈ M (α) ≡ ∃y ∈ L (α) ¬ (|x|  |y|) Vesley points out that one can then prove: Theorem 24 M(α) is a subring of the intuitionistic R.

27 See

Brouwer 1954, p. 4 for Brouwer’s demonstration, and Myhill 1966, p. 295 for the observation that this is KS. Brouwer does not literally state KS; he constructs, from an arbitrary proposition P that as yet cannot be tested, an infinite sequence C(γ , P), and shows that truth of P and rationality of C(γ , P) are equivalent. However, the construction of a witness for KS from C (γ , P) is immediate; and Brouwer’s reasoning towards the existence of C(γ , P) goes through for any P, not only untestable ones.

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Theorem 25 ¬∀α∃x ∈ M(α) ∃n (n · x  1) The species M(α) is called that of the α-infinitesimals. These are elements of (intuitionistic) R, and in this sense the conception is a little like that in Nelson’s IST, where the infinitesimals are elements of classical R. Vesley observes that, although we want to think of the α-infinitesimals as very small, and for that reason give them this suggestive name, should the question whether α is 0 everywhere be decided, M(α) becomes R. This goes against the very idea of an infinitesimal. On the one hand, in light of openendedness of mathematics, there will always be new open problems, so this is not much of an objection against the existence of α-infinitesimals in general. On the other hand, this also means that these infinitesimals only behave as infinitesimals under universal quantification. We cannot prove that all species of αinfinitesimals are non-Archimedean; only that it is not the case that none of them is. Instead of ∀α¬∃x ∈ M(α) ∃n (n · x  1) we only have ¬∀α∃x ∈ M(α) ∃n (n · x  1) And that the latter cannot be strengthened to the former is intrinsic to the whole construction. An individual M(α) will be non-Archimedean as long as it is undecided whether the values of α are 0 everywhere or not, but becomes Archmedean as soon as this has been decided. And there cannot be a particular α for which this is never decided, for that would imply the existence of an absolutely undecidable proposition, which is impossible:28 Theorem 26 (Brouwer, 1907–1908?) There exist no absolutely undecidable propositions. Proof 27 ‘Can one ever demonstrate of a proposition, that it can never be decided? No, because one would have to so by reductio ad absurdum. So one would have to say: assume that the proposition has been decided in sense A, and from that deduce a contradiction. But then it would have been proved that not-A is true, and the proposition is decided after all.’ (Note by Brouwer, as quoted in van Dalen 2001, p. 174 note a; translation mine). 28 One

might think the permanent existence of an α-infinitesimal can be assured by starting a sequence starting with 0’s and stipulating that one will never make the decision between (a) restricting the remaining choices to 0 and (b) making a choice that is not 0. This however will not do, because by choosing 0 until that decision is made, and at the same time resolving always to postpone that decision, the result is that the sequence will be constant 0.

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That ¬(P ∨ ¬ P) is contradictory, and hence that PEM is consistent, was pointed out in Brouwer 1908. The quoted argument was never published by Brouwer, but Wavre 1926, p. 66 and Heyting 1934, p. 16 made the same observation. Van Dalen, who as mentioned made the same connection between Brouwer’s weak counterexample and infinitesimals, considered it ‘highly unsatisfactory to include subjective phrases such as “it cannot be shown” in mathematical texts’ (van Dalen 1988, p. 191), and points out that in Brouwer 1949a this result is strengthened from the weak counterexample ‘there are real numbers α that are greater than 0 yet cannot be shown to be measurably greater than 0’ to the strong counterexample. Theorem 28 (Brouwer 1949a) ¬∀α (α > 0 → α  0) (Note that intuitionistically this does not imply ∃α ¬(α > 0 → α  0), which is contradictory.) Indeed, in the demonstration of this theorem, instead of one open problem and unbounded time to solve it, Brouwer considers the infinity of open and solved problems ‘α ∈ Q’ for all α ∈ [0, 1] with the added condition, which arises from the applicability of the fan theorem to functions defined on that interval, that they should all be solved on the basis of an initial segment of α of uniform length.29 And then not only it cannot be shown that this condition can be met (weak negation), it can be shown that it is leads to contradictions if it could (strong negation). Vesley observes that in this version of nonstandard analysis, ‘the elegance of classical nonstandard analysis is missing for the familiar reason that more distinctions must be recognized intuitionistically’ (Vesley 1981, p. 211), but because of the dependency of his infinitesimals on universal quantification he also sees a similarity to the synthetic differential geometry of Lawvere and Kock,30 which has been developed much further (Kock 2006). From a Brouwerian perspective, Vesley’s approach would philosophically be preferable to synthetic differential geometry (smooth infinitesimal analysis) in that the latter involves a postulation of the existence of a line segment of infinitesimal length, which is certainly not given to us in mathematical intuition.31 Vesley announced a sequel paper to develop the approach further and to see whether it has advantages of its own. Unfortunately, it seems that he gave up on the project.

speaking, Brouwer does not consider the question of rationality of each α ∈ [0, 1], but of each α ∈ J, where J is a fan that coincides with [0, 1]. Note also that our notational use of α is different from that in Brouwer 1949a. 30 There, ‘nilpotents’, which are δ such that δ = 0 but δ 2 = 0, may be cancelled when universally quantified. 31 Compare Brouwer’s objection to Veronese’s postulate above, p. 11. 29 Strictly

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5.8 Reeb: An Intuitionistic Take on IST An approach to infinitesimals that is Brouwerian in a rather different sense than that defined by the three desiderata of Sect. 5.3, and one that would have been of interest to Weyl, was proposed and enthusiastically defended by Georges Reeb at the University of Strasbourg.32 The mathematical content of nonstandard analyis as advocated by him is that of his friend Nelson’s IST,33 described in Sect. 5.2. The originality of Reeb’s approach lies in the fact that, instead of construing IST as an axiomatic theory, in which the predicate ‘standard’ is taken to be implictly defined by the axioms, he proposes a specifically Brouwerian motivation for accepting IST as a formal theory. It is, he notes, a train of thought that starts from intuitionistic observations on ZFC and concludes to ‘the plausibility or the naturalness of IST’ (Reeb 1989, p. 151)34 ; and the article he dedicated to giving his most elaborate account of this, written with Jacques Harthong, was, indeed, titled ‘Intuitionnisme 84’ (Reeb and Harthong 1989).35 The title was, of course, at the same time a reference to Robinson’s ‘Formalism 64’ (Robinson 1965). Just as Robinson asked what formalism could be in 1964, Reeb and Harthong had a view on what intuitionism could be in 1984. In addition, formalism is an essential component of Reeb’s approach, but with a specifically Brouwerian view on it. On the other hand, Reeb was not a Brouwerian intuitionist, and did not aspire to be. Reeb and Harthong acknowledge the difference when they speak of ‘the intuitionistic conception (ours just as much as that of Brouwer) . . . ’ (Reeb and Harthong 1989, p. 52).36 Also Harthong in his afterword 32 For

Reeb’s (philosophy of) nonstandard analysis, see, in French, Reeb 1979, 1981; Barreau and Harthong 1989; Diener and Reeb 1989; Lobry 1989; Reeb and Harthong 1989; L’Ouvert 1994, and Salanskis 1999. There is not much about Reeb’s (philosophy of) nonstandard analysis in English; see Fletcher et al. 2017 for a few recent remarks. 33 Nelson has written: One of the most treasured experiences of my life is my friendship with Georges Reeb. We had many strong discussions together, intuitionist versus formalist. What he created was unique in my experience. His rare spirit, gentle but fiercely demanding of the highest standards, inspired a group of younger mathematicians with an unmatched ethos of collegiality. And their discoveries are extraordinary. Reeb found, and led others to find, not only knowledge and beauty in mathematics, but also virtue. His insights into the nature of mathematics will point the way towards the mathematics of the future. (Nelson 1996, p. 8) 34 In

Reeb 1981, p. 153, he had stated that his notion of naïve numbers leads to ideas that ‘show some analogy with IST’, and this is what one expects of a motivation in constructive reality of a distinction in an idealised, classical formal theory. Note that Reeb in his writings does not much discuss his philosophical differences with Nelson. On Nelson’s philosophy of mathematics, see, besides his own papers, also Buss 2006. 35 1984 is the year in which a first version was written and began to circulate. 36 ‘la conception intuitionniste (aussi bien la nôtre que celle de Brouwer) . . . ’

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to the 1989 reprint of ‘Intuitionnisme 84’ in La mathématique nonstandard is quite forthcoming on this point (Harthong 1989, pp. 265–267). Some differences will be touched upon below. Reeb holds that there is a mathematical reality, constructive, indepent of theory, and which intuitionists aim to describe;37 formal theories such as ZFC and IST as constructive objects that codify ideal(ised) theories of that reality, just as we have idealised theories of physical reality (Reeb and Harthong 1989, sections 2 and 5). Accordingly, the notion of motivation for an axiom takes on a richer sense that relates the formal axiom to mathematical reality. That relation need not be as strong as the axiom being fully interpretable in that reality; it may rather be construed as an idealisation. This sounds Hilbertian, and it is,38 but it must be remembered that it was intuitionistic criticism of his earlier program that led Hilbert to adopt that particular view, which itself goes back to Brouwer’s dissertation (1907).39 The terms of the formal theory do, as such, not refer, but if we construe that theory as an idealisation of reality we must be prepared to act as if they refer to ideal(ised) objects (Salanskis 1994, p. 30). Consistency or conservativeness of the formal theory is therefore not the whole criterion: if ZFC is extended with an independent proposition P, or, alternatively, ¬P, Reeb expresses a preference for the one that seems in a sense closer to mathematical reality than another: writing about Fermat’s Last Theorem in 1989, he says that if it turns out to be an undecidable proposition, one could of course add its negation to ZFC, but ‘the intuitionist will consider this choice . . . far removed from concrete reality’ – in which by then no counterexample had been found (Reeb 1989, p. 158). The importance of Brouwer in Reeb’s view is epistemological, and defined by the insistence that a formal theory, even if shown consistent, cannot, once constructed, replace mathematical reality (Reeb and Harthong 1989, section 4), and that, in particular, accepting the Principle of the Excluded Middle in the formal theory does not mean that every problem in mathematical reality can be solved. Reeb’s main reference for this is ‘Intuitionistic reflections on formalism’ of 1928, in which Brouwer writes: The disagreement over which is correct, the formalistic way of founding mathematics anew or the intuitionistic way of reconstructing it, will vanish, and the choice between the two activities be reduced to a matter of taste, as soon as the following insights, which pertain primarily to formalism but were first formulated in the intuitionistic literature, are generally accepted. The acceptance of these insights is only a question of time, since they are the results of pure reflection and hence contain no disputable element, so that anyone who has once understood them must accept them. Two of the four insights have so far been

37 This

attitude was later described by Sundholm and myself as the intuitionists’ ‘ontological descriptivism’, an attitude they share with Platonists, the disagreement being over the nature of that reality (Sundholm and van Atten 2008, p. 71). If we had known the paper by Reeb and Harthong then, we would surely have taken it into account. 38 Besides the main influence Brouwer, in Reeb one finds quotations or echos from for example Hilbert, Poincaré, Löwenheim, Skolem, and Von Neumann. 39 Brouwer makes the point in Brouwer 1928.

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understood and accepted in the formalistic literature. When the same state of affairs has been reached with respect to the other two, it will mean the end of the controversy concerning the foundations of mathematics. First insight. The differentiation, among the formalistic endeavors, between a construction of the ‘inventory of mathematical formulas’ (formalistic view of mathematics) and an intuitive (contentual)40 theory of the laws of this construction, as well as the recognition of the fact that for the latter theory the intuitionistic mathematics of the set of natural numbers is indispensable. Second insight. The rejection of the thoughtless use of the logical principle of excluded middle, as well as the recognition, first, of the fact that the investigation of the question why the principle mentioned is justified and to what extent it is valid constitutes an essential object of research in the foundations of mathematics, and, second, of the fact that in intuitive (contentual) mathematics this principle is valid only for finite systems. Third insight. The identification of the principle of excluded middle with the principle of the solvability of every mathematical problem. Fourth insight. The recognition of the fact that the (contentual) justification of formalistic mathematics by means of the proof of its consistency contains a vicious circle, since this justification rests upon the (contentual) correctness of the proposition that from the consistency of a proposition the correctness of the proposition follows, that is, upon the (contentual) correctness of the principle of excluded middle. (Brouwer 1928, p. 375, trl. van Heijenoort 1967, pp. 490–491)

The distinction in the First insight Brouwer had made first in Brouwer 1908, in which he had shown that ¬¬(P ∨ ¬ P), and in 1924 this led him to comment on Hilbert’s aim of a consistency proof for formalised classical mathematics that ‘We need by no means despair of reaching this goal’ (Brouwer 1924, p. 3, trl. van Heijenoort 1967, p. 336). In ‘Intuitionistic reflections on formalism’ he added a proof that finite conjunctions of instances of PEM are also consistent, and in his first Vienna lecture he voiced the expectation that ‘An appropriate mechanization of the language of this intuitionistically non-contradictory mathematics should therefore deliver precisely what the formalist school has set as its goal’ (Brouwer 1929, p. 164, trl. mine). Against this background, the question that led Reeb to embrace IST is (as I formulate it) the following. If for example the classical logic in the formal theory is taken to be an idealisation of the constructive logic of mathematical reality, is there, similarly, an aspect of constructive reality that, when idealised, would lead to the notion of a standard number in the formal theory? What is asked for is an intuitive motivation, for introducing the predicate ‘standard’ in the idealised theory IST, not a formal proof (of the existence of formal nonstandard models). Reeb answers that the standard numbers in a formal theory may be seen as an idealisation of what he calls the naïve whole numbers in reality (‘les entiers naïfs’). They are the numbers ‘that exist independently from the theory one uses to describe them’ (Reeb and Harthong 1989, p. 63), and are obtained by just putting units together, Reeb’s (necessarily informal) definition is (Reeb 1979, p. 277, p. 286n3; Reeb 1989, p. 152):

40 [The

role of Brouwer’s ‘contentual mathematics’ corresponds to that of Reeb’s ‘mathematical reality’; but the former is richer than the latter.]

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1. 0 is naïve; 2. if n is naïve, so is n + 1; 3. No other n is naïve. Although the property of being naïve is not in any sense vague, so there is no threat of Wang’s Paradox,41 Reeb resists the argument by induction that the naïve numbers form a set in the classical sense, as we will see in a moment. It should also be noted that Reeb’s position is dissociated from finitism: a naïve number may be constructed in an algorithm or program of any complexity (Reeb and Harthong 1989, section 15). It is implied, then, that there is also a naïve notion of algorithm. This coincides with the idea that formal recursion theory, if to be understood as a theory of computability, presupposes such a pre-theoretical notion.42 At this point Reeb invokes Brouwer’s weak counterexamples to classical theorems. Assume the consistency of the formal theory and suppose that there is a predicate A such that • the formal theory proves ∃x ¬A(x), or it can be shown that ∃x ¬A(x) is independent and we are willing to add it as an axiom; • A and its evaluation at a naïve n can be understood in naïve terms, • but we do not have yet a naïve construction for such a counterexample. In that case ZFC proves the formal existence of a natural number for which we do not have a construction in reality yet. It is a formal number in the formal set N to which corresponds no naïve number in reality (in any case not yet), and which is greater than any naïve number interpreted in the theory. Hence Reeb’s slogan Q: ‘The naïve whole numbers do not fill N’ (‘Les naifs ne remplissent pas N’). At first, Reeb called Q an observation (‘constat’), later a slogan (‘slogan’). As Salanskis points out, the latter is much more appropriate, as seeing things the way Reeb does is not a matter of direct perception but requires accepting a certain philosophical view (Salanskis 1994, p. 29). Furthermore, it requires the conviction that there always will be such predicates A that can be understood and evaluated in naïvely. Reeb supposed that Fermat’s conjecture provided such a predicate A, writing, for example, in 1989: Consider the unique object a in N defined by the formula (not well formalised, but the reader will know how to write a perfect formula): If the statement known as Fermat’s Great Theorem is true, a = 0, otherwise a = x + y + z + n, where n > 3, xn +yn = zn , x, y, z > 1, and x, y, z and n chosen such that a is as small as possible.

41 Wang’s Paradox is: 0 is a small number; if n is a small number, so is n + 1; therefore, all numbers are small. This has generated quite some discussion; the classical papers are Dummett 1975 and Wright 1975. 42 Briefly, the point is that a recursive function is defined by a set of equations, and if the function is to be considered as computable, there must be an effective method to determine that set; but now to understand ‘effective’ as’ recursive’ would be circular. A detailed presentation is given in Heyting 1958, pp. 340–342. For discussion and further references, see Coquand 2014 and Sundholm 2014.

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At the moment I am writing this, it is not possible to convince oneself that a is naïve. (Reeb 1989, p. 152, trl. mine)

It follows from Wiles’ proof, published in 1995, that a = 0; one may think of Goldbach’s conjecture instead. At times, Reeb did not invoke potential examples and limited his motivation to pointing out that the existence of such predicates A cannot be excluded.43 Of course, should a naïve number n be found such that ¬A(n), one looks for another predicate of that type.44 The step to IST is made by idealising the distinction between the (constructive) naïve numbers and the surplus of formal numbers in the formal set N, whose existence is expressed in Q, to that between the classical standard ones and the nonstandard ones.45 Given the differences between the notions of constructive existence in reality and formal existence in a classical theory, it is only to be expected that the idealisation will be one by analogy: It is now a matter of drawing up a suitable list of simple and formal properties verified (or simply suggested) by the naïve numbers, and to consider the formal theory consisting of the statements on this list together with the axioms of classical mathematics. The theory known by the abbreviation IST developed by E. Nelson realises this program efficiently. But like every formal theory, it does not escape observation Q (i.e., ‘The naïve numbers do not fill the standard whole numbers of IST’). (Reeb 1979, p. 287, trl. mine)

To illustrate that last remark: the number a defined in terms of Fermat’s Last Theorem exists classically and is unique, and hence, as was clear also before Wiles’ demonstration, a standard object in IST (Theorem 7, page 6). In ‘La mathématique non standard vieille de soixante ans?’, Reeb presents a beginning of such a list of properties46 of the naïve numbers, which, somewhat abbreviated, runs as follows (Reeb 1979, pp. 278–279): Let ω be a fixed, non-naïve number. • Property 1. If a is naïve, then ω > a. • Property 2 . . . , ω − a, . . . , ω − 2, ω − 1, ω, ω + 1, ω + 2, . . . , ω + a, . . . (where a is naïve) are non-naïve elements of N. Likewise, the following √ elements of N are non-naïve (where a naïve): ω2 , ω3 , . . . , ωa , . . . , ωa , a ω where [x]

43 Personal

communication from Jean-Michel Salanskis, who was a member of Reeb’s group. Reeb, a constructive proof can exist without having been found: ‘ou bien il y a une démonstration constructive, déjà connue ou non . . . ’ (Reeb and Harthong 1989, section 16). For Brouwer, on the contrary, the only sense in which a proof can be said to exist is that it has been constructed. However, for the matter at hand this makes no difference. 45 This idealisation need not lead to IST; it was the theory Reeb knew and liked, but closely related axiomatic nonstandard theories have been developed in the meantime (Kanovei and Reeken 2004). Just as in the natural sciences, different theories of the same phenomena in reality may be developed, and have different theoretical virtues. 46 As Salanskis emphasises (Salanskis 1999, p. 140), Reeb writes ‘properties’, not ‘theorems’, so as to distinguish assertions about reality from provable formulas in a formal system. 44 For

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• • •

•

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stands as usual for the whole part of x; pω , prime number and pω > ω (such pω certainly exist). The number ω! is not naïve and has every naïve whole number as divisor. Property 3. If a > 1 is naïve, then aω > ωa . Property 4. There exists no set X such that ‘x ∈ X’ is equivalent to ‘x is a naïve whole number’. Property 5. Let X ⊂ N be a set such that n ∈ X for every naïve n [respectively, ω ∈ X for every non-naïve ω]. Then there exists a non-naïve α such that α ∈ X [respectively, a naïve a such that a ∈ X]. Property 6. If X is a set of which every element is a naïve whole number, then X is finite.

Reeb’s argument for Property 4 is that, if the naïve whole numbers formed a set, then by induction this set would be identical to N, and that would contradict Slogan Q. If in this list one replaces ‘(non-)naïve number’ with ‘(non-)standard number’, and construes these statements not contentually but formally, one gets theorems of IST. Reeb also notices (current) limitations of this motivation (Reeb 1979, p. 287): 1. He should like to have a notion of ‘naïve object’ that extends beyond natural numbers. In the formal counterpart, IST, the predicate ‘standard’ can be meaningfully applied to any set, hence to any object in its universe (and thereby yield either a truth or a falsehood); but Reeb does not have a correspondingly general notion of construction. (Note that Reeb was aware of, but did not embrace, Brouwer’s wider notion of constructivity.) 2. He has not been able to find a justification for the claim that whenever all naïve whole numbers have a certain internal property, all numbers in N have it. (An analogue to Transfer.) 3. Likewise, while for a given naïve function such as ex it is easily shown that an infinitesimal increase in the argument leads to an infinitesimal increase in the value, it remains to be shown that this is equivalent to (-δ-)continuity of the function, which would require an analogue to Standardization. But Transfer and Standardization are, in their full generality, by and large nonconstructive; see the fine-grained analysis in Sanders 2017. Although Sanders’ analysis is concerned with relations of the formal standard objects with the nonstandard ones, and not with Reeb’s naïve objects, his results strongly suggest that such justifications as Reeb hoped to find will not be forthcoming. That is far from saying that his attempt to find a natural way into IST fails; but it does mean that the idealisations involved in moving from the distinction between naïve and non-naïve numbers to IST are stronger than perhaps was expected.47 47 This

last consideration is one among several that leads to the question of constructive analogues to IST (which was not a particular concern to Reeb, to whom, on the contrary, the idea of using a classical formal theory was attractive). For this, I refer to the papers mentioned in footnote 2.

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To make the transition from the naïve numbers in reality to the standard numbers in IST more explicit, Reeb introduced a middle term, ‘Naïve’ (Reeb 1981, pp. 453– 454; Reeb 1989, pp. 153–154). This term applies to all objects whose existence in ZFC is established by proving a formula of the form ∃!xA(x) This includes all naïve whole numbers, but much more: N, Q, R, P(N), exp, sin, π , . . . The key principle then would be: ‘There exists a finite set F that contains every Naïve object’. One could even decide to do nonstandard analysis using this informal concept. Reeb remarks that, on the one hand, the advantage of doing this is that it allows one to reconstruct non-standard analysis in a way of which the consistency and conservativeness are evident; on the other hand, as an informal concept Naïve may be more difficult to work with than Nelson’s formal theory (Reeb 1981, p. 154). Moreover, it would require quite a sophisticated argument to justify the key principle (Reeb 1989, p. 154). His conclusion is that ‘in this sense, the formalized theory IST is superior to our consideration of Naïve objects, whose interest is limited to the didactical or heuristic sphere’ (Reeb 1981, p. 154). It seems to me that, as a motivation for the introduction of a distinction in an idealised formal theory, a good heuristic will fit the bill. As is clear from the list of four ‘insights’, the idea that one may simultaneously accept (not just finitary but even) intuitionistic mathematics as true and formalized classical mathematics as consistent was shared by Brouwer, who at the time was even optimistic about the prospects of a formal consistency proof. Yet, Brouwer would not have called non-standard analysis in the form of IST an idealised formal theory of the mathematical reality that is the intuitive continuum. After all, IST is a syntactical enrichment of ZFC and in particular of the theory of the classical real numbers, but is not an ontological enrichment of the latter. The objects of IST are the classical real numbers. However, those can not be construed as idealisations of intuitionistic choice sequences. This is clearest from the mathematical contrast provided by Brouwer’s strong counterxamples, which show that the intuitive continuum of mathematical reality, analysed in terms of choice sequences, has properties that in classical analysis with its discrete continuum are contradictory. (As we saw, IST is, on the contrary, conservative over ZFC.) Illustrative are the following: Theorem 29 ¬ ∀ x ∈ R(x ∈ Q ∨ x ∈ Q) (Brouwer 1927). Theorem 30 ¬ ∀ x ∈ R(¬ ¬ x > 0 → x > 0) (Brouwer 1949a). Theorem 31 ¬ ∀ x ∈ R(x = 0 → x < 0 ∨ x > 0) (Brouwer 1949b). For Reeb these theorems are not relevant, as these they depend on intuitionistic considerations that go beyond the finitary mathematics he accepts as mathematical reality.

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5.9 Weyl and Infinitesimal Analysis Overall, one suspects that to Brouwer, the glass that Reeb offers would have seemed to be half empty, what with its essential involvement of a formalism and its limitation of mathematical reality to the finitary. For Weyl this would probably have been different. As is well known, Weyl acknowledged the epistemological superiority of intuitionism in pure mathematics: With Brouwer, mathematics gains the highest intuitive clarity; his doctrine is the culmination of idealism in mathematics.

However, Weyl continues: But with pain the mathematician sees the greater part of his high-rising theories dissolve into the fog. (Weyl 1925, p. 24)

Weyl had come to convince himself that it is necessary to abandon the intuitionistic program because he took it to be a fact that intuitionistic mathematics is not able to found the mathematics required in physics, whereas ‘mathematics should put itself to the service of the natural sciences’ (Weyl 1926).48 This pushed Weyl towards a formalist foundation of classical mathematics.49 In various physical contexts it is, conceptually, natural to apply nonstandard analysis, for example when phenomena are involved at greatly different scales, or where the difference between the observable and the unobservable plays a role.50 In a more foundational spirit, Robert notes that, although tangent vectors can be said to represent infinitesimal displacements at a point in a differentiable manifold, it would be important to have further analyses of differentiability and of continuity, neither of which can be based on the concept of a differential; a theory of infinitesimals would be one, and moreover supply an algebra of differentials (Robert 1988, p.xiv). That last remark brings us to Weyl’s infinitesimal geometry in his theory of spacetime, and also to the quotation from Weyl with which this paper began. For, as Laugwitz aptly observed, And even if Hermann Weyl declared the infinitesimals to have been eliminated, his book Space-Time-Matter, widely available in several editions since 1919, is a perfect example of infinitesimal mathematics in action. (Laugwitz 1986, p. 241, trl. mine)

48 For

a detailed account of that episode, see for example Mancosu and Ryckman 2002, section 6.2.1 and Tieszen 2000, section 7. A recent philosophical discussion on constructive mathematics in physics is Ardourel 2012. 49 But, as we have seen (the four ‘insights’, p. 21), Brouwerian intuitionism does not exclude a formalist foundation of classical mathematics; it includes it as a proper part. However, it is not the part of intuitionistic mathematics that is concerned with the development of contentual mathematics; and the contentual mathematics that Brouwer sought to develop is far richer than the minimum required to get the formalist foundation going. 50 Given the properties of human vision, even at an everyday scale infinitesimal analysis can be useful, as shown by the analysis of the moiré effect in Harthong 1981. Further applications are presented in, e.g., Cutland 1988; Arkeryd et al. 1997; and Lobry and Sari 2008. A recent view from a philosopher of science is Wenmackers 2016.

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Weyl’s infinitesimal geometry is not, in fact, constructive. The fundamental notions are introduced axiomatically, instead of being constructed out of basic intuition, and the proof of uniqueness of Pythagorean metric of 1922 is, in its dependence on the excluded middle, classical.51 Brouwer, no less of an idealist than Weyl, always was a conventionalist about the structure of physical space: ‘a question of convenience, of taste, or of custom’, he wrote in 1909, in a paper to which in a reprint of 1919 he added a note stating that the general theory of relativity ‘would not affect the conclusions on the theory of knowledge’ he had reached (Brouwer 1909, p. 14, trl. Brouwer 1975, p. 116; Brouwer 1919, p.vi, trl. Brouwer 1975, p. 120). Weyl was well aware of the discrepancy between his constructivist philosophy of pure mathematics and his classical practice in mathematical physics. Did Weyl ever hope to give his infinitesimal geometry a constructive foundation later – Brouwer had taken that attitude towards his own theorems in classical topology –,52 and can a notion of subjectivity be motivated that supports, as Weyl’s philosophical foundation of physics requires, the idea of a subject located in a point and whose intuitive space is of infinitesimal size53 without, at the same time, idealising beyond a notion of subject appropriate for constructive mathematics? To investigate these questions would go beyond the scope of the present paper. But if there is no construction allowing to treat infinitesimals as individual objects, then to Weyl, Reeb’s approach, what with its combination of a classical formalism for nonstandard analysis and a nevertheless intuitionistic epistemology, might have made the glass seem at least half full.54 Acknowledgements Earlier versions of this paper were presented at the conference ‘Weyl and the Problem of Space: From Science to Philosophy’, University of Konstanz, May 2015, and at the workshop ‘Workshop on the Continuum in the Foundations of Mathematics and Physics’, University of Amsterdam, April 2017. I am grateful to the organisers for their invitations, and to the audiences for their questions and comments. I have also benefited from exchanges with Julien Bernard (who also shared his instructive, unpublished manuscript ‘New insights on Weyl’s Problem of Space, from the correspondence with Becker’ with me), Dirk van Dalen, Bruno Dinis, Mikhail Katz, Carlos Lobo, David Rabouin, Jean-Michel Salanskis, Sam Sanders, Wim Veldman, and Freek Wiedijk. Gödel’s shorthand notes on the non-Archimedean number system in Brouwer’s

51 On

Weyl’s non-constructive mathematics in physics, see Weyl 1922, p. 146; Weyl 1988, p. 7; Scholz 2001, pp. 95–97; and Eckes 2011, pp. 277, 608–610, 777–778. 52 In a retrospective remark of 1920 Brouwer wrote that, when he had just begun to develop intuitionism, ‘in my contemporary philosophy-free mathematical papers I have frequently also used the old [i.e., non-intuitionistic] methods, trying however to derive only such results as could be hoped to find, after the completion of a systematic construction of intuitionistic set theory, a place in the new system and claim a value, perhaps in modified form.’ (Brouwer 1920, p. 204, trl. mine) 53 See in particular Bernard 2013, pp. 246–248, and Bernard’s instructive, unpublished manuscript Bernard. 54 Palmgren indicates that Nelson’s nonstandard analysis, which corresponds to part but not all of Robinson’s, may well lend itself to constructivisation (Palmgren 1998, p. 234). Weyl, on the other hand, would presumably have been interested in the classical formalism.

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dissertation, mentioned in footnote 1.5, were kindly transcribed by Eva-Maria Engelen. These notes are owned by the Institute for Advanced Study and kept in the Department of Rare Books and Special Collections at the Firestone Library, Princeton University.55

References Ardourel, V. 2012. La physique dans la recherche en mathématiques constructives. Philosophia Scientiae 16 (1): 183–208. Arkeryd, L.O., N.J. Cutland, and C.W. Henson, eds. 1997. Nonstandard analysis. Theory and applications. Dordrecht: Springer. Baron, M. 1969. The origins of infinitesimal calculus. Oxford: Pergamon Press. Barreau, H., and J. Harthong, eds. 1989. La Mathématique non standard. Paris: Éditions du CNRS. Bernard, J. 2013. L’idéalisme dans l’infinitésimal. Weyl et l’espace à l’époque de la relativité. Presses universitaires de Paris Nanterre, Nanterre. Available online at http:// books.openedition.org/pupo/3917. Brouwer, L.E.J. Notebooks, 1904–1907. Brouwer Papers. Haarlem: Noord-Hollands Archief. Available at http://www.cs.ru.nl/F.Wiedijk/brouwer/index.html. ———. 1907. Over de grondslagen der wiskunde. PhD thesis, Universiteit van Amsterdam. ———. 1908. De onbetrouwbaarheid der logische principes. Tijdschrift voor Wijsbegeerte 2: 152– 158. ———. 1909. Het wezen der meetkunde. Amsterdam: Clausen. ———. 1917. Addenda en corrigenda over de grondslagen der wiskunde. Nieuw Archief voor Wiskunde 12: 439–445. ———. 1919. Wiskunde, waarheid, werkelijkheid. Groningen: Noordhoff. ———. 1920. Intuitionistische Mengenlehre. Jahresbericht der deutschen MathematikerVereinigung 28: 203–208. ———. 1924. Über die Bedeutung des Satzes vom ausgeschlossenen Dritten in der Mathematik, insbesondere in der Funktionentheorie. Journal für die reine und angewandte Mathematik 154: 1–7. 1923B2 in Brouwer (1975). ———. 1927. Über Definitionsbereiche von Funktionen. Mathematische Annalen 97: 60–75. ———. 1928. Intuitionistische Betrachtungen über den Formalismus. KNAW Proceedings 31: 374–379. ———. 1929. Mathematik, Wissenschaft und Sprache. Monatshefte für Mathematik und Physik 36: 153–164. ———. 1930. Die Struktur des Kontinuums. Komitee zur Veranstaltung von Gastvorträgen ausländischer Gelehrter der exakten Wissenschaften, Wien. ———. 1948. Essentieel negatieve eigenschappen. Indagationes Mathematicae 10: 322–323. ———. 1949a. De non-aequivalentie van de constructieve en de negatieve orderelatie in het continuum. Indagationes Mathematicae 11: 37–39. ———. 1949b. Contradictoriteit der elementaire meetkunde. KNAW Proceedings 52: 315–316. ———. 1949c. Consciousness, philosophy and mathematics. In Proceedings of the 10th international congress of philosophy, Amsterdam 1948, ed. E. Beth, H. Pos, and J. Hollak, vol. 2, 1235–1249. Amsterdam: North-Holland. ———. 1951. On order in the continuum, and the relation of truth to non-contradictority. KNAW Proceedings 54: 357–358. ———. 1954. Points and spaces. Canadian Journal of Mathematics 6: 1–17.

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Laugwitz, D. 1986. Zahlen und Kontinuum. Eine Einführung in die Infinitesimalmathematik. Mannheim: BI Wissenschaftsverlag. Leibniz, G.W. 1859. In Leibnizens mathematische Schriften, ed. C. Gerhardt, vol. 4. Halle: Schmidt. Lobry, C. 1989. Et pourtant . . . ils ne remplissent pas N. Lyon: Aléas. Lobry, C., and T. Sari. 2008. Non-standard analysis and representation of reality. International Journal of Control 81 (3): 517–534. Mancosu, P. 1999. Philosophy of mathematics and mathematical practice in the seventeenth century. Oxford: Oxford University Press. Mancosu, P., and T. Ryckman. 2002. Mathematics and phenomenology: The correspondence between O. Becker and H. Weyl. Philosophia Mathematica, New Series 10: 130–202. Myhill, J. 1966. Notes towards an axiomatization of intuitionistic analysis. Logique et Analyse 9: 280–297. Nelson, E. 1977. Internal set theory: A new approach to nonstandard analysis. Bulletin American Mathematical Society 83: 1165–1198. ———. 1986. Predicative arithmetic. Princeton: Princeton University Press. Available at https:// web.math.princeton.edu/~nelson/books/pa.pdf. ———. 1988. The syntax of nonstandard analysis. Annals of Pure and Applied Logic 38 (2): 123– 134. ———. 1996. Ramified recursion and intuitionism. Available at https://web.math.princeton.edu/ ~nelson/papers/ramrec.pdf. The year 1996 is that in the date of the TeX file on the same server; the original talk was presented to the Colloque Trajectorien, Strasbourg/Obernai, June 12–16, 1995. ———. 2002. Internal set theory. First chapter of an unfinished book on nonstandard analysis, available at https://web.math.princeton.edu/~nelson/books/1.pdf. The year 2002 is that of the pdf file on the server. Palmgren, E. 1993. A note on mathematics of infinity. The Journal of Symbolic Logic 58 (4): 1195–1200. ———. 1995. A constructive approach to nonstandard analysis. Annals of Pure and Applied Logic 73 (3): 297–325. ———. 1998. Developments in constructive nonstandard analysis. Bulletin of Symbolic Logic 4 (3): 233–272. Reeb, G. 1979. La mathématique non standard vieille de soixante ans ?. References are to the reprint in Appendix A to Salanskis 1999. ———. 1981. La mathématique non standard vieille de soixante ans ? Cahiers de Topologie et Géométrie Différentielle Catégoriques 22 (2): 149–154. ———. 1989. 0, 1, 2, etc . . . ne remplissent pas (du tout) N, 1989. Included as chapter 9 in Analyse non standard, ed. Diener, F., and G. Reeb. Paris: Hermann. Reeb, G., and J. Harthong. 1989. Intuitionnisme 84. In La Mathématique non standard, ed. Barreau, H., and J. Harthong, 213–252. Reprinted in L’Ouvert (1994, pp. 42–77). Robert, A. 1988. Nonstandard analysis. New York: Wiley. Robinson, A. 1965. Formalism 64. In Logic, methodology, and philosophy of science, ed. Y. BarHillel, 228–246. Amsterdam: North Holland. ———. 1966. Non-standard analysis. Amsterdam: North-Holland. Salanskis, J.-M. 1994. Un Maître. In Numéro spécial Georges Reeb, ed. L’Ouvert, 25–32. Institut de recherche sur l’enseignement des mathématiques (IREM) de Strasbourg, Strasbourg. ———. 1999. Le Constructivisme non standard. Villeneuve d’Ascq: Presses Universitaires du Septentrion. Sanders, S. 2017. Nonstandard analysis and constructivism!. https://arxiv.org/abs/1704.00281. ———. 2018. To be or not to be constructive, that is not the question. Indagationes Mathematicae 29 (1): 313–381. Schmieden, C., and D. Laugwitz. 1958. Eine Erweiterung der Infinitesimalrechnung. Mathematische Zeitschrift 69: 1–39.

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Scholz, E. 2001. Weyls Infinitesimalgeometrie (1917–1925). In Hermann Weyl’s Raum-ZeitMaterie and a general introduction to his scientific work, ed. E. Scholz, 48–104. Basel: Birkhäuser. Schubring, G. 2005. Conflicts between generalization, rigor, and intuition. Number concepts underlying the development of analysis in 17–19th century France and Germany. New York: Springer. Skolem, T. 1929. Über die Grundlagendiskussionen in der Mathematik. In Den syvende skandinaviske matematikerkongress i Oslo 19–22 August 1929. Oslo: Broegger. ———. 1934. Über die Nicht-charakterisierbarkeit der Zahlenreihe mittels endlich oder abzählbar unendlich vieler Aussagen mit ausschliesslich Zahlenvariablen. Fundamenta Mathematicae 23: 150–161. Sundholm, G. 2014. Constructive recursive functions, Church’s thesis, and Brouwer’s theory of the creating subject: afterthoughts on a Parisian Joint Session. In Constructivity and calculability in historical and philosophical perspective, ed. Dubucs, J., and M. Bourdeau. Dordrecht: Springer. 1–35. Sundholm, G., and M. van Atten. 2008. The proper interpretation of intuitionistic logic. On Brouwer’s demonstration of the Bar Theorem. In One hundred years of intuitionism (1907– 2007). The Cerisy conference, ed. M. van Atten, P. Boldini, M. Bourdeau, and G. Heinzmann, 60–77. Basel: Birkhäuser. Tieszen, R. 2000. The philosophical background of Weyl’s mathematical constructivism. Philosophia Mathematica 3: 274–301. Troelstra, A. 1982. On the origin and development of Brouwer’s concept of choice sequence. In The L. E. J. Brouwer centenary symposium, ed. A. Troelstra and D. van Dalen, 465–486. Amsterdam: North-Holland. Troelstra, A., and D. van Dalen. 1988. Constructivism in mathematics. Amsterdam: North-Holland. van Atten, M. 2004. On Brouwer. Belmont: Wadsworth. ———. 2015. Essays on Gödel’s reception of Leibniz, Husserl, and Brouwer. Cham: Springer. ———. 2018. The creating subject, the Brouwer-Kripke Schema, and infinite proofs. Indagationes Mathematicae 29: 1565–1636. van Atten, M., and G. Sundholm. 2017. L. E. J. Brouwer’s “Unreliability of the logical principles”. A new translation, with an introduction. History and Philosophy of Logic 38 (1): 24–47. van Atten, M., D. van Dalen, and R. Tieszen. 2002. Brouwer and Weyl: The phenomenology and mathematics of the intuitive continuum. Philosophia Mathematica 10 (3): 203–226. van Dalen, D. 1988. Infinitesimals and the continuity of all functions. Nieuw Archief voor Wiskunde 6 (3): 191–202. ———. 1999. Mystic, geometer, and intuitionist. The life of L. E. J. Brouwer. Volume 1: The dawning revolution. Oxford: Oxford University Press. ———. 2001. L.E.J. Brouwer en de grondslagen van de wiskunde. Utrecht: Epsilon. ———. 2005. Mystic, geometer, and intuitionist. The life of L. E. J. Brouwer. Volume 2: Hope and disillusion. Oxford: Clarendon Press. van den Berg, B., and S. Sanders. 2017. Reverse mathematics and parameter-free transfer. https:// arxiv.org/abs/1409.6881. van den Berg, B., E. Briseid, and P. Safarik. 2012. A functional interpretation for nonstandard arithmetic. Annals of Pure and Applied Logic 163 (12): 1962–1994. van Heijenoort, J., ed. 1967. From Frege to Gödel: A sourcebook in mathematical logic, 1879– 1931. Cambridge, MA: Harvard University Press. Veronese, G. 1891. Fondamenti di Geometria a più dimensioni e a più specie di unità rettilinee, esposti in forma elementare. Padova: Tipografia del Seminario. Vesley, R. 1981. An intuitionistic infinitesimal calculus. In Constructive Mathematics (Lecture Notes in Mathematics, 873), ed. F. Richman, 208–212. Berlin: Springer Wavre, R. 1926. Logique formelle et logique empirique. Revue de Métaphysique et de Morale 33: 65–75.

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Wenmackers, S. 2016. Children of the cosmos. Presenting a toy model of science with a supporting cast of infinitesimals. In Trick or truth?, ed. A. Aguirre, B. Foster, and Z. Merali, 5–20. Dordrecht: Springer. Weyl, H. 1922. Die Einzigartigkeit der Pythagoreischen Maßbestimmung. Mathematische Zeitschrift 12: 114–146. ———. 1925. Die heutige Erkenntnislage in der Mathematik. Symposion 1: 1–32. ———. 1926. Philosophie der Mathematik und Naturwissenschaft. München: Leibniz Verlag. Weyl 1949 (Philosophy of mathematics and natural science. Princeton: Princeton University Press.) is an expanded English version. ———. 1949. Philosophy of mathematics and natural science. Princeton: Princeton University Press. ———. 1988. In Riemanns geometrische Ideen, ihre Auswirkung und ihre Verknüpfung mit der Gruppentheorie, ed. K. Chandrasekharan. Berlin: Springer. Wright, C. 1975. On the coherence of vague predicates. Synthese 30: 325–365.

Chapter 6

Entre phénoménologie et intuitionnisme: la définition du continu Dominique Pradelle

Notre objet est ici de comprendre, à travers l’exemple de Hermann Weyl et de son travail en philosophie des mathématiques, ce qui constitue la spécificité d’une philosophie ou d’une épistémologie des mathématiques qui soit de style proprement phénoménologique, et de ressaisir les traits caractéristiques par lesquels elle se distingue tant d’une épistémologie historique (Brunschvicg, Bachelard, Koyré) qui tente de déchiffrer dans l’histoire les étapes de la construction de la raison mathématique ou les moments de mutation de la rationalité mathématicienne, que d’une philosophie des mathématiques de type logico-syntaxique (Hilbert, Frege, Russell, Carnap) qui tente d’éclairer la teneur eidétique des objets mathématiques grâce à l’examen des systèmes de propositions qui portent sur de tels objets. En quoi, donc, le travail de Weyl en philosophie des mathématiques relève-t-il d’une position proprement phénoménologique ? Quels sont, dans la démarche de Weyl, les traits spécifiquement phénoménologiques qui, par généralisation, permettent de faire le partage rigoureux entre une philosophie des mathématiques qui est bien phénoménologique et une autre qui ne l’est pas ? Si nous posons une telle question, c’est qu’elle est suscitée par la considération de l’œuvre de Husserl lui-même. En effet, d’un côté, Husserl formule dans une terminologie qui lui est propre l’idéal de théorie axiomatisée, c’est-à-dire d’un système de propositions dont les notions primitives sont dépourvues de teneur réale et implicitement définies par les formes d’axiome, dont la validité est réglée par les idées directrices de cohérence syntaxique et de complétude, et dont le domaine d’objets associé se réduit à une forme de domaine satisfaisant à l’ensemble des propositions du système ; il est donc proche des formalistes et de leur recherche des critères métamathématiques de validité des théories envisagées

D. Pradelle () Department of Philosophy, Sorbonne University, Faculté des Lettres, Paris, France e-mail: [email protected] © Springer Nature Switzerland AG 2019 J. Bernard, C. Lobo (eds.), Weyl and the Problem of Space, Studies in History and Philosophy of Science 49, https://doi.org/10.1007/978-3-030-11527-2_6

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comme systèmes formels. De l’autre, cependant, en définissant la méthode de réduction phénoménologique qui met hors circuit toute admission d’être en soi pour régresser aux modalités subjectives par lesquelles tout étant est visé, maintenu et validé en son être, Husserl n’invite-t-il pas, en philosophie des mathématiques, à une mise en question de toute admission de la vérité en soi, qui conduit à poser la vérité comme coextensive à son mode d’atteinte possible, et l’être des objets comme corrélat de procédures de construction subjective explicites et réglées – thèse qui s’apparente bien davantage à la position intuitionniste ? Dès lors, si la philosophie des mathématiques de Weyl peut être dite phénoménologique, à quoi tient, au-delà du rattachement à Husserl et de la revendication expresse, le caractère proprement phénoménologique de sa réflexion ? Un tel caractère tient-il à sa réflexion sur les concepts fondamentaux (Grundbegriffe) des mathématiques et des sciences en général, c’est-à-dire à la tentative de dégager et d’expliciter le fonds ou le fondement eidétique des mathématiques ? Et ce en conformité avec l’assimilation husserlienne des mathématiques à une science de type nomologique, dont toute la teneur se concentre dans un petit groupe de concepts et d’axiomes dont elle dérive ensuite par voie déductive tous les théorèmes ? Ou bien tient-il à son assomption de l’intuitionnisme de Brouwer, c’est-àdire à la mise en question du logicisme (c’est-à-dire la thèse de réductibilité des mathématiques à la logique) et à son souci de scinder de la logique pure ce qui est spécifique aux mathématiques (à savoir, l’intuition proprement mathématique) ? Et à la mise en question de l’applicabilité, aux mathématiques, des principes fondamentaux de la logique (non-contradiction, tiers exclu, double négation), qui va de pair avec l’exigence de construction effective et le refus de la reductio ad absurdum ? Ou bien tient-il encore à l’acceptation du principe fondamental de l’idéalisme transcendantal ou constitutif de Husserl, selon lequel tout objet au sens large (qu’il s’agisse d’un objet sensible de rang inférieur ou d’un objet purement catégorial de degré supérieur) se réduit à une unité de sens intentionnel visé par les actes donateurs de sens de la conscience pure, éventuellement attestée dans des évidences donatrices vécues par cette dernière ? Simplement, en tel cas, quelle est la portée exacte de ce principe de l’idéalisme constitutif husserlien ? Possède-t-il un sens normatif : celui d’une exigence de construction effective réglée par une procédure de choix effectuable, ou encore de l’exigence finitiste d’une procédure limitée à un nombre fini d’étapes ? En d’autres termes, a-t-il le sens d’une stricte corrélation entre l’objectité mathématique et l’indication d’une procédure de construction effectuable de cet objet, et se règle-t-il sur les limites assignées aux activités de toute conscience finie ? Ou bien n’implique-t-il aucun sens normatif pour la pratique mathématicienne, mais seulement une portée descriptive et réflexive au regard des actes et pratiques effectives des mathématiciens – en d’autres termes, requiert-il seulement de dégager réflexivement les actes et procédés de pensée accomplis par les mathématiciens dans leur pratique effective, sans jamais émettre de critique à l’encontre de leur caractère idéalisant ou excessivement formalisant ? À ce titre, est-il indifférent à l’opposition entre finitude humaine et infinité divine, intuitus derivatus et intuitus originarius ? Bref, est-il du ressort de la phénoménologie

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des mathématiques d’édicter ce que peut et ne peut pas faire la conscience mathématicienne – par exemple, qu’elle doit s’en tenir aux objets finis ou à l’indéfini, aux ensembles dénombrables, aux objets constructibles en un nombre fini d’étapes ou selon une procédure réglée, qu’elle doit donc refuser l’infini actuel, les constructions cantoriennes d’échelles de nombres transfinis ou l’axiome du choix ? Ou bien doit-elle simplement prendre acte de l’historicité des techniques mathématiques, de la transformation des procédés et de l’évolution des champs d’objets admissibles ? On retrouve ainsi, dans le champ des mathématiques, le pendant d’une question qui se pose en phénoménologie de l’expérience musicale : mutatis mutandis, est-ce la fonction de la phénoménologie que de décréter, par une analyse de la réceptivité auditive, ce que sont les limites de la perceptibilité musicale, et de déclarer par exemple que la musique sérielle n’est pas conforme aux lois naturelles de la conscience auditive et qu’une série dodécaphonique non effectivement entendue n’existe pas pour l’auditeur comme principe structurant de l’œuvre ? Ou doitelle prendre acte de l’évolution du langage et des techniques musicales, et du fait qu’à rebours de tout postulat d’invariance des structures et potentialités de la conscience finie, les capacités de l’audition musicale subissent un procès de transformation historique qui lui est connexe ? Quel qu’en soit le registre ou le champ d’application, la fonction de la phénoménologie est-elle d’élaborer une eidétique des structures et capacités de la conscience finie, ou bien d’admettre le caractère d’évolutivité historique de telles structures et potentialités, ainsi que leur indifférence à l’opposition entre finitude et infinité, et de tenter de dégager les lois générales qui, au-delà de la diversité des génies et des écoles et de la succession contingente des inventions, régissent le devenir conjoint des champs d’objets et des structures de la conscience ?

6.1 DE HUSSERL À WEYL: LA RÉFÉRENCE AU « SOL ORIGINAIRE DE L’INTUITION LOGICO - MATHÉMATIQUE » Afin de tenter de répondre à ces questions, prenons pour point de départ la correspondance entre Husserl et Weyl, documentée surtout par les lettres adressées à Weyl par le premier. Concentrons-nous sur la lettre qu’il lui adresse le 10 avril 1918, qui suit l’envoi par Weyl de son livre Das Kontinuum. Kritische Untersuchungen über die Grundlagen der Analysis. Husserl y salue l’esprit proprement phénoménologique dans lequel est menée la réflexion de Weyl en philosophie des mathématiques : ce dernier manifeste en effet une véritable compréhension de la « nécessité des modes de considération phénoménologiques dans toutes les questions concernant l’élucidation des concepts fondamentaux » (Notwendigkeit phänomenologischer Betrachtungsweisen in allen Fragen der Klärung der Grundbegriffe), en faisant retour au « sol originaire de l’intuition logico-mathématique

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sur lequel seul s’avèrent possibles une fondation de la mathématique qui remonte effectivement à leur source et une évidence intellectuelle qui pénètre le sens de l’effectuation mathématique ! » (Urboden logisch-mathematischer Intuition, auf dem allein eine wirklich quellenmäßige Begründung der Mathematik und eine Einsicht in den Sinn mathematischer Leistung möglich ist)1 . Qu’est-ce à dire, et que signifie donc l’esprit véritablement phénoménologique que Husserl reconnaît à Weyl ? D’une part, la mathématique doit être ici entendue au singulier, et non au pluriel : ce n’est pas un ensemble disparate de disciplines réunies selon une unité purement nominale, mais une discipline censée posséder une unité réelle, fonction de son domaine d’objets propre et de son type d’activité spécifique ; aussi doit-il être possible de spécifier la nature du domaine d’objets qui appartient à la connaissance mathématique, ainsi que celle des actes de la conscience qui les saisissent et les déterminent. C’est pourquoi il est explicitement fait référence à la question de la fondation de la mathématique (Begründung der Mathematik) : la fonder, c’est régresser vers les sources de la connaissance mathématicienne (quellenmäßige Begründung), en tâchant d’élucider les concepts fondamentaux (Grundbegriffe) des mathématiques, ceux qui désignent des ensembles d’entités élémentaires – ensemble (Mengen-), nombre entier naturel (Anzahl-), ordre (Ordnung-), mesure de grandeur (Größenzahlbegriff )2 : des concepts qui relèvent donc de la théorie des ensembles, de l’arithmétique élémentaire des cardinaux, de l’ordre et de l’arithmétique des ordinaux, ainsi que de l’arithmétique des grandeurs (c’est-àdire la théorie de la mesure, ou des élargissements successifs des ensembles de nombres qui conduisent à l’ensemble des nombres réels). Notable est ici l’absence de tout concept géométrique : aussi bien des figures géométriques élémentaires (point, ligne, plan . . . ) que des espèces d’espaces (espace de courbure nulle, constante, positive et négative, variable) et des transformations de figures (déplacement, translation, rotation, symétries, similitude . . . ). En outre, ces concepts sont baptisés de « concepts fondamentaux authentiquement logiques » (echt logische Grundbegriffe)3 : est-ce à dire que les concepts élémentaires de l’arithmétique sont reconduits à la logique ou, plus largement, à la théorie des ensembles, et que le mode de considération phénoménologique doit être entendu en un sens logiciste, consistant à ramener les concepts de l’arithmétique et de l’Analyse à ceux d’élément, d’appartenance et d’ensemble ? Ou faut-il au contraire entendre que la logique doit être prise en un sens large, qui englobe les concepts arithmétiques ? Pour répondre à cette question, il faut poursuivre la lecture et l’analyse. L’élucidation des concepts fondamentaux doit s’effectuer selon des « modes de considération phénoménologiques » (phänomenologische Betrachtungsweisen), et c’est bien ainsi que la conduit Weyl aux yeux de Husserl. Certes ! Mais au-delà de la référence purement terminologique, quelles sont les modalités précises de ce type de considération phénoménologique ?

1 H USSERL , 2 Ibid. 3 Ibid.

Brief an Weyl, 10. IV. 1918, in Briefwechsel, VII, 287.

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Le premier élément de réponse réside dans la référence au « sol originaire de l’intuition logico-mathématique » (Urboden logisch-mathematischer Intuition), sur lequel est censée s’effectuer la fondation des mathématiques. Prêtons attention à la locution logisch-mathematische Intuition : que désigne-t-elle exactement ? S’agitil d’une intuition à la fois logique et mathématique, c’est-à-dire conjoignant les principes de la pensée logique et les concepts fondamentaux des mathématiques ? Rappelons-nous à cet effet la classification des types d’intuition et de concepts que donne Husserl au § 60 de la Sixième Recherche : les concepts sensibles (rouge en général) sont donnés par une abstraction simple, dont Husserl explicitera ultérieurement la méthode comme étant la variation eidétique ; les concepts catégoriaux mixtes le sont ou par couplage d’une fonction syntaxique et d’une teneur matériale (comme dans être-vu), ou par idéalisation ou passage à la limite (point, droite, vertu) ; enfin, les concepts purement catégoriaux le sont par un acte de formalisation, qui évacue toute teneur réale ou matériale pour ne conserver qu’une pure forme syntaxique (et, ou, nombre cardinal, nombre ordinal)4 . Élucider les concepts fondamentaux de façon phénoménologique, cela ne signifie donc nullement les reconduire à un type d’intuition sensible et inframathématique où ils auraient leur ancrage. Or, cela exclut d’emblée une double tentative de reconduction des concepts arithmétiques à une sphère préalable : d’une part celle, brouwerienne, du concept de nombre cardinal à l’intuition temporelle de la dyade (intuition of two-oneness), c’est-à-dire de la scission temporelle entre le tout juste passé et le maintenant5 ; d’autre part celle, bergsonienne, de ce même concept à la perception spatiale d’une multiplicité coexistante (par exemple, de moutons dans un pré)6 . Est par là récusé tout intuitionnisme extrinsèque7 qui rétrocéderait de la compréhension du sens des concepts mathématiques vers un type d’intuition sensible (temporel ou spatial) précédant la mathématique et lui fournissant la matière pré-mathématique qu’elle aurait à formaliser (multiplicité temporelle ou spatiale) –, et ce, au profit d’un intuitionnisme intrinsèque réglé sur le « principe des principes », qui ordonne le retour à l’intuition originairement donatrice comme unique source de légitimité8 : fonder les concepts arithmétiques, c’est revenir à l’intuition purement catégoriale

4 H USSERL ,

Logische Untersuchungen, VI. Unters., § 60, Hua XIX/1, 713 (trad. fr. H. Élie et alii, Recherches logiques, III, Paris, Puf, 19742 , p. 221). 5 L.E.J. B ROUWER , “Intuitionnisme et formalisme” (Amsterdam, 1912), Bull. Amer. Math. Soc., 20 (1913), p. 85–86, trad. angl. “Intuitionism and formalism” in BROUWER, Collected Works (= CW), vol. 1, Amsterdam-Oxford, North Holland Publishing Company, 1975, p. 127–128 (trad. fr. J. Largeault in Intuitionnisme et théorie de la démonstration, Paris, Vrin, 1992, p. 43–44). Nous vérifions systématiquement, et parfois corrigeons les traductions de J. Largeault. 6 H. B ERGSON , Essai sur les données immédiates de la conscience, Paris, Puf, 1927 (20079 ), p. 57 sqq. 7 Nous empruntons cette dénomination à J. V UILLEMIN qui, dans La philosophie de l’algèbre, (Paris, Puf, 19932 , p. 495), l’applique curieusement à Husserl, et non à Brouwer. 8 H USSERL , Ideen zu einer reinen Phänomenologie, Bd. I, § 24, Hua III/1, 51 (trad. fr. P. Ricœur, Idées directrices pour une phénoménologie pure, Paris, Gallimard, 1950, p. 78–79).

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ou formalisante qui donne les entités arithmétiques comme des objets, sans recourir à l’analogie avec des champs hétérogènes ni à l’importation de structures depuis de tels champs, mais dans une forme d’intuition qui demeure immanente au domaine d’objets thématique. On s’interdira, par exemple, de rapporter la notion de nombre entier à une multiplicité géométrique de points, ou celle de nombre réel à la prétendue intuition du continu spatial, s’imposant au contraire de demeurer sur le plan strictement arithmétique. C’est là une exigence tout à fait similaire à celle que formule Dedekind, lorsqu’il tâche de ressaisir l’essence de la continuité arithmétique en pensant l’essence de la continuité en général, et non à partir de la familiarité perceptive de la continuité de l’espace9 . Allons plus loin : ce « sol originaire de l’intuition logico-mathématique » vat-il jusqu’à désigner une intuition purement logique des concepts fondamentaux de la mathématique, c’est-à-dire une réduction logiciste (ou ensembliste) des concepts élémentaires de l’arithmétique – en particulier, la fondation ensembliste de la notion de nombre cardinal sur celle d’équinuméricité (Gleichzahligkeit) des ensembles ? Si l’on ne saurait attribuer d’emblée à Husserl un tel projet de réduction logiciste de la mathématique, il n’est toutefois guère exclu qu’il soit impliqué par ses conceptions relatives à l’arithmétique. Il reconnaît en effet, dans un texte de 1891 intitulé par l’éditeur « Sur le concept d’opération », que « toutes les déterminations arithmétiques de nombre reposent en dernière instance sur certains types d’activité qui peuvent s’exercer sur des ensembles en général, donc sur des ensembles d’unités, des nombres »10 . Pourquoi cela ? Parce que, dans la formation d’un ensemble E à partir d’une multiplicité d’éléments a, b, c, etc. par réunion de ces éléments en un objet unitaire {a, b, c . . . }, « il faut faire abstraction de toute détermination du contenu de cet ensemble et considérer chacun [de ses éléments] uniquement comme un “quelque chose”, plus précisément comme un “quelque chose” qui se distingue des autres éléments d’une manière quelconque » : réunir des objets donnés en un ensemble, c’est les prendre ensemble et les relier par une pure liaison collective indifférente à leur contenu ; ainsi les opérations fondamentales sont-elles ici celles de la liaison collective, de l’adjonction et de la soustraction d’éléments11 . Partant, le concept de nombre cardinal se réduit à un cas particulier de la notion d’ensemble, un nombre cardinal n’étant en effet qu’un ensemble d’unités 9 R.

DEDEKIND, Stetigkeit und irrazionale Zahlen (1872), § 3, in Gesammelte mathematische Schriften (= GMS), Band III, Braunschweig, Vieweg, 1932, p. 322 (trad. fr. H. Benis Sinaceur, Continuité et nombres irrationnels in DEDEKIND, La création des nombres, Paris, Vrin, 2008, p. 71–72), et la lumineuse Note introductive de la traductrice, p. 36 sqq. 10 H USSERL , ,, “, Hua XII, V. Abhandl., 409 (trad. fr. J. English, « Sur le concept d’opération » in HUSSERL, Articles sur la logique, Paris, Puf, 1975, p. 476). 11 Loc. cit., Hua XII, 385–386 (trad. fr., 454–455). Une telle détermination des ensembles est évidemment insuffisante : seule la théorie abstraite des ensembles thématise ces derniers dans l’abstrait, en laissant indéterminées les propriétés caractéristiques des éléments qui appartiennent à tel ou tel ensemble ; dans la pratique mathématique en revanche, la considération de ces propriétés est primordiale, puisque ces dernières seules définissent un ensemble comme tel. Husserl en était cependant conscient, puisqu’il écrivait dans une note au texte cité : « Il est bien plus utile d’introduire avec Bolzano, au lieu de quelque chose, “quelque chose du genre A”, mais de

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bien distinctes et réunies par un acte de liaison collective : « à tout [ensemble] quelconque E = {a, b, c . . . } appartient donc un [ensemble] E = {1, 1, 1 . . . } qui en dérive si, à chaque élément a, b, c . . . , l’on substitue “quelque chose” ou “un” » [ . . . ]. Dans la formation E = {1, 1, 1 . . . }, c’est-à-dire “quelque chose et quelque chose et quelque chose, etc.”, chaque “quelque chose” désigne un “quelque chose” distinct de tout autre “quelque chose” »12 . Bref, ici s’applique l’exigence frégéenne selon laquelle le nombre un doit être identique à lui-même en toutes ses occurrences, tandis que les unités d’un nombre entier doivent être discernables, afin de ne pas fusionner en une seule unité13 : puisqu’elles sont dérivées de la notion formelle de quelque chose, les unités s’avèrent à la fois non identiques (puisque tout quelque chose est originairement distinct d’un autre quelque chose) et indiscernables (puisque, par formalisation, on a évacué de l’objet tout contenu pour le réduire à un pur et simple quelque chose). Ce faisant, l’acte d’adjonction d’une unité correspond à celui d’ajouter un élément à un ensemble, la soustraction d’unité, à celui d’en retrancher un élément, la multiplication, à la formation du produit de deux ensembles : toutes les opérations élémentaires de l’arithmétique s’avèrent ainsi réductibles à des opérations ensemblistes. Cette thèse du jeune Husserl a pris toute son ampleur avec l’assimilation à l’ontologie formelle, dans Formale und transzendentale Logik, de la mathématique prise sur son versant non apophantique (c’est-à-dire non syntaxique) : une ontologie, c’est-à-dire une doctrine a priori de l’objet ; elle est formelle dans la mesure où, procédant à une formalisation qui exclut des objets toute teneur réale, elle les réduit à de « purs modes du quelque chose en général » ou à « certaines formes dérivées du quelque chose en général » (Ableitungsgestalten des Etwas-überhaupt)14 . Ainsi les cardinaux sont-ils pensables comme des classes d’équivalence de la relation d’équinuméricité entre ensembles d’objets quelconques, les ordinaux, comme des formes de mise en ordre d’ensembles, etc. : les notions d’élément indéterminé et d’ensemble donnent lieu à des déterminations structurelles dans la sphère vide du quelque chose en général.

laisser A indéterminé et pourtant constant, de sorte que son nom n’ait pas à intervenir dans les considérations » (Hua XII, 388, trad. fr., 457). 12 Loc. cit., Hua XII, 389 (trad. fr., 457). 13 G. F REGE , Die Grundlagen der Arithmetik, §§ 38 et 34, Breslau, Koebner, 1884, p. 49–50 et 44–46 (trad. fr. C. Imbert, Les fondements de l’arithmétique, Paris, Seuil, 1969, p. 166–168 et 162–163). 14 H USSERL , Formale und transzendentale Logik, § 24, Hua XVII, 81–82 (trad. fr. S. Bachelard, Logique formelle et logique transcendantale, Paris, Puf, 1957, p. 107–108).

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6.2 LA RÉGRESSION AUX « SOURCES PHÉNOMÉNOLOGIQUES » DES CONCEPTS LOGIQUES: UN RETOUR À L ’ INTUITION ? Le second élément de réponse réside dans la référence de Husserl à ses propres efforts pour « élucider les véritables concepts logiques fondamentaux (parmi lesquels il faut également compter ceux d’ensemble, de nombre cardinal, d’ordre, de mesure de la grandeur) à partir de leurs sources phénoménologiques » (aus ihren phänomenologischen Quellen)15 : non seulement les concepts fondamentaux de l’arithmétique des cardinaux et des ordinaux se laissent caractériser comme des concepts logiques (ce que l’on peut interpréter dans le cadre de l’analyse précédente), mais loin que le travail de définition logique ou de réduction ensembliste soit ultime, il faut encore revenir de ces echt logische Begriffe à leurs sources phénoménologiques, afin qu’acquérir une « intuition intellectuelle du sens de l’activité mathématique » (Einsicht in den Sinn der mathematischen Leistung)16 . Que désignent ces sources phénoménologiques ? On pourrait de prime abord estimer qu’une telle régression des concepts purement logiques à leurs sources phénoménologiques consiste à défaire leur consistance d’idéalités toutes faites pour revenir aux processus subjectifs infralogiques d’où ils dériveraient – c’est-à-dire à rétrocéder du versant noématique des objets au versant noétique des actes producteurs ou donateurs : tel est, par exemple, le renvoi des nombres entiers à la dyade pure qu’est la scission temporelle originaire du maintenant et du tout-juste-passé (Brouwer17 ), ou encore l’intuition de la suite des entiers et la démonstration par récurrence (Poincaré18 ). Or, tel n’est pas le cas ! Loin de renvoyer comme Brouwer ou Poincaré à une forme d’intuition arithmétique élémentaire qui soit irréductible aux concepts logiques et qui en soit la source vive, Husserl fait référence à deux théories résolument situées sur le versant noématique des idéalités. Il mentionne, d’une part, la théorie des fonctions judicatives (Funktionalurteile) qu’il avait élaborée dans son cours de 1917/18 Logik und Erkenntnistheorie, à savoir des jugements comportant une variable libre x, une place vide (Leerstelle) ou un quelque chose vide (leeres Etwas). Ici prévaut la distinction, expressément inspirée de Frege, entre la manière de juger visant un contenu réal et la manière purement formelle (sachhaltige und formale Urteilsweise)19 , entre jugements fermes (feste

15 H USSERL ,

Brief an Weyl, 10. IV. 1918, in Briefwechsel, VII, 287.

16 Ibid. 17 B ROUWER ,

“Intuitionism and formalism”, CW 1, 127 (trad. cit., 43). POINCARÉ, « Nature du raisonnement mathématique » in La Science et l’Hypothèse (1902), Paris, Flammarion, 1968, p. 38–40 ; « Les Mathématiques et la Logique » in Science et Méthode (1908), Paris, Kimé, 1999, p. 130–138. 19 H USSERL , Brief an Weyl, 10. IV. 1918, in Briefwechsel, VII, 287. 18 H.

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Urteile) et fonctions judicatives (Funktionalurteile)20 . Un jugement ferme établit quelque chose, fixe un état de choses réal à propos d’un ou plusieurs objets déterminés, de sorte qu’il se caractérise du côté du sujet par la détermination de la référence, et du côté du prédicat par l’attribution d’un contenu – par exemple, « cette rose est rouge ». Une fonction judicative est au contraire une fonction dont le sujet est une variable (un argument) indéterminée, dont la dénotation demeure ouverte et est susceptible de parcourir un domaine infini, et dont le prédicat est une fonction prenant pour valeur le vrai ou le faux selon que l’objet vérifie ou non la propriété énoncée – par exemple, « x est un satellite de la Terre ». Le propre de la pensée logico-mathématique est d’opérer avec des fonctions : son essence noétique réside dans l’opération d’algébrisation ou de formalisation, c’est-à-dire l’évacuation de toute teneur concrète ou réale (Sachhaltigkeit), qui vise à penser en toute généralité des fonctions ayant un domaine de définition sur un champ – c’està-dire de pures structures catégoriales. Husserl reconnaît ainsi le statut purement formel ou catégorial (au sens strict) du penser mathématique : il n’opère ni avec des concepts sensibles (sinnliche Begriffe) obtenus par variation eidétique et idéation, ni avec des concepts mixtes (gemischte Begriffe) ou des Idées (Ideen) obtenus par idéalisation, mais avec des concepts catégoriaux, vides de contenu (rein kategoriale Begriffe). C’est dire que la tendance inhérente au penser mathématicien réside dans l’orientation vers une généralité toujours plus grande : partant des figures idéalisées du plan euclidien, on passe par généralisation aux formes catégoriales d’espace (de courbure constante, positive ou négative, variable) ; partant de transformations déterminées des figures (translation, rotation, symétries, similitude), on passe aux groupes de transformations sur des domaines indéterminés, etc. En outre, Husserl fait référence à la distinction entre proposition (Satz) et forme propositionnelle (Satzform), qui recoupe en partie la précédente sans s’y réduire : une proposition au sens strict est un jugement, c’est-à-dire une proposition à contenu qui vise des substrats et des états de choses déterminés ; une forme propositionnelle se réduit au contraire à une simple forme de proposition visant des formes de sujet sans dénotation déterminée, et des états de choses eux-mêmes indéterminés. En passant des propositions à de simples formes propositionnelles par une réduction formalisante (formalisierende Reduktion), on passe d’une théorie catégorique portant sur un champ d’objets donné à une forme de théorie (Theorienform) ou forme déductive (Beweisform), c’est-à-dire une forme axiomatisée ou hypothético-déductive de théorie, qui porte sur un champ d’objets indéterminé, sans référence fixe, susceptible de satisfaire l’ensemble des formes d’axiomes posées ; ainsi, partant de la géométrie euclidienne plane qui porte sur des points, droites et plans, peut-on évacuer toutes les teneurs quidditives contentuelles des concepts (alle sachhaltigen Wasgehalte) pour les réduire à de purs modes du « quelque chose en

20 H USSERL ,

Logik und allgemeine Erkenntnistheorie, § 40b, Hua XXX, 181. C’est à dessein que, dans la traduction de la locution centrale, nous inversons les termes, l’expression jugement fonctionnel rendant en effet fort imparfaitement l’idée d’une fonction dont le parcours de valeurs se limite aux seules valeurs de vérité.

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général » (in Modi des leeren ,,Etwas-überhaupt“), c’est-à-dire des types d’objets a, b, c . . . appelés par convention points, A, B, C . . . nommés droites, et α, β, γ . . . baptisés plans – et ce sans que ces objets conservent en eux la moindre référence aux points, droites et plans représentables par idéalisation –, pour obtenir la structure purement syntaxique et déductive appartenant à toute théorie isomorphe à la géométrie euclidienne plane. Ainsi se dégage la seconde orientation de la pensée mathématique, liée au concept de multiplicité définie (definite Mannigfaltigkeit)21 : au lieu de déterminer les propriétés de champs d’objets déterminés, elle axiomatise les théories, afin de dégager conjointement la pure forme syntaxique d’un ensemble de théories apparentées et la forme ontologique d’un domaine d’objets satisfaisant une forme de théorie (c’est-à-dire celle de modèles isomorphes). Sur le plan noétique, le propre du penser mathématicien est donc de procéder par axiomatisation ou réduction formalisante. Concluons. Lorsqu’il complimente Weyl pour avoir fait retour à l’intuition logico-mathématique, Husserl entend-il par là une intuition mathématique prise en un sens antiformaliste ? Nullement. Cette expression sybilline désigne pour l’essentiel l’intuition formalisante, entendue à la fois comme réduction ensembliste des concepts mathématiques, théorie fonctionnelle de la prédication, axiomatisation des théories et tendance vers l’universalité formelle ; une telle orientation téléologique vers la réduction formalisante atteint son acmé avec la théorie des purs systèmes déductifs et des multiplicités associées, c’est-à-dire l’idée d’une domination du champ des formes catégoriales sur le double versant apophantique et ontologique. Ainsi, loin de s’attacher à ce qui pourrait constituer la spécificité de la pensée mathématique par rapport à la pensée logique purement déductive et à la pensée ensembliste purement formelle, Husserl entend au contraire par intuition logico-mathématique une conjonction de pensée ensembliste et de déductivité analytique-formelle – c’est-à-dire un mixte de Frege, Zermelo et Hilbert ! Pour se convaincre du fait que la référence à l’intuition ne comporte ici nulle allégeance à l’intuitionnisme, il suffit de se référer à la lettre qu’adresse Husserl à Weyl le 09 avril 1922, où il écrit : Comme me l’écrit Courant, Hilbert a d’une manière nouvelle fixé le projet d’une fondation de la mathématique [in neuer Weise eine Grundlegung der Mathematik entworfen] – et ce « dans un esprit tout à fait phénoménologique » [,,ganz in phänomenologischem Geiste“] !22 .

Or, à quelle communication précise de Hilbert Richard Courant se réfère-t-il ? Il s’agit de la première conférence donnée en 1922 sur la fondation des mathématiques, où Hilbert pose les fondements du programme formaliste23 . Il y élabore sa propre méthode de formalisation (en un sens bien différent du sens husserlien), afin

21 H USSERL ,

Form. u. transz. Log., §§ 28–32, Hua XVII, 93–102 (trad. fr., 123–134). Brief an Weyl, 09. IV. 1922, in Briefwechsel, VII, 295. 23 H ILBERT , ,,Neubegründung der Mathematik. Erste Mitteilung“, in Abhandlungen aus dem mathem. Seminar der Hamburg. Universität, Bd. 1, 1922, p. 157–177, repris in HILBERT, Gesammelte Abhandlungen (= GA), Band III, Berlin-Heidelberg-New York, Springer, 1970, p. 22 H USSERL ,

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de remédier aux antinomies auxquelles conduisent tant la fondation frégéenne de l’arithmétique sur la logique que la fondation ensembliste effectuée par Dedekind et de rétablir la sécurité en mathématique24 . Pour ce faire, d’une part il adopte la méthode axiomatique, c’est-à-dire la détermination de tout domaine d’objets par un ensemble de relations posées dans les axiomes, le seul problème (cependant de taille !) étant alors de démontrer, sur un plan métamathématique, la noncontradiction du système d’axiomes ainsi posé25 . Par opposition à l’« ultraréalisme du concept » auquel mènent les théories de Frege et de Dedekind, il limite la pensée arithmétique aux seuls signes et suites de signes, considérées comme des « entités discrètes extralogiques » intuitivement données dans une expérience perceptive antérieure à la pensée : « je prends l’exact contrepied des doctrines de Frege et Dedekind, puisque je considère que les objets de l’arithmétique sont les signes eux-mêmes ( . . . ). Au commencement est le signe, telle est ici la loi » (am Anfang – so ist es hier – ist das Zeichen)26 . En d’autres termes, Hilbert adopte ici une position non seulement nominaliste, mais purement terministe, qui restreint l’orientation intentionnelle de la pensée mathématique aux signes (Zeichen) ou expressions (Ausdrücke) tracées sur le papier, sans qu’ils aient pour fonction de dénoter des objets ou contenus idéaux ; il met donc en suspens, au sein de la pensée mathématique, toute visée de significations idéales et tout rapport à des objets idéaux, prenant par là l’exact contrepied de la thèse de l’idéalité de la signification de la Première Recherche logique ! En quoi, par conséquent, un tel formalisme purement terministe est-il conforme à l’esprit phénoménologique ? Eh bien, en rien ! De fait, Weyl reconnaissait lui-même que si Husserl avait rendu à la théorie de la connaissance de si grands services, c’était en assurant « la connaissance que l’“intuition” s’étend largement au-delà du sensible » (,,Anschauung“ weit über das Sinnliche hinausreicht)27 : c’est-à-dire par l’élargissement de la notion d’objet au-delà des seuls objets sensibles et de celle d’intuition audelà de la seule perception sensible, par la reconnaissance conjointe de l’être des objets idéaux et de la possibilité de les intuitionner. Or, en limitant l’intuition mathématique à l’intuition symbolique et en affirmant que les signes arithmétiques « n’ont par ailleurs aucune espèce de signification » (sonst keinerlei Bedeutung)28 , Hilbert récuse frontalement la possibilité d’une intuition des idéalités logicomathématiques ; et, en refusant aussi bien la conception frégéenne que celle de Dedekind, il interdit toute reconduction des concepts mathématiques à des entités logiques ou ensemblistes.

157–177 (trad. fr. J. Largeault, « Nouvelle fondation des mathématiques. Première communication », in Intuitionnisme . . . , p. 111–130). 24 Loc. cit., GA III, 160 (trad. fr., 114–115). 25 Loc. cit., GA III, 158–159 et 161 (113–114 et 115). 26 Loc. cit., GA III, 163 (trad. fr., 117). 27 W EYL , Brief an Husserl, 26/27 III. 1921, in H USSERL , Briefwechsel, VII, 290. 28 H ILBERT , ,,Neubegründung . . . “, GA III, 163 (trad. fr., 117–118).

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6.3 WEYL ENTRE FORMALISME ET INTUITIONNISME : À LA CROISÉE DES CHEMINS DE BROUWER ET HILBERT Si Weyl se réclame expressément de la phénoménologie husserlienne en philosophie des mathématiques, que tire-t-il exactement de la phénoménologie ? Il se réfère, dans la lettre tout juste citée, aux acquis des Recherches logiques : « les aperçus décisifs sur l’évidence et la vérité, la connaissance que l’“intuition” s’étend largement au-delà du sensible » (daß ,,Anschauung“ weit über das Sinnliche hinausreicht)29 – bref, l’élargissement des notions d’objet et d’intuition donatrice au-delà des objets sensibles et de l’intuition sensible, ainsi que la distinction entre les sens noétique et noématique de la vérité. Mais outre cela, Weyl retient-il de Husserl la position formaliste proche de Hilbert et centrée sur le concept métamathématique de multiplicité définie, ou bien la mise en question intuitionniste des principes fondamentaux de la logique, ou encore une position constructiviste liée à la thèse de l’idéalisme transcendantal ? En premier lieu, la référence à la phénoménologie husserlienne permet-elle à Weyl de trancher entre l’intuitionnisme brouwerien et le formalisme hilbertien ? Le témoignage à cet égard le plus précieux est sans doute fourni par les « Remarques en guise de discussion de la seconde conférence de Hilbert sur les fondements des mathématiques », où Weyl écrit ceci : Qu’on me permette de dire quelques mots en faveur de l’intuitionnisme. ( . . . ) Si la conception hilbertienne s’impose et prévaut sur l’intuitionnisme, comme tel est apparemment le cas, j’y vois une défaite décisive de l’attitude philosophique de la phénoménologie pure [eine entscheidende Niederlage der philosophischen Einstellung reiner Phänomenologie], qui se serait alors révélée insuffisante pour comprendre la création scientifique [Verständnis schöpferischer Wissenschaft] dans le domaine même de la connaissance qui est le plus primitif et le plus ouvert à l’évidence – les mathématiques30 .

Qu’est-ce à dire ? Weyl reconnaît-il la défaite globale de l’intuitionnisme en dépit de ses mérites partiels, et l’interprète-t-il comme un échec de la phénoménologie en philosophie des mathématiques ? La question demeure malaisée à trancher. Ainsi, Weyl se réfère à l’exigence brouwerienne que les théories mathématiques soient constituées d’« énoncés réals » (reale Aussagen), de « vérités remplies de sens » (sinnerfüllte Wahrheiten), et à son constat que la pensée mathématique a de toutes parts transgressé les limites de la pensée douée de contenu31 . Pour entendre le sens de ce partage entre ce qui est rempli ou non de sens, reportons-nous à la 29 W EYL ,

Brief an Husserl, 26/27 III. 1921, in HUSSERL, Briefwechsel, VII, 290. ,,Diskussionsbemerkungen zu dem zweiten Hilbertschen Vortrag über die Grundlagen der Mathematik“, Abhandl. aus dem Mathem. Seminar der Hamburger Universität, 8, 1928, repris dans les Gesammelte Abhandlungen, Berlin-Heidelberg-New York, Springer, 1968, Band III (= GA III), p. 147 et 149 (trad. fr. J. Largeault in Intuitionnisme . . . , p. 166 et 169 ; trad. angl. in J. VAN HEIJENOORT, From Frege to Gödel. A source Book in Mathematical Logic, 1879–1931, p. 482 et 484). 31 W EYL , ,,Diskussionsbemerkungen . . . “, GA III, 147 (trad. fr., 167 ; trad. angl., 483). 30 W EYL ,

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conférence donnée par Brouwer à Amsterdam. La seule conséquence, dit-il, qu’aient suscitée pour les formalistes les antinomies de la théorie des ensembles, c’est la suppression des ensembles qui donnent lieu aux antinomies (ensemble défini par application à un ensemble de la propriété qui définit ses éléments, ensemble défini par simple position d’une propriété, etc.) ; en revanche, ils continuent de considérer des ensembles et des objets qui sont dénués de sens pour l’intuitionniste. Ainsi de la théorie cantorienne des puissances ou cardinaux transfinis : ayant défini les concepts de pluralité bien ordonnée (de suite), puis de nombre ordinal dénombrablement infini appartenant à une suite, on se donne le droit de considérer le système de tous ces ordinaux ainsi que sa puissance ℵ1 ; on formule alors l’inégalité ℵ1 > ℵ0 , ainsi que la proposition selon laquelle « ℵ1 est le plus petit cardinal infini supérieur à ℵ0 ». Ainsi encore de la considération de l’ensemble de tous les nombres réels compris entre 0 et 1, soit l’intervalle [0, 1]32 . Le caractère dépourvu de sens de tels objets tient au principe intuitionniste qui régit la raison mathématique, selon lequel « l’intuitionniste ne saurait construire que des ensembles dénombrables d’objets mathématiques » (only construct denumerable sets of mathematical objects) et « sur le fondement de l’intuition du continu linéaire, il admet des suites élémentaires de choix libres comme éléments de construction » (elementary series of free selections as elements of construction)33 : partant, si l’on peut construire des ensembles infinis dénombrables de nombres rationnels compris entre 0 et 1, et si l’on peut associer à chacun de ces ensembles un nombre réel compris entre 0 et 1, il est en revanche illégitime de considérer comme donnée la totalité infinie indénombrable des nombres réels compris entre 0 et 1 ; de même, s’il est possible de construire des ensembles infinis dénombrables d’ordinaux transfinis dénombrables, puis d’associer à chacun de ces ensembles un ordinal transfini qui ne lui appartient pas, il est en revanche illégitime de considérer comme donnée la totalité de ces ordinaux transfinis dénombrables34 . La pensée mathématique est assignée à la limite absolue du dénombrable et des suites de choix libres. Quelle conclusion Weyl en tire-t-il ? Elle s’avère pour le moins ambivalente. D’un côté, Weyl reconnaît la légitimité de la double critique brouwerienne de la pensée sans contenu (sans objet) et de l’application des principes logiques aux ensembles infinis, ainsi que de son exigence de procédures constructives permettant d’atteindre effectivement les objets : la doctrine de Brouwer est en effet « l’idéalisme en mathématique pensé jusqu’à son terme » [der zu Ende gedachte Idealismus in der Mathematik], lui permettant ainsi d’atteindre « le suprême degré de clarté intuitive » [die höchste intuitive Klarheit]35 . En effet, Brouwer a en quelque sorte pensé l’application en mathématique du « principe de tous les

32 B ROUWER ,

“Intuitionism and formalism”, CW 1, 133–134 (trad. fr., op. cit., 48–49). cit., CW 1, 134–135 (trad. fr., 50). 34 Loc. cit., CW 1, 133–134 (trad. fr., 49). 35 W EYL , ,,Die heutige Erkenntnislage in der Mathematik“, IV, Symposion (Berlin), 1, 1925–27, GA II, 533–534 (trad. fr. J. Largeault, « L’état présent de la connaissance en mathématique » in WEYL, Le continu et autres écrits, Paris, Vrin, 1994, p. 154). 33 Loc.

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principes » husserlien, qui fait de l’intuition originairement donatrice de l’objet (originär gebende Anschaaung) la source de légitimité de toute connaissance36 ; pour Husserl, ce principe énonce que l’intuition eidétique qui donne les essences correspondant aux concepts fondamentaux (par exemple, de la mathématique) est source de toute connaissance : vaut ainsi comme objet mathématique attesté ce qui est corrélat de procédures effectives d’atteinte. Or, Brouwer renforce en nous « le sens de ce qui est intuitivement donné en mathématique » [den Sinn für das anschaulich Gegebene]37 , qui est double : d’une part, l’intuition mathématique originaire réside dans la formation de la suite des entiers naturels et dans l’induction complète qui fonde toute démonstration sur l’ensemble des entiers ; d’autre part, l’intuition désigne toute formation de suites de nombres entiers par de libres actes de choix38 . L’intuition eidétique originairement donatrice reçoit par là un double sens : induction mathématique, et procédures constructives par une suite de choix en devenir. Tout ce qui n’est accessible ni à l’une ni à l’autre ne mérite nullement le statut d’objet mathématique ; ainsi les mathématiques doivent-elles être amputées des ensembles infinis non dénombrables. De l’autre, Weyl reconnaît cependant comme un fait capital que Hilbert soit parvenu à sauver les mathématiques classiques amputées par les intuitionnistes –, et ce, « par un changement total d’interprétation de leur sens [durch eine radikale Umdeutung ihres Sinnes], sans amputer leur capital, mais en les formalisant de manière à ce que, d’un système de connaissances intuitives, elles deviennent un jeu sur des formules procédant conformément à des règles fixes »39 . En d’autres termes, s’il adopte la conception brouwerienne de l’intuition mathématique, Weyl ne souscrit pas à l’exigence de limitation de la pensée mathématique à la seule intuition donatrice ; il n’émet nullement, au nom d’une exigence d’intuitivité, de condamnation de la mathématique formalisée de Hilbert, c’est-à-dire de limitation de la pensée mathématique aux seuls signes, mais reconnaît au contraire la « cohérence géniale » avec laquelle Hilbert a complété l’axiomatisation par sa théorie de la démonstration formalisée : « Quant à l’évaluation épistémologique de la nouvelle situation qui en est issue, rien ( . . . ) ne me sépare de Hilbert »40 . Prenons un exemple : les procédures déductives n’acquièrent leur véritable portée que lorsqu’on les applique au domaine du transfini, moyennant l’introduction de variables et de quantificateurs universels ; ce faisant, les mathématiques s’élargissent bien au-delà de la zone de l’intuition donatrice brouwerienne, dans la mesure où il n’est pas d’intuition donatrice des ensembles infinis, mais seulement une intégration de l’infini à des calculs non contradictoires qui, en l’absence d’« interprétation contentuelle raisonnable » [vernünftige inhaltliche Interpretation], permet cependant de

36 H USSERL ,

Ideen . . . I, § 24, Hua III/1, 51 (trad. fr., 78–79). ,,Die heutige Erkenntnislage . . . “, V, GA II, 54 (trad. fr., 160). 38 Loc. cit., V, GA II, 533 (trad. fr., 154). 39 W EYL , ,,Diskussionsbemerkungen . . . “, GA III, 148 (trad. fr., 167 ; trad. angl., 483). 40 Loc. cit, GA III, 148 (trad. fr., 168 ; trad. angl., 483). 37 W EYL ,

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faire « comme s’il était effectivement donné [als wäre es wirklich vorhanden] »41 . Dans cette légitimation de la théorie hilbertienne de la démonstration, Weyl fait ainsi place à la distinction husserlienne entre l’intuitif et le signitif, les domaines de la pensée et de l’intuition, du sens et de l’objet : en mathématique de l’infini, les procédures de calcul symbolique réglées deviennent source de droit pour la connaissance ; la mathématique de l’intuition est relayée par celle du comme si et de la construction symbolique, sous le double signe de Vaihinger et de Leibniz. Enfin, ce règne du symbolique ne signifie pas que Weyl y ait abandonné toute exigence de remplissement intuitif. Au contraire, il maintient en ce domaine l’exigence husserlienne, wissenschaftstheoretisch, de remplissement des jeux de formules hilbertiens par un sens ; et, conformément aux conceptions husserliennes, ce remplissement advient grâce à l’application de la mathématique à la physique théorique – c’est-à-dire à la connaissance du monde effectif, mais une connaissance « d’un type tout autre que là connaissance intuitive ou phénoménale ordinaire » dans la mesure où son sens n’est pas réalisable dans la perception sensible, mais autorise uniquement une confrontation du système tout entier des hypothèses fondamentales avec l’expérience42 . Cette discrépance entre physique théorique et connaissance perceptive tient sans doute au type de testabilité (de réfutabilité) qui est propre à chacune : alors qu’un énoncé perceptif s’atteste ou se réfute sur fond d’intuition donatrice de l’objet perceptif, une expérience négative en physique réfuterait l’ensemble des principes théoriques de la physique, sans cependant que l’on sache lequel de ces principes est touché ; on sent ici l’influence du holisme duhémien. Ces analyses ont pour conclusion le mot d’ordre suivant : « en même temps que le chemin de Brouwer, il faudra aussi nécessairement suivre celui de Hilbert » (neben dem Brouwerschen wird man den Hilbertschen Weg verfolgen müssen)43 . En effet, chaque domaine impose son mode de traitement théorétique propre : le domaine du fini et de l’infini dénombrable est une zone d’intuitivité mathématique qui requiert la voie brouwerienne, celle de l’induction et des procédures constructives ; en revanche, celui de l’infini non dénombrable (de même que celui de la physique théorique) est le règne du calcul et des constructions purement symboliques. Un tel syncrétisme, qui paradoxalement accorde une place conjointe à l’intuitionnisme et au formalisme, respecte un principe fondamental de Husserl : le principe anticopernicien selon lequel tout domaine de connaissance impose son mode d’accessibilité à

41 W EYL ,

,,Die heutige Erkenntnislage . . . “, V, GA II, 538–540 (trad. fr., 158–160). ,,Die heutige Erkenntnislage . . . “, V, GA II, 540 (trad. fr., 160). Cette distinction entre la mathématique symbolique des règles du jeu et l’application qui lui confère un sens épistémique est à rapprocher du § 40 de Form. u. transz. Logik, où Husserl refuse la réduction de la mathématique à un « jeu de symboles », à une discipline « élaborée de façon purement calculatoire », considérant à l’inverse que la « référence à une application possible », qui en fait une « composante de la détermination physicienne », appartient au sens de la mathesis formelle (Hua XVII, 113–115, trad. fr., 148–150). Nota bene : le texte de Weyl est antérieur à celui de Husserl ! 43 W EYL , ,,Die heutige Erkenntnislage . . . “, V, GA II, 542 (trad. fr., 161). 42 W EYL ,

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la conscience, et en conséquence son mode de traitement théorétique spécifique44 ; ainsi certains domaines sont-ils susceptibles de domination par l’induction complète et les suites de choix, tandis que d’autres n’admettent qu’un mode de connaissance par constructions symboliques.

6.4 LA SPÉCIFICITÉ DE LA PENSÉE MATHÉMATIQUE SELON W EYL : FINITISME ET INDUCTION COMPLÈTE En dépit de cette coexistence pacifique et complémentaire des voies brouwerienne et hilbertienne, y a-t-il pour Weyl une spécificité de la pensée mathématique par rapport à la pensée logique et ensembliste ? De fait, sa thèse essentielle est celle de l’irréductibilité de la pensée mathématique à la pensée logique, ensembliste ou analytique-formelle – et ce en vertu du caractère non logique et non ensembliste de ses deux démarches fondamentales, la formation de la suite indéfinie des entiers naturels et celle du continu par une suite de libres choix : Il y a, dans l’édifice des mathématiques, deux places ouvertes où il est possible d’aller vers un abîme sans fond [wo es möglicherweise ins Unergründliche geht] : le progrès dans la suite des nombres naturels, et le continu. Tout le reste ( . . . ) est une affaire de logique formelle [formal-logische Angelegenheit]45 .

De là découle la tâche essentielle d’une phénoménologie de la pensée mathématicienne : élucider, par voie réflexive, ces deux opérations fondamentales de la pensée. En cela, Weyl demeure fidèle à l’inspiration husserlienne par son projet, tout en s’en détachant en vertu de sa thèse propre : il est en effet fidèle au projet méthodique husserlien de saisie réflexive des actes de constitution des objectités mathématiques ; mais il se dissocie de la position husserlienne, qui voyait dans la pensée analytiqueformelle l’essence même de la pensée mathématique. Commençons par l’engendrement de la suite des entiers naturels où, dans le sillage de Brouwer et de Kronecker, Weyl discerne l’un des ultimes fondements de la pensée mathématique : je me convainquis fermement (en accord avec Poincaré, si peu que je partage par ailleurs ses thèses philosophiques) que la représentation de l’itération, de la suite naturelle des nombres [Vorstellung der Iteration, der natürlichen Zahlenreihe], est un fondement ultime de la pensée mathématique [ein letztes Fundament des mathematischen Denkens] – et ce en dépit de la « théorie des chaînes » de Dedekind, qui visait à fonder la définition et la déduction de manière syllogistique, par induction complète [durch vollständige Induktion], sans recourir à cette intuition. S’il est en effet vrai qu’il n’est possible de saisir les concepts fondamentaux de la théorie des ensembles qu’en accomplissant une telle intuition « pure »

44 H USSERL , Cartesianische Meditationen, § 22,

Hua I, 90 (trad. fr. dir. M. de Launay, Méditations cartésiennes, Paris, Puf, 1994, p. 99–100). Ideen . . . I, § 138, Hua III/1, 321 (trad. fr., 467). Ideen . . . III, § 7, Hua V, 36 (trad. fr. D. Tiffeneau, La phénoménologie et les fondements des sciences, Paris, Puf, 1992, p. 44). 45 W EYL , ,,Die heutige Erkenntnislage . . . “, II, GA II, 522–523 (trad. fr., 146).

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[nur durch Vollzug dieser ,,reinen“ Anschauung], il s’avère alors superflu et fallacieux de fonder à son tour le concept de nombre naturel sur la théorie des ensembles [den Begriff der natürlichen Zahl ( . . . ) mengentheoretisch zu fundieren]46 . Le point de départ de la mathématique est la suite des nombres naturels [Ausgangspunkt der Mathematik ist die Reihe der natürlichen Zahlen] ( . . . ). La nature des choses veut que l’intuition eidétique [Wesenseinsicht] d’où proviennent les théorèmes généraux soit toujours fondée sur l’induction complète, l’intuition mathématique originaire [auf der vollständigen Induktion, der mathematischen Urintuition, fundiert]47 . Cette intuition du « toujours un de plus » [intuition of the “even one more”], de l’infinité dénombrable ouverte, est au fondement de toutes les mathématiques. Elle donne naissance à l’exemple le plus simple de ce que j’ai appelé plus haut un domaine de variabilité dominable a priori [an a priori surveyable range of variability]48 .

Que faut-il entendre exactement par là ? Il est nécessaire de distinguer plusieurs niveaux d’analyse afin d’expliciter la notion d’intuition originaire, ainsi que l’irréductibilité de la pensée mathématique à la logique et à la pensée ensembliste. En premier lieu, ce qui est en jeu est la spécificité du raisonnement par récurrence ou par induction complète, qui s’avère irréductible à un principe analytique-formel. C’est là une reprise de la thèse de Poincaré selon laquelle le raisonnement par récurrence – qui consiste à démontrer la validité d’une propriété sur un domaine infini dénombrable et bien ordonné d’objets en montrant qu’elle est vérifiée par son premier élément et que si elle est vérifiée au rang n, elle l’est alors nécessairement au rang n+1 – est « à la fois nécessaire au mathématicien et irréductible à la logique »49 . C’est en effet le « véritable type du jugement synthétique a priori »50 , à savoir la thèse du pouvoir qu’a l’esprit de réitérer indéfiniment un acte, dont l’esprit possède une intuition réflexive directe : c’est bien une synthèse, puisque seul l’enchaînement du même acte d’un rang quelconque au rang immédiatement successif permet de prouver la validité générale de la propriété ; et cette synthèse est a priori, puisqu’elle n’est empruntée à aucune expérience, mais provient du seul pouvoir immanent à l’esprit ; enfin, cet a priori noétique est irréductible à la logique, puisqu’il s’agit d’une règle « irréductible au principe de contradiction »51 . En termes husserliens, la démonstration mathématique portant sur les propriétés dites inductives s’avère irreconductible aux pures lois analytiques de la déductivité de la Konsequenzlogik : le fondement de la mathématique réside ainsi dans une

46 W EYL , Das Kontinuum, Kap. I, Leipzig, von Veit, 1918, p. 37 (trad. fr. J. Largeault, Le continu et autres écrits, Paris, Vrin 1994, p. 83 ; nous modifions la traduction, ici assez gravement défectueuse). 47 W EYL , ,,Die heutige Erkenntnislage . . . “, IV, GA II, 533 (trad. fr., 154). 48 W EYL , “The Mathematical Way of Thinking”, Science 92 (1940), repris in GA III, 713 (« Le mode de pensée mathématique », trad. fr. J. Largeault in WEYL, Le continu . . . , p. 218). 49 P OINCARÉ , « Les Mathématiques et la Logique » in Science et Méthode, p. 130. 50 P OINCARÉ , « Sur la nature du raisonnement mathématique » in La Science et l’Hypothèse, p. 41. 51 Ibid.

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intuition extralogique52 . De là découle le caractère fondamentalement finitiste de la pensée mathématique, y compris dans les procédures démonstratives : Ce qu’il y a de grand dans les mathématiques, je le vois en ce que, dans presque tous les théorèmes, ce qui en son essence est infini est ramené à une décision finie [das seinem Wesen nach Unendliche zu endlicher Entscheidung gebracht wird]53 .

Une telle conception de l’induction complète comme principe extralogique fondant la possibilité de démontrer des propriétés sur des ensembles infinis dénombrables s’oppose à la manière dont Hilbert, dans sa théorie de la démonstration, la réduit à un « principe formel ( . . . ) qui requiert une démonstration, et peut en recevoir une » (eines Beweises bedürftig und fähig ist)54 : ayant évacué tout sens et toute dénotation idéale des mathématiques pour réduire le corrélat de la pensée aux seuls signes, et les démonstrations à des suites de signes conformes à des règles fixes, Hilbert pose en effet un « axiome d’induction complète » qui, sur ce plan formel, régit la démonstration de P(x) par passage du rang n ou au rang n+1 ; et c’est seulement sur le plan supérieur de la métamathématique à contenu, destinée à démontrer la consistance des systèmes formels, que l’on utilise le principe d’induction complète, qui dérive de l’axiome en question55 . À cette conception extralogique et intuitionniste de l’induction se joint en outre une thèse idéaliste et intuitionniste portant sur le statut des entiers naturels : l’ensemble N n’est pas une totalité infinie actuelle d’objets idéaux existant en soi, mais se réduit à une « suite ouverte », c’est-à-dire au strict corrélat intentionnel du procès noétique potentiel d’itération, de répétition indéfinie d’un acte identique (la saisie du successeur, ou l’addition de 1) : Brouwer a rendu évident, et à mon avis indubitable, qu’il n’est pas de preuve attestant la croyance dans le caractère existentiel de la totalité des nombres naturels [no evidence supporting the belief in the existential character of the totality of naturel numbers] ( . . . ). La suite des nombres qui, par passage de chaque nombre à son successeur croît au-delà de toute étape déjà atteinte, est une multiplicité de possibilités ouverte sur l’infini ; elle demeure à jamais in statu nascendi [forever in the status of creation] mais n’est pas un domaine clos de choses existant en soi [a closed realm of things existing in themselves]. Avoir confondu l’un avec l’autre, telle est la source de nos difficultés56 .

Il s’agit de la négation de l’idée d’être en soi pour les ensembles infinis. Or, une telle négation correspond à la thèse husserlienne de stricte corrélation entre l’effectivité du sens objectal (gegenständlicher Sinn) et les actes d’évidence effectuables de la conscience, mais à la condition expresse d’y ajouter une thèse finitiste au regard des actes intentionnels : à savoir que toute évidence donatrice d’objet se laisse identifier à un acte réellement effectuable dans le temps de la 52 W EYL ,

“Mathematics and Logic. A brief Survey serving as a Preface to a Review of “The Philosophy of Bertrand Russell”, § 8, The American Mathematical Monthly, 53 (1946), GA IV, p. 278–279 (trad. fr. J. Largeault, « Mathématique et logique » in WEYL, Le continu . . . , p. 246). 53 W EYL , Das Kontinuum, Kap. I, p. 37 (trad. fr., p. 83). 54 H ILBERT , ,,Neubegründung . . . “ (1922), GA III, 164 (trad. fr. cit., 118). 55 Loc. cit., GA III, 175 (trad. fr., 128–129). 56 W EYL , “Mathematics and Logic”, § 6, GA IV, 275 (trad. fr., 242).

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conscience – qui est toujours fini, bien que potentiellement ouvert. Cette thèse finitiste quant à la possibilité des actes effectuables et des procédures admissibles en mathématiques revient à renouer, en-deçà de Husserl, avec le finitisme kantien : si Kant exclut le nombre infini, c’est en effet parce que sa représentation implique une synthèse d’addition infinie de l’unité à elle-même, laquelle requerrait un temps infini57 ; c’est là replier les représentations de conditions (à savoir les fondements conceptuels impliqués par le sens d’un objet idéal) avec les conditions de la représentation (celles de l’effectuabilité concrète d’un acte dans le temps de la conscience). Par là s’explique l’opposition de Weyl à toute réduction ensembliste de la notion d’entier naturel à celle de nombre cardinal. En effet, alors que l’engendrement de la suite des entiers naturels demeure un devenir, un processus sans cesse in statu nascendi et toujours fini, puisqu’assigné aux limites finitistes de la conscience, d’une part on veut réduire à des concepts logiques ou ensemblistes le processus d’itération irréductible à la logique, et d’autre part on supprime toute limite de principe entre le fini et l’infini58 . Pourquoi cela ? La réduction ensembliste visée par la critique de Weyl est la théorie des chaînes de Dedekind ; celle-ci permet de penser la structure formelle de l’ensemble des entiers naturels, non en partant de cet ensemble comme donné, mais au contraire comme simple cas particulier du type des ensembles simplement infinis. Un ensemble E est infini s’il existe une bijection (application réciproque ou correspondance biunivoque) de E sur l’une de ses parties propres59 ; K est une chaîne par rapport à une application ϕsi ϕ(K) est inclus dans K60 ; et un ensemble E est simplement infini s’il existe une application semblable ϕde E dans lui-même qui fait de E la chaîne d’un élément non contenu dans ϕ(E) – on appellera 1 cet élément primitif, et l’on dira que E est ordonné par cette application ϕ61 . De la sorte, l’ensemble des entiers est pensé à partir des seules notions ensemblistes d’ensemble, d’appartenance, d’application et d’inclusion ; ces notions fondent les concepts de premier élément, de successeur, donc d’ordre, ainsi que le principe d’induction complète, devenu simple théorème démontrable62 . Que reproche exactement Weyl à une telle conception ? Pourquoi parle-t-il de confusion entre le fini et l’infini, alors que Dedekind donne la définition précise des ensembles infinis par l’applicabilité sur l’une de ses parties propres ? Pour le comprendre, partons d’un exemple : comment pense-t-on l’inégalité « n ≥ 5 » ? Dans la conception naturelle fondée sur l’itération, on procède par 57 K ANT ,

Kritik der reinen Vernunft, A 431–432/B 459–460 (trad. fr. Critique de la raison pure, Delamarre-Marty, Paris, Gallimard, 1980, folio, p. 396, A. Renaut, Paris, GF-Flammarion, 20063 , p. 432–434). 58 W EYL , ,,Die heutige Erkenntnislage . . . “, II, GA II, 522 : « Pour la théorie des ensembles, aucune borne de principe [keine grundsätzliche Schranke] ne s’érige entre le fini et l’infini » (trad. fr., 145). 59 D EDEKIND , Was sind und was sollen die Zahlen ?, § 5, 64, GMS III, 356 (trad. fr. H. Benis Sinaceur, Que sont et à quoi servent les nombres ? in DEDEKIND, La création des nombres, 173). 60 Loc. cit., § 4, 36–37, GMS III, 351–352 (trad. fr., 166). 61 Loc. cit., § 6, 71, GMS III, 359 (trad. fr., 178). 62 Loc. cit., § 6, 80, GMS III, 361 (trad. fr., 181–182).

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énumération ou dénombrement pas à pas, et pour tout nombre n donné, on voit par intuition directe que le parcours des nombres de 1 à n passe par 5 : c’est une procédure effectuable, conforme aux limites finitistes de la conscience d’objet. En revanche, dans la conception ensembliste, « n ≥ 5 » revient à dire que n appartient à toutes les chaînes qui contiennent 5 comme élément : or, cela requiert de « parcourir du regard tous les sous-ensembles possibles de N », c’est-à-dire une totalité infinie d’ensembles eux-mêmes infinis et supposés exister en soi63 . À un critère finitiste se substitue ainsi un principe transfini, et la présupposition fondamentale de cette conception ensembliste réside dans un réalisme extrême : en se donnant la possibilité de raisonner sur des relations d’appartenance et d’inclusion rapportées à une infinité d’ensembles actuellement infinis, on présuppose l’existence de ces ensembles en l’absence de toute capacité effective de les intuitionner – car de fait, il est impossible de dominer ou parcourir du regard un infini actuel, et à plus forte raison une infinité d’ensembles infinis. C’est pourquoi Weyl s’oppose à la conception, propre à Dedekind et à Husserl, « d’un ensemble comme d’un “rassemblement” de tous ses éléments par une synopsis de la conscience » (überblickte “Versammlung” aller ihrer Elemente)64 : le rassemblement d’une pluralité d’objets sous le regard de la conscience, puis la conversion de cette visée polythétique de pluralité en visée monothétique, où Husserl voyait les actes constitutifs de tout ensemble, ne sauraient constituer des ensembles infinis, puisqu’il est impossible de réunir sous le regard une infinité actuelle d’objets. En toute rigueur, il n’y a pas de constitution d’ensembles actuellement infinis à titre d’objets unitaires, mais la seule possibilité qui s’offre est celle d’un processus d’itération ou de dénombrement indéfini ; jamais l’infini actuel n’est donné comme objet achevé, mais seul l’infini potentiel se laisse concevoir comme corrélat toujours in statu nascendi d’un procès itératif : « il est dénué de sens de parler d’un parcours achevé à propos d’une suite infinie [fertige Durchlaufung einer unendlichen Reihe] »65 . Weyl réinterprète donc ici en un sens finitiste les idées de constitution transcendantale et d’a priori de corrélation. Dans la thèse husserlienne, en effet, prévaut le motif anticopernicien selon lequel c’est l’essence de l’objet qui prescrit son mode de donnée : ainsi des ensembles transfinis peuvent-ils requérir un mode de visée purement symbolique et leur constitution peut-elle être entendue comme leur définition par récursion ou par une propriété caractéristique, tout en donnant lieu à des raisonnements parfaitement valides – rien n’est édicté sur les limites de ce qui est effectuable par la conscience, laquelle se conforme aux exigences de ce qui est à penser. En revanche, Weyl réinterprète l’a priori de corrélation en un sens normatif : la constitution transcendantale est toujours une construction effectuable et demeure 63 W EYL ,

,,Die heutige Erkenntnislage . . . “, II, GA II, 522 (trad. fr., 145). Das Kontinuum, I. Kap., p. 34 (trad. fr., 80). Telle est la définition donnée par Dedekind d’un System au § 1 de Was sind und was sollen die Zahlen ? (GMS III, 344, trad. fr. cit., p. 154), de même que l’explicitation de la constitution des ensembles par Husserl au § 119 des Ideen . . . I (Hua III/1, 275–277, trad. fr., 405–407). 65 W EYL , ,,Über die neue Grundlagenkrise der Mathematik“, II, § 1, Mathem. Zeitschrift, 10, GA II, 156 (trad. fr. J. Largeault, « Sur la crise contemporaine des fondements des mathématiques », in Intuitionnisme . . . , p. 80). 64 W EYL ,

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arrimée aux capacités finies de la conscience (d’où, par exemple, l’exclusion de l’axiome du choix et des puissances transfinies) – de sorte que l’essence de la pensée mathématique est finitiste, et consiste en un effort pour ramener l’infini à des décisions et choix finis66 .

6.5 LE SECOND IRRÉDUCTIBLE DE LA PENSÉE MATHÉMATIQUE : LA PENSÉE DU CONTINU Le second élément extralogique de la pensée mathématique est le continu ; estil possible de lui appliquer ne varietur ce qui a été dit des entiers naturels ? Le problème qui se pose ici réside dans l’évolution des idées de Weyl – intéressante dans la mesure où Weyl n’a cessé de se réclamer de Husserl, ce qui montre que les frontières exactes de l’intuition mathématique sont malaisées à délimiter. Dans le texte de 1918, Weyl examine le rapport entre continu intuitif (anschauliches Kontinuum) et mathématique (mathematisches Kontinuum), plus précisément arithmétique : ce rapport est de distinction, voire d’opposition – ce qui interdit toute fondation de la pensée mathématique du continu sur le continu intuitif, qu’il soit temporel ou spatial. La continuité mathématique d’une fonction est en effet une propriété transfinie (transfinite Eigenschaft)67 , puisqu’elle prend toutes les valeurs d’un intervalle arithmétique, donc dépend de la « délimitation exacte du concept de nombre réel » ; loin cependant d’adopter le concept de nombre réel comme catégorie fondamentale et de fonder sur elle une « hyper-analyse » (Hyperanalysis), Weyl pose comme exigence (brouwerienne et kroneckerienne) de « résister à la tentation toujours renouvelée de partir d’un niveau plus élevé que celui de la strate fondamentale des nombres naturels » (Grundschicht der natürlichen Zahlen)68 , pour fonder au contraire la continuité sur la théorie des entiers naturels. C’est dire que la conception mathématique du continu est nécessairement atomiste, tâchant de rejoindre le continu à partir de la catégorie fondamentale des entiers naturels : « C’est bien la catégorie des nombres naturels, et non le continu tel qu’il est donné dans l’intuition, qui peut fournir le fondement d’une discipline mathématique »69 . Il n’y a en effet, par principe, aucune coïncidence possible entre les continus temporel et mathématique : alors que ce dernier est analytique et réduit à un ensemble de points, il n’existe au contraire aucun point isolé et indépendant

66 On

remarquera que cette conclusion contredit l’analyse que nous avons précédemment faite du texte de 1926, ce qui semble indiquer qu’en dépit de sa fidélité à Husserl, Weyl a changé de point de vue, oscillant entre un finitisme revendiqué et un élargissement au domaine de l’infini ; c’est dire que la notion d’intuition mathématique n’admet pas d’interprétation fixe. 67 W EYL , Das Kontinuum, II. Kap., § 5, p. 62, et § 6, p. 65–66 (trad. fr., 105 et 108). 68 Op. cit., II. Kap., § 6, p. 72 (trad. fr., 114). 69 Op. cit., II. Kap., § 6, p. 68 (trad. fr., 111).

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(selbständig) au sein du flux temporel, où chaque instant est un point de transition (Durchgangspunkt) caractérisé par la transition fluente (Hinüberfließen von Punkt zu Punkt) ; c’est pourquoi l’essence fluente du temps interdit de rien y fixer avec exactitude (kein exaktes Fixieren), mais ne permet que l’approximation (immer nur ein approximatives)70 . La position de Weyl est donc ici bergsonienne : l’irréductibilité du continu intuitif à un concept mathématique vient de ce que « la conception d’un écoulement consistant en points et, partant, se décomposant en points, se révèle erronée » puisqu’elle laisse échapper la fluence qui caractérise en propre la continuité – celle du temps, aussi bien que de l’étendue ou du mouvement71 . Toutefois, en dépit de cette discrépance d’essence, il est légitime d’« extraire le “logos” immanent à la réalité effective » (den in der Wirklichkeit einwohnenden ,,Logos“ herauszuschälen), c’est-à-dire de franchir cet abîme en procédant à une exactification idéalisante : on constitue alors une théorie analytique du continu, où le concept de nombre réel fournit un « schème abstrait du continu » grâce à son « emboîtement infini de parties possibles » (unendliches Ineinander möglicher Teile), et celui de fonction continue, le « schème de la dépendance des continua qui se recouvrent » (Schema der Abhängigkeit sich ,,überdeckender“ Kontinuen)72 . Essentiel est ici l’acte de schématisation exactifiante : à la continuité intuitive ou fluente se substitue alors le concept d’une suite infinie d’emboîtements qui se resserre autour d’un point. Weyl se montre donc ici fidèle à l’inspiration husserlienne : de même que Husserl oppose dans les Ideen I, les synthèses intuitives continues (spatiales ou temporelles) et les synthèses discrètes et articulées, ainsi que les concepts morphologiques obtenus par description et les Idées issues de l’idéalisation, Weyl refuse de fonder le continu mathématique sur la continuité morphologique ou purement descriptive, prenant ainsi acte de la nature idéalisante de la pensée mathématique ; une chose sont les concepts descriptifs, autre chose les Idées et schèmes mathématiques. Mais tel n’est cependant pas son dernier mot. Dans les textes ultérieurs, en effet, Weyl n’a de cesse de combattre la « conception atomistique du continu », la « conception statique » (neue statische Auffassung) et réaliste qui prévaut dans la construction de la continuité de l’ensemble des nombres réels à partir des coupures (Dedekind) et des suites convergentes de rationnels (Cantor)73 . Cette conception opère une fragmentation du continu intuitif en points ou éléments indépendants pour substituer, à la cohésion ou interpénétration intuitive de ses parties, une « construction logico-arithmétique » à l’aide de relations d’ordre (< et >). Or cette opération repose sur un présupposé réaliste, qui pose l’existence en soi de tous les nombres réels, ainsi que la décidabilité en soi de toute question les concernant –

70 Op.

cit., II. Kap., § 6, p. 70 (trad. fr., 112). cit., II. Kap., § 6, p. 69–70 (trad. fr., 112). 72 Op. cit., II. Kap., § 6, p. 71 (trad. fr., 113). 73 W EYL , ,,Über die neue Grundlagenkrise . . . “, I, § 2, GA II, 149 (trad. fr. in Intuitionnisme . . . , 73) ; ,,Die heutige Erkenntnislage . . . “, II, GA II, 518 (trad. fr. in Le continu . . . , 142). 71 Op.

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scindées de la possibilité effective de définir les uns par une procédure constructive, et de trancher l’autre par une démonstration ostensive. Pourquoi donc ? Référons-nous à la théorie de Dedekind, qui consiste à donner la propriété structurale de tous les domaines continus en général : toute coupure du domaine opérant le partage en deux classes telles que tout élément de la première soit à gauche de tout élément de l’autre détermine l’existence d’un unique élément de l’ensemble74 . Weyl commente : au contraire d’un ensemble fini, qui est déterminable par énumération de ses éléments, un ensemble infini ne peut l’être que par une propriété caractéristique générale de ses éléments ; mais le problème réside alors dans la correspondance entre cette propriété conçue en intension et la donation extensionnelle des éléments qui la vérifient : comment la propriété générale permet-elle de déterminer si tel nombre appartient ou non à l’ensemble ainsi défini75 ? Plus encore, considérer la totalité des nombres réels comme existante implique que l’on puisse considérer comme existant tout sous-ensemble de réels, sans qu’on en ait indiqué de procédure effective d’obtention (par exemple, l’ensemble des réels irrationnels comportant dans leur développement décimal la suite 123456789) ; la position d’existence en soi de tous les sous-ensembles possibles de l’ensemble des nombres réels est ainsi admise a priori, en dehors de toute procédure de construction ou d’atteinte effective76 . La conviction prévaut donc que le continu est un ensemble actuellement infini de points existant en soi, sur lequel il est loisible de découper des sousensembles infinis quelconques – qui nécessairement existent, puisque le tout sur lequel ils sont prélevés existe lui-même. À ce réalisme de l’infini actuel Weyl substitue, dans le sillage de Brouwer, la conception idéaliste du continu comme milieu de libre devenir (Medium freien Werdens)77 : loin d’admettre l’existence en soi de l’ensemble infini non dénombrable des nombres réels, il s’agit de définir chaque nombre réel comme une suite d’intervalle emboîtés dont la mesure tend vers 0, c’est-à-dire par une procédure effective d’encadrement par une suite indéfinie et convergente d’intervalles – qui peuvent être déterminés par une loi, par exemple [(m-1)/2h , (m+1)/2h ]. Tout élément doit donc pouvoir être défini par un procès constructif d’approximation qui peut être poursuivi au-delà de toute limite, conformément au principe d’itération qui déjà valait pour la formation de la suite des entiers, mais qui est ici appliqué à une formule ou loi d’engendrement ; le nombre réel est par conséquent un nombre qui ne peut être donné que de façon approximative (approximativ gegeben), par approximation indéfinie78 – on retrouve ainsi, dans le concept de nombre réel, l’anexactitude ou le caractère approximatif qui appartient à l’essence du continu intuitif.

74 D EDEKIND , Stetigkeit und irrazionale Zahlen, § 3, GMS III, 322–323 (trad. fr. in La création des nombres, 72–73), et l’excellente Introduction de la traductrice, p. 46 sqq. 75 W EYL , ,,Die heutige Erkenntnislage . . . “, II, GA II, 518 (trad. fr., 143). 76 Ibid. 77 W EYL , ,,Über die neue Grundlagenkrise . . . “, II, § 1, GA II, 151 sqq. (trad. fr., 75 sqq.) ; ,,Die heutige Erkenntnislage . . . “, IV, GA II, 531–532 (trad. fr., 152–153). 78 W EYL , ,,Über die neue Grundlagenkrise . . . “, II, § 1, GA II, 152 (trad. fr., 76).

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Au-delà des nombres réels singuliers, le continu est défini comme « milieu de libre devenir » (Medium freien Werdens), c’est-à-dire domaine d’une libre suite de choix : une suite non plus définie par une loi, mais engendrée pas à pas par une libre séquence d’actes libres de choix – « suite en devenir » (eine werdende [Folge]), procès d’engendrement relevant d’une conception dynamique et non statique, et impliquant le primat du devenir sur l’être79 . Le continu est ainsi un domaine d’effectuation où tombent les nombres réels, sans jamais s’identifier à un ensemble infini non dénombrable de nombres80 . Outre le rejet de l’infini actuel comme étant hors de portée des capacités d’atteinte de la conscience finie, quel est le fondement du rejet de la conception ensembliste du continu ? Ce rejet tient une exigence : en dépit de la distinction eidétique entre continus sensible et mathématique, la conceptualité de l’Analyse doit cependant s’adapter à la « nature figurée et sensible du continu », c’est-à-dire exprimer sur le plan mathématique la propriété de cohésion, d’interpénétration ou d’écoulement qui caractérise les continua sensibles (temps, étendue et mouvement chez Aristote). Or, toute décomposition atomistique du continu en points ou assimilation à un ensemble de points laisse justement échapper cette essence intrinsèquement fluente, dynamique et indécomposable du continu : il existe donc une inadéquation foncière de la conceptualité ensembliste à l’essence spécifique du continu. Aussi est-il nécessaire, en vue d’une expression mathématique adéquate du continu, de renoncer à toute assimilation de ce dernier à un ensemble de points, et par conséquent de cesser de vouloir le constituer à partir de la relation fondamentale d’appartenance d’un élément à un ensemble : Comme les points ( . . . ) ne forment pas un ensemble déterminé et délimité en soi [keinen in sich bestimmten und begrenzten Inbegriff bilden], c’est un contresens que de vouloir bâtir la géométrie sur cette catégorie primitive d’objets, d’une manière analogue à celle dont on a procédé ici en édifiant l’Analyse sur le fondement du concept de nombre réel81 .

À la relation d’appartenance doit se substituer, à titre de catégorie primitive de la pensée, la relation méréologique de partie à tout, c’est-à-dire d’inclusion d’un sous-ensemble ou d’un intervalle dans un ensemble : « même si nous le concevons de manière mathématique, nous devons partir non des points, mais des intervalles » (nicht von den Punkten, sondern von den Intervallen ausgehen)82 . Au lieu d’être composé de points, le continu a des parties, et il est par conséquent nécessaire de partir des intervalles qu’il contient ainsi que de leurs rapports d’inclusion et d’emboîtement ; aussi n’admet-on plus d’ensemble actuellement infini de points existant en soi, mais seulement un milieu où l’on puisse librement engendrer des suites indéfinies d’intervalles emboîtés. Cette constructibilité fournit l’expression mathématique d’une propriété essentielle du continu intuitif : à savoir que « chacune 79 Op.

cit., II, § 1, GA II, 152 (trad. fr., 76). cit., II, § 1, GA II, 153 (trad. fr., 77). 81 Op. cit., I, § 2, GA II, 151 (trad. fr., 75). 82 Op. cit., II, § 4, GA II, 173 et 177 (trad. fr., 99 et 103). 80 Op.

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de ses parties se laisse diviser à l’infini » ou qu’il est « quelque chose qui est à l’infini en devenir vers l’intérieur » (ein nach innen hinein ins Unendliche Werdendes)83 .

6.6 CONCLUSIONS Que conclure de ces considérations ? S’il existe des critères permettant de caractériser comme phénoménologique une position en épistémologie des mathématiques, ils se résument dans les termes de Husserl à deux : d’une part le retour au sol originaire de l’intuition logicomathématique, d’autre part la mise en évidence des sources phénoménologiques des concepts fondamentaux. On aurait cependant tort de replier d’emblée ce retour à l’intuition comme une adhésion à l’intuitionnisme brouwerien. Tout d’abord, loin de fonder la pensée arithmétique sur l’intuition inframathématique de multiplicités sensibles (temporelles ou spatiales), Husserl reconnaît la spécificité purement catégoriale de la pensée arithmétique, qui repose sur la formalisation et n’admet aucun contenu ni fondement dans la sensibilité : s’il est ici une forme d’intuition susceptible de donner les objets et d’attester les concepts, elle demeure rigoureusement intrinsèque au domaine d’idéalités considéré et doit procéder de façon strictement arithmétique, sans importation analogique depuis un domaine étranger – ce qui exclut d’emblée toute référence à l’intuition de la dyade et de la succession temporelles pour fonder l’arithmétique élémentaire. En outre, en assimilant la mathématique à une ontologie formelle visant à déterminer les formes dérivées du quelque chose en général et les formes de domaines associés à des formes de théorie déductive, Husserl ouvre la voie à une fondation ensembliste de l’arithmétique : la matrice noétique de l’arithmétique réside dans la formalisation élémentaire qui évacue tout contenu pour passer au quelque chose indéterminé. Enfin, loin d’exclure le point de vue axiomatique et le formalisme hilbertiens, il n’a cessé d’assigner pour telos à la mathématique la domination des formes de théories déductives cohérentes et des formes de domaines d’objets associées, et va jusqu’à accepter l’idée que le point de vue formaliste de la théorie de la démonstration hilbertienne soit d’obédience phénoménologique – en contradiction manifeste avec la thèse de l’idéalité de la signification. C’est dire que les notions centrales d’intuition logico-mathématique et de retour aux sources phénoménologiques ont des frontières fluentes. D’une part, si toute analyse intentionnelle doit être conforme au principe selon lequel toute catégories

83 Op.

cit., II, § 4, GA II, 177 et 172 (trad. fr., 103 et 99) ; cf. Das Kontinuum, II. Kap., § 6, p. 69 (trad. fr., p. 112).

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d’objets prescrit des structures régulatrices au sujet transcendantal84 , c’est chaque domaine d’objets idéaux qui doit prescrire au sujet les actes de pensée qui le rendent manifeste : s’il existe une normativité régissant les actes, elle doit provenir de l’essence du domaine d’objets considéré, et non des limites propres à la sensibilité ou aux actes de la conscience finie. D’autre part, si l’approche phénoménologique est d’essence réflexive et doit acquérir, par la réflexion, une « intuition intellectuelle du sens de l’activité mathématique »85 , elle est condamnée à suivre dans leur effectivité les procédés techniques et les inflexions historiques propres à la praxis mathématicienne, sans jamais vouloir leur imposer de l’extérieur des limites ou des normes : ce qu’est la conscience mathématicienne, c’est là une donnée historique dont la réflexion doit épouser la plasticité. Or, la position de Weyl nous semble fort éloignée de ces thèses husserliennes. Par opposition à Husserl, Weyl défend une thèse qu’à la suite de Vuillemin nous baptiserons intuitionnisme extrinsèque : la conceptualité mathématique étant censée s’adapter à l’essence du continu sensible (temporel, spatial et cinématique), l’activité mathématique est normée du dehors par un champ extramathématique dont elle doit approcher les structures ; ainsi doit-elle exprimer, sur le plan mathématique, la fluence caractéristique du continu intuitif. Cela implique une détermination de la pensée mathématique vis-à-vis d’un double dehors, celui des structures de l’intuition sensible, et celui de leur conceptualisation par la pensée antique, notamment aristotélicienne. D’un côté, loin de se construire à partir de rien, la praxis mathématique présuppose et admet pour fils conducteurs certaines structures de la réceptivité et de l’objectivité sensibles qu’elle s’efforce d’approcher à l’aide de ses outils conceptuels propres86 ; de l’autre, comme ces structures ont été pensées dans le savoir et la philosophie antiques qui a produit les concepts fondamentaux qui les expriment, la pensée mathématique se tient dans un débat de fond avec le savoir antique, notamment la Physique d’Aristote87 . En second lieu, la position de Weyl consiste en ce qu’on peut nommer un intuitionnisme eidétique : c’est en effet l’intuition eidétique du continu, la saisie de son essence et de ses propriétés essentielles (fluence, interpénétration), qui fournit à la pensée mathématique une norme lui interdisant, pour le penser, d’avoir recours aux procédés d’arithmétisation et de reconduction à des concepts ensemblistes, et lui imposant de substituer à la relation fondamentale d’appartenance la relation méréologique d’inclusion ; l’essence inframathématique de l’objet prescrit ainsi

84 H USSERL ,

Cart. Medit., § 22, Hua I, 90 (trad. fr., 99–100), Ideen . . . I, § 138, Hua III/1, 321 (trad. fr., 467), Ideen . . . III, § 7, Hua V, 36 (trad. fr. D. Tiffeneau, La phénoménologie et les fondements des sciences, Paris, Puf, 1992, p. 44). 85 Ibid. 86 H EIDEGGER , Die Frage nach dem Ding, § 18e, GA 41, p. 95 : « Acquérir une détermination plus précise du rapport du mathématique (pris au sens de la mathématique) à l’expérience intuitive de la chose donnée et à celle-ci, voilà qui demeure problématique. » (trad. fr. O. Reboul et J. Taminiaux, Qu’est-ce qu’une chose ?, Paris, Gallimard, 1962, p. 105). 87 Loc. cit., § 10, GA 41, p. 41 : « la science moderne ne devint possible que grâce à un débat prolongé ( . . . ) avec le savoir antique, ses concepts et ses principes. » (trad. fr., 52).

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à la pensée mathématique les voies méthodiques qui permettent de l’exprimer adéquatement. Cela n’est pas sans évoquer les prescriptions inhérentes à la mathesis ᾿ ια, qui interdit de procéder grecque, par exemple le principe d’immutabilité de l’oυσ´ au passage à la limite qui conduit du polygone inscrit dans un cercle à ce dernier88 . Or, n’y a-t-il pas là le risque d’une normativité eidétique excessive qui, délimitant de l’extérieur les possibilités de la pensée mathématique, lui interdit par décret d’emprunter des voies fécondes ? N’est-ce pas souvent la mise en rapport de domaines eidétiquement hétérogènes, et en apparence étrangers, qui s’avère féconde pour la pensée mathématique en lui permettant de ressaisir des structures générales qui sont communes à ces divers domaines89 ? N’est-ce pas également la production de formules étranges, en principe incompatibles avec l’essence d’un type d’objets donné, qui lui permet de produire de nouveaux concepts, et par là d’engendrer de nouveaux objets ? De fait, les limites intrinsèquement inhérentes à l’intuition s’avèrent difficiles à déterminer. Il suffit à cet égard de rappeler les oscillations théoriques de Weyl : en 1921, fidèle à Brouwer, il assigne à l’intuition les limites finitistes ou intuitionnistes qui lui interdisent tout passage à l’infini actuel et toute considération d’ensembles infinis comme des entités achevées ; en 1925 en revanche, il reconnaît que rien ne le sépare de Hilbert, que la mathématique s’étend bien au-delà de la zone d’intuition itérative de Brouwer, et que s’il n’est pas susceptible de donation, le transfini s’intègre néanmoins à des calculs non contradictoires qui permettent de le considérer comme étant quasiment effectif. Comment mieux reconnaître que les frontières de l’intuition d’essence sont inassignables ? Enfin, dans le même ordre d’idées, Weyl donne une interprétation normative du principe husserlien de l’idéalisme constitutif, de l’a priori de corrélation entre actes de la conscience pure et être effectif de l’objet : la constitution de nouveaux objets idéaux s’avère d’emblée arrimée aux capacités limitées de la conscience finie – notamment au principe d’itération ou à l’intuition dite originaire de la suite des entiers naturels, étroitement liée à la structure temporelle de la conscience d’objet : d’où l’exclusion de l’infini actuel, de l’axiome du choix, etc., jugés a priori incompatibles avec les capacités subjectives de construction. Les actes possibles de la pensée mathématicienne sont les seuls actes effectuables, et le champ de ce qui est effectuable est délimité par un ensemble de capacités natives du sujet : retour frappant au principe copernicien de Kant, selon lequel l’essence finie du sujet implique des limitations essentielles quant à ce qui est susceptible de se donner comme objet. Ici, c’est la structure de la succession temporelle de toute conscience finie qui fonde le principe d’itération et la limitation de principe à l’infini potentiel

88 Cf. D ESANTI , « Réflexions sur le concept de “mathesis” » in La philosophie silencieuse, Paris, Seuil, 1975, p. 201–203. 89 Cf. L. B RUNSCHVICG , Les étapes de la philosophie mathématique, § 272, Paris, Blanchard, 1993, p. 446.

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ou en devenir : on reste fidèle au principe kantien selon lequel « le temps est en soi une série (et la condition formelle de toutes les séries) »90 . Or, selon une critique conjointement adressée à la philosophie kantienne des mathématiques par Couturat puis Cavaillès, n’est-ce pas là confondre les conditions psychologiques de la schématisation et les fondements du concept mathématique, ou rabattre les représentations de conditions (de fondements conceptuels) sur les conditions (psychologiques) de la représentation91 ? Décisif s’avère en effet l’infléchissement normatif que Weyl impose à l’a priori de corrélation de Husserl : chez ce dernier règne un principe anticopernicien selon lequel c’est l’essence de l’objet qui impose au sujet son mode de visée et de donnée, sans présupposition de capacités innées ou naturelles de la conscience ; la voie était ainsi laissée ouverte à une historicité de la pensée mathématique où, loin d’être définies sub specie aeternitatis, les capacités de pensée et d’intuition pouvaient accompagner l’engendrement et la maîtrise théorique de nouveaux objets et champs, donc se caractérisaient par une plasticité historique au fil des transformations des champs d’objets. Revenons ici au parallèle avec l’écoute musicale : pas plus qu’on ne saurait décréter qu’une œuvre est inécoutable parce qu’en contradiction avec les capacités naturelles de synthèse des sons, on ne saurait postuler qu’un acte mathématique est à jamais ineffectuable, parce qu’incompatible avec les capacités noétiques du sujet fini ; de même qu’une œuvre musicale novatrice redéfinit les structures de la réceptivité esthétique, de même la position d’un nouveau champ d’objets (par exemple, la série cantorienne des aleph), d’un nouveau concept (par exemple, celui de puissance d’un ensemble), d’un nouveau procédé démonstratif (par exemple, la méthode de la diagonale) ou d’un nouvel axiome (par exemple, l’axiome du choix) implique, du côté subjectif, de nouvelles possibilités noétiques qui, une fois familières, deviennent des capacités naturelles ; le naturel n’est que de l’historique solidifié en structure.

90 K ANT ,

Kritik der reinen Vernunft, A 411/B 438 (trad. fr. DM, 380, AR, 420). COUTURAT, « La philosophie des mathématiques de Kant », in Les principes des mathématiques, Paris, Alcan, 1905, p. 287–292 et 301–302. J. CAVAILLÈS, « Transfini et continu », in Philosophie mathématique, Paris, Hermann, 1962, p. 272 (repris in Œuvres complètes de philosophie des sciences, Paris, Hermann, 1994, p. 470). Nous avons mis en évidence cette convergence des critiques de Kant dans « Le sens de l’antikantisme en mathématiques », Cahiers philosophiques de Strasbourg n◦ 26 (2009), p. 171–199.

91 L.

Chapter 7

From the Problem of Space to the Epistemology of Science: Hermann Weyl’s Reflection on the Dimensionality of the World Silvia De Bianchi

Abstract In analyzing the problem of space from 1917 to 1923, Hermann Weyl confronted with the philosophical underpinnings of the theories of space. Weyl endorsed the distinction between the question of the essence of space and the question of its objective representation, a distinction that many philosophers, such as Ernst Cassirer, inherited from Immanuel Kant’s philosophy. However, Weyl aimed to offer a reliable alternative to Kant’s transcendental idealism of space and time, by means of mathematics and symbolic construction. The consequences of this move will be analyzed in Weyl’s reflection on the epistemology of science after the 1920s and in his late works, with emphasis on his “Why is the World Four-Dimensional?” (1955): a signature of the fact that the problem of space had open questions that engaged the mathematical physicist throughout his entire life.

7.1 Introduction: From the Problem of Space to the Epistemology of Science This paper aims at showing that the philosophical implications of the problem of space are central not only in Weyl’s work until 1923 (Weyl 1918, 1921, 1922a, b, c, 1923) but also in his late writings (Weyl 1940, 1948, 1955b). This contribution will frame Weyl’s “Why is the World Four-Dimensional?” (1955) within the broader picture of his reflection on the problem of space, in particular within that of the

S. De Bianchi () Universitat Autònoma de Barcelona, Barcelona, Spain e-mail: [email protected] © Springer Nature Switzerland AG 2019 J. Bernard, C. Lobo (eds.), Weyl and the Problem of Space, Studies in History and Philosophy of Science 49, https://doi.org/10.1007/978-3-030-11527-2_7

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“new” problem of space (1921–1923).1 I shall do so, by clarifying the terms of one of Weyl’s guiding questions, which consists in showing how and why, among all the infinitely many conceivable metrical spaces, one applies to the physical world. In 1923, Weyl defined a Pythagorean metric on an n-dimensional manifold as given by the following quadratic differential form ds2 ≡

ij

gij (x) dxi dxj ,

  det gij = 0.

(7.1)

In 1923 Weyl searched for the essence of the metric field. Since the latter is one and absolutely determined, in it is expressed the a priori essence of the space-time structure (see Weyl 1923, p. 45).2 It is thus in Weyl (1923) that the reference to the notion of a priori essence of the space-time structure clearly emerges.3 This notion also plays an important role in Weyl’s late writings, but in that context it should be read also as a result of Weyl’s critical interpretation of Husserl’s Wesensanalyse (see Sect. 7.6.3 below). In searching for a philosophical theory of space-time, however, Weyl confronts not only with his contemporaries, such as Edmund Husserl and Ernst Cassirer, but also with previous philosophical debates. More in general, Weyl’s readings range from the classical works of Galileo to Fichte (see Sieroka 2007), including the philosophy of Leibniz (see Scholz 2012) and Kant (see Weinert 2005), as well as Descartes’ reflections on scientific methodology. For not to mention his interest for the classics, such as Plato and Aristotle. In both the first and the second edition of Philosophy of Mathematics and Natural Science, Weyl offers a clear picture of his extraordinary capacity of exploring the history of philosophy and the philosophical questions emerging from the foundations of mathematics, geometry and physics. When dealing with the problem of space and its implications, thus, Weyl confronts with a rich philosophical literature. It is not my intention to deal in detail with this literature in this paper. I shall restrict myself to one question that seems to me crucial in order to understand the philosophical implication of Weyl’s solution to the problem of space. I shall clarify the notion of essence that Weyl attributed to space-time in 1923 in order to show how this notion that Weyl takes from Husserl’s phenomenology changes afterwards and leads to new epistemological questions.4

1 For

the reconstruction of Weyl’s problem of space and the distinction between its first phase (1918–1920) and the second phase (1921–1923), see Scholz (2004). For a discussion on Weyl metric, see Scheibe (1957, 1988). 2 Thus, according to Weyl (1923), the nature of the metric (1) is its being a non-singular quadratic differential form, namely it is an orthogonal group Ok where k is the signature of the corresponding form (1). For the generalization of the quadratic form (1) to its automorphism group, see Scheibe (2001), p. 453. 3 Even if in the 1920s Weyl uses the expression “Wesen des Raumes”, he makes clear that he wants possibly to include time as a fourth dimension. This also happens in Weyl (1955b). 4 That in his late writings Weyl changes his mind with respect to embracing Husserl’s phenomenology is pointed out by J. Bell (2004). He also states that Weyl’s late works are closer to Cassirer, even if it must be recalled that Husserl himself in The Crisis assumes a position, which is closer

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7.2 The Philosophical Implications of the Problem of Space What are then, in Weyl’s view, the philosophical implications of the problem of space? They consist in an epistemological question and a metaphysical one. On the one hand, the resolution of the problem of space is intimately related to Weyl’s notion of objectivity in physics, namely with a process of progressive elimination of the ego from physical theories. Even if a subjective element is always present in them in the form of a coordinate system, the latter can be substituted by a more abstract and general tool, such as an automorphisms group. In (Weyl 1940, pp. 78– 79) the notion of symbolic construction as an objectifying process of natural science is clarified through the following steps: • we reduce the phenomena to simple elements (reduction). • each element varies over a certain range of possibilities (variation) • we can survey this range of possibilities because we construct them a priori (construction) • we construct them in a purely combinatorial fashion from purely symbolic material (symbolic construction). Furthermore, in Weyl’s view, the manifold of space-time points is the most basic of these constructive elements of nature. From an epistemological standpoint, according to Weyl, natural science proceeds via symbolic construction and one mode of symbolic construction is objectification. This means that, in principle, the essence of space-time is something flexible5 and it is not something given, but constructed: its characteristics can be replaced by more general ones thanks to the development of mathematics and the discovery of new physical laws. This not only means that for Weyl the historical development of mathematics and physics is crucial to find an explanation of the dimensionality of the world, but also that what we define as “essence” can change over time. And this leads to the second interesting philosophical implication of Weyl’s treatment of the problem of space, which is a genuine metaphysical question. To recognize that among different possible metrics only the Pythagorean metric embodies real physical space means to attribute to it a necessary status, i.e. that of an a priori essence. However, if this representation of space-time is necessary there must be a deep reason for it and to Weyl a clue can be offered by the understanding of the uniqueness of the Lorentz group. As long as the law imposing this metric is not found, one can appeal to divine action and will, to a Creator that establishes the metric and the dimensionality of space in this world.6 However, Weyl admitted the possibility of recognizing a double status

to Cassirer’s doctrine. That Weyl abandons Husserl’s phenomenology emerges in a clear way in Weyl (1955a). For a comparison between Husserl and Weyl, see Feist (2004). 5 On the philosophical underpinnings of Weyl’s notion of space, see Bernard (2015). 6 In the 1955 paper on the dimensionality of the world, Weyl changes his perspective with respect to his early-1920s writings, where he did not present the problem of justifying the Pythagorean metric through physics. For instance, about the physical laws and the problem of dimensionality

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to the Pythagorean metric, namely that it can be necessary and contingent at the same time: it is an essence that could be replaced by another one. To attribute the status of contingent necessity or necessary contingency to the dimensionality of the world does not rule out in principle theological and teleological arguments for the explanation of space dimensionality, but can lead us to the possibility of rethinking of the conditions under which natural laws “generate” space dimensionality.7 How could the four-dimensionality described by necessary physical laws be also contingent? Is this contingent status to be taken as the product of chance? These are salient aspects of Weyl’s philosophical reflection in his writings on the problem of space from 1923 onward and constitute a sort of engine for his mathematical and physical works on the dimensionality of space throughout his entire career. Indeed, in 1948 he formulates the philosophical problem of space in analogy with the philosophical problem of the essence: It is a fact that many of the known laws of nature can at once be generalized to n dimensions. We must dig deep enough until we hit a layer where this is no longer the case. Our question has this in common with most questions of philosophical nature: it depends on the vague distinction between essential and non-essential. Several competing solutions are thinkable, but it may also happen that, once a good solution to the problem is found, it will be of such cogency as to command general recognition. (Weyl 1948, p. 23)

It is worth noticing that contrary to his 1923 writing in which he claims to have found the essence of space-time and the essential character of the metric field, in 1948 Weyl critically uses the notion of “essence”. He argues that in dealing with the problem of explaining the four-dimensionality of the world, we are in a situation similar to that caused by the vagueness of the distinction between essential and non-essential in the philosophical debate. In analogy with it, he claims that further research in topology will explain why nature decided for a Pythagorean metric, and its essential character will disappear, by opening the question of other essential characteristics determined by fundamental physical interactions. Before showing Weyl’s treatment of the problem of space in the late writings, I shall outline the philosophical influences on Weyl’s reflections.

he says: “Hence in these laws there is no reason to be found for the Creator’s whim to fashion a 4-dimensional world as the scene of our activities. Of course, our present knowledge of the laws of the physical world is incomplete, and one day it might strike a deeper level on which dimensionality ceases to be indifferent, but at the moment this is merely a hope and not a fact” (see Weyl 1955b, p. 211). Also notice the difference with the position expressed in the fourth edition of Space, Time, Matter (see Sect. 7.6 below). 7 The question of the dimensionality of space and arguments related to its explanation are discussed in De Bianchi & Wells (2015). An interesting study concerning the electromagnetic generation of the Lorentz signature of the metric of space-time is Itin and Hehl (2004). Weyl (1921) provides a theorem showing that the space-time metric is already fully determined by the inertial and causal structure of space-time. For a study on the causal theory of space-time and its history, see (Winnie 1977). Important contributions are also Sklar (1974, 1977).

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7.3 The Essence of Space and the Objective Representation of Space The philosophical question of the nature of space engaged philosophers for centuries in the search for an answer that could harmonize physical phenomena and philosophical systems, as well as to characterize the foundations of mathematics. In developing his theory of space, for instance, Kant stressed the importance of the distinction between the notion of space as form of intuition, as a subjective a priori condition for outer experience, from that of space as formal intuition, as any objective representation of space, as it happens in geometry or in physics. In the latter, according to Kant, we represent real empirical or material space by means of a synthesis, which is a construction through concepts. In the Critique of pure Reason, Kant claimed: Space, represented as object (as is really required in geometry), contains more than the mere form of intuition, namely the comprehension of the manifold given in accordance with the form of sensibility in an intuitive representation, so that the form of intuition merely gives the manifold, but the formal intuition gives unity of the representation. In the Aesthetic I ascribed this unity merely to sensibility, only in order to note that it precedes all concepts, though to be sure, it presupposes a synthesis [ . . . ]. (Kant 1787, B160–161 footnote)

With the rising of general relativity theory, it became clear that it was necessary to distinguish the physical from the philosophical question concerning the essence of space. Thus the question “What is space?” could have been read in different ways. Among philosophers, mathematicians and physicists arose the awareness that the ontological questions of the essence of space, be it an absolute real entity, an a priori form of intuition, or net of relations, was a completely different question from the determination of the properties of physical space as an operational concept in a physical theory. Cassirer, for instance, endorsed Kant’s distinction between the question of the nature of space-time and the genuine ontological question of the essence of space and time. In his Einstein’s Theory of Relativity (1923), he raises the following question: What space and time truly are in the philosophical sense would be determined if we succeeded in surveying completely this wealth of nuances of intellectual meaning and in assuring ourselves of the underlying formal law under which they stand and which they obey. The theory of relativity cannot claim to bring this philosophical problem to its solution; for, by its development and scientific tendency from the beginning, it is limited to a definite particular motive of the concepts of space and time. As a physical theory it merely develops the meaning that space and time possess in the system of our empirical and physical measurement. In this sense, final judgment on it belongs exclusively to physics. (Cassirer 1923, p. 456)

In this passage, Cassirer clearly differentiates the philosophical question of the essence of space and time from the space-time structure in physics. At the beginning of the 1920s, he enthusiastically accepts the results of Einstein’s relativity and tries to show that they were not in contrast with philosophical reflections on the ontology of space. The problem rather was to find out a way to harmonize the two in a new

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system, and Cassirer’s system of symbolic forms should have provided precisely such a harmonization. Weyl was familiar with Cassirer’s view of space and agreed to a certain extent with him, as testified even in Weyl’s late works (see for instance Weyl 1955a).8 However, even if Cassirer’s philosophy attracts Weyl’s attention, it is Husserl’s philosophy that exerts a substantial influence on his reflection on the epistemology of the problem of space. Before entering into details, it is worth noticing that in the correspondence between Weyl and Husserl (1918–1921), it emerges that Weyl is genuinely interested in solving the problem of space form a physical and mathematical perspective, but also in developing a philosophical reflection that can be harmonized with that of phenomenology. The literature tends now to distinguish Husserl’s view of the problem of space, Weyl’s interpretation of Husserl and the role played by Oskar Becker in informing Weyl’s received view of Husserl (see Lobo 2009; Mancosu and Ryckman 2002). Even if in the 1930s and the 1940s Weyl is not enthusiastic of Becker’s work, this does not exclude, however, that Weyl decides to go back to Husserl’s philosophy on a later time and reflect upon it, albeit critically. For the scope of the present paper, it is sufficient to consider that Weyl has always been impressed by Husserl’s attempt at constructing a bridge connecting space, time and intentions, namely Weyl appreciated Husserl’s attempt at finding an approach that could have embraced the treatment of perception and its role in the symbolic construction of physical theories. This theme, which is present in the 1922 edition of Space-Time-Matter returns in Weyl’s writing also in the 1940s and, in my view, this is Weyl’s attempt to read Husserl’s philosophy once more, this time within the framework of the development of relativity and quantum theory. However, as we shall see, this attempt leads to a criticism of Husserl’s notion of eidetic variation and his definition of morphological essences.

7.4 Weyl on the Philosophical Problem of Space Before 1940 In the Introduction to Space-Time-Matter, Weyl refers to Kant and Husserl’s idea of the connection of space and time with matter.9 This connection is thought thanks to the concept of motion: Space and time are commonly regarded as the forms of existence of the real world, matter as its substance. A definite portion of matter occupies a definite part of space at a definite moment of time. It is in the composite idea of motion that these three fundamental conceptions enter into intimate relationship. [ . . . ] In the field of philosophy Kant was the first to take the next decisive step towards the point of view that not only the

8 In

(Weyl 1955a), it is argued that if besides physical space one recognizes an intuitive one endowed with an Euclidean structure, this does not necessarily contradict our physical insight, because the latter also holds to the validity of Pythagoras theorem in any infinitely small neighborhood of a point O in which the self is momentarily located. 9 This reference is present also in the fourth edition, see (Weyl 1922a).

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qualities revealed by the senses, but also space and spatial characteristics have no objective significance in the absolute sense; in other words, that space too is only a form of our perception. (Weyl 1922a, p. 3)

Weyl’s epistemological stance is clear: “space, like time, is a form of phenomena” (Weyl 1922a, p. 11; see also p. 141) and can be ascribed to the aprioristic or idealistic conceptions of the essence of space and time. In the realm of physics the theory of relativity clarifies that the “two essences, space and time, entering into our intuition have no place in the world constructed by mathematical physics”. To advance such a statement is possible on the ground of Weyl’s acceptance of the distinction between the question of the nature of space and time, which is a formal one, and the physical real determination of space-time. But how do we reconcile Weyl’s 1918 definition of space and time with those offered from 1930 onwards? Scholz (2011) noticed that in the 1920s Weyl conceived of space as dependent on its material content, but in itself as being free and capable of any virtual changes (“first postulate of freedom”). In this way, Weyl defined the nature of space in such a way that G allows the “widest conceivable range of possible congruence transfers” in one point. This postulate played a crucial role for the definition of the kinematics of quantum physical systems, as well as for their dynamics. In place of free mobility of rigid bodies, Weyl put the idea of a free distribution of matter (Scholz 2011). Weyl demanded from the group G that it took care of a certain coherence of the infinitesimal geometric structure. Such a coherence condition was best expressed by the existence of a uniquely determined affine connection among all the metrical connections which could be generated from one of them by arbitrary infinitesimal rotations at every point.10 According to Ryckman (2005), precisely the postulate of freedom and the requirement that the metric univocally determine the affine connection are fundamental steps allowing Weyl to solve the problem of space in 1923. Now consider what Weyl claimed in the 1930s. In January 1931, Weyl published the article Geometrie und Physik in Die Naturwissenschaften, first given as the Rouse Ball Lecture in Cambridge in May 1930. In this chapter he argues: Space and time are, as Kant said, forms of our intuition. The coordinates are already there to distinguish one from the other the places in the continuum of space and time. They play the same role as names do, through which people can be distinguished from each other and can be named, or as an arbitrary numbering of objects in an object domain that consists of discrete elements. (Weyl 1931a, p. 5)11

In this respect, Weyl’s philosophical conception can be rightly framed within a transcendental approach to the role of space and time. Should we then conclude that 10 According

to Ryckman (2005, p. 155), Weyl followed upon the Helmoltz-Lie tradition when searching for the uniqueness of the quadratic metric determination in an n-dimensional differentiable manifold M, by treating congruence through a continuous group of motions. 11 Translation is mine, the original German text reads: ,Raum und Zeit sind, wie Kant sagt, Formen unserer Anschauung. Die Koordinaten sind dazu da, die Stellen im Kontinuum von Raum und Zeit voneinander zu unterscheiden. Sie spielen die gleiche Rolle wie die Namen, durch welche Personen voneinander unterschieden und nennbar gemacht werden, oder wie eine willkürliche Nummerierung der Objekte in einem aus diskreten Elementen bestehenden Objecktbereich”

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for Weyl space is an a priori form of intuition and possesses a determinate structure because of a process depending on the inner sense of intuition? The answer is no. Things appear to be subtler. In the 1940s Weyl explicitly claims to have found an alternative to Kant’s transcendental idealism of space by means of his conception of symbolic construction and by emphasizing the role of similarity and congruence in constituting the fundamental structure of space-time. In other words, Weyl’s conception of symbolic construction overcomes the distinction between the question of the essence of space and time and their physical representation, space-time, by replacing the coordinate system with a “higher” standpoint that is a topological one (See Sects. 7.5 and 7.6). Therefore, in the late 1920s the successful application of the group-theoretic approach to quantum mechanics changes Weyl’s way of looking at the philosophical problem of space and leads him to reconsider Kant’s doctrine of space in a more refined fashion. In the 1940s Weyl rejected Kant’s transcendental idealism of space and time, by developing an epistemological approach to similarity and congruence. Weyl also emphasized that this approach is also different from Husserl’s phenomenology and the phenomenological interpretation of Kant’s transcendental idealism. However, for the sake of clarity, it should be recalled that Weyl reaches such a result, certainly not as a philosopher, rather as a mathematical physicist.

7.5 Towards an Alternative to Transcendental Idealism In a manuscript dated 194812 “Similarity and congruence: a chapter in the epistemology of science”, ETH-Bibliothek, University Archives, Hs 91a:31, p. 12), Weyl discusses the problem of congruence and similarity with respect to the history of the debate between Clarke and Leibniz on incongruent counterparts and the nature of space. He then proceeds in showing the nature of similarity and congruence in the construction of Einstein’s relativity theories and how these have been modified by means of the development of group theory. In reconstructing what has been the big shock brought about by general relativity theory, Weyl aims at reconciling the deformability of space-time and the fact that the metric structure of it remains the same: The metric structure, and the inertial structure derived from it, exert a powerful influence upon all physical phenomena. But what acts must also suffer. In other words, the metric structure must be conceived as something variable, like matter and the electromagnetic field, which stands with all other physical quantities in the commerce of interaction: it acts and suffers reactions. Only by admitting the metrical field as a variable physical entity among the other physical quantities can the principle of general relativity be carried through. (Weyl 1948, p. 16)

12 The

date is probably 1948, because in the manuscript Weyl mentions that he is writing just 30 years after he presented his theory unifying electromagnetic and gravitational potential, which was in Raum-Zeit-Materie (1918).

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Weyl observes that the Euclidean group of rotations has survived even such radical changes of our concepts of the physical world as general relativity and quantum theory. What then are the peculiar merits of this group to which it owes its elevation to the basic group pattern of the universe? (Weyl 1948, p. 22). Why did God (or Nature) have chosen this group and no others? Weyl has shown that the group of linear transformations that leaves a non-degenerate quadratic form invariant is the only one that ties affine connection to metric in the manner so characteristic for Riemannian geometry and Einsteinian gravitation. The number of dimensions of the world, however, is 4 and not an indeterminate n. In fact, the structure of the group 0 is quite different from the various dimensionalities n. Hence, according to Weyl, it is the group that serves as a clue to discover the reason for the world dimensionality. Thus, the distinctive character of the four-dimensional Lorentz group, either as a group of linear transformations or as an abstract group, must be clarified.13 In Weyl’s view, the problem is not only mathematical. Weyl asks whether another group can replace the four-dimensional Lorentz group in the construction of natural laws by also sharing with the known natural laws those features that are essential for the constitution of physical world and for describing the homogeneity of the four-dimensional world. Weyl’s question appears to be immediately fundamental, because “one cannot claim to have understood nature unless one can establish the uniqueness of the four-dimensional Lorentz group” (see Weyl 1948, p. 23). This question is explicitly posed in 1948 and will be raised also in (Weyl 1955b). However, in Weyl (1948) there is clear account of the way in which he aims at proceeding without endorsing any kind of transcendental idealism of space and time, thus by producing a rupture with the previous works: The divergence between congruence and similarity has often puzzled the philosophers, so Kant. In §13 of his Prolegomena zu einer jener künftigen Metaphysik he claims that ‘by no single concept, but only by pointing to our left and right hand, and thus appealing to intuition (Anschauung) can we make comprehensible the difference between similar yet incongruent objects (such as oppositely wound snails)”. And he is inclined to think that only transcendental idealism is able to solve this riddle. No doubt, the meaning of congruence and similarity is founded in spatial intuition. Kant seems to aim at some subtler point. But just this point is one which can be completely clarified by general concepts, namely by subsuming it under the general and typical group-theoretic situation explained before: a group  of transformations and its normalizer , or a group  and its invariant sub-groups . (Weyl 1948, p. 12)

This means that Weyl aims at offering an alternative to Kant’s transcendental idealism of space, namely he offers an alternative to the Kantian mathematical construction of material space, i.e. matter. Weyl was able to do so, only after the development of his group-theoretic approach to quantum mechanics and the refinement of his notion of gauge invariance in 1929. In a sense, Weyl realizes that what Kant explains as the result of a process of construction of a formal intuition could be viewed as inherent properties subsumed under the concept of real physical space on the ground of a priori principles, i.e. on the ground of invariance. The a 13 For

Weyl’s application of the Lorentz group to quantum mechanics, see Weyl (1931b, p. 147).

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priori principles do not assume the same sense as they did in Kant’s philosophy, even if they can be interpreted in a transcendental fashion.14 They are meant to provide the a priori methodology, the rules of the possible operations that could have been carried over the geometrico-physical objects by means of what Weyl calls “systematic representation”, namely the analysis of their group-theoretical properties. But why is Weyl (1948) so relevant for the present discussion on the problem of space? Weyl concludes his essay precisely with the question of the dimensionality of the world and with the epistemological questions related to the status of the laws of nature determining the four-dimensionality of space-time in such a way that establishing the uniqueness of the four-dimensional Lorentz group assumes a crucial relevance for the development of physics. Thus, contrary to the way in which the question of the dimensionality of the world is presented in current debates (see for instance McCall 2006, pp. 191; 199–200) the problem is not to prove whether 3D and/or 4D are the correct description of the world or whether they are compatible with each other. As Weyl pointed out in 1948 and in the second edition of Philosophy of Mathematics and Natural Science, the crucial question is to explain the uniqueness of the four-dimensional Lorentz group (Weyl 1949, p. 137), by looking for laws of interaction justifying it. Weyl’s approach, indeed, clearly suggests a representation of space-time as varying precisely in the same way in which matter and electromagnetic field do: space-time acts and suffers, it is a mutual interacting structure itself. On the ground of these observations, it can be advanced the hypothesis that the 1948 writing constitutes a preliminary work also to Weyl’s (1955b). I shall come back to the way in which Weyl treats this question in the manuscript in Sect. 7.6.2, when I compare it with (Weyl 1955b).

7.6 Weyl and the Problem of the Dimensionality of Space Now we arrive at the central topic of this paper with a refined view of the philosophical problem of space in Weyl’s late work. As I tried to show, Weyl does not reach an unequivocal solution to the philosophical problem of space in 1923, rather he continues working on it until the end of his career. In “Why is the world four-dimensional?” Weyl offers an explanation of why we are not yet able to justify the 4-dimensionality of space-time. As shown above, Weyl was not only interested by the question of the Pythagorean nature of the metric in the 1920s, but also specifically on the question of the dimensionality of space. Of course, he was already searching for a specific reason for the dimension 4 of space-time in SpaceTime-Matter. His argument is reported by Barrow and Tipler as follows:

14 Even

though Weyl is aware of a certain affinity with them. This awareness is mostly based on Hilbert’s interpretation of Kant’s transcendental ideal of the pure reason with respect to the systematic unity of science, see Weyl (1930, p. 28).

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Weyl pointed out that only in (3+1) dimensional space-time can Maxwell’s theory be founded upon an invariant integral form of the action; only in (3+1) dimensions it is conformally invariant and this ‘does not only lead to a deeper understanding of Maxwell’s theory, but the fact that the world is four dimensional, which has hitherto always been accepted as merely ‘accidental’ becomes intelligible through it’ (Space, Time, Matter 1919 p. 284). (Barrow and Tipler 1986, pp. 260–261)

It should be immediately noticed, however, that the argument that Weyl develops in the 1940s and that explicitly supports in 1955b is different from this one. In what follows, I shall first reconstruct his argument and then will comment on its main features in order to expound Weyl’s philosophical doctrine of space in the mid1950s.

7.6.1 The Structure of the Argument Contrary to what is said in Space, Time, Matter, in 1955 Weyl immediately admits that he is unable to offer an answer to the question “why is the world fourdimensional?”, and rather aims at showing why scientists have been unable to answer. He distinguishes the question of the dimensionality of the world, which includes time, from the question of the dimensionality of space. We have thus a first epistemological distinction: whereas the solution to the problem is not available epistemically in the case of the world, it is not the case for the three-dimensionality of space. That space is epistemically “open” lies on the fact that according to Weyl it can be an object of symbolic construction responding to perfectly known rules. Now, there are two possible interpretations for the statements that a line is 1-dimensional, a plane is 2-dimensional and a space is 3-dimensional. The first is possible in terms of elementary geometry or special relativity, whereas the second in terms of infinitesimal geometry or general relativity. Weyl expresses the metrical structure of Euclidean space as an infinitesimal law: In terms of certain distinguished coordinates, the Cartesian ones ξ, η, ζ, the square ds2 of the distance of two infinitely near points P, P with coordinates ξ, η, ζ, and ξ + dξ, η + dη, ζ + dζ, respectively, is given by ds2 = dξ2 + dη2 + dζ2 (Pythagoras). (Weyl 1955b, p. 206)

Afterwards Weyl replaces infinitesimal quantities such as the differentials dξ, dη by differential quotients in order to avoid the criticism of those rejecting the use of infinitesimal quantities and proceeds in clarifying the Riemannian manifold. Thus any surface in 3-dimensional Euclidean space is a 2-dimensional Riemannian manifold: The 3-dimensional Euclidean space is completely described as a flat Riemannian manifold of dimensionality 3. Indeed all concepts and factors of Euclidean geometry can be defined in terms of, and can be derived from this description. The lawfulness of our space is of such nature that it carries over in an absolutely cogent manner from 3 to any number of dimensions, simply because the notions of quadratic form, of positive-definite quadratic form and of unit form, are clearly not limited to 3 variables. (Weyl 1955b, p. 208)

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Weyl then mentions a typical theological feature of some arguments concerning the dimensionality of space, namely he mentions the question of the reason why God created and modelled actual space in the way he did and concludes that the choice of 3 dimensions appears as a completely arbitrary (contingent) act, because the law describing the structure of space gives no reason for its having 3 rather than 1 or 129 dimensions. Of course, Weyl is not happy with such a weak argument and goes on in evaluating the fact that in the world events happen in space and time, therefore in including time in the discussion. He starts with Minkowski’s definition of the world, according to which it is “the totality of space-time-points, of possible localizations in space and time”, so that having space 3 and time 1 dimension, the world is four-dimensional. In describing the departure from Newton’s absolute space and time, Weyl proceeds in showing the origin of the representation of space in special relativity related to a new formulation of the classical Euclidean geometry. However, from the perspective of the present paper, the case of general relativity theory is more interesting. In general relativity, according to Weyl, Einstein replaced the flat space-time manifold by a more general Riemannian 4-dimensional manifold of index 1; namely by a world with a metric expressed in terms of arbitrary coordinates x1 , . . . x4 by a quadratic differential form ds 2 =



gij dx i dx j

(i, j = 1, 2, 3, 4)

(7.2)

of index 1 with non-constant and a priori unpredictable coefficients gij that determine inertial motion. The fact that gravity of a mass point is always proportional to its inertial mass become understandable if gravitation is considered as the changing part of inertia, so that in the juxtaposition of inertia and force, gravitation belongs to the side of inertia inseparably and not to that of force.15 In Weyl’s view, the success of this move determined the abandonment of the fixed Euclidean metric of the 4dimensional world and led us to replace it by a variable Riemannian metric. Thus, Weyl’s point is that the notion of Riemannian manifold is not bound to the special dimensionality 4, and as far as this metrical structure goes, any other dimension would go. Therefore the question “Why then 4?” rests unanswered. To look at the metrical structure of space is insufficient, we have to look, Weyl suggests, at the laws of the physical phenomena that take place on this stage and ask whether they can be carried out in the same unambiguous way from 4 to any other number of dimensions and we know that they all allow generalization to n dimensions. Thus, there is no reason, until we do not discover other physical laws that work differently in specific dimensionalities, to think that there is something special in the Creator’s will or in the Nature’s choice of modelling a 4 dimensional world. However, Weyl tries to introduce a possible explanation for the dimensionality of space. Things can be clarified by looking at Maxwell’s electromagnetic theory and to Dirac relativistic quantum mechanical equations for the electron. The index allowing the propagation of an effect responds to a topology of causation and must assume a special value 15 For

further details, see Coleman and Korté (1984).

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r = 1, because in a world of index 0 there is no propagation of effect, in a world of index higher than 1 past and future form one connected domain. Only if the index is 1 the interior of the light cone exists and falls apart into 2 disconnected regions: passive past and active feature. Whether a domain is connected or consist of several disconnected parts is a topological difference. Hence in the case r = 1 the world is topologically distinguished by a very simple and decisive feature from all other cases. This is for Weyl the reason for the index of r = 1. Weyl then analyses the case of Dirac and argues that quantum mechanics and its statistical interpretation force the relation e = r + 1 upon a prospective world-builder. After he made r = 1 because of the topology of causation, he has no choice but to take e = 2. These are in Weyl’s view strong arguments for the values r = 1 and e = 2 of inertial index r and of rank e of the electromagnetic field. However, Weyl continues: I wish I could close here, but I am afraid that would be thoroughly dishonest”. It is true that the laws of electromagnetic field in empty space presuppose only a conformal structure for the world, they are invariant with respect to the replacement of the gij defining the metric by · gij . But when we pass to the non-homogenous Maxwell equations which describe the generation of such fields by matter, or to Einstein’s laws of gravitation even in empty space, this is no longer the case. The field laws are not conformally invariant neither for the gravitation field nor for the electromagnetic field in the presence of electric charge and current. (Weyl 1955b, p. 213)

Then Weyl appeals to recent mathematical research that in his view will lead us to an answer for the question of the dimensionality of the world. He wants a deeper reason for why the world has this metrical structure, i.e. the Riemannian structure. In particular the question should be posed as follows: why is the orthogonal group among all groups of homogenous linear transformations the one characterizing the local metric of the world, and why is this metric Pythagorean? A positive-definite quadratic form describes the metric of the 3-dimensional Euclidean space, but this does not hold in Weyl’s view for the metric of the 4-dimensional world, which rather depends on a form of the inertial index 1. Furthermore, the metric of the space-time continuum uniquely determines the gravitational field. Both Helmoltz’s and Weyl’s characterizations of the orthogonal group hold for every number of dimensions and laws are indifferent towards the number of dimensions, but to any group of transformations one can ascribe a structure, which is represented by the corresponding abstract group. The structure of the orthogonal group turns out to be quite different for different dimensions n. Among all orthogonal groups only the four dimensional orthogonal group has a more complex structure, being the direct product of two simple groups. The orthogonal group for a positive definite quadratic form is closely related to the notion of sphere. And Weyl thinks that it will come from topology a solution to this question because it will reveal the elementary and highly significant differences in the behavior of spheres of different dimensions. Unfortunately, Weyl notices that research in mathematics of this deepest level in which properties of spheres change from dimension to dimension is still unable to provide a definite answer. The hope is that “one day physics will discover laws of nature, the mathematical formulation of which takes into account of such structural features as are highly sensitive to dimension. Only this will allow us to explain and

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understand the specific character of the world actual dimensionality 4” (Weyl 1955b, p. 215).

7.6.2 Causality and the Dimensionality of Space: The Role of Topology The first thing to notice is that Weyl distinguishes between the notion of space and that of the world, on the ground of a mathematical consideration. The new problem of space in the 1940s consists for Weyl in defining the principle or the rules according to which independently from the generalization of the laws of physics to n-dimensions, a certain index and thus the four-dimensionality of the world arises. The mathematical underpinning to this question emerges in a clear way when he talks about the topology of causation understood in a pure mathematical way: Locally, that is to say in the infinitesimal neighborhood of a world point P, the equation of the light cone ds 2 =



gij dx i dx j = 0

i,j

describes the world’s causal structure. Knowing the causal structure, we do not know the gij themselves but only their ratios. The causal structure is not changed when the gij defining the metric are replaced by λ · gij with a positive factor of proportionality λ which is an arbitrary function of P. [ . . . ] It would not seem unreasonable to assume that the world is endowed not with a metric, but with a conformal or causal structure only, in other words that only lengths of line elements at the same point are comparable to each other. Then the laws of nature would not be affected by the modification gij → λ · gij of the metrical field. Now it is a fact that the equations characterizing a harmonic linear tensor formula are invariant under this substitution if and only if n = 2e, i.e. if the dimensionality n is twice the rank e of the electromagnetic tensor f. If r = 1, e = 2 we would thus obtain the desired n = 4. (Weyl 1955b, pp. 212–213)

Causality is a conformal structure of the world prior to the dimensionality of space and this structure preserves the laws of physics invariant under the modification of the metrical field.16 We know that Weyl pointed to the difficulties deriving from the fact that not all the physical systems seem to be reducible to conformal structures, but the passage above gives us a hint of Weyl’s conception of causality implied by his philosophical and mathematical treatment of the problem of space. In his view, the progressive erasing of the “subjective” from a physical theory expresses the inevitable objectification of natural science by means of symbolic construction.17 According to Weyl, space-time can be treated as an emergent structure originated

16 For

a study on causal topology as a theory of causal relation and the Zeeman Theorem (1964) according to which causality implies the Lorentz group, see Mittelstaedt and Weingartner (2005, pp. 241–243). See also Bergmann (1992, p. 81). 17 For a clear account of Weyl’s symbolic construction, see (Majer 1998).

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by the conformal or causal structure of the world, rather than by its metric. Already in The mathematical way of thinking (1940), Weyl clarified what is for him the causal structure of the universe: At the ground of the words past, present and future referring to time we find something more tangible than time, namely the causal structure of the universe. Events are localized in space and time; an event of small extension takes place at a space-time or world point, a here-now. (Weyl 1940, pp. 70–71)

This conception of the causal structure depends on the assumption that no effect may travel faster than light (Weyl 1940, p. 71). In Einstein’s representation of the light cone the active future is bounded by the forward light cone and the passive past by its backward continuation, so that “active future and passive past are separated by the part of the world lying between these cones, and with this part I am here-now not at all causally connected” (Weyl 1940, p. 71). Before analyzing Weyl’s argument in favor of topology for revealing the causal structure of the universe and the essence of space-time, let us go back to the question of whether Weyl’s definition of causal structure points to mark a difference with Husserl’s phenomenology. According to Weyl, indeed, “the causal structure is not a stratification by horizontal layers t = const., but active future and passive past are of cone-like shape with an interstice between” (Weyl 1940, p. 72). In other words, the part of the world in between the active future and the passive past of the light cone is not in a causal relation with the origin (here-now), but it is in causal relation with the event-points within the cone.18 According to Husserl, causality is a “functional or lawful relation of dependence which is the correlate of the constitution of persistent properties of a persistent real something of the type, nature” ([HUA IV.132] Ideas II, § 32). In The Crisis Husserl also offers the following definition: “Causality if we remain within the life-world . . . .has in principle quite a different sense depending on whether we are speaking of natural causality or of causality among soulful (seelische) events or between the physical bodily and the soulful (Husserl, The Crisis, § 62). Ryckman (2005, pp. 150ff.) analyzed the role played by topology in Weyl’s work, but to my knowledge he does not analyze (Weyl 1948) and (Weyl 1955b) in relationship with Husserl’s philosophy. For the purpose of the present article it is sufficient to point out that, differently from Husserl and from what Weyl proposes in (Weyl 1922a, pp. 3–4), the development of the notion of symbolic construction, allowed for thinking of the causal structure as something that is not given, but rather constructed within the medium of the four-dimensional world (see Sect. 7.1 above). Thus, if we read Weyl’s late work in comparison with Husserl’s phenomenology, we can conclude that to Weyl in relativity theory the ego cannot be represented by a point and is by no means in direct causal/lawful relation to the space-time structure, because there is no functional or lawful relation of dependence between the “I am here-now” and the passive past and the active future. But why does Weyl believe that topology can offer a solution to the question of the dimensionality of the world? In analyzing a continuum, like space, Weyl 18 I

thank Julien Bernard for pressing me in highlighting this point.

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wants to proceed in “a more general manner than by measurement of coordinates and adopt the topological viewpoint, so that two continua arising one from the other by continuous deformation are the same to us” (Weyl 1940, p. 74).19 In other words, it means that we can construct the connection between here-now points and, according to the rule of isomorphism, we can construct such a connection in view of its systematic exposition, which can lead us to express it in the form of a physical law. As already pointed out in 1940, to Weyl, the topological scheme is what allows us to embrace both special and general relativity: A certain 4-dimensional scheme can be used for the localization of events, of all possible here-nows; physical quantities which vary in space and time are functions of a variable point ranking over the corresponding symbolically constructed 4-dimensional topological space. In this sense, the world is a four-dimensional continuum. The causal structure will have to be constructed within the medium of this 4-dimensional world, i.e. out of the symbolic material constituting our topological space. Incidentally, the topological viewpoint has been adopted on purpose, because only thus our frame becomes wide enough to embrace both special and general relativity theory. The special theory envisages the causal structure as something geometrical, rigid, given once and for all, while in the general theory it becomes flexible and dependent on matter in the same way as the electromagnetic field. (Weyl 1940, p. 77)

However, the causality of topology does not only embrace special and general relativity theories, but also preserves the distinction between them, because in the general theory of relativity the causal structure is something flexible. General relativity theory establishes laws of nature that connect the flexible causal structure with other flexible physical entities, distribution of masses, or the electromagnetic field: These laws in which the flexible things figure as variables are in their turn constructed by the theory in an explicit a priori way. Of course the topological structure can not be flexible as the causal structure is, but one must have a free outlook on all topological possibilities before one can decide by the testimony of experience which of them is realized by our actual world. To that end one turns to topology. (Weyl 1940, p. 81)20

Weyl presupposes here that the dynamical interplay of fields governed by the Einstein equations requires a smooth topologically shaped arena in which to evolve (Torretti 1983, p. 287). If we look at the history of modern physics, Weyl has been a pioneer looking for the enlargement of mathematics by means of general relativity theory and by pursuing the philosophical foundations for the enlargement of this theory. In other words, even if now it is not possible to answer the question of why is the world four-dimensional, Weyl believes that it is just a matter of waiting for the development of mathematics, and of topology in particular, to enlarge our understanding of the essence of space-time structure. As Weyl concluded in 1948:

19 According

to Weyl, a continuous deformation, a one-to-one continuous transformation does not affect local values. 20 According to Weyl (1940, p. 82), the topological scheme is bounded only by certain axioms and wherever axioms occur, they ultimately serve to describe the range of variables in explicitly constructed functional relations.

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Our question has this in common with most questions of philosophical nature: it depends on the vague distinction between essential and non-essential. Several competing solutions are thinkable, but it may also happen that, once a good solution to the problem is found, it will be of such cogency as to command general recognition. (Weyl 1948, p. 23)

What is the vague distinction Weyl is talking about in this passage? The next subsection precisely focuses on this analogy between the question of the dimensionality of the world and the philosophical meaning of the distinction between essential and non-essential.

7.6.3 The Vagueness of the Distinction Between Essential and Non-essential We are now in a position to clarify Weyl’s statement and to deepen the analogy between the philosophical distinction of essential and non-essential and the problem of justifying the four-dimensionality of the world. When Weyl talks about “essence” and refers to the vague distinction between essential and not-essential he is not referring to Aristotle,21 but to Husserl’s analysis of the essence (Wesensanalyse), and one has to understand the meaning of “vague” in this context. In Husserl (1913) the essence can be exemplified for and form intuitions: The eidos, the pure essence (reine Wesen), can be exemplified for intuition in experiential data – of perception, memory and so forth – but equally well from intuitions which are non-empirical, which do not seize upon factual existence but which are instead ‘merely imaginative’. (Husserl, Ideas I, 1913, §4)

Furthermore, according to Husserl: “Everything belonging to the pure eidos must also belong to every corresponding factual occurrence.” (Husserl, Ideas I, 1913, §6). Husserl also distinguishes the exact essence (das exakte Wesen), referred to mathematics, from the morphological essences, which are essential forms of more vaguely defined entities, such as those studied by natural sciences (Husserl, Ideas I §§73–74). Now, morphological essences cannot be replaced by corresponding exact essences, because the latter are ideal and assume a similar function to the ideas in the Kantian sense (Husserl, Ideas I §74), namely they provide ideal limits and can be found as idealizations in the exact sciences, such as geometric figures. According to Husserl, science belongs to the life-world, but at the same time, nature is idealized as a closed domain of exact causal laws. In natural science, thus, individual objects are treated as exemplars. In Ideas I §2, Husserl distinguishes essence from matter of fact. Essence does not relate to what factually exists,

21 In a note to Weyl (1940), p. 83 Pesic recalls that, according to Aristotle, substance (ousia) denotes

the common essence (say of a biological genus), whereas accident (sumbebekos) denotes a quality of an individual member of that genus that does not specifically reflect its underlying essence. Pesic reminds of Aristotle distinction because he thinks that it applies to Weyl’s terminology. However, I argue that it is not the case.

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but defines what is possible. Exact essences can be completely and exhaustively defined, whereas morphological essences have vague boundaries. Essences are grasped by a kind of idealizing abstraction from the concrete individual entity using imaginative variation and possess an unrestricted universality that is different from the kind of generality attached to the laws of nature (Ideas I §6). By “eidetic” (eidetisch) Husserl means essential, namely it is referred to as the thing or property in general (überhaupt). Thus, the question is to determine to what extent in going back to Husserl’s philosophy in the 1940s Weyl eventually rejects it. In 1918, Weyl thinks that his interpretation of Einstein’s relativity can be viewed as an application of Husserl’s phenomenology,22 but from the late 1940s, and in 1955 in particular, he believes that it is no longer the case. In dealing with the problem of space after 1923, Weyl aimed at showing that what is the essence in Husserl’s terminology can become a case of a more general essence. This result, in Weyl’s view, cannot be reached by means of eidetic variation, but can be obtained by means of symbolic construction and by considering mathematical physics in its historical dimension: a position that appears closer to Cassirer than to Husserl’s Ideas. Thus, the theoretical problem underlying the question of the dimensionality of the world is the following: the history of physics has shown that what was believed to be essential, i.e. the Euclidean metric, was not. The study of Pythagorean metric has shown that even Minkowski space-time is not a pure essence. What about the physical law determining the variation of space-time? As shown in 6.1, Weyl believes that the mathematical formulation of physical laws, which takes into account structural features highly sensitive to dimension, are still to be discovered. Therefore, in Weyl’s view, to explain the dimensionality of the world also depends on the historical development of mathematics and on human activity of symbolically constructing generalized formulation of physical laws. In other words, the history of mathematics and science, which describe the empirical genesis of the symbolic constructs, represent a heuristics in view of the systematic exposition of these very same objects of enquiry (Weyl 1940, p. 79). According to a systematic exposition, it is the formulation of physical laws that determines the character of the actual world and its dimensionality. Therefore, Weyl interpreted the vagueness of the distinction between essential and non-essential in the following terms: both essential and nonessential are not at all fixed and well-defined terms once and for all, namely the boundaries of their domains vary. In the same manner, according to Weyl, if one has to search for the foundations of space-time structure, one has to look for a justification outside it. In the 1940s, Weyl suggests to look for a topological scheme that overcomes the separation between the “I am here now” and the past and future of the light cone. This process enabling a far more comprehensive representation of the world cannot be completed without the finding of physical laws corresponding to such a representation. To sum up, from Weyl’s last works it emerges that he is trying to redefine the vague distinction between the contingent and necessary

22 For a reconstruction of causal topology and Weyl’s notion of space-time structure as essence, see

Ryckman (2005, pp. 155 ff.).

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character of the four-dimensionality of the world in analogy with the effort that phenomenologists were doing to clarify what was essential and non-essential by means of eidetic variation. According to Weyl, not only phenomenology cannot provide a univocal distinction, but also symbolic construction alone is not able to account for the contingent necessity of the dimensionality of the world, despite the development of topology. A physical theory to accomplish this task is needed, and the scientific community is still looking for it.

7.7 Conclusion: Space Dimensionality and the Epistemology of Science In the previous sections, it has been shown that the problem of space and its philosophical implications engaged Weyl throughout his career. From the philosophical standpoint, Weyl attempts first at following upon Husserl’s doctrine of essence, then at harmonizing his results with Cassirer’s reflections on the philosophical foundations of geometry and the sciences. However, in the 1940s Weyl was able to advance an alternative view to Kant’s transcendental idealism of space: the foundations of similarity and congruence could be fully treated from the mathematical standpoint and by means of symbolic construction, without appealing to space as a form of intuition. In the late 1940s until 1955, Weyl’s methodology approaching the problem of explaining the dimensionality of the world is dealt with in analogy with the philosophical problem of the difference between essential and non-essential properties, namely with a genuine ontological and metaphysical question raised by Husserl. As shown in the previous sections, Weyl’s methodology, however, is that of symbolic construction and rejects the idea that essences can be given to us and they are rather constructed. Finally, one of the major contributions that Weyl’s work offers to current history and philosophy of science consists in his view of objectivity as a process achieved by means of symbolic construction that led him to ask the question concerning the explanation of the dimensionality of the world, by not limiting the question of the uniqueness of the Lorentz group to a kind of mathematical explanation, rather by heuristically using it in view of a physical explanation and a generalization offered by topology. To conclude, Weyl’s late view of the problem of space and its philosophical implications depict a world in which space dimensionality is a contingent necessity, something in between essential and non-essential properties of a being. In this respect Weyl’s view still describe the current state of affairs in which we are still describing the alphabet through which the language of nature is spelled out. Acknowledgement The research leading to this chapter has been made possible thanks to the fellowship “Research in Paris 2013” offered by the Ville de Paris and to the FP7-COFUND program Beatriu de Pinós (grant n. 2013BP-B00101). The research has been made possible also thanks to the projects 2014 SGR 1410 sponsored by the AGAUR and HAR2014-57776 sponsored by MINECO. I am grateful to Monica Bussmann and to the Staff at the ETH in Zurich, who assisted

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me in visiting the archives in October 2014. I am very thankful to Julien Bernard and Carlos Lobo who invited me to present the earlier draft of this paper at the workshop Weyl and the Problem of Space, From Science to Philosophy (Konstanz, 27-29 May 2015).

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Scholz, E. 2004. Hermann Weyl’s analysis of the “problem of space” and the origin of gauge structures. Science in Context 17: 165–197. ———. 2011. H. Weyl’s and E. Cartan’s proposals for infinitesimal geometry in the early 1920s. Boletim da Sociedada Portuguesa de Matemàtica Número Especial A. da Mira Fernandes, 225–245. ———. 2012. Leibnizian traces in H. Weyl’s “Philosophie der Mathematik und Naturwissenschaft”. In New essays on Leibniz reception, ed. R. Krömer and Y. Chin-Drian, 203–216. Basel: Birkhäuser. Sieroka, N. 2007. Weyl’s ‘agens theory’ of matter and the Zurich Fichte. Studies in History and Philosophy of Science 38: 84–107. Sklar, L. 1974. Space, time, and spacetime. Berkeley: University of California Press. ———. 1977. Facts, conventions and assumptions. In Foundations of space-time theories, volume VIII of Minnesota Studies in the Philosophy of Science, ed. J. Earman, C.N. Glymour, and J.J. Stachel, 206–274. Minneapolis: University of Minnesota Press. Torretti, R. 1983. Causality and spacetime structure. In Physics, philosophy and psychoanalysis: Essays in Honor of Adolf Grünbaum, ed. R.S. Cohen, 273–294. Dordrecht: Reidel. Weinert, F. 2005. The scientist as philosopher: Philosophical consequences of great scientific discoveries. Springer. Weyl, H. 1918. Reine Infinitesimalgeometrie. Mathematische Zeitschrift 2: 384–411. ———. 1921. Zur Infinitesimalgeometrie: Einordnung der projektiven und konformen Auffassung. Nachrichten der Königlichen Gesellschaft der Wissenschaften zu Göttingen; Mathematisch-physikalische Klasse, 99–112. ———. 1922a. Space-Time-Matter (from the 4th German edition), London. ———. 1922b. Das Raumproblem. Jahresbericht der Deutschen Mathematikervereinigung 31: 205–221. ———. 1922c. Die Einzigartigkeit der Pythagoreischen Maßbestimmung. Mathematische Zeitschrift 12: 114–146. ———. 1923. Mathematische Analyse Des Raumproblems. Berlin: Springer. ———. 1930. Weyl levels of infinity. In Levels of infinity, selected writings on mathematics and philosophy, ed. P. Pesic, 17–32. New York: Dover. ———. 1931a. Geometrie und Physik. Die Naturwissenschaften 19: 49–58. ———. 1931b. Theory of groups and quantum mechanics. New York: Dover. ———. 1940. The mathematical way of thinking. In Levels of infinity, selected writings on mathematics and philosophy, ed. P. Pesic, vol. 2012, 67–84. New York: Dover. ———. 1948. ETH-Bibliothek, University Archives, Hs 91a:31, Similarity and congruence: A chapter in the epistemology of science. ———. 1949. Philosophy of mathematics and natural science. Vol. 2, 35–37. Princeton: Princeton University Press. ———. 1955a. Insight and Reflection. In Mind and nature, selected writings on philosophy, mathematics, and physics, ed. P. Pesic, vol. 2009, 204–221. Princeton: Princeton University Press. ———. 1955b. Why is the world four-dimensional? In Levels of infinity, selected writings on mathematics and philosophy, ed. P. Pesic, vol. 2012, 203–216. New York: Dover. Winnie, J.A. 1977. The causal theory of space-time. In Foundations of space-time theories, volume VIII of Minnesota Studies in the Philosophy of Science, ed. J. Earman, C.N. Glymour, and J.J. Stachel, 134–205. Minneapolis: University of Minnesota Press. Zeeman, E.C. 1964. Causality implies the Lorentz group. Journal of Mathematical Physics 5 (4): 490–493.

Part III

From Aprioristic to Physical Foundations of the Metric

Chapter 8

The Changing Faces of the Problem of Space in the Work of Hermann Weyl Erhard Scholz

Abstract During his life Weyl approached the problem of space (PoS) from various sides. Two aspects stand out as permanent features of his different approaches: the unique determination of an affine connection (i.e., without torsion in the terminology of Cartan) and the question which type of group characteries physical space. The first feature came up in 1919 (commentaries to Riemann’s inaugural lecture) and played a crucial role in Weyl’s work on the PoS in the early 1920s. He defended the central role of affine connections even in the light of Cartan’s more general framework of connections with torsion. In later years, after the rise of the Dirac field, it could have become problematic, but Weyl saw the challenge posed to Einstein gravity by spin coupling primarily in the possibility to allow for non-metric affine connections. Only after Weyl’s death Cartan’s approach to infinitesimal homogeneity and torsion became revitalied in gravity theories.

8.1 Introduction According to H. Weyl three aspects have to be taken into account for studying the problem of space (PoS): the extensive medium of the world (“extensives Medium der Aussenwelt”), its metrical structure, and its content by a material quality changing from place to place (“materielle Erfüllung mit einem von Stelle zu Stelle veränderlichen Quale”) (Weyl 1922a, 205). The problem has to be approached from two sides, by a philosophical investigation and by mathematical analysis. In the early 1920s Weyl saw the task of philosophy in clarifying the distinction and the mutual relationships between the three aspects mentioned above. In addition to this mathematics had to “search for correct knowledge of the essence of space and of the spatial structure” as far as it can be “described quantitatively, in logico-arithmetical

E. Scholz () Faculty of Mathematics/Natural Sciences, and Interdisciplinary Centre for History and Philosophy of Science, University of Wuppertal, Wuppertal, Germany e-mail: [email protected] © Springer Nature Switzerland AG 2019 J. Bernard, C. Lobo (eds.), Weyl and the Problem of Space, Studies in History and Philosophy of Science 49, https://doi.org/10.1007/978-3-030-11527-2_8

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relations”.1 Note Weyl’s double description of essence and structure; he considered them as complementary aspects of the concept of space. Such a characterization was given by him in the phase 1921 to 1923 when Weyl developed his program of the mathematical analysis of the problem of space in a well defined, sense. In the following we shall denote it by PoS21 − 23 . It dealt with the question of how to generalize the Helmholtz-Lie analysis of the homogeneity conditions of classical space to the new context of relativistic physics. Weyl insisted on the necessity to reformulate homogeneity in terms of differentiable manifolds endowed with linear groups operating in the infinitesimal neighbourhoods (in modernized language, operating on the tangent spaces), or between them. He gave very general conceptually motivated conditions and analyzed their consequences. His result was an infinitesimal group structure typical for the automorphisms of his generalized differential geometry, “pure infinitesimal geometry”, developed in 1918 (Weylian metric). The contribution of Weyl to the problem of space has found much attention in the history and philosophy of mathematics. Here it will be dealt with from a specific point of view only; for more aspects and finer details the reader may consult the literature.2 This was not the only situation in which Weyl addressed the problem of space. In a more general sense this problem was a recurrent theme in his thought all over his life. Weyl hit upon the PoS (in the wider sense) in different contexts and looked at it from different angles. The present paper puts Weyl’s discussion of the PoS in a wider perspective (Sect. 8.1), but it would be far beyond its scope to deal with all these different facets in some detail. Here we shall concentrate on selected topics which show how Weyl used context dependent relative a priori elements which he considered constitutive for determining the structure of space, or even for grasping its “essence”. Two conceptual features stand out among them: (i) the characterization of homogeneity by means of group structures and (ii) the core role assigned by Weyl to a uniquely determined affine (torsion free) connection among admissible space structures. Both features appear prominently in Weyl’s PoS21 − 23 but also, in different form, in other encounters of him with the problem of space. We shall discuss how Weyl dealt with the problem of homogeneity after the rise of general relativity (Sect. 8.2) and contrast it with Cartan’s answer to the question (Sect. 8.3). That could have given reasons to Weyl to revise the central role of uniquely determined connections as a kind of relative a priori for physical geometry, but it did not (Sect. 8.4). The next challenge was posed by Dirac’s spinor fields lifted to general relativity. In the light of later developments (Einstein-Cartan theory of gravity) one might expect that it 1 “Für

den Mathematiker handelt es sich darum, das quantitativ, in logisch- arithmetischen Relationen Erfaßbare am Wesen des Raumes und der räumlichen Struktur richtig zu erkennen und mit den Hilfsmitteln der Logik, Arithmetik und Analysis auf seine einfachsten Gründe zurückzuführen” (Weyl 1922a, 206). 2 Among many (Scheibe 1988; Sigurdsson 1991; Coleman and Korte 2001; Bernard 2013, 2015; Scholz 2004, 2016). Weyl’s study of the PoS21–23 was the guiding axis of the conference of which this book arose; see in particular the contributions to this volume by A. Roca-Rosell and C. Lobo.

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could have become a problem for Weyl’s affine connection principle already in the 1930/40s. But that was not the case. Why, will be shortly discussed in Sect. 8.5, before we come to a final evaluation (Sect. 8.6). Of course Weyl, like the other mathematicians of the twentieth century, often used the terminology of “space” at other places in a wider sense than above, sometimes in a purely mathematical context. But in the framework of this paper the notion of space is nearly always used in the more restricted sense of a mathematical space structure which serves, or at least may serve, as a candidate for grasping physical space or space-time, the “extensive medium of the external world” in Weyl’s word. For the abbreviation PoS this is always the case.

8.2 The Multiple Faces of the PoS As already mentioned we can give here only a short survey of different contexts and different forms in which Weyl met the problem of space. The following list of topics and contexts may serve as an orientation (main publications indicated in brackets): 1. Modernized presentation of the classical problem of space in the sense of Helmholtz and Lie in the first chapter of Raum – Zeit – Materie (Weyl 1918b) 2. Specification of Riemannian metrics (“Pythagorean nature” of metric) in the wider class of Finsler metrics in Riemann’s approach to geometry (Riemann/Weyl 1919) 3. Mathematical analysis of the problem of space, PoS21–23 , (Weyl 1923a) 4. Characterization of R3 or S3 (the three-dimensional sphere) by combinatorial invariants. Topological space forms and their characterization by discrete groups (operating on the universal covering space) (Weyl 1925/1988, 16ff.) 5. Introduction of a differentiable structure on continuous manifolds, in particular with regard to its justification for the concept of physical space (Weyl 1925/1988, 12) 6. Cartan’s general concept of infinitesimal geometries (Weyl 1925/1988, 38ff.), (Weyl 1929b, Nabonnand 2005) 7. Similarities and congruences as an exemplary case for the distinction of mathematical automorphisms and physical automorphisms of space (Weyl 1949, 1948b/49) 8. Specific role of Lorentz/Poincaré group and the dimension 4 of space-time (Weyl 1948b/49)3 9. Finally Weyl’s considerations on the possible role of non-metric affine connections for the dynamics of spinor fields in general relativity (Weyl 1950) Under the items 1., 3., 6., 7., 8. Weyl dealt with the question of how to characterize the homogeneity of the respective spatial structures. Here different

3 See

the contribution of S. de Bianchi, Chap. 7, this volume.

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versions of automorphism groups played a prominent role. In his discussion of the items 2., 3. and 9. Weyl’s conviction that a proper space structure in the sense of PoS carries a uniquely determined affine (i.e. torsion free linear) connection stood out. In 2., 3. the postulate of a uniquely determined affine connection was not questioned at all; in 9. he subjected it to a check whether it could be defended in the light of relativistic spinor fields. Item 7. and 8. have been discussed elsewhere.4 In the following sections this topic will be dealt in more detail. The topological space problem (item 4) and the problem of differentiability (item 5) have only been brought up at isolated occasions by Weyl; they cannot be discussed in this paper.

8.3 Homogeneity Characterizations Given by Weyl In the first three editions of Raum – Zeit – Materie the classical space problem of the nineteenth century, posed and answered by Helmholtz, Lie, Engel (Weyl added Hilbert, Grundlagen der Geometrie, app. IV) was mentioned by Weyl only in passing (Weyl 1918b, pp. 86, 264, 1st to 3rd ed.). Even in the fourth edition published in 1921, in which he already included a first sketch of his own thoughts on the PoS21–23 , Weyl did not go into more details (Weyl 1921, pp. 86, 289). Only after having finished his own analysis he gave a more extended presentation of the classical solution in the fifth edition (Weyl 1923b, 100). He did not use the concepts of “rigid body” and “free mobility”, which had become problematic with the advent of relativity theory, but expressed Helmholtz’ postulates abstractly in terms of group theoretical constraints for the homogeneity of classical space. He rephrased Helmholtz’s axioms of free mobility by conditions which are now called simple flag transitivity of the homogeneity group.5 Similar in (Weyl 1923a, 5th lecture). Weyl’s presentation stripped Helmholtz’ analysis from the latter’s intention of founding his conditions on supposedly factual conditions (“That sachen”) lying at the basis of any empirical measurement. He did not claim to give a historically precise account of Helmholtz’ thoughts, in fact his passage read as though Helmholtz’ had started from an investigation of the a priori conditions of the homogeneity of physical space.6 Weyl’s reconstruction of Helmholtz’s and Lie/Engels’ approach

4 Scholz

(2018).

5 “Man kommt so zu der folgenden Formulierung des Homogeneitätspostulats im n- dimensionalen

Raum: Es soIl möglich sein, mit Hilfe einer zur Gruppe G gehörigen Abbildung ein System  inzidenter Richtungselemente der 0ten bis (n − 1)ten Stufe in ein gleichartiges, beliebig vorgegebenes System / überzuführen; aber die Identität soll unter den Abbildungen von G die einzige sein, welche ein derartiges System  inzidenter Richtungselemente festläßt” (Weyl 1923b, 100). 6 “Von einem tieferen, gruppentheoretischen Gesichtspunkt aus hat Helmholtz zuerst die Homogeneitätsfrage gestellt. Helmholtz setzt nicht die Gültigkeit des Pythagoreischen Lehrsatzes im Unendlichkleinen, ja nicht einmal die Meßbarkeit der Linienelemente voraus; er spricht allein

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to the space problem was systematic, not historical. It assimilated the classical PoS to a perspective which prepared the way to a type of analysis which Weyl pursued in his own program between 1921 and 1923. From such a perspective the classical PoS seemed to have been the question of an a priori characterization of the homogeneity of space. It led to an answer which allowed to introduce an invariant definite quadratic differential form of constant curvature and thus to the classical spaces of Euclidean and non-Euclidean geometry.7 Such a type of a priori characterization could no longer be apodictic, like Kant’s a priori judgements had been (or, at least, had been claimed to be). It no longer consisted of necessary judgements, but rather of well founded postulates, if possible of the best founded ones, which characterize the conceptually possible in a specified context. In this function it still has an a priori character in distinction from empirical determinations but only relative to the latter and to the theoretical context. With a change of context and/or more refined empirical knowledge, formerly well established a priori conditions can become obsolete and may have to be revised, in agreement with the analysis given by Friedman (1999, 59ff.). Such a revision was the goal of Weyl’s analysis of the problem of space (1921– 1923). By several reasons simple flag transitivity could no longer be upheld as a feature characterizing the homogeneity of space. Firstly special relativity had integrated space proper and time to spacetime as the “extensive medium of the world”. That destroyed flag transitivity because it now became necessary to account for the qualitative difference of timelike and spacelike directions. Moreover general relativity, Einstein’s theory of gravity, broke with the paradigm of constant curvature and made curvature dependent on the distribution of matter and energy, thus giving it an a posteriori character. Therefore Weyl, and a little later Cartan, posed the question of homogeneity of spacetime anew, in forms adapted to the context of general relativity. Both came to different, only partially overlapping answers which we shall discuss in the following. Weyl started from a conceptual analysis of what he considered the most general, minimal conditions for congruence geometry founded on infinitesimal structures like those he had proposed in his purely infinitesimal geometry of 1918. In his investigations 1921–1923 he wanted to dig deeper and to motivate, or even derive, a generalized metrical structure from congruence and similarity concepts in the infinitesimal and a generalized homogeneity principle. In a move which he presented as a conceptual analysis of the idea of congruence in the infinitesimal he established conditions for (linear) groups characterizing congruence by generalized “rotations” in the infinitesimal neighbourhoods of each point of spacetime. An intuitive idea of homogeneity then demanded that the type of group (more precisely von dem wahren Grundbegriff der Geometrie, der Gruppe G der kongruenten Abbildungen des Raumes.” (Weyl 1923b, 100). 7 “Es ist eine wunderbare gruppentheoretische Tatsache, die von Helmholtz, strenger und allgemeiner von S. Lie bewiesen wurde, daß die einzigen dieser Bedingung genügenden Gruppen G die Gruppen Gλ . . . [sind]” ibid. By Gλ Weyl denoted the congruence groups of hyperbolic, parabolic, or elliptic geometry (in the terminology introduced by F. Klein).

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the conjugation class in the general linear group) was equal for any two points. But Weyl did not use the terminology “homogeneous/homogeneity” in this context; he rather spoke of the fixed nature of the group (the conjugation class) which could be expressed by pointwise changing “orientations” of the group (members of the class) (Weyl 1923a, 48).8 Moreover a transition between neighbouring points had to be specified by a linear connection which is compatible with the similarities with regard to the “rotations” (more technically with the normalizer of the rotations). All this was justified by by what Weyl considered an a priori analysis of the concepts involved (Weyl 1923a, 49). But his was not all; Weyl added: I now come to the synthetic part in the KANTian sense. Thetask is now to formulate precisely the postulate, up to now only indicated, which finally determines the type of rotation group which is characteristic for the real world. (ibid.)9

The synthetic component of his a priori justification of infinitesimal congruence structures consisted of a two-part postulate, the first of which demanded a kind of wide adaptability to a posteriori distribution structure of matter (postulate of “free deformability”), the second one was the postulate of unique determination of a compatible affine connection. Later the first part turned out to be mathematically redundant,10 leaving the second part as the mathematical and philosophical core of Weyl’s synthetic a priori of the PoS21–23 .11 Mathematically it was crucial for constraining the groups which could serve as candidates for infinitesimal congruences so strongly that in the end Weyl could show that only the generalized orthogonal groups (of arbitrary signature) satisfy the constraints (main theorem of Weyl’s PoS). A philosophical analogy of this principle with intersubjectivity relations in practical philosophy was expressed and emphasized by Weyl in his Barcelona lectures (Weyl 1923a, 46). The nature of this analogy is being analyzed by N. Sieroka (this volume) and related to Weyl’s exchange with F. Medicus and his reading of Fichte. The existence of a uniquely determined affine connection remained a stable feature in Weyl’s understanding of a good geometric structure designed for representing space from 1919 onward, although the mathematical feasibility of it might have became doubtful after Cartan’s answer to the homogeneity challenge of general

8 Similarly

in the 5th edition of Raum – Zeit – Materie, where he characterized the “nature” of a Riemannian metric by its signature and its “orientation” by the point dependent value of the respective quadratic differential form (Weyl 1923b, 102). In the 4th edition (translated into English and French) he still fought more indirectly with the problem that space as “a form of phenomena . . . is necessarily homogeneous”, while the Riemannian metric is not (Weyl 1922b, 96ff.). 9 “Ich komme jetzt zum synthetischen Teil im KANTischen Sinne. Da gilt es, das früher angedeutete Postulat präzis zu formulieren, das die für die wirkliche Welt charakteristische Art der Drehungsgruppe festlegen soll” (Weyl 1923a, 49), emph. in original. 10 (Scheibe 1957) 11 For more details see the literature cited in fn. 2.

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relativity and even stronger after the advent of relativistic spinor fields. Only much later, in the years between 1948 and 1950, Weyl subjected it to an investigation of its empirical acceptability. His result was that this part of his a priori may be sustained even relative to a context including general relativistic spinor fields (see Sect. 8.5). With regard to homogeneity in the modern (general relativistic) context Weyl performed a tight-rope walk. Clearly, space “as a form of phenomena . . . is necessarily homogeneous” (Weyl 1922b, 96), but Riemannian spaces are not. He solved, or circumvented, the problem by arguing that (simply connected) neighbourhoods of any two points are diffeomorphic and the “nature” of the metric remains the same for all points.12 Both belong to the a priori determinations of space, while the metrical structure is fixed by a posteriori, empirically given factual conditions (or contingent model assumption, we might add).13 For him the diffeomorphism invariant manifold and the nature of the metric expressed the generalized idea of homogeneity in the relativistic context. But he did not speak of “generalized homogeneity”, he rather reserved the terminology of homogeneity for the classical situation of metrically homogeneous spaces (Weyl 1918b, 1923a, 1949). In spite of this terminological decision, the view that the diffeomorphisms of the spacetime manifold are part of the physical automorphisms of general relativistic field theories persisted in Weyl’s thought. At the time of preparing the different editions of Raum – Zeit – Materie (1918–1923) Weyl argued for such an understanding by means of the plasticine analogy discussed in J. Bernard’s contribution, this volume. This was an intuitive, “didactical”, way for expressing the more general postulate that under a physical automorphism the field structures are “dragged along” with the diffeomorphisms (dynamical symmetries in present physicists’ terminology). Otherwise they would not be able to preserve the (a posteriori) field structures, not even the metric induced by them. Weyl insisted on such a generalized understanding of homogeneity in a talk on physical and mathematical automorphisms given in the late 1940s (Weyl 1948b/49).14

8.4 Cartan’s Concept of Infinitesimal Homogeneity Elie Cartan approached the problem of homogeneity in a different way. In 1922 he presented a new type of infinitesimal geometry to the public Sur une généralisation de la notion de courbure . . . (Cartan 1922). He had developed the necessary tools (differential forms and Lie group theory) over a long time and elaborated the basic ideas for his new geometry during the year 1921 in an interplay of differential geometry, Einstein’s gravity theory, and the brothers Cosserat’s generalized theory

12 Speaking

in global terms, Weyl would surely have assumed a transitive diffeomorphism group of the spacetime manifold. 13 Similar in (Weyl 1949, 87). 14 Cf. (Scholz 2018).

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of elasticity.15 In the years to come he would expand his approach to a broad program of generalized infinitesimal geometries later called Cartan spaces. In his survey talk at the International Congress of Mathematicians in Toronto, 1924, Cartan motivated the approach by indicating that general relativity was confronted with . . . the paradoxical task of interpreting in a non-homogeneous universe . . . the multiple experiences made by observers who believed in the homogeneity of this universe (Cartan 1924, 86).

Although general relativity had helped to induce a first step towards bridging the gap between (non-homogeneous) Riemannian geometry and Euclidean geometry (homogeneous in the sense of F. Klein) by motivating T. Levi-Civita’s concept of infinitesimal parallelism (linear connection), he did not yet see the gap closed.16 Cartan alluded to the Kleinean understanding of homogeneity and indicated the idea underlying his approach: . . . while a Riemannian space does not possess absolute homogeneity, it possesses a kind of infinitesimal homogeneity; in the immediate neighbourhood of a point it can be assimilated to a Euclidean space. (ibid.)17

That is, Cartan wanted to implement infinitesimal homogeneity in his new generalized spaces, in addition to their infinitesimal (generalized) rotational symmetries. Weyl, as we have seen, translated homogeneity in the new, relativistic context to the possibility of comparing the neighbourhoods of any two points of spacetime with each other, which boilt down to considering the diffeomorphisms of the underlying manifold as part of the automorphisms of the geometric structure. Cartan’s program thus consisted in an infinitesimalization of the Kleinean concept of geometry as a homogeneous space S in the sense of S ∼ = G/H with a main group G and generalized rotations (isotropy group) H ⊂ G.18 Cartan considered the infinitesimal version of the groups, i.e. the corresponding Lie algebras g = Lie G, h = Lie H and g = l ⊕ h, and assimilated the quotient of the two with the infinitesimal neighbourhoods in the manifold M, such that Tp M ∼ = l, for = g/h ∼ any point p of M. His crucial symbolical tools were ensembles of differential forms, which can be read as differential forms with values in the respective Lie algebras.

15 (Cogliati

2015; Nabonnand 2016; Scholz 2016); for a modern mathematical introduction to Cartan geometry see (Sharpe 1997). 16 “Or, c’est le développement même de la théorie de la relativité, liée par l’obligation paradoxale d’interpréter dans et par un Univers non homogène les résultats de nombreuses expériences faites par des observateurs qui croyaient à l’homogenéité de cet Univers, qui permit de combler en partie le fosse qui separait les espaces de Riemann de l’espace euclidien. Le premier pas dans cette voie fut l’oeuvre de M. Levi-Civita, par l’introduction de sa notion de parállelisme.” (Cartan 1924, 86)) 17 “ . . . si un espace de Riemann ne possede pas une homogeneite absolue, il possede cependant une sorte d’homogeneite infinitesimale; au voisinage immediat d’un point donne il est assimilable a une espace euclidien” (Cartan 1924, 85). 18 In the Euclidean case G ∼ Rn  SO(n, R), with H = SO(n, R), thus S = G/H ∼ Rn . = =

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With their help he introduced a generalized type of connection, now called Cartan connection, which led to two kinds of curvature effects.19 The curvature with respect to the generalized rotations h corresponded to the Riemannian curvature (in more special and slightly different form) which was well known at the time, but the curvature with respect to the generalized translations l was a new effect. Cartan called it torsion because in the context of the generalized elasticity theory in the sense of the Cosserats it could be interpreted as a rotational momentum in the medium. If translated to the coordinate notation of differential geometry (the calculus of Ricci and LeviCivita) Cartan’s connection could, in many λ which cases, be expressed in the form of a linear connection  with coefficients μν λ λ is are no longer symmetric in the lower indices. In fact the condition μν = νμ equivalent to non-vanishing torsion.20 Cartan did not try to argue on a philosophical level as explicitly as Weyl did, but his conceptual analysis of the “paradoxical task” posed by general relativity may be read as establishing a new type of a priori framework for physical geometry, similar to Weyl’s in PoS21–23 although different from the latter. Cartan’s new relative a priori was wider than Weyl’s in two respects. His framework allowed a larger variety of infinitesimal isotropy and homogeneity groups than Weyl’s. Moreover, in his view it would not appear natural to consider the existence and uniqueness of a compatible affine connection (torsion zero) as a “synthetic” a priori of the physical space concept. Cartan reformulated Weyl’s PoS in his framework, but he dealt with it from a mathematical point of view rather then of a philosophical one, and with a slightly different outcome.21 Weyl responded to Cartan’s proposal of a large class of infinitesimal geometries, but at the beginning he was not convinced that the wider perspective was helpful for extending the a priori framework of physical geometry. In the correspondence between him and Cartan he expressed doubts even with regard to the specific geometrical achievements of Cartan’s generalization, although at the end both authors came to a basic acceptance of the other’s viewpoint (Nabonnand 2005).22 In his contribution to the Lobachevsky anniversary volume written in 1925 but published only posthumously, Weyl discussed Cartan’s approach and acknowledged that it allowed a “far-reaching generalization of infinitesimal geometry” (Weyl 1925/1988, 38); in particular:

19 Cf.

(Sharpe 1997). λ = T λ is the torsion tensor, i.e., the translational curvature expressed in coordinate − νμ νμ coefficients. 21 Cf. (Scholz 2016). 22 For a survey see (Scholz 2016); a more refined evaluation of the correspondence is being prepared by C. Eckes and P. Nabonnand. 20  λ μν

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. . . it achieves the natural widest possible range of concept formation which still allows to establish a theory of curvature in analogy to Riemann’s.23

Thus Weyl acknowledged Cartan’s generalization of the curvature concept, but without mentioning that it carries the potential to undermine the central role of the affine connection, which he considered as the most important part of the “synthetic” a priori of the space concept.24

8.5 Affine Connection, Synthetic A Priori or Just A Special Condition Among Others? Shortly after Levi-Civita’s invention of the infinitesimal parallel displacement in Riemannian geometry Weyl introduced affine connections as a concept of its own for differential geometry, which allowed to talk about infinitesimal parallel displacements in allusion to Levi-Civita’s terminology but independent of the structure of a Riemannian metric and without reference to an embedding into a higher dimensional Euclidean space (Weyl 1918c). He simply demanded that (1) for any vector ξ attached to p the change induced by parallel displacement from the −→  point p to an infinitesimal close one p depends linearly on the vector pp , and (2) if for two points p1 , p2 , both infinitesimally close to p, the parallel displacement of − → along − → leads to the end point p and the displacement of − → along − → to pp pp pp pp 1 2 21 2 1 25 p12 , then p12 = p21 : “The result is an infinitesimally small parallelogram.” In the paper directed to mathematicians Weyl argued conceptually, sometimes even in a philosophical style. He presented the task of geometry being “to fathom out the essence of the metrical concepts”.26 He thus understood the conditions (1) and (2) as postulates which arise from an analysis of the concept of infinitesimal parallel displacement. In the philosophical language used in (Weyl 1923a) the generalized affine connection resulted from an a priori conceptual analysis. A little later, in his commentaries to Riemann’s inaugural lecture (Riemann/Weyl 1919), he pondered on the question of how the Riemannian metric could be specified

23 “Und

darauf beruht wohl überhaupt die mathematische Bedeutung seines allgemeinen Schemas: es erreicht den natürlichen weitesten Umfang der Begriffsbildung, welche die Aufstellung einer Krümmungstheorie analog der Riemannschen noch ermöglicht” (Weyl 1925/1988, 39). 24 In the same article he reiterated that he still stood to the content of his PoS21–23: “Das neue gruppentheoretische Raumproblem, das vom Standpunkt der Relativitätstheorie an Stelle des Helmholtz-Lie’schen tritt, glaube ich in meiner Schrift “Mathematische Analyse des Raumproblems” (1923, Vorlesung 7 und 8) formuliert und gelöst zu haben.” (Weyl 1925/1988, 37) 25 “Es entsteht eine unendlich kleine Parallelogrammfigur” (Weyl 1918c, 7). 26 Die Geometrie “ergründet, was im Wesen der metrischen Begriffe liegt” (Weyl 1918c, 2). In the paper presenting his purely infinitesimal geometry to physicists (as the geo- metrical background for his unified field theory) Weyl introduced the affine connection in more concrete form and axiomatically (Weyl 1918a, 32), (Weyl 1918/1997, 26).

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among the wider class of Finsler metrics (which had a striking a priori justification in being homogeneous with regard to rescaling). He conjectured that the Riemannian metrics are just those which are compatible with a uniquely determined affine connection.27 D. Laugwitz would later call this conjecture Weyl’s first problem of space – and answered it positively (Laugwitz 1958). Weyl was convinced of the fundamental import of the principle of a uniquely determined affine connection already in 1919; in the following we shall speak about it as Weyl’s affine connection principle. Its crucial role for deriving the main theorem of the PoS21–23 made it so convincing for Weyl that in 1922/23 he decided to raise it to the status of a “synthetic” a priori. Considered from a wider perspective this was neither self-evident nor imperative (in distinction to a Kantian understanding of synthetic a priori). Only a few weeks after Weyl’s Barcelona lectures in February 1922, E. Cartan presented his first public note on generalized spaces to the Paris Académie des Sciences (Cartan 1922). Of course Weyl could not know about it at the time of his lectures, nor apparently while preparing them for publication, but in hindsight it could have became clear to him that Cartan’s generalized spaces also opened the pathway towards a different relative a priori for relativistic spacetime. For Cartan the difference was not so much of a philosophical nature, but mathematically it was clear to him from the outset. If a parallel displacement,  −→  λ with regard  ξ, pp = ξ  − ξ , is expressed by connection coefficients μν to a coordinate basis of the infinitesimal neighbourhoods (the tangent spaces), the λ = λ . closing condition (2) boils down to the symmetry of the coefficients μν νμ Cartan’s torsion tensor expressed in coordinate coefficients, on the other hand, λ =  λ −  λ ; Weyl’s closing condition (2) thus amounts to vanishing bcomes Tμν νμ μν torsion.28 It was not easy for Weyl and Cartan to disentangle their different points of view on their differences with regard to generalized spaces, although Cartan could treat the mathematical aspects of Weyl’s PoS21–23 quite easily as a special case of his methods and he acknowledged Weyl’s deep philosophical analysis, but without discussing it from his side (Cartan 1923b). Weyl, on the other hand, found it difficult to grasp the subtleties of Cartan’s approach, while he soon understood the wider mathematical generality of the latter’s approach and acknowledged it (see above). In 1929 he even adapted certain aspects of Cartan’s approach for his proposal to

27 “Bei der fundamentalen Bedeutung, die nach den neueren Untersuchungen ( . . . ) dem affine Grundbegriff der infinitesimalen Parallelverschiebung eines Vektors für den Aufbau der Geometrie zukommt, erhebt sich insbesondere die Frage, ob die Mannigfaltigkeiten der Pythagoreischen Raumklasse die einzigen sind, welche die Aufstellung dieses Begriffs ermöglichen und welche dementsprechend nicht bloß eine Metrik, sondern auch affine Zusammenhang besitzen. Die Antwort lautet wahrscheinlich bejahend, ein Beweis dafür ist aber bisher nicht erbracht worden.” (Riemann/Weyl 1919, 27). 28 Cartan discussed this point in his investigation of Weyl’s PoS in slightly different terms (Cartan 1923b, §3).

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a general relativistic version of Dirac’s electron theory (Weyl 1929a). In the same year he gave a survey talk on Cartan’s theory to the Princeton group of differential geometry and mathematical physics. There he argued that, for a proper geometric usage, one has to add certain restrictions to Cartan’s scheme, among them the exclusion of torsion (Weyl 1929b, 210). Cartan had developed and extended his methodology in the course of the 1920s in very general terms, with infinitesimal Kleinean spaces of many different types and even without assuming that, pointwise, the infinitesimal quotient group (Lie algebra) l ∼ = g/h can be identified with the tangent spaces of the manifold (not even the dimension needed to be the same). For Weyl it seemed indispensable that for a proper geometrical usage of Cartan’s general scheme one had to assume pointwise identifications of l, which he called the “tangent plane” denoted by “TP “(sic!), with the infinitesimal neighbourhoods of the manifold, the tangent spaces in the later terminology.29 He called this an “embedding” of “TP ” into the manifold and added additional restrictions motivated by the specific geometrical structure considered. He spoke of “special manifolds”, in particular with regard to projective and conformal structures.30 The specialization conditions contained, in particular, the “invariantive restriction to require that our manifold . . . is without torsion” (Weyl 1929b, 210). In the following correspondence Cartan insisted that his research program did not need such restrictions and deplored that it was not fairly represented in Weyl’s survey. The ensuing exchange of letters centered on the role of “embedding” of l (Weyl’s “TP ”) and the specialization conditions. Although torsion played only a subordinate role, the correspondence shows once again that Cartan considered torsion zero as a technical specialization condition among others without particular conceptual import (Nabonnand 2005).

8.6 The Challenge of Spinor Fields in the 1930–1940s In the light of later developments in Einstein-Cartan theory of gravity (see final section) it ought to be added that, to my knowledge, in the 1930s neither Cartan nor Weyl considered the question whether  the general relativistic Dirac electron field  with non-vanishing spin spin = 12 might undermine Weyl’s affine principle from the physical, perhaps even from the empirical side. Even later, when Weyl started

29 In

the modern understanding of Cartan spaces this is an indispensable property inbuilt in the definition of a Cartan gauge, e.g. (Sharpe 1997, 174). Note that Weyl’s “TP ” stood for for l in the function of what would now be considered the tangent space of the translative subgroup in the fibre direction. 30 For that Weyl drew upon the results of the Princeton school of differential geometry, A.L. Eisenhart, O. Veblen, T.Y Thomas, intending to build bridges between the “French” (E. Cartan) and the “US” (Princeton school) traditions in differential geometry.

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to analyze the consequences of an independent spin coupling of the Dirac field to gravity in the late 1940s, he did not pose the question whether the affine principle had to be given up in favour of Cartan’s view (torsion = 0). He rather chose an approach which Einstein had studied in the 1920s, in which an affine connection and the metric of a generalized Lagrangian were varied independently (Einstein 1925). He thus relaxed the condition of metricity of the connection rather than that of vanishing torsion (calling it a “mixed” theory). For a Lagrangian of the electron field with a Dirac term, a spin term and a mass term Weyl found that “by the infiuence of matter a slight discordance between affine connection and metric is created” (Weyl 1950, 288 and equs. (2), (3)). This interesting observation clearly posed a challenge to Weyl’s affine connection principle. But “by somewhat laborious calculations” (not presented in the paper) he was able to show that by adding a small term of the form -12π G l2 to the Lagrangian (l2 a quartic scalar invariant in the 4-component spinor field ψ) the metric theory became equivalent to the mixed theory (without the additional term).31 He concluded: To this extent then there is a a complete equivalence between the mixed metric-affine and the purely metric conception of gravity. (Weyl 1950, 288)

The metric theory of gravity could be upheld by a small addition to the Lagrangian even in the light of an electron field’s spin coupling to gravity. In this sense, the challenge posed by Dirac spinor fields to Weyl’s affine connection principle was neutralized and the relative a priori of the uniquely determined metric affine connection successfully defended.

8.7 Late Endorsement for Cartan’s Infinitesimal Homogeneity Principle and General Discussion In his PoS21 − 23 Weyl tried to found a new conceptual framework for space and time that lived up to the challenges of the theories of relativity, like the homogeneous spaces of the late nineteenth century had done with regard to classical physics. The challenge arose from physical theories in which Einstein had evaluated both, empirical and theoretical knowledge in a quite specific sense. Regarding physical concepts Einstein was an ingenious innovator (role of time, space, simultaneity, equivalence principle), but with regard to the mathematical theories, he had built with conceptual material inherited from contemporaneous mathematics (Riemannian geometry, Ricci-Levi Civita’s absolute calculus etc.). Weyl intended to go    = ψ 3 ψ1 + ψ 4 ψ2 ψ 1 ψ3 + ψ 2 ψ4 . The “laborious calculations” seem to have been presented in the manuscript (not preserved) for the publication (Weyl 1948a). K. Chandrasekharan, the editor of Weyl’s Gesammelte Abhandlungen remarks about this paper that “due to typographical errors, it is incomprehensible” (Weyl 1968, vol. 4, p. 285, footnote). It was therefore not reprinted in the edition. 31 l

2

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beyond the constraints of the inherited and to “fathom out” (as he said in the above quote) the minimal ingredients of spacetime concepts, necessary for obtaining infinitesimal congruence structures which were able to build a bridge between the general notion of a differentiable manifold and specific metrical determinations. The latter ought to be able to adapt as flexible as possible to contingent distributions of matter and fields. He did so in what he considered an a priori move of conceptual analysis, as he said in open allusion to the Kantian terminology, and found that he had to add the “synthetic” affine connection principle. Although he tried to motivate the synthetic principle by analogy with considerations of practical philosophy (respect for a “common good”) the finally decisive motivation of the principle lay in its success for deriving the main theorem of the PoS, not different from what one would ordinarily expect from the axioms of a mathematical theory. Weyl was well aware that his a priori was different from Kant’s; in particular it was no longer apodictic and made sense only in the context of relativistic physics and the open horizon for new differential geometric structures on manifolds. It thus was relative with regard to the theoretical context and open for potential revisions, like the classical understanding of homogeneity had been. In Weyl’s view this did not make the striving for a well understood a priori obsolete. In his view the role of a priori statements had changed from being necessary judgements to well motivated possible concepts and structures. But their function with regard to more specific theoretical and empirical knowledge remained. In his view “physics projects what is given onto the background of the possible” (Weyl 1949, 220) and mathematics explores the conceptually possible.32 Substituting Kant’s necessary a priori judgements by the “background of the possible” points also towards another shift: the relative a priori need no longer be uniquely determined. In our case study we have come across a possible loss of uniqueness, the underdetermination of the relative a priori, by comparing Weyl’s analysis of the PoS with Cartan’s generalized spaces. The latter’s conceptual motivation was the implementation of infinitesimal homogeneity in addition to the global homogeneity achieved by structure dragging diffeomorphisms. That gave them the potential for becoming a competing a priori structure for relativistic space concepts. In our discussion of Weyl’s different approaches to the space problem this potentiality did not materialize, apparently because Cartan did not like to argue too much on the philosophical level and Weyl did not see imperative reasons to give up his affine connection principle. It needed a change of generations and deep conceptual as well as technical studies of physicists before the a priori potential of

32 “The

dual nature of reality accounts for the fact that we cannot design a theoretical image of being except upon the background of the possible. Thus the four-dimensional continuum of space and time is the field of the a priori existing possibilities of coincidences.” (Weyl 1949, 231).

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Cartan spaces became apparent. A detailed account of this next shift would need a publication of its own (or more). Here we have to content ourselves with an outline.33 The success of conceiving electromagnetism as a U(1)-gauge field induced physicists to study field theoretic consequences of point dependent infinitesimal symmetries of other groups. In the terminology of physics the symmetries were “localized”.34 Most striking and best known is the case of weak isospin SU(2) (Yang/Mills) and the later generalizations to gauge field theories in elementary particle physics. But also the Poincaré group, the symmetry group of special relativity, was “localized” independently and nearly simultaneously by T. Kibble and D. Sciama (Sciama 1962; Kibble 1961). From the point of view of physics the conserved currents of infinitesimal symmetries supplied by the Noether theorems played a crucial heuristic role for this research. For the Poincaré group R4  SO(3, 1) that led to considering the spin current, the Noether current with regard to the Lorentz rotations, as an additional source for the gravitational field supplementing energy-momentum, the current with regard to the translation group.35 The physical idea of “localizing” the translations of the Poincaré group together with the Lorentz rotations was very close to Cartan’s idea to implement infinitesimal homogeneity in addition to infinitesimal isotropy in his concept of generalized spaces, although neither Kibble nor Sciama noticed the kinship of their studies with Cartan’s geometrical framework. This was brought into the open by the work of A. Trautman and F. Hehl.36 Then it also became clear that the simplest Lagrangian in Kibble’s approach, as well as in Sciama’s, is equivalent to the one discussed by Cartan in passing, when he showed what his approach could contribute to understand and to extend Einstein’s theory of gravity (Cartan 1923a, §83). It is now being called Einstein-Cartan gravity (EC).37 Einstein-Cartan gravity modifies Einstein’s general relativity only to a tiny degree; for vanishing spin it reduces to the latter. Moreover, outside of spinning matter field the torsion is zero and the influence of spin on the metric can be taken into account by a small modification of the energy-momentum source of the Einstein equation,38 similar to what Weyl had found for the non-metricity induced by spin in the “mixed” theory. Only for mass densities more than 1038 times the one 33 For

a rich collection of sources with detailed commentaries from the theoretical physics side see (Blagojevic and Hehl 2013). 34 See A. Afriat’s paper, this volume. 35 To be more precise: Sciama presupposed an Einsteinean background and gained spin as an additional current, modifying Einstein gravity to what was later called Einstein-Cartan gravity. Kibble, on the other hand, started from localizing the symmetries of Minkowski space and considered different Lagrangians, the simplest of which led to Einstein-Cartan theory (Blagojevic and Hehl 2013, 106). 36 (Hehl 1970; Trautman 1973; Hehl e.a. 1976) and others. 37 Cf. (Trautman 2006; Hehl 2017). 38 Cf. (Hehl e.a. 1976, 406), (Trautman 2006, 194).

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of a neutron star, respectively a nucleon mass compressed to 106 Planck lengths, which signifies energy densities at the scale of a hypothetical grand unificaton of elementary particle interactions, the experts expect EC “to overtake” Einstein’s general relativity.39 These seemingly technical results of modified gravity are important in our context, because they show that Cartan’s principle of infinitesimal homogeneity has finally become important in foundational studies of gravity. In the last third of the twentieth century it has turned into a relative apriori for relativistic spacetime theories with, at least, the same right as Weyl’s affine connection principle and alternative to the latter. It even would have the advantage of being closer to what Weyl called the “physical automorphisms” of modern physics by fitting well to the Noether current paradigm for infinitesimal symmetries, prominent in contemporary physics.40 On the other hand, there is (still?) no empirical evidence which would force us to revise Weyl’s analysis of the PoS21–23 and to relegate his affine connection principle from the status of a relative apriori to an empirical principle, valid only in “weak” field constellations. In this sense, we seem to be here in the situation of a temporal underdetermination of the relative a priori principles of Weyl and Cartan.41 This seems to be another feature of present a priori structures, at least as important as their being established in the context of wider scientific results and being open to revision with them.

References Bernard, Julien. 2013. L’ idéalisme dans l’ infinitésimal. Weyl et l’ espace à l’ époque de la rélatvité. Paris: Presse Universitaires de Paris Ouest. ———. 2015. La redécouverte des tapuscrits des conférences d’Hermann Weyl à Barcelone. Revue d’ histoire des mathématiques 21 (1): 151–171. Blagojevi´c, Milutin, and Friedrich W. Hehl. 2013. Gauge theories of gravitation. A reader with commentaries. London: Imperial College Press. Cartan, Élie. 1922. Sur une généralisation de la notion de courbure de Riemann et les espaces à torsion. Comptes Rendus Académie des Sciences 174: 593–595. In (Cartan 1952ff., III, 616– 618). ———. 1923a. Sur les variétés à connexion affine et la théorie de la relativité généralisée. Annales de l’École Normale 40: 325–421. In (Cartan 1952ff., III, 659–746).

39 (Trautman

2006, 194) (Blagojevic and Hehl 2013, 108). 1948b/49, Scholz 2018). 41 The so-called teleparallel version of Cartan geometric gravity rearranges the coordi- nation between Noether currents and dynamical equations: Energy-momentum becomes the source of translational curvature and the spin current of the rotational curvature, rather than the other way round as in EC gravity. In oral communications D. Lehmkuhl has indicates that teleparallel gravity may be an interesting example of a principled (in contrast to temporal) underdetermination of empirically equivalent gravity theories. 40 (Weyl

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———. 1923b. Sur un théoréme fondamental de M. H. Weyl. Journal des Mathématiques pures et appliquées 2: 167–192. In (Cartan 1952ff., III, 633–658). ———. 1924. “La théorie des groupes et les recherches récentes de géométrie différentielle” (Conférence faite au Congrès de Toronto). In Proceedings of International Mathematical Congress Toronto. Vol. 1. Toronto 1928, 85–94. L’enseignement mathématique t. 24, 1925, 85–94. In (Cartan 1952ff., III, 891–904). ———. 1952. Oeuvres Complètes. Paris: Gauthier-Villars. Cogliati, Alberto. 2015. Continuous groups and geometry frorn Lie to Cartan. Preprint to appear In Mathematics: Place, production and publication, ed. J. Barrow-Green e.a., 1730–1940. Coleman, Robert, and Herbert Korté. 2001. Hermann Weyl: Mathematician, physicist, philosopher. In Hermann Weyl’s Raum – Zeit – Materie and a general introduction to his scientific work, ed. E. Scholz, 161–388. Basel: Birkhäuser. Deppert, W. e.a. (eds.). 1988. Exact sciences and their philosophical foundations. Exakte Wissenschaften und ihre philosophische Grundlegung. Vorträge des Internationalen HermannWeyl-Kongresses, Kiel 1985. Frankfurt/Main: Peter Lang. Weyl, Kiel Kongress 1985. Einstein, Albert. 1925. Einheitliche Feldtheorie von Gravitation und Elektrizität. In Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu, Berlin, 414–419. Friedman, Michael. 1999. Reconsidering logical positivism. Cambridge: University Press. Hehl, Friedrich. 1970. Spin und Torsion in der allgemeinen Relativi-tätstheorie oder die RiemannCartansche Geometrie der Welt. Habilitations-schrift, Mimeograph. Technische Universität Clausthal. Hehl, Friedrich W. 2017. Gauge theory of gravity and spacetime. In Toward a theory of spacetime theories, ed. D. Lehmkuhl e.a. Basel: Birkhäuser. Hehl, Friedrich, Paul von der Heyde, Kerlick G. David, and J.M. Nester. 1976. General relativity with spin and torsion: Foundations and prospects. Reviews of Modern Physics 48: 393–416. Kibble, Thomas. 1961. Lorentz invariance and the gravitational field. Journal for Mathematical Physics 2: 212–221. In (Blagojevic/Hehl 2013, chap. 4). Laugwitz, Detlef. 1958. Über eine Vermutung von Hermann Weyl zum Raumproblern. Archiv der Mathematik 9: 128–133. Nabonnand, Philippe. 2005. Correspondance Cartan – Weyl sur les connexions. Cartan to Weyl Oct 9, 1922, Jan 5, 1930, Dec. 19, 1930, Weyl to Cartan Nov. 24, 1930. https://hal.archivesouvertes.fr/hal-01095190. ———. 2016. L’apparition de la notion d’espace généralisé dans les travaux d’Élie Cartan en 1922. In Eléments d’une biographie de l’Espace géométrique, ed. L. Bioesmat Martagon, 313–336. Nancy: Editions Universitaires de Lorraine. O’Raifeartaigh, Lochlainn. 1997. The Dawning of gauge theory. Princeton: University Press. Riemann, Bernhard. 1919. Über die Hypothesen, welche der Geometrie zu Grunde liegen. Neu herausgegeben und eingeleitet von H. Weyl. Berlin etc.: Springer. Weitere Auflagen: 2 1919, 3 1923. Scheibe, Erhard. 1957. Über das Weylsche Raumproblern. Journal für Mathematik 197:162–207. Dissertation Universität Göttingen. ———. 1988. Hermann Weyl and the nature of spacetime. In Deppert e.a. 1988, 61–82. Scholz, Erhard. 2004. Hermann Weyl’s analysis of the “problem of space” and the origin of gauge structures. Science in Context 17: 165–197. ———. 2016. The problem of space in the light of relativity: The views of H. Weyl and E. Cartan. In Eléments d’une biographie de l’espace mathématique, ed. L. Bioesmat-Martagon, 255–312. Nancy: Edition Universitaire de Lorraine. arXiv:1412.0430. ———. 2018. Weyl’s search for a difference between ‘physical’ and ‘mathematical’ automorphisms. Studies in History and Philosophy of Modern Physics 61–1. arXiv 1510: 00156. Sciama, Dennis W. 1962. On the analogy between charge and spin in general relativity. In Recent developments in general relativity, ed. Festschrift for L. Infeld, 415–439. Oxford/Warsaw: Pergamon and PWN. In (Blagojevi´c/Hehl 2013, chap. 4). Sharpe, Richard W. 1997. Differential geometry: Cartan’s generalization of Klein’s Erlangen program. Berlin: Springer.

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Sigurdsson, Skúli. 1991. Hermann Weyl, mathematics and physics, 1900–1927. Cambridge, MA: PhD Dissertation, Harvard University. Trautman, Andrzej. 1973. On the structure of the Einstein-Cartan equations. Symposia Mathematica 12: 139–162. Relativitá convegno del Febbraio del 1972. ———. 2006. Einstein-Cartan theory. In Encyclopedia of mathematical physics, ed. J.-P. Françoise, G.L. Naber, S.T. Tsou, vol. 2, 189–195. Oxford: Elsevier. In (Blagojevi´c/Hehl 2013, chap. 4). Weyl, Hermann. 1918/1997. Gravitation and electricity. In The Dawning of gauge theory, ed. L. O’Raifeartaigh, 23–37. Princeton: University Press. (English translation of (Weyl 1918a)). ———. 1918a. Gravitation und Elektrizität. In Sitzungsberichte der Königlich Preußischen. Akademie der Wissenschaften zu Berlin, 465–480. In (Weyl 1968, II, 29–42) English in (O’Raifeartaigh 1997, 24–37). ———. 1918b. Raum, − Zeit – Materie. Vorlesungen über allgemeine Relativitätstheorie. Berlin: Springer. Further editioins: 2 1919, 3 1919, 4 1921, 5 1923, 6 1970, 7 1988, 8 1993. English and French translations from the 4th ed. in 1922. ———. 1918c. Reine Infinitesimalgeometrie. Mathematische Zeitschrift 2: 384–411. In (Weyl 1968, II, 1–28). ———. 1921. Raum, − Zeit – Materie. Vorlesungen über allgemeine Relativitätstheorie. Vierte, erweiterte Auftage. Berlin: Springer. ———. 1922a. Das Raumproblem. Jahresbericht DMV 31: 205–221. In (Weyl 1968, II, 328–344). ———. 1922b. Space – Time – Matter. Translated from the 4th German edition by H. Brose. London: Methuen. Reprint New York: Dover, 1952. ———. 1923a. Mathematische Analyse des Raumproblems. Vorlesungen gehalten in Barcelona und Madrid. Berlin: Springer. Reprint Darmstadt: Wissenschaftliche Buchgesellschaft 1963. ———. 1923b. Raum – Zeit – Materie, 5. Auflage. Berlin: Springer. ———. 1925/1988. Riemanns geometrische Ideen, ihre Auswirkungen und ihre Verknüpfung mit der Gruppentheorie, ed. K. Chandrasekharan. Berlin: Springer. ———. 1929a. Elektron und Gravitation. Zeitschrift für Physik 56: 330–352. In (Weyl 1968, III, 245–267). English in (O’Raifeartaigh 1997, 121–144). ———. 1929b. On the foundations of infinitesimal geometry. Bulletin American Mathematical Society 35: 716–725. In (Weyl 1968, III, 207–216). ———. 1948a. A remark on the coupling of gravitation and electron. Actas de la Academia Nacional de Ciencias Exactas, Fisicas y Naturales de Lima 11: 1–17. (not contained in (Weyl 1968)). ———. 1948b/49. Similarity and congruence: A chapter in the epistemology of science. ETH Bibliothek, Hochschularchiv Hs 91a:31, 23 Bl. In (Weyl 1955, 3rd ed. 2016). ———. 1949. Philosophy of mathematics and natural science. Princeton: University Press. 2 1950, 3 2009. ———. 1950. A remark on the coupling of gravitation and electron. The Physical Review 77: 699–701. In (Weyl 1968, III, 286–288). ———. 1955. Symmetrie. Ins Deutsche übersetzt von Lulu Bechtolsheim. Basel/Berlin: Birkhäuser/Springer. 2 1981, 3. Auflage 2017: Ergänzt durch einen Text aus dem Nachlass ‘Symmetry and congruence’. ———. 1968. Gesammelte Abhandlungen, 4 vols. Ed. K. Chandrasekharan. Berlin etc.: Springer.

Chapter 9

H. Weyl’s Deep Insights into the Mathematical and Physical Worlds: His Important Contribution to the Philosophy of Space, Time and Matter Luciano Boi

Nowhere do mathematics, natural sciences, and philosophy permeate one another so intimately as in the problem of space. Hermann Weyl, PMNS, 1949.

Abstract As it is well-known, Hermann Weyl pioneered two major conceptual trends in the mathematical and physical sciences. The first was the search for a unified theory of the forces of gravity and of electromagnetism. The second, closely related to the previous, was the search for a new geometrical framework appropriate for the elucidation of such a connection. According to Weyl, the first search is essentially dependent on the second, since a new theory of physical forces must rest upon the development of a new kind of geometry capable of explaining the structure of space-time at different scales. Two philosophical ideas underlies the Weyl’s program of geometrization of physics, namely that of emergence and that of the causal power of geometrical objects (see Wheeler JA: Am Sci 74:366– 375, 1986; Penrose R: Hermann Weyl’s space-time and conformal geometry. In: Hermann Weyl 1885 – 1985 centenary lectures. Springer, Berlin/Heidelberg, 1985; Boi L: Le problème mathématique de l’espace, with a foreword of R. Thom. Springer, Berlin/Heidelberg, 1995, Boi L: Synthese 139:429–489, 2004a, Boi L: Int J Math Math Sci 2004(34):1777–1836, 2004b, 2019). The first amount to say that many kinds of physical phenomena in nature emerge out from changes that can occur in the structures and dynamics of space-time itself. The second stresses the fact that geometrical concepts are involved in, rather than applied to, natural phenomena. This new geometric theory, which was first introduced by Weyl in 1918 (Weyl H: Sitzungberichte der Königlichen Preussische Akademie der Wissenschaft, L. Boi () Ecole des Hautes Etudes en Sciences Sociales, Centre de Mathématiques, Paris, France e-mail: [email protected] © Springer Nature Switzerland AG 2019 J. Bernard, C. Lobo (eds.), Weyl and the Problem of Space, Studies in History and Philosophy of Science 49, https://doi.org/10.1007/978-3-030-11527-2_9

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Berlin 26:465–480, 1918a) and thereafter in 1928 (Weyl H: Gruppentheorie und Quantenmechanik. Hirzel, Leipzig, 1928) within the context of quantum mechanics, was grounded on the idea of gauge invariance, or a non-integrable scale factor, which in some formulations of quantum mechanics, especially in those given by Aharonov and Bohm in 1959, can be translated in a phase factor. In 1954, the physicists Yang and Mills rediscovered the Weyl’s gauge principle and developed it within a different physical context and a new mathematical framework. They proposed that the strong nuclear interaction be described by a field theory like electromagnetism, which is exactly gauge invariant. They postulated that the local gauge group was the SU(2) isotopic-spin group. This idea was revolutionary because it changed the very concept of ‘identity’ of an elementary particle. The novel idea that the isotopic spin connection, and therefore the potential, acts like the SU(2) symmetry group is the most important result of the Yang-Mills theory. This concept shows explicitly how the gauge symmetry group is built into the dynamics of the interaction between particles and fields (see Atiyah 1979, 1997). Keywords Geometry · Connection · Gauge theory · Spinors · Orthogonal group · Geometric algebra · General relativity · Quantum mechanics

9.1 Introduction Hermann Weyl was an outstanding mathematician, physicist and philosopher (Wheeler 1986; Penrose 1985). Moreover he thinks that beauty, as a primary aesthetic criterion, plays a fundamental role in the sciences. Science is concerned with the search of dynamical geometric invariants and relationships within the mathematical realm of concepts; besides, according to Weyl, there must be a deep connection between these concepts and physical phenomena. Weyl places this search at the core of his mathematical research and philosophical investigation, which furthermore are profoundly related and merged in some meaningful form of Natural Philosophy. For that reason, Weyl described Einstein’s discovery of general relativity as a supreme example of the power of speculative thought (Weyl 1918b). And Weyl himself has written: “My work always tried to unite the true with the beautiful; but when I had to choose one or the other, I usually chose the beautiful” (PMNS 1949). The example that Weyl gave was his gauge theory of gravitation. Apparently, Weyl became convinced that this theory was not true as a theory of gravitation (1918, 1929); but still it was so beautiful that he did not wish to abandon it and so he kept it alive for the sake of its beauty. But much later, it did turn out that Weyl’s intuition was right after all, when the formalism of gauge invariance was incorporated into quantum electrodynamics (Weyl 1929; Yang 1989). Another example is Weyl’s two-component relativistic wave equation of the neutrino. Weyl discovered this equation and the physicists ignored it for some 30 years because it violated parity invariance. And again, it turned out that Weyl’s intuition were right. According to Weyl, one has to make an effort for showing the profound unity and the likely ever-changing character of knowledge. For him, it is important to

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recognize that “the realm of Being is not closed, but open” (1954). Michel Atiyah has written: “The long-term significance of Weyl’s work is assessed by the important influence that his ideas has had on his successors and by the fact that they helped to shape mathematics and physics in the second half of the twentieth century. In fact, the last 50 years has seen a remarkable blossoming of just those areas that Weyl initiated” (Atiyah 2002, 4). It is worth to stress that the philosophy of Weyl (Weyl 1949; 1918b), deeply inspired by the phenomenological ideas of Husserl, has strongly influenced his mathematical and physical research. More precisely, his deep philosophical views make an important difference to his way of thinking and practicing mathematics and physics. Weyl was one of the rare mathematicians who tackled with profound and significant originality difficult problems in mathematical and physics, as well in philosophy. He gave major contributions to the understanding of the interactions between geometry and physics, especially in relation with group theory and quantum mechanics. (see Weyl 1928, 1931) The most important Weyl’s contribution to physics (1918, 1929) was the idea of gauge invariance. But Weyl work contained a host of other important ideas related to this. One of these was the idea of a connection in differential geometry, as something which can be defined independently of a choice of a metric on the space (see Bourguignon 1992; Regge 1992). An intrinsic property of a manifold can be defined independently of the surrounding space and independently of the particular choice of charts, i.e. of a metric. Given a connection on a manifold, one can define tangent vectors, parallel transport and curvature, which are intrinsic properties (Ricci and Levi-Civita 1901). Moreover, one obtains a covariant differentiation on manifolds. For every Riemannian manifold there exists a connection. The notion of connection, however, can be explained independently of the existence of a metric (affine connection, principal fiber bundles, vector bundles) (Kobayaschi 1957; Husemoller 1966). Weyl proposed a purely infinitesimal geometry. He suggests to separate logically the concept of parallel displacement from metrics and to introduce an affine connection of a differentiable manifold as a linear torsion-free connection (Weyl 1919). The past four decades have seen the rise of gauge theories, Kaluza-Klein models of high dimensions, string theories, and now M theory. This requires sophisticated mathematics involving Lie groups, manifolds, differential operators, all of which are part of Weyl’s inheritance. There is no doubt that he would have been an enthusiastic supporter and admirer of this fusion of mathematics and physics. No other mathematician could claim to have initiated more of the theories that are now being exploited. His vision has stood the test of time. The theory of continuous groups, developed by the nineteenth-century Norwegian mathematician Sophus Lie has been continued and extensively developed by Elie Cartan (Cartan 1908). Weyl took up the topic anew and brought his own point of view, with its emphasis on the global aspect of Lie groups. For his predecessors all the essential formulae were local (leading to the infinitesimal form, the Lie algebra), but Weyl emphasized the whole group, a manifold with, in particular, interesting

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topology (Weyl 1924). Here we see a link with his approach to Riemann surfaces: Weyl liked to see the big picture, the manifold or group in the round. This global view has many technical advantages and in particular for compacts groups (such as the important group of rotations), one could average by integrating over the group (Weyl 1946; Borel 1985). Essentially this made the theory very similar to that of finite groups, which was already well established. One famous consequence of this technique is the Peter-Weyl theorem, which decomposes the space of functions on the group into matrix blocks given by irreducible representations (Peter and Weyl, 1927). In his famous thesis on Lie groups and Lie algebras, Elie Cartan (1869–1951) completed the classification of semi-simple Lie algebras initiated by Wihlem Killing (1847–1923) in the late 1880s. The representation theory for compact Lie groups and its relation to functional analysis was created by Weyl in the 1920s. A highlight is the Peter-Weyl theorem. Finite groups are special cases of compact Lie groups. For example, the rotation group SO(3) of the 3-dimensional Euclidean space of the gauge group (an unitary product group) SU(3) × SU(2) × U(1) of the Standard Model in elementary particle physics are compact Lie groups. Motivated by Dirac’s theory of the relativistic electron (Dirac 1928) and E. Cartan’s geometric spinors (Cartan 1938, 1966), Brauer and Weyl wrote the fundamental paper on spinors in n dimensions (1935). Here they used Clifford algebras in order to construct the universal covering group Spin(n) of the n-dimensional rotation group SO(n), n = 3, 4, . . . This paper represents the geometric-algebraic core of modern spin geometry (see Lawson and Michelsm 1994).

9.2 From Riemann to Weyl and Beyond The vision of geometry has evolved radically from the second half of the nineteenth century until the early twentieth century, thanks to the works of Riemann (Riemann 1854), Clifford (Clifford 1876, 1882), Beltrami, Helmholtz, Klein, Lie and Poincaré, then of Hilbert, Cartan and Weyl. During this period, geometry knows the more fundamental transformation of its history. The relationship of geometry with other branches of mathematics, especially with algebra and analysis, as well as its connection with physics and other natural sciences, has known a very deep change. Therefore we will no longer speak of geometry nor of space, but of geometries and of spaces. The recognition of a plurality of geometries in 3 and n-dimensional spaces constituted a major step for the advance of mathematics in many different yet related directions. For the first time, with the discovery of non-Euclidean geometries, the conception of a single geometry and of an absolute space is completely questioned, in favor of another, radically different. In fact, as science of «pure» forms, geometry belongs to mathematics at the same level as arithmetic and algebra; while as science of real forms, it is intimately linked to physics (Boi et al. 1992a, b; Boi 1995). Bernhard Riemann was undoubtedly the first to mathematically expose the double nature of space. In his landmark 1854 dissertation, “On the hypothesis

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which are at the basis of geometry” (“Über die Hypothesen welche der Geometrie zu Grunde liegen”), he introduced completely new mathematical ideas, whose philosophical value and the meaning to physics appear revolutionary for the time. Following Gauss, but generalizing considerably his intuitions, Riemann shows that the Euclidean space, from a purely mathematical point of view, was no more than a particular case among other possible spaces, and that there was no reason to think that the physical space corresponded to the one described by the axioms of Euclidean geometry. Consequently, there could exist not only several geometries, but also several geometrical spaces (kinds of manifolds) and several different physical spaces. It was certainly a turning point, which deeply and forever changed the landscape of mathematics. Particularly, the concept of manifold (Mannigfaltigkeit) Riemann introduced is at the origin of modern differential geometry as a new field of mathematics. The geometrical concept of manifold maintains an essential link with the functional concept of «Riemann surface», both of which can be explained through a qualitative or spatial conception of mathematics. The concepts of Riemann’s manifold (Riemann 1954) and Clifford’s spatial theory of matter (Clifford 1976) are the starting point of a fruitful movement of geometrization of physics, which culminated in Einstein’s general theory of relativity (Einstein 1916; Hilbert 1924). The existence of several geometries which are also carried out from the point of view of physics (other than the mathematical), was shown in a decisive way thanks to Einstein’s general theory of relativity, even though especially Riemann and Clifford had already admitted that a geometry other than the Euclidean could be applied to our physical space. But to come to such admission, it was first necessary to deeply criticize our conception of space, which could no longer be thought, nor as the place where figures can be constructed, nor as the one where bodies move. In his fundamental work about the hypothesis of geometry, Riemann showed that the property of continuity is linked to the metrical structure of space, which means that each point, as well as its infinitesimal variations, is representable by a continuous function of its differentials. Moreover, he demands such functions to be continually differentiable, which defined the differentiable level of the continuum, after he had recognized the existence of a first topological level of continuity, which could be designated by that of dimensionality – which can equally be expressed by saying that the world we inhabit is a spatial continuum of three dimensions (or a tridimensional manifold). But Riemann sets forth the possibility that there is a third level of the continuum, whose nature isn’t at all analogous to the others we have just mentioned, and whose constitutive principles do not take part in the way we abstractly represent them, as it is the case for the discrete manifolds, like the arithmetic or algebraic manifolds, which are composed by numerable elements, while the continuous manifolds composed by points are measurable through functions of distance. According to Riemann, continuous manifolds (like the differentiable manifolds) could have an origin of dynamic nature, that is, the property of continuity would be linked to the physical content of space. In other words, the physical phenomena and the kind of space in which they take place are

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inseparable: the space imagined by Riemann is non-empty (differently from the one thought by Newton) and endowed with physical effects which would propagate locally (Riemann 1854; Christoffel 1869; Ehlers 1983). Following Riemann, Clifford explicitly theorized a coherent program for a geometric interpretation of the physical phenomena. Clifford takes over Riemann’s intuition and states the hypothesis that physical space (both at macroscopic as microscopic scale) is neither homogeneous nor isotropic, and that it can indeed be curved and not flat (contrary to what asserted by Euclidean geometry) and susceptible of variation under the presence of certain physical effects, and that, furthermore, the behavior of matter depends on how the curvature of space varies. It is worth noting that the spatial theory of curvature and of matter developed by Clifford has played an important role in the development of general relativity until a recent period, particularly in the elaboration of J.A. Wheeler’s Geometrodynamics theory (Wheeler 1962). Riemann’s influence (and Clifford’s indirectly) upon the new physics, and particularly on Einstein’s general relativity, has been tremendous (Bio 2004a, b, 2006a). Actually, the latter’s general relativity is grounded on the concept of Riemann’s differentiable manifold, which was endowed with a nonEuclidean metric (hyperbolic, elliptic or other), and with complicated geometrical objects which we call curvature tensors (Ricci and Levi-Civita 1901). These are geometrical objects that also have a physical meaning, provided that they correspond to the gravitation potential of general relativity (Regge 1992). That means, in other terms, that the properties of the phenomena which occur at the scale of our Universe lie in the geometrical and topological structure of a pseudo-Riemannian manifold, Einstein’s space-time constituting its physical model par excellence.

9.3 The Structure of Geometric Objects In geometry, a lot of efforts have been devoted during last century to provide a “normal form” to certain geometric objects. These work endeavours to understand the structure of geometric objects by subtle investigation of their algebraic and topological properties. Embeddings of surfaces, spaces and maps are the most interesting cases of such a concept. A well-known example is the Weyl’s embedding theorem which says that any abstractly defined closed surface with positive curvature can be isometrically embedded in R3 as a convex surface. Another more recent and beautiful example is the geometric concept of moduli spaces, maybe the most farreaching area of research in algebraic geometry and topology. The whole idea of a moduli problem is related to the search of classifying something new in geometry. Intuitively, Riemann’s moduli space Rp is the space of analytic of equivalence classes of Riemann surfaces of fixed genus p, or, more generally, of the objects that one wants to classify, but with some natural geometrical structure on it that reflects how these objects vary. A very important challenge today is to understand the topology of these moduli spaces that arise in algebraic geometry and a lot of them will have all sorts of nasty singularities. All continuous maps between two

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spaces form themselves an infinite-dimensional space, for example the space of loops on a manifold. Second kind of important spaces are classifying spaces of groups. To understand the homotopy or cohomology groups of mapping or classifying spaces is the key to many problems ranging from differential geometry to theoretical physics (see Atiyah and Jones 1978; Witten 1988). Moduli spaces of Riemann surfaces are historically the first examples of moduli spaces and till today of great importance for differential and algebraic geometry, as well as for theoretical physics (Atiyah and Bott 1982). They are the deformation space of all possible conformal structures on a surface of fixed topological type. If the surface has boundary, then the moduli space is a smooth manifold and the classifying space of the torsion-free mapping class group. The theory of moduli spaces, for those interested in geometry and topology, led to the following fundamental question: “what is the space of spaces?” A particular case of this general question comes to study the space of all the possible surfaces that could exist within a large Euclidean space, what may be called “the space of surfaces”. This mathematical object seems to be profoundly linked to the attempts made especially by string theorists to understand a universe made up of many more dimensions than those we are familiar with. Let us return to the “normal form” concept, which play a fundamental and unifying role in all area of modern geometry. In the process of finding a normal form, there are local problems and global problems. Let us make here few remarks only on the problem of uniformization of geometric structures on smooth manifolds. Classically we have the eminent uniformization theorem that any conformal structure on a surface can be normalized in such a way that it is the quotient of S2 , R2 or D2 by a group of conformal transformations. Other stated, one has the Killing-Hopf theorem, which classified all complete, connected surfaces that can be achieved as quotients of the Euclidean plane R2 by groups of isometries. According to the uniformization theorem, every surface can be realized as such a quotient of one of just three model geometries: the plane R2 , the sphere S2 , and the hyperbolic plane H2 . As a consequence, it can be given a metric of constant curvature. And the model geometry is uniquely determined (in most cases) by the topology of the surface. While the program of “uniformization” originated from function theoretic considerations, here we shall put the accent on a geometric viewpoint. It should, however, be stressed that the function theoretic method is actually an important tool in a major part of the program.

9.4 Origins and Developments of Geometric Algebra For the moment, let us present in an historical vein a brief overview of Clifford algebras and some of their applications (for a detailed presentation, see Lounesto 1996, and Trautman 1997). Clifford developed Grassmann’s ideas into what he called geometric algebras (Clifford 1876; Boi 1997). Clifford algebras entail a new multiplication rule for vectors in Grassmann’s exterior algebra Rn . In the

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special case of R3 this construction embodied Hamilton’s quaternions. Clifford algebra were independently rediscovered by R. O. Lipschitz (1880–1886), who also gave their first application in geometry, namely the representation of rotations in Rn . Spinor representations of rotation group SO(n), or more precisely of the spin group Spin(n), were introduced by E. Cartan (1913) and then generalized by Brauer and Weyl in 1935 (Brauer and Weyl 1935) (for a comprehensive historical and mathematical account, see Akivis and Rosenfeld 1993). Cartan (1908) also discovered the periodicity of 8 in matrix representations of real Clifford algebras (rediscovered by Atiyah, Bott, and Shapiro in 1964), and Brauer and Weyl also presented multiplication of Clifford numbers by binary indices (this was discovered earlier by Veblen in 1897). Veblen and Young (1910) introduced a representation of Möbius transformations of Rn by 2 × 2 matrices in n (2) with entries in the Clifford algebra n of Rn (or more precisely in 0,n of R0, n ). The first one to associate Clifford algebras with quadratic forms was E. Witt in 1937, who determined Clifford algebras of non-degenerate quadratic forms over arbitrary fields of characteristic = 2. The generalization to the exceptional characteristic 2 was given by C. Chevalley (1955) in is construction of (Q) ⊂ End(V). Chevalley went further and gave the most general definition, (Q) = ⊗V/Q , valid not only for fields, but also for commutative rings. Riesz (1957) reconstructed Grassmann’s exterior algebra from the Clifford algebra, in any characteristic = 2, by x∧u=

 1 xu + (–1)k ux 2

where x ∈ V and u ∈ k V. Chevalley related exterior products of vectors to antisymmetric Clifford products of vectors, but his relationship was valid only in characteristic 0 (Chevalley 1946). The modern period in the history of spinors begins probably with the papers by Cartan (1913, 1914) containing a description of the spin representation of the Lie algebras of orthogonal groups. The discovery of the spin of the electron around 1925 forced physicists to find mathematical tools to describe, within the framework of quantum mechanics, this new degree of freedom. When doing this, W. Pauli (1927) and P. Dirac (1928) rediscovered the spin representation and the Clifford algebras associated with three- and four-dimensional vector spaces, respectively. More precisely, a Clifford algebra common in physics is 3 , the Pauli algebra, based on vectors in three-dimensional Euclidean space R3 . A conventional representation uses the 2 × 2 Pauli spin matrices  σ1 =

     01 0 −i 1 0 , σ2 = , σ3 = 10 i 0 0 −1

(9.1)

to represent unit vectors along the Cartesian axes. A vector v with Cartesian components vx , vy , vz , is thus represented by the matrix components vx σ 1 , vy σ 2 , vz σ 3 . An infinite number of different matrix representations may be used, but only the algebra of their products (which they have in common) is physically

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significant. Dirac introduced, at that time, in the context of Minkowski space, the differential operator that now bears his name; the Dirac equation has been very successful in describing the quantum-relativistic behavior of electrons and other particles with spin ½. Soon afterwards, Weyl and Fock developed a local theory of the Dirac operator in curved (Lorentzian) space-time. (For a detailed account on those physical works, see Dirac 1930, Lee 1982, and Itzykson and Zuber 1985).

9.5 The Link Between Group Theory, Geometric Algebras and Quantum Mechanics The investigation by Weyl of the connection between group theory, geometric algebras and quantum mechanics has been one the main contribution to the development of mathematics and theoretical physics in the first half of twentieth century. Let us explain briefly the most important ideas (for a detailed account, see Borel 1985). After its fundamental contributions to the clarification of the deep geometrical structures and the broad physical meaning of general relativity theory, around 1927, Weyl got involved with the applications of group representations to quantum mechanics. He provided a systematic and impressive exposition of this subject in (Weyl 1928). I shall go on confining myself mostly to Lie groups and Lie algebras. As far as Weyl was concerned, the main mathematical contribution stemming from it is the paper on spinors written jointly with R. Brauer and H. Weyl (1935). As we just saw, infinitesimally, Cartan had already described the spinor representations in 1913 by their weights. But Weyl gave a global definition, based on the use of the Clifford algebras, itself suggested by Dirac’s formulation of the equations for the electron. However, the most unexpected fall-out originated with a physicist, H. Casimir, and led to what was viewed at the time, erroneously, as the first algebraic proof of the complete reducibility theorem. In the representations of g = sl 2 (C), or equivalently so 3 (C), an important role in the quantum theoretic applications is played by a polynomial of second degree in the elements of g , which represents the square of the magnitude of the moment of momentum, the sum of the squares of the infinitesimal rotations around the coordinate axes. It commutes with all of g , and hence is given by a scalar in any irreducible representation: this yields an important quantum number j(j + 1), in the representation of degree 2j + 1 (2j ∈ N). Casimir was struck by this commutation property and defined in 1931 an analogous operator for a general semi-simple Lie algebra, called latter on the Casimir operator, and indicated how it would allow one to derive the PeterWeyl theorem from results about self-adjoint elliptic operators (we give a formal definition of this important theorem later on in this paper). A year later, he noticed that in the case of sl 2 , it could be used to give a purely algebraic proof of full reducibility, which was later extended to the general case by B. L. van der Waerden. Afterward, Weyl struggled to develop a purely algebraic theory of Lie algebra, valid at least over arbitrary fields of characteristic zero. In particular:

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(i) He gave a classification of the representations of SLn (C) and GLn (C). After having determined all the holomorphic irreducible representations of SLn (C), he pointed out that the matrix coefficients are in fact polynomials in the matrix entries, and that these representations are the irreducible constituents of the representations of GLn (C) in the tensor SLn (C) algebra over Cn . They are therefore just the tensor spaces, described by means of symmetry conditions on the coefficients. In this Weyl saw the group theoretic foundation of tensor calculus. Weyl tried to extend this algebraic treatment to other classical groups. It later became of even greater interest to him in view of its applications to quantum theory. (ii) He mainly contributed to develop modern “invariant theory” (Weyl 1939). In broad terms its general problem is, given a group G and a finite dimensional representation of G in a vector space V, to study the polynomials on V which are invariant under G (or sometimes only semi-invariant, i.e. multiplied by a constant under the action of a group element). The questions which are usually asked are whether the ring of invariants is finitely generated (first main theorem), and, if so, whether the ideal of relations between elements in a generating set is finitely generated (second main theorem). In concrete situations, one will of course want an explicit presentation of the ring of invariants in terms of generators and relations. One may also look for the dimension of the space of homogeneous invariants of a given dimension (the “counting of the number of invariants”). Such a formulation, however, where G and V are general, emerged at a later stage of the theory, as an abstraction from the classical invariant theory.

9.6 The Classification of Classical Groups The families of linear groups, unitary groups, orthogonal groups and symplectic groups are called classical groups, whose study is one the main objects of Lie group theory. The unitary and orthogonal groups have the fundamental property of being compact. More precisely, this result as been stated and proved as a theorem by C. Chevalley in his fundamental book Theory of Lie Groups (1946). The theorem says that the groups (or spaces) U(n), O(n), SU(n) and SO(n) are compact. Since O(n), SU(n) and SO(n) are closed subsets of U(n), it is sufficient to prove that U(n) is compact. (See below for a detailed explication of these theorem). H. Weyl gave their full classification in the landmark book The Classical Groups (1939). (See also the outstanding books of Chevalley (1946), and Dieudonné (1948) for a complete exposition). Weyl characterized its task precisely as follows: with respect to the assigned group of linear transformations in the underlying vector space, to decompose the space of tensors of given rank into its irreducible invariant subspaces. The main concern is with the various “quantities” obeying a linear transformation law. Such is the problem which forms one of the mainstays of the book, and in accordance with the algebraic approach its solution is sought for not only in the

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field of real numbers, but in arbitrary field K of characteristic zero. Generally speaking, K is any set of elements α, called numbers, closed with respect to the two binary operations: addition and multiplication. Addition and multiplication are supposed to be commutative and associative; however, these two conditions are not required in the case of the fields or normed division algebras of quaternions H and of octanions O, which are non-commutative and non-associative, respectively. Moreover, addition shall allow of a unique inversion (subtraction), i.e., there is a number 0 such that α + 0 = α for every α, and each α has a negative –α satisfying α + (−α) = 0. Multiplication shall fulfill the distributive law with respect to addition: α(β + γ ) = (αβ) + (αγ ), from which one readily deduces the universal equation α · 0 = 0. Multiplication also is required to be invertible (division) with the one expression necessarily imposed by α · 0 = 0: there shall exist a unit 1 satisfying α · 1 = α for all α, and every α except 0 shall have an inverse α −1 or 1/α such that α · α −1 = 1. Were 1 = 0, all numbers α would be = 0 according to the previous; this degenerate case will be here excluded by the axiom 1 = 0. Consequently, as Weyl point out, the notion of an algebraic invariant of an abstract group γ cannot be formulated until we have introduced the concept of a representation of γ by linear transformations, or the equivalent concept of a “quantity of type .” According to Weyl, the problem of finding all representations or quantities of γ must therefore logically precede that of finding all algebraic invariants of γ . The H. Weyl’s theorem for the orthogonal group is of paramount importance since it is strongly connected with several geometrical and topological subjects, in particular with the Bott periodicity theorem and the Atiyah-Singer theorem. We review Weyl’s theorem briefly. Let V be a real vector space with a fixed inner product. Let O(V) denote the group of linear maps of V → V which preserve this inner product. Let ⊗k (V) = V ⊗ . . . ⊗ V denote the kth product of V. If g ∈ O(V), we extend g to act orthogonally on ⊗k (V). We let z → g(z) denote this action. Let f : ⊗k (V) → R be a multi-linear map, then we say f is O(V) invariant if f (g(z)) = f (z) for every g ∈ O(V). By letting g = −1, it is easy to see there are no O(V) invariant maps if k is odd. We let k = 2j and construct a map f0 : ⊗k (V) = (V ⊗ V) ⊗ (V ⊗ V) ⊗ . . . ⊗ (V ⊗ V) → R using the metric to map ⊗ (V ⊗ V) → R. More generally, if ρ is any permutation of the integers 1 through k, we define z → zρ as a map from ⊗k (V) → ⊗k (V) and let fρ (z) = f0 (zρ ). This will be O(V) invariant for any permutation ρ. Weyl’s theorem states that the maps {fρ } define a spanning set for the collection of O(V) invariant maps. For example, let k = 4. Let {vi } be an orthonormal basis for V and express any z ∈ ⊗ k (V) in the form aijkl vi ⊗ vj ⊗ vk ⊗ vl summed over repeated indices. Then after weeding out multiplications, the spanning set is given by f0 (z) = aiijj , f1 (z) = aij ij , f2 (z) = aijj i where we sum over repeated indices. f0 corresponds to the identity permutation; f1 corresponds to the permutation which interchanges the second and third factors; f2 corresponds to the permutation which interchanges the second and fourth factors. We not that these need not be linearly independent; if dim V = 1 then dim(⊗4

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V) = 1 and f1 = f2 = f3 . However, once dim V is large enough these become linearly independent. We are interested in p-form valued invariants. We take ⊗k (V) where k – p is even. Again, there is a natural map we denote by f p (z) = f0 (z1 ) ∧  (z2 ) where we decompose ⊗k (V) = ⊗k–p (V) ⊗ ⊗p (V). We let f0 act on the first k – p factors and then use the natural map ⊗p (V) → p (V) on the last p factors. If ρ is a permutation, we set fρ p (z) = fp (zρ ). These maps are equivariant in the sense that fρ p (gz) = gfρ p (z) where we extend g to act on p (V) as well. Again, these are a spanning set for the space of equivariant multi-linear maps from ⊗k (V) to p (V). If k = 4 and p = 2, then after eliminating duplications this spanning set becomes: f1 (z) = aiij k vj ∧ vk , f2 (z) = aij ik vj ∧ vk , f3 (z) = aij ki vj ∧ vk f4 (z) = aj iki vj ∧ vk , f5 (z) = aj iik vi ∧ vk , f6 (z) = aj kii vj ∧ vk . Again, these are linearly independent if dim V is large, but there are relations if dim V is small. Generally speaking, to construct a map from ⊗k (V) → p (V) we must alternate p indices (the indices j, k in this example) and contract the remaining indices in pairs (there is only one pair i, i here). We are now ready to state the Weyl’s theorem on the invariants of the orthogonal groups. Theorem The space of maps {fρ p } constructed above span the space of equivariant multi-linear maps from ⊗k V → p V. (For the proof and further details, we refer to Weyl 1946, Dieudonné 1948, and Deheuvels 1981). The most important fact about compact groups is the Peter-Weyl theorem, which is essentially the assertion that any compact Lie group is isomorphic to a subgroup of some unitary group Un . A compact Lie group possesses the property of being compact under both left and right translations. When a compact Lie group G acts linearly on a finite dimensional vector space V there is always a positive definite inner product on V which is invariant under G (see Borel 1985; Chevalley 1946). The existence of an invariant inner product, in turn, implies that V is the orthogonal direct sum of subspaces on which G acts irreducibly. Let us apply the preceding remark when V is the Lie algebra of G. The group G acts on by the adjoint representation: the adjoint action of g ∈ G is defined as the derivative of the map x → gxg−1 at the identity element x = 1. We find

(9.2) where G acts irreducibly on each i . It is immediate that the i are sub-Lie-algebra, and that [ i , j ] = 0 when i = j. If Gi is the subgroup of G corresponding to i then G is locally isomorphic to the product G1 × . . . × Gk . The groups Gi into

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which we have decomposed G clearly have no non-trivial connected subgroups. Apart from the circle group T such groups are usually called simple groups – although the terminology is not ideal as the groups can possess finite normal subgroups (necessarily contained in the center). Thus any compact Lie group is locally isomorphic to a product of circles and simple groups. If there are no circles in the decomposition then the group is called semi-simple.

9.7 The Weyl Spinors The semi-spinors ψ ± are called Weyl spinors, or chiral spinors. The Weyl spinors can carry a reducible representation of the real even sub-algebra (Brauer and Weyl 1935; Trautman and Trautman 1994). This occurs when p – q = 0 mod 8. In this case the real even sub-algebra is the direct sum of two real matrix algebras, having the real central idempotents P± = ½(1 ± z). The “Majorana condition”1 can be consistently imposed together with the “Weyl condition”2 to decompose a Dirac spinor into subspaces transforming irreducibly under the real even sub-algebra. The resulting spinors are called Majorana-Weyl spinors. In an odd number of dimensions irreducible representations of the complexified Clifford algebra induce irreducible representations of the even sub-algebra. These can induce a reducible representation of the real, even sub-algebra. Obviously this is the case for p – q = 1 mod 8 where, as we have noted, Dirac spinors carry a reducible representation of the whole real sub-algebra. For p – q = 7 mod 8 Dirac spinors carry irreducible representations of the real sub-algebra and the even sub-algebra. However they carry a reducible representation of the real even sub-algebra. For p – q = 7 mod 8,

1 When

the Dirac spinors carry a reducible representation of the real subalgebra, elements of the irreducible subspaces are called Majorana spinors. This occurs when the real subalgebra is a real matrix algebra, or a sum of two such algebras, and this occurs when p – q = 0, 1, 2 mod 8. In these dimensions the space of Dirac spinors can be decomposed in eigenspaces of the charge conjugation operator. Thus a Majorana spinor is an eigenspinor of the charge conjugation operation  ψ = ±ψ c . This can be written in terms of the Dirac and Majorana conjugates by using ψ c = Cψ  or ψ c = Dψ . In an even number of dimensions the irreducible representations of the complex Clifford algebra induce a reducible representation of the even subalgebra; the spinor representation splitting into two inequivalent semi-spinor representations of the even subalgebra. The central idempotents that project the even subalgebra into simple components are P± = ½(1 ± ž) where either ž = z or ž = iz ensuring z2 = 1, z denoting the volume n-form. 2 If ψ is a Dirac spinor then it may be decomposed into subspaces that transform irreducibly under the even subalgebra, ψ = ψ + + ψ − , where ψ ± = P± ψ. The semi-spinors ψ ± are called Weyl spinors, or chiral spinors. The Weyl spinors can carry a reducible representation of the real even subalgebra. This occurs when p – q = 0 mod 8. In this case the real even subalgebra is the direct sum of two real matrix algebras, having the real central idempotents P± = ½(1 ± z). The “Majorana condition” can be consistently imposed together with the “Weyl condition” to decompose a Dirac spinor into subspaces transforming irreducibly under the real even subalgebra. The resulting spinors are called Majorana-Weyl spinors.

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∼ = C ⊗ M2 (n–1)/2 (R) and, + p,q (R) ∼ = M2 (n–1)/2 (R) where p + q = n. We may thus choose a matrix basis for the Clifford algebra in which the automorphism η simply complex conjugates the components. The complexified algebra C p,q is reducible, with η interchanging the simple components. A Dirac spinor ψ can be decomposed into spinors transforming irreducibly under the real even sub-algebra. Let us now pursue a more geometric approach to spinors. From the Clifford algebra we can define the spin groups, and from the representations of the algebra we induce representations of these groups. One can, however, start from a knowledge of the covering group of the connected component of the orthogonal group and introduce its irreducible representations as spinors (see Lawson and Michelson 1994). Representations of the component of the orthogonal group connected to the identity can then be found from the tensor product of these spinor representations. For the case of four dimensions, and Lorentz signature has been developed into the Infeld–van der Warden formalism, or “two-component spinor formalism”. Given that the double covering of SO+ (3, 1) is SL(2, C) one introduces “two-component spinors” as carrying irreducible representations of SL(2, C). The complex conjugate representations of this group are inequivalent, and a special notation is used to distinguish them. If u is a vector carrying an SL(2, C) representation such that the components of u transform with a matrix m then, say, the components of u are labeled by a Greek superscript. If the vector v transforms with the complex conjugate matrix then the components of v are labeled by a Greek superscript with a dot above it. The vector spaces carrying these representations both admit SL(2, C)-invariant symplectic products, and the adjoint of u, say, with respect to such a product has its components with respect to a dual basis written as subscripts. A similar situation holds for v. Thus indices are “lowered” with the symplectic matrix, which can be taken to have plus one in the top right-hand entry. Because of the antisymmetry of this matrix a convention must be adopted as to which side the matrix is multiplied from to lower an index. The tensor product of these two representations, with themselves and each other, gives a representation of SO+ (3, 1). Thus SO+ (3, 1) irreducible representations are identified without dots, up and down, or a mixture. Of course, starting with SL(2, C) irreducible representations only produces SO+ (3, 1) representations, not O(3, 1) representations. One can extend the representations of SL(2, C) to include other transformations so that the tensor representation extends to a representation of O(3, 1). However, such extensions are not unique and there is certainly no universal convention for complex phase factors (see Yang 1983). p,q (R)

9.8 The Concept of Symmetry and of Broken Symmetry, from Geometry to Topology Weyl has made fundamental contribution to the conceptual extension of the concept of symmetry and to its philosophical clarification (Yang 1989). In particular, the

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concept of symmetry is at the very core of his gauge theory for quantum fields. In an article of 1927, then in his book Gruppentheorie und Quantenmechanik (1928), he proposes developing the mathematical foundations of this newly discovered physical theory by showing its close relationship to group representation theory. In Weyl’s new mathematical approach, the basic question at that time was to explain the properties of particles (protons and electrons) by the properties of the quantum laws: do these laws satisfy the basic symmetries known at that time (right/left, past/future, positive/negative electric charge)? Mathematically, that was equivalent to knowing the structure of certain classes of (continuous) groups and their algebras. These three kinds of symmetry were introduced (under other names) into quantum physics around the l930’s by Weyl himself and by E. Wigner, but no one thought then of unifying the three kinds. In l930 P. Dirac had detected the existence of a particle (positron) with a charge opposite that of an electron, and Weyl then generalized to a universal essential equivalence between positive and negative electricity. This idea was reformulated in l937 as the conjugate invariance of electrical charge. However, in l957 Lee and Yang found that left-right symmetry (or conservation of parity P), which physicists had always found useful to accept, was not entirely satisfied by the laws of nature, particularly in weak interactions, which are responsible for radioactive (beta) disintegration (see Taylor 1976). Since it could be verified theoretically and experimentally that this radioactivity gave a correct description of the neutrino, the conclusion was that the existence of the Weyl-Pauli theory (of the neutrino) violated left-right symmetry (see). This asymmetry seemed to be a consequence of duplication: massless particles (neutrinos) emitted in a beta disintegration existed in only one form (left), while the corresponding anti-particles (anti-neutrinos) could then only exist in the opposite form. Mathematically, this duplication could appear as the existence of two valid solutions for an equation. Some theoretical physicists interprets this phenomenon to speculate that the world did not have to be symmetrical with respect to every operation which left the laws of nature invariant: the loss of symmetry could be ascribed to the asymmetry of the whole universe. Such an explanation raises several questions. It is just as reasonable to believe that the loss of symmetry, as a characteristic of a transitory phase in which the laws of nature apply, could be explained by a richer, more general mathematical symmetry (see Salam 1960). Recent research in this field seems to be oriented toward this second outlook.

9.9 The Birth of Gauge Theory As it is well-known, Einstein never succeeded in constructing a complete system of theoretical physics. The right path which has led to a more complete unification of physical forces arose from Hermann Weyl (Weyl 1918a, 1919). The main idea of Weyl, these of gauge invariance, was of essential geometrical character. It is clear now that its development allowed a major step in the movement of geometrization of physics (Kibble 1979; Derdzinski 1993), and a deeper comprehension of the

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mathematical as well physical nature of spacetime. We shall give now an overview of this theory. The importance of the gauge invariance (Eichinvarianz) can be measured by what wrote the physicist Abdus Salam in its Nobel conference of 1975: “One of the most revolutionary event in the history of science of the last century is the idea of gauge unification of the electromagnetic force with the weak nuclear forces” (Salam 1980). Or by what wrote in 1979 the theoretical physicist T.W.B. Kibble: “Revolutions are hard to recognize till they are past. This is surely true of the changes that have occurred in elementary particle physics over the last two decades. The development of gauge theories may well come to be seen as constituting one of the most fundamental revolutions of this century, rivaling the development of quantum mechanics itself. Yet so far its significance is not widely understood outside the ranks of specialists” (Kibble 1979). Now we have to return back to Weyl. We said that the Weyl’s work was aimed to extend the physical significance of general relativity and consequently to propose a generalization of Riemannian geometry (Weyl 1919). According to Weyl, this generalization ought to be possible by introducing the main idea that length of vectors, and not only direction, must depend on the path. In other words, that of length ceases to be an action-at-distance concept. Mathematically, the idea of local gauge invariance amount to introduce a non-integrable scale factor or a function (Wu and Yang 1975), which should supply the fact that in Riemannian geometry the invariance of the length each two vectors get lost. So Weyl proposes a procedure for recalibrating the displacement of a vector at each point of spacetime, in order to leave the length as well as the direction of these vector locally unchanged. Furthermore, he had the ingenious idea of associating the metric tensor with the strength of the electromagnetic field, and the scale vector with the electromagnetic potential. The idea of Weyl runs as follows. The parallel transport of the two vectors V and  W from x to x + dx and, consequently, around a closed contour is generalized. The angle between the two vectors is still kept fixed under parallel transport, but the assumption of the invariance of the length of both vectors is dropped. The length of a vector—in contrast to the angle between two of them—ceases to be an action-atdistance concept. How should one change the expression  δVk = –

Kl ij

Vj dx i

(9.3)

(which represents the change of the components of a vector Vj (x), if displaced or transported parallely on a Riemannian manifold Mr of r dimensions from the point with coordinates xi to the one with coordinates xi + dxi )? One would like to uphold the bi-linearity of δVk in Vj and dxi , thereby arriving at δVk = –ijk Vj dx i

(9.4)

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with the so far unknown connection coefficients ijk. On the basis of the last expression, we can define once more a covariant differentiation, which we still denote by ∇ i . Then the change of a vector around the contour turns out to be ∇Vl = –

  1/2∇Aij Fijl k Vk ,

(9.5)

this time involving the curvature tensor, and we get l Fijl k = 2∂i jl k + 2im jmk = –Flj ik .

(9.6)

We thus see that in 1918 Weyl enlarged the Riemannian spacetime of general relativity by an independent vector field of geometric origin—in modern terms, a one-form. This additional geometric object is intimately linked with the geometrical structure of spacetime. In addition, the Weyl vector is the compensating potential for allowing invariance with respect to local recalibration of lengths, i.e., with respect to conformal changes of the metric. One can furthermore generalize the Weyl geometry to the metric-affine geometry, which is based on a (symmetric) metric and an independent (non-symmetric) linear connection. In Weyl geometry, one geometrical object, the metric tensor, stands for the gravitational potential, as in general relativity, whereas the other one, the linear connection, was surmised to represent the electromagnetic potential known from Maxwell’s theory. Together with a suitable (gravitational and electromagnetic) field Lagrangian, which turns out to be quadratic in the curvature of the underlying Weyl spacetime, this builds up Weyl’s unified theory of 1918. The idea of gauge invariance (Straumann 1987), or the so-called principle of recalibration, which applies first to length of vectors in spacetime, transmuted to the concept of local gauge invariance of the phase of a wave function in 1929 and represents, in the last form, one of the underlying principles of all modern gauge theories, such as the Weinberg-Salam theory of electroweak interactions (Yang and Mills 1954). The other fundamental contribution of Weyl is related to his gauge theory but concerns quantum mechanics (see Moriyasu 1982; O’Raifeartaigh 1997). In an article (in German) of 1927 on quantum mechanics and group theory, then in his book Gruppentheorie und Quantenmechanik (1928), he proposes developing the mathematical foundations of this newly discovered physical theory by showing its close relationship to group representation theory (Weyl 1925, 1926). In Weyl’s new mathematical approach, the basic question at that time was to explain the properties of particles (protons and electrons) by the properties of the quantum laws: do these laws satisfy the basic symmetries known at that time (right/left, past/future, positive/negative electric charge)? Mathematically, that was equivalent to knowing the structure of certain classes of (continuous) groups and their algebras. These three kinds of symmetry were introduced (under other names) into quantum physics around the l930s by Weyl himself and by E. Wigner (Wigner 1931), but no one thought then of unifying the three kinds. In l930 P. Dirac had detected the existence of a particle (positron) with a charge opposite that of an electron, and

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Weyl then generalized to a universal essential equivalence between positive and negative electricity. This idea was reformulated in l937 as the conjugate invariance of electrical charge. However, in l957 Lee and Yang found that left-right symmetry (or conservation of parity P), which physicists had always found useful to accept, was not entirely satisfied by the laws of nature, particularly in weak interactions, which are responsible for radioactive (beta) disintegration. Since it could be verified theoretically and experimentally that this radioactivity gave a correct description of the neutrino, the conclusion was that the existence of the Weyl-Pauli theory (of the neutrino) violated left-right symmetry. This asymmetry seemed to be a consequence of duplication: massless particles (neutrinos) emitted in a beta disintegration existed in only one form (left), while the corresponding anti-particles (anti-neutrinos) could then only exist in the opposite form. Mathematically, this duplication could appear as the existence of two valid solutions for an equation.

9.10 Weyl and Yang-Mills Gauge Theories: A New Step into the Geometrization of Physics Beginning in the 1970s, it was recognized that mathematically, gauge theory is essentially one branch of differential geometry that uses the new concept of “fibre spaces” with “connections” (Atiyah 1919; Yang 1983a). This notion is absolutely central in the understanding of the relation between mathematical structures and physical theories, and it directly links geometry and physics to the point that it can be said that the two are coextensive. Indeed, consider the mathematical concept of a space with a connection and its curvature. Let f : M → N be a map between spaces M, N, where M, say, represents a model of spacetime, and at each point p of M there is localized a physical system with the space of internal states f–1 (p). A connection on a geometrical object is a rule permitting the transport of the system along the curves in M. In other words, if we know part of the world-lines and the initial internal state of a system in M, then, thanks to the corresponding displacement determined by the connection, we can know the future states of the system. According to recent physical theories, a gravitational field is a connection in the space of internal degrees of freedom of a gyroscope; the connection allows us to follow the evolution of the gyroscope in spacetime. An electromagnetic field is also a connection in the space of internal degrees of freedom of a quantum electron; the connection allows us to follow the evolution of the electron in spacetime. A Yang-Mills field is yet a connection, in the space of internal degrees of freedom of a quark. This geometrical image seems now to be the most universal mathematical model of an ideal universe with a small number of basic interactions (Manin 1988). The state of matter in spacetime, at each point and each moment, is described by a section of an appropriate fibre space N → M. A field is described by a connection on this fibre space. Matter acts on the connection by imposing restrictions on its

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curvature, and the connection acts on matter by forcing it to propagate by “parallel displacement” along world-lines. The famous equations of Einstein, Maxwell-Dirac and Yang-Mills are exactly the embodiment of this idea. The geometrical concept of connection has thus become an essential element of physics. One can see that to each physical entity corresponds a geometrical or global differential concept. For example, field strength is identified with the curvature of the connection; the action integral is but a global measure of curvature. Certain topological and algebraic invariants in the theory of characteristic classes have been seen to be most appropriate to describe the charge of the particle in the sense of Yang-Mills (Chern and Simons 1974). More generally, we can establish a direct correspondence from the concepts of gauge field theory to those of the differential geometry- and topology of fibre spaces. But how can we understand precisely the nature of such a correspondence? Inspired by an idea already proposed by Weyl in another manner (Weyl 1918), we support the thesis that, essentially, physics is but geometry in act. This implies not only that geometry yields mathematical abstract concepts like manifolds, groups, curvature, connections, bundles, but also that it is, in a way, ontologically (or, if you wish, physically) rooted in reality, because it is integral part of the properties of physical entities and the features of phenomena (Boi et al. 1992a, b; Boi 1995). One could go so far as to postulate that there must be a geometrical structure, continuous or discrete according to the theory and the class of phenomena considered, underlying any given physical family of phenomena. Or maybe a topological structure which would encompass at the same time the continuous and discrete characters of space and of nature into a more general mathematical scheme (Atiyah and Jones 1978; Witten 1988, 1989). To convince oneself of this, it suffices to remember that some principles of geometrical symmetry (or, equivalently, some groups) can be transformed into dynamical principles that are in turn responsible for changes in the phenomena. Should we then affirm: “In the beginning was the symmetry or the group . . . ”? However, this concept is not just abstract, and to her related mathematical properties have simultaneously an explanatory power and a capacity to generate a world of forces, interactions and energy . . . , so that the mathematical understanding of this world cannot be separate of the understanding of reality itself. Indeed, at a deeper level, one is increasingly led to believe that symmetry may, in a hidden sense, determine almost everything. Moreover, in view of all this it is not unreasonable to look on topology, like symmetry, as some kind of underlying or unifying principle which helps us to understand natural phenomena at the microscopic as well as at the macroscopic levels. The birth and development of gauge theories has been one of the crucial steps of theoretical physics in the twentieth century (Bourguignon and Lawson 1982; Gross 1995). In this regard, two major geometrical advances of Weyl must be mentioned. In 19l8–19 he outlined what he called a “purely infinitesimal geometry (see: Scholz 1995; Straumann 1987; Vizgin 1994), which should know a transfer principle for length measurements between infinitely close points only, and which should admit a conformal structure (Penrose 1985). The allusion is of course to Levi-Civita’s

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parallel displacement principle in a Riemannian manifold embedded in a sufficiently high-dimensional Euclidean space, locally given by ξi = ξi – i j k ξj dx k

(9.7)

with the dxi to be interpreted as the coordinate representation of a displacement vector between two infinitesimally close points so that the direction vector ξi has been transferred to ξi . According to Weyl, one has to separate logically the concept of parallel displacement from metrics and to introduce what he called an affine connection  on a (differentiable) manifold as a linear torsion-free connection. Thus, Weyl proposes a generalization of Riemannian geometry which seemed to be the most natural mathematical framework for the construction of a unified theory of gravitational and electromagnetic forces. This generalized Riemannian metric, a Weylian metric on a differentiable manifold M, is given by: (i) a conformal structure on M, i.e. a class of (semi-) Riemannian metrics [g] in local coordinates given by gij (x) or gij (x) = λ(x)gij (x), with multiplication by λ(x) > 0 (real valued) representing what Weyl considered to be gauge transformation of the representative of [g], and (ii) a length connection on M, i.e. a class of differential forms ϕ in local coordinates represented by ϕi dxi , ϕi dxi – dlogλ (representing the gauge transformation of the representative of j). This new infinitesimal geometry enfolds in fact the first formulation of a gauge theory. The idea of gauge was introduced by Weyl in a very influential paper of 1918 (“Gravitation und Elektrizität” (see also Pauli 1919). The background of this thinking at that time can be retraced through the preface of the various editions of his landmark book Raum, Zeit, Materie (first edition, 1918). Weyl showed that while Einstein’s gravity theory depended on a quadratic differential form. ds 2 =

ik

(9.8)

gik dx i dx k ,

electromagnetism depended on a linear differential form. φ= (which in today’s notation is



i

φi dx i ,

1 ≤ i, k ≤ 4,

(9.9)

Aμ dxμ ) defined up to the gauge transformation.

ds 2 → λds 2 ,

φ → φ + d logλ.

(9.10)

Thus was born the idea of a non-integrable scalar factor (Wu and Yang 1975). Q

e

p

dφ.

(9.11)

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Weyl argued that the addition of a gradient d(log λ) to dφ = φμ dxμ does not change the physical content of the theory, thus considered as a ‘connection’, which specifies that if the spacetime orientation of a frame at x is given, then the relative orientation of a frame at x + dx can be calculated. Since the frames are in a gravitational field, the connection itself is determined by the strength of the field. In fact, the connection can replace the gravitational field entirely so that all motion can be described in terms of the connection alone. This replacement of the field by a mathematical connection leads to the well-known geometrical picture of general relativity. The familiar ‘curvature’ of spacetime can be calculated directly from the connection. Now Weyl went a step beyond general relativity and asked the following question: if the effects of a gravitational field can be described by a connection which gives the relative orientation between local frames in spacetime, can other forces of nature such as electromagnetism also be associated with similar connections? Generalizing the concept that all physical magnitude are relative, Weyl proposed that the absolute magnitude or norm of a physical vector also should not be an absolute quantity but should depend on its location in spacetime. A new connection would then be necessary in order to relate the lengths of vectors at different positions. This connection is associated with the idea of scale or ‘gauge’ invariance (Yang and Mills 1954). It is important to note that the true significance of Weyl’s proposal lies in the ‘local’ property of gauge symmetry and not in the particular choice of the norm or ‘gauge’ as a physical variable. Actually, the assumption of locality is an enormously powerful condition that determines not only the general structure, but also many of the specific features of gauge theory (O’Raifeartaigh 1997). Thus, after Einstein developed his theory of general relativity, in which a dynamical role was given to geometry, Weyl conjectured that perhaps the scale length, indeed the scale of all dimensional quantities, would vary from point to point is space and in time. His motivation was to unify gravity and electromagnetism, to find a geometrical origin for electrodynamics (see Moriyasu 1982; Gross 1995). He assumed that a translation in spacetime dxμ , would be accompanied by a change of scale or gauge, 1 → 1 + Sμ (x)dxμ . The gauge function Sμ (x) would determine the relative scale of lengths, so that a certain function would transform as f (x) → f (x) + [∂ μ + Sμ (x)] f (x) dxμ . The hope was to identify the connection, Sμ , with the vector potential of electrodynamics, thus unifying this theory with gravity. This did not work, but only temporally! In fact, in 1927, after the development of quantum mechanics, Fock and London noticed that the pμ – eAμ , when pμ is replaced with ∂ μ by ∂ μ – (ie/hc)Aμ , looked very much like Weyl’s change of scale, but with a complex coefficient for the connection. Two years later Weyl (1929) completed the discussion, showing how electrodynamics was invariant under the gauge transformation of the gauge field and of the wave function  of a charged particle, Aμ → Aμ + ∂μ α;

 → eieα/ hc .

(9.12)

The concept of gauge invariance and therefore the principle of local gauge symmetry was born (Yang 1989). Accompanying the translation of charged particle

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there is a phase change. The fact that the physics, at least at Planck scale, remain unchanged with respect to a gauge transformation, lies at the heart of different forms of matter. The most remarkable thing mathematically is that all the objections to the Weyl’s theory disappear if we interpret it, as will be done later, as based on the geometry of a circle bundle over a Lorentzian manifold. Then the form φ above (9.9), subject to the gauge transformation, can be interpreted as defining a connection in the circle bundle and thus the metric remains unaltered. More generally, the characteristic features of gauge theories can be described in terms of the topological and geometrical differential concept of fibre bundles and the connections in them (see Stenrod 1951; Husemoller 1966). The connection is an intrinsic local structure that can be imposed on the bundle; it gives an elementary but fundamental example of a gauge field. Since gauge fields, including in particular the electromagnetic field, are fibre bundles, all gauge fields are thus based on topology and geometry. Starting in the 1970s, 20 years after the discovery by Yang and Mills of a non-Abelian gauge theory for strong force (nuclear interactions) in which the local gauge group was the SU(3) isotopic-spin group, the physicists were able to express the concept of a gauge field in such a way that it could be recognized as an instance of more abstract structures known to mathematicians as connections in fibre bundles.3 The discovery of this equivalence has made it possible to understand why and how powerful mathematical concepts and structures are necessary and suitable for the description and explanation of physical reality. Precisely the mathematical structure of gauge theory is that of a vector bundle E with structure group G over a compact Riemannian manifold M (see Atiyah and Bott 1982; Donaldson 1983). We assumes that G ∈ O(m) and E carries an inner product compatible with G. Let E be the space of G-connections on E, and let G be the space of G-automorphisms of E. Then G acts on E, and we have a quotient space B ≡ E/G. To each connection ∇ ∈ E there is associated a curvature 2-form R∇, and at each point x we can take its norm 3 See

the landmark paper of T.T. Wu and C.N. Yang “Concept of nonintegrable phase factors and global formulation of gauge fields” (1975), in which they introduced the fundamental concept of nonintegrable (i.e., path-dependent) phase factor as the basis of a description of electromagnetism. Further this concept is made to correspond to the definition of a gauge field; to extend it to global problems, they analyzed, in relation with the original Dirac’s result, the field produced by a magnetic monopole. The monopole discussion leads to the recognition that in general the phase factor (and indeed the vector potential Aμ ) can only be properly defined in each of many overlapping regions of space-time. In the overlap of any two regions there exists a gauge transformation relating the phase factors defined for the two regions. The concept of monopole leads to the definition of global gauge and global gauge transformations. A surprising result is that the monopole types are quite different for SU(2) and SO(3) gauge fields and for electromagnetism. The mathematics underlying these results is fiber bundle theory. Furthermore gauge fields, including in particular the electromagnetic field, are fiber bundles, and all gauge fields are thus based on geometry. So maybe all the fundamental interactions of the physical world could be based on these geometric and topological structures.

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 ∇ 2 R  x ≡

i 0 is a smooth manifold of dimension 8 k – 3 (see Atiyah 1978). In physical terms this is the dimension of the space of instantons with topological quantum number k > 0. Instantons can claim a relation to Einstein through the following result. The group SO(4) is locally isomorphic to SU(2) × SU(2), so that a Riemannian metric on a four-dimensional manifold M gives rise through projection to connections in the SU(2)-bundles. M is an Einstein manifold if and only if these connections are self-dual or anti-dual (Donaldson 1996; Witten 1994; Atiyah and Hitchin 1998). The importance of gauge theories in modern theoretical physics is well-known. Yang and Mill’s new gauge theory should especially serve as a model for the study of strong interactions, including the quantum effects on them (Manin 1988; Boi 2011, 2009a, b). The main feature of this gauge theory is the use of a non-Abelian Lie group, the simplest of the non-commutative continuous groups, as its invariance group (Connes 1994). This mathematical property of the symmetry group gives a very rich structure to the theory, whose field equations are more general than Maxwell’s. This already illustrates the fundamental role of both geometrical and internal symmetries in physical problems which can be handled by gauge theories (Bourguignon and Lawson 1982). Already in Weyl’s theory, in addition to the position variables of spacetime, there is an internal space parameter on which the phase group acts. The field identified with the particle’s wave function can therefore be seen as associating to each point of spacetime a point of the internal space, or an angle (of rotation) in the case of electromagnetism. A gauge requires that the coordinates of spacetime be combined with the parameters of the internal space. Weyl’s theory satisfies the “principle of local invariance”: i.e., the field equations are invariant under a gauge shift.

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9.12 Some Reflections About Spacetime and the New Physical Theories A major problem in contemporary physics is the understanding of the structure of space and time. Recent research seems to tend toward a very different conception from everyday experience. The various most current relevant theories (of quantum gravity, supergravity, superstring, non-commutative geometry) have in common a tendency to unify spacetime and dynamics, geometrical structures and interactions of physical theories (Boi 2004a,b, 2018; Connes 1998; Penrose 1985). General relativity and quantum mechanics have already led to a profound revision of our conception of space and time, especially at the cosmological and quantum levels. The first is based on a model of spacetime which is a continuum of 4 dimensions with a finite metric. Quantum mechanics, on the other hand, admits that the space at the subatomic level of particles has a discrete structure and his metric presents infinite terms. The attempts to develop a relativistic (local) quantum theory, that is a quantum field theory, were made in order to find a continuous structure for spacetime that was finer than the model of general relativity. This development of concepts of space and time characterizes the quantum theory of gravity. We can now highlight some new ideas partly responsible for the importance of quantum field’s theories (see Stamatescu 1994): 1. Since quantum field’s theories deal with spacetime distributions, field theories pertain directly to the concept of the spacetime continuum (Derdzinski 1993; Gross 1995). 2. The renormalization program in quantum field’s theories, which has to be seen as a constituent part of them, concerns the definition of the theory at short distances—i.e., its very finite spacetime structure. The results of the renormalization program suggest an intimate connection between the short distance structure of spacetime and interaction, which is a fundamental concept in quantum field theory. Modern physics is dominated by the paradigm of local, relativistic, quantum field theory which represents thus the implementation of a new theoretical scheme. These theories possess an extraordinary potential for the unification of the fundamental phenomena, since: (i) They provide a unique object, the quantum field, to account both for those phenomena which in some well-defined classical limit have a particle appearance as well as for those which in the same limit have field (wave) appearance. (ii) They substitute action at distance by local interaction and solve in this way the old “forces/bodies” dualism. All fields contribute here with the same right—although taking various roles. So interaction between matter fields (e.g., electrons) is transmitted by the gauge fields (e.g., photons) and vice versa. Since particles creation and annihilation is a fundamental feature of these theories, the kinematical concept of particle loses its primordial character,

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while the role of the interaction (dynamics) becomes dominant. We achieve in this way, among other things, a consistent treatment of the “transmutation” phenomena—but also a new picture of the vacuum (see Boi 2011). (iii) They helped recognize a fundamental constructive principle—gauge symmetry—which may help develop the mathematical scheme for a unified view of the various phenomena (electromagnetic, weak, strong, gravitation) (see Salam; Yang 1983). In general, the treatment and understanding of symmetries is well promoted by the quantum field theoretical formalism. (iv) They introduced a construction—the renormalization group (in the sense of Wilson) (Manin 1988)— which allows one to follow quantum phenomena to smaller and smaller scales and trace back in this way the emergence of the diversity of phenomena as induced by phase transitions from a unified, fundamental interaction (Grand Unified Theory) (see Taylor 1976; Gross 1995). Thereby, and in connection with symmetries, an intimate connection between dynamics and the spacetime was suggested (Boi 2009, 2006a; Regge 1992; Bourguignon and Lawson 1982), which then led to a number of new ideas about the structure of the latter, but which may lead to far reaching conceptual consequences in future theoretical developments. One can further outline some new ideas relating to the structure of spacetime in the most recent physical theories, to start with general relativity. 1. The geometric structure of spacetime gives rise to the dynamics of this same spacetime, and in particular of the gravitational field. The other physical fields, electromagnetic and the field of matter, however, remain outside this theory. But some physical theories developed in the last two decades shows quite compellingly that in fact even the electromagnetism and the other physical forces (the nuclear—strong and weak—interactions of elementary particles) emerges as dynamical effects from the (topological) structure of spacetime.4 Conversely, the

4 For

a good overview of this subject, see Morandi (1992). As he pointed out (in the Introduction): “Dirac’s quantization condition (1935) is the first instance in Physics of ‘topological quantization’, i.e., of a quantization of consistency with quantum mechanics, and arising entirely from topology. At about the same time, H. Hopf discovered the fibration S1 → S3 → S2 (1931). That the two structures were actually intimately related became clear only more than 30 years later, when it was realized that the fiber bundle corresponding to the Hopf fibration can be endowed with a natural connection whose curvature can be identified with the field of a magnetic monopole sitting at the center of the sphere S2 . Another instance in which nontrivial topological properties of space-time appear to play a relevant role is provided by the effect discussed by Y. Aharonov and D. Bohm in 1959. Although discussed originally as scattering event, the effect can also be described in different terms by saying that the wave function of a charged particle which is adiabatically dragged around an infinitely long solenoid enclosing a flux  acquires an extra phase of exp[2πi/0 ], where 0 = hc/q. In 1975, in an influential paper, T.T. Wu and C.N. Yang stressed that the proper language to describe quantum mechanics in the presence of electromagnetic couplings is that of U(1) principal fiber bundles, and that wave functions are to be properly seen as sections of such bundles. This paved the way to the development of non-Abelian gauge field theories. In the case of Aharonov-Bohm effect, the bundle is flat, but has nontrivial holonomy, and the phase acquired by the wave function is just a manifestation of the holonomy of the bundle”.

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spacetime itself must be henceforth thought, in some way, as a derived (variable) concept which can be subject to the (quantum) fluctuations of the different fields of matter and of the metric (see Connes 1994, 1998). Moreover, in string theory the auxiliary two-dimensional field theory (which is needed in order to describe a vibrating string) plays in a way a more fundamental role than spacetime, and spacetime exists only to the extent that it can be reconstructed from the twodimensional field theory. So string theory, if correct, entails radical change in our concepts of spacetime (see Witten 1988, 1989). 2. The physical symmetries dictate the different interactions between forces and between particles. This is a very general principle and it is the crucial idea at the heart of quantum field theories. Actually, all the natural phenomena seem to be founded upon such principle. Symmetry, Lie groups and gauge invariance are now recognized, through theoretical and experimental developments, to play essential role in determining the basic forces of nature. Furthermore, and this is very exciting, while great successes have been achieved in these developments, we are still far from a grand synthesis. It is reasonable to believe this is because the full meaning of the word symmetry is not yet understood and key additional concepts are still missing. 3. Gauge invariance has been recognized as an universal physical principle governing the fundamental forces and interactions between particles and matter. All physical theories known so far can be formulated by using this principle. Moreover, there is the hope that all physical theories might be related to each other by a common gauge group of symmetries. There were two stated motivations that lay behind the historical discovery by Yang-Mills (1954) of the first non-Abelian gauge theory. First, they wanted to find a principle that would enable him to select a theory and determine the interactions. The principle was that of a local symmetry. The second motivation was simply to generalize the local gauge invariance of electrodynamics of the non-Abelian symmetry of isotopic-spin. Isotopic-spin was the first symmetry that was evident in the strong interactions, which was introduced by Heisenberg and Wigner. The novel idea—that the isotopic-spin connection, and therefore the potential, acts like the SU(2) symmetry group—is the most important result of the Yang-Mills theory. This concept lies at the heart of local gauge theory. It shows explicitly how the gauge symmetry group is built into the dynamics of the interaction between particles and field.5 In the late 1960s, Yang attempts to give

5 According

to T. Regge (1992), there is no difficulty in writing the modern (gauge) form of electromagnetism (with the compact group SO(1) or U(1) on a Riemannian manifold and it is possible to write à la Cartan general relativity as a SO(3, 1) gauge theory. Besides, it may be useful to recall that Cartan was largely responsible for the introduction of the concept of torsion in Physics. Torsion remains a very interesting idea. We need to use it, even by just declaring it to vanish, if we want to write general relativity as a gauge theory in which all fields and not only the spin connection appear as gauge potentials. The interesting feature of general relativity is that the associate curvature of the vierbein, i.e. torsion, vanishes as a consequence of the variational principle of Hilbert-Einstein-Cartan. And in fact the Lagrangian density is not invariant under

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a new formulation of gauge fields, through the approach of nonintegrable phase factors.6 The remarkable thing here is that the physical contents of gauge theory and the mathematical formalism in post-Riemannian geometry (Atiyah and Jones

all gauge transformations of the Poincaré group but only under those of the Lorentz subgroup. Although nature has prepared the gauge potential for the full group it end up by requiring invariance under a subgroup only. A world with torsion would appear inescapable if we have around enough density of high spin particles which acts as sources, but this density seems at the moment well below the limit of observability. Regards the kind of space in which torsion is supposed to appear, one can remark that it would not be any more a Riemannian manifold or, rather, none of the Riemannian structures existing on the manifold would be directly related to Physics and the theory would not be a geometrical theory in the sense envisaged by Einstein. One could yet consider general relativity as GL(4, R) theory with the Christoffel connection playing the role of a YangMills potential. If the torsion vanishes it follows that the Christoffel symbol is symmetrical into the 2 lower indices whose role is however quite different. The first index is a GL(4, R) gauge index; the second labels instead the differentials on spacetime. We may relate them because of the accidental and marvellous fact that the Jacobian group of derivatives on a differentiable manifold is isomorphic to GL(4, R) and that we use the same indexing for differentials and vectors in GL(4, R). Once the symmetry is established the theory becomes almost by definition geometrical. If there is no symmetry but we can control torsion by introducing suitable norms and bounds then we may still speak of an almost geometrical theory whose exact mathematical definition is still lacking. 6 The exact formulation of the concept of a nonintegrable phase factor depends on the definition of global gauge transformations, i.e., on the choice of the overlapping regions of R (where R is a region of space-time, precisely, all space-time minus the origin r = 0) and of the potential Aμ in this region. Through a certain kind of operations, called distortions, one arrives at a large number of possibilities, each with a particular choice of overlapping regions and with a particular choice of gauge transformation from the original (Aμ )a or (Aμ )b to the new Aμ in each region. Each of such possibilities will be called a gauge (or global gauge). This definition is a natural generalization of the usual concept, extended to deal with the intricacies of the field of a magnetic monopole. Notice that the gauge transformation factor in the overlap between Rα and Rβ does not refer to any specific Aμ . The gauge transformation in the overlap of the two regions is:  (1) S = Sαβ = exp (–iα) = exp (2ige/ hc) φ . Thus two different gauges may share the same characterizations (a) and (b). In the case of the monopole field, one can attach to the gauge any (Aμ )a and (Aμ )b provided they are gaugetransformed into each other in the region of overlap. Thus a gauge is a concept not tied to any specific vector potential. Wu and Yang called the process of distortion leading from one gauge to another a global gauge transformation. It is also a concept not tied to any specific vector potential. The collection of gauges that can be globally gauge-transformed  into each other will be said to belong to the same gauge type. The phase factor exp (ie/hc Aμ dxμ ) (which is nonintegrable, i.e., path-dependent) around a loop starts and ends at the same point in the same region. Thus it does not change under any global transformation, so that we have the, for Abelian gauge fields, the following Theorem 1: The phase factor around any loop is invariant under a global gauge transformation. It follows trivially from this, by taking an infinitesimal loop, that Theorem 2: The field strength fμν is invariant under a global gauge transformation. And Theorem 3: Between two gauge fields defined on the same gauge there exists a continuous interpolating gauge field defined on the same gauge.

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1978; Witten 1988, 1989), which is based on the concept of connection and fibre space with connection, are strictly similar. 4. The principal problem of theoretical physics today, and perhaps of science in general, is that of arriving at a unifying theory of all fundamental physical forces. More specifically, it is a matter of unifying the various quantum fields’ theories with the theory of gravity (Salam 1982; Gross 1995). We know that quantum mechanics and general theory of relativity are mutually incompatible. General relativity fails to comply with the quantum laws that govern the behaviour of elementary particles, whereas on the opposite scale, black holes are challenging the very foundations of quantum mechanics.

References Akivis, M.A., and B.A. Rosenfeld. 1993. Elie Cartan (1869–1951). Translations of Mathematical Monographs 123. Providence: American Mathematical Society. Aharonov, Y., and D. Bohm. 1959. Significance of electromagnetic potentials in the quantum theory. Physics Review 115: 485–491. Atiyah, M.F. 1979. Geometry of Yang-Mills fields. Pisa: Academia Nazionale dei Lincei, Scuola Normale Superiore. Atiyah, M. 2002. Hermann Weyl 1885–1995 (a biographical memoir). National Academy of Sciences Washington 82: 1–17. ———. 1988. New invariants for manifolds of dimensions 3 and 4. In The mathematical heritage of Hermann Weyl, Proceedings of Symposia in Pure mathematics, ed. R.O. Wells, vol. 48, 285– 329. Providence: Am. Math. Soc.

Theorem 4: Consider gauge GD and define any gauge field on it. The total magnetic flux through a sphere around the origin r = 0 is independent of the gauge field and only depends on the gauge:   (2)

fμν dx μ dx ν = (–ihc/e)



  ∂/∂x μ ln Sαβ dx μ ,

where S is the gauge transformation defined by (1) for the gauge GD in question, and the integral is taken around any loop around the origin r = 0 in the overlap between Rα and Rβ , such as the equation on a sphere r = 1. As in the case of electromagnetism, in the non-Abelian gauge fields both the concept of a gauge and the concept of a global gauge transformation are not tied to any specific gauge potentials. The nonintegrable phase factor for a given path is now an element of the gauge group. Since these phase factor do not in general commute with each other, Theorems 1 and 2 for the Abelian case need to be modified as follows. Theorem 5: Under a global gauge transformation, the phase factor around any loop remains in the same class. The class does not depend on which point is taken as the starting point around the loop. Theorem 6: The field strength fk μν is covariant under a global gauge transformation. Theorem 5: defines the class of a loop. This concept is a generalization of the phase factor for electromagnetism around a loop with the magnetic flux as the exponent. It is a gauge-invariant concept.

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———. 1997. Geometry and physics. In Geometry and physics, proceedings, lecture notes in pure and applied mathematics, ed. J.E. Andersen et al., vol. 184, 1–7. New York: Dekker. Atiyah, M.F., and R. Bott. 1982. The Yang-Mills equations over Riemann surfaces. Philosophical Transactions of the Royal Society of London, Series A: Mathematical, Physical and Engineering Sciences 308: 523–615. Atiyah, M.F., and N.J. Hitchin. 1998. The geometry and dynamics of magnetic monopoles. Princeton: Princeton University Press. Atiyah, M.F., and J.D.S. Jones. 1978. Topological aspects of Yang-Mills theory. Communications in Mathematical Physics 61: 97–118. Boi, L. 1995. Le problème mathématique de l’espace, with a foreword of R. Thom. Berlin/Heidelberg: Springer. ———. 1997. Géométrie elliptique non-euclidienne et théorie des biquaternions chez Clifford: l’élaboration d’une algèbre géométrique. In Le nombre: Une hydre à n visages. Entre nombres complexes et vecteurs, ed. D. Flament, 209–238. Paris: Éditions de la MSH. ———. 2004a. Theories of space-time in modern physics. Synthese 139: 429–489. ———. 2004b. Geometrical and topological foundations of theoretical physics: From gauge theories to string program. International Journal of Mathematics and Mathematical Sciences 2004 (34): 1777–1836. ———. 2006a. Geometrization, classification and unification in mathematics and theoretical physics. In Proceedings of the Albert Einstein century international conference, ed. J.-M. Alimi and A. Füzfa, 15. Melville: American Institute of Physics. ———. 2006b. Mathematical knot theory. In Encyclopedia of Mathematical Physics, ed. J.-P. Françoise, G. Naber, and T.S. Sun, 399–406. Elsevier: Oxford. ———. 2006c. The Aleph of space. On some extensions of geometrical and topological concepts in the twentieth-century mathematics: From surfaces and manifolds to knots and links. In What is geometry? ed. G. Sica, 79–152. Milan: Polimetrica, International Scientific Publishers. ———. 2009a. Ideas of geometrization, geometric invariants of low-dimensional manifolds, and topological quantum field theories. International Journal of Geometric Methods in Modern Physics 6 (5): 701–757. ———. 2009b. Clifford geometric algebras, spin manifolds, and group action in mathematics and physics. Advances in Applied Geometric Algebras 19 (3–4): 611–656. ———. 2009c. Geometria e dinamica dello spazio-tempo nelle teorie fisiche recenti. Giornale di Fisica 50: 1–10. ———. 2011. The Quantum Vacuum. A scientific and philosophical concept, from electrodynamics to string theory and the geometry of the microscopic world. Baltimore: The Johns Hopkins University Press. ———. 2018. Some mathematical, epistemological and historical reflection on space-time theory and the geometrization of theoretical physics, from B. Riemann to H. Weyl and beyond. In Foundations of science. (forthcoming). Boi, L., D. Flament, and J.-M. Salauskis, eds. 1992a. 1830–1930 : A century of geometry, mathematics, history and epistemology. Heidelberg: Springer. Boi, L., D. Flament, and J.-M. Salanskis. 1992b. 1890–1990: A century of geometry. Mathematics, history and epistemology, Lecture notes in physics. Vol. 224. Heidelberg: Springer. Borel, A. 1985. Hermann Weyl and Lie Groups. In 1885–1985 centenary lectures, ed. Hermann Weyl, 53–74. Berlin/Heidelberg: Springer. Bott, R. 1988. On induced representations. In The mathematical heritage of Hermann Weyl, Proceedings of Symposia in Pure Mathematics, ed. R.O. Wells, vol. 48, 1–14. Bourguignon, J.-P. 1992. Transport parallèle et connexions en Géométrie et en Physique. In 1830– 1930: A century of geometry, mathematics, history and epistemology, ed. L. Boi, D. Flament, and J.-M. Salanskis, 150–164. Heidelberg: Springer. Bourguignon, J.P., and H.B. Lawson. 1982. Yang-Mills theory: Its physical origins and differential geometric aspects. In Seminar on differential geometry, ed. S.-T. Yau, 395–421. Princeton: Princeton University Press.

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Chapter 10

Logic of Gauge Alexander Afriat

Abstract The logic of gauge theory is considered by tracing its development from general relativity to Yang-Mills theory, through Weyl’s two gauge theories. A handful of elements—which for want of better terms can be called geometrical justice, matter wave, second clock effect, twice too many energy levels—are enough to produce Weyl’s second theory; and from there, all that’s needed to reach the Yang-Mills formalism is a non-Abelian structure group (say SU(N )).

10.1 Introduction This article is an attempt to answer the question(s) “How did we get gauge theory? What’s the logic of the historical process that produced such a theory?” In a sentence, my answer will be something like: Once general relativity was rectified by geometrical justice, a (necessarily relativistic) matter wave was enough—bearing in mind the second clock effect, and the too many energy levels of Dirac’s theory—to produce Weyl’s second theory, which, with a non-Abelian structure group, leads to the Yang-Mills formalism.

That’s a gist, largely unintelligible for the time being. Other answers are of course possible too. It makes sense to begin with general relativity, and I have chosen not to go beyond Yang-Mills theory, concentrating most of my attention on the transition from Weyl’s first gauge theory W18 to his second W29. Surprisingly little is needed, in a purely logical sense, to go all the way from GR to YM—which is not to diminish all the impressive scientific creativity involved in between. What I

A. Afriat () Maître de conférences, Département de philosophie, Université de Bretagne Occidentale, Brest, France © Springer Nature Switzerland AG 2019 J. Bernard, C. Lobo (eds.), Weyl and the Problem of Space, Studies in History and Philosophy of Science 49, https://doi.org/10.1007/978-3-030-11527-2_10

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propose is a necessarily a posteriori1 logical clarification or reconstruction—which may look more anachronistic than it really is—of the historical steps involved, without suggestions of condescending retrospective ‘trivialization’: the point isn’t that now, with the benefit of hindsight, we can see that it was all trivial, or even inevitable, borne along by inexorable historical necessity; but that with the benefit of hindsight a certain logically simplified or even ‘sanitized’ representation of the development can be given, which sheds light on the full evolution, in all its bewildering detail and archaic colour. My purpose is not to replace a close perusal of the texts with misleading anachronisms; but to supplement their study with a clarified representation of the evolution they express, in terms more intelligible to certain modern readers. The usefulness of a map for jungle exploration does not preclude jungle exploration—it can hardly oblige its reader to stay at home with the map. The facts, the texts, underdetermine the logical representation, which cannot be unique; the reconstruction I have chosen is intended to capture the basic structure of the evolution as cleanly as possible. “Logic” and cognates have been used in many different ways—John 1:1, Hegel (1816), Popper (1934) etc. The meaning I have in mind, which should emerge in the sequel, has to do with ‘derivation’ (of a theory for instance), and ‘what elements are needed to derive.’ “History” and cognates have also been used in many different ways; there are many ways to treat, understand, study, describe, approach the past. It is a sociological fact that different communities behave and communicate in different ways. I have left “history” out of the title to avoid confusing a community that uses the word in its own very special manner. There may be a tradeoff between “clear and distinct,” ‘Cartesian’ intelligibility and a kind of historical fidelity. If there is, this paper can be seen as an attempted optimization of sorts, subject to the operative constraints.2 An extreme example illustrating what I mean by “Cartesian intelligibility” might be a good (but perhaps anachronistic) ‘historical’ synopsis in a recent text by a differential geometer. At the other end one can struggle with and try to render, with comprehensive zeal, all the ambiguities, irrelevancies, idiosyncrasies and misunderstandings of the original texts. Such a struggle can provide one special kind of historical understanding; but other kinds or aspects, none the less legitimate, may be left out. Of course

1 I cannot help writing in 2017—not before at any rate. The article will be read no earlier than 2017,

by readers whose habits, mental categories and means of understanding were formed in recent decades, not in the nineteenth century. Any attempt to understand the ideas of 1929 will have to be made using the cognitive resources available to actual or possible readers, not to fancied readers bred in a fictitious past; so-called anachronisms can be as necessary as they are dangerous (the slope is indeed slippery, there’s no denying that). 2 Best of both worlds is more the idea than having one’s cake and eating it—which suggests an attempt to violate inescapable constraints.

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the problem can be eliminated, with discreet but trenchant3 expeditiousness, by carefully adjusting the scope of history to leave out awkward desiderata: “that’s not history (any more)!” The point is not to satisfy all conceivable desiderata, but to avoid the questionable neglect of certain reasonable ends. An appropriate form of Cartesian intelligibility seems a goal worth at least pursuing (success being quite another matter). The importance of explication in historical analysis should not be underestimated; and it is best to explicate in terms intelligible to the (necessarily modern) reader. Questions like “what on earth is meant here?” often arise, and deserve answers, which should be less obscure than the original. Weyl, for instance, can be extremely murky; should that murkiness be faithfully handed on to the reader, to avoid anachronism, in a spirit of historical fidelity? Needless to say, an account that’s already too tidy (again, “anachronistic” is the word that’s used in these cases) for many historians won’t be sanitized enough for many mathematical physicists who use the terms Weyl structure, Weyl tensor etc. every day—but that’s a matter of tribal behaviour and the sociology of scientific communities. ‘Treading a fine line’ isn’t even the right sociological metaphor; far from being fine, it is of considerable—only negative—width. One anachronism is the symbol A I use for the length connection (and electromagnetic potential). Weyl creates W18 for one purpose: the anholonomic propagation of length. So the whole point of the length connection is its curvature, it cannot be an exact differential df. If Weyl’s notation df were taken literally, his theory would be pointless, there would be no electricity, Einstein’s objection would be vacuous and senseless etc. To denote an object that cannot be (necessarily) exact it therefore seems best to avoid a notation expressing exactness. Another anachronism is my use of a single letter W to denote Weyl’s ‘material’ group, for which he uses many letters, in fact several words.4 Notation is bound to be problematic here: too close an adherence to Weyl’s would produce misunderstandings, whereas entirely modern notation, however intelligible today, would just be too remote. I have attempted a compromise (with obvious perils): without losing touch with Weyl I have tried to remain intelligible to modern readers. Many anachronisms are deplorable. But there is something wrong with the common inference some anachronisms are to be avoided therefore anachronisms are all to be avoided. Telling good and bad anachronisms apart can admittedly be hard, it is easiest to proscribe them altogether (baby with bathwater) . . . spares the trouble of distinguishing . . . Since the transition (§10.2.1) from general relativity to Weyl’s first gauge theory (1918, §2) has been amply discussed in Pais (1982), Vizgin (1984), Scholz (1994, 1995, 2001a, 2004, 2011b), Cao (1997), Hawkins (2000), Coleman and Korté

3 An appropriate image might be the discreet amputation of a mischievous or perhaps embarrassing

limb. 4 Weyl

(1929b) p. 333: “man beschränke sich auf solche lineare Transformationen U von ψ 1 , ψ 2 , deren Determinante den absoluten Betrag 1 hat.”

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(2001), Sigurdsson (2001), Ryckman (2003a, b, 2005, 2009), Penrose (2004) and Afriat (2009), I concentrate on the next step, which took Weyl to his second theory (§10.3)—mainly determined by the new undulatory ontology (§10.3.1) introduced by Louis de Broglie (1924), Dirac (1925), Schrödinger (1926) and others. Mainly but not wholly: Weyl had every reason to keep the electricity, gravity and abstract gauge structure of his first theory; but now with three ingredients (matter, electricity, gravity), three different gauge relations were possible, two of which (electricity-matter, electricity-gravity) were more plausible than the third (matter-gravity). Einstein’s objection (1918, §2.3), the second clock effect, ruled out the old gauge relation ((10.3)&(10.5)) between electricity and gravity, leaving the new relation ((10.3) & (10.6)) between electricity and matter. Weyl’s objection that four-component theory provided twice too many energy levels (§10.3.2) is only relevant to his own story, of how he reached his two-component theory of 1929, and not to YM (§10.4)—which by no means favours Weyl’s two-component theory over Dirac’s with four. YM is more relevant here as abstract mathematical physics (like symplectic geometry with differential forms) than as genuine theoretical physics. The story may indeed appear to assume a somewhat unexpected—perhaps even fictitious— character in §10.4. Since the numerous physical details that ultimately did produce YM seem rather foreign to the spirit and purpose of my account, I have chosen to deal only with the ‘purely logical’ transition from W29 to YM.

10.2 Weyl’s First Gauge Theory 10.2.1 Geometrical Justice Again, it makes sense to start with GR.5 Levi-Civita (1917) saw that the connection determined by Einstein’s covariant derivative transported the direction of a vector anholonomically, but not its length, which was left unchanged.6 This was unfair, protested Weyl—length deserved the same treatment as direction.7 To remedy he proposed a more general theory that propagated length just as

5 Einstein

(1916) Ryckman (2003b) p. 80, Ryckman (2009) p. 288. 7 See Afriat (2009) for details of this ‘geometrical justice’—which can also be understood in terms of group extensions (see Scholz (2004) pp. 183, 189, 191–2, Scholz (2011a) pp. 195, Scholz (2011b), third page of the paper): since a Levi-Civita connection subjects direction to SO+ (1, 3) but length to the (group containing only the) identity 1, it is only fair to extend the identity by the dilations, yielding 1 × R = R—which (unlike 1) allows length anholonomies and therefore geometrical justice. The total group, for direction and length together, is the extension SO+ (1, 3) × R giving the relativistic similarities. But the group W Weyl uses in 1929 is (globally) not the extension SL (2, C) × U(1); see §10.3.4.1. Ryckman (2003a, b, 2005, 2009) provides an alternative account of Weyl’s agenda. 6 See

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anholonomically as direction. The resulting congruent transport would also be governed by a connection, which Weyl defined quite generally as a bilinear mapping between neighbouring points: linear in the thing propagated and in the direction of propagation. A connection transporting the (squared) length l from a = γ (0) to its neighbour8 b = γ (1) along the world-line9 γ : J → M, τ → γ (τ ) would therefore be a real-valued10 one-form11 A = Aμ dxμ applied to the tangent vector γ˙ = γ˙ μ ∂μ ∈ Ta M and multiplied by the initial length la , yielding the increment12 δl = lb − la = la A, γ˙  = la Aμ γ˙ μ added to la .13 The final length lb is la (1 + A, γ˙ ) —unless a and b are too far apart for γ to remain straight in between, in which case lb is  la exp

A. γ

Congruent transport can also be expressed by the differential equation ∂τ l = A, γ˙  l, where ∂ τ differentiates along γ˙ . To deal with the geometrical injustice that A was to remedy, the curvature F =

 1 1 Fμν dx μ ∧ dx ν = dA = ∂μ Aν − ∂ν Aμ dx μ ∧ dx ν 2 2

(10.1)

cannot (necessarily) vanish—unlike the three-form dF = d 2 A =

 1 ∂μ Fνσ + ∂ν Fσ μ + ∂σ Fμν dx μ ∧ dx ν ∧ dx σ , 6

(10.2)

which does. Seeing all this, Weyl couldn’t help thinking14 of the electromagnetic four-potential A, the Faraday two-form F = dA and Maxwell’s two homogeneous

8 Which

is so close to a it practically belongs to the tangent space Ta M; see Weyl (1926) p. 28, Weyl (1931a) p. 52. 9 J ⊂ R is an appropriate interval containing 0 and 1. 10 Here the structure group is the multiplicative group R of dilations, generated by the Lie algebra R, + , [·, ·] or rather R, +; the Lie product [·, ·] vanishes since real numbers commute. 11 Einstein’s summation convention will sometimes be used. 12 I often use angular brackets α, X to denote the value of the form or covector α at the vector X. Bras and kets (which presuppose an appropriate natural pairing) will also be useful, especially where inner η| ζ  and outer |ζ η| products both arise. 13 Cf Ryckman (2009) pp. 290–1. 14 See Eddington (1987) p. 175, Scholz (2001a) p. 75, Ryckman (2003a) p. 92, Ryckman (2005) p. 158.

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A. Afriat

equations15 dF = 0: he had unified gravity and electromagnetism—by mistake!16 And indeed Einstein would soon point out the mistake: the anholonomy on which Weyl had built his theory is not found in nature, as we shall see in §10.2.3.

10.2.2 Gauge Weyl sought, then, to rectify general relativity by means of the curvature (10.1), which ensured geometrical justice: vanishing Streckenkrümmung (length curvature) F led to holonomic congruent transport which, alongside anholonomic parallel transport, was manifestly unfair. Differentiation is destructive, or rather irreversible; what d destroys (through its kernel) is the freedom A → A = A + dλ

(10.3)

invisible to F = dA = dA , in the sense that the inverse image [A] = d−1 F of F under d is the whole equivalence class given by the equivalence relation A ∼ (A + dλ), where the function λ assigns a single (and hence path-independent) value to each point of M. If A only served to produce the curvature F, with no other role, (10.3) would be vacuous; but A appears elsewhere too, notably in the law of propagation ∇g = A ⊗ g,

(10.4)

which is not indifferent to (10.3), g being the metric. To make (10.4) invariant, (10.3) therefore has to be balanced by g → g  = eλ g,

(10.5)

whose origin is thus accounted for.17 The point of Weyl’s theory is not the ‘holonomic’18 transformation (10.5), which should, despite much emphasis in the

15 ∇

· B = 0 and ∇ × E + ∂ t B = 0 (2003b) p. 61: “[ . . . ] Weyl did not start out with the objective of unifying gravitation and electromagnetism, but sought to remedy a perceived blemish in Riemannian ‘infinitesimal’ geometry. The resulting ‘unification’ was, as it were, serendipitous.” See also p. 63, Ryckman (2003a) p. 86, Ryckman (2005) pp. 149–54, 158, Ryckman (2009) pp. 287–94. 17 Cf. Weyl (1931a) p. 54: “insbesondere konnte ich nichts a priori Einleuchtendes vorbringen zugunsten der Koppelung des willkürlichen additiven Gliedes ∂λ/∂xp , das nach der Erfahrung in den Komponenten des elektromagnetischen Potentials steckt, mit dem von der klassischen Geometrie geforderten Eichfaktor eλ .” 18 But Dirac (1931) gives infinitesimal (and indeed globally path-dependent, anholonomic) meaning to the similar expression eiβ in his equation (10.3), where β cannot be a single-valued function on space- time. 16 Ryckman

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literature,19 by no means be taken as a premiss, being more of a consequence than a postulate. The real premiss is geometrical justice—direction and length both deserve anholonomic transport. Weyl did define conformally invariant objects, maybe quite a few, but (10.4) is not among them, nor indeed is W18. To see how the metric can correct (only locally) the dilations generated by the connection, we can take a unit length la = ga (Va , Va ) = 1 at point a ∈ M; and a connection which annihilates the tangent vector γ˙ directed towards b nearby, so that A, γ˙  vanishes. As the connection produces no increment δl in this case, the length lb = gb (Vb , Vb ) = 1 remains unaffected and nothing need be done to the metric gb at b. But the increment δl needn’t vanish; in general (10.5) is required to maintain the same numerical value 1 = lb , by adapting the yardstick at b. Any exact term dλ added to the connection will be balanced by a conformal factor eλ , and vice versa; the (holonomic) recalibrations (10.5) of the metric only correct the dilations produced by dλ (and not the anholonomic dilations due to a curved connection). To spell out the implications of geometrical justice in W18 one can write: geometrical justice  anholonomic connection A  curvature F = dA  indifference of F to (10.3); sensitivity of (10.4) to (10.3)  indifference of (10.4) to {(10.3) balanced by (10.5)}. Such compensation is typical of a gauge theory: an invariant expression (here (10.4)) is sensitive to a first transformation, and to a second as well—but indifferent to the two together, if their variations are appropriately constrained, and balance one another.20

19 The

misunderstandings go at least as far back as Eddington (1987, first published in 1920), who first, at the top of p. 169, goes out of his way to explain the anholonomic propagation of length; which then, just a few lines on, gets propagated in the very way that was to be avoided, subject to the very restriction that was to be overcome: “a definite unit of interval, or gauge, at every point of space and time. [ . . . ] when the comparison depends on the route taken, exact equality is not definable; and we have therefore to admit that the exact standards are laid down at every point independently.” Cf. Dirac (1931) p. 63: “We may assume that γ has no definite value at a particular point, but only a definite difference in values for any two points. We may go further and assume that this difference is not definite unless the two points are neighbouring. For two distant points there will then be a definite phase difference only relative to some curve joining them and different curves will in general give different phase differences.” Indeed there are three cases, not two: [1] no variation at all, [2] holonomic variation, [3] anholonomic variation. Eddington has rightly understood that Weyl wants to go beyond [1]. But that leaves the other two—the whole point of W18 being that Weyl wants to go beyond [2] as well. If one’s bent on conflation, the first two cases can be more or less conflated in W18: F vanishes ([1], [2]), or not ([3])—how it vanishes is hardly the point. Eddington seems to feel that conflation is needed somewhere, and duly conflates the last two instead. The tradition he may or may not have founded has had considerable and perhaps growing success—it persists to this day, with an ample, zealous following, and no sign of abating; cf. Afriat (2013), especially footnotes 5 and 9, about an equivalent misunderstanding. 20 See Ryckman (2003a) p. 77.

272

A. Afriat

So far, then, we have two primitive logical elements. 1. GR: general relativity 2. GJ: geometrical justice which together yield Weyl’s theory of electricity and gravity: GR & GJ  W18. The next will be MW: matter wave and SC: (avoid) second clock effect. The latter is essentially Einstein’s objection, which should now be considered.

10.2.3 Einstein’s Objection The tangent of a world-line’s image γ ⊂ M only has a direction, it is a full ray; the length l = γ˙ 2 = g (γ˙ , γ˙ ) of the tangent vector γ˙ = dγ /dτ is given by the parameter rate ∂γ /∂τ . If the values of the parameter are identified with the readings of a clock describing γ , the length l giving the proper ticking rate should remain constant—the hands of a good clock don’t accelerate. But far from remaining constant, lengths in Weyl’s theory aren’t even integrable:  lb (γ ) = la exp

A γ

depends on γ —whereas an exact connection A = dμ would give  lb = la exp

b

dμ = la exp μ

a

along any path, μ being the difference μ(b) − μ(a) between the final and initial values of μ. In addition to the first clock effect (Langevin’s twins) already present in Einstein’s theory, Weyl’s theory therefore involves a second clock effect expressed in the anholonomy of ticking rates.

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Einstein objected that nature provides integrable clocks.21 Two clocks trace out a loop γ = ∂ω enclosing a region ω (without holes): starting from the same point a they describe worldlines γ 1 , γ 2 that meet at b. They tick at the same rate if A is exact, for then   dμ = d 2μ ∂ω

ω

vanishes—since no holes are enclosed it is enough for A to be closed, 

 A= ∂ω

dA ω

vanishes too provided dA does. But if the loop encloses an electromagnetic field F = dA, one of the clocks will tick faster than the other once they’re compared at b. In any case the theory didn’t work: it rested from the outset on an anholonomy not seen in nature.22

10.3 Weyl’s Second Gauge Theory The setback of 1918, Einstein’s objection (his preaching!23 ) put an end to Weyl’s geometrical fantasies, to his ‘wilder days’ as it were, producing a serious new empirical sobriety: “All these geometrical leaps-in-the-air [W18] were premature,

21 Letter to Weyl dated 15 April 1918: “So schön Ihre Gedanke ist, muss ich doch offen sagen, dass

es nach meiner Ansicht ausgeschlossen ist, dass die Theorie der Natur entspricht. Das ds selbst hat nämlich reale Bedeutung. Denken Sie sich zwei Uhren, die relativ zueinander ruhend neben einander gleich rasch gehen. Werden sie voneinander getrennt, in beliebiger Weise bewegt und dann wieder zusammen gebracht, so werden sie wieder gleich (rasch) gehen, d. h. ihr relativer Gang hängt nicht von der Vorgeschichte ab. Denke ich mir zwei Punkte P1 & P2 die durch eine zeitartige Linie verbunden werden können. Die an P1 & P2 anliegenden zeitartigen Elemente ds1 und ds2 können dann durch mehrere zeitartigen Linien verbunden werden, auf denen sie liegen. Auf diesen laufende Uhren werden ein Verhältnis ds1 : ds2 liefern, welches von der Wahl der verbindenden Kurven unabhängig ist.—Lässt man den Zusammenhang des ds mit Massstab- und Uhr-Messungen fallen, so verliert die Rel. Theorie überhaupt ihre empirische Basis.” Another letter to Weyl, 4 days later: “wenn die Länge eines Einheitsmassstabes (bezw. die Gang-Geschwindigkeit einer Einheitsuhr) von der Vorgeschichte abhingen. Wäre dies in der Natur wirklich so, dann könnte es nicht chemische Elemente mit Spektrallinien von bestimmter Frequenz geben, sondern es müsste die relative Frequenz zweier (räumlich benachbarter) Atome der gleichen Art im Allgemeinen verschieden sein. Da dies nicht der Fall ist, scheint mir die Grundhypothese der Theorie leider nicht annehmbar, deren Tiefe und Kühnheit aber jeden Leser mit Bewunderung erfüllen muss.” 22 Cf. Eddington (1987) p. 175, Ryckman (2009) p. 295. 23 Letter to Seelig—quoted in Seelig (1960) p. 274—in which Weyl quotes Einstein: “So – das heisst auf so spekulative Weise, ohne ein leitendes, anschauliches physikalisches Prinzip – macht man keine Physik!”

274

A. Afriat

we return [W29] to the solid ground of physical facts.”24 Impressed at the unexpected transformation, Pauli speaks of “M, revenge”25 ; with the zeal of a convert eager to trumpet his new convictions Weyl even insists that his new theory came straight out of experience26 (and not out of his own hypercreative brain), directly derived from spectrographic data27 . . . For his new theory takes account of the electron’s spin—which in fact got there through the Dirac equation; and in Dirac’s argument (1928) spin does not come straight out of experience28 but out of a mathematical, æsthetic, a priori principle, in much the same spirit as the geometrical justice that produced Weyl’s first gauge theory.

10.3.1 Matter But let us go back a few years. As mentioned in the Introduction, Louis de Broglie (1924), Dirac (1925), Schrödinger (1926) had meanwhile produced an undulatory world. Weyl had no reason to get rid of electricity or gravitation; to those existing ingredients he therefore had to add a matter wave, to update his ontology. As long as there was only gravity and electricity, the gauge relation (10.3)&(10.5) had to hold between them; but now, with a third element, as many compensations were in principle possible, of which only two were plausible: the old relation between gravity and electricity, and a new one between electricity and matter. With (10.3)&(10.5) the theory would have remained subject to Einstein’s objection— which the presence of the electron’s mass m (giving the wavelength29 h/mc and

24 Weyl

(1931a) p. 56: “Alle diese geometrischen Luftsprünge waren verfrüht, wir kehren zurück auf den festen Boden der physikalischen Tatsachen.” Cf. Scholz (2011a) pp. 190–1. 25 “Rache”; Pauli (1979) p. 518: “Als Sie früher die Theorie mit g  = λg machten, war dies ik ik reine Mathematik und unphysikalisch. Einstein konnte mit Recht kritisieren und schimpfen. Nun ist die Stunde der Rache für Sie gekommen; jetzt hat Einstein den Bock des Femparallelismus geschossen, der auch nur reine Mathematik ist und nichts mit Physik zu tun hat, und Sie können schimpfen!” 26 Weyl (1929b) p. 331: “Es scheint mir darum dieses nicht aus der Spekulation, sondern aus der Erfahrung stammende neue Prinzip der Eichinvarianz [ . . . ].” Weyl (1931a) p. 57: “Das neue Prinzip ist aus der Erfahrung erwachsen und resümiert einen gewaltigen, aus der Spektroskopie entsprungenen Erfahrungsschatz.” On Weyl’s ‘empirical turn’ see Scholz (2004) pp. 165, 183, 191–3. 27 Weyl (1931a) p. 57: “Dieses Transformationsgesetz der ψ ist zuerst von P AULI aufgestellt worden und folgt mit unfehlbarer Sicherheit aus den spektroskopischen Tatsachen, genauer aus den Termdubletts der Alkalispektren und der Tatsache, daß die Dublettkomponenten nach Ausweis ihres Zeemaneffekts halbganze innere Quantenzahlen besitzen.” 28 On the logical priority of relativity over spin cf. Weyl (1931b) p. 193: “Da die Möglichkeit einer solchen relativitätsinvarianten Gleichung für ein skalares ψ nicht vorhanden ist, erscheint der Spin als ein durch die Relativitätstheorie notwendig gefordertes Phänomen.” 29 But here Planck’s constant h and the speed of light c—and even charge—are set equal to one.

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frequency mc2 /h) in the Dirac equation made even more convincing,30 by providing an absolute standard of length and time allowing the distant comparisons Weyl sought to rule out in W18.31 The other possibility was left: (10.3) with a quantum version of (10.5),32 of which the most obvious33 was ψ → ψ  = eiλ ψ,

(10.6)

where U(1) replaced the multiplicative group R of (10.5).34 As ψ was now part of a four-dimensional space-time theory, it could no longer obey the Schrödinger equation, which violates relativity by treating space and time very differently.35 Weyl

30 Weyl

(1929c) p. 284: “By this new situation, which introduces an atomic radius into the field equations thereelves—but not until this step—my principle of gauge-invariance, with which I had hoped to relate gravitation and electricity, is robbed of its support.” Weyl (1931a) p. 55: “Die Atomistik gibt unsja absolute Einheiten für alle Maßgrößen an die Hand [ . . . ]. So geht in die DIRACsche Feldgesetze des Elektrons die ,Wellenlänge des Elektrons“, die Zahl h/mc, als eine absolute Konstante ein. Damit fällt das Grundprinzip meiner Theorie, das Prinzip von der Relativität der Längcomessung, dem Atomismus zum Opfer und verliert seine Überzeugungskraft.” See also Penrose (2016) pp. 55–6. 31 See also Weyl (1929c) p. 290. 32 Weyl (1929b) p. 331, Weyl (1929c) p. 284: “this principle has an equivalent in the quantumtheoretical field equations which is exactly like it in formal respects; the laws are invariant under the simultaneous replacement of ψ by eiλ ψ, ϕα by ϕα − ∂λ/∂xα where λ is an arbitrary real function of position and time.” 33 The transformation (10.6) is invisible ‘with respect to position,’ or rather with respect to an observable compatible with the unitary operator determined by (10.6); cf. Weyl (1928) p. 87. The requirement ψ   = ψ is very natural but too weak to determine (10.6), being satisfied by any unitary operator—not just those compatible with the representation (position, momentum or other) in which the wavefunction happens to be written. 34 Weyl (1931a) p. 55: “In dem theoretischen Weltbild bedeutet die Verwandlung von f in p −fp eine objektive Änderung des metrischen Feldes; denn es ist etwas anderes, ob sich eine Strecke bei kongruenter Verpflanzung längs einer geschlossenen Bahn vergrößert oder verkleinert. Nach dem angenommenen Wirkungsgesetz aber ist die Entscheidung über das Vorzeichen der fp auf Grund der beobachteten Erscheinungen unmöglich. Hier enthält darum, in Widerstreit mit einem oben ausgesprochenen erkenntnistheoretischen Grundsatz, das theoretische Weltbild eine Verschiedenheit, welche sich auf keine Weise für die Wahmehmung aufbrechen läßt. P. 57: “Die an der alten Theorie gerügte Unsicherheit des Vorzeichens ±fp löst sich dadurch in das unbestimmte √ Vorzeichen der −1 auf. Schon damals, als ich die alte Theorie aufstellte, hatte ich das Gefühl, daß der Eichfaktor die Form eiλ haben sollte; nur konnte ich dafür natürlich keine geometrische Deutung finden. Arbeiten von SCHRÖDINGER und F. LONDON stützten die Forderung durch die allmählich sich immer deutlicher abzeichnende Beziehung zur Quantentheorie.” See also Weyl (1931b) p. 89. Scholz (2004) p. 193 associates the ‘geometry to matter’ transition from (10.3)&(10.5) to (10.3)&(10.6) with a transition from the a priori fantasies of 1918 to the sober empiricism of 1929. 35 Weyl (1931b) pp. 187–8: “Es ist klar, daß man zu einer befriedigenden Theorie des Elektrons nur kommen wird, wenn es gelingt, das Grundgesetz seiner Bewegung in der von der Relativitätstheorie geforderten, gegenüber Lorentz-Transformationen invarianten Form zu fassen.”

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A. Afriat

adopted what amounted to a Dirac equation,36 but cut in half: deprived of mass and the associated interweaving (σ υμπ λ´εκειν, to use Weyl’s term37 ) of component pairs. We now have four logical elements: 1. 2. 3. 4.

GR: general relativity GJ: geometrical justice MW: matter wave SC: second clock effect;

W29  W18 & MW & SC &? A final element, EL: twice too many energy levels, will be enough to produce W29, its basic structures at any rate.

10.3.2 Spinors The Hamiltonian H of a free classical particle (of mass one-half) is the square p 2 = p, q ˙ of its momentum p; we can loosely write H = p2 . Momentum in quantum mechanics38 is represented by differentiation −i∇ = p , so that the  of a free quantum particle is p = Hamiltonian H 2 = −∇ 2 = −∇ · ∇. Writing H ψ we obtain i∇ t ψ = − ∇ 2 ψ, which shows that Schrödinger’s −∇ 2 in i∇t ψ = H equation violates relativity by differentiating space twice as much as time. But by what should it be replaced? The Klein-Gordon equation39 ( − m2 )ψ = 0 with d’Alembertian  = ∂02 − ∂12 − ∂22 − ∂32 treats space about the same way as time, they have the right transformation properties; but  is ‘squared’ and there are reasons to prefer a √ wave operator and especially a time derivative40 that aren’t. In seeking a square root  Dirac found )∂/ = γ μ ∂μ ,

36 See

Scholz (2006) p. 470. (1939) p. 165 38 See Weyl (1931b) pp. 45–6, 89. 39 Weyl (1931b) p. 186 attributes it to Louis de Broglie. 40 Weyl (1931b) p. 188: “Sie ist nicht im Einklang mit dem allgemeinen Schema der Quantenmechanik, welches verlangt, daß die zeitliche Ableitung nur in der ersten Ordnung auftritt.” P. 193: “Legt man die de Brogliesche Wellengleichung für das skalare ψ zugrunde, in welche die elektromagnetischen Potentiale [Aμ ] durch die Regel [(10.15)] eingeführt sind, so ergibt sich aber für die elektrische Dichte ein Ausdruck, der außer ψ die zeitliche Ableitung ∂ψ/∂t enthält und nichts mit der Ortswahrscheinlichkeit zu tun hat; sein Integral ist überhaupt keine Einzelform. Dies ist nach Dirac das entscheidendste Argument dafür, daß die Differentialgleichungen des in einem elektromagnetischen Feld sich bewegenden Elektrons von 1. Ordnung in bezug auf die zeitliche Ableitung sein müssen.” 37 Weyl

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where the γ μ ’s have the algebraic properties needed to get rid of the cross terms that appear when squaring. He therefore proposed the Dirac equation41 (m − i∂/ ) ψ = 0

(10.7)

which not only treats the three spatial derivatives γ k ∂ k the same way as the time derivative γ 0 ∂ 0 , but differentiates with respect to time only once.42 The γ μ ’s, which do not commute, cannot be numbers; they admit for instance the canonical representations  γ ↔ 0

0 − σ0

σ0 0



 γ ↔ k

0 σk

σk 0

 ,

(10.8)

where all four quaternions σ μ : C2 → C2 are hermitian and unitary; σ 0 is the identity 12 , and the three traceless operators σ k satisfy 2iσ j = εjkl [σ k , σ l ]. The wave ψ on which the γ μ ’s act therefore has four (complex) components— embarras de richesses which Weyl found most troubling: “doppelt zu viel Energieniveaus”! The anti-diagonality of the γ μ ’s govems the embarrassing excess by swapping the two two-spinors making up ψ. As the embarrassment is due to the sign that distinguishes between the different interweavings43 produced by the γ μ ’s, Weyl deals with it by choosing the only mass—none at all—that doesn’t distinguish between plus and minus.44 Without mass and half the components, (10.7) becomes

41 Dirac

(1928) (1931b) p. 190: “Nach dem allgemeinen Schema der Quantenmechanik sollte, wie schon erwähnt, die Differentialgleichung für ψ von 1. Ordnung hinsichtlich der zeitlichen Ableitung von ψ sein. Gemäß dem Relativitätsprinzip kann sie aber dann auch nur die 1. Ableitungen nach den räumlichen Koordinaten enthalten.” 43 Symplectic for time, in the rather standard representation (10.8), but simply ‘NOT’ for space. The inter-weaving produced by four purely NOT γ μ ’s (anti-diagonality with no minuses) would be pointless; Dirac’s unusual hyperbolicity has to be expressed by one or more appropriately placed minuses: if the three spatial gammas have merely NOT anti-diagonality, γ 0 will be symplectically anti-diagonal. 44 Weyl (1929b) p. 330–1: “Die Diracsche Theorie, in welcher das Wellenfeld des Elektrons durch ein Potential ψ mit vier Komponenten beschrieben wird, gibt doppelt zu viel Energieniveaus; man sollte darum, ohne die relativistische Invarianz preiszugeben, zu den zwei Komponenten der Paulischen Theorie zurückkehren können. Daran hindert das die Masse m des Elektrons [...].” Weyl (1929c) p. 292: “The [mass] term (10.5) of the Dirac theory is, however, more doubtful. It must be admitted that if we retain it we can obtain all details of the line spectrum of the hydrogen atom—of one electron moving in the electrostatic field of a nucleus—in accord with what is known from experiment. But we obtain twice too much; if we replace the electron by a particle of the same mass and positive charge +e (which admittedly does not exist in nature) the Dirac theory gives, contrary to all reason and experience, the same energy terms as for a negative electron, except for a change in sign. Obviously an essential change is here necessary.” P. 294: “Be bold enough to leave the term involving mass entirely out of the field equations.” 42 Weyl

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A. Afriat

σ μ ∂μ ψ = 0.

(10.9)

We now have five logical elements: 1. 2. 3. 4. 5.

GR: general relativity GJ: geometrical justice MW: matter wave SC: second clock effect EL: twice too many energy levels;

W18 & MW & SC & EL  W29.45 The foundations having been laid, the rest will fall into place.

10.3.3 Tetrads The new arrival, matter, is doubly unsettling: not only does it produce structural upheaval, leading to a new gauge relation (now between matter and electricity); but the spinors used to represent matter even make Weyl alter the (mathematical) nature of gravity, now represented by tetrads—Achsenkreuze—and no longer by the metric. In W29 we therefore see the appearance of a new material, quantum, complex world alongside the old gravitational, space-time, real world (leaving aside electricity for the time being, which indeed will presently emerge from their relationship). The two worlds are by no means independent, they are well entangled: since matter has to take account of the gravitational curvature of the space-time on which its spinors are defined, the material connection M relies on the gravitational connection Γ —without, however, being exactly the same, as we shall soon see. A differential law like (10.9) compares values at neighbouring points; a background notion of ‘constancy’ is needed for such a comparison to make sense; so a geometrically sensical field equation for spinors has to take account of space-time curvature. Gravity in W18 was represented by the metric g = gμν dxμ ⊗ dxν , the oblique bases dxμ being subject to GL (4, R), which is much larger than SO+ (1, 3) and has nothing to do with the group SL (2, C) often used for relativistic spinors.46 To apply the same laws (or almost) to matter and gravity, Weyl replaces GL (4, R) by the subgroup SO+ (1, 3) locally isomorphic to SL (2, C).47 Why tetrads, then? 45 W18

& MW & SC give something like Dirac-Maxwell theory in curved space-time. spinors will in fact be subject to a slightly larger group but we can think of SL (2, C) for the time being. Relevant group theory will be looked at more closely in §10.3.4. 47 Weyl (1929c) p. 285: “The tensor calculus is not the proper mathematical instrument to use in translating the quantum-theoretic equations of the electron over into the general theory of relativity. Vectors and terms are so constituted that the law which defines the transformation of their components from one Cartesian set of axes to another can be extended to the most general linear transformation, to an affine set of axes. That is not the case for the quantity ψ, however; this kind of quantity belongs to a representation of the rotation group which cannot be extended to the affine group. Consequently we cannot introduce components of ψ relative to an arbitrary coordinate 46 Weyl’s

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Once SO+ (1, 3) is chosen as the gravitational group, one might as well deal with entities it preserves. Of course SO+ (1, 3) would preserve the geometrical relations of oblique bases too; but the orthonormality characterized by ones and noughts is distinguished. It would be pointless to adopt simple geometrical relations only to upset them with a large, disruptive group; but with a group that preserves geometrical simplicity, one might as well adopt that simplicity (orthonormality) from the start. Summing up: relativity & MW  something like the Dirac equation  relativistic spinors  something like SL (2, C)  space-time bases invariant under SO+ (1, 3)  orthonormal tetrads. We’ll now see how Weyl extracts electricity from the relationship between matter and gravity.

10.3.4 Electricity 10.3.4.1

Preliminary Anachronisms

Some geometry48 is needed to understand what Weyl (1929b) is up to on pages 332–4. Hermitian operators on C2 constitute a four-dimensional real vector space with scalar product x, y = 12 Tr (xy). The quaternions σ μ we saw in (10.8) make up a convenient orthonormal basis, σ μ , σ ν  = δ μν The real numbers xμ = σ μ , x can be viewed as the components of a space-time four-vector—whose hyperbolic squared length  x2η = ημν xμ xν is given by det x. The Lorentz group is defined by the isometric condition   O (1, 3) =  ∈ GL (4, R) : x2η = x2η , μ

where the components of x are ν xν An element U of GL (2, C), acting on a Hermitian operator x as x → U xU † , preserves the determinant of x if det U = eiθ , for then det U det(U† ) = 1, where the dagger denotes the adjoint. So the squared length x2η = det x, whose preservation characterizes the Lorentz group, is also preserved by the group49

system in general relativity as we can for the electromagnetic potential and field strengths. We must rather describe the metric at a point P by local Cartesian axes e(α) in toad of by the gpq . The wave field has definite components ψ1+ , ψ2+ ; ψ1− , ψ2− [full Dirac theory] relative to such axes, and we know how they transform on transition to any other Cartesian axes in P. The laws shall naturally be invariant under arbitrary rotation of the axes in P, and the axes at different points can be rotated independently of each other; they are in no way bound together.” 48 See Weyl (2008) pp. 7–15, Weyl (1931b) pp. 128–33, Smirnov (1961) pp. 298–309, Penrose and Rindler (1987) pp. 9–67, Needham (2000) pp. 122–80. 49 Weyl (1929b) p. 333: “U bewirkt an den x eine Lorentztransformation, d. i. eine reelle α homogene lineare Transformation, welche die form−x02 + x12 + x22 + x32 in sich überführt.”

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A. Afriat

W = {U ∈ GL (2, C) : | det U | = 1} acting on C2 .50 The stronger condition det U = 1 gives SL (2, C) = {U ∈ GL (2, C) : det U = 1} . As the Lie algebras are the same, it makes local sense to think of W as an extension51 SL (2, C) × U(1) of SL (2, C) by U(1). But multiplication of the ‘Cartesian factors’ gives (U, eiθ ) → eiθ U, as well as (−U,−eiθ ) → eiθ U. Since U = −U and eiθ = −eiθ , the pairs (U, eiθ ) and (−U, − eiθ ) must be distinct elements of the extension S = SL (2, C) × U(1). So we have a 2-1 homomorphism   f : S → W; U, eiθ → eiθ U with inverse image f−1 (eiθ U) = {(U, eiθ ), (−U, − eiθ )}.52 There is also a 2-1 homomorphism53   h : SL (2, C) → SO+ (1, 3) ; U → σ μ , U σ ν U † between SL (2, C) and the component54 SO+ (1, 3) of O (1, 3) containing the identity 14 ; the inverse image h−1 (14 ) = {±12 } ⊂ SL (2, C) contains −12 too—indeed h(U) = h(−U) for all U. The matrix 

eiω 0  0 eiω

 ∈ U(2) ⊂ W

belongs to SU(2) ⊂ SL (2, C) if ω = − ω. The two eigenvalues would then have opposite phases, so the corresponding O(3) rotation (given by the quotient  eiω/eiω of the eigenvalues, and hence by the difference of the phases) is twice

50 Cf.

Weyl (1929b) p. 334: “Das Transformationsgesetz der ψ-Komponenten besteht darin, daß sie unter dem Einfluß einer Transformation  der Weltkoordinaten x(α) sich so umsetzen, daß die Größen [(10.11)] die Transformation  erleiden.” 51 Cf. Scholz (2004) p. 189; Scholz (2011b), third page of the paper; and footnote 7 above. 52 Here I am indebted to Thierry Levasseur and Johannes Huisman. 53 Weyl (1929b) p. 333: “Man kann ihn normalisieren durch die Forderung, daß die Determinante von U gleich 1 sei, aber selbst dann bleibt eine Doppeldeutigkeit zurück.” 54 Weyl (1929b) p. 333: “wir unter den Lorentztransformationen nur die ein einziges in sich abgeschlossenes Kontinuum bildenden  bekommen, welche 1. Vergangenheit und Zukunft nicht vertauschen und 2. die Determinante +1, nicht −1, besitzen [ . . . ].”

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the SU(2) phase angle ω.55 The homomorphism56 h: SU(2) → SO(3) therefore maps isomorphically on the half-open interval 0 ≤ ω < π ; the isomorphism only breaks down at ω = π , for there the negative identity −12 is reached, which already corresponds to the identity 14 = h(±12 ).57

The double arrows ⇒ denote 2-1 homomorphisms, the hooked arrows → injections. The bottom row is relativistic and hyperbolic whereas the top row is compact, ‘spatial’ and more literally ‘spherical.’ The orbit under SL (2, C) of a timelike, future directed space-time four-vector satisfying det x = x2η = 1

(10.10)

 ⊥ is the unit future hyperboloid H− 1 If x is written in spectral form x = ζ P + ζ P ,  where the eigenvalues ζ and ζ are real and P is the projector |ψψ| along |ψ ∈ C2 ,  normalization (10.10) is given by ζ = 1/ζ . Cutting the hyperboloid by simultaneity surfaces  t orthogonal to σ 0 we obtain the spherical orbits58 H+ 1 ∩ t of SU(2), whose rotations affect only the eigenvectors of x, not its eigenvalues. The spacelike    ray r x1 , x2 , x3 r determines (the polar angles ϕ, θ of) the point on the sphere.  The degenerate Hermitian operator x = 12 (with ζ = ζ = 1) represents rest, at the correspondingly degenerate bottom of the hyperboloid, where the sphere has shrunk to a point. Just as the orbits of SU(2) are horizontal, the pure boosts making up the ‘complementary’ part of SL (2, C) displace vertically, straight up and down the hyperboloid, by affecting only the rest of x: its eigenvalues. A boost B is ‘pure’ with respect to the eigenvectors of x, which it therefore has to share, yielding an operator B = βP + β −1 P⊥ of the same form, where β is also real.59 Generic elements of SL (2, C) which affect all of x, eigenvalues and eigenvectors, produce ‘diagonal’ motions on the hyperboloid.

(1931b) p. 129: “Und zwar ist, wenn ε = eiω = e(ω) gesetzt wird, der Drehwinkel der Drehung um die z-Achse ϕ = − 2ω.” 56 I use the same letter h for the restriction to SU(2). 57 Weyl (1931b) p. 129: “sie bleibt nämlich zweideutig, weil sie durch Multiplikation mit −1, durch Verwandlung von σ in −σ nicht verloren geht. [ . . . ] Man erhält dadurch alle Drehungen und jede genau zweimal.” 58 Weyl (1931b) p. 129: “Jede unitäre Transformation [ . . . ] liefert danach eine Drehung s der Kugel [ . . . ].” 59 Weyl (1931b), bottom of p. 131; a “Zeitachse ändernde Lorentztransformation” is a boost. 55 Weyl

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10.3.4.2

Celestial Sphere



If ζ vanishes we’re left with x = ζ P or just the null ray60 containing ψ; null   because det x vanishes if ζ ζ does—the degenerate case ζ = ζ = 0 being at the origin, the tip of the cones. Rather than xμ = 12 Tr(Pσ μ ) we can write xμ = ψ|σ μ |ψ or61 xμ = ψσ μ ψ.

(10.11)

Since σ 0 is the identity 12 and therefore x0 equals ψ| ψ, one may be tempted to write the Pythagorean expression ψ| ψ = ψ| σ 1 | ψ + ψ| σ 2 | ψ + ψ| σ 3 | ψ,  2  2 which does not hold.62 Pythagorean equations that do hold are x0 = x1 +  2 2  3 2 x + x and ψ| ψ = |ψ 1 |2 + |ψ 2 |2 , where ψ 1 = ϕ1 | ψ, ψ 2 = ϕ2 | ψ, and the basis |ϕ1 , |ϕ2  is orthonormal. Now that we’ve gone from the hyperboloid to the future light cone K+ , the orbit of SU(2) is the celestial sphere St+ = K+ ∩ t , which, choosing t = 1, can be mapped to its equatorial plane C = {(1, x, y, 0)} by stereographic projection from the south pole (1, 0, 0, − 1). I’ve written C because the coordinates x, y of the equatorial plane can be viewed as the real and imaginary parts of the complex number x + iy, which is then identified with the quotient ψ 2 /ψ 1 and hence the ray containing | ψ  = ψ1 | ϕ1  + ψ2 | ϕ2 , where |ϕ2  (or ψ 1 = 0) corresponds to the south pole. If ψ 2 vanishes, the ray from the south pole will be vertical, through the origin of C and the north pole, which therefore corresponds to |ϕ1 . Such a scheme works well for the ‘horizontal,’ in other words ‘purely spatial’ group SU(2); but boosts σ 0 → Uσ 0 U† distort St+ by tilting the surfaces  t perpendicular to σ 0 . Rather than null vectors x ∈ K+ we can take null rays ρ = [rx]r ⊂ K+ . For a given foliation, the ‘space of (future) null rays’ S + is equivalent to a celestial sphere; but S + behaves better under boosts by not relying on foliation.63   To the element U ∈ W which gives ψ = Uψ ∈ C2 and hence x μ = ψ σ μ ψ  ∈ μ  ν 64 + 4 μ R corresponds the element  ∈ SO (1, 3) which returns x = ν x . The

60 Weyl

(1929b) p. 333: “Die Variablen ψ 1 , ψ 2 sowie die Koordinaten xα kommen hier nur ihrem Verhältnis nach in Frage.” 61 Weyl (1929b) equations (2) p. 333 and (3) p. 334, Weyl (1931b) equations (8.12) and (8.16). 62 The terms look appropriately quadratic, the trinions σ k are indeed orthogonal but R3 should not be confused with C2 . 63 See Weyl (1929b) p. 333. 64 See footnote 50 above.

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quadratic form ψσ μ ψ therefore establishes a correspondence between matter (in C2 ) and gravity (in R4 ). The loose angle65 which will produce electricity below can already be seen in ψe−iλ σ μ eiλ ψ = ψσ μ ψ.

10.3.4.3

Electricity, Gravity and Matter

From where, then, does electricity emerge?66 Everything turns on the apparently insignificant choice67 of W over SL (2, C), to propagate matter. One wonders how it can make any difference, for the two groups seem to differ by a mere nuance, but it is out of that nuance that Weyl extracts electricity.68 Various quantities  relevant  are invariant under W, the best example69 being perhaps det U xU † , with U ∈ W; so the choice, however unusual, is by no means insensate. In a sentence: The laws governing matter and gravity are the same up to a detail that makes no difference to matter, but can nonetheless produce electricity. The material connection M is a one-form which, applied to a vector γ˙ ∈ Ta M directed towards nearby b ∈ M, yields a generator M, γ˙  of transport belonging to the Lie algebra w = LieW. The relationship between W and w can be illustrated by looking at the subgroup SU(2) ⊂ W and its Lie algebra su(2) ⊂ w. Taking Uτ = eiE1 τ | ϕ1 ϕ1 | + eiE2 τ |ϕ2 ϕ2 |∈ SU(2) we have Uτ | ψ = ei(E1 τ +η1 ) | ψ1 ϕ1  + ei(E2 τ +η2 ) | ψ2 ϕ2 ,

65 Weyl (1929b) p. 333: “Durch  ist die lineare Transformation U

der ψ nicht eindeutig festgelegt, sondern es bleibt ein willkürlicher konstanter Faktor eiλ vom absoluten Betrage 1 zur Disposition. Cf. Weyl (1931b) p. 131: “Transformationen σ , welche sich nur durch einen Faktor eiλ vom absoluten Betrag 1 voneinander unterscheiden, liefern dasselbe s. 66 See Weyl (1929b) p. 348, Weyl (1929c) p. 291, Afriat (2013, 2015). 67 See footnote 4 above. 68 Weyl (1929c) p. 291: “It is my firm conviction that we must seek the origin of the electromagnetic field in another direction. We have already mentioned that it is impossible to connect the transformations of the ψ in a unique manner with the rotations of the axis system; however we may attempt to accomplish this by means of invariants which can be used as constituents of an action quantity we always find that there remains an arbitrary “gauge factor” eiλ . Hence the local axis-system does not determine the components of ψ uniquely, but only within such a factor of absolute magnitude 1.” 69 Here I am indebted to Ermenegildo Caccese.

284

A. Afriat

which represents a motion,70 infinitesimally generated by H = M, γ˙  = E1 | ϕ1 ϕ1 | + E2 |ϕ2 ϕ2 |∈ su(2), on a two-dimensional torus determined by the two phases β k = Ek τ + ηk ; the angles ηk being the arguments of the coefficients ψ k . Since a = γ (0) and b = γ (1) we have β kb = Eka + β ka and eiβkb = eiEka eiβka , and hence |ψ b  = eiH | ψ a . The generator H acts only on the phase-torus and not on the moduli of |ψ; even infinitesimally, |ψ has to be multiplied by a unitary operator—multiplication by a Hermitian operator H would lengthen the state by acting obliquely (rather than orthogonally). An element of the Lie algebra can be thought of here as an infinitesimal speed on the twodimensional torus. A group element produces an arbitrary rotation, the Lie algebra allows the single rotation to be broken down pointwise into its various infinitesimal rates. The distinction is somewhat obscured by a constant generator H, which produces a constant speed; a variable generator H(τ ) with variable eigenvalues Ek (τ ) would be needed to make full sense of the distinction. The geometrical picture of a motion on a torus suggests the conventional choice of viewing the generators of infinitesimal speed as real, in other words Hermitian, rather than purely imaginary. So the spinors representing matter are propagated by a connection with values in w; whereas gravity, represented by tetrads,71 is governed by a connection with values in o (1, 3). Weyl saw an apparently uninteresting Lie algebra w # o (1, 3)— caught, as it were, between matter and gravity—in which a third connection would surely take its values.72 As in 1918, Weyl identified the real-valued connection with the electromagnetic potential73 A, whose curvature F = dA gave the electromagnetic field. Its derivative dF = 0 in turn provided Maxwell’s two homogeneous equa-

70 Notions of motion, speed and time are clearly metaphorical here, since τ

is an abstract parameter. (1931b) p. 195: “Ferner bedarf man in der allgemeinen Relativitätstheorie an jeder Weltstelle P eines aus vier Grundvektoren in P bestehenden normalen Achsenkreuzes, um die Metrik in P festzulegen und relativ dazu die Wellengröße ψ durch ihre vier [full Dirac theory, with mass] Komponenten ψ ρ beschreiben zu können; die gleichberechtigten normalen Achsenkreuze in einem Punkte gehen durch die Lorentztransformationen auseinander hervor.” 72 Weyl (1929b) p. 348: “Dann ist aber auch die infinitesimale lineare Transformation dE der ψ, welche der infinitesimalen Drehung dγ entspricht, nicht vollständig festgelegt, sondern dE kann um ein beliebiges rein imaginäres Multiplum i · df der Einheitsmatrix vermehrt werden.” Weyl (1929c) p. 291: “Then there remains in the infinitesimal linear transformation dE of ψ, which corresponds to the given infinitesimal rotation of the axis-system, an arbitrary additive term +idϕ · 1.” 73 Weyl (1929c) p. 291, Weyl (1931b) p. 195: “Aus der Natur, dem Transformationsgesetz der Größe ψ ergibt sich, daß die vier Komponenten ψ  relativ zum lokalen Achsenkreuz nur bis auf einen gemeinsamen Proportionalitätsfaktor eiλ durch den physikalischen Zustand bestimmt sind, dessen Exponent λ willkürlich vom Orte in Raum und Zeit abhängt, und daß

infolgedessen zur eindeutigen Festlegung des kovarianten Differentials von ψ eine Li earform α fα dxα erforderlich ist, die so mit dem Eichfaktor in ψ gekoppelt ist, wie es das Prinzip der Eichinvarianz verlangt.” 71 Weyl

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tions.74 Infinitesimally, one can think of electricity as an appropriate ‘difference’ # between matter and gravity (and integrally, as a quotient of sorts; but see §10.3.4.1).

As the ‘bivalency’75 of the homomorphisms represented by the double arrows ⇒ is only global and not local, it is eliminated by the dashed arrows  (from the groups to their Lie algebras), which therefore give rise to the isomorphisms denoted by the arrows ↔ with two heads. All four projections—indicated by the vertical arrows—do away with a real number, which is an angle only for the ‘far’ two, involving groups; an angle being a real number with a global structure to which differentiation (or local linearization) has no access. Since the real numbers eliminated by the two far projections (involving groups) are already present locally, they survive the differentiation denoted by . An abuse of notation is worth pointing out: the dashed arrows map a group, treated as a single element, to its Lie algebra (one sometimes writes → between individuals), whereas the other arrows map, more conventionally, from the domain to the range. The diagram can be extended by the projections.

which are complementary (as in “orthogonal complement”) in the sense that the arrow going down/up omits/keeps what’s kept/omitted by the one going up/down. The ‘angular freedom’76 caught between matter and gravity can be seen in   h (U ) = h eiλ U ∈ SO+ (1, 3) 74 Weyl

(1929b) p. 349, Weyl (1929c) pp. 291–2. Cf. Ryckman (2009) p. 295: “Weyl derived the Maxwell equations from the requirement of local phase invariance, thus coupling charged matter to the electromagnetic field, and so originating the modern understanding of the principle of local gauge invariance (“local symmetries dictate the form of the interaction”) that lies at the basis of contemporary geometrical unification programs in fundamental physics.” 75 Weyl says Doppeldeutigkeit, both homomorphisms are 2-1. 76 See footnote 65.

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A. Afriat

or even     h−1 h (U ) = eiλ U ⊂ W, λ

where h : W → SO+ (1, 3) is given by the same rule U → σ μ , Uσ ν U† , which also shows that the free angle λ never reaches SO+ (1, 3):     eiλ U → σ μ , eiλ U σ ν e−iλ U † = σ μ , U σ ν U † . Again, the fact that the ‘circle-1’ homomorphism h maps eiλ U to a single Lorentz transformation for all angles λ should not be confused with the fact that the 2-1 homomorphism f : S → W maps the two distinct pairs (U, eiθ ) and (−U, − eiθ ) to the same product eiθ U ∈ W. Returning to the Lie algebras o (1, 3) and w = sl (2, C) ⊕ R12 , we have the corresponding expressions h (U) = h (U ⊕ λ12 ) ∈ o (1, 3) and  h−1 (h (U)) = U ⊕ λ12 λ ⊂ w, where h: w → o (1, 3) ; U ⊕ λ12 → U gets rid of λ by projecting the pair (U, λ) to U. The loose phase eiλ ∈ U(1) has become the ‘additive’ freedom λ ∈ R = u(1). So there’s a connection for spinors, another for tetrads, and a third—namely A—for the residual U(1) freedom caught in between.77 The values A, γ˙  belong to the Lie algebra u(1) of the group U(1) caught between matter and gravity. The gravitational connection Γ = Γμ ⊗ dx μ = Γμr Tr ⊗ dx μ

77 Weyl

(1929b) p. 348: “Zur eindeutigen Festlegung des kovarianten Differentials δψ von ψ hat man also außer der Metrik in der Umgebung des Punktes P auch ein solches df für jedes von P −−→ ausgehende Linienelement P P  = (dx) nötig. Damit δψ nach wie vor linear von dx abhängt, muß df = fp (dx)p

eine Linearform in den Komponenten des Linienelements sein. Ersetzt man ψ durch eiλ · ψ, so muß man sogleich, wie aus der Formel für das kovariante Differential hervorgeht, df ersetzen durch df − dλ.” Weyl (1929c) p. 291: “The complete determination of the covariant differential δψ of ψ requires that such a dϕ be given. But it must depend linearly on the displacement PP : dϕ = ϕp (dx)p , if δψ shall depend linearly on the displacement. On altering ψ by multiplying it by the gauge factor eiλ we must at the same time replace dϕ by dϕ − dλ as is immediately seen from this formula of the covariant differential.” Weyl’s notation is confusing: whereas the one-form dλ (which is a differential) is necessarily exact, df and dϕ (my A) aren’t.

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takes its values Γ, γ˙  = Γμr γ˙ μ Tr in o (1, 3), the material connection M = Mμ ⊗ dx μ = Mrμ Tr ⊗ dx μ its values M, γ˙  = Mrμ γ˙ μ Tr in w. The three connections are related by their Lie algebras. Lie W = Lie SL (2, C) ⊕ Lie U(1). For an electron subject to gravity as well as electricity we can write D /ψ = 0

(10.12)

instead of (10.9), where D / = σ μ Dμ and78   ∂μ → Dμ = ∂μ + iMμ = ∂μ + i Γμ + Aμ . Already in W18 one could express the total—rotating and dilating—connection as an appropriate sum Γ + A of a metric (purely gravitational) connection Γ and a dilating (electric) connection A; adding electricity to gravity one obtained the two together, not something new, a third element. In W29 we have the same two terms, but they add up to matter. This account does not render all the historical colour of Weyl’s argument,79 which I can try to express more faithfully as follows. If the phase angle λ were propagated holonomically by A, the curvature F = dA would vanish and electricity with it. To convince himself that λ has to vary anholonomically, Weyl relates its propagation to that of the tetrads representing gravitation. He seems to argue that if the tetrad were ‘constant,’ λ would be too; since the tetrad varies, λ should too. Holonomy is the best meaning I can give to the constancy of a tetrad. Only a flat gravitational connection Γ allows the assignment of the same tetrad to different points; only with flatness can there be global constancy or ‘sameness’; with curvature it becomes meaningless to say that tetrads at different points are the same. Where the constancy of tetrads makes no sense, one can suppose they vary. Since the tetrad’s variation is given by infinitesimal propagation, so is λ’s; in any case there is no reason to confine λ’s variation (holonomically) to a continuous function λ: M → R, which would be too restrictive. The object needed for the infinitesimal propagation of an angle, linearly in the angle and the direction of propagation γ˙ ,

78 This doubly covariant derivative for matter interacting with electricity and gravity is obtained by combining the ‘gravitational covariance’ expressed in Eq. (10.13) of Weyl (1929b) with the ‘electromagnetic covariance’ expressed at the bottom of p. 350 and especially the top of p. 351, same paper. 79 Weyl (1929b) p. 348, Weyl (1929c) p. 291

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A. Afriat

is a real-valued one-form A. A few words about a possible confusion: With a nonAbelian Lie algebra g acting on a (nontrivial) vector space V (such as CN with N > 1), there can be no confusion between the operator M, γ˙  ∈ g and its argument in V, which are mathematical objects of different kinds; the matrix representations are of different shapes and sizes, N × N rather than N. But where A, γ˙  is a real number acting on another real number λ, the operator and its argument are easily confused. Here the angle is acted upon by a ‘scalar’ operator A, γ˙  ∈ u(1). Weyl claims that in special relativity there’s a single tetrad and hence one value of  λ,80 which may mean that the structure groups G = SO+ (1, 3) and G = R (acting at a generic space-time point) coincide with the corresponding gauge groups81 G $ G and82 G  $ G (acting—rigidly here—on all of space-time M).83 In general relativity84 the tetrad can vary, and λ too; the gauge groups G and G  , which no longer act rigidly,85 become much (indeed infinitely) larger than the structure groups.86 But why, one may ask, should the variations of λ and tetrads be at all related in the first place? Are they not independent? Why not a curved A with a flat Γ , or the

80 Weyl

(1929b) p. 348: “In der speziellen Relativitätstheorie muß man diesen Eichfaktor als eine Konstante ansehen, weil wir hier ein einziges, nicht an einen Punkt gebundes Achsenkreuz haben.” Weyl (1929c) p. 291: “In the special theory of relativity, in which the axis system is not tied up to any particular point, this factor is a constant.” 81 A gauge group is made up of vertical automorphisms V : E → E on the (here trivial) fibre bundle E = M × V—vertical inasmuch as each copy Gx of G confines its action to its own Vx , without interfering with the other fibres Vx  . Since horizontal is a metaphor for ‘constancy’ from fibre to fibre along M, vertical means ‘just up the fibre’ (and not along M). However ‘symmetric’ the Cartesian product · × · may look, here it isn’t at all, since the two factors are distinguished as base M and fiber V: a copy Vx gets assigned to each x of the base manifold so that x can be fixed while ψ ∈ Vx is varied, whereas ‘displacement only along the base manifold with constancy in the corresponding fibers {Vx }x ’ makes no sense without further structure, namely a connection. 82 This equivalence with the structure group expresses the ‘constant’ degeneracy of the gauge group, which, having lost all the pointwise freedom to vary its action over the underlying manifold, rigidly applies the same element g ∈ G everywhere. 83 Cf. Kretschmann (1917): general covariance can be countenanced in flat space-time. 84 Weyl (1929b) p. 348: “Anders in der allgemeinen Relativitätstheorie: jeder Punkt hat sein eigenes Achsenkreuz und darum auch seinen eigenen willkürlichen Eichfaktor; dadurch, daß man die starre Bindung der Achsenkreuze in verschiedenen Punkten aufhebt, wird der Eichfaktor notwendig zu einer willkürlichen Ortsfunktion.” Weyl (1929c) p. 291: “But it is otherwise in the general theory of relativity when we remove the restriction binding the local axis-systems to each other; we cannot avoid allowing the gauge factor to depend arbitrarily on position.” 85 Cf. Weyl (1929b) p. 331: “es fällt mir schwer, die Macht zu begreifen, welche die lokalen Achsenkreuze in den verschiedenen Weltpunkten in ihrer verdrehten Lage zu starrer Gebundenheit ancinander hat einfrieren lassen.” 86 Of course the flatness of space-time only imposes holonomy G H ⊂ G , not rigidity G ⊂ GH , which is much stronger; cf. Ryckman (2009) p. 295: “Weyl’s argument for his correct conclusion is, in fact, flawed, resting on an unnecessary assumption about the representation of spinor matter fields within tetrad formulations of arbitrarily curved space-times.” The “flatness” I mean refers to the Levi-Civita connection, which is both metric and symmetric; a metric connection with torsion can produce (torsional) anholonomies, even on Minkowski space-time.

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other way around? We may simply have another case of geometrical justice: since tetrads are allowed to vary anholonomically, why not λ too?87 Even if the length and direction that deserved the same treatment in 1918 were part of a single object,88 they could vary just as independently as λ and tetrads: a flat length connection with a curved directional connection, or even the other way around (which gives electricity without gravity—in flat space-time). In 1918 all Weyl had to identify electricity was the electromagnetic look of the expressions F = dA and dF = 0; but the Hamiltonian, quantum-mechanical content of W29 provides more. Electricity is represented in Hamiltonian theory by adding the electromagnetic potential to momentum89 : p → p + A

(10.13)

or pμ → pμ + Aμ . Again, momentum in quantum mechanics is given by differentiation: p → p  = −id

(10.14)

d → D = d + iA

(10.15)

μ = −i∂μ . The rule90 or pμ → p

or Dμ = ∂ μ + iAμ is obtained by combining (10.13) and (10.14). The compensation of (10.6) by (10.3) can be seen in the Lagrangian    L = ψσ μ Dμ ψ = ψ σ μ Dμ − i∂μ λ ψ  Here electricity is introduced via (10.13), not conjured up; but once we have a real-valued connection A (caught between matter and gravity), whose first two derivatives look electromagnetic, (10.13)–(10.15) provide valuable support. Weyl nonetheless claims to have derived electricity independently (without (10.13)), out of the group-theoretical relationship between matter and gravity.

87 Cf.

Weyl (1929b) pp. 331–2: “Gerade dadurch, daß man den Zusammenhang zwischen den lokalen Achsenkreuzen löst, verwandelt sich der Eichfaktor eiλ , der in der Größe ψ willkürlich bleibt, notwendig aus einer Konstante in eine willkürliche Ortsfunktion; d. h. nur durch diese Lockerung wird die tatsächlich bestehende Eichinvarianz verständlich.” 88 What does or doesn’t constitute a ‘single object’ is rather arbitrary: the direction and length of a vector can be brought apart by taking, instead of a vector, a ray (one object) and a separate number (another object); there are likewise ways of building a single object out of a number and a tetrad. 89 See Weyl (1931b) p. 88. 90 See Weyl (1929c) p. 283, Weyl (1931b) p. 89.

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10.4 Yang-Mills Here the structure group SU(N) replaces U. Weyl is no longer in the foreground, nor is his complaint that Dirac’s theory gave EL: twice too many energy levels. The curved space-time from which W18 arose can now, having done its bit,91 be kept or dropped. Let us go back to (10.6), which is indeed a natural choice to replace (10.5). But is it the only natural choice? The transformation on the infinite-dimensional Hilbert space H containing ψ should no doubt be unitary, but (10.6) is a very special unitary transformation, which ought perhaps to be generalized. One thinks of the function λ as real-valued: it assigns a real number λx to every x ∈ M. Since the wavefunction ψ assigns not a complex number but a spinor ψx ∈ CN x to every x, the value λx is in fact the operator N λx · 1N : CN x → Cx N But then why not take a general Hermitian operator x : CN x → Cx rather than the very special Hermitian operator λx · 1N ? Why stop when one’s almost there? Legitimate question, which is enough to yield YM. The infinitesimal generators  give the Lie algebra su(N ) of the structure group SU(N), whose elements ei act at a single space-time point. Again, the gauge group G acting on all of M will not in general be a rigid copy of the structure group. The pointwise freedom to perform independent rotations at every x ∈ M is obtained by putting together independent copies SUx (N ) of SU(N). The gauge group acts unitarily on H since every eix ∈ SU(N ), point by point, acts unitarily on its fibre CN x . One can think of the vertical character of the automorphism U ∈ G in (appropriately continuous) ‘block diagonal’ terms, where the block eix is identified by x. Mathematically, the Yang-Mills connection A = Akμ Uk ⊗ dx μ can be seen as a unitary (and N-dimensional) version of Weyl’s material connection M, in the sense that SU(N) replaces W. The infinitesimal generators U1 , . . . , UN span the Lie algebra su(N ). Applied to a tangent vector γ˙ ∈ Ta M directed towards nearby b ∈ M, the one-form A gives the generator

  A, γ˙  = Aμ γ˙ μ = Akμ dx μ , γ˙ Uk . In 1929 Weyl would have seen no physical reason to take the step from U(1) to SU(N). Though given to physicomathematical speculation of great imaginative virtuosity, he didn’t take the purely mathematical step either. The details of the physics that ultimately did produce the non-Abelian theory are in Yang and Mills

91 The

geometrical justice of §10.2.1 required a curved length connection A to balance the curved directional connection. By adopting a flat space-time connection alongside a curved isospin connection Yang and Mills (1954) reversed the injustice of Einstein’s theory—which has a curved directional connection with a flat length connection.

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(1954). So one can distinguish between the ‘purely logical’ step (which I’ve called NA: non-Abelian structure group) and the ‘physical’ step that would be taken in the fifties. The logical step, however conceptual or even fictitious, seems just as relevant here.

10.5 Logical Summary Summing up, W18 was given by geometrical justice GJ applied to GR: GR & GJ  W18. To reach W29, matter wave MW, second clock effect SC and twice too many energy levels EL were needed too: W18 & MW & SC & EL  W29. To obtain YM from W29, a non-Abelian structure group NA would have been enough: W29 & NA  YM, or more precisely W18 & MW & SC & NA  YM. Weyl had the greatest creative freedom in 1918, when he applied geometrical justice to GR. The next moves were more constrained. In introducing a matter wave after the discoveries of Schrödinger et al. he had little choice; and it had to be relativistic, hence with spin, which led to the use of tetrads. Weyl’s preference for (10.3)&(10.6) over (10.3)&(10.5) was dictated by Einstein’s objection, the second clock effect. His reaction to the too many energy levels was less constrained, but also less right, less consequential, more idiosyncratic. The adoption of a nonAbelian structure group was mathematically so natural as to be almost inevitable; but ultimately the step was not taken for purely mathematical reasons, and as a physical move it seems more creative, less predictable. Articles are sometimes written to settle priority disputes. Even if I’m aware of none here—my purpose has been entirely different—I’ll end with a few words that may have to do with a kind of ‘priority’: Since the generalization from 1 to N in SU(N) is unremarkable in itself (quite apart from any implications it may have), one possible interpretation of the transition W29 & NA  YM is: it was almost all there in 1929. Of course the mathematical consequences of U(1) → SU(N ) are hardly trivial, and have to be spelled out, which takes doing. But however hard to work out, those consequences are YM.

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I thank Vieri Benci, Julien Bernard, Alexander Blum, Ermenegildo Caccese, Adam Caulton, Radin Dardashti, Jacopo Gandini, Johannes Huisman, Marc Lachièze-Rey, Thierry Levasseur, Jean-Philippe Nicolas, Roger Penrose, Thomas Ryckman, George Sparling and Karim Thébault for valuable conversations and corrections.

References Afriat, A. 2009. How Weyl stumbled across electricity while pursuing mathematical justice. Studies in History and Philosophy of Modern Physics 40: 20–25. ———. 2013. Weyl’s gauge argument. Foundations of Physics 43: 699–705. ———. 2015. Electricity, gravity and matter. In Proceedings of science: FFP14 – Fourteenth international symposium, Frontiers of fundamental physics, Marseilles, 15–8 July 2014. Cao, T. 1997. Conceptual developments of 20th century field theories. Cambridge: Cambridge University Press. Coleman, R. and H. Korté. 2001. Hermann Weyl: Mathematician, physicist, philosopher, pp. 161–388 in Scholz (2001b). de Broglie, L. 1924. Recherches sur la théorie des quanta. Paris: Thèse. Dirac, P.A.M. 1925. The fundamental equations of quantum mechanics. Proceedings of the Royal society A 109: 642–653. ———. 1928. The quantum theory of the electron. Proceedings of the Royal society A 117: 610–624. ———. 1931. Quantised singularities in the electromagnetic field. Proceedings of the Royal society A 133: 60–72. Eddington, A.S. 1987. Space, time & gravitation. Cambridge University Press. Einstein, A. 1916. Grundlage der allgemeinen Relativitätstheorie. Annalen der Physik 49: 769–822. Hawkins, T. 2000. Emergence of the theory of lie groups. Berlin: Springer. Hegel, G. 1816. Wissenschaft der Logik. Nürnberg: Schrag. Kretschmann, E. 1917. Über den physikalischen Sinn der Relativitätspostulate, A. Einsteins neue und seine ursprüngliche Relativitätstheorie. Annalen der Physik 53: 576–614. Levi-Civita, T. 1917. Nozione di parallelismo in una varietà qualunque e conseguente specificazione geometrica della curvatura riemanniana. Rendiconti del Circolo Matematico di Palermo 42: 173–205. Needham, T. 2000. Visual complex analysis. Oxford: Clarendon Press. Pais, A. 1982. ‘Subtle is the Lord ...’: The science and the life of Albert Einstein. Oxford: Oxford University Press. Pauli, W. 1979. Wissenschaftlicher Briefwechsel, Band I: 1919–1929. Berlin: Springer. Penrose, R. 2004. The road to reality: A complete guide to the laws of the universe. London: Jonathan Cape. ———. 2016. Fashion, faith and fantasy in the new physics of the universe. Princeton: Princeton University Press. Penrose, R., and W. Rindler. 1987. Spinors and space-time, volume 1: Two-spinor calculus and relativistic fields. Cambridge: Cambridge University Press. Popper, K. 1934. Logik der Forschung. Berlin: Springer. Ryckman, T. 2003a. Surplus structure from the standpoint of transcendental idealism: The “world geometries” of Weyl and Eddington. Perspectives on Science 11: 76–106. ———. 2003b. The philosophical roots of the gauge principle: Weyl and transcendental phenomenological idealism. In Symmetries in physics: Philosophical reflections, ed. K. Brading and E. Castellani, 61–88. Cambridge University Press (2003).

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———. 2005. The reign of relativity: Philosophy in physics 1915–1925. New York: Oxford University Press. ———. 2009. Hermann Weyl and “first philosophy”: constituting gauge invariance, pp. 279– 298 in Bitbol M., et al. (ed). Constituting objectivity: transcendental perspectives on modern physics, Springer Netherlands. Scholz, E. 1994. Hermann Weyl’s contributions to geometry in the years 1918 to 1923, pp. 203– 230 in Dauben, J., et al. (ed.). The intersection of history and mathematics. Basel: Birkhäuser. ———. 1995. Hermann Weyl’s “Purely infinitesimal geometry”. In Proceedings of the international congress of mathematicians, August 3–11, 1994 Zürich, ed. S.D. Chatterji, 1592–1603. Basel: Birkhäuser. ———. 2001a. Weyls Infinitesimalgeometrie, 1917–1925, pp. 48–104 in Scholz (2001b). ———. ed. 2001b. Hermann Weyl’s Raum-Zeit-Materie and a general introduction to his scientific work. Basel: Birkhäuser. ———. 2004. Hermann Weyl’s analysis of the “problem of space” and the origin of gauge structures. Science in Context 17: 165–197. ———. 2005. Local spinor structures in V. Fock’s and H. Weyl’s work on the Dirac equation (1929), pp. 284–301 in Flament, D. et al. (ed) Géométrie au vingtième siècle, 1930–2000. Paris: Hermann. ———. 2006. Introducing groups into quantum theory. Historia Mathematica 33: 440–490. ———. (2011a) “Mathematische Physik bei Hermann Weyl – zwischen ,Hegelscher Physik“ und ,symbolischer Konstruktion der Wirklichkeit“” pp. 183–212 in K.-H. Schlote and M. Schneider Mathematics meets physics: A contribution to their interaction in the 19th and the first half of the 20th century, Harri Deutsch Verlag, Frankfurt. ———. 2011b. H. Weyl’s and E. Cartan’s proposals for infinitesimal geometry in the early 1920s. Boletim da Sociedada portuguesa de matemàtica, Numero especial A, 225–245. Schrödinger, E. 1926. Quantisierung als Eigenwertproblem (erste Mitteilung). Annalen der Physik 79: 361–376. Seelig, K. 1960. Albert Einstein. Zurich: Europa Verlag. Sigurdsson, S. 2001. Journeys in spacetime, pp. 15–47 in Scholz (2001b). Smirnov, V. 1961. Linear algebra and group theory. New York: McGraw-Hill. Straumann, N. 1987. Zum Ursprung der Eichtheorien bei Hermann Weyl. Physikalische Blätter 43: 414–421. Teller, P. 2000. The gauge argument. Philosophy of Science 67: S466–S481. Vizgin, V. 1984. Unified field theories. Basel: Birkhäuser. Weyl, H. 1918. Gravitation und Elektrizität, pp. 147–159 in Das Relativitätsprinzip, Teubner, Stuttgart, 1990. ———. 1921. Feld und Materie. Annalen der Physik 65: 541–563. ———. 1926. Philosophie der Mathematik und Naturwissenschaft. Munich: Oldenbourg. ———. 1928. Gruppentheorie und Quantenmechanik. Leipzig: Hirzel. ———. 1929a. Gravitation and the electron. Proceedings of the National academy of sciences, USA 15: 323–334. ———. 1929b. Elektron und Gravitation. Zeitschrift für Physik 56: 330–352. ———. 1929c. Gravitation and the electron. The Rice Institute Pamphlet 16: 280–295. ———. 1931a. Geometrie und Physik. Die Naturwissenschaften 19: 49–58. ———. 1931b. Gruppentheorie und Quantenmechanik. 2nd ed. Leipzig: Hirzel. ———. 1939. The classical groups: Their invariants and representations. Princeton: Princeton University Press. ———. 1988. Raum Zeit Materie. Berlin: Springer. ———. 2008. Einführung in die Funktionentheorie. Basel: Birkhäuser. Yang, C.N., and R. Mills. 1954. Conservation of isotopic spin and isotopic gauge invariance. Physical Review 96: 191–195.

Chapter 11

The Plasticine Ball Argument Hermann Weyl, the Homogeneity of Space and Mach’s Principle Julien Bernard

Die einfache Tatsache, daß ich eine Plastelinkugel in meiner Hand zu einer beliebigen Mißgestalt zerdrücken kann, die ganz anders aussieht als eine Kugel, scheint den Riemannschen Standpunkt ad absurdum zu führen. The simple fact that I can squeeze a ball of plasticine with my hands into any irregular shape totally different from a sphere would seem to reduce Riemann’s view to an absurdity. H. Weyl, Space-Time-Matter, first edition, p. 90.1

Keywords Hermann Weyl · Mach’s principle · History of general relativity · Philosophy of space · Gravitational ether

11.1 Introduction 11.1.1 The Plasticine Ball Argument Hermann Weyl’s work is difficult to classify as physics, mathematics, philosophy or history of science. Perhaps because of his wide audience, perhaps also because of his aesthetic preferences, Weyl likes to use analogies and metaphors in order

This article has been translated from French to English by Pascale Pelletier, in collaboration with the author who thanks her for her patient and precise work. 1 As

in the following, we use Bose’s translation, revised if necessary.

J. Bernard () Assistant Professor in Philosophy, Centre Gilles Gaston Granger (CGGG) UMR 7304, Aix-Marseille-University, Marseille, France © Springer Nature Switzerland AG 2019 J. Bernard, C. Lobo (eds.), Weyl and the Problem of Space, Studies in History and Philosophy of Science 49, https://doi.org/10.1007/978-3-030-11527-2_11

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to provide insights about the most difficult and abstract problems of the twentiethcentury science. One of these analogies attracted my attention. This gives its name to what I call the plasticine ball argument (I will abbreviate this as the pba in the following). Weyl uses this argument to think about the relationships between the metric and the material content of space-time. One can find an entire family of Weyl’s texts that develop this argument.2 Weyl wants to legitimate, at the same time, the rise of differential geometry in the domain of physics, the adoption of a dynamical metric and the refusal of flat and fixed spaces (such as Newton’s or Minkowski’s). For these three reasons, he poses the principle of total determination of the metric by matter. This is a radical version of what Einstein was soon to call “Mach’s principle”. The first version of the pba takes the form of an Eleatic aporia. Weyl shows that a too radical principle of determination of the metric by matter could lead to the impossibility of thinking about any kind of motion, or at least any kind of deformation. Slightly rephrasing Weyl, with such a principle, we would no longer understand how it is possible to squeeze a ball of plasticine in order to change its form. Weyl’s argument that leads to this aporia and the way he answers it have some striking formal similarities with Einstein’s famous “hole argument”.3 Nevertheless, the two arguments differ by their functions as well as by the manner in which the cosmic matter is distributed in the respective thought experiments. Einstein considers a hole, that is, a place empty of matter, which is surrounded by a cosmos that is not necessarily empty. On the contrary, Weyl considers a ball of plasticine which is surrounded by a cosmos that is not necessarily full of matter. Since Einstein gave it a name, “Mach’s principle” has never ceased to be the focus of an abundant literature. It is difficult to determine how much this principle is fulfilled in general relativity and to evaluate its contribution to the philosophy of space-time. This difficulty is due not only to the intrinsic mathematical and conceptual complexity of general relativity but also to the usually vague characterization of Mach’s principle itself – at least Mach’s and Einstein’s formulations. Barbour and Pfister enumerated more than 20 meanings to the expression “Mach’s principle”.4 The present article does not tend to review this delicate question5 or to add another meaning again. Rather, I aim at using a precise corpus of Weyl’s texts in order to evaluate his contributions in the debate on Mach’s principle in the first half of the twentieth century and show how his own position is connected to a web of philosophical issues on space. Just like Einstein, Weyl at first totally adhered to

2 See

Sect. 11.1.3. hole argument was conceived by Einstein at the time of the Grossman’s-Einstein’s theory (Einstein and Grossman 1913). Within the rich literature on the hole argument, one can refer in particular to Norton (1999), Iftime and Stachel (2006), Stachel (1993) and Norton (1987). 4 Barbour and Pfister (1995, p. 530). 5 The reader will find a good synthesis on Mach’s principle up until the 1990s in Barbour and Pfister (1995), Reinhardt (1973), Torretti (1983, section 6.2), and Norton (1993, pp. 808–sq.). 3 The

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Mach’s principle and then progressively retreated from that position. His intellectual pathway is not however only a redundant repetition of Einstein’s. Indeed, in Weyl’s specific case, the principle of determination of the metric by matter is considered in the context of a resolutely idealistic philosophy.

11.1.2 The Relativistic Context of the Argument6 The pba appeared in relativistic context. Indeed, during the years 1916–1923, general relativity7 prompted Weyl to develop an important thought on space. An issue was imperative at this moment: how could the foundations of Einstein’s new theory be exhaustively enumerated and precisely characterised? The author of the theory himself met difficulties in dealing with this issue. Through vague and changing characterizations, he spoke about a principle of covariance, a principle of relativity, a principle of equivalence and “Mach’s principle”.8 Already in the first edition of Raum-Zeit-Materie, Weyl gives his own position: the central idea of general relativity cannot be reduced to the mere formal property of generalised covariance,9 but it rather consists in a kind of relationship between matter and the metric, expressing in an indissociable way inertial and gravitational phenomena.10 Afterwards, Weyl remained faithful to this position.11 Therefore, amongst all the different principles expressed by Einstein, Weyl underlined what Einstein called “Mach’s principle” even if Weyl did not use this expression in his first texts.12

6 One

part of this section is identical with a passage of the general introduction of the current volume. 7 Einstein (1916). 8 See Norton (1993, pp. 808–sq.) for a good overview of the foundational debates on general relativity, during the first 80 years of existence of this theory. 9 Amongst the bibliographical references given by Weyl for his chapter IV, we find Kretschmann’s article “Über den physikalischen Sinn der Relativitätspostulate”. Therefore, Weyl had probably been influenced by Kretschmann’s famous argument, according to which the general covariance principle had no physical meaning by itself, since every physical theory can be expressed in a covariant form by a tensorial reinterpretation. See also Norton (1993). 10 Weyl (1918b, p. 181), Weyl (1919a, §26, p. 192), and Weyl (2010, p. 226): A new physical factor appears only when it is assumed that the metrical structure of the world is not given a priori, but that the above quadratic form is related to matter by generally invariant laws. Only this fact justifies us in assigning the name “general theory of relativity” to our reasoning; we are not simply giving it to a theory which has merely borrowed the mathematical form of relativity. 11 For

example, in 1924, see Weyl (1924, p. 197). expression appeared first in Einstein (1918, p. 197). Cf. Barbour and Pfister (1995, p. 10). Weyl does not explicitly refer to Mach within paragraph 12 of Raum-Zeit-Materie. Mach appears however in the bibliographical references of chapter IV, in Weyl (1921, p. 291, bibliographical note 12 The

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Nevertheless, Weyl’s epistemological investigation is not directly characterised as a simple search for the principles of general relativity. This is rather considered as a wide-ranging philosophical investigation on space, only partly directed by Einstein’s theory. Let us develop an analogy. When he conceived his critical philosophy, E. Kant first accepted Euclidean geometry and the bases of Newtonian physics as apodictic sciences; only afterwards did he ask what made these sciences possible. Analogically, Weyl first accepted that Riemann’s mathematical developments13 and Einstein’s general relativity let us enter a new stage of the understanding of the foundations of the notion of space; only afterwards did he develop a conceptual and epistemological theory in order to legitimate these new truths. However neither Kant nor Weyl considered that their respective epistemologies were derived from or were based on the scientific theories they had to account for. This would have been a vicious circle, since the sought-after epistemological justifications are supposed to hold a priori. That is why Weyl, as well as Kant, thought that the respective scientific theories had been simple opportunities to reveal certain a priori epistemological elements. Weyl is peculiarly lucid and subtle when he thinks about the relationships between theory of knowledge, as aiming at a priori claims, and the factual development of positive science.14 Thus Weyl asks: what did Einstein’s theory teach us (or confirm) about the nature of space and the way one can know it scientifically? How can we epistemologically justify that the “correct” notion of space is the one that was finally used by Einstein, after having been announced by Riemann? In order to answer this question, one must deal with two issues that give the global structure of Weyl’s thought on space in the period 1917–1923: 1. The first issue – according to the logical order – consists in justifying that the space-time metric is “of the Pythagorean type”. This means that it has the same properties, in the infinitesimal realm, as the (pseudo-)Euclidean metric. This is the technical meaning of what Weyl calls “the problem of space”.15 2. The second issue consists in justifying the claim that the space-time metric, away from the infinitesimal realm, is a metric the curvature of which is everywhere intrinsically indeterminate. More precisely, the determination of the finite metrical relations is only possible a posteriori, when geometry is articulated with physics. The metric is determined by the manner matter and forces are spatially distributed.

2). Weyl in (1924, p. 198) acknowledged that Mach was the father of the principle of determination of inertia by cosmic matter. 13 In particular the famous text: Riemann (1919). 14 See Weyl’s texts quoted in Michel (2006, p. 198). 15 See the bibliography of Bernard (2015b, p. 198) for the list of Weyl’s works on the problem of space – in its technical meaning – and a historical discussion. Secondary reading on this subject is abundant, see Coleman and Korté (2001, p. 198), Scholz (2004, p. 198), Laugwitz (1958, p. 198), Bernard (2015a, p. 198), Bernard (2018, p. 198), and Weyl (2015, vol. 2).

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11.1.3 Marking Out of the Corpus: Why Is the Argument so Recurring and Polymorphic? This article will address the second of the issues mentioned above. As early as the first edition of Raum-Zeit-Materie, Weyl wants to justify epistemologically the variable and dynamic character of Einstein’s metric and its link with matter. It is in this context that he elaborates his pba.16 It is repeated with notable changes in the third edition of the same work17 and then, with more changes, in the fourth edition.18 The argument disappears from the fifth edition but regularly reappears later in others of Weyl’s texts, in always changing forms: in “Massenträgheit und Kosmos”,19 in both (German and English) editions of Philosophie der Mathematik und Naturwissenschaft20 and in Mind and Nature.21 The pba does not appear in Mathematische Analyse des Raumproblems, but many typical questions can be found there which give rise to the formulation of this argument in the first texts.22 Both in Raum-Zeit-Materie and in Philosophy of Mathematics and Natural Science, the chapter containing the pba is amongst the most modified ones in the different editions.23 In the literature on Weyl, we can find references to or analyses of some of these texts,24 but, to my knowledge, no systematic studies of the entire collection of the occurrences of the pba. Considering all these occurrences, with the pba, we are in front of a thematic which entertained Weyl’s thinking about space during many years and which develops in ever-changing forms in a whole range of texts, as a musical variation of the same theme. Why is this thought experiment so present, and why is it so polymorphic in Weyl’s texts? I already have partly answered this question. Weyl relies on this thought experiment to have an imaginative and conceptual support on which he can base his thought, in order to address the second fundamental epistemological problem enunciated above.25 Weyl first wants to see how adopting a dynamic metric with

16 Weyl

(1918b, p. 90). (1919a, p. 90)]. 18 Weyl (1921, p. 90). 19 Weyl (1924, p. 198). 20 Weyl (1949, 86–87;105). 21 Weyl (1934, p. 129). 22 Weyl (1923, pp. 44–45). We find there the problem of the tension between the homogeneity of space, as a form of appearances, and the heterogeneity of the metric ; and the solution consisting in moving the metric simultaneously with matter. 23 I will specify in this article the most important changes in Raum-Zeit-Materie that I personally noticed. Concerning Philosophy of Mathematics and Natural Science, I have received the information from Carlos Lobo. 24 Namely, Coleman and Korté (2001, pp. 266–267), Coffa (1979, p. 290), Giovanelli (2013a, p. 130, 2013b, p. 290), and Scholz (2019, p. 130). 25 See the end of Sect. 11.1.2. 17 Weyl

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variable curvature can be justified while not abandoning the thesis of space as a homogeneous form of appearances. Consequently the shifts of this argument, from work to work and from edition to edition, partly reflect the evolution of Weyl’s thinking on this key issue. But there is a second reason for this variability of the text. Indeed, to elaborate his thought experiment, Weyl is led to express precisely the way in which the metric properties of space-time are determined or at least correlated with the spatiotemporal distribution of matter. Therefore, even if it was not the determining of the metric by matter which was the problem for Weyl, when he began to elaborate his pba, some technical problems that he met led him to position himself more and more subtly on Mach’s principle. But clarifying this principle is not just a technical problem which would replace the anterior philosophical problem which generated the argument. The critique of Mach’s principle also has a philosophical dimension, which led Weyl to modify his position on the ontology of physics. In the version of Raum, Zeit, Materie of the pba, Weyl supported a form of materialism, inasmuch as all physical phenomena –including gravity which provides its foundation to spatiotemporal geometry – were to be reducible to the relationships between material elements.26 In later versions of the argument, the critique of Mach’s principle led Weyl to become antimaterialistic and argue in favour of a dynamical ether, partly autonomous from fields of matter, thus following an intellectual path close to Einstein’s.27 These two reasons for the variations of the pba provide the two major objectives of the present article. Firstly we are going to use these variations as a means to underscore the decisive stages of the evolution of Weyl’s philosophy of space at that time. Then by studying the technical problems across which Weyl stumbled in the first version of the argument, and by showing how the later versions brought an answer, we will be able to explain which role Weyl played towards clarifying Mach’s principle.28 Then, in the course of the article, we will pursue these two themes of thought simultaneously, going through and commenting on the different versions of the pba, following the order in which they were published. The natural evolution of Weyl’s thought, from text to text, will progressively lead us from the problem of the nonhomogeneity of the metric to the problem of the validity of Mach’s principle and the existence of an ether, the transition being very gradual.

26 The notion of matter which is present in the first four editions of Raum-Zeit-Materie does not form a discrete set of particles but a field. Weyl was then taking up the programme of Gustav Mie which consisted in bringing out the notion of matter from the notion of field. 27 See for example Einstein (1920). 28 We can already find in the literature works that refer to the role of Weyl in the history of Mach’s principle. See in particular Bell and Korté (2016, “Weyl’s Critique of Einstein’s Machian Ideas”), Coleman and Korté (2001, p. 264), and Coffa (1979, p. 290 sq.).

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11.2 Preliminary: The Homogeneity of Space and the Exclusion of the Metric from Space To understand the reasons why Weyl first formulated his pba, we must keep in mind a problem which is at the core of Weyl’s philosophy of space. That is the problematic tension between the homogeneity of space and the nonhomogeneity of the metric. I have dealt with the theme before,29 so I will only mention here the elements that are useful in order to understand the pba.

11.2.1 The Issue of the Nonhomogeneity of the Relativistic Metric30 Geometricians from the beginning of the twentieth century were the inheritors of two crucial developments of the nineteenth century: on the one hand, differential geometry and all the associated analytical tools allowing the study of spaces with freely variable curvature and, on the other hand, the discovery of the founding unifying function of the notion of group towards geometry (Helmholtz, Klein and Lie in particular). But these two legacies are not easily reconciled, inasmuch as Riemannian manifolds and the other related infinitesimal geometries generally have a trivial isometry group. This is why Weyl just as Klein, Poincaré or Cartan considered the tension between the notion of homogeneous space and differential geometry, as a – even for some the – central epistemological question on space raised by the nineteenth century.31 If space is defined by the possibility of defining a group of displacements, must the rich infinite universe of Riemannian manifolds be drastically limited so as to keep only a few homogeneous geometries? Instead cannot the notions of homogeneity and group be transformed in order to become compatible with the perspective of differential geometry? Weyl’s specificity within this group of authors is due to the precise signification that he gives to the homogeneity requisite and his reasons for putting it forwards. In the Erlangen tradition which is not necessarily the one followed by Weyl, the homogeneity of the “spaces” considered is justified from within mathematical practice, by the unified treatment of a vast part of the geometry practices of the nineteenth century it allows (in particular: projective, affine, Euclidean, spherical, Lobatchevskian geometries). In contrast, for Weyl, even though the homogeneity of

29 Bernard (2013, p. 290 sq.), Bernard (2010, p. 290 sq.), and Weyl (2015, vol. 2, 2nd introduction). 30 One

part of this section is identical with a passage of the general introduction of the current volume. There are more details there. 31 See the general introduction of the current volume, Poincaré (1902), Chorlay (2009), Cartan (1925), Scholz (2012) and Scholz (2018).

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space is soon used to legitimate the application of the group theory techniques,32 homogeneity is not originally defined by the notion of group nor justified within a mere mathematical discourse. Instead homogeneity is given as an essential property of space, in the name of a philosophical tradition. For Weyl, because space is ideal, being only a “form of appearances”, it is necessarily homogeneous.33 The homogeneity of space, which follows from its ideality, is defined this way: Space [· · · ] is a form of appearances Form der Erscheinungen. Precisely the same content, identically the same thing, still remaining what it is, can equally well be at some place in space other than that at which it is actually. The new portion of space S  then occupied by it is equal to that portion S which it actually occupied. S and S  are said to be congruent. [· · · ]34

Thus space is defined by Weyl as something the proprieties of which are, by definition, independent from matter (i.e. independent from sensory properties, physical properties and forces induced by matter) which fills it. Space does not yet belong to the domain of physics. It is a form the properties of which can entirely be characterised a priori, precisely because this form and its intrinsic characteristics are given to us prior to any matter which later fills it. In particular, the (topological, projective, affine, conformal, metric) properties intrinsic to space, if any, must be characterisable by a mathematical theory which precedes the study of the forces and the way in which matter occupies space-time. Therefore the phrase “physical space” has no more meaning for Weyl than for Poincaré, for instance. Finally in the context of general relativity, the problem of the tension between the homogeneity of space and the nonhomogeneity of the Riemannian metrics eventually takes the form: how can a non-homogeneous spatiotemporal metric be accepted in physics, when space as such is by nature homogeneous?35 32 For

the shift from the notion of homogeneity to the notion of congruence then to the notion of group of congruences, see Weyl (2010, 5–6;11–15) or Weyl (1923, 44–49). For the use of the theory of groups to found the notion of metric in a context of differential geometry, see Weyl (2010, §18), Eckes (2011, §18) or the texts in relation to the problem of space, in its technical meaning (cf. footnote 15). 33 Weyl (2010, p. 11): Space is a form of appearances Form der Erscheinungen, and, by being so, is necessarily homogeneous. It would appear from this that out of the rich abundance of possible geometries included in Riemann’s conception, only the three special cases mentioned come into consideration from the outset, and that all the others must be rejected without further examination as being of no account: parturiunt montes, nascetur ridiculus mus! Riemann held a different opinion, as is evidenced by the concluding remarks of his essay [· · · ] Only now that Einstein has removed the scales from our eyes by the magic light of his theory of gravitation do we see what these words actually mean. 34 Weyl

(2010, 11, 1919a, p. 10). See also Weyl (2015, p. 1), in which the opposition between form and matter becomes of a more psychological nature, inasmuch as “matter” refers to the sensory content of perception, and it is connected to Kant. 35 In the specific case of general relativity, we have (Weyl 2015, p. 44): According to Einstein, the metric structure of the universe is not homogeneous. How is this possible, given that space and time are forms of appearances?

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However, this question does not only address the theory of general relativity but also any physical theory based on differential geometry, i.e. on a variable curvature metric. This is why Weyl just like Poincaré addresses this problem to Riemann.

11.2.2 Solution: The Metric Is Excluded from the Intrinsic Properties of Space In the first edition of Raum-Zeit-Materie,36 Weyl provides part of solution to the problem expressed in the previous section. In the context of an infinitesimal geometry – in Riemann’s manner – the homogeneity of space must be expressed by the fact that a portion of matter can be moved from a region S of the spatial manifold towards any other region S  while keeping all its properties invariant. Matter must be represented by a field, in which material qualities,37 like mass or electric charge, are distributed: To simplify this examination of the underlying principles we assume that the material content can be described fully by scalar phase quantities skalaren Zustandsgrößen such as mass-density, density of charge, and so forth. We fix our attention on a definite moment of time.38

Thus the simplification offered by Weyl is twofold: (1) reducing matter to a few scalar properties – he then keeps only one of them – and (2) eliminating the time factor. The text that follows actually considers two different distributions of matter (which I will express as ρ before and ρ after ), but each of them is considered as static, at its point of equilibrium; we do not consider the transitional stage. This eventually led Weyl to represent matter at first with a simple scalar function which depends only on position: ρ before :

36 See

f (x1 x2 x3 );

Weyl (1918b, pp. 88–90) and the corresponding parts in the three editions of Raum-ZeitMaterie that follow. 37 In Raum-Zeit-Materie, Weyl does not refer to material qualities. He only refers to “the material” das Materiale. However in other texts of the same time, such as Weyl (1923, introduction), Weyl calls the material content “qualitative” and describes the homogeneity of space by the fact of being able to move these qualities towards any point. 38 Weyl (1918b, p. 88). The fact that Weyl choses as an example the density of electric charge Elektrizitätsdichte is significant. Perhaps he has already in mind his own theory (to be published in 1918, see Weyl (1918a)) in which the metric field is the carrier of the gravitational and electromagnetic interactions, simultaneously. So, if something like a principle of metric determination by matter is to be considered in such a conceptual framework, it cannot take the form “mass determines the metric” any longer but, instead, “mass and electric charge determine the metric”.

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the coordinates x1 x2 x3 vary so that they take all the values corresponding to the region S of space where matter initially is. The change of position to which we want to subject matter is expressed in our system of coordinates by a transformation: xi = φi (x1 x2 x3 ). Thus the region S  towards which we move matter is represented, always in the same coordinate system, by the set of xi (corresponding to xi of S). The movement of matter is technically expressed by the fact that the field ρ is pulled forward39 on space by the transformation φ. Weyl writes this in coordinates: ρ after :

f (x1 x2 x3 ).

Here we must understand that the letter f represents the same mathematical function as above. In other terms, the distribution of matter after the displacement must be expressed (still in the same coordinate system) by the function (f ◦ φ). By this process, we have moved matter, keeping all its intrinsic qualities unchanged. But, Weyl goes on, in order to assert that space is homogeneous, the metric properties of the material body that was moved must have been kept. These metric properties before the moving were given by a metric field defined on S 40 :

39 The “pull-forward” (“pull-back”) terminology is posterior to Weyl. See Iftime and Stachel (2006,

p. 1243). Besides Weyl does not mention here the fact that a region S cannot be moved towards any region S  . Instead, as shown by the process used, one must take a region S  diffeomorphic to the first one. For instance a simply connected region cannot be transformed into an annular region. Weyl is explicit about it further on in the text (Weyl 2010, p. 98). 40 Weyl insists on the fact that, in order to determine the visual shape of a portion of matter, one must not only know the metric coefficients for the portion S of space-time where the matter is but also for all the space-time points through which the light rays which, emitted from S, will reach the observer. The body of the latter is represented by a point-eye set on a point outside S. The necessity to take into account the metric on the intermediate trajectory is clear as soon as we think of phenomena such as light rays deflection by gravity or, in an anachronistic way, the gravitational lenses phenomena. Weyl will come back to the necessity to take into account the intermediate metric field to differentiate the rotation of the stellar compass from the rotation of the stars themselves in Weyl (1924, p. 198, left hand column). For Weyl, having the visual observer intervene in order to define the shape of a material object is an important step, in view of his attachment to the Husserlian phenomenology during the years which we are considering; Weyl, here, uses significantly the term “experiences of consciousness” Bewußtseinserlebnissen (Weyl 1918b, p. 89). Concerning this point-eye idea or Ego-centre IchZentrum, and its phenomenological function, see in particular analyses in Ryckman (2009, p. 286), Mancosu and Ryckman (2005, p. 89), and Bernard (2013, 241–sq.). See also Kerszberg’s article in the present volume. However concerning the specific issue with which we are dealing, we do no need to discuss it further. The “pulling forward” of the metric in the space-time region which separates the observer from the element of matter that is observed is technically expressed in the same way as the pulling forward of the region S itself. The point-eye representing the observer is also pulled forward.

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before gik (x1 x2 x3 )dxi dxk ,

(i,k=1)

Since our space is supposed to be Riemannian, there is no reason to think that the before (x  x  x  ) of the metric at the point S  are initially the same as in S. values gik 1 2 3 This is why, Weyl concludes, if the metric was fixed a priori once and for all, then space homogeneity could not be preserved, since we would have: after    before    before gik (x1 x2 x3 ) = gik (x1 x2 x3 ) = gik (x1 x2 x3 ).

Any material body moved in space would generally be metrically deformed. However, Weyl continues, space homogeneity can be preserved if we say that metric is dynamic and determined by matter. Indeed according to this supposition, after having moved matter, the metric will change so as to conform with it. Once the equilibrium between matter and the metric has been re-established, the body shall have recovered its metric properties. So we will have: after    before gik (x1 x2 x3 ) = gik (x1 x2 x3 )

before    ( = gik (x1 x2 x3 )).

To justify this equality, Weyl plays with the twofold active/passive interpretation of φ, as per a process also at work in Einstein’s hole argument.41 Once matter has been moved (transformation φ actively interpreted as a pulling forward of ρ on the manifold), Weyl changes the coordinates. The point P  previously had the coordinates: (x1 x2 x3 ). It will now have the new coordinates: (x1 x2 x3 ), that P had in the first coordinate system before the displacement. Thus Weyl now uses φ −1 as a passive transformation. The intrinsic properties of matter, after the displacement, and in the new coordinate system will be expressed again by the function f = f ◦φ ◦φ −1 . So, if the metric functions gik are perfectly determined by the function f which represents matter, the conclusion shall be that metric will be moved exactly in the same way as matter. More precisely, it will have taken exactly the same values, in the new coordinate system, as it had before matter was moved, in the first coordinate system. Thus the displaced body has kept its metric properties, and the space homogeneity is preserved! Weyl’s argument can be transcribed in a more modern mathematical language which avoids the coordinate systems just as Stachel and Iftime did for Einstein’s hole argument.42 Such a rewriting may hide some of the problems met by Weyl and Einstein, but it can also clarify some aspects of the problem. To outline the problem briefly, it is supposed that matter is represented by a function ρ before : M → R which associates its density to any point of the manifold M. A moving in

41 See

Norton (1987, pp. 164–165, 1993, pp. 801–sq., 1999, Appendix: “Active and Passive Covariance”). 42 Iftime and Stachel (2006, Appendix: “Active and Passive Covariance”).

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the manifold is simply a diffeomorphism φ : M → M. Moving matter by means of φ amounts to producing a new distribution ρ af ter obtained by “pulling forward” the preceding distribution. So we have: ρ after = φ ∗ ρ before = ρ before ◦ φ. Let us suppose, in Weyl’s manner, a law of total determination of the metric by matter. If the metric g before is associated with the distribution of matter ρ before , and if our law is generally covariant, then, to the distribution of matter φ ∗ ρ before , we must necessarily associate the metric g after = φ ∗ g before ; in which the pulling forward φ ∗ g of a metric by a diffeomorphism is defined by: x , y%); φ ∗ g(φ ∗ x%, φ ∗ y%) = g(%   and the pulling forward of a vector x% at P x% ∈ TP (M) is in turn defined by φ ∗ x% = Dφ|P (% x ) (it is a vector of Tφ(P ) (M)). Thus, in this rewriting, the metric invariance that Weyl is aiming for is directly encoded in the fact that the law of the determination of the metric by matter is generally covariant. Weyl in view of the technical solution that we have reported concludes: [· · · ] Space in itself is nothing more than a three-dimensional manifold devoid of all form; it acquires a definite form only through the advent of the material content filling it and determining its metric relations.[· · · ] the metrical groundform will alter in the course of time just as the disposition of matter in the world changes. [· · · ] We shall illustrate in greater detail [· · · ] that any two portions of space which can be transformed into one another by a continuous deformation, must be recognised as being congruent in the sense we have adopted, and that the same material content can fill one portion of space just as well as the other.45

The beginning of the text shows that the solution suggested by Weyl consists in excluding the metric from the intrinsic properties of space. It is rather part of the content of space, in the same manner as matter.46 So when he refers to Riemann’sEinstein’s dynamic metric, Weyl is careful not to call it “space” Raum. At least, space, when it is endowed with Einstein’s metric, has already ceased to be space in itself, space with only its intrinsic properties, but is already space as being informed by matter. Therefore, its onto-epistemological status has changed: These metric relations are not the outcome of space being a form of phenomena Form der Erscheinungen, but of the physical behaviour of measuring rods and light rays as determined by the gravitational field.47

43 Added

in the fourth edition. this issue, see also the analogy of flexible sheet metal in Weyl (2015, p. 44). 45 Weyl (1918b, p. 88, 1921, pp. 87–88). 46 A question will remain whether matter could even totally emerge from matter itself. It is the question raised by Mach’s principle below. 47 Weyl (1918b, p. 91, 1919a, p. 91, 2010, p. 102). 44 On

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The context of this quotation clearly shows that Weyl, here, does not aim at making the notion of space as a form of appearances obsolete but at taking out the metric determinations. Thus, space(-time) the homogeneity of which Weyl can keep on ascertaining, and which can still continue to act as “form of appearances”, is eventually reduced to the naked spatiotemporal manifold, i.e. deprived of any metric. That the spatiotemporal manifold, with regard to its only topological and differential properties, is homogeneous is of course correct from a mathematical perspective.48 However, this solution, in spite of often appearing in the literature of that time and in spite of the fact that it effectively captures an important aspect of general relativity, is insufficient, from both mathematical and epistemological perspectives. We will not develop here what is the nature of these difficulties.49 Let us just mention here that Weyl will qualify this thesis when he further develops his philosophy of space and his “infinitesimal geometry” or “contact geometry” Nahegeometrie. He will then specify that the infinitesimal metric properties are indeed part of the essence of space and can be a priori characterised, in contrast with the fortuitous variations of metric relations in a finite space-time region which alone has the status of an a posteriori determined physical field. The second a priori notion of space which we have just characterised, which is metric and infinitesimal, does not replace the global and topological notion which we have characterised above (the naked manifold). Instead, both notions are at work in the foundational discourse of general relativity or of any physical theory based on an infinitesimal geometry. In Weyl’s work, the concomitance of these two a priori notions of space is expressed as follows. Weyl retains the idea that the naked spatiotemporal manifold plays the role of a (globally) homogeneous “space”, used as an individuation principle, but he adds the idea that the infinitesimal metric structures, identical everywhere, are part of the “essence of space” Wesen des Raumes. By integrating both ideas, the space of general relativity will appear not as formless but as multiform, its changing form being illustrated by Weyl with the image of a snail shell which is built at the same time as the matter that fills it and adapts to it.50 It is that type of image that must be kept in mind as a basis for thinking about the pba. Even though an infinitesimal structure can still be characterised a priori, as being part of the essence of space, the exact form of the metric in a finite region remains intrinsically indeterminate, waiting to be completely in-formed (shaped) by matter.

n–dimensional manifold that is (arcwise) connected and of class Cp for p = 0, 1, · · · , ∞ is not only homogeneous but even maximally isotropic. Given any two points P , P  of M, and  given {v1 , · · · , vn−1 }, {v1 , · · · , vn−1 } two families of linearly independent vectors, respectively, taken in TP (M) and in TP  (M). Then there is a diffeomorphism of class Cp which sends P to P  , sends the infinitesimal straight line < v1 > to < v1 >, sends the infinitesimal plane < v1 , v2 > to < v1 , v2 >, · · · and finally sends the infinitesimal hyperplane < v1 , · · · , vn−1 > to  < v1 , · · · , vn−1 >. 49 On that question, see Bernard (2013, Chap. III, 3.). 50 Weyl (2015, p. 44). 48 Any

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11.3 The pba in Raum-Zeit-Materie 11.3.1 The pba as an Eleatic Aporia We are now technically armed to enter the pba. We shall start from the first text in the series, from a chronological perspective, namely, the first edition of RaumZeit-Materie. The other versions shall then be considered in contrast with the first one. In our Sect. 11.2.2, we have shown how Weyl, to preserve the homogeneity of space, was led to presuppose that “matter determines the metric”. We can identify the idea later known as one of the many forms of “Mach’s principle”.51 However it is introduced by Weyl in his own manner and without any reference to Mach. The totally Machian idea that inertia is determined by the masses of the cosmos is not clarified. The principle of equivalence and the generalised principle of relativity which will be closely related to Mach’s principle in Einstein’s thought are absent too.52 Weyl is more directly interested in the link between the metric and matter. Weyl chooses Riemann instead of Einstein as a symbolic figure of the idea of a metric determined by matter. This attribution is justified by a small passage of Riemann’s habilitation text which is enough, according to Weyl, to make him a prophet of general relativity.53 Whatever the relevance of this attribution to Riemann, in any case it is significant. It shows that the conceptual framework to which Weyl belongs is indeed broad. The point is not to look for the bases of an individual physical theory (Einstein’s) but to work on a philosophical issue which more generally addresses all physical theories of fields based on an infinitesimal dynamic metric. To meet both the current conventions and the specificity of Weyl’s point of view, let us attribute the idea to the triplet Riemann-Mach-Einstein (RME): RME Principle : the values of the metric relations are perfectly determined by the distribution of matter and of its intrinsic qualities ρ (charge, mass· · · )

The strategy of §1254 of Raum-Zeit-Materie to preserve the homogeneity of space is problematic because it is too simplistic. In addition to the philosophical problems evoked at the end of the previous section, concerning the foundations of the infinitesimal metric structures, Weyl stumbles on another difficulty linked to his exaggeratedly strong interpretation of this principle. It will lead him to an

51 See

our Footnote 12. point partly follows from the structure of Raum-Zeit-Materie. The text which we have explained is taken from §12, therefore from chapter II, while general relativity is mentioned only in chapter IV. See what we have said p. 297 about Weyl’s standpoint towards the generalised covariance principle. 53 See our comments in Bernard (2018, p. 3). 54 I refer to the paragraphs numbering in the fourth edition. 52 This

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aporia. While the intrinsic properties of matter are automatically kept in the course of time and are expressed by scalar fields ρ, and if metric is perfectly determined by matter in the sense described above, then it is the very possibility of some sort of transformation of matter which seems to be hampered, so that Weyl is led to state: Riemann’s point of view seems to be reduced to the absurd by the simple fact that I can shape a plasticine ball in my hands, and give it any irregular shape, totally different from the initial spherical shape.55

The plasticine ball represents any portion of matter, and the kneading represents any physical force able to move elements of matter in relation to others, in order to produce a deformation. Thus, what has become difficult to consider here is the very possibility that portions of matter may modify their reciprocal distances in the course of time. That is why I think that the difficulty on which Weyl stumbles can be reformulated as an aporia of the Zeno’s paradoxes type. It is the very ability to understand the possibility for any change which seems compromised in such a framework. Reformulation of the pba as an aporia of the Eleatic type Given a physical theory based on the following assumptions: 1. Space is a Riemannian manifold, the metrical relations of which are not a priori fixed. 2. Matter which occupies this space is intrinsically characterised by one (or several) scalar field(s) ρ defined on the manifold. 3. Matter completely determines the metric. It means that a system of coordinates being fixed, if ρ(x) is determined for any x, then the gμν (x) must also be determined for any x. In other words, there are n2 functions Fμν such as gμν = Fμν (ρ), where we must understand that gμν (x) is not necessarily only dependent of ρ(x) but of the data of the entire field ρ. Then, the value of the metric field associated with any point cannot evolve in time, a point of the manifold being identified by the element of matter that fills it. All the distances between the elements of matter are therefore invariant. Thus, as in Zeno’s paradoxes, we come to the conclusion that any change in the universe is impossible. To paraphrase Weyl: we can no longer even understand how it is possible to knead a plasticine ball to change its shape. Thus, we have achieved a physical theory in which any motion and therefore, doubtless, any change can no more be conceived. It is not exactly a logical contradiction but close to it. For what is a physical theory if not a theory of change? In this regard Aristotle’s reaction to the Eleatic arguments which aimed at showing the impossibility for any motion is famous. For the Stagirite, a philosophy that 55 Weyl

(1918b, p. 90, 1919a, p. 90, 1921, p. 90).

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negates what our senses teach us with the strongest evidence – i.e. the existence of motion – does not even deserve to be called “physics”.56 So faced with such an argument, an argumentative refutation seems pointless. Like Diogenes the Cynic, we can dismiss such a standpoint with the simple gesture which shows in an immediate intuition the possibility of motion, for instance, walking.57 It is somehow Weyl’s starting point. He shows the absurdity to which his own standpoint was leading with the simple gesture consisting in kneading a plasticine ball. Luckily Weyl does not stop there but tries to identify where exactly the error lies which led to this unsustainable situation. A paradox meaning a manifestly wrong proposition but deduced from a plausible argument can only be useful for knowledge if it is analysed so that the flaw can be isolated and deconstructed. Showing that it is false is not enough, as Aristotle himself eventually admitted.58 How, starting from Weyl’s text, have we elaborated our aporia? In order to interpret the assumption 2. above, we had to make two relevant choices concerning the text: (2 bis) we have established that the function ρ operating in the hypothesis (2) was a scalar in the formal sense, i.e. a variable represented by a number that is independent from the location of this element in space, and from the choice of the coordinate system. (2 ter) We have assumed (this is not explicit in the text) that the functions ρ were constants in time. In spite of the formal analogies between Weyl’s pba and Einstein’s hole argument, we can notice that they follow clearly different intellectual paths. For Einstein the problem was to reach a physical theory in which the metric coefficients are perfectly determined by matter. The problem met by Einstein consisted in the fact that, whatever the equation of type: Gμν = T μν chosen as the fundamental law,59 a total determining of the metric coefficients in a coordinate system where the factors T μν are known is impossible. Einstein only shows it in the case of the existence of regions absolutely empty of matter (holes). In these empty regions, the factors T μν are absolutely cancelled out and therefore do no more vary during the application of a diffeomorphism on the manifold, while gμν continues to covary according to its tensorial nature. In Weyl’s text, the problem is not reaching a theory in which metric is totally determined by matter. On the contrary this is an accepted assumption, posited to

56 Aristotle, 57 Diogenes

Physics, book I, Chap. 2. Laercius, Lives, Doctrines and Sentences of Famous Philosophers, VI, Chap. 2

[Diogenes]. 58 After the passage evoked previously, Aristotle eventually admitted that even if the Eleatic opinion

of the immobility and unity of the world is obviously false, dedicating efforts to refute Parmenides and Melissus may be physically instructive. It is of course also the case for Zeno. 59 T μν is the energy-momentum tensor and Gμν a tensor only dependant of the metric field and its derivatives, which would still need to be determined. See Norton (1987, pp. 162–sq.).

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try to solve Weyl’s own issue (cf. Sect. 11.2). It only becomes an issue because of its unexpected consequences, leading to negating the very possibility of any movement.

11.3.1.1

Covariance Problem in the Formulation of the pba

Before trying to find a solution to this aporia, let us notice that the way in which the problem is set down is open to doubt, because it seems incoherent at the level of the properties of covariance. Let us show that this nevertheless does not invalidate the problem set down by Weyl. Weyl supposes that a specific scalar field ρ, representing matter, would totally determine the metric field gμν . But this hypothesis seems absurd, since a scalar field and a metric field (i.e. a field of tensors with two covariant indices) do not have the same covariance properties. Starting from two such fields, we can find, at least in some cases, a change of coordinates (in modern language a diffeomorphism) which leaves the field ρ invariant, while modifying the values of gμν . By contrast, the Einstein equations Gμν = T μν seem to be free from this covariance problem, both sides of those equations being of the same tensorial nature. Therefore, with Einstein, one needed to consider the very specific case of a “hole”, in order to find an application which modifies g without modifying T . In general relativity, the distribution of matter is represented by the energymomentum tensor T μν . Yet, in very simple situations, this tensor is reduced to one or two scalars. Indeed if we consider the approximation of a perfect fluid60 , then, in a system of coordinates that is comoving with the fluid, the tensor T μν is reduced to two scalars: ρ the density of matter at rest and p the hydrostatic pressure: ⎡ T μν :

ρ ⎢0 ⎢ ⎣0 0

0 p 0 0

0 0 p 0

⎤ 0 0⎥ ⎥ 0⎦ p

if we choose the signature + − −− for the metric and take c = 1. This tensor is absolutely invariant for a purely spatial transformation of the coordinates, i.e. as long as we remain in a system of co-mobile coordinates. In this simplified case, the tensor T μν is indeed eventually reduced to two scalars. But, the fully covariant definition remains: T μν = (ρ + p) uα uβ − p.g αβ , uα being the fluid’s four-velocity

60 Weyl

uses himself this approximation in Weyl (2010, p. 205).

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This equation is coherent at the covariance level but requires to show explicitly the metric.61 Weyl specifies that generally the most natural form for the energy62 which in that case has the momentum tensor is the mixed tensorial density Tμ ν value: Tμ ν =

!  −det(g) (ρ + p) uα uβ − p.δαβ

and that includes the metric, even in the absence of pressure. Schwarzschild’s pioneering works63 use that type of simple matter characterisation. The simplification made by Weyl, when he says that matter is reduced to a few scalars, is therefore not at all incongruous. We can actually use Schwarzschild’s solution to interpret the pba. Let us consider a general-relativistic space-time (M, gμν , Tμν ), which fulfills the Einstein equations without the cosmological term. Matter is supposed to be concentrated in a region S, the remaining of space being empty. This matter is admittedly an incompressible perfect fluid at rest which, in an adequate system of coordinates (co-mobile with the fluid), admits a spherical symmetry. Finally, we suppose that, in that same system of coordinates, the metric admits also a spherical symmetry and tends towards the Minkowski flat metric at infinity. Then, Schwarzschild’s metric (interior and exterior) is required.64 The form of our “sphere” of matter is therefore well determined. Let us now suppose that the elements of matter have been moved so that matter, after equilibrium is reestablished, can still be characterised as a perfect fluid at rest with the same mass and the same uniform density. Finally, let us suppose that the spherical symmetry is resumed. According to general relativity, the metric must resume the form imposed by Schwarzschild in the system of coordinates adapted to the new location of the “ball” at rest. It is a simple and precise interpretation of Weyl’s thought experiment, which can also be used as a very simple model for the moving of a “spherical” body with a uniform density in a flat cosmic environment.

11.3.2 Is the Temporal Variability of Matter Properties Sufficient to Get Out of the Aporia? Weyl’s strategy to get out of the aporia in Raum-Zeit-Materie consists in abandoning (2 ter). He keeps the idea that matter keeps intrinsic pre-metric properties ρ, which in turn completely determine metric (hypothesis 3). But these properties do not need to be constant. Here are the reasons why a plasticine can however be deformed:

61 Weyl

(2010, p. 205; 262–263). (2010, 229). 63 Schwarzschild (1916a,b, p. 205; 262–263). 64 Here, we leave aside the issues arising at the boundary, when joining the two solutions. 62 Weyl

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So that the form that was squeezed may seem spherical to an observer from any perspective, we should need, among other things, a deformation of the internal atomic structure of the plasticine different from the one I can actually produce with my hand.65

Thus the type of physical change that we can induce on a plasticine ball, by kneading it, would be of another nature than a simple pulling forward of its intrinsic properties on the spatial manifold. In kneading the plasticine ball, we fundamentally change its intrinsic properties, which in turn allows the modification of its metric properties. So the variability of the magnitudes ρ allows restoring the possibility of motion. Weyl’s assertion teaches us that the properties ρ in his general formulation of the RME principle are not physical magnitudes which would be fundamental constants of matter, absolutely invariant like the charge or mass density of the electron, for instance. Rather, they are magnitudes capable to take different contingent values in the course of time for the same element of matter; as is the case for energy-momentum density which appears in the Einstein equations. However, to get out of this aporia, we have been compelled to adopt a physical theory of a very specific type, in which no motion is possible without modifying the intrinsic properties (the ρ) of matter. Something like “pure motion” has become impossible. We can, nevertheless still imagine that the form of the ball can be modified without the intrinsic qualities ρ of its matter being affected. It is indeed possible if we modify the properties ρ of matter outside the S region corresponding to the ball. let us remember that the expression of the RME principle is non-local. The problem then takes a cosmological turn. This is why in the 4th edition of RaumZeit-Materie, Weyl adds a phrase to the precedent sentence: [to restore the spherical form, we would need to consider] a deformation of the internal atomic structure of the plasticine, or a rearrangement of all the cosmos masses

(Here I emphasise the added part of the sentence). I will however add that the use of cosmology, introduced by Weyl in the 4th edition, is dubious. Indeed let us suppose that we kneaded our plasticine without changing its internal properties ρ. If its form has changed, Weyl’s text suggests, it is because we have modified the properties of matter outside the region filled with the ball. However, we could modify the field of matter only in the immediate environment of the ball. The need to place ourselves at a cosmological level does not seem very relevant. Besides, if the immediate environment of the ball is empty or nearly empty, it is represented by ρ = 0, and we cannot really see how the change of shape of the ball could be ascribed to it. Finally, last difficulty, let us recall that the pba appeared in the context of a philosophical challenge specific to Weyl: saving the homogeneity of space understood as the possibility of moving a material content without modifying its nature. When the material content that is moved is finite, such a move can mean: keeping the properties of the ball, including the metric properties, while changing its metric relations with the other bodies of the cosmos. This has a

65 Weyl

(1918b, p. 90).

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clear physical sense. But if the matter that is to be moved is the cosmic matter as a whole and that all the cosmic metric relations are kept by such a “motion”, it then seems that motion can only have a purely ideal meaning.66 To summarise, Weyl only gets out of his aporia by allowing a variability of the magnitudes ρ either inside or in the cosmic environment of the plasticine ball. However, the text leaves in abeyance several fundamental problems, which will come up again in later texts. We are going to discuss the two main ones.

11.3.3 Why Is Weyl so Focused on Finding a Physical Interpretation for Diffeomorphisms? A reader familiar with modern literature on the covariance principle and Einstein’s hole argument may look back on Weyl’s text suspiciously. Let us recall that, indeed, Einstein, after elaborating his hole argument, had been temporarily led to reject all the generally covariant formulations of gravitation. It is generally accepted that Einstein had made a conceptual error while elaborating his argument: he had wrongly thought that a system of coordinates had, per se, a physical meaning. In fact it is per se only a mathematical artefact, as long as it is not linked with physical entities (matter and metric fields). So, two ordered pairs (gμν (x), T μν (x)) and (gμν (x  ), T μν (x  )) representing a field of matter and a metric field, related by a simple active diffeomorphism (using a modern language), would in fact only be two mathematical representations of the same physical situation.67 A reader who is aware of these developments may be surprised at the apparent naivety with which Weyl tries at all costs to give a physical significance to the operation of pulling forward the matter and the metric field on the manifold. Why does he not conclude, with a spirit close to Einstein’s, that the spatiotemporal manifold has lost all objectivity and, therefore, the operation consisting in pulling forward simultaneously the metric and matter on the manifold has no physical significance but is only the expression of a mathematical latitude in representing the same physical situation? Bluntly accepting it would lead to a form of ideality of spatial (or spatiotemporal) manifold. It would not be a physical reality but a mere mathematical artefact used to label spatial points. Only matter and metric would have a physical reality, but not the manifold. That form of ideality had been considered by certain forms of neo-Kantism of the time.68 Why does Weyl, in spite of his idealism, stop before 66 Giving

some sense to such a moving of matter would require setting the problem in a really dynamic framework, without only considering the initial state and the final equilibrium state. In Massenträgheit und Kosmos, Weyl will be able to give sense to such a global movement with his Boats-Lake Analogy. See further Sect. 11.4.7. 67 Iftime and Stachel (2006, p. 1243), Norton (1987, pp. 170–171;177, 1993, pp. 804–805, 1999, pp. 804–805), and Stachel (1989, pp. 170–171;177). 68 In particular, Cassirer, in the line of the Marbourgh school, criticises Kant’s philosophy of space, in so far as it is too strongly connected to perception data, leading to an exclusive focus

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reaching that position? This question can only give rise to speculation. He was perhaps looking for a form of idealism closer to the original Kantian form, which makes space a “form of our intuition” and not a mere analytical mathematical artefact. It seems that Weyl construes the idea of space as a form of intuition involving the possibility of a “real” motion (not only a mathematical transformation) taking a body from one point to another “real” point, without modifying its properties. It is clear that, when Weyl speaks of moving elements of matter from a region S towards a region S  , he has in mind much more than a mere transformation of the mathematical representation of a physical situation otherwise left unchanged. In fact, in order to give form to his thought experiment, Weyl takes as his model the idea of an electrically charged body in equilibrium with an electrical field; moreover, he considers that the moving of matter that is considered generates a temporary physical perturbation. We are tempted to ask Weyl: how, in a physical theory based on a dynamical geometry, could we physically identify, in the course of time, a point of the manifold? We can a priori see only two solutions: • Either we consider a simplified framework in which metric is static and the system of coordinates is chosen relatively to that metric. This solution can indeed allow for physically identifying points, for instance, to give sense to the moving of a test particle in a static field. But we must assume that the displacement of the body (the test particle) does not perturb the fundamental metric. Such a solution is therefore inapplicable in the pba case. • Or we attach the coordinates to elements of matter taken as physical markers of the position. It is therefore a system of coordinates which is co-mobile towards a specific material background. In that case, moving the plasticine ball means: leading its constitutive points to coinciding with new elements from the “material background”. In that case we only fall into the pba aporia if we suppose that the material background has a negligible influence on the determination of the metric, compared with the ball that has been moved. It is in this cosmological fault of the pba aporia that Weyl rushes when he adds the sentence element quoted above.

11.3.4 On the Impossibility of a Primitive Separation of Matter and Metric Another difficulty emerges as early as the first version of the pba: the impossibility to radically differentiate, in the formulation of the RME principle, matter from the metric. on Euclidean geometry. See Cassirer (1910, chapitre III, particularly p 106) and Cassirer (1923, chap. V). According to Cassirer, the a priori notion of space which must be incorporated to science needs rather to be based on the driving forces of mathematical analysis and numerical symbols.

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Several features of Weyl’s text seem to confirm that Weyl – just as Einstein in his first formulation of Mach’s principle – was initially driven by the idea of a total emergence of the metric from matter (ontological anteriority of matter). This provides reasons for researching a manner to characterise matter totally independently from any metric consideration, the metric being supposed to then emerge from this originally nonmetrical matter. This is suggested by the fact that Weyl formulates the RME principle in a very radical manner, according to which matter is to completely determine metric. This idea is still reinforced when Weyl concludes, at the end of the passage on the pba, that space-time, prior to any considering of matter and forces, is absolutely deprived of any form. However it is on this formless space that the field of matter is originally defined. Moreover by making the qualities of the matter, ρ, mere scalars, Weyl seems to be trying to suppress any dependence of matter towards a prior metric structure. Yet, this supposed total anteriority of matter on the metric is at the core of the problems met by Weyl in his aporia. We have seen that, even assuming that the quantities ρ characterising matter are variable, we can only get out of the aporia by adopting a specific type of physical theory in which a “pure” motion, leaving matter properties untouched, has become impossible. Besides that, we spontaneously wonder what kind of variability of the properties ρ is concerned. We are led to think that the different possible values of the ρs for the same elements of matter refer to different manners, for matter, to distribute spatially. This is clear if we have in mind the nature of the energy-momentum tensor in general relativity, which is a specification of the general idea of matter ρ which is at work in the Weyl’s RME principle. It is also clear in Weyl’s text, since he refers to (matter or charge) density as the prototype of what must be understood by the functions ρ describing matter. However the idea of density clearly has a metric meaning and not only a topological one. If we did not presuppose a metric, it seems that we could not provide meaning to the simple “rest-mass density” scalar. The properties ρ seem to be modes of matter spatialisation, presupposing a metric. This questions the very possibility to distinguish, within matter properties, between pre-metric qualities, supposed to be primitive, and metric properties, supposed to be derived. It seems therefore that we are forced to abandon the ontological anteriority of matter upon metric in order to get out of the pba aporia. Weyl does not reach that conclusion in Raum-Zeit-Materie. In the first edition he informs the reader, in a footnote, that things will get clearer as regards the pba in chapter IV.69 However, Weyl does not explicitly come back to it, and, moreover, his footnote is deleted from the third edition onwards. It is only in Massenträgheit und Kosmos that Weyl will develop all the consequences of his aporia.

69 Weyl

(1918b, p. 88): “Genaueres hierüber in Kap. IV.”.

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11.4 The pba in Massenträgheit und Kosmos 11.4.1 A New Issue In the Massenträgheit und Kosmos article published in March 1924, we find numerous elements from Raum-Zeit-Materie which we have discussed. These elements are numerous enough for us to consider that it is indeed the same argument, the “plasticine ball” one, which comes back in an altered form. These common elements are the metaphor of the plasticine ball itself,70 the issue of the relations between the metric and matter, and many technical notions and considerations. However the issue which motivates the argument is now different. The idea of space as the form of appearances, the homogeneity71 requisite that goes with it and the tension triggered by this homogeneity towards adopting variable curvature metrics do not appear anymore. Instead, the pba is motivated by a questioning on the legitimacy of the principle of determination of inertia and metric by matter. What was only part of the argumentation in Raum-Zeit-Materie, vaguely expressed, has now become the very core of the questioning. The problematic is now closer to Einstein’s focus on the legitimacy of “Mach’s principle”, the reference to Mach being newly introduced by Weyl p. 197. We will begin this section by discussing the dialogue form of this text and the function of the characters. Then we shall proceed with analysing the pba in the first part of the dialogue. Finally we will address the cosmological aspect of the debate which corresponds to the second part of the dialogue.

11.4.2 The Dialogue Form of Massenträgheit und Kosmos: Who Are Paulus and Petrus? In the dialogue, both characters, Paul and Peter , meet to resume a discussion that was interrupted in 1915, on the foundations of general relativity. We are led to think that Paul impersonates Weyl as the one who leads the dialogue and takes it to its conclusion. Moreover, the intellectual stages through which Paul tells us he went remind us of the ones Weyl actually experienced. Paul introduces himself as somebody who initially strongly believed in Mach’s principle, in its most radical form, similar to the RME principle in §12 of Raum-Zeit-Materie, before retracting. Now, he no longer believes in the validity of Mach’s principle, and the dialogue unfolds as Paul explains to Peter the reasons for his change of mind. Paul says (1924, p. 198): Plastelinmasse. notion of homogeneity appears punctually in Massenträgheit und Kosmos; however it is not the homogeneity as an a priori property required from space “per se”, but, instead, the (local or global) homogeneity of some configurations of matter, considered as particular cases, or the metric homogeneity of some specific solutions to the Einstein equations, particularly the de Sitter’s one. Homogeneity has become the exception rather than the a priori rule. 70 Weyl 71 The

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that if the belief in Mach’s principle forms the “stone base on which the relativity Church lies”,72 then he has become an apostate, a heretic. He has changed from Saul to Paul.73 It is difficult to precisely follow the manner in which Weyl wants to use the episode of Paul’s conversion to illustrate, apparently, his own intellectual journey. The limit of the image used by Weyl comes from the fact that the intellectual journey of Paul in Massenträgheit und Kosmos is in fact a double conversion then reconversion movement. By detaching himself from the old “religion”, Newton’s and the belief in absolute space, Paul has temporarily joined the new Einstein’s-Mach’s Church. Then, he has discovered the error of this point of view and renounced Mach’s principle, thus becoming an apostate, this time from the perspective of the new religion. As we will develop below, this reconversion is not however a mere return to the former belief (Newton’s). Paul will believe again that space cannot be reduced to matter, that it does not simply emerge from it. Nevertheless, this space with an autonomous existence will no longer be an absolute space with fixed properties, as for Newton, but a dynamical “ether”, interacting with matter, without any ontological hierarchy between them. Paul does not consider this reconversion as just leaving the general relativity “Church” but, instead, as a deviation from the orthodox interpretation of the theory – this is 1923 – based on Mach’s principle. From this point of view, the word “heresy” is rather well-suited. We can note that Einstein followed an intellectual path very similar to Weyl’s, at first subscribing without any restriction to Mach’s principle, before retracting.74 So he could also very well be the person represented by Paul in the dialogue.75 72 Weyl

(1924, p. 197). There is an implicit reference to the famous biblical sentence “And I tell you that you are Peter (Céphas = Rock), and on this rock I will build my church”. 73 Saul of Tarsus was the Jewish name of the man later known as the Apostle Paul or Saint Paul in the New Testament. It is said that he was initially a Pharisee, violent towards Christians, before converting and joining Jesus Christ. He changed his name from Saul to Paul to mark this conversion. There is a German phrase “change from Saul to Paul” sich vom Saulus zu Paulus wandeln, used to describe a radical change of personality or behaviour. 74 About Einstein’s abandonment of Mach’s principle, see letter of 02.02.1954 to F. Pirani, the extract of which is reported and translated into English in Renn (2007, p. 61). In Norton (1987, pp. 180–sq.), it explains that this abandonment by Einstein of Mach’s principle takes the form of a shift from an overt antirealism towards space (then identified with the naked manifold) to a realism towards space (then identified with the metric field, called “ether”). Other references about Einstein’s position concerning Mach’s principle in: Barbour and Pfister (1995, P. 10; 67–90), Norton (1993, pp. 808–sq.) and Torretti (1983, section 6.2). Before abandoning Mach’s principle, Einstein gave it very variable forms. According to Norton and Torretti (1983, p. 201), Einstein’s change of mind about Mach’s principle began in the years 1918–1919. 75 The question whether Weyl had real persons in mind behind his characters is minor. What is important is to underscore the fact that Paul’s intellectual evolution is close to Einstein’s and Weyl’s. Paul and Peter, in the dialogue, say that they first met in the United States in 1915. Einstein and Weyl met as early as 1913 at the E.T.H. of Zurich. Weyl arrived at the institute when Einstein was there, working with his friend Grossman at elaborating general relativity. In the dialogue, Paul tells Peter that the latter should well know the axial symmetry solutions of the theory of general

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11.4.3 Typology of the Principles of Determination of the Metric by Matter, Rejection of Mach’s Principle Weyl and his relativist contemporaries – starting with Einstein – had met many difficulties in applying Mach’s principle in general relativity. This is highlighted by the technical difficulties met by Weyl while developing his first versions of the pba. Weyl seems to have become gradually aware that the encountered technical problems were not contingent – only due to an oversimplified conceptual framework – but on the contrary fell within the range of the actual difficulties inherent to Mach’s principle itself. This leads Weyl, in Massenträgheit und Kosmos, to refine his thought about the adequate expression and the relevance of the principle which links matter, inertia and metric. The problem is all the more important since Weyl remains convinced that such a principle (and not the covariance one) must be at the core of our understanding of general relativity.76 While §12 of Raum-Zeit-Materie only contained a radical and vaguely expressed version of Mach’s principle (“RME principle” above), the dialogue from 1924 includes a series of more or less important variations of Mach’s principle and of related principles. The function of this plurality is to successively isolate the difficulties that make Mach’s principle inapplicable in general relativity or more widely in any field theory that adopts a dynamic metric. Weyl starts from the following principle which he classically attributes to Mach: (M) the inertia of a body comes to existence Zustande kommt due to the interactions of all the universe masses.

Weyl specifies that Mach’s principle is a particular case of an absolutely general principle, which he calls “causality principle”: (C) all the [physical] events are causally univocally determined by matter, that is by charge, mass and the state of motion of matter constitutive elements.

Considering the way Weyl uses this principle, the name is rather ill-chosen. For it is not a question of opposing causality to causeless phenomena. Instead the debate is about knowing whether we can relate the ultimate causes of physical phenomena to pure relations between elements of matter or whether we are led to adopt an immaterial physical entity like Newton’s absolute space or ether such as it appears at the end of the nineteenth century, deprived of any material consistency. The principle (C) opts for the first alternative and should therefore be called a materialism. So, Weyl turns Mach’s principle into a restricted version of a principle which concerns the ontology of physics (the word “ontology” is however not used by Weyl). The point is to postulate that all physical events – starting with inertia – would come to existence Zustande Kommen from masses and their interaction. relativity, since he raised the problem of their existence. Weyl (with Lense and Thirring) is amongst the first scientists who published such solutions (see Weyl 2010, §32; and bibliography note 22 of chapter IV; Weyl 1919b). 76 Weyl (1924, p. 187, right hand column). See our Footnote 10 above.

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It is therefore a reductionist principle which, if it turned to be true, would lead to a materialistic ontology for physics, in which matter (and its “interactions”) would be the only primitive entity. The formulation (C) develops the idea, subjacent to (M), of an ontological reduction, in the terms of a univocal causal determination. Paul will then develop some arguments which lead to abandon (M) and a fortiori (C). At first we note that the principle (M), contrary to the one at work in Raum-ZeitMaterie, does not directly refer to the metric notion but only to the notions of inertia and mass. In this it is close to Mach’s original thought. However, Weyl demonstrates that, here, the metric is an inevitable element of thought. In fact, let us suppose that we attribute to cosmic masses the causal origin of an inertial phenomenon, for instance, the flattening of the Earth at the poles. The simple presence (in the sense of the determination of the positions) of cosmic masses could not be sufficient to be used as the causal origin to the phenomenon. We would have to say that it is the motion of the Earth relatively to the big cosmic masses which is the cause of its flattening.77 Yes, but, Weyl goes on, general relativity teaches us that: (A) [Independently from the metric field], the concept of the relative motion of several bodies separate from one another is as untenable as the absolute motion of a single one.

To understand this radical affirmation, we must clarify the meaning of the word “separate” getrennter. Separate bodies are bodies that are located in topologically disjoint regions. What can allow us to assert that two such distant bodies are in motion relative to each other? Observation will not suffice. While we commonly say that we can see fixed stars turn around us, in reality what we see turning is the stellar “compass” Sternenkompaß, that is, the beam of all the light rays which reach our eyes from the stars. But we cannot make any inference from the rotation of the stellar compass (relatively to us) to the rotation of the stars themselves without making hypotheses relative to a metric field which occupies the intermediate region between the stars and us and which determines light trajectories as geodesic. Mach did not consider the necessity to take the metric into account, because in his time everybody believed in the static nature of the cosmos and in the rigid Euclidean body which could ideally extend all over the cosmos. Thus, Mach’s criticism of Newton’s absolute space mainly targeted its ontological independence towards matter, but the Euclideanity of the metric was hardly questioned.78 With general relativity, remarks Weyl, we become aware of the contingency of the hypothesis of the indefinitely extended rigid Euclidean body. Just as inertial phenomena, metric phenomena are physical and can vary contingently. Thus, to give sense to (M), we need to break free from any metric hypothesis in defining matter. Just as in §12 of Raum-Zeit-Materie, we are led to characterise matter 77 Thus,

since at the time one consensually believed in the static nature of the cosmos, Newton’s bucket experiment was explained by Mach by the fact that the bucket is in motion (rotating) relatively to the frame of reference defined by the static whole of cosmic masses. 78 However, there is no consensus on Mach’s real purpose. See Barbour and Pfister (1995, 9–65;90–sq.).

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only through its intrinsic features (charge, mass) and its distribution (in a purely topological sense) within space-time. It is in that context that we must understand Weyl’s assertion (A). The notion of motion – even relative motion – between two separate bodies loses all its meaning if we overlook the metric as medium. That is what the fibred plasticine ball argument (I will abbreviate this as the fpba in the following) will show. Before going into this argument in detail, let us remark that its conclusion is strikingly akin to Weyl’s discourse in the introduction of Mathematische Analyse des Raumproblems. Indeed, Weyl has just shown to us that one cannot inquire into the relationships between matter and inertia without involving a third intermediate element: the metric. Similarly, in the Mathematische Analyse des Raumproblems, Weyl had exposed the impossibility to correctly conceive the problem of space only on the basis of the “space/content (=matter)” duality favoured by Kant. Instead, one must consider the triplet “space/matter/metric”.79 So, the metric must be introduced as a third irreducible element, between matter and space, in both the infinitesimal sphere which is addressed in Mathematische Analyse des Raumproblems and the finite sphere which is addressed in the texts about Mach’s principle.

11.4.4 The Fibred Plasticine Ball Argument The fibred plasticine ball argument (the fpba) is a variant of the pba which, to our knowledge, appears in Massenträgheit und Kosmos: [On can] think about the four-dimensional universe as a plasticine mass penetrated by fibres, world-lines of particles of matter, which cannot converge into a single one, but which can otherwise spread arbitrarily [· · · ]

In Raum-Zeit-Materie, the pba concerned space rather than space-time. The idea of motion was there reduced to a transformation from an initial static equilibrium state into a final state of the same type. In contrast, in the present text, the fibred plasticine represents a four-dimensional Lorentzian manifold endowed with a still undetermined metric. The fibres in the plasticine represent the world-lines of the material points, which are one-dimensional submanifolds of space-time.80 At the end of the text, Weyl specifies in a literary therefore imprecise way that this family of lines must constitute what is nowadays called a foliation (at least local and generally not unique) of space-time of the type R × R3 (with one-dimensional sheets). Weyl’s formulation is not precise enough to indicate whether he has in mind only a local foliation (which exists in all Lorentzian manifolds) or, in a more restrictive manner, a global foliation. In that case, the hypothesis would not be trivially verified in general relativity but would require an additional cosmological hypothesis, similar to what

79 Weyl

(2015, p. 1). a metric is attributed to space-time, we shall expect the world-lines to be time-like.

80 When

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is nowadays known as “Weyl’s principle”.81 In other words, it should be assumed that space-time can accept a globally defined (not necessarily unequivocal) temporal orientation. Local or global, Weyl’s hypothesis is in any case sufficient for him to develop his argument: [One can] continuously deform the plasticine so that, not only one fibre, but all the fibres become straight and vertical. If the vertical axis represents the axis of times, it is construed as follows: each body remains in its place in space.

The technical background is quite clear. Let us take any system of coordinates compatible with our foliation. It will transform each line in our family into a “vertical line” of R4 , that is, a set of the form: {(t, a, b, c)|t ∈ R} in which (a, b, c) ∈ R3 is a fixed triplet, the coordinate t representing time, the three others representing space. Thus, by choosing coordinates co-mobile with matter particles, we have simultaneously “put at rest” all matter, this notion of rest being understood only in a topological (pre-metric) sense. So, the fpba shows us that, as long as we do not provide ourselves with a metric, the very difference between rest and motion is only illusory since a motion defined only in a topological way, on the spatiotemporal manifold, can be destroyed by a mere change of coordinates. This justifies the affirmation (A) above. We can instantiate this idea on the concrete example of a matter reduced to a perfect fluid without pressure (“dust”), represented by the twofold covariant tensor T μν = ρuμ uν . In this particular case,82 the tensor T μν has no metric feature, since it is definable as a simple tensor on the naked manifold. It shows therefore that differentiating between matter at rest and matter in motion is impossible only on the basis of this tensor. Whatever the initial value of this tensor, one can ensure that the four-velocity field is identically (1, 0, 0, 0) (matter “at rest”) by shifting to co-mobile coordinates. Then, the value of the tensor T μν is everywhere ⎡ T μν :

81 Kerszberg

ρ ⎢0 ⎢ ⎣0 0

⎤ 000 0 0 0⎥ ⎥. 0 0 0⎦ 000

(1986, p. 1). case contrasts with the tensorial density Tμν which always depends on the metric (or at least its determinant) and with tensors of more complex matter (with pressure, etc.) which we shall consider later.

82 This

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The fpba shows us that Mach’s principle, in some radical form, is not only contradicted by experience, it is even absurd; moreover, in contrast with RaumZeit-Materie, Weyl explicitly says so.83 The fibred plasticine argument forbids us to differentiate motion and rest, when we originally refuse to provide ourselves with the metric, in conformity with the full interpretation of (M). We can therefore attribute to matter only static properties such as charge or mass.84 The belief that the metric could be univocally derived from such static properties is what Weyl (Paul) qualifies as absurd. This qualification confirms backwards our interpretation of the §12 of Raum-Zeit-Materie as developing an aporia of the Eleatic type.85 In summary, the fpba shows that, at a simple topological level, motion cannot be differentiated from rest. But Mach’s principle, in its strongest versions, requires that the metric be entirely determined by the field of matter, the latter being described at a purely topological level, as the simple data of world-lines of matter and/or scalar fields. Such a principle can only lead to immobility (this time, immobility in its full sense, that is metrical). It is an absurd requirement for a physical theory. It seems to us that the too radical forms of Mach’s principle which are denounced here encompass in particular the RME principle in §12 of Raum-Zeit-Materie. It is doubtless partly the numerous difficulties met by Weyl with the first versions of the pba which led him to reach this conclusion.

11.4.5 The Existence of the Inertial-Gravitational Ether Weyl (Paul) then unfolds the philosophical consequences of the radical impossibility to give sense to Mach’s principle. A metric must be originally given, at the same ontological level as matter itself. It is on this metric field that the inertial motion shall be based (which is identified to gravitational or perhaps gravitationalelectromagnetic motion86 ). Only then shall we be able to physically differentiate rest and movement. To designate this field, Weyl either uses Einstein’s word “ether” or speaks of a “guide field” Führungsfeld. Since without ether motion cannot have any significance, Weyl challenges the materialistic interpretation (in the sense of the principle (C) above) of general relativity, which de-substantiates space and alleges that space can emerge from pure relations between material elements: 83 Weyl (1924, p. 198, 2nd column): Da dies offenbar ist absurd. See also earlier in the text (Weyl

1924, p. 197, right hand column), in which Weyl (Paul) says that he understands a priori that the principle (M) is inapplicable. Let us insist here on “a priori”. 84 See the formulation of the principle (C) above, in which the properties considered in order to characterise matter were motion, charge and mass. If, according to the fpba, motion disappeared, there would only be mass and charge left, construed as simple scalars, as in §12 of Raum-ZeitMaterie. 85 Section 11.3.1. 86 The reader is referred to the literature on the evolution of Weyl’s belief in his theory of unification of gravitation with electromagnetism (1918). See Afriat (2009, p. 197, right hand column).

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The theory of relativity, well construed, does not attempt to eradicate absolute motion in favour of relative movement, but it destroys the concept of kinematic motion and replaces it with dynamic motion.87

General relativity has thus not replaced absolute motion – i.e. relative to space – with a motion only relative to matter. Instead, it has kept absolute motion but has fundamentally changed its nature. We have shifted from kinematic motion (Newton) to dynamic motion (Einstein).88 In fact, the notion of space on which this absolute motion is based, ether, has no longer the same status as Newton’s absolute space had. It does not have a rigid structure fixed a priori from all eternity. Instead, the ether has dynamic metric properties interacting with the field of matter. In this dynamic coupling of matter and ether, none of the two partners can entirely be deducted from the other one. Matter and ether are co-original. Finally, we must accept to abandon Mach’s principle (M),89 and adopt a weaker principle for the relationships between matter and the metric, which subsumes Einstein’s equivalence principle: (G) the guide (ether) is a field of physical state (like the electromagnetic field) which interacts with matter. Gravitation belongs to the guide and not the force; and it is only in that way that we have an in-depth understanding of the equality between the inertial mass and the weight mass.90

11.4.6 Cosmological Consequences of the pba 11.4.6.1

Preliminary to the Discussion: The Einstein Equations and the Cauchy Problem

In the second part of Massenträgheit und Kosmos, the protagonists of the dialogue give a cosmological dimension to the problem posed,91 coming still closer to the problem as it was formulated by Mach and Einstein. In fact, even though he does not see any fault in the a priori argument which demonstrates the absurdity of Mach’s principle, Peter cannot be convinced because: [it seems that] Einstein has already done what you refute [, realise Mach’s principle], in the work in which he has generalized his original gravity laws, by [introducing the] “cosmological term”. In view of this fact, any proof of its impossibility is therefore invalid.

The fact that general relativity is on the right track to express Mach’s principle will be supported further down in the text, by Thirring’s work. The latter demonstrated that, according to general relativity, a massive and hollow sphere, rotating

87 Weyl

(1924, p. 199, right hand column). (1924, p. 199). 89 See our Footnote 74 about the similar rejection found in Einstein’s thought. 90 Weyl (1924, p. 199, left hand column). 91 This late arrival of cosmology in the debate had already been noticed in Raum-Zeit-Materie. See our Footnote 43. 88 Weyl

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about itself, exerts, on a mass inside it, an effect comparable to the centrifugal force.92 Thus, general relativity seems to confirm Mach’s answer to Newton’s bucket experiment.93 Paul then accepts to re-examine Mach’s principle not on the basis of an a priori thought but in criticising more precisely the way in which Einstein and his immediate successors tried to realise it, within general relativity, and more specifically in the 1917 cosmological article. The idea of a determination of the metric – therefore of inertia – by masses is expressed in general relativity by the Einstein equations. They are partial differential equations which pose the proportionality between Einstein’s tensor   1 Gμν = R μν − g μν .R +.g μν , 2 which has a metric significance, and the energy-momentum tensor T μν which represents matter. If knowing T μν is enough to univocally determine the metric, then, according to Einstein, we could be allowed to assert that Mach’s principle is indeed theoretically verified.94 These are second-order equations in regard with the metric, Einstein’s tensor expressing a form of spatiotemporal curvature of the metric. More specifically Gμν is a determined function of the metric and its first and second derivatives. Therefore, an infinite number of non-isometric g μν may correspond to a single Einstein tensor Gμν . In particular, there are an infinite number of “Einsteinian metrics” which are the solutions of the Einstein vacuum equations Gμν = 0. Because of the nature of these equations, we can approach the issue of the determination of the metric by matter in the form of a “Cauchy problem”. Typically we start from a space-like hypersurface, on which, besides the matter Tμν , we set as initial conditions the metric and its first derivatives.95 The Einstein equations being of the second order we then may expect to develop one single solution (up to an isometry) for the couple (gμν , Tμν ) at least on a neighbourhood of the initial hypersurface.96 Thus construed as providing a solution to the Cauchy problem, the Einstein equations do not directly express a radical genesis of the metric but only

92 Weyl

(1924, pp. 199–200).

93 It is a classic in relativistic literature. See Barbour and Pfister (1995, “bucket experiment” p. 531). 94 It

is a way to express Mach’s principle which is most often used by Einstein. See Einstein (1918, pp. 241–242) and Barbour and Pfister (1995, 67–sq.). We will discuss later the apparent circularity of the process, the metric seeming necessary in order to interpret the tensor Tμν . 95 Typically, along the lines of Choquet-Bruhat and Geroch, the first derivatives are not given but, instead, a second-order tensor giving the external curvature of the initial hypersurface within the manifold which is to be generated. 96 The difficult problem of the existence and unicity (up to a diffeomorphism) of the solutions to the Einstein equations, with several types of initial conditions and regularity hypotheses, has been subject since the 1950s to major progress, thanks to Choquet’s work. See Choquet-Bruhat (1952, 1969, 67–sq.). She has shown the existence and unicity (up to a diffeomorphism) of a local solution to the Einstein equations, within the neighbourhood of a space-like hypersurface on which the Cauchy boundary conditions were given. The existence and unicity results are valid for

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indicate how a metric initially given must evolve, in view of the (metric) distribution of matter on the initial hypersurface. Rather than expecting a metric to entirely emerge from a purely a non-metric field of matter, we simply expect the (metric) distribution of matter at a specific time to determine the metric (and the distribution of matter) for any ulterior time. The works discussed by Weyl in Massenträgheit und Kosmos, whether they are Einstein’s, Thirring’s or Schwarzschild’s, do not pose the Cauchy problem in general relativity in all its generality but are limited to the static case. It means that they assume a distribution of matter Tμν and a metric gμν that are invariant in relation to the time coordinate. In this simplified framework, the Einstein equations correlate a form of spatial curvature of the metric to the properties of matter “like its density ρ” and possibly its pressure p) as do the Poisson equations for classic static gravitation. Let us suppose that we know the values of these properties of matter and we are trying to determine the metric on a domain D of the spatial manifold. The Einstein equations, as well as Poisson’s, enable us to univocally determine a solution only if we give ourselves the “boundary conditions”, i.e. the values of gμν on the spatial boundary of the domain ∂D. Taking an infinite domain does not make a difference since we shall always need to know the values at spatial infinity to univocally solve the equations. The Einstein equations, in this case, lead us to a Dirichlet problem instead of a Cauchy problem. 11.4.6.2

Return to the Idea That Tμν Makes Sense Only in an Already Metrical Context

Einstein tried to solve his equations without having to impose any boundary conditions for the metric. Having such a goal, was he under the range of Weyl’s a priori argument which doomed some formulations of Mach’s principle to being absurd? Einstein asserts that, if his project was realised, then the metric would be entirely determined by the tensor Tμν . It seems that we must suppose the tensor Tμν

the Einstein equations without sources but also with sources like perfect fluids or electromagnetic fields. See Choquet-Bruhat and Geroch (1969, p. 331). To obtain global results, we must add hypotheses such as the global hyperbolic character of the manifold. See Choquet-Bruhat and Geroch (1969, p. 331). The problem becomes more complicated, sometimes with no solution or no unicity, if the correct regularity conditions are not posed, if the initial conditions are ill-defined or if hypotheses similar to global hyperbolicity are not available (which has everything to do with Weyl’s principle in cosmology). In the 1920s, we do not know whether results, at least partial, similar to the ones obtained by Choquet were already available. The most ancient reference given by Choquet is Darmois (1927, p. 331). Einstein’s and Weyl’s convictions on the possibility to correctly pose the Cauchy problem for the Einstein equations could be based on the similarity of these equations with the Laplace and Poisson equations, for which the results of existence and unicity were well known, and on some successful attempts to the univocal determining of a metric in a few specific cases (in the first place those considered by Schwarzschild). For a presentation by Weyl of the theorem of existence and unicity of the solution to a system of partial differential equations, see Weyl (2015, Appendix 3, 2nd part).

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to be deprived of metric properties, before the solving of the field equations. Can this idea be given any sense? It is clear that the tensor Tμν , once the equations have been solved and a metric has been determined, takes a clearly metric signification, since it encompasses in particular the metrical notions of density and momentum. In contrast, before the correlation with the metric, the tensor Tμν admits no definite interpretation. It may be one of the reasons why Weyl prefers mixed tensorial density Tμ ν , which explicitly involves the determinant of the metric. It is also why, in the Cauchy problem, the tensor Tμν is immediately correlated to the metric on the initial hypersurface. This leads Ehlers97 to assert that Tμν can in no case correctly describe the state of the field of matter, until it has been correlated to a metric. However, following Einstein, we can try to see whether the values of the Tμν may be determined before a determined metric has been given. In the simplest cases, as in the hypothesis of a matter reduced to a perfect fluid without pressure, we saw that the tensor Tμν could be defined independently from the metric.98 It is doubtless these simple cases that Einstein had in mind when he formulated Mach’s principle by the request for a determining of the metric field by the tensor Tμν . Nevertheless, in the most general cases (when the matter-energy comprises a pressure factor, or an electromagnetic field, etc.), the metric appears on both sides of the Einstein equations. It is actively involved in the general form given to the tensor Tμν . In that case, the metric seems both determined (by the integration of Gμν ) √ μ −det (g)Tν ). and determining (as an ingredient to give sense to Tμν or to Tμ ν = Therefore in the most general Cauchy problem, we cannot firstly calculate the tensor Tμν (outside the initial hypersurface) and then determine gμν . Instead both fields are simultaneously co-calculated, except in a few cases which were studied later.99

97 In

Barbour and Pfister (1995, p. 93), Ehlers notes that the energy-momentum tensor, until it is coupled with a metric, does not properly describe the field of matter, and to this day, no physical theory can describe the field of matter before a metric is given. Thus he agrees that Mach’s principle, if it stipulated that “matter in itself [i.e. prior to any metric consideration] determines the metric” would be neither true nor false but even pure nonsense. He mentions that Einstein eventually admitted it in his letter to Pirani of 02.02.1954: “the Tμν which must represent “matter”, always presupposes gμν ”, quoted from Einstein’s letter to Pirani of 02.02.1954 in Torretti (1983, p. 202). Einstein then suggests to avoid from now on speaking of Mach’s principle in regard to general relativity. 98 See p. 322 above. 99 Stachel (1969) showed that, when we limit ourselves to a field of matter with restricted properties, then we can find dynamic variables describing sources, independently from any metric data. In these particular cases, the tensor T μν has the properties of a simple tensorial field that can be defined on the naked manifold. Stachel illustrates it with three cases: • a scalar field without a mass, • an incoherent matter ([dust]), • an incoherent radiation. We can then calculate the Tμν outside the initial Cauchy surface (by solving the conservation equations), before solving the Einstein equations to have gμν . Imposing from the start, the conservation equations enable then to obtain the conditions to the integration of the Einstein

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So the initial data of the Cauchy problem for general relativity usually include information on the metric and its first derivatives. The Tμν alone is insufficient.100 We can illustrate it with a simple case. In the numerical space of coordinates, we note S the set x12 + x22 + x32 = 1. We then consider the following initial distribution of matter: μ

T μν (x1 x2 x3 ) = ρδ0 δ0ν if (x1 , x2 , x3 ) ∈ S, T μν (x1 x2 x3 ) = 0 otherwise, (δ is the Kronecker symbol, ρ a fixed positive constant). This represents a cosmos with a differentiated region, where lies a perfect fluid without pressure with a constant density (a cloud of dust), the rest being void. Even so, can we say that our tensor has univocally determined a distribution of matter? Depending on the metric that is correlated to that Tμν , the matter that is described may be a small sphere which can attain a stable state, or, if it is given a large enough radius, a black hole,101 or again a body without a spherical symmetry, if the metric adopted does not itself have this symmetry. The circularity exposed does not however immediately invalidates Einstein’s idea. Here we only reproach him with the unfortunate slogan “the Tμν determines the gμν ” which can work to describe field equations only in the elementary cases where the Tμν does not explicitly contain the metric. Even here, we do not know all the properties of matter from the beginning, but only some global properties,

equations, whatever the metric ultimately retained. In that sense, we can calculate the dynamic of the sources before knowing the space-time geometry. In these particular cases, the metric only appears on one side of the Einstein equations, contrary to the general case. However, this does not invalidate the fact that a given field Tμν , even of one of these very simple types, shall take totally different physical significations according to the specific metric to which it is correlated. Moreover, Stachel shows that this early calculation of Tμν on the whole manifold does not generate any extra restriction on gμν , which still fully depends on the initial conditions that can be freely chosen. 100 Afriat and Caccese in Afriat (2010, pp. 16–17) argue that we can sometimes attain a notion of matter without using any metric. After having considered various types of metric tensor, they conclude: Generally, then, the reliance of matter on the metric seems to depend on the kind of matter; in particular on how rich, structured and complicated it is. The simplest matter –absent matter– can do without the metric; the more frills it acquires, the more it will need the metric. Of course, these affirmations do not raise any problem if we replace everywhere the word “matter” by “tensor Tμν ”. However what is precisely debatable is the possibility that the tensor Tμν alone, before being coupled with a metric, represents a well-determined state of matter. So, for example, even if the tensor Tμν = ρ.uμ uν (“dust”) does not depend on gμν , it will represent a very different state of the matter, depending on the metric to which it is correlated on a considered hypersurface. 101 Let us remember that the Schwarzchild radius of a massive body is proportional to its mass, while the geometrical radius of the object grows much √ more slowly based on the mass (if geometry was Euclidean, this radius would of course grow as 3 m). Therefore, the initial density being fixed, a ball of matter will become a black hole as soon as its radius is large enough. The hypothesis of a constant density then loses its coherence.

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independent of the precise determination of the metric, such as the scalar fields ρ and p.102 So, from the perspective of differentiating between what is a starting data, and what is deduced, the usual separation of the Einstein equations in two terms, Gμν and Tμν , is misleading. Both terms of the Einstein equations need dismantling. The starting data –i.e. before solving the field equations – are actually limited to some general hypotheses on the nature of the matter involved and to the data of the scalar fields (ρ, p · · · ) which describe some of its properties, independently of the precise metrical distribution. The gμν (and its derivatives), which is involved in both the term Gμν and the term T μν , is then only obtained by integrating the equations. This is confirmed by the study of the texts of Einstein’s contemporaries.103 Finally, in general, the only necessary metric data, to solve the Einstein equations, are the initial or boundary conditions. Einstein is therefore right to be concerned with it.

11.4.6.3

Does Mach’s Principle Require to Eliminate Boundary Conditions?

To solve the Einstein equations as a Cauchy problem, we cannot do in general without the metric data on the Cauchy hypersurface representing the initial time. Let us call it “the initial metric”. It correlates to the tensor Tμν in order to give it a full sense. Is it sufficient to invalidate the fact that these equations can realise Mach’s principle? Several specialists of Mach’s principle of the second half of the twentieth

102 This

is in particular the case when we solve the Einstein equations in the peculiar case of a stationary solution, with specific symmetries, as in the calculus made by Schwarzschild for his “interior metric”. In this kind of simple situation, the application of “Mach’s principle” takes a form that is reminiscent of Weyl’s formulation of the RME principle in Raum-Zeit-Materie, that is: g = F (ρ · · · ). 103 Of course, for any relativistic calculation of the metric on a void region, the problem does not arise since the energy-momentum tensor is simply null, therefore, independent per se from any metric. This particularly includes the approximate derivation, by Einstein (1915), of the metric which surrounds a point mass, the exact solution suggested by Schwarzschild (1916a), or again the metric interior to Thirring’s hollow sphere (Thirring 1918, 1921) or the metric near a LenseThirring rotating massive body (Lense and Thirring 1918). However in the case of the calculation by Schwarzschild of the metric interior to a spheric mass of perfect fluid, incompressible and at rest (Schwarzschild 1916b, pp. 16–17), the problem arises since Tμν should appear under the general form: T μν = (ρ + p) uα uβ − p.g αβ which explicitly depends on the metric. In Schwarzschild’s text, it is however clear that the T μν is not a starting data of the problem. In a significant manner, Schwarzschild starts with the mixed tensor T11 = T22 = T33 = −p and T44 = ρ0 (ρ0 is a constant, since the fluid is incompressible, p depends on the radial coordinate as per a function which will be determined by the stability hypothesis). The presence of symmetry hypotheses indeed enables Schwarzschild to specify the general form of the metric, before calculating. But it is clear that the metric (therefore the Tμν ) is only perfectly determined after the field equations have been solved.

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century, of whom J.W. Wheeler104 is an illustrious representative, argued that the existence and unicity of a solution to the Einstein equations, for given Cauchy data, is enough to express Mach’s principle, in spite of the inevitably metric status of these initial data. It is perhaps possible to support the fact that the initial metric does not represent an autonomous entity, in the manner of Newton’s absolute space, but would be reducible to the set of all the metric relations in concreto between the elements which constitute the field of matter. In any case, this position was not Einstein’s. He thought that no one could legitimately pretend that Mach’s principle is actually realised in relativistic cosmology, as long as we need to use initial conditions or boundary conditions on the metric, to solve the equations. Einstein’s programme, in the years 1917–1918, consists in looking for all possible subterfuges to avoid boundary conditions. Weyl shows in the last part of Massenträgheit und Kosmos that this programme is in principle unrealisable. All cosmological subterfuges imagined by Einstein and others105 to get rid of the boundary conditions are doomed to failure. These conditions, whether they concern the spatial infinite or the past (infinitely distant or not), are inevitable.106 Weyl ends his argumentation by specifying that a choice has to be made between all the possible universes without matter, which are the solutions to the Einstein equations with Tμν = 0. This choice is a real physical hypothesis, since this solution

104 See

in particular Barbour and Pfister (1995, pp. 188–sq.). Mach’s principle construed as a as a rule imposed on the boundary conditions to select some cosmological models 333, see Reinhardt (1973, pp. 531–534) and Barbour and Pfister (1995, pp. 39;77;79–83;95;97;148;190–195;228;238–239;443). About the idea of realising Mach’s ideas within the limits of initial relational data between elements of matter, in a context wider than general relativity, see Barbour and Pfister (1995, 107;111–112;204–207;218–222;443–444). 106 In Weyl (1924, p. 201), Weyl remarks that the hypothesis of a static universe, like in Einstein (1917, p. 201), is equivalent to determining the state of the metric in the past: 105 About

The difficulty that arises from the spatial horizon is evidently resolved by [the choice of a] closed space ; but it remains nevertheless, since it is located everywhere in the universe continuum which can deform [· · · ] in the same manner as a mollusc. The restriction to static conditions is indeed opaque and debatable. Weyl then develops an analogy with electromagnetism and asks how Coulomb’s equations, in the static case, derive from Maxwell’s equations. Then he concludes: The formation of this field F inevitably results from the variable electromagnetic field laws, if we add the hypothesis that space was deprived of a field at the beginning of the sequence. If so, it is not because the field is fixed on the infinitely distant spatial horizon, but, instead the link comes from the world boundary of the past which goes back to an infinitely distant [time]. This argument is also developed in Weyl (1949, §23 C). Besides the hypothesis of a static nature, on which Weyl insists, it seems to me that the homogeneity and symmetry hypotheses on which Einstein and cosmologists usually rely in their derivations have also a metric significance.

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will represent the ether in its “normal state”, when it is not disturbed by any matter. Concerning this choice, Weyl prefers de Sitter’s hypothesis107 for: • The spectral galaxy redshift suggests an expansion of the universe.108 • De Sitter’s universe has good properties (infinitely distant past and future are topologically disjoint) in order to prevent any time loop.109 • This universe solves the dark night sky paradox and avoids repeating the images of galaxies, eon after eon, as we are compelled to do in Einstein’s model.110

11.4.7 Weyl’s Boats-Lake Analogy Let us conclude our discussion of Massenträgheit und Kosmos with a precise study of the passage in which Weyl develops the following analogy: matter is to the ether what boats are to the surface of a lake. This analogy is important for us inasmuch as it shows an evolution of Weyl’s position on some issues of the pba of Raum-ZeitMaterie: Your objection being based on the principle of continuity I can doubtless weaken it in the best manner, intuitively, with an analogy in which I compare the ether with the surface of a lake, and matter with boats that plough it. The different possibilities that you have mentioned lie here in the fact that what can be materially realised in an infinity of different manners is the same form of the surface of the lake, the same qualitative state; the “material state” is in fact considered as determined only when it has been established in which point of the lake basin each particle of water is. Here, the arbitrary marking (for instance, by numbering) which helps differentiating the identical individual particles of water corresponds to the setting of a system of coordinates in the Ether, [that is], to the relations to a medium. If the water is at rest in the evening, when all the boats are in port, then the qualitative state is exactly the same as in the morning before the boats plough it: the surface of the lake is a “homogeneous” smooth plane. But the material state hidden behind it may have completely shifted. It is impossible (as it happened for the guide-field before Einstein) to recompose the actual position of all the particles of the water in the lake that were stirred by the boats, starting from a rest position fixed once and for all and from an elongation caused by the boats.111

This analogy is developed by Weyl/Paul in reply to an objection raised by Peter.112 The latter wonders how it is possible to attribute a metric – the Minkowski one, for instance,113 – to a portion of the universe void of matter. He then develops 107 For

an extensive development of the subject: Kerszberg (1989, §23 C) and Bergia and Mazzoni (1999). 108 Weyl (1924, p. 202, right hand column). 109 Weyl (1924, p. 202, left hand column). 110 Weyl (1924, p. 202, between both columns). 111 Weyl (1924, p. 203), my translation. 112 Weyl (1924, p. 202, right hand column). 113 In Massenträgheit und Kosmos, it is de Sitter’s. In Philosophy of Mathematics and Natural Science, Weyl will return to the same argument using Minkowski metric as the rest metric of the ether. This changes nothing to the argument that follows.

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an argument against that type of possibility, which is a reminiscence of Einstein’s hole argument.114 There is only one way to describe the void (i.e. by the vanishing of the energymomentum tensor), while there is an infinity of ways to position a given metric structure on a manifold by means of diffeomorphic pulling forward.115 Let us suppose that the metric structure tends towards the structure we have chosen (for us: the Minkowski one), where matter tends to vanish. We are embarrassed for we do not know which concrete realisation of this structure we must choose, within the infinite class obtained by the pulling forward on the manifold. So that it seems absurd to Peter that a determined metric may be associated to a void region of the universe. Weyl/Paul answers by denying that the different manners to position the metric on the void region respond to situations that are physically different: A difference [between two ways to position de Sitter’s metric on a void region of the spatiotemporal manifold] would exist only if the four-dimensional world were a subsisting environment, in which, in some manner, traces of the material processes were discernible. And it is only then that one could acknowledge as distinct the possibilities of realisation that you have mentioned. But this subsisting environment will be completely rejected by the theory of relativity, probably with your applause.116

In this text, Weyl does not just assert that the naked spatiotemporal manifold, with neither matter nor metric, is deprived of a form. He goes as far as to say that it has no physical existence. The different realisations (obtained by diffeomorphic pulling forward) of a single metric do not represent different physical situations but rather different representations of a single physical state. This position is quite different from the one he had in §12 of Raum-Zeit-Materie in which Weyl seemed to be desirous, at all cost, to give a physical significance to the system of coordinates. In reality, it is not a total reversal on Weyl’s part. For, in the next lines of the text, we are told that the system of coordinates (or the manifold which it enables to describe) may acquire a physical significance if we connect it to the trajectories of some determined elements of matter. For a better perception, let us develop the analogy to its end. Let us suppose that our starting point is a region of the universe, R0 , void of matter, where the metric takes the form: gμν (t, x, y, z) = ημν = diag(+1, −1, −1, −1) which we consider as characteristic of the naked ether. This state is compared with the plane surface of a lake where water is at rest. The specific form gμν = ημν is

114 Weyl

explicitly refers to Einstein’s article from 1914. (1924, p. 202, right hand column): “though [de Sitter’s metric] is, per se, qualitatively totally determined, there are however an infinity of possible ways for this state to be realised in the continuum of the world”. 116 Weyl (1924, p. 203, left hand column). 115 Weyl

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dependent on the fact that we have chosen an adapted natural system of coordinates (t, x, y, z) (which Weyl calls, in his analogy, a specific “numbering of the molecules of water” which compose the surface of the lake). This system of coordinates is naturally related to a congruence of geodesic time-like world-lines, which we can imagine as being traversed by small free-falling label-particles of negligible mass. This congruence is given by the lines: Lx,y,z {(t, x, y, z)t ∈ R} . The coordinate t which parameterises each of these geodesic lines corresponds to the proper time measured along it. The simultaneity hypersurfaces are orthogonal to these geodesic lines and are provided with Euclidean distance. The spatial distance between two label-particles do not evolve with t. In other words, the value of the distance between (t, x, y, z, t) and (t, x  , y  , z ) is: −(x − x  )2 − (y − y  )2 − (z − z )2 , independently from t. We can then graphically represent the congruence of the Lx,y,z by vertical lines and the simultaneity surfaces by horizontal lines (we delete two spatial dimensions in order to give a simple representation). It is the usual foliation of Minkowski space defined by an inertial reference frame. Here, the verticality of the lines has a stronger sense than in the fibred plasticine argument, for it designates a true metric invariance. Let us suppose that these metric properties are valid for a first interval of time when our geodesic lines remain in the region R0 . Let us now suppose that the boats come to disturb the surface of the water. The lines Lx,y,z are extended (as geodesic lines) in a new region R1 of space-time where a mass M curves the metric. Minkowski metric is no longer valid: we no longer have gμν (t, x, y, z) = ημν either in the system of coordinates defined by the congruence of the Lx,y,z , or in any other one. What happens now when the boats return to port, i.e. if the Lx,y,z are again extended into a third region R2 which is void like the first one? The surface of the water gradually becomes plane again. That is, the metric will converge again towards a Minkowski metric. However, Weyl insists, the position of each particle of water will not necessarily be the same, within this new plane surface of the water, as before the passing of the boats. To what does Weyl analogically refer? The Minkowski metric, which will be restored in the new environment devoid of mass, will generally be “orientated”117

117 Here

we use Weyl’s terminology. He very often uses the word “orientation” to refer to the different manners in which a same geometrical object may be expressed in coordinates, that is the different manners in which it can unfold on the manifold. See, for instance, Weyl (2015, pp. 44–45) or Weyl (1921, p. 126). The orientation of a geometrical object, in that sense, may have a purely subjective status, resulting from an arbitrary choice, as when we consider the orientation of the Riemannian metric at a singular point of the manifold or, on the contrary, have an objective invariant sense, as when we consider the variation of the orientation of the metric throughout an open domain.

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Weyl’s analogy of the lake and boats, reinterpreted with congruences of geodesics. In a region R0 (here in yellow), void of matter, a congruence of geodesic lines Lx,y,z have been selected

(represented in blue) which materialise the Minkowskian structure of space-time in that region. That is, these geodesic lines, to which negligible label-particles are associated, remain at a constant spatial distance in the course of time. The simultaneity hypersurfaces (represented here by green horizontal lines) are provided with a constant Euclidean structure in the course of time. Then, these geodesic lines enter the region R1 (not coloured) where a mass responsible for a nonnull curvature lies (we did not represent the world-lines of the elements of matter). The geodesic lines Lx,y,z start to converge. When these geodesic lines reach the region R2 (in orange) void of matter again, they are no longer adapted to reveal the recovered Minkowskian structure. According to Weyl’s analogy: “the position of the particles of water was disturbed by the passing of the boats”. To reveal the recovered Minkowskian structure, we must change geodesic congruences and take the Lx  ,y  ,z (represented in red). So, even if the absolute orientation of the Minkowskian structure of a void region of the universe has no sense, by considering the intermediate region, we could give a sense to the relative change of orientation of the metric of a region in relation to the metric of another region.

In the same manner, here Weyl says that the absolute orientation of Minkowski metric in one region considered in isolation has no physical significance. But the relative change of the orientation of the metric in passing from a region to another one makes sense. Weyl generally illustrates this type of behaviour by referring to the discussions on the differentiation between right hand and left hand in Kant’s and Leibniz’s works.

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differently. That is, it is not necessarily expressed by gμν (t, x, y, z) = ημν but more generally by gμν (t, x, y, z) = φ ∗ ημν (φ ∗ : diffeomorphic pulling forward) for a specific φ). Actually, the congruence of the geodesic lines Lxyz which was defined in the region R0 , then extended to R2 , will no longer be adapted to the recovered Minkowskian structure. While these particles were at rest in relation to one another at the beginning of the process, these particles are no longer at (metric) rest, relative to one another after passing near the mass M. A new system of coordinates (t  , x  , y  , z ) should then be established, by choosing a new family of label-particles, so that the congruence Lx  ,y  ,z of their geodesic lines enables us to re-establish gμν (t  , x  , y  , z ) = ημν in R2 . With this analogy, Weyl suggests a subtle answer to the question: has the spatiotemporal manifold a physical significance? The changes of coordinates may be defined in a purely mathematical manner. Thus two fields gμν and φ ∗ gμν can be seen as different representations of a single metric. But as soon as a system of coordinates is associated to a material reality – as the congruence of the geodesic lines of material label-points – then a diffeomorphism becomes physically significant. So, the change of orientation to which Minkowski metric was subjected in our example has a precise physical significance. To materialise the (flat) recovered Minkowskian structure, once we have penetrated into the region R2 , the family of label-particles used to define our system of coordinates must be changed. Finally we can see that if we consider two flat regions topologically disjoint of space-time, it is meaningless to question whether the Minkowski metrics of the two regions appear with the same orientation. This is a merely subjective matter which is based on a choice of coordinates. But as soon as the two regions are connected by an intermediate region, the geodesic lines can be extended from one of the regions towards the other one, and the orientation acquires a physical significance. This enlightens more effectively the position, surprising at first sight, that Weyl had adopted in Raum-Zeit-Materie about the physical significance of a change of coordinates.

11.5 The pba in Later Texts To our knowledge, Weyl revisited the pba, after 1924, in two texts only: his vast philosophical monograph Philosophy of Mathematics and Natural Science and in Mind and Nature.

11.5.1 In Philosophy of Mathematics and Natural Science Philosophy of Mathematics and Natural Science was first published in German in 1927, then, in English in 1949. As early as 1927, we find paragraph 16, called “The Structure of Space and Time in their Physical Effectiveness”. It goes over

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the fpba and the analogy of the lake and boats again. It is the first paragraph of the chapter “Space and Time, the Transcendental External World”. The objective of the whole paragraph is to question the content and the origin of the space-time structure, namely, its metric structure. Part of the content of the paragraph already appeared in Weyl’s previous literature, but its arrangement is new, and, as in the other chapters of Philosophy of Mathematics and Natural Science, we can find definite references to Leibniz. The purpose of the passage in which we are particularly interested118 is to solve the problem of the nonequivalence between the kinematic perspective and the dynamic perspective on the analysis of motion. From the kinematic perspective, any reference frame should be equivalent to describe motion. But from the dynamic point of view, it seems that there are privileged reference frames, in which solely physics laws can be expressed with simplicity. It was Newton’s point of view, which was not outdated by special relativity. Is it possible to go beyond this apparent limitation, on a dynamical level, and to enunciate the laws of physics independently from referring to a privileged point of view? Mach’s ideas are described here as an attempt to reach the largest generality from the point of view of the dynamical reference frames, without having to assume the existence of anything excepted matter. Weyl names Huygens as a predecessor in this regard and of course Einstein as a successor on this path, at least for some time. It seems however that the issue of general relativity is partly back projected on Huygens and Mach by Weyl. Weyl then refutes as absurd Huygens’s and Mach’s path. This refutation closely follows the one in Massenträgheit und Kosmos. Weyl repeats the fpba to justify the fact that the naked spatiotemporal manifold, without metric, cannot be used to support a difference between the motion and the rest of two separate elements of matter. Ultimately, it is on the inertial structure (itself included in the metric structure) that the existence of a (local) dynamically privileged reference frame is based. According to Weyl, a reasonable solution to our problem could not be found as long as we had not understood that the inertial structure itself was a physical field, “a real thing which not only exerts effects on matter, but undergoes effects from it”. Here, as in the very first version of the pba, Weyl suggests that Riemann is a precursor of that idea, which Einstein would have only developed – though in a decisive manner – nearly 70 years later, by introducing the principle of equivalence between inertia and gravitation. Weyl does not dwell on the autonomous physical reality granted to the “ether” as much as in the previous text. This derives however from the fpba: this metric field, the existence of which was intuited by Riemann, and which Einstein’s theory rightly identified with the inertial gravitational field, could in no way be a pure emergence of the field of matter, as it is defined on the naked spatiotemporal manifold. It must be granted a particularly autonomous existence. In the 1949 version of our paragraph, Weyl then takes up the analogy of the lake and boats. The content of the analogy is not different from the version of

118 Weyl

(1949, pp. 104–107).

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Massenträgheit und Kosmos. However, the function given to the argument is much more important. In the 1924 version, it seemed that this analogy only had a technical function, insuring that the ether had the potential to recover its “rest shape”,119 away from any matter, without falling into an aporia due to the indetermination of the “orientation” of the metric of the ether. In Philosophy of Mathematics and Natural Science, Weyl sees in the analogy of the lake and boats an illustration of the essential difference between the Newtonian and the relativistic points of view, concerning the relations between gravity and inertia. In the Newtonian physics, there was a rigid structure, fixed once for all, inertia. When bodies gravitationally interacted, they left the tracks of the inertial structure and returned to them as soon as they were far enough so that gravitation could be ignored again. In contrast, in general relativity there is no structure of ether at rest, fixed once for all from the start. Instead, one knows how to qualitatively characterise which metric structure the ether must adopt in the absence of any matter. But the new “orientation” taken by this structure will depend on the dynamical history followed by the ether in its relation to matter, in the intermediate region connecting the two regions void of any gravitating matter.

11.5.2 In Mind and Nature The pba appears again in Mind and Nature in 1934.120 The context is different again. The general purpose of this article is to show that the subjects, by their body actions and passions and by their conscious minds, are inevitable constitutive elements of physical science. This is how he concludes the end of chapter IV: I dare hope that we will have made the following point intelligible: how and up to what point the structure of our scientific knowledge is conditioned by the circumstance that the world, which is the purpose of all our scientific research, is not something that exists per se, but only exists and occurs from the encounter between the subject and the object.

To reach that conclusion, Weyl, in the different sections of the article, gradually moves up in the hierarchy of the knowledge related to the world. He moves from the perception data to the primitive physical concepts (Locke, Descartes) in which the sole sensory qualities are questioned with regard to their objectivity. Finally, he comes to the questioning of the objectivity of space and time. Now the subject has no direct relation with the physical properties of the object. Instead he is necessarily led to reach objectivity through symbolic representation.

119 In

Philosophy of Mathematics and Natural Science, contrary to Massenträgheit und Kosmos, Weyl uses Minkowski’s metric (and not de Sitter’s) for the ether, when he develops the analogy. The cosmological preference for de Sitter’s metric will nevertheless be reasserted (and justified in the same manner) by Weyl a few pages further. 120 It is therefore posterior to the German edition of Philosophy of Mathematics and Natural Science, but not to the English edition.

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In Chapter IV, Weyl decides to illustrate that with the particular case of the relativity of space and time. The problem which he reaches, after a detour through the correspondence between Leibniz and Clarke, is the same one as in Philosophy of Mathematics and Natural Science, namely, the “antinomy between kinematics and dynamics”.121 To explain how general relativity solved this problem, every system of coordinates must firstly be deprived of any objectivity: We are led to see the concept of coordinates in an essentially more fundamental manner. Coordinates are no longer measured, but are nothing more than arbitrary numberings of the universe [i.e. of space-time]; they are only symbols used to label and differentiate the universe points from one another.122

Thus Weyl enunciates more neatly than in the previous texts the absence of physical objectivity of the naked spatiotemporal manifold (without the metric or any other structure) or systems of coordinates that represent it. But this assertion will be qualified further in the text in a passage in which the spatiotemporal manifold is illustrated by a pba: The sole relations [that can be expressed on the naked manifold] which have an objective signification are the ones that are preserved by any deformation of the plasticine. The intersection of two world-lines is, for instance, of that kind.

We see that Weyl uses the plasticine to encode the naked manifold, as in the two previous occurrences (fpba versions), even though he is less explicit here concerning the fibration as such. The pure subjectivity of coordinates, mentioned above, is now qualified. The topological invariants keep a form of objectivity. Thus, assuming that space-time is numerically locally represented by an open set of R4 , we can give as examples of non-objective properties of the naked manifold: • The individual identity of a point. • The fact for a world-line of being straight or curved, vertical or not • Etc. However, two lines that intersect in a system of coordinates will continue to do so in any other system. These are the only kinds of objective data actually encoded by the manifold. This argument was also put forward by Einstein as early as 1916.123 In a second stage, Weyl introduces the metric structure on the manifold. It is a field which has a physical significance, which is broken down in two component fields. On the one hand, we have the inertial structure, which determines the trajectories of the bodies not subjected to any influence other than gravity. On the other hand, we have the causal structure, which corresponds to the data of the light cones (one at every point) and which determines which events of the universe can be

121 Weyl

(1934, p. 125). See above p. 336. (1934, p. 128). 123 This is what the Einsteinian literature has called the “point-coincidence argument” since Stachel’s suggestion, see Norton (1999, p. 128). 122 Weyl

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causally linked to which ones and in which order. This breaking down of the metric did not appear in the texts in which the pba appeared previously. They however were long established by Weyl.124 Weyl then shows that the inertial and causal structures, respectively, replace Newton’s absolute space and time, to account for the gap between the total kinematic homogeneity of space-time, and its dynamical nonhomogeneity. Physical realities were indeed responsible for this gap. However, they were not immutable and fixed entities but dynamical fields interacting with matter. These developments clarify the original idea, present in the first edition of Raum-Zeit-Materie, stating that the variable coefficients of the metric are not properties inherent to space but the correlate of a physical reality which determines the behaviour of rulers and clocks.125 The elements that are really new in this text126 are present in the conclusion of Chapter IV. The subject, because of its singular place in the world and because of its consciousness, appears as a necessary mediation in order to root the knowledge of the physical world in something absolutely given. The singularity of the subject is expressed by the contingent form given to the “plasticine” (the system of coordinates). This singularity is then neutralised, to reach objectivity, through the principle of relativity. This neutralisation is somehow an impoverishment. The same subject, in the same conditions and facing the same objective situation, will be led to feel the same conscious experiment of moving forms expanding in time and space. But this space and time experience goes much further than the sole objective spatiotemporal structure, which is indeed only a poor formal skeleton. Finally, the objective inertial and causal structures are only measured by means of sending test bodies and light rays in free fall. But, even though minimally, this necessarily disturbs the structure to be measured. Therefore, the system of coordinates (the “plasticine” pattern) is not the only way the subject takes part in the determining of spatiotemporal forms. It is also an entity which can only know the metric structure by operating on it and therefore by disturbing it. This consideration prepares, in Weyl’s text, the evaluation of the position of the observing subject in quantum mechanics.

11.6 Summary and Conclusion We have shown that the pba, by its richness, its recurrence and its polymorphic character in Weyl’s work, is a very valuable material. The text corpus which includes the different versions of the argument shows the long evolution of a complex and audacious thought on space, with several reversals. They are a consequence of

124 See

Weyl (2015, pp. 17–19) in which the inertial and causal structures respectively correspond to the projective and conformal structures. 125 Weyl (2010, p. 125). See above p. 306 126 They are in great number in comparison with the text discussed previously .

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Weyl’s confrontation with physical reality, with his contemporaries’ thought and with the difficulties inherent to his own philosophical standpoints. The original problem which led to the emergence of the argument consists in justifying the adoption of a metric with variable coefficients. This problem, which has been at the core of geometry since the middle of the nineteenth century and which Einstein’s theory has made even more pressing, is addressed by Weyl under the very specific angle of his idealism: how could we adopt a nonhomogeneous metric, while space, as a form of our intuition, is necessarily homogeneous? According to Weyl, in Raum-Zeit-Materie, adopting a dynamical metric solves this tension. In fact, if we pose a principle of determination of the metric by matter, then a material body may be displaced in space while keeping its properties – metric properties included. This comes to no longer considering the metric as a property inherent to space, but like a property emerging from its content. This argument, however, leads Weyl into a series of technical and philosophical difficulties. Firstly his solution, by excluding the metric of the essential properties of space, is too radical. The infinitesimal metric structure does not emerge contingently from matter, but is part of the essence of space, and should be justified a priori. On that issue, Weyl will rectify his standpoint when developing his infinitesimal geometry, the “problem of space” – in its technical sense – and the epistemological discourse which goes with it. Secondly, the principle of determination of the metric by matter is difficult to formulate coherently. At each step, we may fall into an aporia. For if matter is characterised by a simple scalar field, and if it totally determines the metric, we may come to a theory in which every deformation, and thereby any change, have become impossible. It is the argument which is illustrated by the thought experiment of the plasticine ball and that I have interpreted as an Eleatic aporia, negating the possibility of change. The modifications made in Raum-Zeit-Materie, one edition after the other, show the difficulties met by Weyl in order to avoid this aporia. Is it possible to accept a theory in which any displacement generates a modification of the field of matter? Is it acceptable to hide behind a cosmological argument, the plasticine ball only recovering its shape after a cosmological rearrangement? How is it possible to give sense to the idea of a density of matter and to the idea of a change of the field ρ of matter while supposedly standing at a pre-metric level? In Massenträgheit und Kosmos , Weyl’s stance towards Mach’s principle changes radically. This principle, which was accepted in Raum-Zeit-Materie, under Riemann’s patronage, as an assumption used to solve a philosophical problem, becomes problematic itself and the subject of a critical investigation. It is as if the difficulties anticipated by Weyl, while developing his pba, had gradually gnawed at his belief in the pertinence of this principle. Perhaps Einstein’s own disappointment towards this principle also played a part in Weyl’s reversal. In this new text, Weyl offers a truly spatiotemporal version of his argument, using a four-dimensional and fibred plasticine. If matter is characterised only by a simple congruence of trajectories, it is still possible for a simple diffeomorphism to rectify all the trajectories, that is, put matter at rest. Then, the absurdity of claiming

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to define the motion of matter prior to any metric – in order to be able, then, to define the metric – becomes blatant. Therefore, Weyl provides another signification to Einstein’s failure. If, in general relativity, Einstein could not realise Mach’s principle, it is not a contingent fact, related to the lessons of experience. Rather, it was an inevitable fact since an a priori investigation on the signification of the most radical version of Mach’s principle reveals its absurdity or, at the very least, brings it back to an Eleatic negation of motion. This is why a simple thought experiment, like the pba, is sufficient. Weyl, in the second part of the dialogue, still engages in a criticism specifically targeting Einstein’s attempt at realising Mach’s principle in the context of relativistic cosmology. Mach’s principle must be expressed through the Einstein equations which correlate Einstein metric tensor Gμν with the energy-momentum tensor T μν . By slightly departing from Weyl’s text, we have been able to highlight that T μν itself generally depends on the metric. As such, in opposition to Einstein’s assertions, even if Mach’s principle could be realised in general relativity, it could not be in the form “the T μν is sufficient to determine the metric” (which entails a circularity). The only coherent starting point, for a total determination of the metric, should be the Tμν modulo its correlation with a metric. This retrospectively justifies the form taken by Mach’s principle in Weyl’s work, as early as the first edition of Raum-Zeit-Materie, as referring to a notion of matter determined by scalar fields. However, once the problem is rightly posed, the Einstein equations being equations with second-order partial derivatives, it cannot be solved without giving some initial (Cauchy or Dirichlet) conditions. This expresses a form of unsurpassable autonomy of the metric in relation to matter. Weyl shows that all cosmological subterfuge imagined by Einstein to break free from these conditions is illusory. Equilibrium or symmetry hypotheses always hide metric determinations. And how could it be otherwise since Mach’s principle which Einstein is researching, when it is correctly formulated, falls within the range of Weyl’s a priori demonstration: in no case could a metric univocally emerge from a purely topological notion of matter. Weyl then draws the consequences for the ontology of physics. In order to reestablish the possibility of motion, hence the elaboration of physics, one must accept the existence, alongside matter, of a metric field, the ether, which is partly autonomous. This metric field does not have immutable properties, like Newton’s absolute space, but is in dynamical interaction with matter. Due to its partial autonomy, when we move away from any matter, the ether will go back to its “rest state”, which is specific to it. Several possibilities being open regarding the metric properties of this state, a choice must be made which shall be a true physical hypothesis of a cosmological nature, to be evaluated in connection with the observational data. Weyl, through his character Paul, then says that he is in favour of choosing the de Sitter metric (this is 1923), because it has good topological properties and seems compatible with the astronomical observations. Weyl then returns to a question left pending since Raum-Zeit-Materie: can a spatiotemporal diffeomorphism which acts upon the metric have a physical meaning, or does it only express a simple mathematical licence in expressing a single physical reality? Because of the modern point of view, in relation to

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covariance in general relativity and to the Einstein hole argument, we would tend to choose the second option. Weyl offers a more nuanced vision. His analogy of the surface of the lake enables him to give a physical sense to a global diffeomorphism. Two regions of space-time, distant from all matter, will be isometric, but the metric can be expressed with a “different orientation”, following the disturbance generated by a mass filling the intermediate region. In Philosophy of Mathematics and Natural Science, Weyl endorses the conclusions of Massenträgheit und Kosmos; he reconstructs the history of the principle of determination of the metric by matter, making Huygens and Mach the initiators of the contradictory version (because it is too strong) of the principle and Riemann and Einstein the moderators who gradually succeeded in expressing a coherent version of it. Finally in Mind and Nature, Weyl places the pba in the global context of the subject/object interrelations in the construction of physical knowledge. Ultimately, we can see how a single argument, the pba, was repeated in Weyl’s work with substantial modifications which not only reflect the evolution of the technical apparatus and the precision in the expression of Mach’s principle but also the evolution of the philosophical problems which guided Weyl’s thought and led him to reuse the same argument for very different purposes. The study of the corpus, in its evolving continuity, shows that Weyl’s reversals concerning his philosophical positions are not the consequences of an unstable nature or a deeply volatile temperament of the German mathematician. Instead, these changes result from the time needed by Weyl to clarify and solve the difficulties which appeared as early as the first edition of Raum-Zeit-Materie. Weyl’s philosophy is not a frozen system, constructed prior to science, but it is constructed with a time consuming reflection, on the more and more complex scientific theories of his time. Therefore, if Weyl’s philosophy may at times seem very unstable, in the light of the great lasting secular systems of the tradition, it is the result of an interaction with science which develops rapidly in that lively era of the beginning of the twentieth century.

References Afriat, Alexander. 2009. How Weyl Stumbled across electricity while pursuing mathematical justice. Studies in History and Philosophy of Science Part B 40(1):20–25. Afriat, Alexander. 2010. The relativity of inertia and reality of nothing. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 41:9–26. Barbour, Julian, and Herbert Pfister. eds. 1995. Mach’s Principle. From Newton’s Bucket to Quantum Gravity. Einstein Studies. From the colloquium in Tübingen, July 1993. Boston: Birkhäuser. Bell, John L., and Herbert Korté. 2016. Hermann Weyl. In The Stanford encyclopedia of philosophy, ed. Edward N. Zalta, Winter 2016. Metaphysics Research Lab, Stanford University. Bergia, Silvio, and Lucia Mazzoni. 1999. Genesis and evolution of Weyl’s reflections on de Sitter’s universe. In The expanding worlds of general relativity. Einstein Studies 7, ed. Hubert Goenner and et al., 325–342. Boston/Basel/Berlin: Birkhäuser. Bernard, Julien. 2010. Les fondements épistémologiques de la Nahegeometrie d’HermannWeyl. Ph.D. thesis. Université de Provence, http://tel.archives-ouvertes.fr/tel-00651772

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Bernard, Julien. 2013. L’idéalisme dans l’infinitésimal. Weyl et l’espace à l’époque de la relativité. Prix Paul Ricœur 2012. Presses Universitaires de Paris Ouest, 2013. ISBN:2840161311. Bernard, Julien. 2015a. Becker-Blaschke Problem of Space. Studies in History and Philosophy of Modern Physics, Part B 52:251–266. Bernard, Julien. 2015b. Les tapuscrits barcelonais sur le problème de l’espace de Weyl. Revue d’Histoire des Mathématiques 21:147–167. Bernard, Julien. 2018. Riemann’s and Helmholtz-Lie’s problems of space from Weyl’s relativistic perspective. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 61:41–56. (S. De Bianchi and G. Catren, Elsevier Bv). Bitbol, Michel, Pierre Kerszberg, and Jean Petitot. 2010. Constituting objectivity: Transcendental perspectives on modern physics. The Western Ontario series in philosophy of science, vol. 74. Dordrecht: Springer. Cartan, Élie. 1925. La théorie des groupes et les recherches récentes en géométrie différentielle. Ens. Math. 24. Conférence faite au Congrès de Toronto en 1924, 1–18. Cassirer, Ernst. 1910. Substance and Function. Dover Books on Mathematics Series. New York: Dover Publications. Cassirer, Ernst. 1923. Einstein’s theory of relativity, considered from the epistemological point of view. In Substance and function and Einstein’s theory of relativity, ed. W.C. Swabey, and M.T. Swabey. Chicago: Open Court. Choquet-Bruhat, Yvonne. 1952. Théorème d’existence pour certains systèmes d’équations aux dérivées partielles non linéaires. In Acta Mathematica, 1st ser. 88, 141–225. Choquet-Bruhat, Yvonne. 1969. Problème de Cauchy global en relativité générale. Séminaire Jean Leray 2:1–7. Choquet-Bruhat, Yvonne, and Robert Geroch. 1969. Global aspects of the Cauchy problem in general relativity. Communications in Mathematical Physics 14(4):329–335. Chorlay, Renaud. 2009. Passer au global: le cas d’élie Cartan, 1922–1930. Revue d’histoire des mathématiques 15:231–316. Coffa, J. Alberto. 1979. Elective affinities: Weyl and Reichenbach. In Hans Reichenbach: Logical empiricist, ed. Wesley C. Salmon, vol. 132, 267–304. Dordrecht: Springer Netherlands. Coleman, Robert, and Herbert Korté. 2001. Hermann Weyl: Mathematician, physicist, philosopher. 4.11 the laws of motion and Mach’s principle. In Hermann Weyl’s Raum Zeit Materie and a general introduction to his scientific work, ed. Erhard Scholz. §II. 4.5–4.7, 262–270. Georges Darmois. 1927. Les équations de la gravitation einsteinienne. Mém. Sc. Math. 25. Eckes, Christophe. 2011. Groupes, invariants et géométries dans l’œuvre de Weyl. Ph.D. thesis, Université Jean Moulin Lyon 3. Einstein, Albert. 1915. Erklärung der Perihelbewegung des Merkur aus der allgemeinen Relativitätstheorie. In Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften (Berlin), 831–839. Einstein, Albert. 1916. Die Grundlage der allgemeinen Relativitätstheorie. Annalen der Physik und Chemie. Einstein, Albert. 1917. Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie. In Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften (Berlin), 142–152. Einstein, Albert. 1918. Prinzipielles zur allgemeinen Relativitätstheorie. Annalen der Physik 55:241–244. Einstein, Albert. 1920. Äther und Relativitätstheorie, 18. Berlin: Springer. Einstein, Albert, and Marcel Grossman. 1913. Entwurf einer verallgemeinerten Relativitätstheorie und einer Theorie der Gravitation. Leipzig: B.G.Teubner. Giovanelli, Marco. 2013a. Erich Kretschmann as a proto-logical-empiricist: Adventures and misadventures of the point-coincidence argument. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 44:115–134. Giovanelli, Marco. 2013b. Leibniz equivalence. On Leibniz’s (Bad) influence on the logical empiricist interpretation of general relativity. From a preprint, Unpublished. Iftime, Mihaela, and John Stachel. 2006. The hole argument for covariant theories. General Relativity and Gravitation 38:1241–1252.

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Chapter 12

Intuition and Conceptual Construction in Weyl’s Analysis of the Problem of Space Francesca Biagioli

Abstract Hermann Weyl adopted the Kantian definition of space as a form of intuition and referred to Edmund Husserl’s phenomenological approach for the philosophical characterization of space in the introduction to Raum-Zeit-Materie (1918) and other writings from the same period (1918–1923). At the same time, Weyl emphasized that subjective factors are completely excluded from the mathematical construction of physical reality in Albert Einstein’s general theory of relativity, with the sole exception of the setting of a coordinate system, which for Weyl is what remains of the original perspective of the self in becoming aware of one’s own intuitions. This paper reconsiders Weyl’s philosophical position as a possible response to the earlier debate on the relation between intuition and conceptual construction in the foundation of geometry, key figures of which, besides Husserl, included Hermann von Helmholtz, Felix Klein, and Moritz Schlick. Keywords Classical and relativistic problems of space · Form of intuition · Hermann von Helmholtz · Hermann Weyl

12.1 Introduction Weyl adopted the Kantian definition of space as a form of intuition and referred to Husserl’s phenomenological approach for the philosophical characterization of space in the introduction to Raum-Zeit-Materie (1918) and in other writings from the same period (1918–1923). At the same time, Weyl emphasized that subjective factors are completely excluded from the mathematical construction of physical reality in Einstein’s general theory of relativity, with the sole exception of the setting of a coordinate system, which for Weyl is what remains of the original perspective of the self in becoming aware of one’s own spatial intuition. With regard to physical

F. Biagioli () Department of Philosophy, University of Vienna, Vienna, Austria e-mail: [email protected] © Springer Nature Switzerland AG 2019 J. Bernard, C. Lobo (eds.), Weyl and the Problem of Space, Studies in History and Philosophy of Science 49, https://doi.org/10.1007/978-3-030-11527-2_12

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space or space-time, the problem arose to establish the conditions for determining the underlying geometrical structure, which became known as the problem of space, in its different formulations.1 Weyl’s approach to the problem of space in Mathematische Analyse des Raumproblems (1923) reflects his philosophical considerations, insofar as he indicates increasingly higher levels of abstraction in the mathematical analysis of the structure of space. According to Weyl, a decisive step in this direction was taken by Helmholtz and Lie, with the group-theoretical deduction of the conditions for such a structure to satisfy the law of homogeneity. Given the heterogeneity of spacetime in general relativity, Weyl’s own solution of the problem of space required him to reformulate the invariants of the space-time continuum in differential geometry. In this respect, Weyl relied on Riemann. I believe that, nevertheless, the debate about the geometrical structure of space initiated by Helmholtz (1868, 1870) played an important role in Weyl’s understanding of the relation between intuition and conceptual construction in the foundation of geometry. In order to better appreciate this aspect of Weyl’s work, this paper contrasts his reconstruction of the classical problem of space with Helmholtz’s empiricist arguments against the Kantian theory of pure intuitions.2 Helmholtz’s arguments were twofold: as a physiologist of vision, he argued that the formation of spatial intuitions deserved an empirical explanation; as a physicist and an epistemologist, he urged a generalization of the notion of the form of intuition to all possible contents that can enter the relevant form of perception. In Helmholtz’s view, such a generalization ought to include the different hypotheses about metrical geometry formulated by Riemann. The corresponding (classical) problem of space was to establish necessary and sufficient conditions for a Riemannian metric of constant curvature, which included non-Euclidean geometries as special cases. Therefore, in 1878, Helmholtz reformulated his second argument by distinguishing between the general properties of the form of space (which he identified as a threefold extended manifold of constant curvature) and the specific axiomatic systems of metrical geometry, which correspond to the assumption of a flat or positively or negatively curved space. As pointed out by Ryckman, Helmholtz argued against the Kantian philosophy of geometry while retaining an inherently Kantian theory of space, according to which the former properties provide us with conditions of the possibility of geometrical measurement.3 In other words, Helmholtz distanced himself from the received view of geometrical axioms as evident and necessary truths and reformulated the notion of the form of intuition in terms of a conceptual and hypothetical construction that lies at the foundation of measurement.

1I

refer to Scholz (2013) for a survey of different problems of space from the classical problem posed by Helmholtz to Weyl’s and Cartan’s problem of reformulating the older criteria for determining the structure of space in the light of differential geometry and the general theory of relativity (i.e., the modern or relativistic problem of space). 2 On Helmholtz’s empiricist philosophy of mathematics, see esp. DiSalle (1993). 3 Ryckman (2005, pp. 73–74).

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It is well known that the required mathematical tools for the solution of Helmholtz’s problem of space were developed only later, in Lie’s theory of continuous transformation groups. But this mathematical development showed another aspect of the epistemological problem as well: Is it still possible to draw a distinction between general and specific properties of space, given the fact that both kinds of properties can find an exact expression in axiomatic terms? An additional problem, after general relativity, is that the space-time structure under consideration would deserve a further generalization in order to include the hypotheses of variably curved spaces disregarded by Helmholtz. Therefore, commenting on the centenary edition of Helmholtz’s epistemological writings of 1921, Moritz Schlick maintained that, in order to uphold Helmholtz’s original distinction, the intuitive and indescribable factors of spatial perception ought to be distinguished from all kinds of axioms. Schlick ruled out the synthetic aprioricity of geometrical knowledge in Kant’s sense by sharply distinguishing between acquaintance with sense qualities, which for Schlick pertain to the contents rather than the form of perception, on the one hand, and geometrical knowledge, which for Schlick is analytic, on the other. However, it has been questioned whether Schlick’s distinction does justice to Helmholtz’s account of localization as a construction of the form of space, which presupposes the interdependence of intuition and understanding.4 With reference to this debate, I address the question whether the forms of intuition in Kant’s sense admit of a mathematical characterization, which is crucial to Weyl’s connection between the philosophical and the mathematical aspects of the problem of space. One of the problems is that is not always possible to determine which of the earlier philosophical positions were known to Weyl. Ryckman has offered a detailed reconstruction of Weyl’s relation to Husserl.5 I will give evidence of the fact that even the wider debate finds an echo in Weyl’s discussion, although sometimes more indirectly. Furthermore, I believe that some of the motivations for Weyl’s defense of the phenomenological approach go back to the same debate. My suggestion is that the analysis of essence in Husserl’s sense provided a plausible solution to the problem concerning the possibility of a mathematical characterization of space. The fact that essences in Husserl’s sense are not given in isolation but only in their entanglement with external factors enabled Weyl to account for the fact that the new insights into the structure of space-time resulted from a long and complex historical development of geometry from Euclid to Riemann. At the same time, the clarification of the notions of space and time in general relativity provided a principled distinction between essential and accidental aspects in the geometrical representation of space, time, and matter. Whereas as a matter of fact the notion of the a priori form of intuition emerged from a conceptual construction of different hypotheses about the structure of space, as Helmholtz suggested, the form thus obtained retained the role of synthetic a priori knowledge

4 See

Friedman (1997). (2005, Ch. 5).

5 Ryckman

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in Kant’s sense, namely, as knowledge that is independent of experience, because it lies at the foundation of the possibility of experience itself. The first part of the paper provides a general account of the shift from the classical to the modern analysis of the problem of space, with a special focus on the Kantian framework of Weyl’s analysis. The second part offers a discussion of the epistemological problems concerning the distinction between intuitions and conceptual constructions by reconsidering different arguments for and against the possibility of a mathematical characterization of what Helmholtz called the form of spatial intuition in group-theoretical terms. I argue in the concluding section that Weyl’s interpretation of the analysis of essence, considered against its philosophical background, offers a vantage point for discerning the form of spatial intuition from the external factors of perception.

12.2 From the Classical to the Modern Analysis of the Problem of Space It is well known that Weyl was one of the first to account for the fact that the problem of determining the geometrical structure of space, as addressed in different ways from Kant to Helmholtz–Lie, deserved a new formulation after general relativity. In other words, Weyl recognized that, mathematically speaking, there are different problems of space, and worked on a rigorous solution of the relativistic problem of space from 1918 to 1923 and again from 1928 to 1929, in the context of the discussion about the foundations of quantum physics.6 At the same time, Weyl emphasized some continuity in the shift from the classical to the relativistic analysis of the problem of space with regard to the philosophical dimension of the problem. As Weyl put it, the philosophical problem of space consists in establishing the relations between the form and the matter of appearances, which he conceived of in Kant’s sense as the extensive medium that makes it possible to individuate different qualitative contents.7 The philosophical problem guides the mathematical analysis, insofar as this deals with the structure of space and its generalization from the classical to the relativistic view. This section offers a brief reconstruction of the Kantian framework of Weyl’s analysis of the problem of space in his lectures of 1923 and investigates the question: In what sense and to what extent does the 1923 formulation of the problem integrate Weyl’s earlier relativistic account of metrical geometry in Raum-Zeit-Materie? In order to highlight the philosophical aspect of the problem, the second part of the section draws a comparison with the earlier generalization of the Kantian form of spatial intuition by Helmholtz.

6 See

Scholz (2004). (1923, p. 1).

7 Weyl

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12.2.1 The Kantian Framework of Weyl’s Analysis After analyzing the mathematical form of Euclidean space and its role in classical physics in Raum-Zeit-Materie, Weyl emphasized that the shift from classical to relativistic physics entails a corresponding transformation of geometry, which was foreshadowed by Riemann. Weyl compared the transition from Euclidean to Riemannian geometry to the shift from action at distance to infinitely near action in physics. The fundamental fact of Euclidean geometry is Pythagoras’s theorem, according to which the square of the distance between two points is a quadratic form of the relative coordinates of the two points. Intuitively, the fundamental idea of Riemann’s geometry was to interpret the same law as strictly valid only when the points under consideration are infinitely near. The mathematical expression of this fact gives us a Riemannian metric of constant curvature, that is, the condition that is satisfied by both Euclidean geometry (when the curvature equals 0) and by the other two classical cases of such manifolds (when the curvature is greater or less than 0). The latter case was proved by Beltrami (1869) to correspond to the non-Euclidean geometry of Bolyai-Lobachevsky and played a central role in Helmholtz’s imagination of the observations on a convex mirror that would be compatible with the homogeneity of space.8 Therefore, Helmholtz argued that the particular value of curvature is a matter for empirical science, which in the late nineteenth century became known as the “Riemann-Helmholtz” theory of space.9 However, in the context of Weyl’s interpretation of general relativity, Weyl’s emphasis is on the novelty of Riemann’s approach, even when compared to his contemporaries: Space is a form of phenomena, and, by being so, is necessarily homogeneous. It would appear from this that out of the rich abundance of possible geometries included in Riemann’s conception only the three special cases mentioned [i.e., spherical and BolyaiLobachevsky] come into consideration from the outset, and that all the others must be rejected without further examination as being of no account: parturiunt montes, nascetur ridiculus mus! Riemann held a different opinion, as is evidenced by the concluding remarks of his essay. Their full purport was not grasped by his contemporaries, and his words died away almost unheard (with the exception of a solitary echo in the writings of W. K. Clifford). Only now that Einstein has removed the scales from our eyes by the magic light of his theory of gravitation do we see what these words actually mean.10

8 Helmholtz

(1870). We have already mentioned that Helmholtz had posed the classical problem of space as the problem of establishing the necessary and sufficient conditions for obtaining a Riemannian metric of constant curvature in Helmholtz (1868). However, it was only after his correspondence with Beltrami, in 1869, that he became aware of the fact that his previous characterization of space included non-Euclidean manifolds of constant curvature. The relevant correspondence between Beltrami and Helmholtz is now available in Boi et al. (1998, pp. 204– 205). 9 See, for example, Erdmann’s (1877) then popular exposition. 10 Weyl (1921/1952, pp. 96–97).

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Not only was Riemann the first to consider the hypothesis of variably curved spaces in Riemann (1854/1867), but it followed from his distinction between discrete and continuous manifolds that only a manifold of the former kind entails the principle of its metrical relations in itself or, as Weyl put it, “a priori.”11 According to Riemann, under the assumption that space is continuous, we would have to seek the ground of its metric relations outside it, in the binding forces that act upon it. The importance of the latter claim, in Weyl’s view, lies in the fact that Riemann, acknowledged the possibility that the form of space depends on its material contents, insofar as this determines metrical relations. In doing so, Riemann distanced himself from the received view that the metrical structure of space is fixed and inherently independent of the physical phenomena for which it serves as a background. More precisely, the equivalence between gravitational and inertial phenomena established by Einstein ruled out the view that space is necessarily homogeneous, and therefore the conception of space as a form of appearance in Kant’s sense or even its generalization to the class of the manifolds of constant curvature in the works of Helmholtz, Klein, Lie, and Poincaré. Nevertheless, there is evidence that one of Weyl’s motivation in dealing with the relativistic problem of space was to vindicate the Kantian view of space in some respects. The fourth edition of Raum-Zeit-Materie contains a new section on “Space Metric from the Group-Theoretical View,” which is characteristic of the tradition inaugurated by Helmholtz.12 But the most comprehensive treatment of the problem of space from this point of view is found in Weyl’s 1922 lectures at the Institut d’Estudios Catalans in Barcelona and at the Universidad Central in Madrid. Weyl held these lectures in French and Spanish. A German version of the lectures appeared in 1923 under the title “Mathematische Analyse des Raumproblems.”13 In the Preface, Weyl presented this work as an “integration” of Raum-Zeit-Materie: “The deeper, group-theoretical conception of the problem of space was sketched only briefly in the latter book, because the main focus there was on physics and relativity theory and their immediate presuppositions.”14 Weyl’s goal in 1923 was to catch up with the problem of space. Weyl used his infinitesimal geometry to generalize the problem of space so as to encompass general relativity. Firstly, he allowed for indefinite metrics (“postulate of freedom”). Secondly, he postulated a metric connection according to which to each congruent transfer there exists exactly one equivalent affine connection (“postulate of coherence”). Weyl’s goal was to show that these principles are necessary and sufficient to individuate the subgroups of the general linear group that correspond to 11 Weyl

(1921/1952, p. 97). (1921/1952, pp. 138–148). The question whether the group-theoretical view provides a suitable interpretation of Helmholtz’s thought experiments, as Weyl assumes, is discussed in the next section. 13 A French translation of this text, along with notes drawing the relevant comparisons with the original version of Weyl’s lectures, has been made available by Audureau and Bernard (Weyl 2015) 14 Weyl (1923, Vorrede). 12 Weyl

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rotations in relativistic physics. He identified the geometry that satisfies the principle of freedom with the type of Finsler metrics also called by him Pythagorean.15 Without entering into the details of Weyl’s analysis of the problem of space, the relevant point for my present purpose is that Weyl’s new formulation of the problem enabled him to emphasize some continuity in the historical development from the classical to the relativistic view. Weyl described such a development as a progressive generalization of the a priori component of the representation of space (or space-time) that provides us with preconditions for the possibility of measurement. In the context of Newtonian physics, Euclidean geometry satisfies this purpose by postulating the homogeneity of space via Pythagoras’s theorem. Kant’s Transcendental Aesthetic provides a philosophical account of the role of Euclidean geometry in physics, according to which homogeneity pertains to the nature of space as a form of appearance. Therefore, the spatial properties of a physical object do not depend on its location and the metrical structure of space can be established a priori, independently of its material contents. The classical problem of space that occupied scientists and philosophers after the development of non-Euclidean geometry was to individuate the class of metrical spaces that are homogeneous in the same sense. Therefore, Helmholtz postulated that a rigid body in space possesses the degree of free mobility that corresponds to a Euclidean metric. We have already mentioned that Helmholtz in 1868 restricted his consideration to Euclidean geometry. After his correspondence with Beltrami, in Helmholtz (1870), he acknowledged that the Riemannian metric of constant curvature thus characterized included different possible metrical geometries as special cases and maintained that the measure of curvature is a matter for empirical investigation. Weyl argued that, mathematically speaking, Helmholtz’s requirement of homogeneity received a precise interpretation only in Sophus Lie’s theory of continuous transformation groups.16 Lie identified the spaces that satisfy this requirement as the group of congruent transformations. It followed that Helmholtz’s conjecture was confirmed, at least insofar as the specific value of curvature was left undetermined by the mathematical analysis. Regarding the philosophical problem of space, Weyl concluded his discussion of the classical view as follows: Under these circumstances, one is led almost necessarily to ask oneself whether in the end actual space could be not Euclidean at all, but rather a spherical space with non-vanishing curvature λ. Owing to its metrical homogeneity, such a space would be equally suitable as the Euclidean to serve as a form of the appearances. With this question, we turn back from the mathematical analysis to reality.17

Weyl went on to present the relativistic view as a further generalization of the homogeneity requirement. In terms of Weyl’s infinitesimal geometry, the same requirement corresponds to the fact that the metric at a point P and the metrical con-

15 For

a detailed discussion of Weyl’s approach to the problem of space and its development from 1921 to 1923, see Coleman and Korté (2001). 16 Lie (1893, pp. 437–471). 17 Weyl (1923, p. 43).

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nection of P to its neighboring points is everywhere the same, essentially singular and absolutely determined. The mutual orientation of the metrics in different points, on the other hand, is contingent and capable of continuous variations depending on the material content. Weyl used the Kantian framework of his analysis to identify the former requirement as the “a priori essence of the space-time structure” and the variability of metrics as a posteriori or “impossible to fully grasp in a rational manner, but only approximately and with the help of immediate and intuitive reference to reality.”18 Weyl identified the postulate of freedom in particular (i.e., the possibility that the metric field is subject to arbitrary changes while the nature of the metric remains fixed) as a sort of relativistic equivalent of Helmholtz’s homogeneity requirement.19 Insofar as both of Weyl’s postulates (i.e., freedom and coherence) provided an interpretation of the conceptual analysis of the problem of space, he called this the “the synthetic part of the investigation in Kant’s sense.”20 To sum up, Weyl relied on Lie for the mathematical analysis of the classical problem while looking at Helmholtz’s approach to the philosophical problem. The Kantian theory of space need not be rejected in the light of the later scientific developments, insofar as the requirement of homogeneity can be generalized accordingly. The most explicit indication of Weyl’s reliance on the Kantian conception of space is his attempt to redraw the line between the a priori and the a posteriori components of the structure of space-time as follows: Even Einstein upholds the view that the metrical structure of the world is everywhere of the same kind, as our general metrical infinitesimal geometry presupposes it to be. It is not simply denied that something in the structure of the extensive medium of the external worlds is a priori; however, the borderline between what is a priori and what is a posteriori is set somewhere else.21

This passage suggests that the appropriate strategy for dealing with the relativistic problem of space lies in nuce in Helmholtz’s interpretation of the form of spatial intuition as dependent on its content for the determination of the specific metrical structure of space. In order to better understand Weyl’s position, the second part of this section contrasts it with Helmholtz’s own considerations on the form of intuition.22 The following section turns back to Weyl’s further claim that the group-theoretical view provides a conceptual analysis of spatial intuition, and therefore the appropriate means to implement the strategy above.

18 Weyl

(1923, p. 45). (1923, p. 46). 20 Weyl (1923, p. 49). 21 Weyl (1923, pp. 44–45). 22 See Biagioli (2014b) for a more thorough discussion of Helmholtz’s stance towards the Kantian theory of space. 19 Weyl

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12.2.2 Helmholtz’s and Weyl’s Accounts of the Form of Intuition As suggested by Scholz (2013), Weyl’s connection with Helmholtz becomes apparent in the light of a manuscript from the 1840s “On the General Natural Concepts,” which was made available in the second volume of Königsberger’s intellectual biography of Helmholtz.23 Helmholtz’s aim was to determine the concept of space “in such a way that it can comprehend all possible changes of matter, which are obviously to be considered here only as changes in spatial relations, i.e., as motions.”24 The same requirement received a precise formulation in Helmholtz’s geometrical papers as the free mobility of rigid bodies.25 However, as Scholz admits, there is no information about the possibility that Weyl had read this text, nor is it possible to determine precisely which of Helmholtz’s works Weyl might have known. It might be added that the debate on non-Euclidean geometry and the Kantian theory of space at that time focused on Helmholtz’s geometrical papers. To my knowledge, Helmholtz’s earlier manuscript did not receive attention until only recently. Since there is no evidence that Weyl had read this text, my suggestion is to restrict the consideration to the works cited by Weyl. At that time, there were several editions of Helmholtz’s geometrical papers available, including the centenary edition of Helmholtz’s Schriften zur Erkenntnistheorie (1921) by Paul Hertz and Moritz Schlick. However, Weyl quotes Helmholtz’s “Ueber die Thatsachen, die der Geometrie zugrunde liegen” (which first appeared in 1868) from the earlier edition of Helmholtz’s Wissenschaftliche Abhandlungen (vol. 2, 1883). Amongst Helmholtz’s epistemological works, the same volume included “Ueber die thatsächlischen Grundlagen der Geometrie”26 and “Ueber den Ursprung und Sinn der geometrischen Sätze,” the German translation of Helmholtz (1878b), which Helmholtz had originally published in Mind in reply to Land (1877). Subsequently, the German translation of Helmholtz (1878b) was incorporated in “Die Tatsachen in der Wahrnehmung” (1878a). All this to say that, arguably, Weyl was acquainted with the latter paper. This might shed some light on Weyl’s reading of Helmholtz’s stance on the classical problem of space, as this paper contains Helmholtz’s most comprehensive discussion of the Kantian themes already addressed in the 1840s manuscript. In particular, the concluding paragraph of the German version of the paper describes the notion of the form of intuition as follows:

23 Königsberger

(1903, pp. 126–138). (1903, p. 34). The English translation of this and other passages is found in Hyder (2009, pp.140–146). 25 Scholz (2013). 26 Here dated 1866. However, this paper first appeared in 1868. 24 Königsberger

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Kant’s doctrine of the a priori given forms of intuition is a very fortunate and clear expression of the state of affairs; but these forms must be devoid of content and free to an extent sufficient for absorbing any content whatsoever that can enter the relevant form of perception. But the axioms of geometry limit the form of intuition of space in such a way that it can no longer absorb every thinkable content, if geometry is at all supposed to be applicable to the actual world. If we drop them, the doctrine of the transcendentality of the form of intuition of space is without any taint. Here Kant was not critical enough in his critique; but this is admittedly a matter of theses coming from mathematics, and this bit of critical work had to be dealt with by the mathematicians.27

Not only does this passage contain the same claim that Weyl compared to the postulate of freedom, but it clearly suggests the above strategy for dealing with the philosophical problem of space. Helmholtz’s starting point was the consideration that a generalization of Kant’s form of intuition was made necessary by the mathematical discovery of non-Euclidean geometry as a possible application of the free mobility of rigid bodies. In order to show this possibility, in Helmholtz (1870), Helmholtz described an imaginary world in a convex mirror in line with Beltrami’s interpretation of hyperbolic geometry. Helmholtz constructed a thought experiment by assuming that for every measurement in our world, a hypothetical inhabitant of the mirror would be able to carry out a corresponding measurement. However, this would appear to us to be subject to some contraction in the vicinity of the border of the mirror. This fiction shows that, in principle, it would be possible to interpret the same operations in both ways. Helmholtz concluded that it is impossible to make a choice on purely mathematical grounds, regardless of some further knowledge of the laws of mechanics. The axioms of geometry as propositions concerning physical quantities – he concluded – must be empirical propositions and can be subject to revision, contrary to Kant’s view of (Euclidean) axioms as evident propositions that lie at the foundation of the theory of motion. In 1878, Helmholtz expressed the same idea by calling the type of knowledge that results from analytic geometry, on the one hand, and from observations on rigid bodies, on the other, “physical geometry.”28 In response to Land and other orthodox Kantians, who defended the aprioricity of Euclidean geometry as grounded in the form of spatial intuition, Helmholtz emphasized that the metrical structure of space is a posteriori in the sense of physical geometry. Nevertheless, he drew on his physiological work to explore the possibility of deriving a more general form of spatial intuition from the laws governing the localization of particular objects in space. It is in this sense that Helmholtz deemed the mathematical formulation of the free mobility of rigid bodies a fundamental “fact” induced by repeated observations on solid bodies. The same fact generalized provides us with a precondition for the possibility of measurement that is compatible with different metrical geometries, as in Helmholtz’s thought experiment.

27 In

Helmholtz (1883, p. 660). I quoted from the English translation of the same passage in Helmholtz (1921/1977, pp. 162–163). 28 Helmholtz (1921/1977, p. 153).

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Summing up, Weyl’s relativistic approach to the problem of space can be compared to Helmholtz’s in several respects. Firstly, both Helmholtz and Weyl rely upon a conceptual analysis of the mathematical problem to show that not all the possible hypotheses had been considered in the previous definition of the a priori form of space. Consequently, they argue for a generalization of the previous view. Secondly, given the variability of metrics established a priori, they maintain that the specific metrical structure of space is a matter for empirical science. Thirdly, they deal with a philosophical problem of space that goes back to Kant and consists in accounting for the possibility of physico-geometrical knowledge. More specifically, the form of intuition figures in both accounts as the a priori component of the investigation that lies at the foundation of synthetic principles such as the free mobility of rigid bodies or its relativistic equivalent. To conclude this section, however, it is important to notice that there is an important difference between Helmholtz’s and Weyl’s accounts. Whereas Helmholtz proposed an explanation of how the form of intuition is acquired, Weyl defended a Kantian view of a priori intuition also with regard to its evidence. The difference is apparent in the following passage from von Helmholtz (1878a): I have [ . . . ] frequently emphasized in my previous studies the agreement between recent physiology of the senses and Kant’s doctrine, although this admittedly does not mean that I had to swear by the master’s words in all subordinate matters too. I believe the resolution of the concept of intuition into the elementary processes of thought as the most essential advancement in the recent period. This resolution is still absent in Kant, which is something that then also conditions his conception of the axioms of geometry as transcendental propositions.29

Such an interpretation of intuition in terms of more fundamental conceptual processes was considered to be an advancement over Kant by most of Helmholtz’s contemporaries.30 So the question arises whether Weyl’s sort of return to intuition as a (distinct) basis for conceptual construction is a regressive view. I will argue that this is not the case in the second part of the paper by showing that Weyl’s appreciation of the phenomenological aspects of intuition offered a plausible solution to the problem of giving a more precise mathematical characterization of Helmholtz’s form of intuition. My suggestion is that the background for Weyl’s group-theoretical approach is provided by an earlier debate on the same subject. In particular, I will consider two opposed readings of Helmholtz by Klein and Schlick.

29 Helmholtz

(1921/1977, p. 143). This passage was added in the full paper “Die Tatsachen in der Wahrnehmung,” but not in the shortened version of the paper that appeared in Helmholtz (1883). 30 Arguably, this was one of the main reasons for Helmholtz’s influence on the renewal of Kant’s transcendental philosophy proposed by different directions of neo-Kantianism (Biagioli 2014a, 2016, Ch. 1–2).

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12.3 Group Theory in the Reception of Helmholtz’s Work As it emerges from Weyl’s historical excursus in Weyl (1923), by the end of the nineteenth century it was a commonplace that group theory provided a rigorous analysis of Helmholtz’s problem of space. However, Weyl did not discuss the fact that Helmholtz himself made no use of the concept of group. The mathematical analysis considered by Weyl emerged only later, in the context of Lie’s theory of continuous transformation groups. My suggestion is that, nevertheless, the debate on the origin of the grouptheoretical representation of space played an important role in Weyl’s defense of a phenomenological approach to Kant’s spatial intuition. A more comprehensive account of the relevant positions would require us to take into consideration a much wider debate from Helmholtz to Poincaré. For reasons of space, this section is restricted to the more specific question whether group theory provides a plausible interpretation of Helmholtz’s view. Klein (1898) was one of the first mathematicians to answer this question affirmatively by relying on Lie’s theory of transformation groups. Twenty-four years later, commenting on the centenary edition of Helmholtz’s epistemological writing, Schlick called into question this reading by pointing out that it remained unclear how the conceptual constructions of mathematics should represent something intuitive. Weyl avoided this difficulty by adopting the phenomenological approach.

12.3.1 Klein It is well known that Felix Klein was one of the first to envision a unified approach to geometry based on group theory in his “Vergleichende Betrachtungen über neuere geometrische Forschungen” (1872). This pamphlet was distributed during Klein’s inaugural address as newly appointed Professor at the University of Erlangen, and therefore is best known today as the “Erlangen Program.” Although the ideas of the Erlangen Program have been considered to be very influential in retrospect, there is evidence that Klein himself did not draw much attention to it until only after the development of essential requirement for the implementation of such a project by other mathematicians.31 One of this contributions was Lie’s theory of transformation group, which appeared in three volumes between 1888 and 1893. In the same years, Klein resumed his work on non-Euclidean geometry and promoted his early ideas by publishing a revised version of the Erangen Program in Mathematische Annalen (1893) and other writings and lectures on related subjects. His most detailed discussion of Helmholtz is found in Klein’s review of the third

31 The

latter aspect has been emphasized by several historical studies (see esp. Hawkins 1984; Rowe 1992). Cf. Birkhoff and Bennett (1988) for the opposing view that the Erlangen Program was very influential.

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volume of Lie’s Theorie der Transformationsgruppen, where Lie dealt with the classical problem of space.32 Klein delivered his review at the University of Kazan in 1897 when Lie was awarded the first Lobachevsky prize.33 On this occasion, Klein emphasized the broader significance of the grouptheoretical approach for the theory of measurement. Klein began with a general consideration about the numerical representation of spatial relations. Such a representation presupposes the use of axioms, which Klein here and elsewhere defined as “the postulates by which we read exact assertions into inexact intuition.”34 The need to introduce postulates corresponds to the fact that there is a lower limit to the precision of empirical measurements. Klein’s approach to the problem of space emerges from the following consideration about the upper limit: Correspondingly, when it comes to take into consideration the topologically different forms of space for the determination of the geometry of actual space, we are faced not so much with an arbitrary but with an inner consequence. Our empirical measurement has also an upper limit, which is given by the dimensions of the objects accessible to us or to our observation. What do we know about spatial relations in the infinitely large? To begin with, nothing. Therefore, we have to formulate postulates.35

The problem under consideration consists in determining the class of all surfaces in elliptic, hyperbolic, and parabolic space that are locally isometric to the Euclidean plane.36 Similar to Helmholtz before him, Klein restricted the consideration to the three classical cases of manifolds of constant curvature to then single out Euclidean geometry for reasons of convenience. The relevant aspect for Klein’s comparison with Helmholtz’s problem, however, lies in his methodological consideration about the need of introducing postulates to cope with the limits of intuitions. Similarly, Klein introduced the distinction between metrical and projective geometry “not as arbitrary or indicated by the nature of the mathematical methods, but as corresponding to the actual formation of our spatial intuition, in which mechanical experiences (concerning the movement of rigid bodies) are combined with experiences of visual space (concerning the different kind of projection of intuited objects).”37 The introduction of numbers in projective space (e.g., the construction of a numerical scale on a projective line) presupposes that the indefinite divisibility of empirical intuition is replaced by the axiom of continuity, which was implicit in Christian von Staudt’s foundation of projective geometry as an autonomous branch of geometry. As showed by Klein in Klein (1871), the numerical

32 Lie

(1893, pp. 437–471). the following, I refer to the reprinted version of Klein’s review in Mathematische Annalen (1898). 34 Klein (1890, p. 572). 35 Klein (1898, p. 595). 36 Klein’s solution to this problem is found in Klein (1890). This is now known as a distinct problem of space, which is called “Clifford-Klein” or “the problem of the form of space” (see Torretti 1978, p. 151). 37 Klein (1898, p. 593). 33 In

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representation of space opens the door to the study of a projective metric, which contains elliptic, hyperbolic, and parabolic metrical geometries as specific cases. In other words, projective metrical geometry provides a model of non-Euclidean geometry or “Klein’s model.” The classification thus obtained is generalized further to the discussion of all the possible topological forms of space in the infinitely large.38 To conclude, Klein maintained that his model of non-Euclidean geometry provides an adequate interpretation of Helmholtz’s thought experiment about free mobility in a hyperbolic space. The fundamental ideas of Helmholtz’s argument find a precise expression in projective and group-theoretical terms by saying that the measurements in our world and the corresponding measurements in the mirror belong to the larger group of collineations.39 Klein based this interpretation on the following consideration about the mathematical aspects of Helmholtz’s physiology of vision: Helmholtz was presumably far from the typical projective way of thinking (in the sense of von Staudt). It must be added that, in the years of Helmholtz’s mathematical work, projective geometry was usually considered to be a specialty area of research; the insight into its foundational meaning for every geometrical speculation was not widespread at all. Or maybe Helmholtz, as a natural scientist, was fundamentally reluctant towards the abstraction that lies at the foundation of projective geometry. In the introduction to his Göttingen notice from 1868 he distances himself from a foundation of geometry that would put forward the properties of visual space, because even the blind can acquire correct representations of space. Interestingly, this is in contrast though with the fact that Helmholtz himself is continuously led to deal with projective questions by his extensive optical investigations. He deals with these questions by auxiliary means of his own invention, but also sometimes by means of general reasoning.40

Not only did Helmholtz foreshadow the projective and group-theoretical view of geometry according to Klein, but the passage above suggests that there is some continuity between Helmholtz’s psychological considerations on spatial intuitions and the generalization to the mathematical reasoning. This example shed further light on Klein’s view of geometric knowledge as based on axioms, where axioms impose conceptual constraints on fundamentally imprecise intuitions. The plausibility of this reading notwithstanding,41 it remained unclear about the status of what Helmholtz, referring to Kant, called form of spatial intuition. If identified as imprecise intuitions, the form of intuition would lose its general character. If identified with the full-blown mathematical analysis of a projective metric, on the other hand, such a form would lose its immediacy and would coincide fundamentally with conceptual thinking. The latter option is suggested by Helmholtz’s remark about the resolution of intuition into intellectual processes and

38 Klein

(1898, p. 597). (1898, p. 599). 40 Klein (1898, p. 598). 41 Helmholtz himself drew attention to the relevance of his psychological investigations to his mathematical considerations in Helmholtz (1870/1977, p. 15). 39 Klein

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inspired several strategies for an intellectualization of intuition in neo-Kantianism. The next section deals with a fundamental objection to this tradition by Moritz Schlick.

12.3.2 Schlick Schlick distanced himself from the received interpretation of Helmholtz in his comments on the centenary edition of Helmholtz’s Schriften zur Erkenntnistheorie (1921). In particular, in what follows, I will refer to Schlick’s comments on “Die Tatsachen in der Wahrnehmung,” namely, the central work for the articulation of Helmholtz’s view of the form of intuition.42 On the one hand, Schlick and Paul Hertz, the coeditor of Helmholtz (1921), referred to Lie for a rigorous solution of the mathematical problem of space posed by Helmholtz. On the other hand, Schlick distanced himself from the philosophical interpretation of this result as providing a generalization of Kant’s form of intuition. Schlick reconsidered Helmholtz’s distinction between the general properties of the form of spatial intuition and its narrower specifications by means of geometrical axioms. Helmholtz’s examples suggest that the latter correspond to Euclidean axioms as follows.43 However, it remains unclear what properties exactly count as general and if they would admit an axiomatic formulation in the light of later mathematical developments. Schlick admitted that modern geometers (in particular Lie and Klein) tended to answer this question affirmatively, although not everyone agreed on the particular axiomatization.44 However, he advocated a different interpretation, for the following reasons. Firstly, in order to uphold Helmholtz’s distinction, “the ‘general form’ will have to be understood as the indescribable psychological component of spatiality which is imbued in sense perception.”45

42 Arguably,

Schlick commented on Helmholtz’s most philosophical papers (i.e., Helmholtz 1870, 1878a) while leaving to Paul Hertz the comments on Helmholtz’s mathematical papers (Helmholtz 1868, 1887). It might be objected that such a division obscures the connection between the philosophical and the mathematical considerations in Helmholtz’s work. In the following, I suggest that this partly depends on Schlick’s own attempt to clarify the different aspects of Helmholtz’s notion of space. 43 Helmholtz examples include such propositions as: Between two points only one straight line is possible; through any three points a plane can be placed; through any point only one line parallel to a given line is possible (von Helmholtz 1878a/1977, p. 128). 44 Schlick contrasts the received interpretation via a projective metric with Poincaré’s identification of a purely qualitative geometry as his development of “analysis situs,” which became known as “topology” (Schick in Helmholtz 1921/1977, pp. 172–173). It might be added that even more recent axiomatic interpretations of Helmholtz’s distinction differ slightly, although most interpreters identify Helmholtz’s general characterization of space as a differentiable, three-dimensional manifold of constant curvature (Cf. Torretti 1978; Lenoir 2006). 45 Schlick in Helmholtz (1921/1977, p. 172–173).

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Secondly, Schlick called into question Helmholtz’s distinction between form and matter of spatial intuition. In Schlick’s view, a more consistent naturalization of Kant’s forms of intuition would reduce them to sensuous contents. The result of such a reduction is what Schlick identified as the subjective spatial intuition or acquaintance. Insofar as the physical concept of space can be characterized mathematically, on the other hand, Schlick identified it with a formal, conceptual construction.46 As pointed out by Friedman (1997), Schlick’s reading of Helmholtz presupposes a very different scientific and philosophical context after the development of the axiomatic method in geometry, on the one hand, and general relativity, on the other. Furthermore, Schlick tends to ascribe to Helmholtz his own philosophical assumptions about causal realism, which are sometimes inconsistent with Helmholtz’s remarks on the status of scientific knowledge as restricted to the phenomena in Kant’s sense. Nevertheless, it is true that, by advocating the interpretation above, Schlick formulated a compelling objection against the received view about the form of spatial intuition. Whereas the mathematical tradition associated with Helmholtz underpinned the view that a mathematical description of what Kant called form of intuition would be made possible by the developments of nineteenth-century geometry, Schlick showed the impossibility of such a description, insofar as space had to retain both intuitive and a priori character in Kant’s sense. The following section reconsiders Weyl’s position in this debate as an original response to the same problem within a Kantian framework.

12.3.3 Weyl As in the case of Weyl’s reception of Helmholtz, it is not always possible to reconstruct what Weyl might have known directly or indirectly from the debate above. Nevertheless, it is safe to assume that Weyl was well acquainted with the Göttingen mathematical tradition, of which Klein was one of the leading figures.47 Weyl studied in Göttingen from 1904 to 1908, when he completed his doctoral theses under Hilbert’s supervision. Subsequently, Weyl habilitated in Göttingen in 1910 and was a Privatdocent there until 1913, when he was offered a professorship at the Eidgenössische Technische Hochschule in Zürich. Klein was Weyl’s teacher in Göttingen and corresponded with him even later, while Weyl was working on his relativistic analysis of the problem of space.48 Furthermore, Weyl’s extensive references to Klein in Weyl (1923) are evidence

46 Schlick

in Helmholtz (1921/1977, p. 167). the development of this tradition from Riemann to Klein and Hilbert, see Rowe (1989). For further details about Weyl’s relationship to the Göttingen community, see Sigurdsson (1994). 48 A letter dated December 28, 1920, in which Weyl informs Klein about the group-theoretical treatment of metrical space in the fourth edition of Raum-Zeit-Materie is found in Klein’s Nachlass. An extract of this letter is quoted by Scholz (2001, p. 87). 47 On

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that Weyl was well acquainted with Klein’s works on non-Euclidean geometry, even though Weyl did not provide any detailed references. At other times, Weyl’s considerations are clearly reminiscent of Klein’s views, even when there is no explicit reference. For our present discussion, for example, it is worth noting that Weyl reproduced Klein’s definition of geometrical axioms almost word for word in his own discussion about how numbers are introduced in measurement in Das Kontinuum (1918). The broader aim of this work was to provide an alternative foundation of analysis that would overcome the paradoxes of the standard, settheoretical foundation. Therefore, Weyl embraced an intuitionist approach close to Brower’s.49 Although Klein was very far from such a view, Weyl clearly relied on Klein’s concept of a projective metric when he maintained that the geometry of continuity proper can be developed only analytically, that is, by articulating analysis as a part of the pure theory of numbers in order to transfer its propositions to geometry. Weyl identified geometrical axioms as “the formulation of such transfer principles out of particular relations, which have to be regarded as immediately given.”50 As we saw in Sect. 3.1, a very similar definition of geometrical axiom is found in Klein (1890). In the same passage, Klein went on to say that “the theory of irrationals should be developed and delimited arithmetically, to be then transferred to geometry by means of axioms, and hereby enable that precision which is required for the mathematical consideration.”51 However, the imprecise spatial intuition according to Klein provides us at best with a trigger for the formulation of axioms. Therefore, he denied that intuition can suffice for establishing the existence of a mathematical object in the lack of conceptual determinations.52 By contrast, Weyl’s emphasis is on the inevitably subjective origin of what is immediately given for the formulation of axioms. Weyl developed his view in the following years by reconsidering the a priori function of spatial intuition. This does not imply that we are aware of what lies in our intuition before any geometric knowledge about space. Once the relevant axioms have been formulated, we can nonetheless ground such knowledge in what pertains to the form of intuition in opposition to its empirical contents. In other words, it does not follow from the subjectivity of spatial intuition that such a form would either coincide with its heterogeneous contents or be indescribable, as assumed by Schlick. Weyl did not discuss the aforementioned objection of Schlick against the possibility of generalizing the form of intuition in line with Helmholtz’s homogeneity requirement. Nevertheless, Weyl was certainly acquainted with Schlick’s argument against the possibility of a pure intuition in Kant’s sense, which found one

49 Weyl

distanced himself from his earlier approach in the course of his mathematical analysis of the problem of space – which was largely based on the set-theoretic account of analysis – and later, more explicitly, in Weyl (1949, p. 54), on account of the unjustified limits that Brower’s intuitionism would impose on mathematical practice. 50 Weyl (1918, p. 73). 51 Klein (1890, p. 572). 52 Ibid.

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of its clearest formulations in Schlick’s Allgemeine Erkenntnislehre: “The validity of geometrical propositions cannot be grounded in a pure intuition, for the simple reason that the space of geometry is not intuitive at all.”53 As we have mentioned in Sect. 3.2, Schlick used the same distinction between intuition or acquaintance with the spatiality of sense qualities and the physico-geometrical construction of space to rule out a mathematical interpretation of the form of intuition. Weyl wrote a review of Schlick’s work in 1923, in which he especially criticized the lack of intuition as a mediating term between the semiotic character of cognition and the mere modality of sense experience. Referring to this passage, Ryckman argued that Weyl’s motivation was to respond to Schlick’s attack against one of the fundamental concepts of Husserl’s phenomenology, that is, evidence as a source of insight.54 Weyl’s distinction between evidence and acquaintance enabled him to argue that the conceptual articulation of intuition is always possible in principle, although the historical study of the problem of space shows that in fact it took centuries and great efforts to correctly identifying the essential characteristics of space. Weyl believed to have proved such a characterization to be possible in the second chapter of Weyl (1921) on the metrical continuum. Therefore, he called this chapter a good example of analysis of essence in Husserl’s sense and concluded the analysis thus provided with the following historical consideration: The historical development of the problem of space teaches how difficult it is for us human beings entangled in external reality to reach a definite conclusion. A prolonged phase of mathematical development, the great expansion of geometry dating from Euclid to Riemann, the discovery of the physical facts of nature and their underlying laws from the time of Galilei, together with the incessant impulses imparted by new empirical data, finally the genius of individual great minds – Newton, Gauss, Riemann, Einstein – all these factors were necessary to set us free from the external, accidental, non-essential characteristics which would otherwise have been held us captive. Certainly, once the true point of view has been adopted reason becomes flooded with light, and it recognises and appreciates what is of itself intelligible to it. Nevertheless, although reason was, so to speak, always conscious of this point of view in the whole development of the problem, it had not the power to penetrate into it with one flash. This reproach must be directed at the impatience of those philosophers who believe it possible to describe adequately the mode of existence on the basis of a single act of typical presentation: in principle they are right: yet from the point of view of human nature, how utterly they are wrong!55

53 Schlick

(1918, p. 301). (2005, pp. 113–114). Weyl’s quotation, in the English translation provided by Ryckman, reads: “In Schlick’s opinion, the essence [Wesen] of the process of cognition is exhausted by [the semiotic character of cognition]. To the reviewer, it is incomprehensible how anyone, who has ever striven for insight [Einsicht], can be satisfied with this. To be sure, Schlick also speaks of ‘acquaintance’ [‘Kennen’, in opposition to cognizing, Erkennen] as the mere intuitive grasping of the given; but he says nothing of its structure, also nothing of the grounding connections between the given and the meanings giving it expression. To the extent that he ignores intuition, in so far as it ranges beyond the mere modalities of sense experience, he outrightly rejects self-evidence [die Evidenz] which is still the sole source of all insight.” 55 Weyl (1921/1952, pp. 147–148). 54 Ryckman

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Weyl elaborated on this view in Weyl (1923) by proposing a mathematical description of the form of space in group-theoretical terms in line with Klein’s reading of Helmholtz. At the same time, Weyl escaped the charge of relying on arbitrary assumptions in the distinction between the general and the special properties. The historical shift from the classical to the relativistic view reveals the true a priori component of the form of intuition, insofar as the mathematical analysis of the problem of space presents the latter view as a generalization of the previous one. In this sense, the mathematical problem is an essential development of the philosophical problem of space and offers a principled solution against Schlick’s empiricist objection.

12.4 Concluding Remarks The question remains whether Weyl offered a plausible account of Helmholtz’s view. Our previous considerations suggest that Weyl captured well Helmholtz’s emphasis on the wider scope of his homogeneity requirement when compared to Kant’s. I argued in Sect. 12.2 that the first analogy between Helmholtz and Weyl is a similar strategy for dealing with the philosophical problem of space: instead of rejecting any a priori intuition, the borderline between the a priori and the a posteriori is set somewhere else. As argued in the present section, a second analogy derives from Weyl’s reliance on the Göttingen mathematical tradition initiated by Riemann. Weyl was very clear about the wider scope of Riemann’s geometry, despite the common way to refer to the “Riemann-Helmholtz” theory of space. It remains true that nevertheless, as Weyl suggested, both Riemann and Helmholtz belong to a tradition that looked at the conceptual and hypothetical constructions of mathematics as a necessary presupposition of physical geometry.56 In line with the same tradition, which can be traced back to Gauss, Riemann and Dedekind, Klein maintained that conceptual postulates are necessary to overcome the limits of our spatial intuitions.57 Weyl departed from this tradition, insofar as he grounded the possibility of measurement in an inevitable rest of subjectivity. In this sense, he contrasted the intuitive and the mathematical continuum in 1918.58 He called into question the full-blown objectivity of physical geometry in Raum-Zeit-Materie by saying that “the objectivity of things conferred by the exclusion of the ego and its data derived directly from intuition, is not entirely satisfactory; the co-ordinate system which can 56 In

the literature, a closer connection between Riemann and Helmholtz as the proponents of an empiricist view has been emphasized by DiSalle (2008, p. 91): “The empiricist view [ . . . ] was that dynamical principles – principles involving time as well as space – could force revision of the spatial geometry that had been originally assumed in their development. We might say that this view acknowledges the possibility, at least, that space-time is more fundamental than space.” 57 See Sect. 3.2. 58 Weyl (1918, pp. 70–71).

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only be specified by an individual act (and then only approximately) remains as an inevitable residuum of this elimination of the percipient.”59 Weyl’s work provides a subtle analysis of the conceptual preconditions of measurement in the relativistic context, and therefore an important development of the classical theory of measurement. However, his fundamental assumptions clearly contradict Helmholtz’s view that physical equivalence is “a completely determinate, unambiguous objective property of spatial magnitudes,” insofar the results of measurement hold true in any frame of reference.60 In other words, in Helmholtz’s view, complete objectivity extends insofar as the laws of (Newtonian) physics apply. He acknowledged that the domain of validity of these laws cannot be fixed once and for all. Nevertheless, he argued for the possibility of a progressive extension of this domain to all known physical phenomena according to the demand of the comprehensibility of nature. By contrast, Weyl retained a foundational role of some subjective structures of cognition while restricting the domain of a priori knowledge to the highly generalized homogeneity requirement of his infinitesimal geometry. I argued that Weyl’s phenomenological standpoint enabled him to overcome Schlick’s empiricist objection against the mathematical representation of spatial intuition in the period between 1918 and 1923. A further question is whether Weyl himself reconsidered his aprioristic arguments in favor of a more empiricist approach to the problem of space in his later work on Dirac’s wave functions.61 But that is a question for another paper. Acknowledgements This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 715222). Prior to this, research leading to this paper was carried out within the project “Mathematical and Transcendental Method in Ernst Cassirer’s Philosophy of Science”, funded by the Marie Curie Actions in co-funding with the Zukunftskolleg at the University of Konstanz. I wish to thank Julien Bernard, Carlos Lobo, Silvia De Bianchi, Paola Cantù, Thomas Ryckman and Georg Schiemer for helpful comments and discussions of this material.

References Beltrami, Eugenio. 1869. Teoria fondamentale degli spazi a curvatura costante. In Opere Matematiche 1, 406–429. Milano: Hoepli. 1902. Biagioli, Francesca. 2014a. Hermann Cohen and Alois Riehl on geometrical empiricism. HOPOS: The Journal of the International Society for the History of Philosophy of Science 4: 83–105. ———. 2014b. What does it mean that ‘space can be transcendental without the axioms being so’? Helmholtz’s claim in context. Journal for General Philosophy of Science 45: 1–21. ———. 2016. Space, number, and geometry from Helmholtz to Cassirer. Cham: Springer.

59 Weyl

(1921/1952, p. 8). Helmholtz (1878a/1977, p. 158). 61 See Scholz (2005). 60 von

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Birkhoff, Garrett, and Mary Katherine Bennett. 1988. Felix Klein and his Erlanger Programm (Please reintroduce capital letters as shown in my original manuscript. This and the other instances marked below are German nouns, which are always written with capital letters, whether they occur in a title or in a sentence). In History and philosophy of modern mathematics, ed. William Aspray and Philip Kitcher, 144–176. Minneapolis: University of Minnesota Press. Boi, Luciano, Livia Giancardi, and Rossana Tazzioli, eds. 1998. La découverte de la géométrie non euclidienne sur la pseudosphère: Les lettres d’Eugenio Beltrami à Joules Hoüel; (1868–1881). Paris: Blanchard. Coleman, Robert A., and Herbert Korté. 2001. Hermann Weyl: Mathematician, physicist, philosopher. In Hermann Weyl’s Raum-Zeit-Materie and a general introduction to his scientific work, ed. Erhard Scholz, 161–386. Basel: Birkhäuser. DiSalle, Robert. 1993. Helmholtz’s empiricist philosophy of mathematics: Between laws of perception and laws of nature. In Hermann von Helmholtz and the foundations of nineteenthcentury science, ed. David Cahan, 498–521. Berkeley: University of California Press. ———. 2008. Understanding space-time: The philosophical development of physics from Newton to Einstein. Cambridge: Cambridge University Press. Erdmann, Benno. 1877. Die Axiome der Geometrie: Eine philosophische Untersuchung der Riemann-Helmholtz’schen Raumtheorie. Leipzig: Voss. Friedman, Michael. 1997. Helmholtz’s Zeichentheorie and Schlick’s Allgemeine Erkenntnislehre: Early logical empiricism and its nineteenth-century background. Philosophical Topics 25: 19– 50. Hawkins, Thomas. 1984. The Erlanger Programm of Felix Klein: Reflections on its place in the history of mathematics. Historia Mathematica 11: 442–470. Helmholtz, Hermann v. 1868. Über die Tatsachen, die der Geometrie zugrunde liegen. In Helmholtz (1921/1977), 38–55. ———. 1870. Über den Ursprung und die Bedeutung der geometrischen Axiome. In Helmholtz (1921/1977), 1–24. ———. 1878a. Die Tatsachen in der Wahrnehmung. In Helmholtz (1921/1977), 109–152. ———. 1878b. The origin and meaning of geometrical axioms. Part 2. Mind 3: 212–225. ———. 1883. Wissenschaftliche Abhandlungen. Vol. 2. Leipzig: Barth. ———. 1887. Zählen und Messen, erkenntnistheoretisch betrachtet. In Helmholtz (1921/1977), 70–97. ———. 1921/1977. Schriften zur Erkenntnistheorie, ed. Paul Hertz and Moritz Schlick. Berlin: Springer. English edition: Helmholtz, Hermann von. 1977. Epistemological writings, Trans. Lowe, Malcom F., ed. Robert S. Cohen and Yehuda Elkana. Dordrecht: Reidel. Hyder, David. 2009. The determinate world: Kant and Helmholtz on the physical meaning of geometry. Berlin: De Gruyter. Klein, Felix. 1871. Über die sogenannte Nicht-Euklidische Geometrie. Mathematische Annalen 4: 573–625. ———. 1872. Vergleichende Betrachtungen über neuere geometrische Forschungen. Erlangen: Deichert. ———. 1890. Zur Nicht-Euklidischen Geometrie. Mathematische Annalen 37: 544–572. ———. 1898. Gutachten, betreffend den dritten Band der Theorie der Transformationsgruppen von S. Lie anlässlich der ersten Vertheilung des Lobatschewsky-Preises. Mathematische Annalen 50: 583–600. Königsberger, Leo. 1902–1903. Hermann von Helmholtz, vols. 3. Braunschweig: Vieweg. Land, Jan Pieter Nicolaas. 1877. Kant’s space and modern mathematics. Mind 2: 38–46. Lenoir, Timothy. 2006. Operationalizing Kant: Manifolds, models, and mathematics in Helmholtz’s theories of perception. In The Kantian legacy in nineteenth-century science, ed. Michael Friedman and Alfred Nordmann, 141–210. Cambridge, MA: The MIT Press. Lie, Sophus. 1893. Theorie der Transformationsgruppen. Vol. 3. Leipzig: Teubner. Riemann, Bernhard. 1854/1867. “Über die Hypothesen, welche der Geometrie zu Grunde liegen.” Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen 13: 133–152.

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Rowe, David E. 1989. Klein, Hilbert, and the Göttingen mathematical tradition. Osiris 5: 186–213. ———. 1992. Klein, Lie, and the Erlanger Programm. In 1830–1930: A century of geometry, epistemology, history and mathematics, ed. Luciano Boi, Dominique Flament, and Jean- Michel Salanskis, 45–54. Berlin: Springer. Ryckman, Thomas. 2005. The reign of relativity: Philosophy in physics, 1915–1925. New York: Oxford University Press. Schlick, Moritz. 1918. Allgemeine Erkenntnislehre. Berlin: Springer. Scholz, Erhard. 2001. Weyl’s Infinitesimalgeometrie, 1917–1925. In Hermann Weyl’s Raum-ZeitMaterie and a general introduction to his scientific work, ed. Erhard Scholz, 48–104. Basel: Birkäuser. ———. 2004. Hermann Weyls analysis of the problem of space and the origin of gauge structures. Science in Context 17: 165–197. ———. 2005. Local spinor structures in V. Fock’s and H. Weyl’s work on the Dirac equation (1929). In Géométrie au XXe siècle, 1930–2000: histoire et horizons, ed. Joseph Kouneiher, Philippe Nabonnand, and Jean-Jacques Szczeciniarz, 284–301. Montréal: Presses internationales Polytechnique. ———. 2013. The problem of space in the light of relativity: The views of H. Weyl and E. Cartan. E-Print: https://arxiv.org/abs/1310.7334, accessed on March 8, 2019) Sigurdsson, Skuli. 1994. Unification, geometry and ambivalence: Hilbert, Weyl and the Göttingen community. In Trends in the historiography of science, ed. Kostas Gavroglu, Jean Christianidis, and Efthymios Nicolaidis, 355–367. Dordrecht: Springer. Torretti, Roberto. 1978. Philosophy of geometry from Riemann to Poincaré. Dordrecht: Reidel. Weyl, Hermann. 1918. Das Kontinuum: Kritische Untersuchungen über die Grundlagen der Analysis. Leipzig: Veit. ———. 1921/1952. Raum-Zeit-Materie: Vorlesungen über allgemeine Relativitätstheorie, 4th ed. Berlin: Springer. English edition: Weyl, Hermann. Space-time-matter. Trad. Henry L. Brose. New York: Dover, 1952. ———. 1923. Mathematische Analyse des Raumproblems. Berlin: Springer. ———. 1949. Philosophy of mathematics and natural science. Princeton: University Press. Trad. Helmer, Olaf. An expanded English version of Philosophie der Mathematik und Naturwissenschaft, München, Leibniz Verlag, 1927. ———. 2015. L’analyse mathématique du problème de l’espace, ed. Éric Audureau and Julien Bernard. Aix-en-Provence: Presses universitaires de Provence.

Part IV

Weyl’s Methodological Issues: Intuition, Symbolic Thought and Manifolds of Possibilities

Chapter 13

Espace et variété de possibilités chez Hermann Weyl Benoît Timmermans

En quoi la découverte de Hermann Weyl sur les conditions de représentation des groupes de Lie complexes semi-simples a-t-elle infléchi, modifié ou peut-être fixé sa philosophie des mathématiques, voire sa conception du monde et de la façon de le penser ? La question peut paraître étrange, et même doublement. D’abord parce que le problème de l’espace auquel est consacré ce volume, problème qui a absorbé Weyl principalement entre 1918 et 1923, n’est pas directement lié à sa découverte sur la représentation des groupes de Lie. Ensuite, et surtout, parce que Weyl n’invoque pratiquement jamais explicitement cette découverte pour justifier sa philosophie des mathématiques ou du monde. Cependant le moment où advient cette découverte1 – que Weyl considère comme « l’une des plus extraordinaires que l’on puisse faire en mathématiques »2 – fait immédiatement suite à celui où il parvient à résoudre mathématiquement le problème de l’espace, et appartient à la période où apparaît pour la première fois dans son œuvre une description assez précise, en trois étapes, du processus de la connaissance « constructive » ou « mathématique ». Cette

1 1924–1926 si l’on s’en tient à la publication des trois articles fondamentaux sur ce sujet : « Theorie

der Darstellung kontinuierlicher halbeinfacher Gruppen durch lineare Transformationen. I-III », Mathematische Zeitschrift, vol. 23, 1925, p. 271–301 [reçu le 21 janvier 1925]; vol. 24, 1926, p. 328–376 [reçu le 11 février 1925]; vol. 24, 1926, p. 377–395 [reçu le 23 avril 1925]. 2 Lettre du 12 janvier 1925 au Président de l’ETHZ, citée par Thomas Hawkins, Emergence of the theory of Lie groups, New York – Berlin, Springer, 2000, p. 456. B. Timmermans () Faculté de Philosophie et Sciences sociales, Université Libre de Bruxelles (ULB), Bruxelles, Belgique Fonds National de la Recherche Scientifique, Bruxelles, Belgique e-mail: [email protected] © Springer Nature Switzerland AG 2019 J. Bernard, C. Lobo (eds.), Weyl and the Problem of Space, Studies in History and Philosophy of Science 49, https://doi.org/10.1007/978-3-030-11527-2_13

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description, Weyl va s’y fixer, s’y tenir de manière remarquablement ferme et stable jusqu’à la fin de sa vie. Brièvement dit, le processus de « connaissance constructive », associé parfois à l’approche arithmétique de l’infini, se décompose en trois étapes (parfois quatre, j’y reviendrai). La première est une opération de liaison entre différentes données. La deuxième est l’expression symbolique des liaisons entre ces données. La troisième étape, dans laquelle Weyl aperçoit « la marque distinctive de la science théorique »3 , est celle sur laquelle il insiste le plus, revient régulièrement, et qui retiendra toute notre attention. Les symboles sont projetés sur un arrière-plan de possibles (Hintergrund des Möglichen) caractérisé aussi comme « variété de possibilités » (Mannigfaltigkeit von Möglichkeiten) générée par un « procédé fixe » (festem Verfahren) et « ouverte à l’infini » (ins Unendliche offenen). Jusqu’à la fin de sa vie Weyl continue donc de décrire la connaissance « constructive » sous la forme d’un processus en trois étapes et souligne en particulier l’importance de la troisième. J’ai compté, sans viser l’exhaustivité, cinq occurrences de la description en trois ou quatre étapes (1927, 1929, 1931, 1934, 19534 ) auxquelles s’ajoutent neuf évocations 3 « Über den Symbolismus der Mathematik und mathematischen Physik », Studium Generale, vol. 6, 1953, p. 225, trad. fr. in H. Weyl, Le Continu, trad. Jean Largeault, Paris, Vrin, 1994, p. 255. 4 • 1927 : Philosophie der Mathematik und Naturwissenschaft, München, R. Oldenbourg, 1927, p. 30; [Philosophy of Mathematics and Natural Science, revised and augmented english edition based on a translation by Olaf Helmer, Princeton, Princeton University Press, 1949, p. 37:] « Schon bei der Zahl treten uns also die folgenden Grundzüge des konstruktiven Erkennens entgegen: 1. Das Resultat gewisser geistiger Operationen am Gegebenen, die für allgemein ausführbar gelten, wird, sofern es durch das Gegebene eindeutig bestimmt ist, als ein dem Gegebenen an sich zukommendes Merkmal aufgestellt (...). 2. Durch Einführung von Zeichen wird eine Aufspaltung der Urteile vollzogen und ein Teil der Operationen durch Vershiebung auf die Zeichen vom Gegebenen und seinem Fortbestand unabhängig gemacht. (...) 3. Sie werden in ihrem aktuellen Vorkommen nicht einzeln herausgehoben, sondern auf den Hintergrund einer nach festem Verfahren herstellbaren geordneten, ins Unendliche offenen Mannigfaltigkeit von Möglichkeiten projiziert. [3. Characters are not individually exhibited as they actually occur, but their symbols are projected on the background of an ordered manifold of possibilities which can be generated by a fixed process and is open into infinity.] » • 1929 : « Consistency in Mathematics », The Rice Institute Pamphlet, vol. 16, 1929, p. 247–8, trad. fr. in H. Weyl, Le Continu, trad. Jean Largeault, op. cit., p. 166–167 : « En développant l’arithmétique il y aurait peut-être à y distinguer quatre étages en ce qui concerne le rôle joué par l’infini. Au premier appartient un jugement concret particulier, tel 2 + 3 = 5. Au second, par exemple le jugement de généralité hypothétique ( . . . ). Au troisième étage, les signes numériques qui figurent sont plongés dans la suite de tous les nombres possibles ( . . . ). Ici l’existant est projeté sur le fond du possible, sur le fond d’une multiplicité de possibilités qui est produite et ordonnée suivant un processus fixé mais qui est ouverte dans l’infini (Here the existent is projected into the backgound of a manifold of possibilities which is produced and ordered according to a fixed process but is open into infinity). ( . . . ) Je crois que nous touchons là le fond de la méthode mathématique en général ( . . . ). En quatrième lieu, et c’est là que commence, d’après Brouwer, la faute des mathématiques, la théorie des ensembles a déclaré que la suite des nombres naturels, laquelle est ouverte dans l’infini, est un agrégat clos et achevé d’éléments en soi. » • 1931: « Die Stufen des Unendlichen », Vortrag gehalten am 27 Oktober 1930 bei der Eröffnung der Gästetagung der Mathematischen Gesellschaft an der Universität Iena, Iena,

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de la seule troisième étape (Hintergrund des Möglichen ou manifold of possibilities) (1924, 1926, 1927, 1932, 1940, 1949, 1949, 1952, 19535 ). Ceci dément la légende d’un esprit « changeant », « Protée qui se transforme sans cesse pour se dérober aux

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•

•

Gustav Fischer, 1931, p. 3–4, trad. fr. in H. Weyl, Le Continu, trad. Jean Largeault, Paris, Vrin, 1994, p. 294–295 : « Dans le développement de l’arithmétique on distingue touchant le rôle de l’infini quatre degrés. Au premier appartient un jugement concret individuel comme 2 < 3 ( . . . ). Au second degré appartiennent par exemple l’idée de a. ( . . . ) Du tout nouveau arrive cependant avec le troisième degré, lorsque je plonge les chiffres qui se présentent en acte, dans la suite de tous les nombres possibles, suite qui naît par un processus génératif conforme au principe que d’un nombre présent n peut toujours être engendré par un nouveau nombre, le successeur immédiat n’ de n. Là l’étant est projeté sur l’arrière-plan du possible, d’une multiplicité de possibles, ordonnée d’après un procédé fixe et ouverte dans l’infini (Hier wird das Seiende projiziert auf den Hintergrund des Möglichen, einer nach festem Verfahren herstellbaren geordneten, ins Unendliche offenen Mannigfaltigkeit von Möglichkeiten) ( . . . ) [Dans le] quatrième degré de l’arithmétique ( . . . ) le possible est erronément transformé en un être absolu ( . . . ).» 1934 : Mind and Nature, Philadelphia, University of Pennsylvania Press, reed. in Mind and Nature. Selected Writings on Philosophy, Mathematics, and Physics, edited and with an introduction by Peter Pesic, Princeton – Oxford, Princeton University Press, 2009, p. 118– 119 : « I hope you will understand, if I now describe the essential features of constructive cognition as follows : 1. Upon that which is given, certain reactions are performed by which the given is in general brought together with other elements capable of being varied arbitrarily. If the results to be read from these reactions are found to be independent of the variable auxiliary elements they are then introduced as attributes inherent in the things themselves (...). 2. By the introduction of symbols, the judgments are split up ; and a part of the manipulations is made independent of the given and its duration by being shifted on to the representing symbols which are time resisting (...). 3. Symbols are not produced simply ‘according to demand’ wherever they correspond to actual occurrences, but they are embedded into an ordered manifold of possibilities created by free construction and open towards infinity. » 1953 : « Über den Symbolismus der Mathematik und mathematischen Physik », Studium Generale, vol. 6, 1953, p. 226–227, trad. fr. in H. Weyl, Le Continu, trad. Jean Largeault, op. cit., p. 256–257 : « Instruits par l’exemple des nombres naturels, nous pouvons réunir les linéaments fondamentaux de la connaissance symbolique-constructive, qui domine la science entière, et les décrire ainsi ( . . . ) : α) Le résultat de certaines opérations appliquées au donné ( . . . ) est un caractère qui appartient au donné ( . . . ). β) L’introduction de signes permet de décomposer des jugements ( . . . ). γ) Les signes ne sont pas individuellement mis pour quelque chose de donné en acte chaque fois ; ils sont tirés de la réserve potentielle d’une multiplicité ordonnée de signes ouverte à l’infini, et produite selon un procédé fixe. (Die Zeichen werden nicht einzeln für das jeweils aktuell Gegebene hergestellt, sondern sie werden dem potentiellen Vorrat einer nach festem Verfahren herstellbaren, geordneten, ins Unendliche offnen Mannigfaltigkeit von Zeichen entnommen.) » 1924: « Was ist Materie ? Zwei Aufsätze zur Naturphilosophie », Naturwissenschaften, vol. 12, 1924, p. 81: « Dass wir das Wirkliche zum Zwecke seiner theoretische Beschreibung auf den Hintergrund des Möglichen stellen müssen (des Raumzeitkontinuums mit seiner Feldstruktur). » 1926 : « Die heutige Erkenntnislage in der Mathematik », Symposion (Berlin), vol. 1, 1925– 1927, p. 23, trad. fr. in H. Weyl, Le Continu, trad. Jean Largeault, op. cit., p. 154 : « la mathématique représente en fin de compte l’impossibilité où nous sommes de tracer une

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prises de l’adversaire » selon les termes d’Alain Chevalley et André Weil6 . Il semble que, sur le plan en tout cas de la conception de la connaissance « constructive », ou « mathématique », un noyau d’invariance se met en place à partir de 1926 et se maintient jusqu’à la fin. La question ici posée est donc en quoi ce noyau, cette description « philosophique » de la connaissance constructive est-elle tributaire de la découverte mathématique de Weyl concernant la représentation des groupes de Lie ? Mais aussi en quoi se rattache-t-elle éventuellement à sa conception de l’espace ? Ou encore, de façon plus générale et peut-être plus engagée, qu’est-ce que cette image théorique de l’être, sauf sur le fond du possible (ein theoretisches Bild des Seins nuf auf dem Hintergrund des Möglichen) ». • 1927 : Philosophie der Mathematik und Naturwissenschaft, op. cit., p. 30 : « Hier wird das Seinde projiziert auf den Hintergrund des Möglichen, einer nach festem Verfahren herstellbaren geordneten, wenn auch ins Unendliche offenen Mannigfaltigkeit von Möglichkeiten. » Ibid., p. 94 : « In der Doppelnatur des Wirklichen ist es gegründet, dass wir ein theoretisches Bild des Seienden nur entwerfen können auf dem Hintergrund des Möglichen. » (English ed., op. cit., p. 131 : « The dual nature of reality accounts for the fact that we cannot design a theoretical image of being except upon the background of the possible. ») • 1932 : The Open World. Three lectures on the metaphysical implications of science, Woodbridge, Ox Bow Press, 1989, p. 69 : « ( . . . ) I want now to discuss the attempts to convert the field of possibilities that is open to infinity into a closed domain of absolute existence. » • 1940, « The mathematical way of thinking », Science, vol. 92, 1940, p. 441, trad. fr. in H. Weyl, Le Continu, trad. Jean Largeault, op. cit., p. 218 : « Comme je l’ai déjà dit maintes fois, cela projette l’être sur le fond du possible, plus précisément sur une multiplicité de possibles (manifold of possibilities) qui se déploie par itération et est ouverte sur l’infini. » • 1949 : Philosophy of Mathematics and Natural Science, op. cit., Appendix A: The structure of mathematics, p. 234: « For any attempt to get at the real source of the antinomies Russell’s analysis ( . . . ) the deepest root of the trouble lies elsewhere : a field of possibilities open into infinity has been mistaken for a closed realm of things existing in themselves. » • 1949 : Address at the Princeton Bicentennial Conference, in Mind and Nature. Selected Writings on Philosophy, Mathematics, and Physics, op. cit., p. 185: « The sequence, naturally, is never completed; rather it is a field of possibilities open into infinity. Thus Being is projected onto the background of the Possible, or, more precisely, into an ordered manifold of possibilities producible according to a fixed procedure and open towards infinity. » • 1952 : Symmetry, Princeton, Princeton University Press, 1952, trad. fr. Symmétrie et mathématique moderne, trad. fr. G. Th Guilbaud, Paris, Flammarion, 1964, p. 106 : « ( . . . ) notre affirmation qu’il n’existe pas plus que 17 groupes ornementaux différents demande quelque explication. ( . . . ) Le parallélogramme fondamental, construit sur les deux vecteurs de base du réseau, peut avoir n’importe qu’elle forme et n’importe quelles dimensions ; nous avons le choix parmi un ensemble infini continu de possibilités (continuously infinite manifold of possibilities). Or, pour aboutir au nombre 17, nous comptons toutes ces possibilités pour un cas seulement. » • 1953 : « Über den Symbolismus der Mathematik und mathematischen Physik », Studium Generale, vol. 6, 1953, p. 225, trad. fr. in H. Weyl, Le Continu, trad. Jean Largeault, op. cit., p. 255 : « Ici le réel est projeté sur l’arrière-plan du possible (Hintergrund des Möglichen), lequel est une multiplicité librement créée par l’esprit suivant une procédure fixe et ouverte sur l’infini. ( . . . ) Je vois précisément dans la projection du réel contingent sur un possible obtenu à priori par un procédé de construction, la marque distinctive de la science théorique. » 6 Claude Chevalley et André Weil, « Hermann Weyl (1885–1955) », L’enseignement mathématique, vol. 3, 1957, repris dans H. Weyl, Gesammelte Abhandlungen, éd. K. Chandrasekharan, Berlin – New York, Springer, 1968, vol. 4, p. 659.

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description de la connaissance mathématique en trois étapes a de si important, de si exigeant et sans doute de si enthousiasmant pour que Weyl y ait tant tenu ? Le chemin qui a mené Weyl à sa découverte sur la représentation des groupes de Lie complexes semi-simples a été bien étudié notamment par Thomas Hawkins7 , Christophe Eckes8 et Renaud Chorlay9 . On sait que c’est un faisceau d’éléments qui y a concouru. Aux travaux de Weyl sur les surface de Riemann (191310 ), sur la manière dont elles peuvent se connecter et sur leur lien à la théorie des groupes de Galois (191611 ) s’ajoutent les recherches d’Élie Cartan (1894, 191312 ) sur les algèbres de Lie, mais aussi l’intérêt marqué de Weyl pour la physique relativiste en général13 et plus spécifiquement sa réflexion, en écho aux remarques d’Eduard Study à ce sujet14 , concernant la possibilité de fonder le calcul tensoriel à partir de la théorie algébrique des invariants (192415 ). L’analyse mathématique du problème de l’espace, qui requiert la théorie des groupes de Lie et algèbres de Lie linéaires (192316 ), marque également une étape, sans oublier l’article de Schur (192417 ) sur les représentations du groupe orthogonal et du groupe spécial orthogonal. Par-delà cette diversité on pourrait se demander s’il n’y a pas une intuition générale qui unifierait en quelque sorte les différentes perspectives et rendrait sensible une sorte de continuité entre, disons, les travaux géométriques de Weyl axés sur l’analyse infinitésimale et ses travaux algébriques orientés vers la saisie ou la représentation des invariants de groupes. Dès la première édition de Espace-temps-matière (1918) Weyl est à la recherche d’un principe qui 7 Op.

cit.

8 Christophe

Eckes et Amaury Thuillier, Les groupes de lie dans l’œuvre de Hermann Weyl : Traduction et commentaire de l’article « Théorie de la représentation des groupes continus semi-simples par des transformations linéaires » (1925–1926), Nancy, Editions universitaires de Lorraine, 2014. 9 Renaud Chorlay, « Passer au global : le cas d’Élie Cartan, 1922–1930 », Revue d’histoire des mathématiques, vol. 15, 2009, p. 231–316. 10 H. Weyl, Die Idee der Riemannschen Fläche, Leipzig, B.G. Teubner, 1913. 11 H. Weyl, « Strenge Begründung der Charakteristikentheorie auf zweiseitigen Flächen », Jahresbericht der Deutschen Mathematikervereinigung, vol. 25, 1916, p. 265–278. 12 E. Cartan, Sur la structure des groupes finis et continus, Thèse, Paris, 1894 ; « Les groupes projectifs qui ne laissent invariante aucune multiplicité plane », Bulletin des sciences mathématiques, vol. 41, 1913, p. 53–96. 13 H. Weyl, « Relativity theory as a stimulus in mathematical research », Proceedings, American Philosophical Society, vol. 93, 1949, p. 535–541. 14 E. Study, Einleitung in die Theorie der Invarianten linearer Transformationen auf Grund der Vektorenrechnung, Vieweg, Branunschweig, 1923. 15 H. Weyl, « Randbemerkungen zu Hauptproblemen der Mathematik », Mathematische Zeitschrift, vol. 20, 1924, p. 131–150; « Über die Symmetrie der Tensoren und die Tragweite der symbolischen Methode in der Invariantentheorie », Rendiconti del Circolo Matematico di Palermo, vol. 48, 1924, p. 29–36. 16 H. Weyl, Mathematische Analyse des Raumproblems : Vorlesungen, gehalten in Barcelona und Madrid, Berlin, 1923. 17 Issai Schur, « Neue Anwendungen der Integralrechnung auf Probleme der Invariantentheorie », Sitzungsberichte der Akademie der Wissenschaften zu Berlin, 1924, p. 189–208.

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rende possible la comparaison de deux éléments linéaires à des distances autres qu’infiniment voisines. À partir, semble-t-il, de 1924 il se demande dans quelle mesure les propriétés d’invariance révélées par une algèbre de Lie concernent aussi la totalité des transformations du groupe correspondant, ou sous quelles conditions une représentation infinitésimale, locale, peut-elle constituer une représentation irréductible d’un groupe considéré dans sa globalité ? La continuité de l’intuition de recherche qui semble s’exprimer ici serait, pour reprendre une formule qui apparaîtra plus tard dans le livre sur Les groupes classiques, leurs invariants et leurs représentations (1939), « the ever-recurring topological device by which one links events in the small and in the large »18 . Comment, donc, relier le petit au grand, on le local au global ? Cette question, cette intuition qui semble perdurer dans différents travaux de Weyl justifie peut-être une philosophie d’ensemble, une conception unitaire de l’acte de connaître ou de construire en mathématiques. Mais justement, une telle intuition n’est pas particulièrement visible, centrale, dans la description en trois étapes qui se fixe à partir de 1926. Peut-être précisément parce que la question qui oriente les recherches propres de Weyl n’embrasse pas, n’inclut pas toute démarche théorique ou mathématique en général. Qu’est-ce que la description de la connaissance mathématique en trois étapes apporte donc de neuf dans ces conditions ? Je distinguerai trois traits caractéristiques, trois aspects singuliers de cette nouvelle conception.

13.1 Le glissement de la question de l’homogénéité vers celle de la possibilité L’homogénéité joue un rôle clé dans les quatre éditions de Espace-temps-matière (1918, 1919, 1921, 1923) et encore dans l’Analyse du problème de l’espace (1923). Julien Bernard19 a bien montré que, tout en occupant une position centrale, cette notion est aussi transversale et en devenir dans la mesure où elle s’avère être relative à certains critères, propriétés, relations. Par exemple les corps ou objets peuvent être dits homogènes si on peut les déplacer d’un lieu quelconque à un autre sans rien changer à leurs relations de grandeurs. Cette homogénéité-là, relative à la propriété très « forte » de conservation des relations métriques, n’est vérifiée que dans un espace à courbure constante. À l’autre extrémité du spectre, l’homogénéité peut être relative à la propriété beaucoup plus faible de tridimensionnalité. Les objets sont alors « homogènes » simplement s’ils conservent leurs trois dimensions. L’homogénéité caractérise ici l’espace « en soi », conçu comme « variété tridi-

18 H.

Weyl, The Classical Groups, their Invariants and Representations, Princeton, Princeton University Press, 1939, p. 259. 19 Julien Bernard, L’idéalisme dans l’infinitésimal. Weyl et l’espace à l’époque de la relativité, Paris, Presses universitaires de Paris Ouest, 2013.

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mensionnelle sans forme »20 . Quelque part entre ces deux extrêmes Weyl dégage un nouveau type d’homogénéité relative, cette fois, à la manière dont les objets ou vecteurs conservent leurs relations métriques infinitésimales sous l’action de groupes de rotations opérant localement autour de points. Ce qui intéresse Weyl, c’est la manière dont les espaces tangents à ces points sont connectés entre eux, c’est-à-dire modifient la grandeur et la direction du vecteur. On a donc affaire ici à une homogénéité définie d’abord localement, qu’on cherche ensuite à étendre à l’espace dans sa globalité. Comme il a été dit, on pourrait considérer que c’est d’une certaine manière la poursuite de cette intuition qui a amené Weyl à sa découverte sur la représentation des groupes de Lie. Mais force est de constater que, en 1926, Weyl ne place plus au centre de son vocabulaire ou de sa réflexion la question de l’homogénéité, que ce soit pour caractériser sa découverte sur les groupes de Lie ou pour évoquer le processus de connaissance en général. Le début de la Philosophie des mathématiques et des sciences de la nature (1927) contraste bien l’homogénéité de l’espace et la nature spéciale ou caractéristique des nombres (les nombres sont « singuliers », « individuels », les points ne le sont pas21 ), mais pour faire voir, une vingtaine de pages plus loin, que, par-delà ce contraste ou cette opposition, toute connaissance constructive, qu’elle concerne l’espace homogène ou les nombres hétérogènes, consiste à « projeter les symboles sur un arrière-fond de possibles, une variété de possibilités générée par un procédé fixe et ouverte vers l’infini»22 . À bien le considérer, ce passage de la question de l’homogénéité vers celle des possibles n’apparaît pas comme un basculement ou une rupture. La transition est plus unie et souple. Commençons par dire ce qu’elle ne signifie pas. On n’assiste pas ici à un renversement des priorités entre ce que Weyl appelle « l’abstraction originaire » et « l’abstraction mathématique », c’est-à-dire entre l’acte d’isoler une possible propriété des choses (par exemple la couleur) et l’acte de poser l’égalité ou la similitude entre ces choses (ces fleurs ont la même couleur, ou ces entiers appartiennent à la même classe de congruence). La deuxième section de la Philosophie des mathématiques et des sciences de la nature (consacrée à la définition constructive mathématique) rappelle que, d’un point de vue mathématique, l’acte d’isoler est subordonné à celui de poser l’égalité ou la similitude23 . Le déplacement de la question de l’homogénéité vers celle des possibles n’implique pas non plus, semble-t-il, un recentrage massif des problèmes autour du discret plutôt que du continu. Certains passages pourraient certes le laisser penser, comme lorsque Weyl remarque que « chaque continu a son propre schème arithmétique »

20 H.

Weyl, Raum-Zeit-Materie, Berlin, Springer, 1919, p. 87. Weyl, Philosophie der Mathematik und Naturwissenschaft, op. cit., p. 7 (Philosophy of Mathematics and Natural Science, op. cit. , p. 7–8). 22 H. Weyl, Philosophie der Mathematik und Naturwissenschaft, op. cit., p. 30 (Philosophy of Mathematics and Natural Science, op. cit. , p. 37). 23 H. Weyl, Philosophie der Mathematik und Naturwissenschaft, op. cit., p. 10 (Philosophy of Mathematics and Natural Science, op. cit. , p. 11). 21 H.

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lié au « squelette » ou à la « charpente » topologique (topologische Gerüst) de la variété qui détermine la manière dont « ses pièces élémentaires se bordent l’une l’autre »24 . Mais comme le processus de connaissance ici visé concerne aussi bien la géométrie que l’arithmétique25 , il semble que le mouvement qui se dessine doive plutôt s’interpréter comme un élargissement, une généralisation de la perspective. Ce qui est dit de la connaissance constructive doit valoir aussi bien pour les ensembles discrets que pour les ensembles continus, aussi bien pour ceux où la métrique est déjà donnée que pour ceux où elle ne l’est pas, et que pour ceux où cette métrique varie. Plus spécifiquement, les deux concepts clés autour desquels s’articule l’élargissement sont déjà partiellement présents dans l’évocation du problème de l’espace. Le premier est d’origine riemannienne : c’est le concept de variété. Le second est d’origine leibnizienne : c’est celui de la combinatoire des possibles. On peut aller vite sur le concept de variété. L’idée très générale est que la « multiplicité » (Mannigfaltigkeit), le champ ou « l’arrière-fond » (Hintergrund) qui intéresse le mathématicien peut bien être un espace continu, mais ne l’est pas nécessairement. Ce peut être un ensemble discret (par exemple de nombres entiers) déjà muni d’une métrique, comme le signalait déjà Riemann dans son mémoire sur Les hypothèses qui servent de fondement à la géométrie (1868) en un passage cité aussi bien dans Espace-temps-matière (1919) que dans la Philosophie des mathématiques et des sciences de la nature (1927) : « Dans une variété discrète, le principe des rapports métriques est déjà contenu dans le concept de cette variété, tandis que, dans une variété continue, ce principe doit venir d’ailleurs. Il faut donc ou que la réalité sur laquelle est fondée l’espace forme une variété discrète, ou que le fondement des rapports métriques soit cherché en dehors de lui, dans les forces de liaison qui agissent en lui. »26

Avec le concept de variété le glissement de l’homogène vers le possible apparaît donc comme très progressif. Un passage de Qu’est-ce que la matière (1924) marque le moment où, pour la première fois, la variété continue d’espace-temps est caractérisée comme « arrière-plan du possible » (Hintergrund des Möglichen) : « Pour obtenir une description théorique du réel (das Wirkliche), nous devons le poser sur l’arrière-plan du possible (auf den Hintergrund des Möglichen) [qui est ici] le continuum d’espace-temps muni de sa structure de champ »27 .

24 H.

Weyl, Philosophie der Mathematik und Naturwissenschaft, op. cit., p. 63 (Philosophy of Mathematics and Natural Science, op. cit. , p. 90). 25 H. Weyl, Philosophie der Mathematik und Naturwissenschaft, op. cit., p. 30 (Philosophy of Mathematics and Natural Science, op. cit. , p. 36–37). 26 Bernhard Riemann, « Über die Hypothesen, welche der Geometrie zu Grunde liegen », Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen, vol. 13, 1867, p. 297. Cité par H. Weyl dans Raum-Zeit-Materie [1919], op. cit., p. 87 et dans Philosophie der Mathematik und Naturwissenschaft, op. cit., p. 35. 27 H. Weyl, Was ist Materie[1924], op. cit., p. 81

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Or ce passage nous conduit tout naturellement vers l’idée d’une combinatoire des possibles dans la mesure où il réapparaît sous une forme analogue dans la Philosophie des mathématiques et des sciences de la nature (1927) mais immédiatement suivi, cette fois, d’un commentaire renvoyant à Leibniz : « C’est pourquoi Leibniz appelle ‘espace abstrait’ l’ordre de toutes les positions supposées être possibles, et ajoute que en conséquence c’est quelque chose d’idéal »28 .

Weyl renvoie ici à la cinquième lettre de Leibniz à Clarke (1715–171629 ). D’autres textes de Leibniz expriment sans doute encore plus clairement cette idée de rassembler les ordres du discret et du continu sous un même registre de « possibilités idéales ». Par exemple la Réplique au dictionnaire critique de Bayle (1702) : « L’étendue ( . . . ), le continu ( . . . ) ne sont que des choses idéales ; c’est-à-dire qui expriment les possibilités, tout comme font les nombres ( . . . ), ces ordres cadrent non seulement avec ce qui est actuellement, mais encore à ce qui pourrait être mis à la place, comme les nombres sont indifférents à tout ce qui peut être énuméré (res numerata)»30 .

Peut-être pourrait-on, en prolongeant encore la piste leibnizienne, interpréter le glissement de la question de l’homogène vers celle des possibles comme un glissement du « continu » vers le « qualitatif » ou, plus précisément, de « l’homogène » vers « l’homogone » pour parler comme Leibniz. Le « qualitatif » désigne pour Leibniz ce qui peut être connu dans les choses quand elles sont observées isolément, sans requérir de comprésence 31 . Or il peut arriver que, à la limite, une entité qualitativement distincte bascule dans une autre catégorie, ce qui revient à l’inclure dans un système plus vaste de notions ou entités possibles. Ainsi deux entités peuvent ne pas être homogènes entre elles, c’est-à-dire ne pas pouvoir « être rendues similaires entre elles par le moyen de transformations », et néanmoins « venir l’une à l’autre par un changement continu » 32 : elles sont alors appelées « homogones ». Par exemple « un terme n’est pas homogène à ce qu’il termine, ni une section à ce qu’elle coupe »33 mais l’un et l’autre, comme l’instant et le temps ou le point et l’espace, peuvent être homogones. S’il est vrai que Weyl ne fait pas usage de

28 H.

Weyl, Philosophie der Mathematik und Naturwissenschaft, op. cit., p. 94 (Philosophy of Mathematics and Natural Science, op. cit. , p. 131). 29 Cinquième écrit de Mr Leibniz, ou réponse à la quatrième réplique de Mr Clarke, § 104 (Leibniz, Opera philosophica quae exstant latina, gallica, germanica omnia, Johann Eduard Erdmann éd., Scientia Verlag Aalen, 1975, p. 775) : « Je ne dis point que l’espace est un ordre ou une situation qui rend les choses situables ( . . . ) mais un ordre des situations, ou selon lequel les situations sont rangées, et que l’espace abstrait est cet ordre des situations, conçues comme possibles. Ainsi c’est quelque chose d’idéal. » 30 Réplique aux réflexions contenues dans la seconde édition du dictionnaire critique de Mr Bayle, article Rorarius sur le système de l’harmonie préétablie [1702] (Leibniz, Opera philosophica, op. cit., p. 189). 31 Leibniz, Initia rerum mathematicarum metaphysica, in Mathematische Schriften (C.I. Gerhardt ed.), Hildesheim, Olms, 1962, vol. 7, p. 19. 32 Ibid., p. 20 : « dum unum in alterum cntinua mutatione abire potest ». 33 Ibid.

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cette distinction leibnizienne entre homogène et homogone dont il est peu probable, au demeurant, qu’il ait eu connaissance, il ne fait pas de doute, en revanche, que la philosophie de Leibniz représente pour lui le sol même dans lequel a germé cette nouvelle mathématique des possibles. La théorie des groupes, qui est peutêtre, écrit-il, « le concept le plus caractéristique du XIXe siècle », « appartient au cadre de l’ars combinatoria de Leibniz »34 . Ainsi le glissement de la question de l’homogène vers celle des possibles renvoie comme en écho à l’œuvre de Leibniz dont on sait que Weyl s’est véritablement imprégné entre l’été 1925 et le début de l’année 192635 . À ce stade, cependant, il faut bien reconnaître que le procédé qui consiste à dégager les représentations irréductibles d’un groupe de Lie semi-simple complexe ne semble pas jouer un grand rôle. C’est pourquoi un deuxième aspect de la description en trois étapes de la connaissance mathématique peut à présent retenir notre attention.

13.2 La notion de « procédé fixe » générant une « variété ordonnée de possibilités » De quelle nature est ce procédé ? Que nous apprend-il sur notre façon de connaître le monde ? Weyl propose régulièrement deux exemples pour l’illustrer. Le plus fréquent est le procédé de construction de la suite des entiers naturels par addition réitérée de l’unité. Qualifié de « fondement ultime de la pensée mathématique » en 1918 dans Le continu36 , il est par la suite rétrogradé au rang de simple exemple, certes parmi les plus courants, du processus d’ordonnancement des possibles37 . L’autre exemple souvent mobilisé est le procédé de localisation de points ou d’événements dans un espace ou une variété. Dans le style de pensée mathématique (1940) Weyl décrit précisément ce procédé de décomposition d’une variété en simplexes permettant de « localiser ( . . . ) tous les ici-maintenant possibles »38 : par une « suite de schémas dérivés »39 on parvient, écrit-il, à « saisir n’importe quel point », bref à « décrire la variété »40 considérée ici comme « champ de possibilités

34 H.

Weyl, Philosophie der Mathematik und Naturwissenschaft, op. cit., p. 23 (Philosophy of Mathematics and Natural Science, op. cit. , p. 27–28). 35 Erhard Scholz, « Leibnizian Traces in H. Weyl’s Philosophie der Mathematik und Naturwissenschaft », in Science and Philosophy of Science 1800–2000. New Essays in Leibniz Reception (R. Krömer and Y.C.-Drian eds.), Basel, Springer, 2012, p. 203–216; Herbert Breger, « Leibniz, Weyl und das Kontinuum », Studia Leibnitiana Suppl., vol. 26, 1986, p. 316–330. 36 H. Weyl, Le Continu, trad. Jean Largeault, op. cit., p. 83. 37 H. Weyl, « Consistency in Mathematics » [1929], op. cit., trad. fr. in H. Weyl, Le Continu, trad. Jean Largeault, op. cit., p. 167 ; « The mathematical way of thinking »[1940], op. cit., trad. fr. in H. Weyl, Le Continu, trad. Jean Largeault, op. cit., p. 218. 38 Ibid., p. 224. 39 Ibid., p. 223, 222. 40 Ibid., p. 225.

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que nous pouvons dominer à priori parce que nous construisons ces possibilités à priori de façon purement combinatoire à partir d’un peu de matériel symbolique »41 . D’autres passages plus ténus ou allusifs suggèrent que ce sont aussi des procédés de ce type (procédés fixés a priori de mise en ordre des possibles) qui entrent en jeu aussi bien dans la résolution mathématique du problème de l’espace que dans la représentation des groupes de Lie. Ainsi le fait que la métrique fondamentale d’une variété riemannienne à quatre dimensions possède une seule dimension négative (non pas deux, ni trois, ni zéro) détermine la structure causale du monde réel ou encore participe, écrit Weyl en 1926, d’un « schème d’ordonnancement des choses réelles » (Ordnungschema der wirklichen Dinge) dans la mesure où il détermine la structure causale du monde réel. Ce schème d’ordonnancement, poursuit-il, « intervient comme un composant intégral de la construction théorique du monde »42 . La notion de procédé est également évoquée dans le cadre de la représentation des groupes de Lie. En 1928 le grand livre-synthèse sur la Théorie des groupes et la mécanique quantique propose, sans doute en écho au processus

de Cayley utilisé en théorie des invariants comme opérateur différentiel agissant sur le groupe linéaire général43 , de nommer « processus gamma » (Γ Prozess) une « méthode générale » pour obtenir une représentation nouvelle d’un sous-groupe à partir de celle du groupe plus large qui le contient44 . Autre exemple, tiré cette fois du cours de Princeton sur Les groupes classiques, leurs invariants et leurs représentations (1939), après avoir décrit l’opération consistant à faire la moyenne des valeurs mesurables dans un certain espace sous l’action d’un groupe fini, Weyl précise : « mais ce que nous avons vraiment en tête est d’appliquer ce procédé de moyenne (averaging process) à un groupe compact continu plutôt qu’à un groupe fini »45 . L’objectif de cet article n’est pas de recenser tous les passages où la représentation des groupes de Lie serait décrite ou approchée comme un procédé de mise en ordre des possibles, mais plutôt de demander qu’est-ce qu’une telle description nous apprend sur notre façon de connaître « mathématiquement » ou « théoriquement » le monde. Il semble que le type même de « procédé fixe », et ce que Weyl nous en dit, plaident cette fois en faveur d’un véritable déplacement, une redéfinition voire un bouleversement des lignes de frontières entre diverses catégories philosophiques et mathématiques qui sont ici l’a priori et l’a posteriori, l’abstrait et le concret, l’algébrique et le transcendant, le transcendantal et l’empirique. Partons de l’a priori. En de multiples endroits Weyl insiste sur le fait que le « procédé fixe », ou « l’arrière-fond de possibles » que ce procédé ordonne, est

41 Ibid.,

p. 224. Weyl, Philosophie der Mathematik und Naturwissenschaft, op. cit., p. 97 (Philosophy of Mathematics and Natural Science, op. cit. , p. 134. 43 Arthur Cayley, « On linear transformations », Cambridge and Dublin mathematical journal, vol. 1, 1846, p. 104–122. 44 H. Weyl, Gruppentheorie und Quantenmechanik, Leipzig, Hirzel, 1928, p. 118. 45 H. Weyl, The Classical Groups, their Invariants and Representations, op. cit., p. 187. 42 H.

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« obtenu a priori ». Encore faut-il entendre ce que signifie ce terme sous la plume de Weyl. L’« a priori » ne signifie jamais, en ces passages, ce qui appartient à la forme pure de notre subjectivité, mais n’exclut pas non plus complètement la notion d’ego (ou de « Je-esprit » comme l’écrit parfois Weyl) dans la mesure où le champ des possibles peut aussi être fixé par « les possibles états objectifs d’un ego percevant »46 . « le déploiement de la construction mathématique appliquée à la réalité repose en fin de compte sur la double nature, subjective et objective, de la réalité : elle n’est pas un être en soi, mais un apparaître pour un Je-esprit. »47 Ainsi la frontière, la délimitation du sens de l’a priori ne passe pas par la distinction sujet-objet mais plutôt par la distinction entre descriptif et manipulatoire : « Le passage de la description a posteriori du donné en acte à la construction a priori du possible ( . . . ) [s’opère parce que ce dernier est] ordonné non pas sur la base de caractéristiques descriptives, mais sur la base de manipulations et de réactions intellectuelles ou physiques effectuées sur ce donné –par exemple du processus de compter »48 .

De même à propos du troisième mode d’approche de l’infini mathématique : « Nous touchons là le fond de la méthode mathématique en général : la construction a priori du possible, par contraste à la description à posteriori du donné en acte. »49

L’a posteriori est descriptif, l’a priori est constructif et « manipulatoire », y compris au sens physique du terme. D’où ce paradoxe: l’a priori est d’une certaine manière plus « réel » que le monde donné. Le monde réel n’est « pas seulement ce qui est donné actuellement en ce moment, mais aussi les perceptions possibles ( . . . ) d’un ego »50 . Ou, comme le dit encore joliment Weyl, « la construction théorique naît du souci de compléter le donné dans l’intérêt de la totalité »51 . Il en résulte que « l’a priori » est, par certains aspects, très concret. Il y a certes quelque chose de très concret dans le fait de compter des objets ou de réitérer une opération visant à localiser un point, mais aussi dans le fait de chercher à représenter un groupe : « La notion générale de groupe est sortie par abstraction de celle de groupe de transformation ( . . . ) [Mais] on doit aussi, à partir d’un schéma abstrait, retomber sur les groupes concrets de transformations. La réalisation ou représentation d’un groupe abstrait consiste

46 H.

Weyl, Philosophie der Mathematik und Naturwissenschaft, op. cit., p. 85 (Philosophy of Mathematics and Natural Science, op. cit. , p. 119). 47 H. Weyl, « Die Stufen des Unendlichen »[1931], op. cit., trad. fr. in H. Weyl, Le Continu, op. cit., p. 294. 48 Ibid., p. 295. 49 H. Weyl, « Consistency in Mathematics »[1929], op. cit., trad. fr. in H. Weyl, Le Continu, op. cit., p. 167. 50 H. Weyl, Philosophie der Mathematik und Naturwissenschaft, op. cit., p. 85 (Philosophy of Mathematics and Natural Science, op. cit. , p. 119). 51 H. Weyl, Philosophie der Mathematik und Naturwissenschaft, op. cit., p. 88 (Philosophy of Mathematics and Natural Science, op. cit. , p. 122).

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en ceci qu’à chacun de ses éléments s on fait correspondre, dans l’espace des variables, une transformation E. »52

Toutefois ce retour au concret est plus complexe qu’il n’y paraît. Dans une modification apportée à l’édition anglaise (1948) de la Philosophie des mathématiques et des sciences de la nature, Weyl note que si un groupe de transformation 0 53 « est considéré comme la représentation d’un groupe abstrait, alors l’accent doit être mis plus sur les traits distinctifs de la structure de ce groupe abstrait que sur sa représentation spéciale concrète 0 »54 . Autrement dit, il y a bien dans la représentation d’un groupe un retour au concret, mais ce concret doit être « abstraitement intéressant ». Ce qu’il y a d’intéressant dans la représentation du groupe 0 n’est pas sa forme spécifique en elle-même mais la structure qu’elle permet d’élucider. Par là on voit que ce ne sont pas seulement les distinctions entre l’a priori et l’a posteriori ni entre l’abstrait et le concret qui sont à revisiter mais aussi celles entre l’algébrique (au sens de la théorie générale des groupes abstraits) et le transcendant (au sens de l’analyse infinitésimale, qui peut paraître plus « concrète »). Weyl s’est souvent montré plus proche des méthodes transcendantes et topologiques que des méthodes algébriques. « Je n’ai jamais réussi à assimiler complètement la manière algébrique abstraite de raisonner, écrit-il en 1939, et éprouve constamment le besoin de traduire chaque étape sous une forme analytique plus concrète. »55 Mais il n’ignore pas non plus combien ses propres travaux sur la représentation des groupes ont contribué à répandre et populariser l’outil « algébrique abstrait » notamment dans le cadre de la mécanique quantique qui, à partir de 1928, affronte l’irruption de « la peste des groupes »56 dont il fut l’un des principaux vecteurs57 . Dans ce contexte, la tension entre l’algébrique et le transcendant devient presque anecdotique. Comment ne pas percevoir la tranquille ironie de Weyl observant, dans le même texte de 1939, que « le serpent est entré dans le paradis »58 parce que les techniques classiques d’intégration par rapport à un même élément de volume ne peuvent pas s’étendre si facilement de la représentation de n’importe quelle algèbre de Lie à celle de son groupe correspondant. C’est précisément cette entrée du serpent dans le paradis qui rend nécessaire le procédé

52 H.

Weyl, « Sur la représentation des groupes continus », L’enseignement mathématique, vol. 26, 1927, p.76. 53 Ici le groupe des « rotations euclidiennes », correspondant au groupe spécial orthogonal tridimensionnel dans R. 54 H. Weyl, Philosophy of Mathematics and Natural Science, op. cit. , p. 137. 55 H. Weyl, « Invariants », Duke Mathematical Journal, vol. 5, 1939, repris dans H. Weyl, Gesammelte Abhandlungen, éd. K. Chandrasekharan, Berlin – New York, Springer, 1968, vol. 3, p. 682. 56 Lettre de septembre 1928 de Paul Ehrenfest à Wolfgang Pauli, citée par Erhard Scholz, « The introduction of groups into quantum theory », Historia Mathematica, vol. 33, 2006, p. 441. 57 Voir la préface de H. Weyl à la seconde édition de son ouvrage Gruppentheorie und Quantenmechanik, Leipzig, Hirzel, 1931. 58 H. Weyl, « Invariants »[1939], op. cit., p. 676.

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de représentation mis au point par Weyl59 . Dans ces conditions, ce dernier ne semble pas tellement désolé que « l’ange de la topologie et le démon de l’algèbre »60 soient désormais voués à cohabiter et à se battre ensemble. Enfin la redéfinition, le déplacement des lignes de frontières atteint également le couple empirique – transcendantal. L’empirique, dans sa plus large part, appartient bien à la « description a posteriori du donné en acte »61 . Mais il semble jouer aussi un rôle dans la construction mathématique même en tant que celle-ci peut, comme on l’a vu, prendre une forme concrète, manipulatoire, parfois même physique. Le transcendantal, quant à lui, est exclu dans la mesure où ce qui est à explorer n’est pas les formes pures de la subjectivité. Mais, d’un autre côté, ce que le procédé construit ou exhibe, précisément parce qu’il y met de l’ordre, s’apparente bien aux conditions de possibilité a priori de résolution d’un problème. Par exemple les conditions de représentations irréductibles d’un groupe de Lie semi-simple complexe peuvent être saisies abstraitement et a priori, à condition de ne pas aller les chercher dans les formes pures d’une sensibilité mais dans le concret de la représentation (algébrique ou topologique) du « champ de possibilités » ouvert par la variété à laquelle est associé le groupe. Ceci nous conduit à un troisième trait caractéristique de la connaissance constructive selon Weyl.

13.3 Ouverture de la variété sur l’infini ? À plusieurs reprises Weyl insiste avec une vigueur, une intensité particulière sur l’ouverture de la variété (Mannigfaltigkeit ins Unendliche offenen ; manifold open into infinity). Dans la Philosophie des mathématiques et des sciences de la nature la phrase qui suit immédiatement la présentation de la connaissance constructive en trois étapes est en fait un avertissement : si quelqu’un transforme la variété ouverte vers l’infini en un agrégat clos d’objets existants en eux-mêmes, il opère un « saut dans l’au-delà »62 consistant à conférer une existence absolue à des objets qui « ne sont pas de ce monde »63 . Par exemple, la croyance en l’existence de l’infinie totalité des nombres conduit, explique Weyl dans les Degrés de l’infini, à une quatrième étape, un quatrième degré (ici de l’arithmétique) qui, cette fois, prend la forme d’un « pas dangereux »64 . Weyl a des mots très durs pour ceux qui sont passés 59 La

réduction du groupe complexe G à ses transformations unitaires G(U) et la mise en évidence du groupe fini de permutations des racines de l’algèbres de Lie associée au revêtement universel de G(U). 60 H. Weyl, « Invariants »[1939], op. cit., p. 681. 61 H. Weyl, « Die Stufen des Unendlichen »[1931], op. cit., trad. fr. in H. Weyl, Le Continu, op. cit., p. 294.Ibid., p. 295. 62 Sprung ins Jenseits, leap into the beyond : H. Weyl, Philosophie der Mathematik und Naturwissenschaft, op. cit., p. 31 (Philosophy of Mathematics and Natural Science, op. cit. , p. 38). 63 Ibid. 64 H. Weyl, « Die Stufen des Unendlichen »[1931], op. cit., trad. fr. in H. Weyl, Le Continu, op. cit., p. 296.

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dans cette « autre sphère »65 et ont transformé « erronément le possible en un être transcendant et absolu » : ce sont des « métaphysiciens » animés par la « haine de la liberté »66 . Le seul auteur cité est Hegel, partisan d’un « univers conceptuel dialectique pétrifié »67 . Mais le nom de « Hegel » semble plutôt utilisé ici comme un épouvantail ou un costume dont on cherche à affubler l’adversaire avec d’autant plus d’ironie que Hilbert, partisan des infinis cantoriens, accusait lui-même Weyl dans les années 1919–1920 de pratiquer une « physique hégélienne »68 . Méfiance, donc, vis-à-vis des agrégats clos, mais en même temps la question qui se pose immédiatement est de savoir pourquoi Weyl insiste tant sur l’ouverture de la variété alors qu’une grande partie de son travail de mathématicien consiste justement, très souvent, à établir ou vérifier la fermeture des variétés auxquelles il a affaire. Pour prendre un exemple classique qu’il affectionne dans ses présentations didactiques, la localisation de points dans un continuum se fait souvent, remarque-til, sur une droite. « Mais je préfère considérer un continu unidimensionnel fermé, le cercle »69 . Cela permet en effet de diviser le cercle en arcs, ceux-ci en demi-arcs, et de progresser ainsi dans l’affinement de la localisation de nombre ou de points sans craindre de tomber dans des partitions indéfinies puisqu’on s’assure ici qu’on peut diviser la variété, la partitionner en un nombre fini de simplexes. Dans le domaine plus spécialisé de la représentation des groupes de Lie, l’article de 1927 sur la représentation des groupes continus résumant les résultats nouveaux obtenus par Weyl dans ce domaine avertit d’emblée que cette synthèse ne vaut que « pour les groupes continus dont les éléments forment une variété close »70 . En mécanique quantique, un autre article de 1929 sur la symétrie sphérique des atomes montre que le calcul du moment cinétique de l’électron ne peut se faire « indépendamment de la structure dynamique du système physique considéré » que parce que le groupe des rotations virtuelles de cet électron autour de son centre « construit une variété close, comme les points à la surface d’une sphère »71 . 65 H.

Weyl, « Consistency in Mathematics »[1929], op. cit., trad. fr. in H. Weyl, Le Continu, trad. Jean Largeault, op. cit., p. 167. 66 H. Weyl, « Die Stufen des Unendlichen »[1931], op. cit., trad. fr. in H. Weyl, Le Continu, op. cit., p. 295. 67 Ibid. 68 D. Hilbert, Natur und mathematisches Erkennen. Vorlesungen, gehalten 1919–1920 in Göttingen, Basel, Birkhäuser, 1992, p. 100, cité par Erhard Scholz, « The changing concept of matter in H. Weyl’s thought, 1918–1930 », in The Interaction between Mathematics, Physics and Philosophy from 1850 to 1940, J. Lützen (éd.), Dordrecht, Kluwer, 2006, p. 281–305. 69 H. Weyl, « The mathematical way of thinking »[1940], op. cit., trad. fr. in H. Weyl, Le Continu, trad. Jean Largeault, op. cit., p. 220. 70 « Pour les groupes ouverts, le problème fondamental ne consiste pas à établir les circonstances compliquées qui, faisant échec au théorème de la réductibilité complète, viennent remplacer les lois simples et claires valables pour les groupes clos, mais bien de chercher à sauver ces lois par des restrictions appropriées apportées à la notion de fonction. » (H. Weyl, « Sur la représentation des groupes continus », op. cit., p. 89). 71 H. Weyl, « The spherical symmetry of atoms », The Rice Institute Pamphlet, vol. 16, 1929, repris dans H. Weyl, Gesammelte Abhandlungen, éd. K. Chandrasekharan, Berlin – New York, Springer, 1968, vol. 3, p. 274.)

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Comment comprendre, donc, cet apparent paradoxe : une variété, un champ de possibilités dont on cherche le plus souvent à établir la clôture pour des raisons d’efficacité mathématique mais qui, en même temps, doit absolument rester « ouvert » dans l’ultime étape de la connaissance constructive ? Plusieurs pistes d’interprétation peuvent être suggérées. Dans l’article de 1929 sur la symétrie sphérique des atomes Weyl remarque qu’à côté des rotations virtuelles de l’électron dans l’espace, ses déplacements actuels dans le temps (relativement à l’espace des phases de Hilbert) obéissent à un tout autre régime : « Parce que ces déplacements constituent un groupe ouvert, les différentes représentations unidimensionnelles possibles ne peuvent pas être réduites a priori à un ensemble discret défini »72 . Il existe en effet, poursuit-il, « une distinction essentielle entre les deux cas : la rotation dans l’espace est un processus virtuel, le déplacement dans le temps décrit ce qui arrive réellement. Et cela a pour conséquence que la loi d’addition de l’énergie ne tient que quand les systèmes demeurent dynamiquement, aussi bien que cinétiquement, indépendants, c’est-àdire tant qu’ils n’interagissent pas entre eux. »73 Le temps, par ailleurs inséparable de l’espace, semble émerger ici comme le facteur, le paramètre par lequel la variété (correspondant dans ce cas au groupe des déplacements réels de l’électron) tend à rester ouverte, à empêcher ses représentations irréductibles ou ses recouvrements finis. L’importance, constamment réitérée par Weyl, de la variété ouverte serait donc liée à l’importance du temps ou du devenir dans la connaissance constructive. Celleci doit en effet permettre d’approcher non pas seulement ce qui est mais ce qui arrive réellement : « Le ‘il y a’ nous attache à l’être et à la loi, le ‘chaque’ nous lance dans le devenir et dans la liberté. »74 Le mot « liberté » qui apparaît ici suggère peut-être une autre piste. Comme il existe plusieurs ordres ou partitions possibles pour découper, structurer les ensembles ou les variétés sur lesquels nous travaillons, on pourrait dire que nous sommes « libres » de choisir ou imaginer l’une de ces partitions. « La mathématique n’est pas le schéma rigide et rigidifiant que le profane croit si volontiers ; elle est plutôt ce carrefour de sujétion et de liberté où résiderait la condition même de l’homme »75 , écrit Weyl dès 1926. La mathématique, notons-le bien, est un carrefour, non un pur espace de libre créativité. La liberté n’efface pas la nécessité, l’ouverture ne se substitue pas à l’ordre. Nous avons certes toujours la liberté d’envisager d’autres ordres possibles, mais la construction de ces possibles est ellemême soumise à certaines contraintes, certaines lois de cohérence. D’une certaine

72 Ibid.,

p. 275. p. 276–277. 74 « Die heutige Erkenntnislage in der Mathematik »[1926], op. cit., trad. fr. in H. Weyl, Le Continu, op. cit., p. 152. 75 Ibid., p. 154. 73 Ibid.,

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manière ceci nous ramène au problème de l’espace. L’analyse mathématique du problème de l’espace (1923) mettait bien en évidence deux principes : principe de liberté (la structure géométrique doit autoriser la plus large gamme possible de distribution de matière dans l’espace) et principe de cohérence (chaque transfert congruent ne doit avoir qu’une seule connexion affine équivalente), principes que Weyl comparait, en 1923, à la liberté d’agir des citoyens pour le premier, et au fait que cette liberté ne doit pas entraver le bien-être général pour le second76 . Ainsi l’expression « variété ordonnée de possibilités ( . . . ) ouverte à l’infini » serait porteuse d’une tension déjà perceptible dans les années précédant la découverte sur la représentation des groupes de Lie. Tension entre l’ordonnancement de la variété, et son ouverture à l’infini, comme le marquent les prépositions « wenn auch » (wenn auch ins unendliche offenen) et « yet » (yet open towards infinity)77 . Ce que cette tension exprimerait, c’est que comprendre, résoudre, ordonner et clore la variété n’est pas tout. Le processus de connaissance constructive ne se résume pas à cela. « Nous, mathématiciens, ne sommes pas un Ku Klux Klan avec un rituel secret de la pensée ( . . . ), on ne s’attendra pas que je donne du style de pensée mathématique une description plus claire que celle qu’on peut donner du style de vie démocratique. »78 La connaissance constructive n’est pas (seulement) une méthode de structuration des possibles ou de résolution des problèmes. À la fin du mois d’octobre 1954, Weyl donne à la Columbia University une conférence sur « l’unité de la connaissance ». Il y distingue non plus trois, mais quatre étapes de la connaissance. La troisième étape (succédant à 1◦ l’intuition et 2◦ l’expression et la compréhension), intitulée « penser le possible », rappelle bien la projection sur l’arrière-plan des possibles dont il a été question. Mais la quatrième, celle qui achève la séquence, n’évoque plus, cette fois, le « pas dangereux » qui caractérisait le geste de clôture des métaphysiciens. Il ne suffit pas de laisser s’étendre sous nos yeux le domaine de toutes les spécifications possibles. Il faut encore fixer, choisir ou parfois deviner celle qui, parmi ces spécifications, sera privilégiée. Cela peut se faire par la construction de symboles, formules ou dispositifs de mesure si l’on est scientifique mais aussi, écrit Weyl, par une « puissance herméneutique d’interprétation qui en définitive naît d’une conscience et d’une connaissance intérieure de soi »79 . Bien sûr, cette puissance-là n’est nullement réservée aux sciences « exactes » ou mathématiques. C’est, écrit-il,

76 H.

Weyl, Mathematische Analyse des Raumproblems, op. cit., p. 46. Weyl, Philosophie der Mathematik und Naturwissenschaft, op. cit., p. 30 (Philosophy of Mathematics and Natural Science, op. cit. , p. 37). 78 H. Weyl, « The mathematical way of thinking »[1940], op. cit., trad. fr. in H. Weyl, Le Continu, trad. Jean Largeault, op. cit., p. 212. 79 H. Weyl, « The unity of knowledge »[1954], Address at the Columbia University Bicentennial celebration, reed. in Mind and Nature. Selected Writings on Philosophy, Mathematics, and Physics, op. cit., p. 202. 77 H.

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la puissance de l’historien capable de se connecter avec son objet, puissance d’entrer en sympathie « non pas émotionnelle mais intellectuelle et imaginative »80 avec ce qu’il étudie. Une puissance qui, peut-être, fait écho à cette réflexion de Weyl sur Descartes en 1932 : « Si la conviction de Descartes en la liberté n’est pas illusoire, c’est-à-dire, si le royaume de l’être considéré par rapport à ce qui le détermine n’est pas clos, mais ouvert, alors cette ouverture doit également se manifester dans la nature et dans sa science. »81

80 Ibid. 81 H.

Weyl, The Open World [1932], op. cit., p. 53.

Chapter 14

Husserl and Weyl on the Constitution of Space The Role of Symbolic Knowledge Jairo José da Silva

Abstract I discuss in this paper Husserl and Weyl’s views on the role of intuition and symbolization in empirical science and their reactions to purely symbolic extensions of mathematical representations of empirical reality. Although both accept that physical space as considered in mathematical physics, for instance, is an intentional construct out of perceptual space that eliminates subjective content of perceptual experience in favor of objective form, thus transforming space as perceived in an empty mathematical manifold, they differ as to the freedom allowed to mathematics to further elaborate this and other mathematical representatives of perceptual reality. Husserl puts serious restrictions to non-eliminable, non-denoting symbolic extensions of representing manifolds, which Weyl, in his more holistic approach, is willing to accept.

The 1952 Nobel Prize winner Felix Bloch (Bloch 1976) reported the following conversation with Werner Heisenberg, of whom he had been the first doctorate student: “We were on a walk and somehow began to talk about space. I had just read Weyl’s book Space, Time and Matter, and under its influence was proud to declare that space was simply the field of linear operations. ‘Nonsense’, said Heisenberg, ‘space is blue and birds fly through it’”. Blue with birds flying through it is, of course, how space appears to our immediate perceptual experience, for space can only be perceived through the things that are in it, with all their qualities. However, the space of perception is not yet a mathematical manifold, or not to the extent that it must be to serve the purposes of the mathematical science of the empirical world. To become one, it must be somehow acted upon. Sensorial qualities are the first things to be left behind in the process of mathematizing perceptual space. Bodies, however, remain, if only as pure extensions, maintaining with one another “spatial” relations; they occupy different

J. J. daSilva () State University of São Paulo, UNESP, São Paulo, Brazil © Springer Nature Switzerland AG 2019 J. Bernard, C. Lobo (eds.), Weyl and the Problem of Space, Studies in History and Philosophy of Science 49, https://doi.org/10.1007/978-3-030-11527-2_14

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relative positions with respect to one another, and to us, are bigger or smaller in comparison to each other, etc. Reducing physical bodies to their extension is the work of abstraction, formal abstraction, more specifically. The field of linear operations, on the other hand, is a structured mathematical domain of abstract “points”, shadows of departed entities, in which abstract operations are defined obeying certain formal properties. It is a skeleton, a formal mold that can be “filled” by no matter which material content. In particular, perceptual space, but not without a serious amount of cutting (abstraction) and polishing (idealization). In order to receive a mathematical structuring, perceptual space must be first reduced to its abstract form, that which remains when material content is removed leaving behind only extensions in relation, and then decomposed into “points”, atoms of corporeal forms to which nothing corresponds in perception. Atomizing space into mathematical “points” is a further intentional act. Abstracted of its material content and idealized as a domain of mathematical points, space becomes a mathematical manifold. Kant believed that this manifold was intrinsically structured mathematically. For him, the structure of space was a priori and Euclidean. We no longer hold that view, but we have not abandoned Kant completely. To say with him that space is the form of external experience is to say nothing beyond that our experience of bodies in the empirical world external to us is necessarily the experience of bodies in space. This is a far cry from saying that the structure of abstract idealized perceptual space has a mathematical structure intrinsic to it independently of what goes on in it. The amorphous idealized abstraction that mathematical physics calls space, or better, physical space, does not come with a mathematical structure, or at least not completely (in particular, the metric of space is not determined a priori). As we believe today, surpassing Kant, the structure of physical space depends on what is and happens in it. This structure has many layers, topological, affine, projective, conformal and metric; some, we may argue, may be a priori (and the a priori can be either logical-conceptual, i.e. analytic, or material, i.e. synthetic)1 and some a posteriori. Since mathematics is the science of abstract structures in general, instantiated in some material domain or posited by fiat (i.e. defined by axioms), that is, since mathematics is a chapter of formal ontology, the structure of physical space is of interest to mathematics. Any structure, any conceivable way of structuring whatever domain of entities, intuitively given or only emptily intended, is in principle an object of mathematical interest. The mathematical theory of perceptual space, in particular, has no concern for the material content of spatial perception, only the abstract idealized structure of space as perceived matters. By stripping perceptual space of its material content, (formal) abstraction reduces it to its pure form. Abstract perceptual space, however, is not a manifold of ideal points, but a proper continuum, whose parts and parts of parts are also continua. There are lines, surfaces and bodies in abstract perceptual space, but not the 1-dimensional lines, the 2-dimensional surfaces and the 3-dimensional bodies of the exactification (or idealization) of it we call

1 For

the notion of material a priori see da Silva 2016.

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mathematical-physical space. By idealizing possible positions in perceptual space as collections of extensionless points, we do violence to perception; points do not correspond to anything in perceptual space for nothing can exist therein that does not occupy space. Points are mathematical limits of sequences of nested spatial regions shrinking beyond any finite limit. Idealization consists precisely in “taking the limit” of such sequences. However, by reducing the form of perceptual space to a manifold of points where relations can be defined which capture in idealized form the spatial structure of perception one allows mathematics to come in with full force. Now, since for mathematics only structure matters, and since the same structure can be instantiated in materially different domains, there is, from a mathematical perspective, no reason why one should confine oneself to the domain of points, we can move to any other that have the same structure. Descartes and Fermat saw this clearly: we can instantiate the geometrical structure of space in numerical domains and carry out geometrical investigations by analytical methods. In other words, instead of synthetic we can do analytic geometry. Only now linear operations can come in and one can characterize mathematical-physical space in terms of groups of linear operations. As we see, Bloch simply ignored the extent to which perceptual space must be intentionally elaborated before it can be characterized in mathematical terms. Heisenberg reminded him of that. To use an expression dear to Weyl, perceptual space must be reconstructed in symbols so mathematics can have anything to do with it. However, no symbolic reconstruction is possible before a series of intentional acts take place. Now, once mathematical physics has a symbolic reconstruction of perceptual space in its hands, two questions impose themselves naturally: (1) can mathematical-physical space, the mathematical surrogate of perceptual space be, for methodological purposes, further enriched with mathematical structures that do not necessarily represent, via abstraction and idealization, anything experienceable in perceptual space? (2) Can such enrichment be epistemologically justified? How? Mathematical-physical space is not essentially different from other cases of symbolic representation of experience and the problem can be immediately generalized: if a symbolic domain represents in idealized form abstract aspects of a perceptual domain, are we logically and epistemologically justified in extending the symbolic domain with “empty”, non-representing symbols? I will present here two very different answers to these questions, Weyl and Husserl’s.

14.1 The Intuitive and the Symbolic For Weyl, “all knowledge, while it starts with intuitive description, tends towards symbolic construction” (Weyl 1963, 75). Although subscribing to Husserl’s “principle of all principles” – knowledge must necessarily be founded on intuition – Weyl detaches himself from the radical thesis that knowledge must be confined to intuition. Intuition is the beginning but not the end of knowledge, which tends to symbolization as a requirement of objectivity. In order to leave the subjective domain of intuitions and gain the objective realm and become a communal posses-

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sion knowledge must rely on language. The problem, however, is that language cannot, by itself, independently of extra-linguistic means (such as pointing, for example) fix the reference of its terms.2 By and in itself all that language can express are formal aspects of intuition, capable of being filled by no matter which material content. Those who want to confine knowledge to the intuitive are, then, right to mistrust language as a means for conveying knowledge; language, by itself, can only grasp the formal aspects of experience. Let us examine this more closely. Weyl’s commitment to Husserl’s brand of transcendental idealism comes out clearly in a well-known line in The Continuum (Husserl 1994, Chap. 2 §6): “Being is and can only be given as the intentional content of the mental experiences of a pure I that bestows meaning (Dasein nur gegeben ist und gegeben sein kann als intentionaler Inhalt der Bewusstseinserlebnisse eines reinem, sinngebenden Ich)”. In Space-Time-Matter he says (Weyl 1952, 4): “the real world, and every one of its constituents with their accompanying characteristics, are, and can only be given as, intentional objects of acts of consciousness”. Sensorial perception is the most fundamental form of intentional experience; as phenomenologically understood, percepts are not simply given but constituted out of the hyletic material of sensations by the action of pre-categorial, pre-reflexive, pre-logical intentional acts. Perceptions, as Weyl says, “are not composed of mere stuff of sensation, as many positivists assert. In perception there is indeed an object standing there before to which sensations relate”. Consider the following metaphysical theses: (1) perception gives the subject access to an objective world existing independently of him, (2) different subjects can perceive the objectively and independently existing world in essentially the same way, (3) subjects can communicate their perceptions about the world completely, semantic content and logical form, by means of a common language, whose semantic domain, the world, is available to anyone via perception, and (4) on the basis of successful communication, a community of subjects can come to general agreement on what in perception effectively correspond to a transcendent reality (by eliminating cases of misperception, illusion or deception). If all these theses were true, there would be no insurmountable gap between the private realm of perception and the objective realm of science, the world out there. According to these theses, by telling someone “there is a red apple hanging from that tree over there” I am not simply communicating the content of my personal experience, but also and more importantly, if I am not mistaken, deceitful or given to hallucinations, a fact of the world. From the metaphysical perspective embodied in these theses, perception gives the subject and his fellow men access to reality itself ; perception is the presentation of reality out there to consciousness. In normal circumstances, no man of the pre-scientific life-world would doubt that an honest account of a competent perceptual experience does indeed open a gate to objective reality. The idealist philosopher, however, particularly the transcendental idealist philosopher that Weyl was cannot subscribe to these theses. As Weyl says,

2 Provided,

of course, pointing happens in a context where it is interpreted as denoting.

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“philosophical reflection probably begins in every one of us [ . . . ] when he first becomes skeptical about the world-view of naive realism” (Weyl 1952, p. 3). For the idealist, who under the influence of the epoché suspends his commitment to the thesis of reality of the world that appears in perception, the perceived world is only a phenomenon, an intentional object for the I.3 Objectivity is constituted from, and hence must be confronted with subjective experience. As Weyl says in Philosophy of Mathematics and Natural Science (Weyl 1953, 71–2): “our knowledge stands under the norm of objectivity”, but “objectivity is an issue decidable on the ground of experience only”. The question then presents itself: how can an objective, communal world be constituted? How can the I, the individual I, whose world does not extend beyond his experiences, which by the action of epoché are not supposed to merely represent an independently existing objective reality, can go beyond himself and constitute an objectively valid reality? At this point Weyl turns his back, or believes to be so doing, to the Husserlian Ego, which he believes to be irredeemably confined to his own interiority, embracing Fichte’s dialectics (by then, Husserl had not yet turned to the problem of intersubjectivity and the constitution of an objective transcendent reality, despite some passing remarks, for example, Ideas I §151). For Weyl, the constitution of an objective reality demands that the individual subject join other individual subjects in a communal subject: “The thou”, he says, “is required of the ego and the ego has to be extended to include the whole of mankind . . . ” (Weyl 1963, 98); or still: “ . . . the unique ‘I’ of pure consciousness, the source of meaning, appears under the viewpoint of objectivity as but a single subject among many of its kind [ . . . ] Thou art for thyself once more what I am for myself, conscious-existing carrier of the world of phenomena” (Weyl 1963, 124). Objective reality is by definition that which can in principle be experienced by any normally perceiving subject, it is the objective intentional correlate of the persistently coherent system of experiences that are in principle possible for a subject in general.4 Communication is the natural way by which a communal world can be constituted out of shared experiences. This world cannot, however, claim independent existence, for it remains an intentional object for the I, a communal I this time. But it can claim transcendence, that is, to be a source of ever new experiences, and objectivity, that is, to be a world for all to experience, a communal world. The communal world would be constituted all the way down to its material content if it were not for a fundamental shortcoming of language: reference cannot be fixed descriptively independently of a communal semantics (like, for example,

3 “[ . . . ]

this world does not exist in itself, but is merely encountered by us as an object in the correlative variance of subject and object. The world exists only as that met with by an ego, as an appearing to consciousness; the consciousness in this function does not belong to the world, but stands out against the being as the sphere of vision, of meaning, of image, or however else we may call it” (Weyl 1934, 83). 4 For Weyl, objective reality, as an intentional object, relates to experience in a manner that (1) there cannot be incoherent experiences of reality and (2) there cannot be anything in reality that cannot in principle correspond to something in experience (see Weyl 1963, 121–24).

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that presupposed by realists). Let us examine this more attentively. Suppose that a language L has a fixed semantics S that is, like L itself, shared by the community of L-users. A new term t can then be introduced in L by a definition that singles out an element in S; L-users can introduce new terms in the language referring to objects in S and make assertions about them that other L-users can recognize as either true or false based on their own experiences. In short, if experiences of Lusers are confined to S, L can be used to communicate the content of experiences to the community of L-users. However, if a communal world is not yet available, for it is under constitution, no amount of detail a subject A succeeds in offering in the effort of characterizing his experiences can secure that they are the same experiences of another subject A’, even if A’ recognizes his own experiences as adequately characterized by A’s descriptions. This simply because any isomorphic copy of the system of A’s experiences satisfies A’s descriptive account of his experiences. Suppose that a subject A describes his experience of the world in a language L, but that this time we do not fix a semantics for L. Suppose that another subject A’ reacts assertively to A’s descriptions; that is, that A’ recognizes the truth of A’s assertions based on his own experiences. Can we say that the experiences of A and A’ have the same material content? The answer is no, of course; although the situation is highly unlike for naturalistic reasons, it is in principle possible that A’ and A associate different semantics to L but that there is an isomorphic correspondence between them; both will agree on what is true and what is false, but each would be referring to a different world. This is a logical possibility. A question now imposes itself: what is that upon which A and A’ both agree? Since it cannot be the semantic content expressed by the sentences both agree to be true, it can only be the common abstract form of the worlds to which these sentences refer. So, a conclusion imposes itself, by itself language can only convey the abstract form of the domain to which it refers. The domains where A and A’ interpret the sentences of their common language have, by being isomorphic, a common form, and it is this form that the subjects agree upon. In short, the argument is this: intuitions are private, language is communal; linguistic meaningfulness is regulated by publicly surveyable rules of use, which fall short of fixing semantic content. To the extent that meaningfulness is independent of a fixed reference for linguistic terms, successful communication cannot secure access to but the abstract form of experience. This is why science, a communal task that involves an open community of co-workers spread in space and time relying essentially in language as a means of sharing experiences, can only grasp the abstract form of the world, which can be symbolically expressed. Symbolic systems, particularly mathematical ones, have the power of expressing the objectively valid form of the world.5 This is not, of course, an argument that Weyl himself presents, but it is clear that, for him, objectivity can only be granted by relinquishing the material content of experience in benefice of their formal aspects, which can, and only it can be adequately expressed in symbols. I believe that, for Weyl, this is the only

5 “ . . . this

objective world, representable only in symbols . . . [my emphasis]” (Weyl 1963, 120).

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possible way the tension between an epistemology centered on the subject’s intuitive experiences under the action of phenomenological epoché, on the one side,6 and, on the other, science as a communal activity of investigation of an objective transcendent reality to which our experiences give us access can be eased.7 In Mind and Nature (Weyl 1934, 95–96), for example, Weyl says that: A science can never determine its subject-matter except up to an isomorphic representation. The idea of isomorphism indicated the self-understood, insurmountable barrier of knowledge. It follows that toward the ‘nature’ of its objects science maintains complete indifference. This [ . . . ] one can only know in immediate alive intuition. But intuition is not blissful rest in itself from which it may never step forth, but it urges on toward the variance and venture of cognition. It is, however, fond dreaming to expect that by cognition a deeper nature than that which lies open to intuition should be revealed – to intuition.

In Raum-Zeit-Materie, referring to our experience of space vis-à-vis geometry, the science of objective space, he says (Weyl 1952, p. 26): Geometry contains no trace of what makes the space of intuition what it is in virtue of its own entirely distinctive qualities which are not shared by “states of addition-machines” and “gas mixtures” and “systems of solutions of linear equations”. [ . . . ] We as mathematicians have reasons to be proud of the wonderful insight into the knowledge of space which we gain, but, at the same time, we must recognize with humility that our conceptual theories enable us to grasp only one aspect of the nature of space, that which, moreover, is most formal and superficial.

Geometry, he says, can only express the formal aspects of our experience of space, those that space shares with other isomorphic domains, such as states of adding-machines, gas mixtures and systems of linear equations, which are all 3dimensional affine manifolds. In Philosophy of Mathematics and Natural Science (Weyl 1963, 113) he says also: Intuitive space and intuitive time are thus hardly the adequate medium in which physics is to construct [my emphasis] the external world. No less that the sense qualities must the intuitions of space and time be relinquished as its building materials; they must be replaced by a four-dimensional continuum in the abstract arithmetical sense.

An objective world, “capable only of representation by symbols” (Weyl 1952, 113), he says, is to be distilled from “what is immediately given to my intuition” (id. ibid.) A bit further (ibid, 116–117) he insists still: “the immediate experience is subjective and absolute [ . . . ], this objective world is of necessity relative”. [ . . . ] Objective reality, however, “is not given but constructed (nicht geben sondern aufgegeben)”.

6 “The

datum of consciousness is the starting-point at which we must place ourselves if we are to understand the absolute meaning as well as the right to the supposition of reality” (Weyl 1952, 5). 7 “It cannot be denied that a theoretical desire, incomprehensible from the merely phenomenal point of view, is alive in us which urges toward totality. Mathematics shows that with particular clarity; but it also teaches us that that desire can be fulfilled on one condition only, namely, that we are satisfied with the symbol and renounce the mystical error of expecting the transcendent ever to fall within the lighted circle of our intuition” (Weyl 1963, 66).

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As the following quote (Weyl 1952, 237) shows, Weyl believed that the symbolic character of scientific knowledge is the price science pays for aiming at objectivity: Perhaps the philosophically most relevant feature of modern science is the emergence of abstract symbolic structures as the hard core of objectivity behind – as Eddington puts it – the colorful tale of subjective storyteller mind

Let us consider, in particular, the perceptual experience of space.

14.2 The Objectivity of Space Perceptual space is fundamentally subjective; it is the form in which the subject experiences manifolds of co-existing bodies. Spatial experiences have content and form; the form is the particular system of relations by which the content of spatial experiences presents themselves to the subject merely by being spatial. Now, as already discussed, the objectivation of subjective experiences of space requires that they be abstracted of their material content in favor of form. As pure forms, conveniently idealized, spatial experiences reduce to systems of relations among “points”. Space, as pure form, is nothing but a structured manifold of points. Physical space is the objective, and then necessarily formal space of perception conveniently idealized; by necessity, the structure of physical space imposes itself as form on any spatial experience conveniently (formally) abstracted and idealized. Physical geometry, the mathematical science of physical space, is as all sciences objective. This means that only what is objective in our experience of space is of scientific interest. As already explained, only the formal aspect of spatial experience can be objectified, but not all formal aspects of subjective spatial experiences are objective, only those that have intersubjective validity. But, as already mentioned, intersubjective validation depends on linguistic communication. The first “problem of communication” is how to refer and distinguish among points that have no individuality. This can be accomplished by means of a system of reference, “the necessary residue of the ego-extinction”, in Weyl’s words (Weyl 1963, 75). Subjectivity is manifested in the structures the subject discerns in spatial experience (which may not always have objective validity) and the particular system of reference he chooses to describe them. Objectivity requires that the subjectively discerned structures of space have objective validity, i.e. that they can in principle manifest themselves to any subject and, moreover, that they be independent of the particular system of reference used to describe them; i.e. they cannot be “illusions” created by the system itself. Weyl adopts the following criterion of objectivity for abstract relations: a point relation is objective if it is invariant with respect to every automorphism of space. In other words, objective structures (systems of relations among points) must survive arbitrary automorphic rearrangements of the points of space. Of course, the notion of automorphism requires that a set of basic point relations be selected: automorphisms are the 1–1 rearrangements of points in space that preserve the basic relations. In Weyl’s words (Weyl 1963, 73):

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. . . we select as our basic relations some of the point relations of which we are convinced that they have an objective significance.

The objective structure of space, to which all subjects have access, is then the system of relations definable from basic point relations in a common language. The process, however, is usually turned around; instead of choosing the basic relations and define the automorphisms accordingly, a group of transformations is chosen and the basic (objective) relations defined as those that are invariants under those groups. Instead of coming to an agreement on which relations are objectively valid (valid for all), the subjects can instead agree on which transformations count as structure preserving. This opens the possibility of determining different structures of space (topological, projective, affine, conformal, and metric) by choosing different groups of basic transformations (resp. continuous, projective, affine, conformal and congruent). Now, given two systems of reference, two different ways of “naming” points in space, how can we be sure that they describe the objective structures of space in equivalent ways? According to Weyl, only if both systems are similar, that is, if they correspond to each other by the automorphisms of the group of transformations in terms of which objectivity is characterized. A given class of systems of reference is, according to Weyl, objectively distinguished if the following is the case: two system of the class are similar and any system that is similar to a system of the class is also in the class. Now, suppose that a rule A is given that associates with any point P of space its coordinates x in a system f of an objectively distinguished class C: x = A(P, f ); for a given f there is a one-to-one correspondence between points and coordinates. Let now i be an automorphism, then x = A(i(P), (i(f )); i.e. in similar systems of reference similar points have the same coordinates; in other words, coordinates are objectively determined by f and P. From this it follows that to any group  of point transformations there corresponds in f an isomorphic group S of transformations of coordinates. Now, if a different frame f1 is chosen in the class C that corresponds to f by an automorphism, then the coordinates x of P in f and the coordinates x1 of P in f1 are such that x = Sx1 .8 The upshot is the following, whereas points and point transformations can only be exhibited intuitively, coordinates and coordinate transformations, which correspond isomorphically to them, are distinguished symbols capable of objectively valid symbolic manipulations. In this manner, subjectivity is definitely overcome, but at the cost of preserving from intuition only its form, which only can be symbolically represented.9

8 See

Weyl 1963, 75–78. can easily distinguish an hierarchy: (1) morphological, quality-filled perceptual space; (2) an abstract and idealized, properly mathematical physical space; and (3) an analytic surrogate of physical space, isomorphic to it and only remotely representing in ideal form the structure discernible in perceptual space.

9 One

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14.3 Purely Symbolic Extensions of Symbolic Reconstructions of Perceptual Reality Now, once perceptual reality is reconstructed in symbols or, more precisely, a symbolic domain is devised that is isomorphic to an idealization of perceptual reality, there is no reason why science should stop there. Symbolization can be made into an instrument for investigating reality beyond the possibilities afforded by intuition. Both Weyl and Husserl realized the immense possibilities symbolization offers as an instrument of scientific inquiry, a fact that alone is enough to relativize their supposed “intuitionist” penchants. In mathematics, as Weyl recognizes (Weyl 1963, 75), the superiority of symbolic methods is evident, and there is no reason why it should be different in the mathematical sciences of nature. After all, in science, as Weyl says: “we are in contact with a sphere that is impervious to intuitive evidence; here cognition necessarily becomes symbolical construction” (Weyl 1932, 80). But there are no natural bounds for symbolic constructions; the objective perceptual world, reduced to pure form expressible in symbols can be extended symbolically indefinitely. This is possible by extending the language of science, enriching the structure to its domains or both; in this manner, structures discernible at a certain level of description can be better investigated through the investigation of richer structures, discernible at a more refined level of description. Analogously to exploring the domain of real numbers by adding extra structure to it, say, by immerging it in the domain of complex numbers – and then, for example, calculating integrals of real-valued functions by the method of residues – one can, for instance, explore the quantum realm by means of complex-valued functions that have no analogue in experience. This is a well-tested and successful methodological strategy extensively used in mathematics; empirical science, once it became mathematical, quickly incorporated the technique.10 Symbolic extensions, however, although methodologically useful, may not have correspondents in perceptual world. In fact, no symbolic representation of the world can, as Weyl says, “seriously pretend to be the true real world” (Weyl 1949, 184). “It is evident”, Weyl insists, “that now the words ‘in reality’ must be put between quotation marks; who could seriously pretend that the symbolic construct is the real world?” (Weyl 1954, 198–99). However, as we have seen, although not the real world, isomorphic copies of the idealized real world can be said to represent it somehow. This is not so with non-representing symbols. So, the pressing epistemological issue cannot be avoided: how can a symbolic reconstruction of the world that may contain more than what can by a process of “reactivation” of experience be found in experience be epistemologically justified? Weyl and Husserl answered this question in radically different ways; Husserl more conservatively, insisting on the epistemological primacy of intuition over

10 Husserl

was quite aware of this possibility: “The solution of problems raised within a theoretical discipline, or one of its theories, can at times derive the most effective methodological help from recourse to the categorial type or (what is the same) to the form of the theory, and perhaps also by going over to a more comprehensive form or class of forms and to its laws” (Husserl 1900, §70).

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symbolization; Weyl more daringly, recognizing the epistemological limitations of intuition. This may serve as an illustration of Weyl’s belief that knowledge does indeed begin with intuition, but cannot be confined to it, as well as of the fact that despite obvious influences, Weyl was aware of the scientific limitations of Husserl’s epistemology founded on intuition.

14.4 The Intuitive and the Symbolic in Science For Husserl, symbolization is always a source of alienation. Alienation, however comes in degrees, from the mild and methodologically acceptable one intrinsic to the logistic of arithmetic and symbolic logic, to the strong variety, involving symbolization that can only be epistemologically justifiable if eliminable in principle. Alienation is essentially the loss of meaning and direct personal responsibility (the responsibility of the Ego) that, for Husserl, characterizes the “crisis” that he, at the end of his philosophical carrier, believed to be identified at the core of European science and humanity.11 Symbolization and symbolic manipulation of signs according to rules, in short, symbolic thinking, is essentially thinking without resorting to the meaning associated with the symbols being manipulated. Alienation is mild or strong depending on whether meaning can or cannot be “reactivated”. For Husserl, there are three levels of alienation involved in symbolic thinking: mild, serious and disastrous. Mild when symbols have meanings that in principle can be reactivated. This is, for instance, the case of Descartes’ algebraizing of geometry. By manipulating algebraic symbols we can do geometry, but by so doing we alienate ourselves from the meaning of what we are doing, which however can be reactivated. Or still the arithmetical calculus, which Husserl treated in his Philosophy of Arithmetic (Husserl 1891, 2003), Or finally the logical calculus of symbolic logic (Husserl 1891b, 21): The proper task of a calculus is to be, for an entire domain of knowledge, a method of symbolic deduction of consequences; hence, an art for substituting, by means of an appropriate designation of ideas, a calculus for effective deductions, that is, a conversion and a substitution according to rules of signs by signs and then, by virtue of the correspondence between signs and ideas, for obtaining from the final formulas the desired judgments.

In all these cases symbolization plays a surrogating role; symbols substitute objects and concepts proper and intuitive thinking can in principle be reactivated by going back from symbols and symbolic manipulation to objects, concepts, ideas and intuitive reasoning. The symbolic domain is an isomorphic copy of the intuitive domain. From the perspective of epistemology, symbolization becomes a serious affair when this is no longer the case, that is, when symbols no longer stand for anything that can be, at least in principle, intuitively given. This is the case, for example, 11 See

Husserl 1936/1954.

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of the calculus presented by Schroeder in his Lectures of the Algebra of Logic of 1890, reviewed by Husserl (Husserl 1994, 52–91), or still “false” numbers such as 0 and 1 in the arithmetical calculus. Husserl’s conservative approach to this brand of symbolic reasoning was presented – and never dismissed to the end of his career – in the double talks delivered in Göttingen in 1901. According to him, in cases such as these, symbols have, or should have, only psycho-technical relevance, that is, they may serve as instruments for thinking, aides to memory, purely technical devises, but should be, at least in principle, unnecessary and superfluous; the symbols of the calculus whose sense cannot be reactivated must, at least in principle, be done without. If this is not possible, empty, meaningless symbolization becomes essential, intuitive content is no longer accessible and alienation is irreversible and disastrous.12 Unless, of course, symbolization is being used solely for the purposes of formal ontology. Formal ontology – the investigation, logical in nature, of possible realms of things exclusively as to form – is not concerned with the knowledge of intuitable things, only with empty logical forms that can possibly give form to possible domains of things. In this case, symbols and symbolic reasoning stand for possible intuitable things and possible intuitive reasoning, respectively; they play a logical, not epistemological role. Summarizing, Husserl is definitively not an enemy of symbolization. He says (Husserl 1970, 349): Without the possibility of symbolic representations substituting for more abstract proper representations, difficult to distinguish and handle, or even representations that are not proper [my emphasis], there would not exist a higher spiritual life, and even less science.

Symbolization, he thinks, is unavoidable and useful, essential for thinking, but must be kept under strict surveillance and control. Weyl had a completely different approach to symbolization in science. Once a formal substitute of experienceable reality is available, the scientist is free to improve on it the way he sees more fit; individual assertions of the theory do not necessarily represent experienceable facts and only as a whole a theory can be empirically tested by means of its testable consequences, standing or falling as a whole. About adding extra symbols to the language of science, Weyl says (Weyl 1949, 183–4): Thus we had better not commit ourselves to any definition and rather develop the theory as a symbolic construction with unexplained symbols and only at the end indicate in which way certain derived quantities may be checked by observation. The theory then becomes a connected system that only as a whole may be confronted with experience. Only this entire connected theory [he is referring to physical theory, my note], into the texture of which geometry also is interwoven, is capable of being checked by observation. An individual law isolated from this theoretical structure simply hangs in the air. Ultimately all parts of physics including geometry coalesce into an individual unit

12 For

details see da Silva 2012a, b and 2013a, b.

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Or still (Weyl 1954, 199): In this manner a theory of nature emerges which only as a whole can be confronted with experience, while the individual laws of which it consists, when taken in isolation, have no verifiable content. This discords with the traditional idea of truth, which looks at the relation between Being and Knowing from the side of Being, and may perhaps be formulated as follows: ‘A statement points to a fact, and it is true if the fact to which it points is so as it states’. The truth of physical theory is of a different brand.

14.5 Conclusion Both Husserl and Weyl believed that knowledge should be based on intuition, they however differed on whether and to what extent intuition could, in an epistemologically justified way, be supplemented with symbolization. What Weyl named “cognition” goes further than intuition, it is an adventure of the spirit that beginning with intuition gives free rein to formal creativity. Only the final product of the process must face the highest court of experience. Conceptual analysis lie, for both Husserl and Weyl, at the hearth of philosophical reflection, but at least with respect to empirical concepts Weyl doubted the value of Husserlian eidetic intuition within the limits of phenomenological epoché. For Weyl, conceptual intuition of scientific concepts cannot be accomplished independently of established scientific knowledge, for science is also, and primarily, conceptual clarification. Weyl’s doubts as to whether to include Husserl’s intuition of essences among the valid means of establishing the foundations of knowledge comes out clearly in this quote (Weyl 1954, 202): At the basis of all knowledge there lies [ . . . ] intuition, mind’s act of ‘seeing’ what is given to it, limited in science to the Aufweisbare [something to which we can point in concreto13 ] but in fact extending far beyond these boundaries. How far one should include here the Wesensschau of Husserl’s phenomenology, I prefer to leave in the dark.

Husserl, on the other hand, laid all his chips on intuition as the provider of the foundations of knowledge, including the intuition of essences. For him, symbolization had a role to play in the dynamics of knowledge, although not the leading role. Despite the influence Husserl had on him, Weyl, as the great thinker he was, never pledged unrestricted allegiance to the basic tenets of Husserlian epistemology.

13 See

Weyl 1954, 200.

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References Bloch, F. 1976. Heisenberg and the early days of quantum mechanics. Physics Today 29 (12): 23–27. da Silva, J.J. 2012a. Husserl on geometry and spatial representation. Axiomathes 22: 5–30, reprinted in Hill & da Silva 2013, 31–60. ———. 2012b. Away from the facts, symbolic knowledge in Husserl’s philosophy of mathematics. In Symbolic knowledge from Leibniz to Husserl, ed. A.L. Casanave, 115–135. London: College Publications. ———. 2013a. Husserl and the principle of bivalence. In Hill & da Silva, 285–298. ———. 2013b. Mathematics and the crisis of science. In Hill & da Silva, 345–367. ———. 2016. The analytic/synthetic dichotomy. In Husserl and analytic philosophy, ed. G.E. Rosado Haddock, 35–54. Berlin/Boston: De Gruyter. Husserl, E. 1891. “Besprechung von E. Schröder, Vorlesungen über die Algebra der Logik I”, in Husserl 1890–1910, 3–43, published in English as “Review of Ernst Schröder’s Vorlesungen über die Algebra der Logik” in Husserl 1994, 52–91. Originally published in Göttinger Gelehrte Anzeigen, no 7, 243–278. ——— 1970. Philosophie der Arithmetik, mit ergänzenden Texten (1890 – 1901), Hua XII, The Hague: Martinus Nijhoff. ———. 1994. Early writings in the philosophy of logic and mathematics. Dordrecht: Kluwer. ———. 2003. Philosophie der Arithmetik. Logische und psychologische Untersuchungen, Bd. I, Pfeffer, Halle a.d.S. 1891. English translation: Philosophy of arithmetic, psychological and logical investigations with supplementary texts from 1887 – 1901 (D. Willard, ed.). Dordrecht: Kluwer. Weyl, H. 1932. The open world: Three lectures on the metaphysical implications of science. In Weyl 2009, 34–82. ———. 1934. Mind and nature. In Weyl 2009, 83–150. ———. ca, 1949. Man and the foundations of science. In: Weyl 2009, 175–193. ———. 1952. Space, time, matter. New York: Dover. (H. L. Brose, Trans.). from the 4th ed. of Raum, Zeit, Materie, 1921. ———. 1953. Axiomatic versus constructive procedures in mathematics. In Weyl 2012, 191–202. ———. 1954. The unity of knowledge. In Weyl 2009, 194–2003. ———. 1963. Philosophy of mathematics and natural science. New York: Atheneum. Revised and augmented English edition based on a trans. by Olaf Helmer. Parts of the book originally published in Handbuch der Philosophie as Philosophie der Mathematik und Naturwissenschaft, 1927.

Chapter 15

The Scientific Implications of Epistemology: Weyl and Husserl Pierre Kerszberg

15.1 Theory and the Real The title of this article refers to the subtitle of the section “Subject and Object” of the major work of Hermann Weyl.1 We can immediately remark the surprising order of the terms: should we not expect from a man of science that he preferably discuss the epistemological implications of science? Using epistemology as the vector of scientific thought, Weyl examines how, under the pressure of an issue inherent in science, the classical philosophical doctrine of the subjectivity of sensible qualities eventually rooted out any trace of intuition in the now purely symbolic representation of the objective world. The precedence of epistemology over physics could be justified by this conceptual revolution initiated by special relativity: physics is now geometry in action. Through gradual improvement geometrical concepts have become more and more involved in phenomena taking place in physical space-time: geometry is not only ontologically rooted in the real through its ability to produce semantic universals (curvatures, manifolds, groups, connections, etc.), but, besides, it is actively involved in its becoming by the fact that its symmetry principles have a constitutive action. In short, while mathematics is still in Galileo’s physics only a language appropriate to the study of nature, a remarkable ontological

Translated from French by Pascale Pelletier, revised by the author. 1 H.Weyl,

Philosophy of Mathematics and Natural Science (Princeton: Princeton University Press, 1949), Section 17. Hereafter referred to as PMNS.

P. Kerszberg () Université Toulouse- Jean Jaurès, Toulouse, France e-mail: [email protected] © Springer Nature Switzerland AG 2019 J. Bernard, C. Lobo (eds.), Weyl and the Problem of Space, Studies in History and Philosophy of Science 49, https://doi.org/10.1007/978-3-030-11527-2_15

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extension of the founding project of mathematical physics led to a deep upheaval of its sense: the means of knowledge cannot be separated from the knowledge of the objects themselves. Weyl’s scientific work abounds in philosophical references amongst highly sophisticated formal and technical reflections. Considering the revolutionary developments of the theory of relativity and quantum mechanics, which necessitated an in-depth revision of the fundamental principles of science, he has no qualms about trusting the internal progress of these theories. They have become sufficiently reflective to justify their development by integrating their own conditions of possibility. This demands a kind of dogmatism reasonably motivated by their good faith.2 Such dogmatism is nevertheless sufficiently enlightened to allow philosophical thought to contribute in its own way to the meaning of fundamental issues about the nature of space, time and matter. Philosophy and science do follow radically different paths, but the path of philosophy is justified by the very fact that, from the start, science is subject to specific methods; moreover, the domain of objects covered by science is limited to the concrete objects of experience (thus excluding living creatures, persons, values, etc.). Now, at the time of the birth of relativistic and quantum theories, philosophy too underwent a revolutionary change: namely, the breakthrough accomplished by Husserl’s transcendental phenomenology, the relevance of which is vindicated by Weyl in view of the fundamental concepts at work in these theories. The significance of epistemology for science begins to emerge, since the purpose of phenomenology is precisely to explore methodically the entire range of experience related to the “things themselves”, as they are concretely given in experience. According to Husserl, as Weyl explains, the starting point we must adopt in order to have a chance to understand the claim of physics to grasp the real is nothing else than the absolute data of consciousness (STM p. 5). In other words, the philosophy of science appropriate to the new context shall aim at the phenomenological form of idealism. Weyl sees no reason why the symbolic theoretical construction should stop at the threshold of the facts of life and psyche (PMNS p. 214). The perceptive sketches of things accumulate and progress from an lower level of objectivity to a higher level; this progression does not remain confined to the objects of the world as phenomena, for it eventually invests in the objective world represented by pure symbols. The symbolic world enjoys the autonomy of a domain per se, and its relationship with the corresponding data of consciousness must now be described (PMNS p. 113). Thus, going further than Husserl, the question concerning the reality of the world cannot be left behind in phenomenological brackets as long as no reason is given to account for its existence in accordance with lawful mathematical harmony. What is in fact the function allotted to idealism in natural science, which adopts in good faith the mathematical form in order to probe into concrete experience? In which way is this experience indebted to mathematics, which is a mind-dependent

2 H.

Weyl, Space-Time-Matter, trans. H. Brose (New York: Dover, 1952), p. 2. Hereafter referred to as STM.

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abstract structure? As soon as reason confronts the reality of the world, it meddles with it to impose lawful mathematical harmony on it. But, in spite of being inseparably mingled following this process, reason and the world continue to oppose each other concerning the sense of transcendence they aim at it: if the real world eventually remains transcendent to reason because it falls on us ready-made, reason looks for sense in whatever it can do by itself to master an origin that nevertheless remains hidden from it (PMNS p. 125). Hence the astounding difficulty created by the phenomenological exploration of a scientific theory. In order to come to the ultimate core of sense, the return to the origin must undo what is done in the product of the origin: in other words, it must endeavour to hold together two transcendences which repel each other, see what actually happens in the tension of holding them together, without tipping the balance in favour of one of the two poles. H. Weyl, Philosophy of Mathematics and Natural Science, p. 118, “Schematic representation of a theory with a redundant part Z.”

The tendency to separation inside a unit in which mingling prevails is obvious in a theory like general relativity. On a plane Weyl draws a straight line which separates it into two distinct regions (PMNS p. 118): hatched in diagonal lines, the lower region represents the real; the upper region, which is supposed to represent the theory, contains a kind of bubble E hatched in horizontal lines, surrounded by void. The bubble seems to be lying on the real as if it was sticking to it. At first sight this could mean that, provided that some transformation is effected (the diagonal lines into horizontal lines), theory constructs a small-scale model of the real through these shortcuts that are mathematical symbols. But looking attentively we are struck by the fact that, somehow in the manner of a brush, theory touches the line of the real only on a few points; there are free gaps between these points, which belong neither to theory nor to the real. Nevertheless, contrary to the void outside, theory encloses these void spaces. It thus looks as if this enclosed void represented the neuralgic point of the tension between these two opposed transcendences which are reason and the real world. Weyl does not justify this manner to represent the relation between theory and reality. To follow his reasoning, one must accept it without question, as an evidence specific to the knowledge of the world according to relativity. Clearly enough, nothing prevents us from generalising it as the inescapable epistemological starting point of any theory: it is a matter of seeing that, as an imaginary system, theory is not up to understanding the real as such, though it touches some real – or part thereof – by virtue of the sheer fact that there are experiences of something that are “obviously” not meaningless. This simple fact accounts for the most elusive

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enigma of scientific knowledge, an enigma that science takes with it in all its symbolic constructions.3 Reflecting upon the impact of the new theories which seem to impose new norms for philosophy, Eddington shows great wisdom when he writes: “it is not a question of asserting a faith that science must ultimately be reconcilable with an idealistic view, but of examining how at the moment it actually stands in regard to it.”4 Weyl is less wise: in light of the intangible results of science, he is ready to put the idealism in which he is interested (Husserl’s idealism, and therefore implicitly also Kant’s) to the test. Consequently, the idealistic question par excellence which shall get his attention is the question of the irreducible role of intuition in the formation and critical assessment of mathematical physics. Intuition deals with “something” of the real, but in a mathematical theory of nature, what has been thus attested is not obvious. As it turns out, Weyl’s diagram is completed by another bubble Z which floats freely above the region of the real, away from the bubble representing theory, but yet also horizontally hatched. This representation echoes a tension inherent in general relativity, namely, the retrospective efforts to maintain the reality of Euclidean geometry for bodies supposed to be rigid in a gravitational field. Arbitrary quantities are needed in order to compensate for the lack of direct perception of their deformations. Now, non-Euclidean geometry in the theory of general relativity is a phenomenological progress as much as a scientific one, since it is based on the direct comparison of length standards. As long as the arbitrary quantities are maintained against non-Euclidean evidence, they form a theory redundant with Einstein’s theory. In this example, we can see how a phenomenological criterion is likely to disqualify a theory which does not comply with it. This is why Weyl wants to understand the role of intuition both at the starting point of the symbolic representation of the real and at the arrival point, when one wonders what remains of concrete intuition once it has passed in the symbolic realm. Let us start with the remainder of intuition in a mathematical theory of nature like special relativity (STM pp. 3–4).

15.2 Critical Reflection on the Basis of the Theory of Relativity The scientific revolution initiated by Galileo and Descartes considers the possibility of knowing material bodies in themselves by means of the constructive mathematical method, which demands that the sensible qualities be detached from these bodies

3 The

generalization of Weyl’s diagram to scientific knowledge in general is suggested by J.Ortega y Gasset, L’Evolution de la théorie déductive, transl. J.P. Borel (Paris: Gallimard, 1970), p. 26. Between the two regions, there is indeed a correspondence, guaranteed by experiments, but not a similarity. 4 A.S. Eddington, The Nature of the Physical World (Cambridge: Cambridge University Press, 1928), pp. 343–344.

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and sent back to the realm of subjective intuition. Denying the possibility to know things in themselves, while maintaining the validity of the scientific ideal, Kant’s critical philosophy extends to space and time the field of qualities which have no absolute objective significance. It is precisely space as the pure form of sensibility which blocks the knowledge of things in themselves, so that the object of science is properly understood as the realm of phenomena. Next, in uniting space and time in the four-dimensional space-time continuum, the theory of special relativity confirms that space and time separated from one another are intuitions which have no validity in the symbolically constructed world. This new stage in the process of discarding intuition is quite singular, since it consists in the fusion of two intuitions which abolishes them as intuitions. The effect of this process is to explicitly raise the question of the role of intuition in the symbolic world. When he formalised the Einsteinian concept of four-dimensional continuum by means of the idea of symmetry in group theory, Minkowski rejected the separation of space and time as two absolutes in pre-relativistic physics by calling upon the lesson of immediate intuition: nobody has ever observed a place outside time or a time outside a place.5 However, does not immediate intuition also teach us the irreducible qualitative differences between the original experiences of space and of time? Minkowski specifies that he still “respects” what he calls the “dogma” of an independent meaning for space and for time. But respecting a dogma does not necessarily mean believing in it . . . As a matter of fact, nothing more is said about this independent meaning, and furthermore the meaning of the four-dimensional continuum is purely formal. In Minkowski’s interpretation, the “world” simply designates the multiplicity of all the value systems for the space variables and the time variable. The qualitative distinction between space and time is nothing more than an inessential consequence of the fact that we continue to measure with rigid rods and clocks; he says that it is a question of “projection” from the four-dimensional world considered in itself, in which space and time lose their individuality. Nothing would be changed to this in-itself if measurement disappeared altogether or if it was effected with other concrete means. In that way, recalling the original experience, physics immediately reduces its specific originality. This conclusion is in line with a statement by Husserl on the accomplishment of special relativity. Husserl observes that in the theory of special relativity Einstein modifies the space and time formula inherited from Galileo to form the concept of fourdimensional continuum, but since space and time in Galileo’s physics are already forms of idealized physis, this modification does not at all affect the space and time of our original life.6 Space-time in the idealised world, as the fusion of two intuitions, must however reflect the rights of intuition in the original experience of the world in which we actually live. This reflection is made possible by the

5 H. Minkowski, ‘Space and Time’, The Principle of Relativity, transl. W. Perrett and G.B. Jeffery (New York: Dover, 1952), p. 76. 6 E. Husserl, The Crisis of European Sciences and Transcendental Phenomenology, trans. D.Carr (Evanston: Northwestern University Press, 1970), p. 295.

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consciousness of the observer. Weyl argues quite aptly that “only the consciousness that passes on in one portion of this world experiences the detached piece which comes to meet it and passes behind it, as history, that is, as a process that is going forward in time and takes place in space.” (STM p. 217). Consciousness is in possession of all imaginable concrete means to step in the four-dimensional continuum and project this continuum onto the stage of experience; with respect of consciousness and its sense of history, the clock does not say anything essential. The resistance of conscious life to the relativistic world is a good example of the manner in which intuition undoes what is done in theory, allowing a qualitative distinction between the two worlds to be regained as something now essential to both – even though the idealized world masks it. The resistance of intuition to the theoretical constructions of special relativity is the prelude to a new deployment of conscious life in a world specific to it, namely, the life-world. Let us consider what remains of intuition in the theory of general relativity, in which invariance with respect to an arbitrary transformation of coordinates is required as a matter of principle. In this case, the remainder will make us swing into a new dimension of experience, for this theory now completely breaks free from intuitive space and time (PMNS p. 115). Consider the observation of two or more stars, from a consciousness which apprehends them as if it were a “point eye” going over its world-line. The angle under which the stars appear to the observer at moment O in his light cone is determined by the world-lines of these stars which enter the cone. Thanks to the light signals which connect O to the point of entry into the cone, the angle can be constructed in the four-dimensional number space as an invariant which resists any arbitrary deformation of the entire image. The angle keeps the same numerical value if it is calculated according to the same prescription for any deformed image compared with the initial image: as such it is, Weyl says, an objective reality. What is left to intuition in an objective reality, if the latter is now purely numerical? Weyl suggests this answer: “The angles θ between any two stars of a constellation determine the objectively indescribable, only intuitively experienced, visual shape of the constellation, which appears under the equally indescribable assumption that I myself am the point eye at O”. Two intuitive remainders must be retained: the visual shape of a thing, on the one hand, and myself capable of vision of this thing from a point, on the other hand. Weyl draws this conclusion: “The objective world simply is, it does not happen. Only to the gaze of my consciousness, crawling upward along the life-line of my body, does a section of this world come to life as a fleeting image in space which continuously changes in time” (PMNS p. 116). According to the invariant construction, the objective world is a world that is, but paradoxically it is a dead existence, indifferent to sense, as long as it is not penetrated by consciousness. Despite their opposition, the extension of consciousness onto the world succeeds insofar as consciousness exposes the world to its own life. To be sure, the world described by the theory of relativity is a set of world-lines blindly followed by physical bodies, but a world-line makes sense only as a “life-line” for a seeing consciousness. The seeing consciousness is carried by the living body of the observer, so that it joins the body to bring life to the world-line; however, as

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this living body is also a physical body, this “life-line” sneaks in the physical world to bring it to life. But in the final analysis, the life in question is at most a semblance of life. Indeed, the consciousness which is aware of its own living body is also aware of its own free motion in the original space and time of its life, with no other reference system than itself. Thus the being of physical objects and events is apprehended only through an image; moreover this is a fleeting image for no transformation makes it possible to fix it as the invariant of something. In reaction to the symbolic construction of the being of the world by theoretical consciousness, living consciousness aware of its passage through the universe constructs a spatiotemporal image of it which is an irreducibly singular image. Consequently, since the evidence of vision will remain forever inaccessible to natural science, exact knowledge of the world will be necessarily symbolic.7 The symbol concentrates within itself an amount of intuition which will always exceed its own capacity to represent itself. Hence the problem: given the passage from image to symbol, will mathematical physics ever be able to do more than presuming the existence of the world? Indeed, Weyl emphasizes that subjective experience “is given in its very haziness thus and not otherwise”: it is therefore both subjective and absolute, finally more reliable in its absoluteness than the supposedly objective world. Actually, the objective experience of the objective world aimed at in science is relative: “it can be represented by definite things (numbers and other symbols) only after a system of coordinates has been arbitrarily carried into the world.” Even though it is given absolutely thus and not otherwise, the content of consciousness is hazy because it is not preserved after an arbitrary transformation from a state of consciousness to another. This is also true for the passage from a consciousness to another, so that even intersubjectivity does not vouch for objectivity. The argument is a powerful critique of all widespread and finally naive beliefs in the omnipotence of the phenomenon perceived as a primitive entity; the intersubjective consensus as a prerequisite to science is subject to the same critique.8 On the other hand, things arbitrarily represented as symbols remain well defined after an arbitrary transformation. This opposition of the subjective/absolute against the objective/relative would be the most important epistemological lesson to be drawn from the contemporary development of science. Now, the mind cannot be satisfied with merely registering the opposition. The precariousness of sense in constituted knowledge deserves to be questioned at the level of the will to know which precedes it. Whatever can be said of it, this will must ensure that a specific operation of consciousness provides a bridge between the subjective and the objective poles in spite of the fact that an abyss separates them. What is this operation? Husserl did not ask the question so directly, insofar as he explored the lifeworld as a subjective/relative world, opposed (though linked in a paradoxical

7 H.

Weyl, « Über den Symbolismus der Mathematik und mathematischen Physik », Studium Generale, 6 (1953), p. 219–228. 8 See for instance R. Thom, Paraboles et catastrophes, Paris, Flammarion, 1983, p. 35.

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manner) to the objective/absolute world. In the wake of the revolutionary scientific theories of the beginning of the twentieth century, Husserl writes: “the paradoxical interrelationships of the ‘objectively true world’ and the ‘life-world’ make enigmatic the manner of being of both. Thus the true world in any sense, and within it our own being, becomes an enigma in respect to the sense of this being.”9 Since the scientific revolution of the seventeenth century, mathematical theories of nature construct next to nature as given to us a second nature, in which there are laws that could very well exist even if we did not. Theories endow this world with an ontological reality which may have a value different from the one it has for me, or even have a significance without me. The objective world is supposed to be constructible without a living subject “at home” in it. On the other hand, the subjective life-world is relative because it is twofold: it refers to both the world in opposition to the quantified world according to the mathematical science of nature, and the world of primary evidence out of which this ideal nature emerges. The subject of the life-world is entangled in unceasing egological mobility, between the “simple experience” of real entities intuitively felt and “the ways in which their validity is sometimes in suspense.”10 The floating modality of what seems to be fixed forever characterises the life-world as a world essentially and always only possible. Reflecting on what connects the mathematical idealities of science with the living world of intuition, Husserl uncovers a sense of totality valid for this intuitionworld: all natural beings display by themselves, independently from any concept, a universal causal regulation which is rather vague, but structured enough to connect them to each other in an all-encompassing unity (Alleinheit).11 In the scientific worldview, these experienced causal connections are determined accurately by means of mathematical tools. This consists in “constructing systematically and in a sense in advance, the world, the infinitude of causalities, starting from the meagre supply of what can be established only relatively in direct experience.”12 According to Husserl, however abstract it may be, construction is bound to remain consistent with the general style originally impressed on the world by the intuition of an allencompassing unity. Weyl’s strategy is different. Its motivation rests on the eventually inevitable break within this universal consistency. How can the physicist project himself into a world that transcends sense-experience, even though the latter can never be completely cancelled? Must not idealized nature, as the object of physical theory, eventually go against that nature which merely appears? As it is more and more concerned with the empirical wealth of the world, science aims at postulating principles able to support the ever growing weight of experience. What will happen if the empirical realm were to grow to the point of falling into the world of mathematical idealities and symbols, a world that sensibility cannot retrieve? That is what actually happens

9 Crisis,

§34, p. 131. §44, p. 156. 11 Ibid. §9b, p. 31. 12 Ibid. p. 32. 10 Ibid.

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to any mathematical science of nature. If the objective state of things such as Weyl described it makes it possible to account for all that is necessary to have subjective appearances of it, the question of the passage of these appearances into the symbolic world generates the reversed question: what happens in intuition which justifies symbolic construction? Is not the intuitive remainder a constant threat on the integrity of the objective/relative world? If we were limited to what is given at one instant to a seeing consciousness, and even if this consciousness relies on its memory to acknowledge the equivalence of the data of consciousness for other minds, there would be no reason for constructing an objective world (PMNS p. 117). If need be it would still be possible to talk about an objective world without constructing it if each consciousness could be compared to a Leibnizian monad. Each monad is actually like a spiritual universe, a separate and self-sufficient world, impenetrable to any external action. At all times, each monad expresses the whole universe in its own way. To be sure, this monadic universe could be subjected to a construction: imagine a certain kind of geometrical transformation which would regulate the passage between the images formed by several different minds. But besides the fact that this procedure would be cumbersome, Weyl simply does not believe in it: each consciousness as it experiences itself is already penetrated by “limitations and gaps” which are not found in the “complete real world.” In short, the suggested construction is akin to the impossible reconstruction of a pre-established harmony between minds. Hence the question: what is the condition that the “complete real world” must satisfy in order not to be simply an illusion? The bridge to the complete real world is opened by the domain of the possible (PMNS p. 121), which is precisely the domain of the life-world according to Husserl. Limitations and gaps of consciousness still affect the memory of the ego as well as the intersubjective agreement tacitly admitted by this ego. But these deficiencies are made up for by the possible perceptions of the ego; these perceptions are themselves dependent on the possible objective states of the ego (for example the world-line of its body). Weyl here agrees with the Husserlian theory of thingperception by successive sketches of the thing as a whole. This perception does not proceed by means of images. The thing is perceived in concrete unity as the noematic correlate of the acts of consciousness directed towards it. The unlimited development of concordant intuitions of a thing is an ideal yet concrete possibility for the confirmation of the existence of this thing outside consciousness. But while the image-consciousness sees a floating image, somewhere between being and nonbeing, the constituting consciousness is eventually not able to assume to the end the heritage of the possible. The heritage is such a burden for consciousness that its claim to reach the transcendent real is in fact threatened from the start. Any beginning of concordance in the order of possible intuitive acts shall never be constraining enough to prevent future discordance as another possibility. Between the real world and the given in lived experience, there is in fact a correspondence in the sense of a mathematical projection, but as this operation is now under the control of the phenomenological condition of certified intuition, it is at all times undermined by possible disintegration.

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Giving up the hope for a true theory, Weyl lays down the less ambitious conditions of theory which is at least correct. A correct theory is one that from the start respects in principle two requirements, which makes it comply with two criteria: one of logical consistency (concordance), one of theoretical coherence (non-redundancy). Here concordance means the same numerical determination of a quantity through different means, while non-redundancy stipulates the possibility to determine by observation a definite value for a quantity appearing in the theory.13

15.3 Mathematics and Time Consciousness If the mathematical theory of nature takes into account the phenomenological conditions of knowledge, the following question arises: is a mathematical representation of phenomenological intuition, if not feasible, at least thinkable? Weyl addresses this question in his essay on the continuum.14 There is an abyss between the intuitive and the mathematical continuum. And yet there are reasonable motives to try to get over this abyss. Is it not by a similar motivation that physics goes beyond daily reality in order to understand objective reality behind sensible qualities? How could motivation still be reasonable in front of such an abyss? Let us keep in mind the fundamental problem of all knowledge according to phenomenology: “in how far the delimitation of the essentialities perceptible in consciousness expresses the structure peculiar to the realm of presented objects, and in how far mere convention participates in this delimitation.” (STM p. 148). This double question takes a special turn when consciousness considers itself as an object in order to understand how it relates to the real. To be sure, everything that is given to consciousness is absolute, given absolutely as this or that. But this consciousness is also open on the real, which in turn means absolutely something for it. The real “is” in proportion to my capacity to ascertain that it is indeed such as it appears to me. The verification constantly calls for new experiences, some of which will certainly be contradictory, but the whole of which always constitutes, at least approximately, a harmonious relation: all things qualified as real melt into a world that is always there. Now, the first clue which opens consciousness to what it is conscious of is nothing else than the most primitive form of consciousness: its own temporal stream. The most primitive content of consciousness is not “this is”, because it would not give rise to knowledge, but “this is now”, which has the strange peculiarity to no longer be at the following instant. If the now is the first object of consciousness which observes its own stream, this now is dragged into the stream as a set of relations between a before and an after. In this

13 A

good example of redundancy is the Euclidean manner to “unfold” non-Euclidean geometry in a gravitational field by renouncing the absolute rigidity of the bodies moving in it. 14 H.Weyl, Das Kontinuum (Leipzig: Veit, 1918), Ch.2 §6.

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way, knowledge is awakened to itself when time consciousness directly confronts the destructive yet creative collision between being and non-being. Husserl developed a reflective theory of inner time consciousness. Reflection brings out a twofold intentional structure which accounts for the fact that the present comes from a past that was retained, even though it continues to flow. The theory takes up in its own manner the famous analogy between the flow of time and the flow of a river. If time flows like a river, in order to see this flow one must be positioned on a fixed spot of the bank. This makes it possible to fix the flow for an instant, while thinking that this capture does not stop the motion. Hence the two intentionalities at work: the transversal intentionality which relates the present to what has already flowed (all retentions piled up on one another by successive modifications), versus the longitudinal intentionality according to which what seems fixed for an instant is in fact a flow. Husserl emphasizes that in principle none of the lived experiences gets lost in their description. We are permanently at the outpost of a fantastic number of sleeping retentions, and with an effort of attention or will we could in principle recover any of the buried retentions. The continuous embedding of retentions in the stream of lived experience is such that the latter is an infinite unity, the form of which “necessarily comprises all mental processes pertaining to a pure ego.”15 Is this interpretation of the time flow in agreement with the famous “principle of all principles,” that Husserl defines as the requirement that everything originarily given to us in intuition is to be accepted simply as what it is presented as being, without going beyond the limits within which it is given.16 Weyl reveals himself a more authentic phenomenologist than Husserl, since he demonstrates that the Husserlian theory of the stream of consciousness remains too close to a mathematical conception. “Let us tear ourselves away from this river by reflection,” Weyl asks, implying reflection in Husserl’s sense of the term. As it turns out, this reflection cannot refrain from idealising beyond what concrete experience should allow. The transversal intentionality makes it possible to say: “this alone is,” namely, the whole of experience at each point of time. How does this whole partake in the concretely lived duration? The answer is: “the more or less distinct memory” of retentions. It is important to understand that this “more or less” will never be upgraded to something fixed. An experience of short duration (like the perception of a flash of light) is sufficient to realize that a more or less clear distinction of memory is fatal to a representation of the flow in terms of successive points. An instant of this perceptual experience contains more than it can contain: to the perceptual experience of the instant are added not only the memory of other past perceptual experiences in the recent past of the short time-length, but also, for a past instant, the global experience of that instant. And so forth for all prior instants, including those that preceded the beginning of the perceptual experience. Husserl refers to this process as the total

15 E.

Husserl, Ideas pertaining to a pure phenomenology, Book I, transl. F.Kersten (The Hague: M.Nijhoff, 1982), §82, p. 196. 16 Ibid. §24, p. 44.

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field of phenomenal time for a pure ego. Weyl breaks this argument in a striking manner. Continuous perception, as he puts it, would consist of infinitely many mutually related systems of infinitely many memories, one system packed inside another. But clearly we never experience anything like this; moreover, such a system of point-like moments of experience fitted endlessly into one another in the form of a completely apprehended unity is something quite absurd. In temporal consciousness there is nowhere to be found the equivalent to a world always there and concordant with itself: this equivalent would be a “temporal world,” capable of harmonizing the relations between instants. As for the flow actually felt, we cannot describe it because of the genuine primitiveness of phenomenal time. While the origin is for the physicist’s reason the transcendence at which it is aiming, time consciousness takes hold of this origin and delves into the heart of the enigma. Now, the distinctive feature of the original phenomenal time which hinders its description is continuity. Any reduction of the flow to a set of points contradicts the continuity of the flow. If Weyl had pursued this thought as a phenomenologist, he could have questioned the apparently untouchable principle that he eventually shares with Husserl: the phenomenal time is continuous. In order to justify again the attempt to get over the abyss between lived experience and the supposedly objective world, he reacts more as a physicist when he declares in a footnote that our inability to connect continuity with the schema of the whole numbers is not just a matter of personal preference. Who knows, Weyl concludes, what the future theory of quanta has in store. Let us remember that these thoughts were jotted down in 1918 . . . Once the description of the essential features of its temporal stream has reached a limit, consciousness has no other resource than entering the domain of convention, and assuming it as far as possible. This amounts to proceeding constructively by means of points in the mathematical sense. In relativistic space-time, consciousness could be reduced to a point eye. How does Weyl proceed with the new construction, assuming that the inadequacy to intuition cannot be fully corrected? As we follow his procedure, we will see how close he comes to criticizing his own unquestioned bias in favour of continuity. Though a phenomenological mathematics is impossible, let us ask if a mathematics conformable with definite phenomenological conditions of knowledge is at least thinkable. Time as intuitively given could be treated mathematically in that manner. Considered as such, the stream leaves aside any real process of the psychophysical or material type: phenomenal time as effectively lived is not the objective time. What happens to the intuitive data when the mathematical treatment is applied to it: does this operation reach the essence of phenomenal time? The question is answered in two steps: (i) the lived experience looks for its own form; (ii) the contents correlated with this form lead intuition into abstraction. The flow of time as a lived experience of successive instants can be fixed in points of time, and these points are not alien to lived experience since they are ordered in accordance with the relation before/after. However banal it may seem, this operation is a reduction which right away distorts intuition. In fact, the point-like representation of the flow is not adequate to the continuous flow of the present which sinks into the past. The mathematical determination of temporal position can take into account neither

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its sinking, nor its recapture in a form other than simple repetition: memory. The point isolated in time, Weyl argues, is “pure nothingness” which comes to existence only as a transitional point; such a point cannot be grasped mathematically. As the physico-geometrical universe is drawn out from the intuitively given, the ego is destroyed in favour of the objectivity of things. If nevertheless, assuming the discrepancy between original intuition and essence, mathematical construction is pursued in spite of its fundamental incongruity with respect to intuition, this is how it proceeds. What does this construction still capture from intuition? A temporal segment in a given time is delimited by two instants, and the stream of consciousness itself will have a form if its content remains unchanged when located elsewhere in time. This journey in time is doubtless the formal equivalent of memory in intuitive life; its material correlate is nothing else than the clock. Thus, at the foundation of the intuition of time lies the equality between two temporal segments: this is the condition of possibility for the measure of a time-length by means of real numbers. This is how a temporal segment becomes the origin and the unit of measurement for a coordinate system. However, this system does not provide an exact measure, but only an approximate one, because an individual act of the ego is still needed in order for the system to be specified. How can the determination of an object by measurement free itself from a purely individual specification in order to reach the full objectivity requested by exact knowledge? (STM pp. 8–10). The answer is that a system of coordinates must be given, the choice of which is now itself arbitrary. The subordination of the system of coordinates to an individual action is deleted and overtaken in favour of a conceptual determination ensured by the arbitrary choice of such a system. The conceptual determination as such is provided by the coordinates of the object in the system, i.e., numbers. By contrast, my ego can never be experienced arbitrarily as the ego of another. The price to pay for this strategy is that the objects of the physical world are no longer constituted by consciousness: they must be defined directly. Any physical theory which makes use of mathematics in order to overcome the singularity of the ego is bound to define its objects relatively to the chosen conceptual means. A relation in which a given instant is associated with a real number can be constructed, but the intuition of time is unable to validate the correspondence. To expect such a proof from intuition is literally a nonsense. Our consciousness, Weyl says, is incapable to jump over its shadow, yet it tries to do so and goes as far as possible in that effort. For practical and utilitarian purposes, as well as for the sake of the economy of thought, we could have been satisfied with the exact concept of a real number associated to phenomenal time, and go no further. Husserl did not believe that phenomenal time is measurable, because the essential character of phenomenal time can be grasped only if objective time is suspended. But it is not thus according to Weyl. There are reasonable motives which implicitly justify the attempt of consciousness to go beyond itself in order to reach the real “as such,” if not “in itself.” They reflect the authenticity of reason as it lays bare the “logos” dwelling in reality. The mathematical construction of the temporal continuum is the first step of the scientific mind as it apprehends the real and tries to uncover the objective hidden behind qualitative appearances. Reason struggles as far as it

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can against the nonsensical, and the first expression of this struggle shows in time consciousness. Here Weyl owns up to being unable to address more deeply the question of the motivation of science.

15.4 From Time to Space But just how does mathematical physics build its edifice on the basis of this precondition? The first contact of time consciousness with the objective world occurs through the idea of perseverance. Consider the conservation of mass or charge. Classical physics accounts for it by a tendency to perseverance, namely, inertia. The founding principle is indeed the principle of inertia: the world perseveres in its being owing to a tendency of the bodies to persevere in their state, whether this state is original or acquired. Now, the theoretical conception of inertial forces in Newton’s mechanics is logically incoherent. Logically, the inertial forces should be subjected to the law of equality between action and reaction. That is to say, if a physical entity such as absolute space generates inertial forces as observable effects, there is no reason why it would not undergo the reaction of the objects on which it acts. The only way to be faithful to this requirement is to substitute the Riemannian metric field to absolute space. To be sure, the metric field is no more observable than absolute space; but its fusion with gravity endows it with an observable origin, namely, the distribution of matter and energy in the universe (PMNS p. 105). Since all knowledge deals with two transcendent data, namely, the origin of things and the products of this origin, connecting the physical laws to an observable origin brings them together. That is why the idea that a physical cause must be affected by that on which it exerts effects expresses a general principle of universal reciprocity between the constitutive elements of nature. The epistemological priority of this principle reflects the evolution of physics since Galileo (PMNS p. 288). Whereas classical physics asked how the world continues in holding together its basic constituents, the question raised by the theory of relativity concerns more radically what makes the world hold together. If a quantity depending upon inertia is preserved, its initial value can always be set arbitrarily; nothing in the laws of classical physics constrains the value of mass or charge, since these laws are compatible with arbitrary values. Moreover, the preserved quantities are not really preserved, since perturbations will certainly occur in the course of the evolution of a system, and deviations from the initial values are expected in accordance with the past history of the universe. Now, this past history is not included in the laws, which reflect a cosmic order implicitly regarded as immutable. Is it possible to include the perturbations in universal reciprocity, so that the initial values would agree with the a priori contingent history of the universe? The answer to this question requires a critical progress over both classical physics and the Riemannian basis of general relativity.

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In the Riemannian metric field of the relativistic theory of gravitation, the fundamental metric quantity ds2 is measured by rods supposed to be rigid and clocks the period of which is supposed to be constant. The perseverance of mass and charge for the elementary particles in the constitution of rods and clocks justifies these two suppositions. That is why Riemannian geometry is still dependent on an ideal representation of the world, according to which two rods which coincide at some point will continue to do so after they have been transported to another point, independently of the path which is followed. This is nothing other than an undesirable residue of the ancient concept of action at a distance, which is at odds with the conception of natural laws as differential laws. A purely infinitesimal geometry is required, in which this condition of equality is dropped and indetermination is allowed for the comparison of lengths at a distance. In the new theory, comparison is restricted to points separated by infinitesimal intervals. Technically speaking, a purely infinitesimal geometry is constructed in terms of gauge invariance, which allows for connections only between adjacent points infinitely close to one another (PMNS p. 213). How is it possible to extend the infinitesimal connections in order to comply with universal reciprocity? This question echoes a long tradition of inquiry into the reciprocal reflection of microcosm and macrocosm. In order for the ds2 to remain a fundamental quantity, the behaviour of rods and clocks needs no longer to be preserved. Particles now adjust themselves in accordance with a definite proportion to a primitive field quantity. This quantity could not be as arbitrary as is the nature of the gauge system. Bodies adjust spontaneously to a natural gauge, which is in fact the radius of the universe. An elementary form of universal reciprocity was already present in the union of space and time achieved by the special theory of relativity. The higher level of universal reciprocity is achieved by the general theory, in which the spatiotemporal metric depends on the contingent distribution of matter and energy as much as this distribution depends on the metric. The full realisation of universal reciprocity implies that the cosmic order is not the result of a perturbation on a pre-existent configuration, as a persistent metric field can be. Rather, the cosmic order itself is to be accounted for without persistence as an ideal model. Adjustment is the tool needed to understand the world as reality. Indeed, if magnitudes persisted in their egoism, indifferent to their inclusion in one universe, perturbations due to their own past history ought to be observed. From the standpoint of adjustment, as Weyl explains, a definite value is not arbitrary because “it reasserts itself after any disturbances and any lapse of time as soon as the old conditions are restored” (PMNS p. 288). However ideal it may be, a common origin includes chance factors which “are never missing in a concrete development,” and indeed statistical physics demonstrates “how chance is by no means incompatible with ‘almost’ perfect macroscopic regularity of phenomena” (PMNS p. 294). To be sure, infinitesimal geometry allows us to think adjustment and reciprocity together, and in this way it is certainly an essential step toward the unification of all natural interactions. Things are proportional to each other, but Weyl acknowledges that the theory is not capable yet to fix the values for the factors of proportionality.

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Independently from the unfinished struggle against the nonsensical, the larger scope of the principle of the relativity of magnitude over the other formulations of relativity raises epistemological questions beyond the theory as such. It is precisely by keeping these questions in mind that Weyl addresses the principle. They go back to the living subject who constructs science: is this subject dependent on an analogous principle? The subject “sneaks in” the world, as it were: does it persevere in its unnatural transcendentality, or do subject and world adjust to each other? The principle of reciprocity is so universal that it applies to rival physical theories, assuming that each of them takes a step toward truth. Thus, the separation between geometrical (a priori) and material (a posteriori) characters of the world can never be absolute in any theory: while Einstein discovers the fundamental metric form from a physical behaviour, Weyl starts from this form to recover the behaviour. The inversion of the starting point shows the reciprocity of the theories. Similarly, if consciousness does not exhaust and can never exhaust the world, the world is not the residue left by consciousness after it was deleted from it. The ego’s relation to the world is best understood in terms of the relation between intuitive and physical space (PMNS p. 135). Suppose the metrical structure of intuitive space is Euclidean. Infinitesimal geometry restrains this Euclidean quality to an infinitely small neighbourhood around a point O. Let us now imagine an observer (an ego) located at this point. The relation of intuitive space to physical space can be represented by means of the image of a tangent plane (intuitive space) touching a curved surface (physical space). In the immediate vicinity of the point, the two spaces coincide inasmuch as the intrinsically curved space can still be intuited as a curved surface in the Euclidean space. As we move away from the ego centre, however, the correspondence between the two spaces becomes hazier as the curved surface is more and more Riemannian. The correspondence becomes more arbitrary as the distance from the ego increases. In the final analysis, the ego’s relation to the world is something like an intrusion that physics is bound to reduce to a minimum, so that the eidetic necessity of the natural laws is not completely contaminated by arbitrariness. The living subject of science is constituted by its own intuitive forces, as a transcendental subject whose adjustment to nature, if it is possible at all, is definitely singular.