Structures Mères: Semantics, Mathematics, and Cognitive Science [1st ed.] 9783030518202, 9783030518219

Table of contents :
Front Matter ....Pages i-xi
Reflections on Bourbaki’s Notion of “Structure” and Categories (John L. Bell)....Pages 1-17
Bourbaki and Foundations (Gabriele Lolli)....Pages 19-35
Forms of Structuralism: Bourbaki and the Philosophers (Jean-Pierre Marquis)....Pages 37-57
Ladders of Sets and Isomorphisms (Claudio Bartocci)....Pages 59-90
The Wrapped Dimension of Bourbaki’s Structures Mères (Alberto Peruzzi)....Pages 91-117
The Basic Structures of Motor Cognition (Silvano Zipoli Caiani)....Pages 119-134
Topological Aspects of Epistemology and Metaphysics (Thomas Mormann)....Pages 135-152
The Difficulty of Neutrality (Caterina Del Sordo)....Pages 153-163
Structures, Archetypes, and Symbolic Forms. Applied Mathematics in Linguistics and Semiotics (Wolfgang Wildgen)....Pages 165-185

Citation preview

Studies in Applied Philosophy, Epistemology and Rational Ethics

Alberto Peruzzi Silvano Zipoli Caiani   Editors

Structures Mères: Semantics, Mathematics, and Cognitive Science

Studies in Applied Philosophy, Epistemology and Rational Ethics Volume 57

Editor-in-Chief Lorenzo Magnani, Department of Humanities, Philosophy Section, University of Pavia, Pavia, Italy Editorial Board Atocha Aliseda Universidad Nacional Autónoma de México (UNAM), Mexico, Mexico Giuseppe Longo CNRS - Ecole Normale Supérieure, Centre Cavailles, Paris, France Chris Sinha School of Foreign Languages, Hunan University, Changsha, China Paul Thagard University of Waterloo, Waterloo, Canada John Woods University of British Columbia, Vancouver, Canada

Studies in Applied Philosophy, Epistemology and Rational Ethics (SAPERE) publishes new developments and advances in all the fields of philosophy, epistemology, and ethics, bringing them together with a cluster of scientific disciplines and technological outcomes: ranging from computer science to life sciences, from economics, law, and education to engineering, logic, and mathematics, from medicine to physics, human sciences, and politics. The series aims at covering all the challenging philosophical and ethical themes of contemporary society, making them appropriately applicable to contemporary theoretical and practical problems, impasses, controversies, and conflicts. Our scientific and technological era has offered “new” topics to all areas of philosophy and ethics – for instance concerning scientific rationality, creativity, human and artificial intelligence, social and folk epistemology, ordinary reasoning, cognitive niches and cultural evolution, ecological crisis, ecologically situated rationality, consciousness, freedom and responsibility, human identity and uniqueness, cooperation, altruism, intersubjectivity and empathy, spirituality, violence. The impact of such topics has been mainly undermined by contemporary cultural settings, whereas they should increase the demand of interdisciplinary applied knowledge and fresh and original understanding. In turn, traditional philosophical and ethical themes have been profoundly affected and transformed as well: they should be further examined as embedded and applied within their scientific and technological environments so to update their received and often old-fashioned disciplinary treatment and appeal. Applying philosophy individuates therefore a new research commitment for the 21st century, focused on the main problems of recent methodological, logical, epistemological, and cognitive aspects of modeling activities employed both in intellectual and scientific discovery, and in technological innovation, including the computational tools intertwined with such practices, to understand them in a wide and integrated perspective. Studies in Applied Philosophy, Epistemology and Rational Ethics means to demonstrate the contemporary practical relevance of this novel philosophical approach and thus to provide a home for monographs, lecture notes, selected contributions from specialized conferences and workshops as well as selected Ph.D. theses. The series welcomes contributions from philosophers as well as from scientists, engineers, and intellectuals interested in showing how applying philosophy can increase knowledge about our current world. Initial proposals can be sent to the Editor-in-Chief, Prof. Lorenzo Magnani, [email protected]: • A short synopsis of the work or the introduction chapter • The proposed Table of Contents • The CV of the lead author(s). For more information, please contact the Editor-in-Chief at [email protected]. Indexed by SCOPUS, ISI and Springerlink. The books of the series are submitted for indexing to Web of Science.

More information about this series at http://www.springer.com/series/10087

Alberto Peruzzi Silvano Zipoli Caiani •

Editors

Structures Mères: Semantics, Mathematics, and Cognitive Science

123

Editors Alberto Peruzzi University of Florence Florence, Italy

Silvano Zipoli Caiani University of Florence Florence, Italy

ISSN 2192-6255 ISSN 2192-6263 (electronic) Studies in Applied Philosophy, Epistemology and Rational Ethics ISBN 978-3-030-51820-2 ISBN 978-3-030-51821-9 (eBook) https://doi.org/10.1007/978-3-030-51821-9 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are solely and exclusively licensed 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

During the twentieth century, the term “structure” became commonplace in almost every scientific domain as well as in the humanities. In order to understand what a thing is, so went the structuralist slogan, study its structure, or at least, the structure of the system of which it is a part. In the present day, the term occurs frequently, even in ordinary speech, although its meaning typically varies with the context. This is a striking linguistic fact, since before the twentieth century the term was used only infrequently. More importantly, as the term “structure” spread, it came to be associated with a number of related claims posited for the “foundational” purpose of eliminating the commitment to the idea of “substance” which had long dominated philosophy. While the new use of the term was not entirely novel, it led to the emergence of a general structure-based doctrine. In its turn, this engendered a well-defined body of theoretical commitments and methods, in which basic assumptions of an ontological and epistemological character play a central role. As applied to a given system of whatever sort, the term “structuralism” is intended to convey the view that attention should be directed primarily toward the system’s overall structure, rather than to its constituents. Now the concept of structure can be defined in a variety of ways. Occasionally, this fact is exploited in a naïve, intuitive, sense. While harmless enough in itself, the resulting ambiguity in meaning of the term “structure” can give rise to divergent consequences. Similarly, the doctrine of structuralism itself can also be formulated in a number of different ways. Here clarification is needed. The upshot is that, as a doctrine, structuralism, far from being monolithic, is susceptible to a number of interpretations. Nevertheless, certain features of structuralism remain constant across its various domains of interpretation. These range from general linguistics, where a structuralist view was first introduced by Ferdinand de Saussure, to “structural” anthropology by Claude Lévi-Strauss, from cognitive psychology, in the work of Jean Piaget to… mathematics. Within mathematics, structuralism is mainly associated with the program of thoroughly (that is, with generality and rigor) “cleaning up” language and theory in every area of mathematical research, from analysis (where the project began) to algebra and topology. This program was undertaken in earnest in the 1930s by a v

vi

Introduction

group of mathematicians (French for the most part), who adopted as their collective pseudonym “Nicolas Bourbaki”—after an actual French general, Charles Bourbaki, who took part to the 1870–71 war, which saw the victory of Prussia over France. In Bourbaki’s hands, structuralism took on a very specific shape, the features of which are the subject matter of the greater part of the present volume. Some of the contributed papers are devoted to reconstructing (at least, in part) the historical genesis of Bourbaki’s structuralism; others describe the consequences, and non-consequences, of formal characterizations of the concept; still others explore the philosophical import of positing as primary building blocks one kind of structure rather than another. The French expression used in the title of this book, meaning “mother-structures”, comes from Bourbaki’s manifesto “L’architecture des Mathématique” (1948) and is intended to make clear that the core of the present book lies in Bourbaki’s approach to mathematics. However, as the reader will soon see, the theme of “mother-structures” recurs in different ways and with different emphases throughout the papers which, in the second part of this volume, deal with structuralism in areas of investigation beyond mathematics. The idea of producing this book was stimulated by a short but memorable conference held in May 10–12, 2017 at the University of Florence. The intended purpose of the conference was to showcase the importance of Bourbaki’s legacy for current research on open questions of both mathematics and philosophy; in fact the book’s title is almost identical with that of the meeting, inverting just the word order in the subtitle, which was originally “semantics, mathematics and cognitive sciences.” The editors of this volume did not originally intend to collect, for publication, the talks given at the conference. What convinced us that it was a viable option that was the constructive spirit in discussion, the atmosphere of mutual interest which emerged among the meeting’s participants, and the importance they assigned to its topic. It soon became clear as the conference unfolded that the importance of Bourbaki’s structuralism is not confined to its history, but also extends to current research in the foundations of mathematics and, more generally, to the use of mathematical notions linked to the structuralist perspective introduced to deal with central problems of epistemology and ontology. A particular and recurring theme of interest was the controversial issue of how strictly structuralism, in Bourbaki’s or other mathematical forms, requires the use of category theory and in what sense category theory itself calls for a structuralist interpretation, since the way this issue is dealt with affects both the “architecture” of mathematics and the specific form structuralism can take in relation to other fields. During the conference, we realized that other aspects of the subject called for examination, and accordingly further contributions had to be added to the talks given on the occasion. In particular, we wanted to include an analysis of the evolution of Bourbaki’s views and their actual realization in the epoch-making effort embodied in the many volumes of the Éléments de mathématique. We were happy that a number of expert scholars accepted our invitation to deal with these aspects, so filling the main gaps. As a result, a sufficiently wide collection of essays

Introduction

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emerged, though no doubt there are further core aspects which deserve to be examined, in particular, Bourbaki’s form of mathematical structuralism, with the place to be assigned (or not) to structures mères, and structuralist ideas which, along more or less similar lines of research, have emerged in areas such as linguistics, semiotics, phenomenology, ontology, and philosophy of mind. The contents of this volume are accordingly divided into two groups of chapters. The first group focuses on aspects of Bourbaki’s notion of structure (chapters “Reflections on Bourbaki’s Notion of “Structure” and Categories”–“The Wrapped Dimension of Bourbaki’s Structures Mères”), while the second group focuses on concepts of a structuralist nature which have evolved in a more general epistemological and ontological setting (chapters “The Basic Structures of Motor Cognition”–“Structures, Archetypes, and Symbolic Forms. Applied Mathematics in Linguistics and Semiotics”). While connections between the first and second groups are scattered throughout the arguments worked out in almost all the essays, these connections are made explicit when the notion of structure mére is taken at face value (see particularly chapter “The Wrapped Dimension of Bourbaki’s Structures Mères”). The volume opens with the contribution of John Lane Bell titled “Reflections on Bourbaki’s Notion of ‘Structure’ and Categories”. Here Bell traces a path from the first appearance of Bourbaki’s Éléments de mathématique to the rise of category theory. In doing this, Bell introduces category theory as a set of concepts concerning mathematical forms that marks a fundamental advance over Bourbaki’s ideas of structures and species. In particular, Bell describes how the central notion of “mother structure” in Bourbaki’s mathematics becomes identified with ordered and topological structures within the category-theoretic formulation. According to Bell, the transition from Bourbaki’s account of mother-structures in terms of set theory, to the categorical formulation transcends the “purely cosmetic”, and offers a conceptual gain unavailable from the classical view. This is well illustrated by the “smooth infinitesimal analysis” made possible by category theory, which does not rest on the classical set-theoretic formulation of the Cauchy–Weierstrass account of the continuum of real numbers and avoiding the controversial idea that the continuum is composed of discrete atoms. The second chapter entitled “Bourbaki and Foundations”, by Gabriele Lolli, provides an analysis of the intrinsic issues arising in Bourbaki’s foundationalist account of mathematics. Lolli accepts Dieudonné’s view that traces back to Hilbert the idea of a purely syntactic approach to mathematics, but sees difficulties in the uncritical assumption that all mathematical structures are ultimately extensions of set theory. According to Lolli, this assumption was one of the principal reasons that Bourbaki’s foundational program failed. Ironic as it may seem, the very set theory which Bourbaki took as the embodiment of absolute formal rigor, is revealed to be too imprecise and sloppy to be used as the basis of all mathematical knowledge. In the third chapter, titled “Forms of Structuralism: Bourbaki and the Philosophers”, Jean-Pierre Marquis challenges the widespread idea that Bourbaki’s structuralism is not relevant for the philosophy of mathematics. Marquis argues that Bourbaki’s technical analysis of structures illuminates the ultimate

viii

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nature of mathematical structuralism. After introducing the main argument used by philosophers to dismiss the importance of Bourbaki’s structuralism, Marquis considers several aspects of mathematical structuralism, from the axiomatization program to the very definition of isomorphism and structure, and shows that Bourbaki’s ideas provide a general formal framework for building up abstract concepts. The fourth chapter by Claudio Bartocci, entitled “Ladders of Sets and Isomorphisms”, provides a detailed analysis of the historical and conceptual path leading from early attempts to develop a structuralist approach to algebra, to the recent development of category theory. Bartocci begins by acknowledging the pioneering efforts of Emmy Noether and her colleagues in introducing a structuralist account of algebra. Bartocci goes on to describe Bourbaki’s early attempts to formalize the notion of structure and the related issues they encountered during their work. Finally, Bartocci explains the relationship between Bourbaki’s program and the subsequent development of category theory, showing how the rise of the latter strengthened the idea of a purely formal approach to mathematics. The fifth chapter, by Alberto Peruzzi, represents the link between the first and the second part of this volume. In his contribution, entitled “The Wrapped Dimension of Bourbaki’s Structures Mères”, Peruzzi discusses the theoretical relevance of Bourbaki’s view for the general understanding of the notion of structure. In particular, Peruzzi focuses on Bourbaki’s assumption that axiom systems constitute implicit definitions of the concepts axioms express and argues that if the concept of structure is only implicitly defined, then part of its explanatory power is inevitably missed. Accordingly, Peruzzi addresses what is known as the “structure grounding problem”, suggesting the role of spatial interaction patterns as generators of all varieties of possible structures. According to Peruzzi, such a topologicalschematic approach can benefit from the conceptual tools provided by category theory and allows for a generalization of the structuralist program outside the boundaries of mathematics, to wit, semantics and cognitive sciences. In the sixth chapter, entitled “The Basic Structures of Motor Cognition”, Silvano Zipoli Caiani shows how some fundamental concepts of category theory can be employed in the cognitive modeling of action planning and execution. Zipoli Caiani focuses on the representational formats that are involved in action cognition, that is, the structures that informational contents can take to deliver information. In particular, according to Zipoli Caiani the content of the action concepts that constitute an intention to act can be interpreted as a function linking an initial bodily state to a goal state. Such a structure resembles that of a morphism, which configures a change in the agent’s body (or a part of it), moving it from an initial shape to a final shape. Such a morphism represents the execution of a motor act, which in turn may be suitable or unsuitable for prescribing the execution of an intended action. In the seventh chapter, entitled “Topological Aspects of Epistemology and Metaphysics”, Thomas Mormann maintains that elementary topology is a useful tool for dealing with classical philosophical issues. In particular, according to Mormann, a topological approach is of crucial importance in dealing with questions

Introduction

ix

that cannot be addressed by the mere manipulation of symbols, such as the problem of the origin and meaning of logical terms and rules. To support this claim, the author forges a link between topology and epistemology, showing how the conceptual system of knowledge emerges from spatial representations and structures, the understanding of which is best expressed topologically. The eight chapter, by Caterina Del Sordo, entitled “The Difficulty of Neutrality. A Graph-Theoretical Solution”, deals with the philosophical theory of neutral monism, and shows how the definition of neutral entities can be obtained by using graph-theoretical structures. Del Sordo divides neutral monism into three main currents and provides the problem of defining neutral entities with an analytical systematization. After presenting the main issues that plague classical theories of neutral entities, the author introduces a graphic-theoretical model in the spirit of Carnap’s early idea of relational reconstruction of empirical reality. Finally, in the ninth chapter entitled “Structures, Archetypes and Symbolic Forms, Applied Mathematics in Linguistics and Semiotics”, Wolfgang Wildgen addresses the issue of whether an elaboration of the mathematics of structure is a rewarding strategy. In order to answer this question, Wildgen draws on Cassirer’s philosophy of symbolic forms according to which a tremendous variety of structures are postulated as basic principles of the human capacity of producing language, art and science. Although it may seem difficult to support Cassirer’s intuition with empirical data, Wildgen finds significant cases in the fields of linguistics, visual art and music, where mathematics exhibits amazing explanatory power. In conclusion, it should be remarked that, taken as a whole, this collection of nine chapters does not manifest perfect convergence of any chapter with any other, nor is it intended to. Each chapter sheds light on a specific topic and, if on certain issues there is a remarkable coherence, or even a sort of unestablished harmony, of other issues the same cannot be said. The Editors hope that the reader will benefit from such a plurality of perspectives, by being invited to weigh the arguments, pros or cons, for specific claims about each of the manifold aspects of Bourbaki’s structuralism examined in the first group of chapters, and, in the second group, the arguments exploring structuralist approaches to nonmathematical topics. Alberto Peruzzi Silvano Zipoli Caiani

Contents

Reflections on Bourbaki’s Notion of “Structure” and Categories . . . . . . John L. Bell

1

Bourbaki and Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gabriele Lolli

19

Forms of Structuralism: Bourbaki and the Philosophers . . . . . . . . . . . . Jean-Pierre Marquis

37

Ladders of Sets and Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Claudio Bartocci

59

The Wrapped Dimension of Bourbaki’s Structures Mères . . . . . . . . . . . Alberto Peruzzi

91

The Basic Structures of Motor Cognition . . . . . . . . . . . . . . . . . . . . . . . . 119 Silvano Zipoli Caiani Topological Aspects of Epistemology and Metaphysics . . . . . . . . . . . . . . 135 Thomas Mormann The Difficulty of Neutrality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Caterina Del Sordo Structures, Archetypes, and Symbolic Forms. Applied Mathematics in Linguistics and Semiotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 Wolfgang Wildgen

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Reflections on Bourbaki’s Notion of “Structure” and Categories John L. Bell

Abstract The article is a discussion of the nature of structure in Bourbaki’s “Elements de Mathematique”, contrasting it with the account of structure arising in category theory. It is also explained how Bourbaki’s concept of “mother structure” is given category-theoretic form.

1 Structure in Bourbaki’s Éléments de Mathématique A remarkable passage in Edward Gibbon’s Decline and Fall of the Roman Empire goes The mathematics are distinguished by a particular privilege, that is, in the course of ages, they may always advance and can never recede.

Gibbon’s assertion could serve as the starting point of an absorbing discussion of whether “the mathematics” has in fact never made a backward step. But I quote it here only to draw attention to the fact that Gibbon uses the plural form “mathematics”, even with the (now obsolete use of) the definite article. This may be because classical Greek mathematics—the quadrivium—was a plurality, divided into arithmetic, geometry, astronomy and music. In English the singular form “mathematic” does not exist as a noun, but in French the singular form la mathématique and the plural form les mathématiques are both acceptable, even if the singular has a whiff of the archaic. Bourbaki adopted the singular form in entitling his masterwork Éléments de Mathématique.1 I have a particular affection for Bourbaki’s Élements because it—more specifically, the chapters on Topologie Génerale, Théorie des Ensembles, 1 Bourbaki 2 Mashaal

(1939). (2006).

J. L. Bell (B) University of Western Ontario, Ontario, Canada e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 A. Peruzzi and S. Zipoli Caiani (eds.), Structures Mères: Semantics, Mathematics, and Cognitive Science, Studies in Applied Philosophy, Epistemology and Rational Ethics 57, https://doi.org/10.1007/978-3-030-51821-9_1

1

2

J. L. Bell

and Algèbre opened my undergraduate eyes to the world of what I was pleased to identify as “real” mathematics. It has been suggested, quite plausibly, by Maurice Mashaal2 that the use of the singular “mathématique” in the title of the Élements is tendentious, intended to convey the authors’ conviction that mathematics is a unity, contrary to what the use of the plural form of the term “mathematics” might suggest. Mashaal also claims that the use of the plural form in Bourbaki’s Éléments d’histoire des mathématiques3 is intended to indicate that, before they appeared on the scene, mathematics was a set of scattered practices, and that the modern notion of structure enabled these practices to become fused into a single discipline. This claim also has a certain plausibility. Bourbaki’s article The Architecture of Mathematics (ghostwritten by J. Dieudonné),4 begins with a question: Mathematic or mathematics? The article continues: To present a view of the entire field of mathematical science as it exists, - this is an enterprise which presents, at first sight, almost insurmountable difficulties, on account of the extent and the varied character of the subject. As is the case in all other sciences, the number of mathematicians and the number of works devoted to mathematics have greatly increased since the end of the 19th century. The memoirs in pure mathematics published in the world during a normal year cover several thousands of pages. Of course, not all of this material is of equal value; but, after full allowance has been made for the unavoidable tares [weeds], it remains true nevertheless that mathematical science is enriched each year by a mass of new results, that it spreads and branches out steadily into theories, which are subjected to modifications based on new foundations, compared and combined with one another. No mathematician, even were he to devote all his time to the task, would be able to follow all the details of this development. Many mathematicians take up quarters in a corner of the domain of mathematics, which they do not intend to leave; not only do they ignore almost completely what does not concern their special field, but they are unable to understand the language and the terminology used by colleagues who are working in a corner remote from their own. Even among those who have the widest training, there are none who do not feel lost in certain regions of the immense world of mathematics; those who, like Poincaré or Hilbert, put the seal of their genius on almost every domain, constitute a very great exception even among the men of greatest accomplishment. It must therefore be out of the question to give to the uninitiated an exact picture of that which the mathematicians themselves cannot conceive in its totality. Nevertheless it is legitimate to ask whether this exuberant proliferation makes for the development of a strongly constructed organism, acquiring ever greater cohesion and unity with its new growths, or whether it is the external manifestation of a tendency towards a progressive splintering, inherent in the very nature of mathematics, whether the domain of mathematics is not becoming a tower of Babel, in which autonomous disciplines are being more and more widely separated from one another, not only in their aims, but also in their methods and even in their language. In other words, do we have today a mathematic or do we have several mathematics?

Here Bourbaki/Dieudonné uses the phrase “tower of Babel” with its usual connotation of “place of confusion”. But it is worth pointing out that, according to Genesis, it was the fact that human beings possessed only a single language that enabled them to embark on the construction of the tower of Babel. God frustrates these aims by the 3 Bourbaki 4 Bourbaki

(1994) (1950).

Reflections on Bourbaki’s Notion of “Structure” and Categories

3

introduction of linguistic diversity. Before God’s intervention, the Tower of Babel was actually a place of order, not confusion. Thus the tower of Babel might be seen, not as representing the jumble of separate practices that Bourbaki deplores, but rather as the unity that they wished to impose on mathematics. In that case, Bourbaki’s Élements would, ironically perhaps, amount precisely to the attempt to build a mathematical “tower of Babel”. The fact that Bourbaki failed—as is wellknown—to complete his grandiose project as originally conceived was not, as in Genesis, the result of God sowing linguistic confusion - the Bourbaki members, after all, still spoke a common mathematical language. Rather, Bourbaki’s project was simply too ambitious to be brought to completion. Nevertheless, the Éléments, unlike the tower of Babel, remains a magnificent plinth. In the Architecture of Mathematics Bourbaki/Dieudonné asserts that the unity of contemporary mathematics rests on the axiomatic method, and that the latter, in mathematics at least, rests in turn on the notion of structure. Bourbaki identifies three basic types of mathematical structure—structures mères”—or “mother structures”. These are algebraic, order, and topological structures, which can be summed up as the “three C’s”: Combination, Comparison and Continuity. The group concept is presented as a simple, fundamental kind of mathematical structure: One says that a set in which an operation … has been defined which has the three properties (a), (b), (c) is provided with a group structure…. (or, briefly, that it is a group); the properties (a), (b), (c) are called the axioms of the group structures.

Here we see that a group is a set, while group structure is a “something”, specified by axioms, imposed on the set. The use of the term “axiom” to specify structure is analogous to the Definitions of geometric objects (as opposed to the axioms and postulates) in Euclid’s Elements. The text continues: It can now be made clear what is to be understood, in general, by a mathematical structure. The common character of the different concepts designated by this generic name, is that they can be applied to sets of elements whose nature has not been specified; to define a structure, one takes as given one or several relations, into which these elements enter, then one postulates that the given relation, or relations, satisfy certain conditions (which are explicitly stated and which are the axioms of the structure under consideration.) To set up the axiomatic theory of a given structure amounts to the deduction of the logical consequences of the axioms of the structure, excluding every other hypothesis on the elements under consideration (in particular, every hypothesis as to their own nature).

Now this passage does not make it entirely clear what is to be understood by a “mathematical structure”. It would seem that a structure is to be taken as a definite set having some prescribed form. This impression is reinforced by the fact that in the Théorie des Ensembles Bourbaki uses the phrase “structure of the species T ”. A species is thus a collection of structures sharing a common form. As to the notion of set itself, we read in a footnote:

4

J. L. Bell We take here a naive point of view and do not deal with the thorny questions, half philosophical, half mathematical, raised by the problem of the “nature” of the mathematical “beings or “objects.” Suffice it to say that the axiomatic studies of the nineteenth and twentieth centuries have gradually replaced the initial pluralism of the mental representation of these “beings” thought of at first as ideal “abstractions” of sense experiences and retaining all their heterogeneity-by a unitary concept, gradually reducing all the mathematical notions, first to the concept of the natural number and then, in a second stage, to the notion of set. This latter concept, considered for a long time as “primitive” and “undefinable,” has been the object of endless polemics, as a result of its extremely general character and on account of the very vague type of mental representation which it calls forth; the difficulties did not disappear until the notion of set itself disappeared (my boldening) and with it all the metaphysical pseudo-problems concerning mathematical “beings” in the light of the recent work on logical formalism.

The observation that the difficulties did not disappear until the notion of set itself disappeared is striking. What Bourbaki seems to mean is that while the notion of a mathematical structure in the strictest sense is dependent on the concept of set, in using mathematical structures the intrinsic properties of the sets (whatever these are) from which the structures are actually built can be safely ignored. In handling structures all one needs to know is that [the structure in question] “can be applied to sets of elements whose nature has not been specified”. Accordingly, the departure from the scene of the concept of set opened the way for Bourbaki to maintain that the unity of mathematics stems, not from the set concept, but from the concept of structure. Bourbaki’s manifesto can be seen as a declaration, avant la lettre, of what has come to be termed mathematical strcturalism. Yet, as Leo Corry5 has pointed out, the concept of mathematical structure as such plays a very minor role in Bourbaki’s development of mathematics in the Éléments de Mathematique. True, Chap. 4 of the Théorie des Ensembles is devoted to the presentation of a theory of structures in which, roughly speaking, a structure is defined to be a collection of sets together with functions and relations on them. Similar structures are organized into what are called species. (This theory cane to be described by Pierre Cartier, a later Bourbaki member, as “a monstrous endeavor to formulate categories without categories”.) But the cumbersome mechanism fashioned there is never again called forth. All of the succeeding volumes of the Éléments can be read in complete ignorance of what Bourbaki terms a “structure”. In particular, no explanation is provided of the importance of the “mother structures”. The hierarchy of structures as presented in the Elements is best understood as an (unconscious) version of simple type theory. For, give a collection C of base sets (types), Bourbaki’s “structures” are essentially just the members of the universe C* of sets obtained by closing C under the operations of power set and Cartesian product. (Since Bourbaki takes ordered pairs as primitive, rather than defining them as sets, the operation of Cartesian product is required.) The types of simple type theory can, analogously, be obtained by starting with a collection T of base types and closing under the operations of power type, product, and subtype. Church’s definitive 1940 5 E.g.

in Corry (1992).

Reflections on Bourbaki’s Notion of “Structure” and Categories

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formulation of simple type theory is actually based on functions rather than relations or classes, and incorporates certain features of the λ-calculus which he had already developed. It seems unlikely, given Bourbaki’s well-known distaste for logic, that he would have known of Church’s contributions, and even if he had, he would likely have regarded it as irrelevant to his concerns. In any case, as already remarked, Bourbaki’s general concept of structure, organized into species, plays only a very minor role in his actual development of mathematics. By and large, only specific kinds of structure are discussed e.g., topological spaces, algebraic structures, and combinations of the two such as topological groups. In practice, the role of structure in general is played by the defining axioms of the various species of structures.

2 Category Theory as a Theory of Mathematical Structure and Form In the middle 1940s, a decade after the launch of the Éléments, Eilenberg (later to become a Bourbaki member) and Mac Lane invented category theory.6 Their original intention was to systematize the construction of homology theories, a procedure in algebraic topology which involves the correlation of topological spaces and groups—two of Bourbaki’s mother structures. This correlation between different sorts of structure—the key idea underlying category theory—was termed by Eilenberg and Mac Lane a functor. The idea of a category was introduced to underpin the notion of functor by furnishing it (like a function) with a definite domain and range. They conceived functors as acting not just on the structures themselves but also on the ‘structure-preserving’ maps, or morphisms, between structures. Accordingly categories would have to contain these latter as well. The recognition that categories would have to incorporate as basic constituents not just structures but morphisms marks the fundamental advance of the category concept over Bourbaki’s idea of species.7 Eilenberg and Mac Lane rather played down the notion of category, stating: It should be observed … that the whole concept of a category is essentially an auxiliary one; our basic concepts are essentially those of a functor and of natural transformation (…). The idea of a category is required only by the precept that every function should have a definite class as domain and a definite class as range, for the categories are provided as the domains and ranges of functors.

But this view was to change. Starting in the 1950s, the category concept came to be perceived as a nascent embodiment of the idea of mathematical structure in general, in which Bourbaki’s conception of mathematical structure as individual structures, defined in set-theoretic terms and only then organized into species, is replaced by 6 Eilenberg 7 Roughly

and Mac Lane (1945). speaking, Bourbaki’s species of structures correspond to “mapless” categories.

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the category of all such structures given in advance. The Bourbaki fraternity became uncomfortably aware that their program of structuralist grounding of mathematics might be better realized through the systematic use of category theory, but by then it was too daunting a project to reconstruct the whole of their oeuvre in categorytheoretic terms.8 In any case it is far from clear—even today when category theory has assumed a commanding place in confirming the unity of mathematics- how this could actually have been done.9 Category theory offers an account of mathematical structure far transcending that pioneered by Bourbaki, opening doors of conception whose very existence was previously undreamt of. What is a category? Formally, a category C is determined by first specifying two assemblies Ob(C), Arr(C)—the of C-objects and C-arrows, C- morphisms, or C -maps. These are subject to the following axioms: • Each C-arrow f is assigned a pair of C-objects dom(f ), cod(f ) called the domain and codomain of f , respectively. To indicate the fact that C-objects X and Y are f

respectively the domain and codomain of f we write f : X → Y or X −→ Y . The collection of C-arrows with domain X and codomain Y is written C(X, Y ). • Each C-object X is assigned a C-arrow 1X : X → X called the identity arrow on X. • Each pair f, g of C-arrows such that cod(f ) = dom(g) is assigned an arrow g ◦ f : dom(f ) → cod(g) called the composite of f and g. Thus if f : X → Y and g: Y → Z f

g

then g ◦ f : X → Z. We also write X −→ Y −→ Z for g ◦ f. Arrows f, g satisfying cod(f ) = dom(g) are called composable. • Associativity law. For composable arrows (f, g) and (g, h), we have h ◦ (g ◦ f ) = h ◦ (g ◦ f ). • Identity law. For any arrow f : X → Y, we have f ◦ 1X = f =1Y ◦ f. The concept of category may be regarded as vastly generalized and streamlined, yet richer version of Bourbaki’s concept of species of structures. While a Bourbakian species is composed solely of structures, the structures (objects) of a category comprise just half of its constituents, the structure-preserving maps (arrows) between them furnishing the other half. In this spirit we may think of a category as an explicit presentation—an embodiment—of a mathematical Form or Structure, together with the various ways in which that Form is preserved under transformations. The objects of a category C are then naturally identified as instances of the associated Form C and the morphisms or arrows of C as transformations of such instances which in some specified sense “preserves” the Form. If we identify categories with Forms, then the specification of a Form requires us to specify, along with its instances, the 8 It

should be noted, however, in Chap. 4 of the Théorie des Ensembles Bourbaki does formulate versions of certain concepts—such as universal arrows and the solution set condition for their existence—which were later to become central to category theory. Mac Lane (1971) remarks that Bourbaki’s formulation “was cumbersome because [their] notion of ‘structure’ did not make use of categorical ideas”. 9 A pioneering first step in this regard at an elementary level was undertaken by Lawvere and Schanuel in their work Conceptual Mathemtics (Lawvere and Schanuel 1997).

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transformations which “preserve” it. This opens up the possibility that two Forms may have the same instances but different Form-preserving transformations. This is illustrated in the first three of the examples below: Category/Instances of Form

Form

Transformations

Sets

Pure discreteness

Functional correlations

Sets with a Distinguished Point (DP)

Pure discreteness

DP— preserving functional correlations

Sets with partial maps

Pure discreteness

Functional correlations on parts

Groups

Composition/Inversion

Homomorphisms

Topological spaces

Continuity

Continuous maps

Differentiable manifolds

Smoothness

Smooth maps

In this spirit a functor between two categories may be identified as a pair of correlations (satisfying certain simple conditions): the first between instances of the two Forms embodied by the given categories and the second between Formpreserving transformations of these instances. In short, a functor is a Form-preserving correlation between (the instances of) two Forms. To be precise, a functor F: C → D between two categories C and D is a map that “preserves commutative diagrams”, that is, assigns to each C-object A a D-object FA and to each C-arrow f: A → B a D-arrow Ff : FA → FB in such a way that:

When categories are regarded as Forms, a functor between two Forms is a correlation between instances (transformations) of the first Form with instances (transformations) of the second which preserves composites of morphisms and identity morphisms. Functors considered as acting on Forms will be called formorphisms. If the objects of a category are the instances of a given Form, when should two of these instances be regarded as identical? Precisely when they are isomorphic (Greek: equal form). In a category two objects, are seemed isomorphic, written ∼ =, if there is an invertible morphism, an isomorphism, from one to the other. In Bourbaki’s formulation isomorphisms and isomorphic structures are defined set-theoretically in terms of bijections. One of the principal aims of the structuralist approach to mathematics is to take seriously the idea that isomorphic structures should be regarded as in a fundamental sense identical. On the set-theoretic account of structures, this is not literally possible. In category theory, however, each category is equivalent (in a sense to be introduced

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below) to a skeletal category, one in which isomorphic objects are always identical. So, if we take the further step of identifying Forms with skeletal categories, isomorphic instances of Forms are literally identical. If isomorphism is construed as identity of instances of a given Form, then how should expression be given to the idea of identity of Forms themselves? In Bourbaki’s set-theoretic account of structures this question is never raised, nor would there seem to be any reasonable answer in Bourbaki’s framework. But category theory deals with this question most elegantly, through the idea of equivalence of categories. Given two categories C and D, a functor F: C→ D is an equivalence if it is “an isomorphism up to isomorphism”, that is, if it is • faithful: Ff = Fg ⇒ f = g. • full: for any h: FA → FB there is f : A → B such that h = Ff. • dense: for any D-object B there is a C-object A such that B ∼ = FA. If categories are regarded as Forms, then an equivalence between two Forms C and D is a form orphism from one Form to the other which is bijective on transformations and is such that each instance of D is isomorphic to the correlate of an instance of C. Two categories, or Forms are equivalent, written ≈, if there is an equivalence between them. Equivalence of Forms means that, considered purely as Forms, they can be taken as identical. The idea of equivalence of Forms afforded by category theory is rich and deep. As a simple example, consider the two categories: Sets with Partial Maps (SPM) and Sets with a Distinguished Point (SDP). Objects of SPM are pairs of sets (X, U) with U ⊆ X and an arrow (X, U) → (Y, V) between two such objects is just a function f: U → V. Objects of SDP are pointed sets, i.e. pairs (X, a) with X a set and a ∈ X. An arrow (X, a) → (Y, b) between two such objects is a function f : X → Y such that f(a) = b. These two categories are equivalent. The equivalence correlates each object (X, U) of SPM with the pointed set (X ∗, l) where X* = X ∪ {l} is the set obtained by adding a distinguished “point at infinity” (l) to X. Each arrow f: (X, U) → (Y, V) of SPM is correlated with the arrow f*: (X*, l) → (Y*, l) in SDP defined by f*(x) = / U. This is depicted below. f (x) for x ∈ U, f*(x) = l for x ∈

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f (X,U)

(Y, V ) V

U

f* (X*,  )

(Y*,  )

U

V





Adopting the language of Forms, the objects of the category SPM can be considered as instances of the Form Whole and Part, with transformations strictly between parts. (This is to be distinguished from transformations between wholes which preserve parts, which leads to a different category and Form.) The objects of SDP can be considered as instances of the Form Whole and Distinguished Element, with transformations between Wholes preserving distinguished elements. The equivalence of the two categories, and so of the associated Forms, means that Whole-Part Forms

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are Formally the same as Whole-Individual Forms



in which Parts of Wholes are, so to speak, ‘shrunk’ to points. Another important concept in category theory which has a satisfying formulation in terms of Forms is that of opposite, or mirror category. Given a category C, the opposite, or mirror category is defined to be the category C⇓ whose objects are those of C but whose arrows are those of C “reversed”, or “viewed in a mirror’, if you will. That is, the arrows X → Y in C⇓ are the arrows Y → X in C. Simple examples of mirror categories are obtained by considering preoredered sets. A preorder on a set P is a reflexive transitive relation on P. A preordered set is a pair P = (P, ≤) consisting of a set P and a preorder ≤ on P. Preordered sets can be identified with categories in which there is at most one arrow between each pair of objects. Consider the preordered set N = (N, ≤) where N is the set of natural numbers and ≤ is the usual equal to or less than relation on it. Regarding N as a category, its mirror category N⇓ may be identified with the ordered set of negative numbers. Given a category C with associated Form C, the mirror Form C⇓ is the Form associated with the mirror category C⇓ . The Form N associated with the category N is limitless succession. Its mirror N⇓ may be called limitless precession. For a given category (Form), the associated mirror category (Form) is usually difficult to identify as an autonomous category. But in certain important cases, mirror categories can be shown to be equivalent to naturally defined categories. Given two categories C and D, a duality between C and D is an equivalence between C⇓ and D (or, what amounts to the same thing, between C and D⇓ ).

3 Duality Theory for Commutative Rings This sort of duality is the core of the important representation theory for commutative rings. Here is a brief history.10 The concept of commutative ring (with identity) provides a basic link between algebra and geometry. Commutative rings arise naturally as algebras of values of 10 See

Johnstone for a full account of the representation theory for commutative rings.

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(intensive) quantities over topological spaces. For example, consider the earth’s atmosphere A. There are many intensive quantities defined on A—temperature, pressure, density, (wind) velocity, etc. The real number values of these quantities varies continuously from point to point. In general, we can define a (continuously varying value of an) intensive quantity on A to be a continuous function on A to the field R of real numbers. Intensive quantities construed in this way form an algebra in which addition and multiplication can be defined “pointwise”: thus, given two intensive quantities f, g, the sum f + g and the product fg are defined by setting, for each point x in A, ( f + g)(x) = f (x) + g(x)( f g)(x) = f (x)g(x). In general, given a topological space X, we consider the set C(X) of continuous real-valued functions on X, with addition and multiplication defined pointwise as above. This turns C(X) into a commutative ring, the ring of real-valued (continuously varying) intensive quantities over X. We can also consider the subring C*(X) of C(X) consisting of all bounded members of C(X), the ring of bounded intensive quantities over X. When X is compact, C*(X) and C(X) coincide. More generally, given any commutative topological ring T, the ring C(X, T) of continuous T-valued functions on X is called the ring of T-valued intensive quantities on X. Given a commutative ring, it is natural to raise the question as to whether it can be represented as a ring of intensive quantities (with values in some commutative topological ring) on some topological space. It was M. H. Stone who provided the first answer to this question. In the celebrated Stone Representation Theorem, proved in the 1930s, he showed that each member of a certain class of commutative rings, the so-called Boolean rings, is representable as a ring of intensive quantities—with values in a fixed simple topological ring (the two-element ring 2 = {0, 1}) over a certain class of spaces—the Boolean or Stone spaces. A Boolean ring is defined to a ring in which every element is idempotent, x 2 = x for every x. A totally disconnected compact Hausdorff space is called a Boolean space. The Stone Representation Theorem establishes the duality between the category of Boolean rings and the category of Boolean spaces. In 1940 Stone established what amounts to the duality between the category of compact Hausdorff spaces and the category of real C*-algebras—commutative rings equipped rings with an order structure and a norm naturally possessed by rings of bounded real-valued intensive quantities. The Russian mathematician I. Gelfand and his collaborators proceeded in another direction, replacing the real field by the complex field |, so introducing rings (or algebras) of complex-valued intensive quantities. The abstract versions of these are called commutative complex C*-algebras. Gelfand and Naimark established a duality between the category of commutative complex C*-algebras and the category of compact Hausdorff spaces.

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The representation of rings as rings of intensive quantities to has been extended to arbitrary commutative rings, leading to new dualities. In these representations the given commutative ring is represented as the globally defined elements of a collection of varying rings of intensive quantities in which the ring of values of the quantities varies with the point in the space—a so-called ringed space—at which the quantity is defined. This idea leads to the so-called Grothendieck duality: the category of commutative rings is dual to a certain category of ringed spaces: the category of affine schemes.

The idea of duality, illustrated so beautifully in the case of commutative rings, and naturally and precisely expressed within category theory, transcends Bourbaki’s setbased structuralist account of mathematics.

4 The Fate of the “Mother Structures”: Algebraic and Ordered Structues In the transition from Bourbaki’s account of mathematics to its category-theoretic formulation, what is the fate of the “mother structures”? It is remarkable that, given Bourbaki’s distaste for logic, their mother structures came to play a key role in establishing the connection between category theory and logic. To begin with, algebraic structures become algebraic theories as introduced by Lawvere. Here the key insight was to view the logical operation of substitution in equational theories as composition of arrows in a certain sort of category. Lawvere showed how models of such theories can be naturally identified as functors of a certain kind, so launching the development of what has come to be known as functorial semantics.11 An algebraic theory T is a category whose objects are the natural numbers and which for each m is equipped with an m-tuple of arrows, called projections, πi : m → 1 i =1, …, m making m into the m-fold power of 1: m = 1m . (Here 1 is not a terminal object in T.) In an algebraic theory the arrows m → 1 play the role of m-ary operations. Consider, for example, the algebraic theory Rng of rings. To obtain this, one starts with the usual (language of) the first-order theory Rng of rings and introduces, for each pair of natural numbers (m, n) the set P(m, n) of n-tuples of polynomials in the variables x 1 , …, x m . The members of P(m,n) are then taken to be the arrows m → n in the category Rng. Composition of arrows in Rng is defined as substitution of polynomials in one aother. The projection arrow πi : m → 1 is just the monomial x i considered as a polynomial in the variables x 1 , …, x m . Each polynomial in m variables, as an arrow m → 1, may be regarded as an m-ary operation in Rng. In a similar way every equational theory—groups, lattices, Boolean algebras— may be assigned an associated algebraic theory. 11 See,

e.g. Bell (2018).

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Now suppose given a category C with finite products. A model of an algebraic theory T in C, or a T-algebra in C, is defined to be a finite product preserving functor A: T → C. The full subcategory of the functor category CT whose objects are all T-algebras is called the category of T-models or T-algebras in C, and is denoted by Alg(T, C). For example, if GRP is the theory of groups, then a model of GRP in the category of topological spaces is a topological group; in the category of manifolds, a Lie group; and in a category of sheaves a sheaf of groups. In general, modelling a mathematical concept within a category amounts to a kind of refraction or filtering of the concept through the Form associated with the category. When T is the algebraic theory associated with an equational theory S, the category of T-models in Set, the category of sets, is equivalent to the category of algebras axiomatized by S. Lawvere later extended functorial semantics to first-order logic. Here the essential insight was that existential and universal quantification can be seen as left and right adjoints, respectively, of substitution. To see how this comes about, consider two sets A and B and a map f : A → B. The power sets PA and PB of A and B are partially ordered sets under inclusion, and so can be considered as categories. We have the map (“preimage”) f −1 : PB → PA given by: f −1 (Y ) = {x : f (x) ∈ Y }, which, being inclusion-preserving, may be regarded as a functor between the categories PB and PA. Now define the functors ∃f, ∀f : PA → PB by ∃ f (X ) = {y : ∃x(x ∈ X ∧ f (x) = y} f orall f (X ) = {y : ∀x( f (x) = y⇒x ∈ X }. These functors ∃f (“image”) and ∀f (“coimage”), which correspond to the existential and universal quantifiers, are easily checked to be respectively left and right adjoint to f −1 ; that is, ∃f (X) ⊆ Y ⇔ X ⊆ f −1 (Y ) and f −1 (Y ) ⊆ X ⇔ Y ⊆ ∀f (X). Now think of the members of PA and PB as corresponding to attributes of the members of A and B (under which the attribute corresponding to a subset is just that of belonging to it), so that inclusion corresponds to entailment. Then, for any attribute Y on B, the definition of f −1 (Y ) amounts to saying that, for any x ∈ A, x has the attribute f −1 (Y ) just when f (x) has the attribute Y. That is to say, the attribute f −1 (Y ) is obtained from Y by “substitution” along f. This is the sense in which quantification is adjoint to substitution. Lawvere’s concept of elementary existential doctrine presents this analysis of the existential quantifier in a categorical setting. Accordingly an elementary existential doctrine is given by the following data: a category T with finite products—here the objects of T are to be thought of as types and the arrows of T as terms—and for each object A of T a category Att(A) called the category of attributes of A. For each arrow f : A → B we are also given a functor Att(f ): Att(B) → Att(A), to be thought

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of as substitution along f, which is stipulated to possess a left adjoint ∃f —existential quantification along f. The category Set provides an example of an elementary existential doctrine: here for each set A, the category of attributes Att(A) is just PA and for f : A →B, A(f ) is f −1 . This elementary existential doctrine is Boolean in the sense that each category of attributes is a Boolean algebra and each substitution along maps a Boolean homomorphism. Functorial semantics for elementary existential doctrines is most simply illustrated in the Boolean case. Thus a (set-valued) model of a Boolean elementary existential doctrine (T, Att) is defined to be a product preserving functor M: T → Set together with, for each object A of T, a Boolean homomorphism Att(A) → P(MA) satisfying certain natural compatibility conditions. This concept of model can be related to the usual notion of model for a first-order theory T in the following way. First one introduces the so-called “Lindenbaum” doctrine of T: this is the elementary existential doctrine (T, A) where T is the algebraic theory whose arrows are just projections among the various powers of 1 and in which Att(n) is the Boolean algebra of equivalence classes modulo provable equivalence from T of formulas having free variables among x 1 , …, x n . For f: m → n, the action of Att(f ) corresponds to syntactic substitution, and in fact ∃f can be defined in terms of the syntactic ∃. Each model of T in the usual sense gives rise to a model of the corresponding elementary existential doctrine (T, A). Ordered structures become identified with categories having at most one arrow between any pair of objects. But they have a further significance, as we shall see.

5 The Fate of the “Mother Structures”: Topological Structures Topological structures become associated, not with the category of topological spaces, but with the category of sheaves over a topological space, the archetypal example of a topos12 in the sense of Grothendieck. Grothendieck saw toposes as “generalized spaces”. A sheaf over a topological space may be thought of as a set ‘varying continuously’ over the space. The construction of the topos of sheaves (or presheaves) over a space T depends not on the elements of the underlying set of T, but only on the topology of T, that is, the partially ordered set of opens of T—a so-called locale or pointless space. In this way ordered structures come to replace topological structures in the construction of sheaf toposes. Observing that ordered structures are themselves categories, Grothendieck generalized these to the concept of a site, a (small) category together with a notion of covering, and further extended the concept of a sheaf over a topological space to that of sheaf over a sit. A Grothendieck topos is the category of sheaves over a site. 12 See,

e.g. Mac Lane and Moerdijk (1992) for an account of topos theory.

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Lawvere and Tierney later generalized Grothendieck toposes to elementary toposes. These are the categorical counterparts to higher-order logic. An (elementary) topos may be defined as a category possessing a terminal object, products, exponentials, and a truth-value object. Here a truth-value object is an object S such that, for each object A, there is a natural correspondence between subobjects of A and arrows A → S. (Just as, in set theory, for each set X, there is a bijection between subsets of A and arrows A → 2.) The system of higher-order logic associated with a topos is a generalization of classical set theory within intuitionistic logic: intuitionistic type theory, or as it is sometimes called local set theory. The category of sets is a prime example of a topos, and the fact that it is a topos is a consequence of the axioms of classical set theory. Similarly, in a local set theory the construction of a corresponding “category of sets” can also be carried out and shown to be a topos. In fact any topos is obtainable (up to equivalence of categories) as the category of sets within some local set theory. Toposes are also, in a natural sense, the models or interpretations of local set theories. Introducing the concept of validity of an assertion of a local set theory under an interpretation, such interpretations are sound in the sense that any theorem of a local set theory is valid under every interpretation validating its axioms and complete in the sense that, conversely, any assertion of a local set theory valid under every interpretation validating its axioms is itself a theorem. The basic axioms and rules of local set theories are formulated in such a way as to yield as theorems precisely those of higher-order intuitionistic logic. These basic theorems accordingly coincide with those statements that are valid under every interpretation. Once a mathematical concept is expressed within a local set theory, it can be interpreted in an arbitrary topos. This leads to what I have called ilocal mathematics: here mathematical concepts are held to possess references, not within a fixed absolute universe of sets, but only relative to toposes. Absolute truth of mathematical assertions comes then to be replaced by the concept of invariance, that is, “local” truth in every topos, which turns out to be equivalent to constructive provability. In category theory, the concept of transformation (morphism or arrow) is an irreducible basic datum. This fact makes it possible to regard arrows in categories as formal embodiments of the idea of pure variation or correlation, that is, of the idea of variable quantity in its original pre-set-theoretic sense. For example, in category theory the variable symbol x with domain of variation X is interpreted as an identity arrow (1X ), and this concept is not further analyzable, as, for instance, in set theory, where it is reduced to a set of ordered pairs. Thus the variable x now suggests the idea of pure variation over a domain, just as intended within the usual functional notation f (x). This latter fact is expressed in category theory by the “trivial” axiomatic condition f ◦ 1 X = f, in which the symbol x does not appear: this shows that variation is, in a sense, an intrinsic constituent of a category.

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In a topos the notion of pure variation is combined with the fundamental principles of construction employed in ordinary mathematics through set theory, viz., forming the extension of a predicate, Cartesian products, and function spaces. In a topos, as in set theory, every object—and indeed every arrow—can be considered in a certain sense as the extension {x: P(x)} of some predicate P. The difference between the two situations is that, while in the set-theoretic case the variable x here can be construed substitutionally, i.e. as ranging over (names for) individuals, in a general topos this is no longer the case: the “x” must be considered as a true variable. More precisely, while in set theory the rule of inference P(a) for every individual a ther e f or e ∀x P(x) is valid, in general this rule fails in the internal logic of a topos. In fact, assuming classical set theory as metatheory, the correctness of this rule in the internal logic of a topos forces it to be a model of classical set theory: this result can be suitably reformulated in a constructive setting. In Bourbaki’s Éléments set theory provides the “raw materials” for the fashioning of mathematical structures, just as stone or clay constitute the materials from which the sculptor’s creations are fashioned. In category theory, on the other hand, mathematical structures are not built from sets: they are given ab initio. For this reason category theory does much more than merely reorganize the mathematical materials furnished by set theory: its role far transcends the purely cosmetic. This is strikingly illustrated by the various topos models of synthetic differential geometry or smooth infinitesimal analysis. Here we have an explicit presentation of the Form of the smoothly continuous incorporating actual infinitesimals which is simply inconsistent with classical set theory: a form of the continuous which, in a word, cannot be reduced to discreteness. In these models, all transformations are smoothly continuous, realizing Leibniz’s dictum natura non facit saltus and Weyl’s suggestions in The Ghost of Modality and elsewhere. Nevertheless, extensions of predicates, and other mathematical constructs, can still be formed in the usual way (subject to intuitionistic logic). Two startling features of continuity then make their appearance. First, connected continua are cohesive: no connected continuum can be split into two disjoint nonempty parts, echoing Anaxagoras’ c. 450 B.C. assertion that the (continuous) world has no parts which can be “cut off by an axe”. And, even more importantly, any curve can be regarded as being traced out by the motion, not just of a point, but of an infinitesimal tangent vector—an entity embodying the (classically unrealizable) idea of pure direction—thus allowing the direct development of the calculus and differential geometry using nilpotent infinitesimal quantities. These near-miraculous, and yet natural ideas, which cannot be dealt with coherently by reduction to the discrete or the notion of “set of distinct individuals” (cf. Russell, who in The Principles of Mathematics roundly condemned infinitesimals as “unnecessary, erroneous, and self- contradictory”), can be explicitly formulated in

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category-theoretic terms and developed using a formalism resembling the traditional one.

6 Conclusion Finally, let me return to Bourbaki’s Élements. In writing their masterwork the fraternity’s members saw themselves qs both sculptors and architects of mathematics. As sculptors, they used set theory to provide the “clay” from which the individual mathematical structures—the “sculptures”, so to speak—which were to be exhibited within the grand mathematical edifice (museum, even) they aimed to build. As architects, they constructed this edifice from the same set-theoretic “clay” as the sculptures. Yet at the same time they insisted that, once sculptures and edifice had been formed, the raw materials used in their production could be consigned to oblivion. This is the lordly attitude of the master architect concerned only with the grand design, who in the end ignores the constitution of the bricks from which his edifices are built, as opposed to the sculptor who has a much more intimate relationship—“hands on”, so to speak—with the materials from which her creations are shaped. By contrast, category theorists—those, at least, who are sensitive to such issues (and those constitute the majority)—regard set theory as a kind of ladder leading from pure discreteness to the depiction of the mathematical landscape in terms of pure Form. Categorists are no different from artists in finding the landscape (or its depiction, at least) more interesting than the ladder, which should, following Wittgenstein’s advice, be jettisoned after ascent.

References Bell, J. L. (2018). Categorical logic and model theory. In E. Landry (Ed.), Categories for the Working Philosopher. Oxford University Press. Bourbaki, N. (1939) Éléments de Mathématique (Vol. 10). Paris: Hermann. Bourbaki, N. (1950). The architecture of mathematics. American Mathematical Monthly, 67, 221– 232. Bourbaki, N. (1994). Éléments d’histoire des mathématiques. Berlin: Springer. Corry, L. (1992). Bourbaki and the concept of mathematical structure. Synthese, 92(3), 311–348. Eilenberg, S., & Mac Lane, S. (1945). General theory of natural equivalences. Transactions of the American Mathematical Society, 58, 231–294. Reprinted in Eilenberg, S., & Mac Lane, S. (1986). Eilenberg-Mac Lane: Collected Works. New York: Academic Press. Lawvere, F. W., & Schanuel, S. (1997). Conceptual mathematics. Cambridge University Press. Mac Lane, S. (1971). Categories for the working mathematician. Berlin: Springer. Mac Lane, S., & Moerdijk, I. (1992). Sheaves in geometry and logic; a first introduction to topos theory. Berlin: Springer. Mashaal. M. (2006). Bourbaki: a secret society of mathematicians. American Mathematical Society.

Bourbaki and Foundations Gabriele Lolli

Abstract Bourbaki saw his project of the Éléments as the culmination and synthesis of two trends leading to the state of mathematics in the thirties of last century: axiomatics and set theory. We argue that, in Bourbaki’s rendering, it was an unholy marriage. To stay together, set theory had to be reduced to a language and logic to the grammar of that language, and Hilbert to the caricature of a formalist. Both set theory and mathematical logic have suffered from this misrepresentation; they couldn’t force their entrance in the house, as other disciplines did, capitalizing on their necessity, since they were already, nominally and distorted, inside.

Bourbakism has been the last—presumed—harmonious synthesis of the whole of mathematics. It dominated, for better or worse, for about thirty years, from the fifties to the eighties of the twentieth century;1 it claimed to drop the curtain over a period of unruliness and to offer the ultimate organisation of mathematics. According to Pierre Cartier (1932) Bourbaki’s aim in the fifties and the sixties was “to create what Thomas Kuhn called normal science”;2 it succeeded, or could be deemed to succeed, but against his hopes only for a short time. Bourbaki’s approach entailed strong constraints on what was worth of pursuing, of being called mathematics, on how to do it and, for the overzealous followers, how to teach it. The new paradigm has been short-lived because, even admitting that the Éléments de mathématique had provided a foundation for the existing mathematics, “if you 1 Nicolas

Bourbaki was born in 1935 and the first publication appeared in 1939, Fascicule de Résultats de la Théorie des Ensembles, but there was the war: “Bourbaki survived during the war with only Henri Cartan and Jean Dieudonné—according to Cartier. But all the work that had been done in the thirties blossomed in the fifties”. 2 Cartier (1998). Cartier himself was a bourbakist. We shall refer grammatically to Bourbaki as a single individual, male of course, although Cartier describes four generations of members of the family. G. Lolli (B) Scuola Normale Superiore, Palazzo della Carovana, Pisa, Italy e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 A. Peruzzi and S. Zipoli Caiani (eds.), Structures Mères: Semantics, Mathematics, and Cognitive Science, Studies in Applied Philosophy, Epistemology and Rational Ethics 57, https://doi.org/10.1007/978-3-030-51821-9_2

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have such a rigid format it is very difficult to incorporate new developments” (Cartier). In truth, not all of mathematics known in those times was included, only those parts that came under Bourbaki’s definition and idiosyncrasies, and were suitable for a systematic presentation (stable et bourbachisable was the mark of the approved drafts).3 Bourbaki has been reproached for ignoring physics and applied mathematics;4 we’ll come back later, with hindsight, to the many new budding researches that were ignored, but it stands out that classic subjects such as hard analysis, measure theory,5 group theory,6 number theory have no place in the treatise. A few projects in the seventies, concerning e.g. spectral theory of operators, homotopy theory and complex variables, didn’t materialise, both for doubts about their treatment, and because triumphant Bourbakism produced already textbooks written in its style by new adepts.7 Nowadays omissions are excused for the reason that “Bourbaki […] had a dogmatic view of mathematics: everything should be set inside a secure framework”;8 he is absolved with the justification that the value laid in the very project, the first volumes serving only as an illustration; omissions cannot be helped and should be excusable, since “you can think of the first books of Bourbaki as an encyclopædia of mathematics, containing all the necessary information. That is a good description. If you consider it as a textbook, it’s a disaster” (Cartier).

1 Bourbaki’s Foundations Bourbaki’s dogmatism however was not confined to the generic quest for a “secure framework”. We are not concerned with practical shortcomings, rather we would like to understand the overall vision of Bourbaki. Apparently Bourbaki’s position as presented by Jean Dieudonné (1906–1992) is disdainful and dismissive with respect to the foundations of mathematics and would endorse an absolute lack of interest: 3 From

the minutes of the Bourbaki’s meetings, see later n. 32. Hermann (1986). Cartier recalls how André Weil (1906–1998), staying in Göttingen in 1926 and studying with David Hilbert (1862–1943), was unaware of the inception of quantum mechanics. 5 Only Radon measure is presented; it is not by chance, because the primary interest was oriented to the theory of integration (we are not suggesting that Bourbaki was ignorant on this topic). The grand project was born from the idea of writing an up to date analysis textbook. 6 Only crystal groups are mentioned, thanks to Cartier’s insistence; it was him who pointed to their relationship with Lie groups and introduced Coxeter’s work to the others bourbakists. 7 “There were various attempts within the group to focus on new projects. For instance, for awhile the idea was that you should develop the theory of several complex variables, and many drafts were written. But it never matured, I think partly because it was too late. There were already many good textbooks on several complex variables in the seventies, by Grauert and other people. […] There was a whole generation of textbooks, and books, which were under his influence”, Cartier (1998). 8 Cartier (1998). 4 See

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On foundations, we believe in the reality of mathematics, but when the philosophers attack us with their paradoxes we most certainly go and take cover behind the formalism and answer: “Mathematics is nothing else than a manipulation of symbols devoid of any meaning”, then we write Chap. 1 and 2 of the Éléments presenting set theory. At last we are left in peace and may go back and do our mathematics as we have always done, with the feeling of working on something real. Probably this feeling is a delusion, but it is very comfortable. This is Bourbaki’s attitude with respect to foundations.9

Actually Bourbaki has moulded a precise idea of what is mathematics reality and how it must be dealt with, not just “as we have always done”. He does not evade the foundational question, but has his own original answer, structured with several components. 1. First and foremost the demand for rigour, which is satisfied by a controversial formalist method: “The principal benefit of the formalist method will be that of dissolving for good the obscurities that still burdened the mathematical thought”. The term “formalist” in Bourbaki refers to an implicit lesson of David Hilbert’s work, although a bit twisted: clarity is gained by the adherence to the principle that the correctness of a piece of mathematics does not depend on its interpretation but is guaranteed by strictly following a set of rules. We will see later the inherent ambiguity of the term “formalist”. 2. The first demand is met in the Éléments by the formal, rigorous introductory presentation of the syntax of the mathematical language. The language is unique and, we will see why in 3, it is the set theoretic language. Echoing a neo-positivistic stance, Bourbaki affirms: “Logic, so far as we mathematicians are concerned, is no more and no less than the grammar of the language we use, a language which had to exist before the grammar could be constructed”.10 Hence “if logic (as grammar) is to acquire a normative value, it must, with proper caution, allow the mathematician to say what he really wants to say”, and not try to make him conform to some elaborate and useless ritual. “After the logician has properly discharged such duties, he may set himself further objectives”, for instance that of consistency. However, “it is of course quite untrue that mathematics is free from contradictions”, consistency “is a goal to be achieved not a God-given quality that has been granted us once for all. […] Contradictions do occur; but they cannot be allowed to subsist”. Absence of contradiction is an empirical fact, to be dealt with in a practical way, and the contribution of logical analysis cannot be but modest: “Let the rules be so formulated, the definitions so laid out, that every contradiction may most easily be traced back to its causes, and the latter either removed or well surrounded by warning signs as to prevent serious trouble”.11 3. The grammar does not concern only the logical alphabet but governs also the set theoretic language; the reason is that for Bourbaki, “as every one knows, all mathematical theories can be considered extension of the general theory of sets”. Given this amazing, all but popular premise, Bourbaki thinks that “in order to clarify 9 Dieudonné

(1939), reprint p. 544. The translations from Dieudonné’s essay are of the author. (1949), p. 1. 11 ivi, pp. 2–3. 10 Bourbaki

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my position on the foundations of mathematics, it only remains for me to state the axioms which I use for that theory”.12 They are essentially those of Zermelo’s theory Z. “On these foundations, I state that I can build up the whole of the mathematics of the present day”.13 4. Once the language is fixed, the presentation of a theory should obey some criterion; Bourbaki is not reductionist. Reductionism in the philosophy of mathematics means that the objects of a theory T are defined in set theory (e.g. the natural numbers as the von Neumann ordinals) and the theorems of T are proved using the axioms of set theory. It is a refinement of the so called “genetic method” to which Hilbert contrasted the axiomatic one in 1900.14 Bourbaki’s criterion is axiomatic, without demur: Every mathematician who cherishes intellectual probity is forced at this point by an absolute necessity to present his reasonings in an axiomatic form.15

We will presently see the exact meaning of this statement. We must warn that on the subject of the axiomatic method and of formalism bourbakists are in two minds, or have been. We will examine two essays, by Dieudonné (1939) and by Bourbaki (1948). They are separated by a few years, but they were both included in Le Lionnais (1962); (the first already in Le Lionnais 1948); Dieudonné was Bourbaki, or he was the spokesman for Bourbaki,16 especially in those years, so one would expect at least consistency; this is not so, but the surprising difference between the two manifestos faded into oblivion. 5. The axiomatic method according to Bourbaki (1948) is presented in the following way. Every one knows that mathematical reasonings are long chains of small steps, as described by Descartes, but these chains are only a transformation mechanism applicable to any set of premises, so it cannot serve to characterise these premises. “It is therefore a meaningless truism to say that this ‘deductive reasoning’ is a unifying principle for mathematics. So superficial a remark can certainly not account for the evident complexities of mathematical theories, not any more that one could, for example, unite physics and biology into a single science on the ground that both use the experimental method”.17 What should be put at its place? Although proliferation of new disciplines at the beginning of the twentieth century, each with its own concepts, seemed to jeopardize the unity of mathematics, “[t]oday we believe […] that the internal evolution of mathematical sciences has, in spite of appearance, brought about a closer unity among its different parts, so as to create something like a central nucleus that is more coherent than it has ever been. [This evolution has included] a systematic study of the relations existing between

12 ivi,

p. 7. p. 8. 14 Hilbert (1900). 15 Dieudonné (1939), reprint p. 544. 16 And probably wrote this essay; “Dieudonné was the scribe of Bourbaki”, according to Cartier. 17 Bourbaki (1948), English transl. p. 223. 13 ivi,

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different mathematical theories, […] which has led to what is generally known as the ‘axiomatic method’ ”.18 Concerning the latter, Bourbaki warns that also the labels “formalism and formalistic method” are improperly employed to refer to it, and it is important for him to make clear the meaning of these “ill-defined” words, which are often used by the opponents of the axiomatic method to throw discredit upon it. A precise regimentation of vocabulary, rules and syntax is useful (implying that such regimentation is covered by the word “formalism”), but it is only an aspect of the axiomatic method, “indeed the less interesting one”. “Many [mathematicians] have been unwilling for a long time to see in axiomatics anything else than futile logical hairspilling not capable of fructifying any theory whatever”,19 because they have confused it with “logical formalism”. “The unity which it [axiomatic method] gives to mathematics is not the armor of formal logic, the unity of a lifeless skeleton. […] What the axiomatic method sets as its essential aim, is exactly that which logical formalism by itself can not supply, namely the profound intelligibility of mathematics”.20 The intelligibility of mathematics is provided by the concept of “structure”: to introduce the concept, Bourbaki considers the addition of real numbers, the multiplication of integers modulo a prime number p and the composition of displacements in three dimensional Euclidean space, observing that all of them have properties that can be expressed in a common notation; he chooses the axioms for groups (with the operation symbols τ and  and the individual symbol e), and shows that others properties, which could be proved in each of the corresponding theories by reasonings peculiar to them, can be obtained by a method applicable in all cases, as logical consequences of the axioms, with a deduction which leaves the nature of elements completely out of account (details are given for the left cancellation law). Then follows the well known classification of the mother-structures (algebraic, ordered, topological), the multiple structures and the particular ones. “The organising principle will be the concept of a hierarchy of structures, going from the simple to the complex, from the general to the particular”.21 6. The axiomatic method according to Dieudonné (1939). The paper begins with a clear and shareable description of the Euclidean axiomatic method; summing up: “until the 19th century mathematicians reasoned on notions of which they had a rather vague idea, a kind of idealisation of experimental notions”; the same remained true also when things worsened, for instance with the controversial use of infinitesimals; the reason for such stability was that the conclusions of the reasonings were always intuitive and made reference to experimental facts. Then came the teratologies of analysis, the distrust of intuition, the awareness that the route from experience to mathematical concepts was not so simple and anodyne.

18 ivi,

p. 222. p. 230. 20 ivi, p. 223. In the last two quotations it is noticeable Poincaré’s influence. 21 ivi, p. 228. We shall not dwell on the concept of structure, which is dealt with in other contributions to this volume. See also Corry (2003). 19 ivi,

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Here Dieudonné inserts the statement on the “absolute necessity to present […] reasonings in an axiomatic form”, which continues: that means in a form where the proposition are enchained thanks only to the rules of logic, abstracting willingly from every intuitive “evidence” suggested to the mind by the terms occurring in them.22

Dieudonné’s agreement with Hilbert goes so far as to observe that in geometry the terms “point, line, plane” may continue to be used, but one could substitute them with arbitrary ones, provided that the relations involving these words are kept invariant. Then, “if the statements obtained in this way have still an intuitive sense, one has a new intuitive interpretation of the geometric propositions”. He gives also an example with a sheaf of lines and planes all passing for a point: sheaf line of the sheaf angle of two lines dihedral angle trihedral

plane point of the plane distance of two points angle of two lines triangle

obtaining from this dictionary the propositions of a two-dimensional non-Euclidean geometry (Riemann). Given the above described mathematicians’ new awareness, the first task was that of revising the theories already developed; Hilbert did it for geometry and many other mathematicians axiomatised several classical theories.23 Dieudonné then wonders what this new point of view entails “for the intelligibility of the mathematical language”, that is “which mental representations have to correspond to the words used”; the answer is that “it is useless to have a definite mental image of the objects one reasons about”: these mysterious objects have only a quality, that of “being”, and being means to be elements of sets. Sets appear out of the blue in Dieudonné’s aperçu, as the objects (with connected notions such as function, subset, union etc.) whose study belongs to “the young Set Theory” due to the genius of Cantor. However, reasoning with infinite sets, which were by now “the foremost notions of mathematics”, besides being distasteful to many for its non-constructive character produced the paradoxes. Set theory and all the other theories which in greater or lesser measure presuppose the infinite seemed to crumble. Mathematics had never experienced such a crisis, but at last a “coherent and stable point of view” has emerged and seems to be agreed upon by the majority of the new generation of mathematicians, “the formalist conception”, due again to Hilbert, who has thus brought to completion the work begun twenty years earlier. 22 Dieudonné

(1939), reprint p. 544. does not mention Giuseppe Peano (1858–1932) and the work of his school, even on new theories such as vector spaces. Could be for disdain of Peano’s artificial language, or national rivalry. 23 Dieudonné

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“The essential thesis of Hilbert, which we have embraced here, is that clear thinking is possible only of objects determined and given in a finite computable number”.24 But, even abstracting from every intuition concerning the real numbers, one cannot refrain from thinking that a proposition such as x + y = y + x is of a different order from all its instances obtained by substituting the variables with specified numbers. It is with similar sentences that the introduction of an idea of infinite appears inevitable. Now Hilbert gets out of the impasse by claiming that “it is impossibile to elude the concept of infinite as long as one believes that the essential of a proposition is its content, that is the mental representation of which it is a symbol; but every difficulty disappears if one acknowledges that the essential of a proposition is its form, or in other words, that it is useless that a proposition expresses a mental representation different from the perception of the signs from which it is assembled”.25 “Strange and paradoxical position [as it sounds] this however is Hilbert’s fundamental idea”: in the statement of mathematical propositions, signs are devoid of every meaning. Up to know Dieudonné presentation could be almost acceptable,26 if only he would have mentioned that this was Hilbert’s idea of formalised mathematics, not of a formalistic mathematics (to Hilbert formalised mathematics is an image of true mathematics, built with a precise aim, that of proving consistency), and that the manipulation of the signs is done according to meaningful operations in concrete mathematics, namely metamathematics. Instead Dieudonné dwells on the analogy with chess game, and concludes that “mathematics becomes a game”. Finally for Dieudonné “it remains to be shown that Hilbert’s conception is realisable”, and the task can only be accomplished “by realising it effectively”. Dieudonné’s pragmatism shows that Hilbert’s purported conception is not Hilbert’s, since to realise it the latter needed a consistency proof. On the contrary, Dieudonné denounces that Hilbert and his collaborators didn’t contribute to the solution of realising the formalistic conception, having chosen to entirely devote themselves to “proof theory, or metamathematics”. As to the task, many equivalent solutions are possible for Dieudonné; he of course sketches Bourbaki’s proposal of “the rules system of formalist mathematics”, referring the reader to the first book of the Élements, Chaps. 1 and 2 of Théorie des Ensembles. With Bourbaki’s conception “the famous ‘paradoxes’ disappear”. But “a closing remark” on metamathematics is offered.27 It still could happen, although nobody ever noticed such occurrence up to now, that a proposition and its negation were both true (in the sense of the game). Such possibility is the problem of the consistency [non-contradiction] of mathematics. To face this problem there 24 Dieudonné

says “nombre fini expérimental”, meaning a number upon which “one can perform the operation of counting”. 25 Dieudonné (1939), reprint p. 556. 26 Alonzo Church (1903–1995) reviewed Dieudonné (1939) in Church (1939) and his short report was rather appreciative, with only one criticism for the exclusive role assigned to Hilbert, and the neglect in particular of Frege and type theory. 27 Dieudonné (1939), reprint p. 553.

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are two attitudes: the first is to observe that the probability of such an event, considering the intuitive origin of mathematics, is comparable to that of the sun missing a sunrise; the second is that in case of a contradiction it would be possible to isolate the rule responsible, modifying only or reorienting some parts of mathematics, as it happened with the discovery of the irrationals; it was not considered the unveiling of a contradiction but a new beginning. C’est tout, on metamathematics. Dieudonné does not take a position, but it is apparent that the first attitude is at variance with that of Bourbaki (1949) as discussed in Sect. 2; the second attitude is compatible, and is more realistic (in adding the need to modify some parts of mathematics) about the effects of removing the causes. Some specific comments on this first Bourbaki’s public foray: notice that the word “structure” still does not appear anywhere. In 1939 it hadn’t yet occurred to refer to structures to describe the axiomatic method. Moreover, and consequently, nothing is said on the intelligibility of mathematics, but only of language; intelligibility is credited to the specific set theoretical language. Notice also: no mention of Hilbert’s program, although the task in which Hilbert and his collaborators were engaged is called “proof theory, or metamathematics”, and the last “closing remark” is offered as a (rather foolish) comment on it. No mention therefore of Kurt Gödel (1906– 1978), and his 1930 refutation of the feasibility of Hilbert’s program.28 But more amazing is the omission of Ernst Zermelo (1871–1953), as if he had not contributed to systematise set theory, he who was the first to axiomatise it in 1908. Dieudonné lets the reader surmise that the first was Bourbaki, while the latter instead simply loaned Zermelo’s axioms.29 In conclusion, although Dieudonné claims that formalism is only a defence against the attack of the philosophers, he gives a fairly detailed and fair account of it, making however a mockery of Hilbert on his way. If we try to confront synoptically the two introductions to Bourbakism by Dieudonné (1939) and Bourbaki (1948), we see that they are rather at variance. As noticed, Dieudonné’s descriptions of formalism and of the axiomatic method are accurate, as well as the attribution of the principal axiomatics’ paternity to Hilbert; the latter however was no formalist as there depicted; or he was so only in a few occasional slips of the pen; for Hilbert formalisation, not formalism, was an action applied to mathematics to transform its statements in an array of texts amenable to a consistency proof by finitistic methods.30 In Bourbaki on the contrary, the axiomatic method is something different, as we will presently explain, while formalism is reduced to precise syntax, a minimalist feature shared by every philosophy of mathematics. 28 The lack of reference to Gödel is forcefully denounced and lamented by A. R. D. Mathias (1944) in Mathias (1992). 29 See Rosser’s comment in n. 34. 30 In Hilbert (1922) for example, he describes how through formalisation “real mathematics o mathematics in a strict sense, becomes an array of provable formulae”; by applying the contentual [inhaltlich] finitary methods one obtains “copies of the transfinite propositions of usual mathematics”.

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A different description and appraisal of metamathematics is given in the historical notes added to the volumes, and collected in Bourbaki (1960): metamathematics is said there to be an “autonomous science” which provides “useful knowledge of the mechanisms of mathematical reasoning”.31 Gödel’s first incompleteness theorem is also sketched, albeit with the usual idiosyncracies: the conclusion of the theorem is that a theory satisfying the hypotheses is not categorical, which is correct only because he defines as categoricity what is and was already universally known as completeness (see more in the next section). Moreover to Bourbaki the main result of Gödel’s tour de force is that of the second theorem, the impossibility of proving consistency. The curiosity to know who wrote the historical note, and when, is legitimate.32 In a meeting of Bourbaki group in 1951 it was decided (p. 4) that Jacques Dixmier (1924) would write the final version of logic within two months after the imprimatur of John Barkley Rosser (1907–1989); that the list of axioms for set theory would be sent to the same Rosser, and André Weil would write the historical note for set theory, again with the help of Rosser (C’est ici le meilleur endroit pou traiter de sujets polémiques. Weil se fera tapiriser per Rosser) (p. 9). Rosser approved the axioms in october (Rosser les trouve kosher). In the 1952 meeting, upon a suggestion of Chevalley and Weil, it was decided that the concept of application, until then simply a set of ordered pairs, would be better expressed by a triple, called correspondence, with domain and codomain (while graph remained for the old definition); in october they would look at the new version of chapter II on sets written by Dieudonné; as for the historical note Pierre Samuel (1921–2009) would be instructed by Rosser at Cornell (Samuel se fera tapiriser par Rosser à Ithaca) (p. 3); apparently Weil had renounced or turned down his engagement. Rosser was thus Bourbaki trait d’union with logic (discussions on matters of logic in the group were limited to minor questions, such as whether equality should be primitive or defined, and notations).33 Rosser reviewed the paper Weil read at a meeting of ASL in december 1948, probably upon recommendation of Saunders MacLane

31 Bourbaki

(1960), English transl. p. 40. The historical notes were the best place to treat polemical issues (des subjets polémiques) as it is said in La Tribu, n. 25, p. 9, see next foootnote. 32 I am indebted to A. R. D. Mathias for directing me to the minutes of Bourbaki’s meetings (Congrès), written in the form of a journal, La Tribu. Bulletin œcoumémique, apériodique et bourbachique. The relevant issues for our concerns are n. 25, 5/26-7/8 1951 and n. 28, 5/25-7/8 1952 (Congrès de la motorisation de âne qui trotte). The conclusions one can draw have been already signalled in Mathias (2014). Mathias also notes that Claude Chevalley (1909–1984) had read parts of the 1940 monography of Gödel on constructible sets and urged for a stronger system modelled on Gödel’s. Page numbers in the rest of this section refer to the typewritten La Tribu minutes of these two meetings. 33 The exception was the discussion for the acceptance of Hilbert’s -formalism, endorsed by Weil and Chevalley.

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(1909–2005), that became Bourbaki (1949).34 Rosser was a friend of Weil, known in USA, a relation certainly not based on logic, given Weil’s scant interest. According to Mathias, Rosser was probably concerned to have Bourbaki not committing errors, but not to advance or impose a personal vision.

2 Bourbaki’s Axiomatics Why Bourbaki’s axiomatic method is not the method that was moulded by Pasch, Peano, Enriques, Poincaré and Hilbert (among others)? Let us start from the statement that “all mathematical theories can be considered extension of the general theory of sets”. What Bourbaki has in mind is that e.g. group theory is the theory of the structures of type τ, · , e which satisfy the properties (1) xτ (yτ z) = (xτ y)τ z, (2) eτ x = xτ e = x, (3) xτ x  = x  τ x = e. In this way the axioms are a masked definition of a class of structures. Still nowadays in college textbooks written under Bourbaki’s influence one finds the introduction, say of the natural numbers, given in the following way, called an “axiomatic definition”: “N is a set satisfying the following properties” followed by a list of statements that are Peano’s axioms.35 This however is an explicit definition, the very definition given by Richard Dedekind (1831–1916), who was not an axiomatiser but an avant-garde logicist (also considered a forerunner of structuralism). Similarly in Bourbaki the axioms, say of group theory as above, are the definition of a class of structures, those satisfying the definition of “group” given by the conjunction of the axioms. Bourbaki’s thesis that “all mathematical theories can be considered extension of the general theory of sets” means that every theory is presented by adding to set theory the definition of a class of structures consisting of formulae he calls “axioms”. This move, far form being an extension of set theory, is rather an extension of its language, a conservative extension of course. The truth is that set theory and the axiomatic method can coexist as the devil and holy water.36 Set theory as the basis of an encyclopædic treatment of mathematics makes sense only in a reductionist program. The axiomatic method is pluralist, it doesn’t aim at unity nor to give “profound intelligibility” to mathematics. Its outlook is a product of modernity: “Unum bonum est; plura vero malum” was a belief of Gerolamo Cardano (1501–1576) in the sixteenth century.37 34 Rosser

(1950). His remarks: the axioms are a weak version of the original Zermelo system; the union of two sets exists only if they are subsets of a same set; individuals without elements are not sets, as in Zermelo, but only one can be proved to exist, the null class specifically taken as non-set; there is only one reference to the relevant literature; the notion of synonymity seems to formalise some personal feeling, but should probably better be left out. 35 With possibly some variants; for instance the well-ordering property instead of induction. 36 An Italian figure of speech easily understood. 37 Cardano (2009).

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There is only one condition for the coexistence of the axiomatic and the set theoretic souls, and it is that set theory be the handmaid of axiomatics, that it serves as a generale semantic metatheory providing the language for the structures which are the interpretations of the theories.38 Mathematical structures have, as is well known, the following definition: they are sets endowed with operations, relations and special elements. Such definition, accredited in broad sense to Alfred Tarski (1901–1983),39 was meant to provide a precise characterisation of the idea of “interpretation” of axiomatic theories and subject them to a rigorous treatment. Since it is impossible to consider all the infinitely many interpretations of a theory, nor to confine oneself to a purely deductive development, one thinks (pretends to think) of a generic model; this is a collection of entities whose nature is irrelevant to the mathematical treatment; the generality is well rendered by the word “set” and if also relations and functions, extensionally conceived, are imagined as sets of ordered pairs, the semantical metalanguage becomes set theoretic. It is not however sufficient to talk of structures to have an axiomatic conception. An axiomatic theory is, in principle, devoid of any interpretation, as a necessary condition to have several of them. A specific interpretation of a mathematical theory is given by means of other, already familiar, mathematical concepts, hence in another language. Bourbaki never says, as Hilbert did, that the words “point”, “line”, “plane” could be substituted by “love”, “law”, “chimney sweep” and with the definition of suitable relations between such entities the axioms and theorems of geometry become valid for them. Dieudonné said so in 1939, but only once and as an isolated point in the bourbakist manifold. Bourbaki’s notion of structure does not come from the semantics of formal languages but from algebra. The model is set by Bartel L. van der Waerden (1903–1996) in Moderne Algebra,40 with its hierarchy of groups, rings, fields. The structures united under the same definition have already their justification and mathematical meaning, their existence. Intelligibility precedes and dictates Bourbaki’s axiomatics. Structures conceived as arbitrary interpretations of languages do not provide any profound intelligibility; they open the way for taking advantage of different mathematical intuitions, “a thousand spiritual eyes” for the in-depth analysis of theories, as Federigo Enriques (1871–1946) said of the riches of the axiomatic method.41

38 To this end it is sufficient in fact a weak theory, of strength comparable to that of Z, in which to justify a sparing construction of sets and operations on sets inside a given set. See e. g. Mathias (2001). 39 Actually Tarski did not use the term “structure” but that of “model”, defining it as a pair K , R of a set and relation such that the axioms of a simple toy theory with a single relational symbol could be interpreted and become true. See Tarski (1941). 40 van der Waerden (1931). “What van der Waerden had done for algebra would have to be done for the rest of mathematics”, according to Cartier. 41 Enriques (1922), p. 140.

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Algebraic theories in particular are seen, in the axiomatic method, as the study of the properties of operations: the focus of the interest is on the operations, not on the structures in which they can materialise. A clue about the divergence of structuralism and axiomatic method is given by the remark made by Bourbaki to make a reason of the misgivings some mathematicians had with respect to axiomatics: “The first axiomatic treatments, and those which caused the greater stir (those of arithmetic by Dedekind and Peano, those of Euclidean geometry by Hilbert) dealt with univalent theories,42 i.e. theories which are entirely determined by their complete system of axioms […] (quite contrary to what we have seen for instance, for the theory of groups)”.43 Apart from inaccuracies, of which forthwith, it is obvious that to prove theorems of categoricity, that is that any two models whatsoever of a theory are isomorphic, it is necessary to confront two infinite models, so this confirms that set theory is unavoidable for semantics. But inaccuracies lie in the characterisation “entirely determined by their complete system of axioms”. First of all it is not the theory which is completely determined, under the univalent condition, but its models, which reduce to one. Moreover, if by “entirely determined” Bourbaki means to suggest that the theory cannot be consistently extended with new axioms, this property is the completeness property, which does not coincide with categoricity. “Completeness” is an essentially deductive property, “categoricity” deals with models (although it implies, without being equivalent to, completeness). Completeness concerns formal sentences and deductive power, so mathematicians under a Bourbaki spell, being distrustful of logic, prefer to deal with the other stronger but less fine property. More to the point, Bourbaki is missing the deep significance of the existence of univalent and polyvalent theories. Arithmetic and geometry were axiomatised with the express aim to obtain categorical theories for the two concepts that are since its historical origins the building blocks of mathematics; the other axiomatisations on the contrary sought deliberately to build theories with many models and they are a more genuine role model of the pluralistic axiomatic method. But there is a logic behind these two trends of axiomatisation: there are two logics. The axioms systems for arithmetic, the theory of real numbers and geometry are categorical only if formulated in second order logic (i.e. induction in Peano and the completeness axiom in Hilbert are written with quantifiers ranging over all subsets of the domain). If these axioms are substituted by schemata, one instance for each definable property,

42 A

term coined gratuitously, given the by then universally acceptance of “categorical”. It is true that in the historical notes Bourbaki defines categoricity as completeness, as we have seen; but then we have three terms, “univalent, categorical, complete” to account for. If univalent means complete, the two words “univalent and categorical” name the same concept; if univalent means categorical it cannot mean complete. We’ll propose below a reading coherent with the accepted meanings of these logical terms, but any reading turns out ambiguous. 43 Bourbaki (1948), p. 230.

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definable by a first order formula, it is proved in the semantic metatheory that there are always non isomorphic models, so called non standard models.44 Structuralism is a compromise, original perhaps, but still a compromise, logically shaking, between reductionism and axiomatics. It took a foothold only because Bourbaki’s reductionism is not a real reductionism and Bourbaki’s axiomatics is not the axiomatic method. Bourbaki doesn’t want to be a reductionist, and in a sense he is not, because he does not define the classical mathematical entities, only the structures to which they belong: entities are characterised by the subsumption of their containers in classes defined by a list of properties. On the other hand, to be a reductionist he should rely on a stronger metatheory, ZF or ZF plus an inaccessible cardinal (a universe).45 In another sense however he could be called reductionist since only the structure are mathematical objects, and structures are treated exclusively in the set theoretic language. Similarly the method glorified by Bourbaki is not the axiomatic method. Let us ignore the fact that Bourbaki disregards the primary role and the study of languages (in the plural) and of their interpretations, downplaying the linguistic component of the method. The capital sin is that he forgets that set theory itself is an axiomatic theory, and as such it has infinitely many models of different cardinalities, even denumerable ones, as is known since 1922 and the theorem of Thoralf Skolem (1887– 1963). It follows that the concepts Bourbaki believes to be defined in his language, for instance the structures of univalent theories, are not absolute but relative to each particular model of set theory, isomorphism doesn’t cross over models. No authority can pass an ordinance the the effect that certain axioms systems have to be treated differently from the others. It is not allowed to cancel mathematical results merely by aversion to logic. It would have been better if Bourbaki had not “formalised” the semantic metatheory with set theory, but had managed with an informal semantics. That of Bourbaki is an aged foundation with deep creases; not because of the passing of time, but because it was born old. The trends Bourbaki tried to compose, reductionism and axiomatics, were heritages of the rigour period of nineteenth century: reductionism was a child of the genetic method, by which the hierarchy of numerical system had been built; axiomatics descended from the proliferation of algebras and the discovery of non-Euclidean geometries; it preceded by a great deal the foundational crisis, it was not affected by it, on the contrary it was a factor of clarification. Bourbaki is a blasphemous mixture of Hilbert, Zermelo and van

44 The sequence of events which brought to the solution of the apparent paradox of two contradictory theorems for a while cohabitant has been tortuous but in the end the puzzle has been worked out, and is now easily explained, of course with some knowledge of modern logic. All the ingredients were ripe and well known when Bourbaki was born. But again, the puzzle requires the acknowledgment of the presence of logic in the mathematical thought. 45 Without the replacement axiom of ZF the theory of ordinals and transfinite induction could not be justified. Bourbaki could rebut that the latter is useless, or does not belong to mainstream mathematics, but it seems bizarre to exclude from mathematics the theory of cardinality.

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der Waerden. His success confirms that often in the history of thought it is not the most limpid ideas that prevail but those sustained by the strongest authority and best rhetoric.

3 Far from the Madding Straitjacket There is no doubt that all the mathematical work done under the aegis of Bourbaki’s authority is a rich and precious patrimony that mathematicians are not able to do without; it still inspires and shapes mathematical thought also in fields not perused by Bourbaki. But the overall mathematical landscape today is quite different by what Bourbaki imagined when he said: “It is quite possible that the future development of mathematics may increase the number of fundamental structures, revealing the fruitfulness of new axioms, or of new combinations of axioms”.46 We have already hinted that in the Éléments “[t]here is essentially no analysis beyond the foundations: nothing about partial differential equations, nothing about probability. There is also nothing about combinatorics, nothing about algebraic topology, nothing about concrete geometry. And Bourbaki never seriously considered logic”.47 There is also no geometrical figure, according to the prestigious model of Lagrange, and it is known how much this attitude has been a damper on the teaching of geometry. Vision had another function: it is patent that the outward look of the more widely circulating products is crucial for shaping their image in the customers mind, even more that content. Bourbaki was well aware of this, witness the pedagogic care he put in the graphics of his volumes to impress the sense of austerity and necessary precision, not without an aesthetic complacency: just look at the cover and to the first page of a volume, with a cascade of titles and subtitles in several fonts, sizes and styles, followed by arabic and roman numberings, different alphabets, decorative symbols.48 In the Éléments the computer was obviously missing, and everything falling into its orbit, discrete mathematics, the theory and practice of the algorithms. Obviously all that appeared after the seventies was missing, one can not pretend that anybody could foresee then the outburst of researches in the field of ict or cryptography; but computability theory (recursion theory as was named at the beginning) was starting to blossom in the thirties, coeval with Bourbaki. Anyway the hard fact is that in today mathematical research prevail topics that one could not even imagine by studying the Éléments: chaos theory, mathematical finance, biomathematics, algo46 Bourbaki

(1948), English transl. p. 230. (1998). As for analysis, even in the engineering schools reached by Bourbaki’s influence the calculus textbooks began with a chapter in set theory, and often developed analysis not on the base of real or complex numbers but on abstract structures such as metric or topological spaces. 48 Roubaud (1997), pp. 158–9. Jacques Roubaud (1932) is a bourbakist and a poet. To him the outline of the first pages of the topology volume of the Éléments conjured up the inescapable structure of Wittgenstein’s Tractatus. 47 Cartier

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rithms, networks, complexity.49 Or in a lighter vein let us review the mathematical topics that occur in the tv series Numb3rs;50 in the episodes come in succession: criminal geographic profiling, statistics, bayesian inference, reliability of witnesses, data mining, dna profiling, image enhancement and reconstruction, neural networks, risk analysis (the prisoner dilemma), several algorithms (tree pruning, squish-squash algorithm to detect weak signals in a noisy environment), small world nets, Fourier analysis, wavelets, brownian motion, Fokker-Planck equations for describing the chaotic motion of a body subjected to certain forces, percolation theory, predatoryprey model, the mathematics of fluid flow, and others. A special mention is due to categories: “It is amazing that category theory was more or less the brainchild of Bourbaki. The two founders were Eilenberg and MacLane. MacLane was never a member of Bourbaki, but Eilenberg was, and MacLane was close in spirit. The first textbook on homological algebra was CartanEilenberg, which was published when both were very active in Bourbaki. Let us also mention Grothendieck, who developed categories to a very large extent. I [i.e. Cartier] have been using categories in a conscious or unconscious way in much of my work, and so had most of the Bourbaki members. But because the way of thinking was too dogmatic, or at least the presentation in the books was too dogmatic, Bourbaki could not accommodate a change of emphasis, once the publication process was started”.51 It is not just a question of emphasis. Categorical concepts simply do not fit in the Procrustean bed of Tarskian structures. Bourbaki, son of his times, or better of his fathers’ times, had not in himself the surge to propose a really new vision. But the best thing that happened to category theory was that of not being inserted in the Bourbaki picture, thus escaping from being dragged down with its fall. It could thus apply, besides other developments, in fields that did not pass the Bourbaki eligibility scrutiny. Its use in computability theory, for instance, dates back to 1970 with the book Eilenberg and Elgot (1970). According to the authors, the algebraic approach allows to discover the connections between the fundamental concepts of recursion theory and those that play a major role in the theory of programs, in that of finite automata and in linguistic theory. Consequences were of great import because computability was a young theory in discrete mathematics that became fertilised by algebra. Categories are genuinely faithful to the spirit of the axiomatic method much more than Bourbakism, regardless of how they are introduced (MacLane opts for an axiomatic definition);52 the reason is that category theory pays attention to concepts and operations that occur in several areas with different meaning, e.g. the product in algebra, in set theory, in logic. The irony of history is that bourbakist “modern mathematics”, overflown in the mathematical education thanks to enthusiastic and unprepared followers, and also to 49 See

Cipra and MacKenzie (2015). It is a series of booklets began by Barry Cipra in 1993 (Dana Mackenzie joined as co-author from vol. 6 on) offering reports on recent results and researches. 50 Devlin and Lorden (2007). 51 Cartier (1998). On the history of category theory see Krömer (2007). 52 MacLane (1972).

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political contingencies,53 gave origin to the “New Maths”, a curriculum that however has been identified with, or seen as a product of logic. George Polya (1887–1985) for instance, according to Sawyer, said regretfully of Max Beberman (1925–1971), the originator of the Illinois project, that “he was a nice person, fallen under the influence of logicians”. It is a fact that the axiomatic presentation of structuralism, without a previously acquired familiarity with the mathematical structures, and in absence of the explanation given by logical theorems, which show the relations between sentences and models, easily degraded to pure formalism; moreover, instead of focusing on the mathematical content the teaching led to questions of sophistic logic, such as the distinction of “use” and “mention”, and similar formal conundrums.54 The difficulties of the first impact with structuralism are wittily narrated by Jacques Roubaud: when in the classrooms of the Institut Henri-Poincaré in 1954–1955 he took the first bourbakist courses given at Paris University by Gustav Choquet (1915–2006), not a strict bourbakist, and Laurent Schwartz (1915–2002), students were terrified since they didn’t understand anything. “In a few places [the Institut, rue d’Ulm, …] the very nature of what was taught as mathematics had been completely changed”. But the concours for the agrégation hadn’t changed. Students of the modernist educational path dreaded that they would have been sent to squash against an insurmountable wall.55 What actually Choquet was teaching in the first lectures was nothing else that the algebra of sets, with the help of typical oval figures. But: “We looked, we did not understand. We did not understand what was there to understand, in what sense those were part of mathematics”.56 The final irony is that the very set theory that for Bourbaki had to dictate absolute rigour ended up as an imprecise sloppy blackboard dialect, meaning nothing not because formal but on the contrary because of vagueness. Mathematical education, having to work only on received wisdom, was the field that most suffered the incompleteness, partiality and befuddlement of Bourbaki ideology; for this reason, the most relevant findings of the considerations on Bourbaki’s attitude to foundations expounded in the present essay are those expressed at the end of Sect. 2: the damages of obsolete and what is more undigested, uncritical and conceited knowledge.

53 In the fifties in the USA the panic caused by the success of the soviet space program, with the urge to bridge the gap with URSS in scientific and technological knowledge, encouraged to accept all the projects for a better teaching which were ready at hand, even if conceived with other concerns; this was the case with the Illinois project from which the new maths originated: see the remarks by W. W. Sawyer (1911–2008) in http://www.marco-learningsystems.com/pages/sawyer/sawyer.htm. 54 A famous general criticism with detailed denunciations is due to Kline (1961). 55 Roubaud (1997), p. 96. 56 ivi, p. 21.

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References Bourbaki, N. (1948). L’architecture des mathématiques, in Le Lionnais 1948, 35–47. American edition. The architecture of mathematics. American Mathematical Monthly, 57 (4), 221–232. Bourbaki, N. (1949). Foundations of mathematics for the working mathematician. Journal of Symbolic Logic, 14(1), 1–8. Bourbaki, N. (1998). Éléments d’histoire des mathématiques. Paris: Hermann. English edition: Bourbaki, N. (1960). Elements of the History of Mathematics. Berlin and Heidelberg: Springer. Cardano, G. (2009). De uno. Florence: Olschky. Cartier, P. (1998). The continuing silence of Bourbaki. The Mathematical Intelligencer, 20(1), 22– 28. www.ega-math.narod.ru/Bbaki/Cartier.htm. Church, A. (1939). Review of Dieudonné (1939). The Journal of Symbolic Logic, 4, (4), 163. Cipra, B., & Mackenzie, D. (2016). What’s happening in the mathematical science (Vol. 10). Providence R. I: AMS. Corry, L. (1996). Modern algebra and the rise of mathematical structures (Vol. 17, 2nd ed.). Birkhäuser: Science Networks, 2003. Devlin, K. J., & Lorden, G. (2007). The numbers behind NUMB3RS: solving crimes with mathematics. Plume Books. Dieudonné, J. (1939). Les méthodes axiomatiques modernes et les fondements des mathématiques. Revue Scientifique, LXXVI, 224–232; reprinted in [Le Lionnais 1962, pp. 543–555]. Eilenberg, S., & Elgot, C. C. (1970). Recursiveness. New York: Academic Press. Enriques, F. (1922). Per la storia della logica. Bologna: Zanichelli, 1987. American edition: The historic development of logic: the principles and structure of science in the conception of mathematical thinkers. New York: Holt, 1929. Hermann, R. (1986). Mathematics and Bourbaki. The Mathematical Intelligencer, 8(1), 32–33. Hilbert, D. (1900). Über den Zahlbegriff. Jahresbericht der DMV, 8, 180–184. Hilbert, D. (1922). Neubegründung der Mathematik. Erste Mitteilung. Abhandlungen aus dem mathematichen Seminar der Hamburgischen Universität, 1, 157–77. Kline, M., & Add, W. J. C. (1961). The failure of the new mathematics. New York: Random House. Krömer, A. (2007). Tool and object: a history and philosophy of category theory. Historical Studies, Birkhäuser, Basel: Science Network. Le Lionnais F. (ed.) (1948). Les grands courants de la pensée mathématique. Cahiers du Sud. Le Lionnais F. (ed.) (1962). Les grands courants de la pensée mathématique. Paris: A. Blanchard, second augmented edition of Le Lionnais (1948). MacLane, S. (1972). Categories for the working mathematician, graduate texts in mathematics (2 ed. 1998). Berlin-New York: Springer. Mathias, A. R. D. (1992). The ignorance of Bourbaki. The Mathematical Intelligencer, 14(3), 4–13. Mathias, A. R. D. (2001). The strength of Mac Lane set theory. Annals of Pure and Applied Logic, 110, 107–234. Mathias, A. R. D. (2014). Hilbert, Bourbaki and the scorning of logic. In C. Chong, Q. Feng, W. H. Woodin, & T. A. Slaman (Eds.), Infinity and truth (pp. 47–156), Singapore: World Scientific. www.dpmms.com.ac.uk Rosser, J. B. (1950). Review of [Bourbaki 1949]. Journal of Symbolic Logic, 14, 258–259. Roubaud, J. (1997). Mathématique: Éd. Paris: du Seuil. Tarski, A. (1941). Introduction to logic and to the methodology of deductive sciences. New York: Oxford Univ. Press; first published in Polish in 1939; many translation and additions in later printings. van der Waerden, B. L. (1930–31). Moderne algebra (Vol. 2). Berlin: Springer.

Forms of Structuralism: Bourbaki and the Philosophers Jean-Pierre Marquis

Abstract In this paper, we argue that, contrary to the view held by most philosophers of mathematics, Bourbaki’s technical conception of mathematical structuralism is relevant to philosophy of mathematics. In fact, we believe that Bourbaki has captured the core of any mathematical structuralism.

1 Introduction In the recent philosophical literature on mathematical structuralism, it is often declared that there is a sharp separation between the philosophical brand of mathematical structuralism and what mathematicians take to be structuralism. The standard position is that mathematicians practice a form of methodological structuralism and that the latter has very little to offer to philosophical discussions regarding the semantics, the ontology and the epistemology of structuralism. Ian Hacking, for instance, articulates this position explicitly in his book on philosophy of mathematics: One of the most vigorous current philosophies of mathematics is named structuralism, but it has only a loose connection with the mathematician’s structuralism of which we have spoken. (...) No label fit well, so let call structuralism of the Bourbaki type mathematician’s structuralism, and that of recent analytic philosophy philosopher’s structuralism. (Hacking 2014, 237)

In his book Rigor and Structure, John Burgess arrives at the same conclusion: ... this [Bourbaki] kind of “structuralism” is now widely regarded as a trivial truism, so long as it is separated from the particulars of Bourbaki’s attempt to pin down the relevant notion of “structure” in a technical definition. The contentious issue debated under the heading “structuralism” in contemporary philosophy of mathematics is not to be confused with Bourbaki’s claim. (Burgess 2015, 17)

J.-P. Marquis (B) Département de philosophie, Université de Montréal, Montréal, Canada e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 A. Peruzzi and S. Zipoli Caiani (eds.), Structures Mères: Semantics, Mathematics, and Cognitive Science, Studies in Applied Philosophy, Epistemology and Rational Ethics 57, https://doi.org/10.1007/978-3-030-51821-9_3

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In this paper, I argue that, on the contrary, once mathematical structuralism is properly understood and developed, that is when an adequate metamathematical analysis is provided, it ought to serve as a springboard to any philosophical discussion of mathematical structuralism. Let me be clear: I am not saying that the practice of mathematicians is directly relevant to philosophical issues tied to structuralism (although, the practice certainly cannot be ignored altogether by philosophers). Rather, I claim that Bourbaki’s technical analysis and general views on mathematics include essential components illuminating the very nature of mathematical structuralism and should guide thinking on philosophical issues springing from structuralism. Bourbaki’s analysis contains the main elements of the metamathematical analysis required, but it has unfortunately been misunderstood and misread. Surprisingly perhaps and as I will briefly indicate, Bourbaki himself is partly to blame for this situation. The paper is organized as follows. In the first part, we take a quick look at how the philosophical literature has treated Bourbaki’s structuralism and why it has been dismissed as being philosophically irrelevant. In the second part, we go back to Bourbaki’s paper The Architecture of Mathematics, which is a popular exposition of Bourbaki’s position on mathematics and, in particular, on mathematical structuralism and present its main theses. We believe that some of the central claims made by Bourbaki have been forgotten and that they can easily be extended to the contemporary mathematical context. We then move to Bourbaki’s technical discussion of the notion of mathematical structure. In this section, our goal is to clarify the technical notion, in particular to emphasize its metamathematical character. In the last section, we sketch how the work by the logician Michael Makkai, namely First-Order Logic with Dependent Sorts, FOLDS for short, captures Bourbaki’s technical notion adequately and allows us to extend it to categories and other structures that are inherently different from set-based structures. We then discuss why such a metamathematical analysis ought to be taken seriously, at least as proper conceptual analysis, by anyone who wants to develop a philosophical stance towards structuralism.

2 Discarding Bourbaki’s Structuralism: A Practical Guide Bourbaki’s books have had from very early on a clear and undeniable influence on the mathematical community, on the development of mathematics, its exposition and its goals. It is extremely easy to find testimonies by mathematicians that attest this influence. Here is a sample: All mathematicians of my generation, and even those of subsequent decades, were aware of Nicolas Bourbaki, the Napoleonic general whose reincarnation as a radical group of young French mathematicians was to make such a mark on the mathematical world. ... Many of us were enthusiastic disciples of Bourbaki, believing that he had reinvigorated the mathematics of the twentieth century and given it direction. (Atiyah 2007, 1)

Two elements have to be underlined in this quote. First, that many young mathematicians throughout the world considered themselves disciples of Bourbaki. This

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demonstrates Bourbaki’s influence and impact. Second, Atiyah emphasizes the fact, and rightly so, that Bourbaki had given mathematics a direction. This has to be emphasized, for it indicates in what sense young mathematicians considered themselves disciples of Bourbaki. The French mathematician and philosopher of mathematics, Frédéric Patras, details this last claim as follows: Mathematicians, like all scientists, need teleological purposes that allow them to structure and justify their discourse. Bourbaki had succeeded in giving an indisputable aura to mathematical thought, by giving it a spring—structuralism—and a finality—the search of a successful and hierarchical architecture of its concepts and results.1 (Patras 2001, 119)

In some sense—and this point would need to be developed, but I will not do so here—Bourbaki more or less consecrates the scientific character of mathematics by giving it a global, unified, theoretical organization, as opposed to a haphazard bunch of results, tools and methods. Mathematics’s unity is revealed by abstract structures. These are ultimately what mathematics is about and what mathematicians should be looking for. Mathematical research, at least when it is concerned with the understanding of mathematics, aims at disclosing abstract structures, their properties and their combinations. Despite this influence and despite the fact that, as we will see in the next section, Bourbaki defends and develops a thoroughly structuralist conception of mathematics, philosophers of mathematics have systematically ignored Bourbaki’s work.2 As we have already seen above, it is usually claimed that Bourbaki’s work simply does not address the semantical, ontological and epistemological issues central to the contemporary philosophical disputes.

2.1 Bourbaki’s Evaluation of His Own Analysis Bourbaki is partly responsible for this state of affairs, but for reasons that are different from those invoked by philosophers of mathematics. Bourbaki came to the conclusion that his original version of structuralism failed simply because it was, according to him, superseded by the notions of categories and functors. Again, here are typical claims made by, first, one of the founding fathers of Bourbaki and, second, by one of the most influential members of the second generation of Bourbaki. Let us immediately say that this notion [that is, Bourbaki’s notion of structure] has since been superseded by that of category and functor, which includes it under a more general and 1 Here is the original

text in French: “Les mathématiciens, comme tous les scientifiques, ont besoin de visées téléologiques qui leur permettent de structurer et de légitimer leurs discours. Bourbaki avait réussi à donner une aura indiscutable à la pensée mathématique, en lui procurant un ressort— le structuralisme—et une finalité—la recherche d’une architecture aboutie et hiérarchisée de ses concepts et résultats.”. 2 There is one important exception to this, namely the so-called structuralist school in philosophy of science of the 1970’s and 1980’s, led by Suppes, Suppe, Moulines and Balzer. Some of them explicitly based their work on Bourbaki’s technical notion of structure. See, for instance, Erhard Scheibe’s work in philosophy of physics.

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J.-P. Marquis convenient form. It is certain that it will be the duty of Bourbaki, ..., to incorporate the valid ideas of this theory in his works. (Dieudonné 1970).

It is not that Bourbaki’s analysis was wrong or useless, but simply that a better notion was available when the volume on structure finally came out. Pierre Cartier says essentially the same thing: Bourbaki’s volume on structure, which finally appears in 1957, includes all these evolutions. On the one hand, it is a shameful treatise on categories, where the key notions (categories, functor, etc., ...) appear in filigree, but not in the “official” text. On the other hand, it is a grammar of structures: analyzing his own style, Bourbaki describes a certain number of reasoning-types, which return whenever a certain structure shows up. Ironically, the science of structures does not go beyond the descriptive stage and does not access the structural stage. ((Cartier 1998, 22), my translation)

The fact is: Bourbaki did not know how to fit categories, functors, etc., in his general analysis of mathematical structures. This is deeply ironic for two reasons. First, it is undeniable that a category is a mathematical structure and, as such, it should be describable in Bourbaki’s language of structures. Second, category theory can itself be used directly to describe mathematical structures and links between mathematical structures. According to Krömer’s analysis of Bourbaki’s archives on the subjects, Bourbaki envisaged adding a chapter of categories and functors after the chapter on structures. Somehow and for reasons that are not entirely clear, that chapter never made it to publication. (See Krömer 2006, 143 for details.) Be that as it may, Bourbaki came to the conclusion that his analysis of the general notion of abstract mathematical structure was somehow too short, for it did not encompass the notions of categories, functors, etc. This does not mean that Bourbaki concluded that his structuralist standpoint was refuted. Having worked on his volume on sets and structures for nearly 20 years and having published a series of influential books which were based on the latter, there were also practical reasons underlying his decision. Bourbaki could not rewrite all these volumes from a different starting point. Moreover, although categories, functors and natural transformations had been introduced in 1945 and used quickly after in algebraic topology and homological algebra, it was not a proper theory before the publication of Bourbaki’s volume on sets and structures, namely 1957. Indeed, the concepts of adjoint functors and of equivalence of categories, to mention but the most two important notions, were introduced in print precisely in 1957 by Kan and Grothendieck respectively. Functor categories play a central role both in Kan’s and in Grothendieck’s works, thus raising pressing issues on the foundations of category theory itself. Grothendieck introduced the concept of representable functor only in 1961. Abstract categories, that is categories given by a list of axioms, for instance additive categories, abelian categories, etc. were also introduced at that time. The point is simply that when Bourbaki had finally decided to publish its volume on sets and structures, he only had an incomplete understanding of the nature and impact of category theory, both on mathematics and its foundations. It would thus make sense to use Bourbaki’s own evaluation of his work on the general notion of abstract mathematical structure to push it aside and adopt a categorical point of view, assuming that such an analysis is available. However, this is

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not what one finds in the philosophical literature. The reasons given in the latter are of a different nature altogether.

2.2 The Philosophers’ Evaluation of Bourbaki’s Analysis To be fair, one has to remember that the contemporary philosophical discussion surrounding mathematical structuralism in the anglo-american world originated from Benacerraf’s papers published in 1965 and 1973. In those papers, Benacerraf does not refer to Bourbaki. It is interesting to note, however, that Benacerraf’s first paper opens up with a long quote taken from Richard Martin: The attention of the mathematician focuses primarily upon mathematical structure, and his intellectual delight arises (in part) from seeing that a given theory exhibits such and such a structure... (...) But ... the mathematician is satisfied so long as he has some “entities” or “objects” (or “sets” or “numbers” or “functions” or “spaces” or “points”) to work with, and he does not inquire into their inner character or ontological status. (R. M. Martin, quoted by (Benacerraf 1965, 272))

Martin simply reflects the attitude adopted by the mathematical community, the structuralist stance that was pushed and developed by Bourbaki. Be that as it may, the fact is that most of the papers and books that have been written afterwards by philosophers were targeted at Benacerraf’s arguments on truth and knowledge. Resnik is one of the first philosophers that have tried to develop a structuralist philosophy of mathematics. Nowhere in his writings does one find a reference to Bourbaki. In fact, Resnik even suggests to use the word “pattern” instead of the expression “structure” and qualify mathematics as the science of patterns. As Resnik readily admits, he is not so much interested in understanding structuralism as such, but rather he wants to defend and develop a form of (Quinean) realism about mathematics. Basically the same can be said about Shapiro’s well-known work on the subject. Shapiro has introduced what is now the standard terminology in the philosophical literature, namely the distinctions between ante rem structuralism, in re structuralism and modal structuralism. The philosophical literature focuses mostly on the qualifiers and less on structuralism as such. Like Resnik, Shapiro’s goal is to articulate a form of structural realism about mathematical knowledge. Interestingly enough, Shapiro does mention Bourbaki in his book (Shapiro 1997). He cites key passages from Bourbaki’s The Architecture of Mathematics. However, at the end of the day, Shapiro relies on Leo Corry’s evaluation of Bourbaki’s contribution to mathematical structuralism. Although their Theory of sets [1968] contains a precise mathematical definition of “structure”, Corry [1992] shows that this technical notion play almost no role in the other mathematical work, and only a minimal role in the book that contains it. The notion of “structure” that underlies the work of Bourbaki and contemporary mathematics, is inherently informal... (Shapiro 1997, 177)

Shapiro quotes Corry at this point. We will reproduce only the last part of that quote, since it illustrates an important point that needs to be underlined.

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J.-P. Marquis [T]he rise of the structural approach to mathematics should not be conceived in terms of this or that formal concept of structure. Rather, in order to account for this development, the evolution of the nonformal aspects of the structural image of mathematics must be described and explained. (Corry 1992, 342)

This book [namely Shapiro’s book] is a contribution to the program described by that last sentence. (Shapiro 1997, 342)

Thus, Shapiro’s work, as well as the work of many others, aims at describing and explaining the nonformal aspects of mathematical structures, since it is assumed that no such formal analysis can be provided. Thus, Bourbaki’s work is irrelevant to the philosophical enterprise. Hellman, in his work (Hellman 1989, 1996), does not mention Bourbaki, although in (Hellman 2003), he considers category theory as a candidate for structuralism and, as such, could be said to be faithful to Bourbaki’s spirit. However, he dismisses category theory as providing an autonomous framework for structuralism.3 Chihara in (Chihara 2004) does not refer to Bourbaki. The whole work is a conversation with Resnik, Shapiro, Hellman and others we have not mentioned. Finally, in his review of contemporary mathematical structuralism (Cole 2010), J. Cole does mention Bourbaki in a footnote, and declares that Bourbaki defended a form of settheoretical eliminative structuralism. Thus, Bourbaki is acknowledged, but confined to a precise type of mathematical structuralism.4 As we have seen, Leo Corry’s analysis of Bourbaki’s work played an important role in the recent literature on the subject. Corry has presented a detailed analysis of the evolution of the notion of structure in the first half of 20th century mathematics and has given a thorough and documented analysis of Bourbaki’s work.5 In the end, and as we have already seen in Shapiro’s quote, his evaluation of Bourbaki’s notion of structure is critical and negative. We will restrict ourselves to a unique quote that, we believe, faithfully represents Corry’s take on Bourbaki. The central notion of structure, then, had a double meaning in Bourbaki’s mathematical discourse. On the one hand, it suggested a general organizational scheme of the entire discipline, that turned out to be very influential. On the other hand, it comprised a formal concept that was meant to provide the underlying formal unity but was of no mathematical

3 To

be fair, although Dieudonné and Cartier, as well as others, have claimed that category theory provided a more convenient framework for structural mathematics, to use Dieudonné terminology, he never offered a proper and general analysis of the notion of abstract mathematical structure in the language of categories nor can we find a clear claim made by a member of Bourbaki that the latter provides foundations for mathematics. 4 As we will see, this is not unfounded and unjustified. 5 The main source here is his book: (Corry 1996). We agree with many claims made by Corry, in particular the claim that the volume on set theory stands apart in Bourbaki’s output, and not because of its clarity and rigor. We disagree on one particular, but important point about Corry’s approach, as we will make clear.

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value whatsoever either within Bourbaki’s own treatise or outside it. But Bourbaki’s theory of structures6 was only one among several attempts to develop a general mathematical theory of structures, ... . (Corry 2009, 32)

Corry makes three distinct claims in this passage. First, structuralism as an organizational scheme was put forward by Bourbaki from very early on and was very influential. Everybody agrees that this is essentially correct. Second, the technical notion presented by Bourbaki was simply worthless, even for Bourbaki. Corry is far from being the only one saying this. Saunders Mac Lane, one of the creators of category theory, went so far as to say that Bourbaki’s presentation “is the ugliest piece of writing to have come from Bourbaki’s pen.7 ” (Lane 1986, 5). This assertion can only be made if one takes Bourbaki’s analysis as being essentially mathematical and not, as it was, metamathematical. Third, Bourbaki’s technical notion is but one among many and one that is particularly bad. Corry’s general position with respect to a technical notion of structure is that such a notion simply cannot exists, for the notion of structure has evolved and will always evolve beyond the boundaries of a formal analysis. It is worth giving another explicit quote on this particular point: I will claim [in the article] that the “structural character of contemporary mathematics” denotes a particular, clearly identifiable way of doing mathematics, which can however only be characterized in nonformal terms. After that specific way of doing mathematics was crystallized and became accepted in the 1930s, diverse attempts were made to prodide a formal theory within the framework of which the nonformal idea of “mathematical structure” might be mathematically elucidated. Many confusions connected to the “structural character of mathematics” arise when the distinction between the formal and the nonformal senses of the word is blurred. (Corry 1992, 316)

We fundamentally disagree with Corry on this claim. He is partly right when he claims that Bourbaki’s influence was not a consequence of his technical notion of structure. But only partly so. It is always dangerous to claim that a way of doing mathematics can only be characterized in nonformal terms. We believe that Bourbaki’s proposal contains the main ingredients of a formal analysis and that Makkai’s FOLDS provides a completely general formal analysis of what structuralism means for abstract mathematics. Notice how Corry’s claims resemble Burgess’s evaluation quoted at the beginning or this paper. Corry recognizes the influence Bourbaki had within the mathematical community, but that influence had nothing to do with the technical notion of structures expounded by Bourbaki in his volume on set theory. This is indeed partly correct, to the extent that Bourbaki’s formal notion of structure did not have a direct impact. However, a certain component of his notion did have a tremendous impact. We will get back to this point in due course. This is how Bourbaki’s analysis is dismissed and, in the end, ignored.

6 When

Corry uses the italics, he refers to the technical notion of structure. that this is an aesthetic evaluation. Elsewhere, Mac Lane called it “a cumbersome piece of pedantry.”(Lane 1996, 181). This is also an aesthetic judgment. It might nonetheless still be essentially correct, despite being ugly.

7 Note

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3 Mathematical Structuralism: Its Architecture 3.1 Structuralism and the Axiomatic Method Before we look at the specific formal analysis presented by Bourbaki, we believe it is worth our while to rehearse some of the claims he made in the paper The Architecture of Mathematics, originally published in French in 1948, and then translated into English in 1950. For in the heads of Bourbaki’s members, structuralism is more than the claim that abstract structures are what mathematics is about. Of course, it includes the latter claim. As Bourbaki asserts himself in a footnote of that paper: “From this new point of view, mathematical structures become, properly speaking, the only “objects” of mathematics.” (Bourbaki 1950, 11, footnote †). Notice that structuralism is not taken as a starting point. It follows from a “new point of view”, with an emphasis on the novelty of the method here. Bourbaki clearly claims that its structuralist stance is a consequence of the axiomatic method. It is crucial to understand what he meant by the latter and how he saw its function or what its role is in the organization and development of mathematics. Bourbaki has thoroughly assimilated Hilbert’s and his school’s directives to use the axiomatic method systematically. He also adopted the idea that the whole of mathematics could be based on the principles of set theory and that the latter could and should be presented in an axiomatic fashion.8 Needless to say, this does not mean that doing mathematics merely amounts to deriving theorems from the axioms of set theory. Let us quote Henri Cartan, one of the founding members of Bourbaki, who published a paper in 1943 and which can be read as a companion to The Architecture of Mathematics. Let us suppose that these axioms [of set theory] are chosen once and for all. Our mathematical theory cannot limit itself to be a dreary compilation of truths, ... . For mathematics to be an efficient tool and, for us, mathematicians, to be really interested in it, it must be a living construction; one must clearly see the sequence of theorems, group partial theories. ((Cartan 1943, 11) [our translation])

How does one see “the sequence of theorems, group partial theories”? With the help of ideas, concepts. According to Bourbaki, concepts constitute the core of mathematics and concepts are captured by the axiomatic method. To wit: What the axiomatic method sets as its essential aim, is exactly that which logical formalism by itself cannot supply, namely the profound intelligibility of mathematics. [...] Where the superficial observer sees only two, or several, quite distinct theories, ..., the axiomatic method teaches us to look for the deep-lying reasons for such a discovery, to find the common ideas of these theories, buried under the accumulation of details properly belonging to each of them, to bring these ideas forward and to put them in their proper light. (Bourbaki 1950, 223) 8 Bourbaki’s

axiomatic set theory was definitely odd, idiosyncratic and arguably inadequate. The details of Bourbaki’s set theory has no impact on our main claim. However, for more on Bourbaki’s set theory, see for instance (Mathias 2014).

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The last sentence clearly indicates what the axiomatic method is used for: to abstract common ideas from different theories. Thus, one could and should perhaps talk about the abstract method in this particular case.9 The emphasis is on how mathematics develops, not on how it is founded or justified. Thus that the axiomatic method is used to study the relations existing between different mathematical theories, hence to abstract from these mathematical theories. It is therefore grounded in classical mathematics, has its roots in the latter, it is nourished by its elements. But it yields new fruits, fruits that are of a different type than the soil it comes from. And in turn, the abstract structures reorganize the landscape completely: the seeds produce new fields, new crops.10 From the axiomatic point of view, mathematics appears thus as a storehouse of abstract forms—the mathematical structures; (...) It is only in this sense of the work “form” that one can call the axiomatic method a “formalism”. The unity which it gives to mathematics is not the armor of formal logic, the unity of lifeless skeleton; it is the nutritive fluid of an organism at the height of its development, the supple and fertile research instrument to which all the great mathematical thinkers since Gauss have contributed, all those who, in the words of Lejeune-Dirichlet, have always labored to “substitute ideas for calculations”. (Bourbaki 1950, 231)

These forms are not generated randomly, nor are they mere generalizations. They are original ideas in both sense of the word: not only are they new, but in the logical order, the classical ideas are special cases of these innovations, and in that sense, the classical ideas originate from them. Furthermore, at the time, Bourbaki believed that three families of basic forms could be identified, families whose surname started with a ‘c’: composition, continuity and comparison, corresponding respectively to algebraic structures, topological structures and order structures. He even suggested they be called “mother structures”.

3.2 Mother Structures and Their Descendants It is important to understand the organization of mathematics that naturally follows from Bourbaki’s usage of the axiomatic method. First, the mother structures are seen as tools: It should be clear from what precedes that its most striking feature [of the axiomatic method] is to effect a considerable economy of thought. The “structures” are tools for the mathematician; as soon as he has recognized among the elements, which he is studying, relations which satisfy the axioms of a known type, he has at his disposal immediately the entire arsenal of general theorems which belong to the structure of that type. (Bourbaki 1950, 227) 9I

have looked at this method in more details in the following two papers: (Marquis 2014, 2016). has to be pointed out that when Bourbaki writes and decides to systematically develop this standpoint, he finds resistance among contemporary mathematicians who believed that the axiomatic method was unable to produce any genuinely new concepts and results. At the risk of repeating ourselves, Bourbaki sees the axiomatic method as a creative method in mathematics. For a discussion of the creative role of axioms in mathematics, see also (Schlimm 2011). 10 It

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This probably sounds entirely trivial today, but was not when Bourbaki was writing. One of the byproducts of this economy of thought is that the solution to certain problems do not depend on the personal talent of a mathematician. The latter now has a toolbox of concepts, theories, results that can be applied directly to various cases, apparently unrelated. One only needs to verify that this particular problem is indeed a case of the abstract structure and apply the relevant theorems to the problem at hand. Thus, these mother structures modify the very practice of mathematics. But they also transform the very fabric of mathematics. In place of the sharply bounded compartments of algebra, of analysis, of the theory of numbers, and of geometry, we shall see, for example, that the theory of prime numbers is a close neighbor of the theory of algebraic curves, or, that Euclidean geometry borders on the theory of integral equations. The organizing principle will be the concept of a hierarchy of structures, going from the simple to the complex, from the general to the particular. (Bourbaki 1950, 228)

One starts from the most general, the simplest, in other words the structures with the smallest number of axioms. It is then possible to add axioms to these mother structures to obtain more specific structures, e.g. Hausdorff topological spaces, uniform spaces, abelian groups, linearly ordered sets, etc. Thus, each mother structure already has an impressive family tree. Of course, these structures also combine together, these combinations yielding more than the simple addition of the original structures. Beyond the first nucleus, appear the structures which might be called multiple structures. They involve two or more of the great mother structures simultaneously not in simple juxtaposition (which would produce nothing new), but combine organically by one or more axioms which set up a connection between them. (Bourbaki 1950, 229)

Bourbaki exemplifies the latter by comparing topological algebra to algebraic topology.11 We are still at the level of abstract structures, although no longer in the original trees of the mother structures. It is possible to go down further still and end up working on the classical structures of mathematics, for instance the real numbers.12 Farther along we come finally to the theories properly called particular. ... At this point we merge with the theories of classical mathematics, the analysis of functions of a real or complex variable, differential geometry, algebraic geometry, theory of numbers. But they have no longer their former autonomy; they have become crossroads, where several more general mathematical structures meet and react upon one another. (Bourbaki 1950, 229) 11 This is a point where the introduction of category theory would have changed the picture considerably. Indeed, a whole section should be written on the development of the axiomatic method in the language of categories, as was done by Grothendieck in his paper on homological algebra (Grothendieck 1957). It can be argued that Grothendieck’s way of doing mathematics is a natural extension of Bourbaki’s presentation in The Architecture of Mathematics. Thus, one aspect of Grothendieck’s style is not that surprising when seen in this light. 12 We are not saying that this is original with Bourbaki. The idea, at least restricted to algebra, was already implicit in van der Waerden’s Moderne Algebra and explicit in a paper written by Helmut Hasse in 1931. See Hasse (1986). Since some members of Bourbaki have worked with Hasse and others of the German school in these years, it is not entirely ridiculous to believe that they had discussed these matters with them.

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There are numerous epistemological gain to this way of working. We have already mentioned simplicity. Cartan is a bit more explicit about these in his paper: Thus, not only the axiomatic method, based on pure logic, gives an unshakable basis to our science, but it allows us to better organize it and to better understand it, it makes it more efficient, it substitute general concepts to “computations”, which, done haphazardly, would likely lead to nothing, unless done by an exceptional genius. (Cartan 1943, 11) [our translation]

Better organization, better understanding, more efficiency are obvious epistemological virtues. Better understanding is obtained by the separation of the abstract components involved in proofs. Indeed, it is possible to identify the role played by the various abstract components in a given proof.

4 Metamathematical Structuralism: Its Nature As we have said, Bourbaki’s presentation contains two components that are somehow confused, even by Bourbaki himself.13 The first component can be presented in a purely mathematical manner if one wants, although Bourbaki himself does not. It is the notion of echelon of structure. The second component is clearly metamathematical and is a formal requisite on any mathematical theory that pertains to talk about abstract mathematical structures. Joined together they yield the notion of species of structure. Most of the literature and Bourbaki himself have concentrated on the first aspect, on the mathematical notion of structure. The metamathematical has been more or less evacuated. We want to reverse completely this tendency and put the metamathematical component at the forefront.

4.1 Confusing the Tree for the Forest: The Notion of Mathematical Structure In The Architecture of Mathematics, Bourbaki gives an informal presentation of the notion of mathematical structure. It can now be made clear what is to be understood, in general, by a mathematical structure. The common character of the different concepts designated by this generic name, is that they can be applied to sets of elements whose nature has not been specified; to define a 13 This claim might sound silly, but it is fairly easy to explain how it happened by looking at the Bourbaki archives and the evolution of the project. It has to be kept in mind that Bourbaki had no logician among its members. Claude Chevalley was the only founding member who was interested in logic and metamathematics and there is clear evidence that he was responsible for the presence of logic and the metamathematical standpoint in the various versions of the notion. Other members, like Dieudonné and Weil, thought that logic and metamathematics were peripheral and secondary to the whole enterprise.

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J.-P. Marquis structure, one takes as given one or several relations, into which these elements enter; then one postulates that the given relation, or relations, satisfy certain conditions. To set up the axiomatic theory of a given structure, amounts to the deduction of the logical consequences of the axioms of the structure, excluding every other hypothesis on the elements under consideration. (Bourbaki 1950, 225–226)

It is important to notice that Bourbaki attempts to give a totally general notion of mathematical structure. At the time, various particular cases were well-known, leading Bourbaki to introduce the idea of mother structures in his popular paper.14 To give a general analysis is another enterprise altogether, bound to be somewhat opaque and mysterious at first. It is, of course, the result of an analysis, which is done with the various examples in mind. One of the problems that Bourbaki encountered is that these examples kept popping up and kept being somewhat different from the original ones, forcing him to adjust his analysis—for instance with the case of modules which forced Bourbaki to introduce fixed parameters in the analysis—, and in the end, with the advent of categories, even to give it up! The core of the analysis is simple enough.15 We assume the language of first-order logic. Notwithstanding the cumbersome notation used by Bourbaki, the underlying ideas are simple enough.16 One starts with a finite list of set variables, which we will denote by A = A1 , A2 , . . . , An and a finite list of set constants, or parameters, which we will denote by B = B1 , . . . , Bm . Of course, in many cases, there are no parameters, but in others, they are indispensable, for instance for vector spaces, where the parameter is a field k, or modules, where the parameter is a ring R. Bourbaki first defines what he calls an echelon construction: it is a collection of E of terms defined inductively by the following simple rules: 1. each of A1 , . . . , An , B1 , . . . Bm is in E; 2. If X and Y are in E, so is X × Y ; 3. If X is in E, so is ℘ (X ). Thus, an echelon construction E provides us with all possible basic terms that are required for a structure of a given kind to be defined. This is how Bourbaki intends to cover all possible types of abstract structures to start with. Once this is done, it is necessary to introduce ways to restrict this echelon to get back to actual structures. For instance, for most algebraic structures, various products and powers of a certain type will be necessary, and for topological structures, powers, and powers of powers, etc., will be indispensable, and some other products and powers will be necessary to specify certain properties of the structure. Let us denote an element of an echelon construction E by Si and we call such an element a sort. 14 The

notion of mother structure is nowhere to be found in the official texts. have to point out, as many have done, that Bourbaki’s presentation of logic and set theory is very idiosyncratic and it is difficult to understand why he clung to his vocabulary and axioms. One obvious example is his choice to talk about assemblage to designate what any other logician calls a formula. Most commentators would focus on his choice of the τ operator and his axioms for set theory, and rightly so. He could easily have used standard notation and notions at that point, since after all, Kleene’s monumental Introduction to Metamathematics was published in 1952, to mention but the most famous textbook available at the time. Bourbaki was well aware of Hilbert and Ackermann’s book published in 1928, but he unfortunately did not adopt its conventions. 16 We are not following Bourbaki’s conventions, which we find unnecessarily complicated and we simplify both the notation and the presentation. 15 We

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Thus, given a echelon construction E, the next step is to pick elements S1 , . . . , S p of E, which we can call specified sorts, that is those that are necessary for the definition of the type of structure one has in mind. One then adds a list R1 , . . . , Rk of sorted relation symbols.17 This yields what is commonly called a signature or a  B,  S,  R).  By interpreting these symbols in the obvious way similarity type L = L( A, in the domain of sets, one obtains the notion of an L-structure. We should hasten to add that this is not yet the notion of structure we are driving at, nor is it yet Bourbaki’s notion. Bourbaki defines what he calls ‘a species of structure’ and for that a key component is still missing: the notion of isomorphism or, in Bourbaki’s terminology, transport of relation. Before we introduce it, it should be noted that in most presentations of Bourbaki’s analysis, the distinction between the mathematical and the metamathematical levels is usually blurred. For instance, here is how Corry presents the notion of a species of structure in one of his articles: Now to define a ‘species of structure’  we take: 1. n sets x1 , x2 , . . . , xn ; the ‘principal base sets’; 2. m sets A1 , A2 , . . . , Am ; the ‘auxiliary base sets’ and 3. a specific echelon construction scheme: S(x1 , . . . , xn , A1 , . . . , Am ). This scheme will be called the ‘typical characterization of the species of structure ’. The scheme is obviously a set and the structure is now defined by characterizing some of the members of this set by means of an axiom of the species of structure. This axiom is a relation which the specific member s ∈ S(x1 , . . . , xn , A1 , . . . , Am ) together with the sets x1 , . . . , xn , A1 , . . . , Am must satisfy. The relation in question is constrained to satisfy the conditions of what Bourbaki calls a ‘transportable relation’, which means roughly that the definition of the relation does not depend upon any specific property of s and the sets in themselves, but only refers to the way in which they enter in the relation through the axiom. (Corry 1992, 323–324)

Notice how Corry presents everything directly in terms of sets. This simply reads like we are dealing with a generalization of the presentation of any usual mathematical structure, e.g. the group structure, the topological structure, etc. We have sets, certain basic set-theoretical operations on them necessary to define specific relations and operations on them and, finally, axioms specifying the specific properties that these relations have to satisfy. Except for the following bit on transportable relation, which is not explained and seem simple enough. It is very easy to lose track that we are firmly in a metamathematical context. The fact that in the volume on sets, the chapter immediately preceding the section on structure treats the notions of ordered sets, cardinals and integers does not help. We are squarely in set theory. The reader is thus asked to move back to the metamathematical mode without any specific warning at this moment. Furthermore, the foregoing presentation can, in fact, be presented directly in terms of sets and opera17 This

can easily be translated into purely formal requirements of the usual kind.

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tions on sets, functions on sets, etc.18 In some of the earlier versions of the chapter on structures, Bourbaki himself does not carefully make the distinction. In the final volume, we are clearly in metamathematics, but the presentation oscillates between sets and formal expressions. What one has to understand, is that Bourbaki also introduces a completely general notion of isomorphism between instances of a species of structure and that the latter determines directly and precisely what one can write down and prove about these structures. And this, in our mind, is the key element.

4.2 Seeing the Forest: The Notion of Isomorphism In the same way that Bourbaki had to give a completely general notion of structure, he had to give a completely general notion of isomorphism of structure as well. At the time, once again, numerous particular examples were clear: various specific notions of isomorphism for algebraic structures were known, similarly for topological structures and order structures. It should be emphasized that the identification of the correct notion of isomorphism for a structure of a given type is not a trivial business and that, in some cases, it took quite some time before the community of mathematicians finally settled on the right notion.19 It also has to be pointed out that there is no mention of isomorphism in The Architecture of Mathematics. This omission is perhaps attributable to the fact that the latter paper was aimed at a general audience and that Dieudonné might have felt that explaining even particular cases of isomorphisms would simply be too technical in such a short paper.20 A quick examination of the Bourbaki archives shows clearly that the notion of isomorphism was nonetheless incorporated in the analysis right from the start. Ten years after The Architecture of Mathematics, when Cartan reflects on Bourbaki, the emphasis is crystal clear: 18 It is, indeed, very tempting to start the analysis in the category of sets and define the notion of species of structures directly there. That would yield a perfectly acceptable mathematical analysis of that latter notion. It is probably what Bourbaki would have done, had he agreed on a way to do it. It was done by Ehresmann in (Ehresmann 1965) and, more recently and in a different context, by Joyal in (Joyal 1981). Our goal, and we believe Bourbaki’s goal too, is to provide a genuine metamathematical analysis, something that is required to anchor a structuralist standpoint about the whole of abstract mathematics. 19 Two specific and surprising cases have to be mentions: the notion of homeomorphism for topological spaces and the notion of equivalence of categories. In the case of topological spaces, mathematicians did not see immediately what the right notion was and there was some confusion in the literature for quite some time. See Moore (2007) for details. As for categories, Eilenberg and Mac Lane introduced the notion of isomorphism of categories in 1945, thinking that it was the proper criterion of identity for them. The right notion, namely the notion of equivalence, was introduced by Grothendieck in his paper in homological algebra in 1957, thus twelve years after the publication of Eilenberg and Mac Lane’s original paper. See Marquis (2009). 20 Dieudonné does not give the definition of a topological space either, believing that “the degree of abstraction required for the formulation of the axioms of such a structure is decidedly greater that it was in the preceding examples; the character of the present article makes it necessary to refer interested readers to special treatises” (Bourbaki 1950, 227).

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Bourbaki’s decision to use the axiomatic method throughout brought with it the necessity of a new arrangement of mathematics’ various branches. It proved impossible to retain the classical division into analysis, differential calculus, geometry, algebra, number theory, etc. Its place was taken by the concept of structure, which allowed the definition of the concept of isomorphism and with it the classification of the fundamental disciplines within mathematics. (Cartan 1979, 177)

It is remarkable to see that the concept of structure allows the definition of isomorphism. They go hand in hand. The fact that they form a pair is obfuscated by the emphasis on the axiomatic method. But in Bourbaki’s presentation, the notion of isomorphism is an intrinsic part of the axiomatic method. It should also be pointed out that the title of the first section of the fourth chapter is ‘Structures and isomorphisms’. As we have seen in Corry’s presentation, the transport of relation—which is quickly shown by Bourbaki to be equivalent to the notion of isomorphism—is often thought of as a constraint which merely guarantees that, to quote Corry again, ‘the definition of the relation does not depend upon any specific property of s and the sets in themselves, but only refers to the way they enter in the relation through the axiom’.21 Of course, this is entirely correct, and as a comment aimed at expressing a key feature of transport of relations, it is entirely acceptable. However, it is necessary to be a little more precise here. The crucial element is that transportable relations are built-in Bourbaki’s notion of a theory of a species of structure. This means, literally, that the only theorems provable in a theory of a species of structure are those that are invariant under isomorphism. Bourbaki proceeds as follows. We go back to our basic set variables, but we now suppose that we have two lists of terms A = A1 , A2 , . . . , An and A = A1 , A2 , . . . , An . We now add terms f 1 , . . . , f n to our theory such that the relations “ f i is a function from Ai to Ai ” are theorems of the theory, for all 1 ≤ i ≤ n. It can then be shown that these f i s can be canonically extended to the echelon construction. Moreover, if the original f i s are bijections, the canonical extensions are bijections too.22 We can now introduce the notion of transport of structure. We have our specified sorts and sorted relations. For a relation R to be transportable in a theory T in the language L means that it can be proved in the theory T that the relation R holds for the sorts defined over Ai if and only if R holds for the sorts over f i (Ai ) = Ai and their canonical extensions in the echelon construction. Again, this claim can be made very precise with the proper notation, but it would occupy an unnecessarily long portion of this paper. The crucial point to notice is the 21 Corry is very well aware of the fact that there were no general analysis of the notion of isomorphism before Bourbaki. Indeed, in a different paper, he says “None of these concepts, however, is defined in a general fashion so as to be a priori available for each of the particular algebraic systems [in van der Waerden’s textbook]. Isomorphisms for instance, are defined separately for groups and for rings and fields, and van der Waerden showed in each case that the relation “is isomorphic to” is reflexive, transitive and symmetric” (Corry 2001, 172). 22 Notice the ambiguity here. We are still in metamathematics, but it is all too easy to fall back on a purely set theoretical reading of the definitions and notions given. We are talking about set theoretical formulas all along. We have to write down formulas such that, when they are interpreted in sets, then the f i s are bijections.

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fact that one has to prove in the theory that the relations are transportable and that the notion of species of structure has not been defined yet. We are now ready to do so, for Bourbaki requires that the relations used in the axioms of a structure provably be transportable. Bourbaki is very clear: a species of structure  in the language of set theory is a text, i.e. a series of first-order formulas of set theory in the given signature L together with a transportable relation R, the latter called the axiom of the species of structure.23 In a terminology which is more in tune with contemporary logical conventions, one can say that a Bourbaki species of structure is given by the L-structures whose relations are transportable, in other words, as Bourbaki himself shows, whose relations satisfy the condition of isomorphism invariance. Thus, Bourbaki incorporates in the axiomatic method itself the requirement that the defining properties of a structure be invariant under isomorphism. It is primordial to understand that the latter is a metamathematical requirement, in the sense that it has to be provable in the theory that the relation is invariant under isomorphism. And this gives us what deserves to be called the structuralist motto: for a logical theory to be considered a structuralist theory for abstract mathematics, it has to satisfy the following property: for objects X and Y of the theory T , the only properties P that are legitimate in T are those that satisfy invariance under isomorphism: If P(X ) and X ∼ = Y , then P(Y ). Clearly, Bourbaki’s development of mathematics satisfies this motto. Remember that the volume on set theory is the only volume in which Bourbaki takes a metamathematical standpoint. When one looks at the other volumes, say on topology or algebra, the definitions proceed in the standard fashion. Bourbaki does not state that the axioms of a topological space are given by a text or by formulas in the language of set theory. He simply states them. But what he does, and in fact systematically, is to introduce the proper notion of isomorphism by invoking the general construction given in the chapter on structures.24 The general notion of isomorphism is used as it should be in the remaining volumes. And that is the whole point. This is why and how one knows what is, for instance, a topological property or a group property, etc. And this is precisely why mathematicians do not fall prey to Benacerraf’s problem. It is striking to see that the remaining sections of the chapter on structures reads more and more as a standard mathematical text and less and less as a metamathematical analysis. It is also striking that the notions presented in the published version are 23 This is very surprising. Of course, Bourbaki does not literally mean that a species of structure is a text, for it has to be an interpretation of that text in a mathematical domain. There is no doubt that Bourbaki understood that, but this is what one reads. I suspect that the emphasis on the text was deliberate in order to underline the metamathematical nature of the analysis. 24 See, for instance definition 3 in the first section of the volume on general topology. The reference is explicit. The definition given by Bourbaki of isomorphism of topological spaces is not the standard definition. Bourbaki then immediately shows that the definition that follows from the general notion of isomorphism of structure is equivalent to the standard definition of homeomorphism. Corry is absolutely correct to point out that “the verification of this simple fact (which is neither done nor suggested in the book) is a long and tedious (though certainly straightforward) formal exercice”(Corry 1992, 330). However, it misses the main point, which is essentially metamathematical.

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indeed concepts that are afterwards presented in the language of categories: initial structure, product structure, final structure, quotient structure and universal map. And even as such, the presentation is clumsy. But as we have said, had Bourbaki decided to present these ideas in the language of categories, he would probably not have done a better job, since the latter had not reached its maturity in the mid 1950s.25

5 FOLDS: Encompassing All Forms of Structures Bourbaki’s structuralism faced two difficulties. First, the theory one starts with, namely set theory, is not itself structural. This might or might not be a serious problem, but it certainly deserves a discussion. For one thing, the question whether it is possible to build a set theory that would satisfy the structuralist motto is worth investigating. Second, as we have already mentioned, categories do not fit easily and immediately in Bourbaki’s scheme. The problem arises with the very definition of a category. Is a category a set? Clearly, the category of all set cannot be a set. This was the first problem faced by Bourbaki (and, in fact, all those using categories). Second, the criterion of identity for categories is given by the notion of equivalence of categories, not the notion of isomorphism. Bourbaki could not have known this when he published his volume on sets and structures, for the notion of equivalence of categories was introduced by Grothendieck in 1957. Thus, the notion of category raises a new metamathematical challenge, for the notion of transport of relation as given by Bourbaki is inadequate for categories. Both of these problems now have a solution that is consistent with the structuralist motto. These solutions rely on the logical framework developed by Michael Makkai more than twenty years ago, namely First-Order Logic with Dependent Sorts, or FOLDS for short.26 ,27 It is impossible to do justice to FOLDS in such a short paper. We will summarize its mains features and emphasize how it captures the fundamental idea underlying 25 Thus, we disagree with Corry’s evaluation that by 1957, category theory had reached the status of an independent discipline that enabled generalized formulations of several recurring mathematical situations. Mac Lane had further developed some central ideas in his article on ‘duality’ (Corry 1992, 332). Category was not yet an independent discipline and although it did enabled generalized formulations of several recurring mathematical situations, these were restricted to algebraic topology and homological algebra. Mac Lane’s paper was not very influential and it is with hindsight that one sees into it some of the ideas that will become central after 1957, once they will be shown to be systematically related to central concepts of the theory. 26 Makkai published only one “official” paper on FOLDS, namely (Makkai 1998). There is much more available on his web site: http://www.math.mcgill.ca/makkai/. For an informal presentation of FOLDS and some aspects of its motivation, see (Marquis 2018). Emily Riehl pointed out to me that the claim made in the footnote 37 of the latter that the simplex category with morphisms restricted to the injective functions is the FOLDS-signature for simplicial sets is wrong. It is the FOLDS-signature for semi-simplicial sets. 27 It could be argued that homotopy type theory with the univalence axiom also provides a solution. See, HoTT (UFP 2013). We will not discuss HoTT here.

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Bourbaki’s approach and how to generalize it to more abstract forms. First, FOLDS is a revision of first-order logic. For one thing, it is a multi-sorted syntax. Second, these sorts are dependent sorts. For instance, when one writes f : X → Y , the underlying syntax is that one has X, Y : Object, that is X and Y are declared to be variables of sort ‘Object’ and then f : Arr (X, Y ) is declared to be a variable of sort ‘Arrow’ and f depends on X and Y . One has to provide syntactical rules, which will necessarily be more complicated than the usual syntactical rules for first-order logic, for such a system and it is precisely what is done in FOLDS. FOLDS has to be seen as a general theory of mathematical identity. Indeed, in FOLDS, mathematical identity is not given a priori and it is not a relation, it is derived from a given signature in the language and it is a structure. Thus, when FOLDS is used to develop set theory, the criterion of identity is not given by the axiom of extensionality, but by the structure between functions that defines the notion of bijections between sets. When FOLDS is used to develop (1-)category theory, one gets the notion of equivalence of categories. When FOLDS is used to develop bicategories, the notion of isomorphism one gets is the notion of biequivalence of categories. When FOLDS is used to develop homotopy theory, one gets the notion of homotopy equivalence. As a consequence of the way it handles identity, that is the notion of ‘isomorphism’ for the type of structure obtained from a signature, it is possible to prove that the invariance principle holds: given a language L with its notion of L-isomorphism and L-structures M and N , then if  M φ and M L N , then  N φ. It turns out that Bourbaki’s way of dealing with invariance of isomorphism can be described in this set up. Thus we believe that FOLDS does provide a formal, metamathematical analysis of what it is to be an abstract mathematical structure.

6 From Metamathematical Structuralism to Philosophical Structuralism If structuralism for abstract mathematics is to hold any water, it ought to be based on a metamathematical analysis that captures the fundamental intuition underlying it. I claim that this is precisely what FOLDS provides. In contrast with what one finds in the paper The Architecture of Mathematics, we do not end up with three kinds of mother structures. This is not the point and there is nothing in the framework itself that points towards some privileged structures. We have a completely general formal framework that allows us to see how a purely structural mathematical framework can be developed. It does indicate an architecture of mathematics, how certain abstract structures are build one upon others. Furthermore, it is clearly open ended, with levels of abstractions and interplays between and within these levels. In a way, Bourbaki already had a glimpse of the evolution of species of structures.

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It is quite possible that the future development of mathematics may increase the number of fundamental structures, revealing the fruitfulness of new axioms, or of new combinations of axioms. We can look forward to important progress from the invention of structures, by considering the progress which has resulted from actually known structures. On the other hand, these are by no means finished edifices; it would indeed be very surprising if all the essence had already been extracted from the principles. (Bourbaki 1950, 230)

Thus, Bourbaki certainly envisaged the possibility that new fundamental structures might emerge. What he did not see coming was the possibility that these new structures would be more abstract than the ones he was familiar with.28 How is this related to philosophical structuralism? I believe that it is directly related to it. The usual questions about reference, meaning and truth, for instance, can now be put in their proper formal context. The epistemological and the ontological issues can also be formulated in the proper framework. As a bonus, mathematical structuralism and philosophical structuralism are now aligned along the same lines. This is what foundational research is all about: revealing explicitly what underlies the practice of a kind of mathematics and allowing for a better reflection of the philosophical content of that practice. Acknowledgments The author gratefully acknowledge the financial support of the SSHRC of Canada while this work was done. This paper is part of a larger project on Bourbaki and structuralism which would not have seen the day without Michael Makkai’s influence and generosity. I want to thank him for the numerous discussions we had on the subject. I also want to thank Alberto Perruzzi and Silvano Zipoli for inviting me to present this work at the conference on Bourbaki, mother structures and category theory.

References Atiyah, M. (2007). Bourbaki, A secret society of mathematicians and the artist and the mathematician-a book review. Notices of the AMS, 54(9), 1150–1152. Benacerraf, P. (1965). What numbers could not be. The Philosophical Review, 74(1), 47. Bourbaki, N. (1950). The architecture of mathematics. American Mathematical Monthly, 57, (221– 232). Burgess, J. P. (2015). Rigor and structure. Oxford: Oxford University Press. Cartan, H. (1943). Sur le fondement logique des mathématiques. Revue scientifique, LXXXI, 3–11. Cartan, H. (1979/80). Nicolas Bourbaki and contemporary mathematics. Math. Intelligencer, 2(4), 175–180. Cartier, P. (1998). Le structuralisme en mathématiques: mythes ou réalité?. Bures-sur-Yvette: Technical report. Chihara, C. S. (2004). A structural account of mathematics. New York: The Clarendon Press, Oxford University Press. 28 One

of the things that Bourbaki did not see is the importance of the relations between structures, in particular between algebraic and topological structures, which are made explicit and possible by category theory. The most striking example of that interaction that was available from early on is the equivalence between the category of affine schemes and the (opposite) of the category of commutative rings. It is this interaction that allowed Grothendieck to develop new foundations for algebraic geometry, foundations that are not given by the axiomatic method.

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Cole, J. C. (2010). Mathematical structuralism today. Philosophy Compass, 5(8), 689–699. Corry, L. (1992). Nicolas Bourbaki and the concept of mathematical structure. Synthese, 92(3), 315–348. Corry, L. (1996). Modern algebra and the rise of mathematical structures (Vol. 17). Science Networks Basel: Historical Studies. Birkhäuser Verlag. Corry, L. (2001). Mathematical structures from Hilbert to Bourbaki: the evolution of an image of mathematics. In U. Bottazzini, & A. D. Dalmedico (Eds.), Changing images in mathematics: from the french revolution to the new millenium, studies in the history of science, technology and medicine (pp. 167–186). New York: Routledge. Corry, L. (2009). Writing the ultimate mathematical textbook: Nicolas Bourbaki’s ‘Éléments de mathématique’. In E. Robson & J. Stedall (Eds.), The Oxford handbook of the history of mathematics (pp. 565–588). Oxford: Oxford University Press. Dieudonné, J. A. (1970). The work of Nicholas Bourbaki. The American Mathematical Monthly, 77(2), 134. Ehresmann, C. (1965). Catégories et structures. Paris: Dunod. Grothendieck, A. (1957). Sur quelques points d’algèbre homologique. Tôhoku Mathematical Journal, 2(9), 119–221. Hacking, I. (2014). Why is there philosophy of mathematics at all?. Cambridge: Cambridge University Press. Hasse, H. (1986). The modern algebraic method. The Mathematical Intelligencer, 8(2), 18–23. Hellman, G. (1989). Mathematics without numbers: towards a modal-structural interpretation. Clarendon Press. Hellman, G. (1996). Structuralism without structures. Philosophia Mathematica. Series III, 4(2), 100–123. Hellman, G. (2003). Does category theory provide a framework for mathematical structuralism? 11(2), 129–157. Joyal, A. (1981). Une théorie combinatoire des séries formelles. Advances in Mathematics, 42(1), 1–82. Krömer, R. (2006). La “machine de Grothendieck” se fonde-t-elle seulement sur des vocables métamathématiques? Bourbaki et les catégories au cours des années cinquante. Revue d’histoire des mathématiques, 12, 119–162. Mac Lane, S. (1986). Letters to the editor. The Mathematical Intelligencer, 8(2), 5–5. Mac Lane, S. (1996). Structure in mathematics. Philosophia Mathematica. Series III, 4(2), 174–183. Makkai, M. (1998). Towards a categorical foundation of mathematics. In J. A. Makowsky & E. V. Ravve (Eds.), Logic Colloquium ’95 (Haifa) (Vol. 11, pp. 153–190), Lecture Notes in Logic. Berlin: Springer. Marquis, J.-P. (2009). From a geometric point of view: a study in the history and philosophy of category theory. In: Logic, epistemology, and the unity of science, vol. 14. Springer. Marquis, J.-P. (2015). Mathematical abstraction, conceptual variation and identity. In: P.-E. Bour, G. Heinzmann, W. Hodges, & P. Schroeder-Heister (Eds.), Logic, methodology and philosophy of science, proceedings of the fourteen international congress, pp. 299–322. London: College Publications. Marquis, J.-P. (2016). Stairway to heaven: the abstract method and levels of abstraction in mathematics. The Mathematical Intelligencer, 38(3), 41–51. Marquis, J.-P. (2018). Unfolding FOLDS: a foundational framework for abstract mathematical concepts. In: E. Landry (Ed.), Categories for the working philosophers, pp. 136–162. Oxford University Press. Mathias, A. R. D. (2014). Hilbert, Bourbaki and the scorning of logic. In: Infinity and truth, pp. 47–156. Hackensack, NJ: World Sci. Publ. Moore, G. H. (2007). The evolution of the concept of homeomorphism. Historia Math, 34(3), 333–343. Patras, F. (2001). La pensée mathématique contemporaine. PUF.

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The Univalent Foundations Program. (2013). Homotopy type theory—univalent foundations of mathematics. Princeton, NJ: The Univalent Foundations Program; Princeton, NJ: Institute for Advanced Study (IAS). Schlimm, D. (2011). On the creative role of axiomatics. The discovery of lattices by Schröder, Dedekind, Birkhoff, and others. Synthese, 183(1), 47–68. Shapiro, S. (1997). Philosophy of mathematics. Structure and ontology. New York: Oxford University Press.

Ladders of Sets and Isomorphisms The Shortcomings of Bourbaki’s Notion of “Structure” Claudio Bartocci

Abstract By examining not only the printed texts of Nicolas Bourbaki’s Éléments de mathématique but also the huge collection of primary sources held by the Archives Bourbaki, I shall try to illustrate the development of the concept of “structure” in “Bourbaki’s woek”, and its shortcomings, on a time span of approximately twenty years (1936–1957)

1 Towards a Structural Conception of Algebra (1921–1939) The name of Bourbaki is, almost unavoidably, associated with the concept of structure. Yet this concept was not an invention of Bourbaki (who, as stressed by Dieudonné (1982), p. 619, “anyway disclaim[ed] ever having invented anything”). Actually, around the mid 1930s, when Bourbaki started planning his multi-volume project (at first simply titled Traité d’analyse and only later Éléments de mathématique),1 the use of the term “structure,” understood in a plurality of senses, was already fairly widespread in the mathematical literature. More than that, it appears that a novel “structural” conception of algebra started gradually emerging, as witnessed to by the research programmes of some mathematicians like Øystein Ore and Saunders Mac Lane. The reshaping of “classical algebra” into a somewhat different discipline— called “modern” or “abstract” algebra—is to be regarded, of course, as a quite long and definitely nonlinear process, which is rooted in the pioneering work of Dedekind 1 In

1936 the members of the group Bourbaki—namely, Szolem Mandelbrojt, Jean Delsarte, Henri Cartan, André Weil, Jean Dieudonné, René de Possel, Jean Coulomb, Claude Chevalley, and Charles Ehresmann—addressed a letter to Jean Perrin, at that time “sous-secrétaire d’État à la Recherche Scientifique,” applying for financial support for the writing of a Traité d’analyse mathématique (Archives Bourbaki 1936b). According to Beaulieu (1994), “the subtitle and the title Éléments de mathématique were chosen, at latest, in 1938.”

C. Bartocci (B) Università di Genova, Genova, Italy e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 A. Peruzzi and S. Zipoli Caiani (eds.), Structures Mères: Semantics, Mathematics, and Cognitive Science, Studies in Applied Philosophy, Epistemology and Rational Ethics 57, https://doi.org/10.1007/978-3-030-51821-9_4

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and Hilbert. Although several mathematicians—e.g. Hurt Hensel, Alfred Loewy, or Ernst Steinitz2 —gave substantial contributions to specific research domains, there is little doubt that the key figure in this process was Emmy Noether, who—both in her writings and in her teaching—played an instrumental role in moulding a new image of algebra.3 Noether’s 1921 groundbreaking article Idealtheorie in Ringbereichen (Noether 1921), for example, generalised the result obtained by Emanuel Lasker and (independently) by Francis Macaulay in the case of polynomial rings, by proving the existence of a primary decomposition of ideals in any commutative ring that satisfies what is now called the ascending chain condition: the notions of ring and ideal are introduced in a purely axiomatic fashion,4 and the whole treatment of the subject, including proofs, is only dependent on the “necessary relations” between concepts which are determined by the axioms. At the risk of oversimplifying a more nuanced state of affairs, one could argue that, in this and other analogous papers (e.g. Noether 1927), Noether was, on the one hand, developing some algebraic ideas ultimately attributable to Dedekind and, on the other, casting them into Hilbert’s axiomatic conceptiont. As is well known, the core of this conception is clearly expressed by what Hilbert himself wrote in his famous letter to Frege of 29 December 1899: But it is surely obvious that every theory is only a scaffolding or schema of concepts together with their necessary relations to one another, and that the basic elements can be thought of in any way one likes. If in speaking of my points I think of some system of things, e.g. the system: love, law, chimney-sweep …and assume all my axioms as relations between these things, then my propositions, e.g. Pythagoras’ theorem, are also valid for these things.5

The 21-year-old Bartel Leendert van der Waerden came to Göttingen from Amsterdam in 1924: according to his own recollections, “a new world opened before [him]” and he “started learning abstract algebra” (van der Waerden 1975, pp. 32–33). Six years later, in 1930, van der Waerden published the first volume of his textbook Moderne Algebra (van der Waerden 1931), whose title was tellingly followed by 2 In

Ernst Steinitz’s important article Algebraische Theorie der Körper (1910), the notion of prime field was introduced and the fact that every field has a unique up to isomorphism algebraical closure was proved. In the introduction, Steinitz claimed that his main interest was “the concept of field [Körperbegriff] in itself” and that his aim was “to achieve an overview of all possible types of fields and to establish the basic features of their mutual relations” (Steinitz 1910, p. 167). 3 Leo Corry’s book Modern Algebra and the Rise of Mathematical Structures (Corry 2004) offers a detailed and systematic treatment of the subject. 4 Noether’s definition of ring essentially followed that given by Abraham Adolf Fraenkel in his (Fraenkel 1914) Inaugural-Dissertation Über die Teiler der Null und die Zerlegung von Ringen (Fraenkel 1914). However, Fraenkel’s rings are not necessarily commutative and have a multiplicative unit, whilst Noether’s rings are always commutative but may fail to have a multiplicative unit. 5 “Ja, es ist doch selbstverständlich eine jede Theorie nur ein Fachwerk oder Schema von Begriffen nebst notwendigen Beziehungen zu einander, und die Grundelemente können in beliebiger Weise gedacht werden. Wenn ich unter meinen Punkten irgendwelche Systeme von Dingen, z.B. das System: Liebe, Gesetz, Schornsteinfeger …, denke und dann nur meine sämmtlichen Axiome als Beziehungen zwischen diesen Dingen annehme, so gelten meine Sätze, z.B. der Pythagoras auch von diesen Dingen” (Frege 1976, p. 67; English transl. p. 40).

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the caption “unter Benutzung von Vorlesungen von E. Artin und E. Noether” (“with the use of lectures by E. Artin und E. Noether”). This text “formulated clearly and succinctly the conceptual and structural insights which Noether had expressed so forcefully” (Mac Lane 1997, p. 321) and rather quickly became a standard reference in the field, both in Europe and in the United States. In particular, as pointed up by Dieudonné, it exemplified the “style of proofs which most influenced Bourbaki” (Dieudonné 1982, p. 619). An enthusiastic reviewer of van der Waerden’s Moderne Algebra was Øystein Ore, who too had spent a period in Göttingen in the early 1920’s and had imbibed the new conception of algebra from Emmy Noether. In his overtly laudatory review (1932), Ore remarked that “modern algebra is a subject quite different from the classical algebra built up in its last golden era by Dedekind, Weber, Frobenius, and Kronecker” and that “[i]ts fundamental principles are closely related to Hilbert’s ideas of a formal foundation of mathematics, reducing all theories to an axiomatic basis consisting of relational properties of undefined elements” (Ore 1932, p. 327). From this standpoint—Ore argued—“the main problem of abstract algebra is however the determination of all systems with a given operational basis, i.e. to find the structural properties of all such systems” (Ore 1932, p. 327). He himself addressed this problem a few years later in the ambitious papers On the foundations of abstract algebra, I and II (Ore 1935, 1936a). His starting point was the observation that In the discussion of the structure of algebraic domains one is not primarily interested in the elements of these domains, but in the relations of certain distinguished sub-domains like invariant sub-groups in groups, ideals in rings and characteristic moduli in modular systems. (Ore 1935, p. 406)

Since “[f]or all of these systems there are defined the two operations of union and cross-cut satisfying the ordinary axioms,” he was lead to introduce “new systems […] having these two operations,” which he named “structures” (Ore 1935, p. 406). A “structure” in Ore’s sense corresponded to what we now call a lattice,6 and was by no means a novel notion. Indeed, under various forms and for various purposes, the lattice concept had been around in mathematics at least since the last quarter of the 19th century: first introduced by Ernst Schröder and, independently, by Richard Dedekind (who coined for it the term Dualgruppe), later used by Karl Menger in his axiomatisation of projective geometry, it eventually gave rise to a rich theory whose foundations were laid down in the 1930s by Garrett Birkhoff and, in a different perspective, by Ore.7 Ore regarded his theory of “structures” as a “general theory attempting to unify the various theories of algebraic systems and their decomposition theorem,” and he 6 According to today’s standard definition, a lattice is a set equipped with two operations, denoted by

∨ (join) and ∧ (meet), which satisfy the idempotent, commutative and associative laws, as well as the absorption laws: x ∨ (x ∧ y) = x and x ∧ (x ∨ y) = x. Equivalently, a lattice may be defined as a partially ordered set in which least upper bounds and greatest lower bounds of any two elements exist. 7 On the development of lattice theory see (Mehrtens 1979; Schlimm 2011); a detailed exposition of Ore’s theory of “structures” may be found in (Corry 2004, Chap. 6).

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stressed that “[t]his was achieved, to put it paradoxically, by the elimination of the elements from the algebraic theories” (Ore 1936a, p. 265). As a fundamental tool to implement this programme he introduced the notion of “Dedekind structure,” that is a modular lattice8 in today’s terminology: examples of “Dedekind structures” which naturally arise in the study of “algebraic systems” are lattices of subspaces of vector spaces, lattices of ideals of a ring, lattices of normal subgroups of a group, and lattices of submodules of a module over a ring. Ore proved a theorem about “Dedekind structures” which constitutes a generalisation of the classical Jordan-Hölder theorem for groups9 and showed that his formalism was “sufficient to obtain all the central algebraic theorems of decomposition” (Ore 1936a, p. 265). A pivotal role in Ore’s approach is played by the concept of “homomorphism of structures”, which corresponds to what would today be called a surjective lattice homomorphism.10 In his mostly expository 1936 booklet Algèbre abstraite (Ore 1936b)—issued in the series “Actualités scientifiques et industrielles” published by Hermann11 —, he remarked that this concept made it possible to state several important algebraic results in a more elegant and intrinsic way; for instance, the fundamental theorem of Galois theory could be rephrased in the language of “structures” as follows: Il existe une isomorphie de structure (inverse) entre les sous-corps d’une extension normale K d’un corp k, et les sous-groupes d’automorphismes de K par rapport a k. (Ore 1936b, p. 50)

Noether’s innovative approach to algebra exerted a decisive (albeit indirect) influence also over Saunders Mac Lane, who, after having attended Ore’s first graduate courses at Yale in 1929 and spent one year in Chicago, moved to Göttingen to write a thesis on logic (under Paul Bernays’s guidance) during the years 1931–1932. As he was to write much later on in his autobiography, “[i]n her [viz. Noether’s] view, algebra should deal with concepts and not just manipulation. It was only in Göttingen that I came to understand these things […] (Mac Lane 2005, p. 50). Back in the United States, Mac Lane wandered through different universities and different research subjects. In the fall of 1938, he and Garrett Birkhoff both became assistant professors at Harvard and quite soon they started working on a joint book assembling and expanding on the lecture notes of the courses on algebra they delivered. The resulting volume, A Survey of Modern Algebra, published by Macmillan in 1941, was—according to the second author—“the first American undergraduate text lattices are lattices that satisfy the further condition (x ∧ (y ∨ z)) ∨ z = (x ∨ z) ∧ (y ∨ x). Such “structures” are called Dedekind structures by Ore because Dedekind had been the first to state the modularity condition in the paper (Dedekind 1900). It should be pointed out that Ore had an in-depth knowledge of Dedekind’s work, for he had been the co-editor, together with E. Noether and R. Fricke, of Dedekind’s Gesammelte mathematische Werke (3 vol., 1930–1932). 9 More precisely, Ore managed to generalise (Ore 1935, Theorem III.22) the extension of the JordanHölder theorem obtained by Otto Schreier in 1928. It is not unlikely that Ore drew some inspiration from Birkhoff’s paper (Birkhoff 1934). 10 See (Ore 1935, p. 416). 11 The same series where Bourbaki’s Éléments de mathématiques would have later been published in instalments. 8 Modular

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in algebra that wholeheartedly presented the abstract ideas of Emmy Noether and B.L. van der Waerden” (Mac Lane 2005, pp. 81–82). A certain particular notion of “structure” appears to be central to Mac Lane’s conception of algebra. “What is algebra?”—he wondered in the last section of a survey giving an account of a conference on algebra held at the University of Chicago in the summer of 1938 (Mac Lane 1939). His answer was that “algebra concerns itself with the postulational description of certain systems of elements in which some or all of the four rational operations are possible”; however—he added—“[t]he abstract or postulational development of these systems must […] be supplemented by an investigation of their ‘structure’” (Mac Lane 1939, pp. 17–18). Mac Lane steered clear of providing a precise definition of the concept of “structure” and contented himself with saying that under such a term one should include: (a) the number and interrelations of the subsystems of a given system, either subsystems just like the whole system (lattice of subgroups), or subsystems with especially characteristic properties (sets of integers, maximal orders, ideals, subfields of an algebra, etc.); (b) the group of automorphism of a system, and connections between the subgroups of this group and the subsystems of the given system (Galois theory, class field theory); (c) the construction of all systems of specific types out of simpler systems of the same or other types (the construction of cyclic algebras and matrix algebras, the reduction of a given surface to a birationally equivalent surface without singularities, construction of Lie algebras); (d) alternatively, the description of given systems as subsystems of larger systems (complete fields, power series fields); (e) criteria or invariants to determine when two explicitly but differently constructed systems are abstractly the same or isomorphic (the canonical generation of a cyclic algebra; the genus as an invariant defined by the differentials of a function fields). (Mac Lane 1939, p. 18)

Mac Lane “ventured” to conclude from these premises that Algebra tends to the study of the explicit structure of postulationally defined systems closed with respect to one or more rational operations. (Mac Lane 1939, p. 18)

He was fully aware, however, that his statements were “hyper-generalizations [that] fit the facts only when the facts are first slightly distorted.” He pointed out, in particular, that the concluding summary quoted above did not “account well for the use of topological operations in algebra” (Mac Lane 1939, p. 18), of which valuations provide a relevant example.12 The “structural” approach—Mac Lane seems to hint at—does not appear to be the most appropriate to elucidate mathematical concepts that are not purely algebraic.

12 In Mac Lane’s terminology (which later will be Bourbaki’s one), a valuation on a field K is a function V : K \ {0} → R+ such that V (x y) = V (x) + V (y) and V (x + y) ≥ min(V (x), V (y)). Notice that, if V is valuation on K, then the function e−V defines a non-Archimedean (or ultrametric) absolute value on K. On the topic of valuation theory Mac Lane had published the paper (Mac Lane 1938).

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2 Bourbaki’s Early Attempts to Formalise the Notion of Structure (1936–1939) As Bourbaki is not an individual, but a group of mathematicians, whose composition has been changing in time, it is even too obvious that there can be no such things as “Bourbaki’s idea of algebra”, “Bourbaki’s structuralist approach to mathematics”, “Bourbaki’s opinion on functors” and the like. The only meaningful way to say something about Bourbaki is to examine and compare the printed texts bearing his name as author, and to make, in parallel, an extremely cautious and judicious use of other available primary sources (manuscripts, typed or mimeographed documents, etc.) related to the activity of the Bourbaki group.13 By adopting this method, I shall try, in this and the following sections, to illustrate the development of the concept of “structure” in “Bourbaki’s work” on a time span of approximately twenty years (1936–1957). In September 1936, in the course of the congress held at Chançay and known as “Congrès de l’Escorial,”14 Bourbaki approved a “projet de laïus scurrile,”15 intended as a first draft of a preamble to the planned book on set theory: L’objet d’une théorie mathématique est une structure organisant un ensemble d’éléments: les mots “structure”, “ensemble”, “éléments” n’étant pas susceptibles de définition, mais constituant des notions premières communes à tous les mathématiciens. Ils s’éclaireront d’eux-mêmes dès qu’on aura eu l’occasion de définir des structures, comme il va être fait dès ce chapitre même. Grâce à une structure, on a le droit de dire que des éléments ou des parties de l’ensemble considéré dans une théorie ont entre eux certaines relations ou possèdent certaines propriétés: les mots “parties”, “relations”, “propriétés” ne sont pas susceptibles de définition non plus, et constituent également des notions premières. (Archives Bourbaki 1936a)

Yet, despite such a firm statement, a definition of “structure” is given in the same document, just two pages after the “laïus scurrile.”16 First, very sketchily, the essential notions of set theory are outlined. In particular, the following basic facts are stated: a set equipped only with the “structure” [sic] ∈ is called an ensemble fondamental; 13 A

great part of this material is accessible online: http://sites.mathdoc.fr/archives-bourbaki/. his Souvenirs d’apprentissage, Weil explains that the members of Bourbaki had planned to convene in the Spanish town of San Lorenzo de El Escorial, but the civil war prevented them from holding their congress there: “au dernier moment la mère de Chevalley nous offrit l’hospitalité de sa belle propriété de Chançay en Touraine, non loin de Vouvray” (Weil 1991, p. 118). 15 As for the word “scurrile”, in the short and quite informal report of Bourbaki’s first congress at Besse (only later, in 1961, named Besse-en-Chandesse, and now Besse-et-Saint-Anastaise) it was specified that “les termes scurrile et futile, ainsi que leurs superlatifs, dont la nécessité est mise hors de doute par les remarques de Cartan, sont adjoints au vocabulaire mathématique et seront de la compétence de la commission dite ‘Lautmann’ [sic]” (Archives Bourbaki 1935). It should be reminded that André Weil strongly disliked Albert Lautman’s approach to philosophy of mathematics; see Weil’s letter to his sister Simone, March 26, 1940, in (Weil 2012) (notice that, several years later, Weil decided to expunge all negative references to Lautman from the partial reproduction of this letter he included in (Weil 1979)). 16 Quite incongruously, it is claimed that this definition is given “à la suite du laïus scurrile” (Archives Bourbaki 1936a). 14 In

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the power set of any fundamental set is a fundamental set, and so is the product of two fundamental sets. Then, the notion of “structure” is defined as follows: On dit qu’on a défini une structure dans un ensemble fondamental quand on s’est donné des propriétés des (ou relations entre les) éléments de cet ensemble, ou de l’un de ceux qu’on peut l’en déduire par une combinaison des opérations ci-dessus, et éventuellement d’ensembles fondamentaux auxiliaires préalablement donnés. Plus on s’est donné de propriétés, plus la structure est forte (example: la structure la plus faible de toutes est la structure ∈). (Archives Bourbaki 1936a)

Immediately after this quite unsatisfactory definition, the crucial notion of transport de structure is introduced: quand deux ensembles sont en correspondance biunivoque et que l’un possède une structure, celle-ci peut être transportée à l’autre: isomorphie. (Archives Bourbaki 1936a)

The attempt to interpret the concept of isomorphism of structures exclusively in terms of transport of structures, however, was far from being as straightforward as at least some members of Bourbaki seem to have thought: in fact, as we shall see, it opens up a Pandora’s box of difficulties. Also the philosophical remark on the notion itself of isomorphism that was endorsed by Bourbaki at the “Congrès de l’Escorial” appear to be somewhat hasty, if not naïve: Si on s’en tient au point de vue dit ontologique, on fait aussitôt après l’isomorphie un laïus général, pour dire que dans la théorie qui étudie un type de structures, deux ensembles isomorphes au point de vue de cette structure peuvent être considérés comme identiques et recevoir le même nom, quelles que soient d’ailleurs leurs origines respectives (p. ex. en topologie il est d’usage d’appeler sphère toute variété simplement connexe, etc.) (Archives Bourbaki 1936a)

This view—which is perhaps reminiscent of Poincaré’s famous definition of mathematics as “l’art de donner le même nom à des choses différentes” (Poincaré 1908, p. 29)—can be acceptable (or better, in accordance with the usual practice) in topology, but certainly not in other domains of mathematics, and in particular in algebra. For example, although a vector space V and its dual V ∗ are always isomorphic as vector spaces, there is no way “to regard them as identical,” because every isomorphism between V and V ∗ depends on the choice of a basis: so, there is no reason to give them the same name. In constrast, if V is of finite dimension, one shows that there is a canonical or natural isomorphism between V and its double dual V ∗∗ , that is an isomorphism independent of the choice of a basis: V and V ∗∗ , therefore, may be given the same name. A more historically significant example one may consider is that of an abelian group G and its character group Ch(G), i.e. the group of group homomorphisms χ : G → C× , where C× = (C \ {0}, ·) is the multiplicative group of the field of complex numbers. If G is a finite group, then it can be proved that Ch(G) is isomorphic (as a group) to G, but there exists no canonical isomorphism between G and Ch(G). The evaluation map ev : G → Ch(Ch(G)) is instead a natural isomorphism17 : actually, as we shall see in the next section, a careful examination of character χ ∈ Ch(G) is a group homomorphism χ : G → C× , so χ(gh) = χ(g)χ(h) for all g, h ∈ G. The group structure of Ch(G) is given by pointwise multiplication, that is

17 A

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this example led Eilenberg and Mac Lane, in their 1942 pioneering paper (Eilenberg & Mac Lane 1942a), to introduce the notions of functor and of natural transformation. The notion of structure outlined at the “Congrès de l’Escorial” by the Bourbaki group found an almost immediate realisation in André Weil’s memoir Sur les espaces à structure uniforme et sur la topologie générale (Weil 1937), published in 1937, where he provided an axiomatic definition of topological space in terms of a family of subsets (the open sets) of the underlying set,18 as well as the equivalent characterisation in terms of neighbourhoods (voisinages). The author used the symbols ∩, ∪ for the set-theoretic operations of intersection and union, respectively, and emphasised that Ces notations, de même que toutes notations et définitions de ce mémoire sont conformes à l’usage de N. Bourbaki et des ses collaborateurs.19 (Weil 1937, p. 6, fn (1))

In September 1937 the Bourbaki group convened, once again, at Chançay. They established the now usual set of axioms defining a topological structure,20 decided to adopt the symbol ∅ for the empty set,21 and set up the fundamentals of the theory of filters, jocularly dubbed boumologie (Archives Bourbaki 1937c). The notion of filter (filtre)—an original idea of Henri Cartan22 —allowed to develop a theory of convergence more general than, and independent of, the usual one based on the notion of (countable) sequence. In this way, the countability assumption was banned from (χχ  )(g) = χ(g)χ  (g) for all χ, χ  ∈ Ch(G) and for all g ∈ G. For every g ∈ G the evaluation ev(g) : Ch(G) → C× given by χ → χ(g) is a group homomorphism, hence a character of the group Ch(G). It is straightforward to check that ev : G → Ch(Ch(G)) is a group homomorphism; when G is a finite abelian group (or, more generally, a topological locally compact abelian group), this map is proved to be an isomorphism. 18 Besides the standard properties of the family of open sets defining a topology on a given set, Weil makes the further assumption that the points are closed (equivalent, as well known, to the fact that the topological space is T1 ). 19 Bourbaki had already made his first academic appearance as the author of the note (Bourbaki 1935). 20 The two axioms defining the family of open sets of a topological were the same as those already given by Weil in (Weil 1937), whereas the closeness condition for points was abandoned. 21 According to Weil, he was personally responsible for the adoption of this symbol: “[l]e Ø appartenait à l’alphabet norvégien, et j’étais seul dans Bourbaki à le connaître” (Weil 1991, p. 119). 22 Since the notion of filter is now somewhat out of fashion (though still of some use, especially in functional analysis), we remind its definition (Bourbaki 1987, Ch. 1, Sect. 6.1): A filter on a set X is a set F of subsets of X which has the following properties: (FI ) Every subset of X which contains a set of F belongs to F ; (FII ) Every finite intersection of sets of F belongs to F ; (FIII ) The empty set is not in F . An example of filter is given by the set of all neighbourhoods of a point in a topological space. A filter F converges to a point x in a topological space X if and only if the neighbourhood system of x is a subset of F in the power set of X . An ultrafilter on X is a filter that is properly contained in no filter on X . One can prove that a topological space is compact if and only if every ultrafilter is convergent. Cartan expounded his theory of filters and ultrafilters in two short notes published in 1937 (Cartan 1937a, b). For the story of the invention of the notion of filter at the Chançay congress (including an explanation of why the words boum and boumologie were at first used) see (Audin 2012, pp. 8–11).

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the principles of topology, as Weil had expressed, quite vehemently, his wish to do in the memoir quoted above, making reference to the second countability axiom: on voit apparaître ici cette hypothèse du dénombrable […], malfaisant parasite qui infeste tant de livres et de mémoires dont il affaiblit la portée tout en nuisant à une claire compréhension des phénomènes. […] Naturellement, lorsqu’on quitte le dénombrable, il n’est plus légitime de faire des notions de suite et de limite l’outil essentiel, et on doit les remplacer par d’autres dont le champs d’action soit moins restreint. (Weil 1937, p. 3)

Thank to the concept of filter, this “evil parasite” seemed to be definitively wiped out.23 The notion of topological space provided an instance of “structure” which realises the spirit of the preliminary plan approved by Bourbaki at the “Congrès de l’Escorial”24 better than the notion, say, of group or ring. Indeed, the axioms defining a topological structure on a set X are not imposed on the elements of X , but on a family elements of the power set P(X ) of X ,25 hence they are conditions imposed on an element in P(P(X )). In a certain sense, it could be argued that the concept of “structure” more or less collectively elaborated by the members of Bourbaki in the early years of their activity was more akin to Ore’s viewpoint than to Mac Lane’s one. There was, however, a crucial difference, not just of a technical nature: whilst Ore’s approach was limited to algebra, the Bourbaki group had the much wider ambition that the notion of structure could encompass the whole of mathematics. According to the “engagements” agreed on by Bourbaki in the congress held at Dieulefit in 1938, Dieudonné had to complete the writing of the “nearly definitive”

23 In his Collected Papers, publishe in 1979, Weil briefly commented upon the strong dislike he had expressed for the dénombrable 40 years earlier: “Avec le recul que donnent les quarante dernières années, on sourira sans doute du zèle que j’apportais alors à l’expulsion du dénombrable: chassé par la porte, il a fini pour rentrer par la fenêtre, avec les espaces paracompacts, les espaces polonais, etc. Cette occasion ne fut pas la seule où j’aie pris (le plus souvent par réaction contre une orthodoxie dominante) une position qui par la suite s’est révélée trop dogmatique” (Weil 1979, p. 536). 24 This plan was modified, though not substantially, at the meeting which took place in Nancy on March 15, 1937; see (Archives Bourbaki 1937b). Those present (Mandelbrojt, Chevalley, de Possel, Dieudonné, Ehresmann, Delsarte) discussed Delsarte’s rédaction des ensembles (Archives Bourbaki 1937a). According to Delsarte, a mathematical theory results from the combination of two distinct components: “D’une part, les ensembles fondamentaux qui sont l’object de la théorie, d’autre part, la structure qui forme le sujet de la théorie et qui en est la partie vivante et essentielle. […] Les structures possibles sont d’une infinie diversité, mais toutes sont des perfectionnements, dans des directions extrêment variées, de deux structures primitives qui, à vrai dire, constituent le fond commun de toutes les cogitations humaines” (Archives Bourbaki 1937a). These “primitive structures” were the structure ∈ and the structure “contenu dans.”. 25 The Bourbakist conception of topological space was, to a certain extent, anticipated by ideas put forward by Maurice Fréchet a decade earlier: an abstract topological space “n’est pas seulement un ensemble d’éléments auxquels on aura donné le nom de points; c’est un ‘système’ (P, K ), P désignant un certain ensemble d’éléments abstraits et K désignant l’opération de dérivation des ensembles d’éléments de P” (Fréchet 1928, p. 167).

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version of the résultats d’ensembles by December 1st of that year (Archives Bourbaki 1938).26 It was parenthetically specified that the fascicule should have been “avec figures and sans logique,” for Dieudonné had already produced an introduction to set theory (Archives Bourbaki 1937–1938), which was actually a rather informal (almost colloquial) exposition of the principles of mathematics, illustrating, among others, the logical rules of mathematical reasoning and including several remarks on the truth of mathematical theories and on the nature of mathematical objects. It is to believe that Dieudonné met his commitment, since a handwritten annotation indicates his name as the author of an undated typescript (Archives Bourbaki 1938– 1939) that matches very closely, except for a few slight changes, the text of the Fascicule de résultats (Bourbaki 1939) of the book Théorie des ensembles published by Hermann in 1939, as the first issue of Éléments of mathématique.27 In particular, with regard to “structures”, the corresponding sections in the two texts are essentially identical. To survey this matter, therefore, it will be enough to refer only to (Bourbaki 1939). Dieudonné considers, as an example, three distinct sets E, F, G and observes that one can build 12 new sets by taking their powers sets and by taking all their (ordered) pairwise products. On these 3 + 12 = 15 sets one may perform the same operations as above, discarding the possibly repeated sets, and so on. Any set obtained through this procedure is said to be part of the “échelle des ensembles ayant pour base E, F, G” (Bourbaki 1939, p. 41). Then, he gives the rather convolute definition of the notion of “kind of structure”: D’une façon générale, considérons un ensemble M d’une échelle, dont la base est formée, par exemple, de trois ensembles E, F, G; donnons-nous un certain nombre de propriétés explicitement énoncées d’un élément générique de M, et soit T l’intersection des parties de M définies par ces propriétés; on dit qu’un élément σ de T définit sur E, F, G une structure de l’espèce T ; les structures d’espèce T sont donc caractérisées par le schéma de formation de M à partir de E, F, G, et par les propriétés définissant T , qu’on appelle les axiomes de ces structures; on donne un nom spécifique à toutes les structure de même espèce. (Bourbaki 1939, p. 42)

To better understand this definition, one may take the example of a set E equipped with a partial order relation ω{x, y}. The usual axioms for partial order are equivalent to the assignment of an element C = {(x, y) ∈ E × E | ω{x, y}} ∈ P(E × E) satisfying the conditions a) C ◦ C ⊂ C ; 26 As

−1

b) C ∩ C =  ,

it was humorously remarked at the beginning of the engagements de Dieulefit, “[i]l est entendu que Dieudonné fera toutes les rédactions presque-définitives, et se retirera quand on les aura repoussées” (Archives Bourbaki 1938). On that same occasion, the first two chapters of the mégarédaction of set theory were assigned to Weil. 27 Besides stylistic changes and other minor modifications, the main difference between the two texts consists in the creation of Sect. 6.—Ensembles ordonnés of (Bourbaki 1939), which supplies the basic definitions and incorporates some parts already included in Sect. 4 of (Archives Bourbaki 1938–1939).

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where  is the diagonal in E × E, C is the image of C under the map E × E → E × E switching x and y, and ◦ denotes composition of relations.28 So, the échelle for “order structures” has one base set E and two rungs (regarding E as ground floor). In an analogous manner, one may characterise the “structures” of group, ring, topological space, uniform space (Bourbaki 1939, p. 43). It is true, nonetheless, that the books of Éléments de mathématique devoted to topology and algebra do not formalise these notions in terms of the concept of échelle d’ensembles. So, in the first chapter of Topologie générale (Bourbaki 1940), published only one year after (Bourbaki 1939), a “topological structure” is defined as an “ensemble O de parties d’un ensemble E” whose elements (i.e. “les ensembles de O”) satisfy the usual axioms, and not by explicitly stating the properties of the set O ∈ P(P(E)) itself (as it is done for “order structures”). Likewise, even the definition of a very basic “algebraic structure” such as that of associative magma on a set E is not given by specifying a set in P(E × E × E) and its properties, but through the more intuitive notion of composition law (Bourbaki 1942b). As trenchantly remarked by Mac Lane, Bourbaki […] in one of his very first volumes (Bourbaki 1939) gave a cumbersome definition of a mathematical structure in terms of what he called an “échelle d’ensembles”; though he did not say so, this is close to the notion of a type theory in the sense of Bertrand Russell, and has the same cumbersome characteristics. (Mac Lane 1988, p. 337)

Almost en passant, and in a completely informal way, Dieudonné introduces the troublesome notion of “auxiliary set.” In most cases—he remarks— quand on utilise une échelle ayant une base composée de plusieurs ensembles E, F, G, l’un de ces ensembles, E par exemple, joue dans les structures qu’on considère un rôle prépondérant; aussi dit-on, par abus de langage, que ces structures sont définies sur l’ensemble E, les ensembles F et G étant considérés comme ensembles auxiliaries. (Bourbaki 1939, p. 43)

This is the case, for example, of a “structure” of R-module on a set E, where the set underlying the commutative ring plays the role of auxiliary set. In general, as we shall see, the problem of formalising “structures” involving auxiliary sets in terms of échelles is a more difficult one than Dieudonné seems to have imagined in 1939. A key concept defined in (Bourbaki 1939) is that of “transport of structures.” Let E, F, G, E  , F  , G  be sets and assume we are given bijections (applications biunivoques) E → E  , F → F  , G → G  . Since one can extend bijections to power sets and to products, it follows that the given bijections can be extended, step by step, to the sets M, M  , built according to the same scheme (schéma)29 in the échelles of E, F, G be any (possibly non distinct) three sets, A ∈ P (E × F), B ∈ P (F × G). Then, the composition B ◦ A is the element

28 Let

B ◦ A = {(x, z) ∈ E × G | ∃y ∈ F s.t. (x, y) ∈ A ∧ (y, z) ∈ B} ∈ P (E × G) . The condition C ◦ C ⊂ C is therefore equivalent to the familiar transitivity condition for ω: ω{x, y} and ω{y, z} implies ω{x, z}. 29 Dieudonné did not supply any definition of “schéma” in (Bourbaki 1939). As we shall see below, a definition of “schéma de construction d’échelon” is given in (Bourbaki 1957).

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sets whose bases are, respectively, E, F, G and E  , F  , G  . Let f : M → M  the bijection obtained in this way: Si σ est une structure sur E, F, G, élement d’une partie T de M, on dira que f (σ ) est la structure obtenue en transportant la structure σ sur E  , F  , G  , au moyen des applications biunivoques de E sur E  , de F sur F  et de G sur G  . (Bourbaki 1939, p. 44)

Proceeding in accordance with the “projet de laïus scurrile” approved at the “Congrès de l’Escorial”, Dieudonné defines the notion of isomorphism by means of that of transport of structures: […] étant données une structure σ sur E, F, G, et une structure σ  sur E  , F  , G  , on dira qu’elles sont isomorphes (ou qu’il y a isomorphie entre ces structures) si σ  peut être obtenue en transportant σ par des applications biunivoques de E sur E  , de F sur F  et de G sur G  respectivement; ces applications sont alors dites constituer un isomorphisme de σ sur σ  . (Bourbaki 1939, p. 44)

As I have already noted, such a definition of isomorphism does not allow to recognise natural isomorphisms among all others. Dieudonné’s idea that, when there is an isomorphism f from a set E equipped with a structure σ onto a set E  equipped with a structure σ  , il est souvent commode […] d’identifier E et E  , c’est de donner le même nom à un élément d’un ensemble M de l’échelle de base E, et à l’élément qui en est l’image par l’extension convenable de f à l’ensemble M (Bourbaki 1939, p. 44)

is a repetition of the “ontological” remark included in the Décisions Escoriales and quoted above. What is really disconcerting is that Dieudonné’s Fascicule de résultats de la théorie des ensembles does not contain any definition of the notion of “morphism” between sets equipped with structures of the same kind. One finds, of course, a definition of continuous maps between topological spaces in the first chapter of Topologie générale (Bourbaki 1940), published only one later, and likewise one finds definitions of several kinds of algebraic homomorphisms in the first chapter of Algèbre (Bourbaki 1942b) (1942). Nonetheless, both homeomorphisms and algebraic isomorphisms are characterised in the usual way, and not as extensions of bijective maps to an échelle d’ensembles. The Fascicule was undoubtedly a long-pondered result of numerous, often animated, discussions among the members of Bourbaki. Yet, the convoluted notion of “structure” defined in terms of échelles d’ensembles appears quite unsatisfactory under many respects, and, moreover, it would have proved to be of absolutely no use in the subsequent parts of Éléments de mathématique, with the only exception of Chap. 4 of Théorie des ensembles (Bourbaki 1957), published 18 years later.30

30 The

notion of échelle d’ensembles was renamed échelon de schéma S sur les ensembles E1 , . . . , En .

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3 Natural Transformations and Categories (1942–1945) The notion of functor was introduced by Eilenberg and Mac Lane in their short report Natural isomorphisms in group theory (Eilenberg & Mac Lane 1942a), which appeared in 1942.31 As the title suggests, their main purpose was to give a precise meaning to the word “natural” in such expressions as “natural” isomorphism between two groups or between two complexes, or “natural” homeomorphism between two topological spaces, and the like. A covariant functor T is defined as a mapping (a “function” in their terminology) which determines for each group G a group T (G) and for each group homomorphism γ : G 1 → G 2 a group homomorphism T (γ ) : T (G 1 ) → T (G 2 ) in such a way that T (id G ) = id T (G) and, whenever the composition γ1 γ2 is defined, T (γ1 γ2 ) = T (γ1 )T (γ2 ). Analogously, a contravariant functor U maps each group G to a group U (G) and each group homomorphism γ : G 1 → G 2 to a group homomorphism U (γ ) : U (G 2 ) → U (G 1 ) in such a way that U (id G ) = idU (G) and U (γ1 γ2 ) = U (γ2 )U (γ1 ). Some basic examples they had in mind were, for abelian groups G and H , the group Hom(G, H ), which determines a functor covariant in H and contravariant in G,32 and the tensor product, introduced by Whitney in 1938 (Whitney 1938), G  H , which determines a functor covariant in both arguments. The key concept defined by Eilenberg and Mac Lane in (Eilenberg & Mac Lane 1942a) was that of natural transformation (that they also named a natural equivalence). Given two covariant functors S and T as above, a natural transformation τ from S to T is the assignment, for every group G, of a group homomorphism τ (G) : S(G) → T (G) in such a way that, for every group homomorphism γ : G → H , one has T (γ ) ◦ τ (G) = τ (H ) ◦ S(γ ); equivalently, the following diagram is commutative: S(G)

τ (G)

S(γ )

S(H )

T (G) T (γ )

τ (H )

T (H )

An analogous definition is given for contravariant functors. Now, if Ch(G) is the character group of G, the mapping G → Ch(G) is a contravariant functor, and, by composition of functors, the mapping G → Ch(Ch(G)) is a covariant functor. Eilenberg and Mac Lane remarked that, if G is a locally compact abelian group, there is a natural transformation τ from the identity functor I (i.e. I (G) = G and I (γ ) = γ ) to the functor Ch ◦ Ch; in other words, for every two locally compact abelian groups G, H and for every continuous group homomorphism γ the diagram

31 For

an overview of the outset of category theory the reader is referred to (Krömer 2007) and (Marquis 2009). 32 To be precise, Eilenberg and Mac Lane considered the group of continuous homomorphisms between two locally compact abelian groups.

72

C. Bartocci τ (G)

G

Ch ◦ Ch(G)

γ

Ch ◦ Ch(γ ) τ (H )

H

Ch ◦ Ch(H )

is commutative (cf. Eilenberg & Mac Lane 1942a, p. 541). It is in this precise sense that the isomorphism G  Ch(Ch(G)) is “natural.” On the contrary—they explained very neatly— When G is finite, the isomorphism G → Ch(G) cannot be “natural” according to our definitions, for the simple reason that the functor I on the left is covariant, while the functor Ch on the right is contravariant. (Eilenberg & Mac Lane 1942a, p. 542)

The notion of natural transformation was largely employed by Eilenberg and Mac Lane in their technically impressive paper Group extensions and homology (Eilenberg & Mac Lane 1942b), published in 1942 (and submitted one year earlier).33 Group extensions had been previously described, in 1934, by Reinhold Baer in terms of “factor sets” (Baer 1934). Let us assume, for simplicity’s sake, that G, H are abelian groups. Then, Ext(G, H ) denotes the group of isomorphism classes of extensions of G by H , where: i) an extension (E, α, β) is given by an abelian group E, an injective group homomorphism α : G → E, and a surjective group homomorphism β : E/α(G) → H ; ii) two extensions (E 1 , α1 , β1 ), (E 2 , α2 , β2 ) are isomorphic if there exists a group homomorphism ω : E 1 → E 2 such that α2 = ω ◦ α1 and β1 = β2 ◦ ω (cf. Eilenberg & Mac Lane 1942b, p. 767).34 Eilenberg and Mac Lane were interested in showing that “the theory of group extensions forms a natural and powerful tool in the study of homologies in infinite complexes and topological spaces” (Eilenberg & Mac Lane 1942b, p. 759). Their main result, under simplifying hypotheses and in slightly modernised notation, can be summarised as follows. Let K • be a complex ···

di+1

di

K i+1

Ki

di−1

K i−1

di−2

···

d0

K0

0

of free abelian groups; we remind that this means that di−1 ◦ di = 0 for all i > 0, or equivalently that Im di ⊂ Ker di−1 . The i-th homology group of the complex K • 33 The 34 In

word functor does not appear in this paper. other words, in today’s mathematical parlance, an extension E is a short exact sequence α

G E 0 isomorphic if the diagram

β

0

0 of abelian groups and two extensions E 1 , E 2 are

H

G

α1

E1

β1

H

0

H

0

ω

0 commutes.

G

α2

E1

β2

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is defined as the quotient Hi (K • ) = Im di / Ker di−1 . If G is an abelian group, we let K Gi be the group Hom(K i , G) and define the group homomorphism δi : K Gi → K Gi+1 by setting, for each homomorphism μ ∈ Hom(K i , G), δi (μ) = μ ◦ di . Since δi ◦ δi+1 = 0, we get a complex K G• 0

K G0

δ0

···

δi−2

K Gi−1

δi−1

K Gi

δi

K Gi+1

δi+1

···

whose i-th cohomology group is H i (K G• ) = Im δi−1 / Ker δi .35 It is not hard to show that there are natural homomorphisms i : H i (K G• ) → Hom(Hi (K • ), G): Eilenberg and Mac Lane proved that Ker i is naturally isomorphic to Ext(Hi−1 (K • , G)) (cf. Eilenberg & Mac Lane 1942b, Theorem 31.3). Quite manifestly, a piece of information was lacking in the definition of functor: which is precisely the “domain” of a functor if it is regarded—as Eilenberg and Mac Lane did in their note (Eilenberg & Mac Lane 1942a)—as a “function” whose arguments may be not just mathematical objects, but also mappings between those objects? Eilenberg and Mac Lane wanted to be able to deal with vector spaces and their linear transformations, as well as with groups and their homomorphisms, with topological spaces and their continuous mappings, with simplicial complexes and their simplicial transformations, with ordered sets and their order preserving transformations. (Eilenberg & Mac Lane 1945, p. 234)

“In order to deal in a general way with such situations”, in the groundbreaking paper General theory of natural equivalences, they introduced the notion of category.36 According to the original definition, a category A consisted of “an aggregate of abstract elements A”, called objects of the category, and “abstract elements α,” called mappings (currently, arrows, or morphisms) of the category. A mapping e is an identity of A if and only if the existence of any product eα or βe implies that eα = α and βe = β. “Certain pairs of mappings α1 , α2 ∈ A determine uniquely a product α2 α1 ∈ A”, subject to the following three axioms: (1) The triple product α3 (α2 α1 ) is defined if and only if (α3 α2 )α1 is defined. When either is defined, the associative law α3 (α2 α1 ) = (α3 α2 )α1 holds. This triple product will be written α3 α2 α1 . (2) The triple product α3 α2 α1 is defined whenever both products α3 α2 and α2 α1 are defined. K • to the complex K G• all arrows are reversed; this happens because we have applied the functor Hom(–, G), which is contravariant. 36 Eilenberg and Mac Lane emphasised that “that the whole concept of a category is essentially an auxiliary one; our basic concepts are essentially those of a functor and of a natural transformation […]. The idea of a category is required only by the precept that every function should have a definite class as domain and a definite class as range, for the categories are provided as the domains and ranges of functors.” (Eilenberg & Mac Lane 1945, p. 247) By adopting this “intuitive” viewpoint, they sought to steer clear of foundational antinomies (“leaving the reader free to insert whatever type of logical foundation (or absence thereof) he may prefer.” (Eilenberg & Mac Lane 1945, p. 246)). 35 Note that, in passing from the complex

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(3) For each mapping α ∈ A there is at least one identity e1 such that αe1 is defined, and and at least one identity e2 such that e2 α is defined. Moreover, (4) For each object A there is a corresponding unique mapping e A , which is an identity. (5) For each identity e ∈ A there is a unique object A ∈ A such that e A = e.37 (Eilenberg & Mac Lane 1945, pp. 237–238) It is worth pointing out that Eilenberg and Mac Lane, adopting what we may call a historical stance, regarded their novel approach […] as a continuation of the Klein Erlanger Programm, in the sense that a geometrical space with its group of transformations is generalized to a category with its algebra of mappings.38 (Eilenberg & Mac Lane 1945, p. 237)

In Chap. 5 of (Eilenberg & Mac Lane 1945), titled Applications to topology, Eilenberg and Mac Lane considered homology and cohomology functors on various categories ˇ homology of complexes,39 and they presented an original “treatment of the Cech theory in terms of functors” by using the theory of direct and inverse limit they had developed in the preceding Chap. 4. A purely axiomatic definition of homology theories of a topological space was provided by Eilenberg and Steenrod in the short paper (Eilenberg & Steenrod 1945), which too was published in 1945. In this case, the relevant category is that whose objects are pairs (X, A), where X is a topological space and A a subset of X ,40 and whose morphisms f : (X, A) → (Y, B) are continuous maps f : X → Y such that f (A) ⊂ B. A homology theory is defined as a sequence of covariant functors Hn (n ≥ 0) from this category to the category of abelian groups, along with natural natural transformations (called boundaries operators) ∂n−1 : Hn (X, A) → Hn−1 (A, ∅) = Hn−1 (A), satisfying the following set of axioms: (1) the sequence ···

∂n

Hn (A)

Hn (i)

Hn (X )

Hn ( j)

Hn (X, A)

∂n−1

Hn−1 (A)

···

is exact, where i : A = (A, ∅) → X = (X, ∅) and j : (X, ∅) → (X, A) are the canonical inclusion maps; 37 The now usual definition of category is slightly different, though equivalent: one assigns to each arrow α two objects, dom α and cod α (its domain and its codomain, resp.), and it is required that the product of two arrows α, β exists if and only if cod α = dom β. 38 Eilenberg and Mac Lane’s claim is thoroughly discussed in (Marquis 2009, Chap. 1). 39 They made use of the notion of chain complex originally introduced by Walther Mayer in (Mayer 1929); their fundamental reference on algebraic topology was Solomon Lefschetz’s treatise (Lefschetz 1942). In 1944 Eilenberg had provided a “precise and systematic treatment of the singular homology theory” in his masterly paper (Eilenberg 1944). 40 In the original formulation, the subset A is required to be closed (Eilenberg & Steenrod 1945, p. 117).

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(2) if f, g : (X, A) → (Y, B) are homotopic as maps of pairs, then Hn ( f ) = Hn (g) for all n; (3) if U is an open set of X whose closure U is contained in the interior of A (i.e. U ⊂ V ⊂ A for some open set V of X ), then the inclusion i : (X \ U, A \ U ) → (X, A) induces isomorphisms Hn (i) : Hn (X \ U, A \ U ) → Hn (X, A) for all n; (4) If P is the space consisting of a single point, then Hn (P) = 0 for all n ≥ 1.41 The group G = H0 (P) is called the coefficient group of the homology theory. The dependence of the homology theory on its coefficient group is functorial, in the sense that, if {( H˜ n , ∂˜n )}n≥0 is another homology theory such that H˜ 0 (P) = G˜ and if ϕ : G → G˜ is a group homomorphism, then there are induced natural transformations ϕ n : Hn → H˜ n for all n ≥ 0, such that ∂˜n−1 ◦ ϕ n = ϕ n−1 ◦ ∂n . In particular, this implies that the coefficient group G uniquely determines the homology theory {(Hn , ∂n )}n≥0 up to isomorphism.

4 Structural Difficulties (1940–1948) Bourbaki’s work was slowed down, but not interrupted by World War II, as attested by the ten issues of “La Tribu (Bulletin œcuménique, apériodique et bourbachique)” more or less regularly released from March 1940 through April 1944.42 During that period, Weil—who was to narrate, with affected detachment, his misadventures during the War in his Souvenirs d’apprentissage—was able to attend only the “Congrès croupion” (“Rump congress”) held in Clermont in December 1940, the other members present being Delsarte, De Possel, Dieudonné, Ehresmann, and, with the status of “chrysalis” (“comme chrysalide”), Laurent Schwartz. But four more Bourbaki meetings were convened before 1945: twice again in Clermont in April 1941 and in August 1942, at Liffré (Brittany) in September 1943, and in Paris in April 1944. After the end of the war the activity of the Bourbaki group resumed quite intensively: at least ten congresses took place between 1945 and 1948,43 and a Séminaire Bourbaki was organised in the years 1945–1946 and 1948–1949. During the same period 41 I have listed, in a slighlty modified form, the four last axioms stated by Eilenberg and Steenrod in (Eilenberg & Steenrod 1945), where the words “functor” and “natural transformation” are not used. Actually, Eilenberg and Steenrod’s first two axioms amount to say that Hn are functors, while their axiom 3 is equivalent to the fact that the boundary operators are natural transformations. The same authors provided a much more extensive and detailed exposition of their theory in the classical book Foundations of Algebraic Topology (Eilenberg & Steenrod 1952). 42 The issues of “La Tribu” can be accessed online: http://sites.mathdoc.fr/archives-bourbaki/ feuilleter.php. I shall follow the numeration provided in the table of contents (Archives Bourbaki n.d.); in some cases this numbering does not match the numbering indicated on the digitalised copy. Some issues of “La Tribu” are unnumbered. 43 Congrès de Paris, June 22–July 4, 1945; Congrès de Nancy, February 9–11, 1946; Congrès de Strasbourg, June 8–11, 1946; Congrès de Paris, January 18–20, 1947; Congrès de Paris, March 15–18, 1947; Congrès de Paris, November 8–11, 1947; Congrès de Noël, Paris, December 19–26, 1947; Congrès de Nancy, April 9–13, 1948; Congrès œcuménique de juin 1948, Paris, June 1–8 and Strasbourg, June 15–20, 1948; Congrès du fil directeur, Nancy, October 27–30, 1948.

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of time, in addition to the parts of Éléments de mathématique already published (viz., Fascicule de résultats of set theory (Bourbaki 1939), Chaps. 1–4 of Topologie générale (Bourbaki 1940, 1942a) and Chap. 1 of Algèbre (Bourbaki 1942b), four more parts went to print: Chaps. 5–8 and 9 of Topologie générale (Bourbaki 1947a, 1948b) and Chaps. 2 and 3 of Algèbre (Bourbaki 1947b, 1948a). Several issues discussed by the Bourbaki group in those years were more or less closely related to the problem of characterising natural isomorphisms within the conceptual framework of “structures” defined in terms of échelles d’ensembles. For example, tensor products were dealt with at the second Clermont congress (1942) and then at the Liffré congress (1943). At Liffré it was decided to define the product of two modules F, G according to Whitney’s method, as improved and adapted by Cartan, that is by means of a propriété caractéristique (what we now would call a “universal property”) ensuring the uniqueness of F ⊗ G only up isomorphism (“à une isomorphie près”) (Archives Bourbaki 1943, p. 3).44 What is meant here by the word “isomorphism”? Though in principle not impossible, it would be quite odd to use the definition stated in the Fascicule de résultats and regard the isomorphism between two different solutions, say F ⊗ G, F ⊗ G, of the given universal problem as extension of a bijective map between sets to an échelle d’ensembles, which, in this case, involves also an auxiliary set (namely, the ring over which the modules are defined). On the contrary, it appears evident that the existence of an isomorphism of modules between F ⊗ G, and F ⊗ G is a direct consequence of the propriété caractéristique of tensor products. Similar difficulties arise when one tries to interpret in the light of the the Fascicule de résultats the “isomorphie de E ⊗ F et de F ⊗ E” (Archives Bourbaki 1943, p. 3) or, in the case of a vector space V , the“isomorphisme p V )∗ , and the canonique” between the dual of the space  p of∗ p-forms on V , i.e. ( ∗ (V ) (Archives Bourbaki 1943, p. 5). space of p-forms on the dual V , i.e. The Bourbaki group started discussing the subject of algebraic topology at the congress held in Paris in April 1944: Grâce à l’important travail préparatoire accompli depuis 3 mois par Cartan, et à la compétence technique de Charles [Ehresmann], le Congrès a pu se faire une première idée d’ensemble des matières à traiter, et dresser un plan assez détaillé des premiers chapitres à rédiger; la mise en train de cette rédaction est prévue pour le début de la prochaine année scolaire, au plus tard.45 (Archives Bourbaki 1944, p. 1)

Topics to be covered included duality theory for topological groups, abstract complexes, homology groups, dimension theory, the Poincaré group and coverings.46 44 Cartan had previously tried to adopt a different method for defining tensor products of vector spaces, starting from the dual of the vector space of bilinear forms; see Décisions du 3e` me Congrès de Clermont (Archives Bourbaki 1942, p. 2). 45 The prediction was just wishful thinking: the first four chapters of Bourbaki’s Topologie algébrique (Bourbaki 2016) have been first published only in 2016. 46 The plan was as follows : “Chapitre I. Dualité dans les groupes topologiques. Chapitre II. Complexes euclidiens et complexes abstraits. / Chapitre III. Groupes d’homologie. / Chapitre IV. Classes d’applications continues et prolongements d’applications continues. Théorie de la dimension (1 ou 2 chap.). / Chapitre V. Variétés. Chapitre VI. Groupe de Poincaré. Revêtements (1 ou 2 chapitres).” (Archives Bourbaki 1944, p. 2).

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Talks on papers by Eilenberg, Steenrod, J. H. C. Whitehead and Ralph Fox dealing with some of these topics were subsequently included in the programme of the 1945– 1946 Séminaire Bourbaki.47 Now, the quintessential features of homology theories are captured by the Eilenberg-Steenrod axioms, which are intrinsically functorial in nature. This machinery appears hardly translatable in the language of “structures” so as shaped in the Fascicule de résultats, where a definition of homomorphism was missing: without this definition, clearly, even the very notion of invariant could not be even formulated in purely “structural” terms. The issue of duality was no less problematic in that framework. Just a couple of years later, in his paper Duality for groups (an address delivered in 1948 and published in 1950), Mac Lane was to state a general “duality principle”: if any statement S about a category is deducible from the axioms for a category, the “dual statement”, i.e. the statement obtained from S by reversing all arrows, is likewise deducible (Mac Lane 1950, p. 498). For example, in the category of groups, the direct product A × B equipped with the two projections p A : A × B → A, p B : A × B → B, can be characterised by means of the following universal property: given any group C and any group homomorphisms α : C → A, β : C → B, there exists one and only one group homomorphism γ : C → A × B making the diagram C α

A

pA

γ

A×B

β

pB

B

commutative. By reversing all arrows, one obtains the dual universal problem which “conceptually describes” the free product A ∗ B equipped with the two canonical injections i a : A → A ∗ B, i B : B → A ∗ B: given any group C and any group homomorphisms α : A → C, β : B → C, there exists one and only one group homomorphism γ : A ∗ B → C making the diagram C α

A

iA

γ

A∗B

β

iB

B

commutative. Mac Lane pointed out that […] the proof that the direct product is unique up to an isomorphism can be phrased so as to be exactly dual to the proof of the uniqueness of the free product up to an isomorphism. (Mac Lane 1950, pp. 490–491)

Such an argument was not even conceivable in the formalism set up by Dieudonné in 1939. There is strong evidence suggesting that at least a few members of Bourbaki were keenly aware of these difficulties already in the second half of 1940s. In 1946, 47 See

(Archives Bourbaki 1945b).

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significantly, it was decided that Eilenberg was to be invited to join the group and charged with the task of the writing of the part on algebraic topology.48 Another relevant example is represented by the Rapport sur les applications universelles (Archives Bourbaki 1947a), a text which is to be dated to 1947.49 In the first few lines of this report we find the following clear-cut statement: Le rédacteur […] pense (ce qui n’est pas nouveau! programme d’Erlangen!) qu’une structure T n’est jamais complètement definie sans le “T -applications”. (Archives Bourbaki 1947a, Commentaire)

(It could be noticed that these words seem to echo those of Eilenberg and Mac Lane in the remark quoted above.) Given two sets E 1 , E 2 endowed with a T -structure, T -mappings (T -applications) are then defined axiomatically: “A1 : Tout T -isomorphism est une T -application. A2 : L’application composée de deux T -applications est une T -application. A3 : Pour qu’une application biunivoque f de E 1 sur E 2 soit un T -isomorphisme, il faut et suffit que f et f −1 soient des T -applications.”50 (Archives Bourbaki 1947a, p. 1) With this definition at hand, the notion of “induced structure” (structure induite) is introduced by analogy with the notion of induced topology: given two T -structures 48 See “La Tribu” n◦ 12, Compte-rendu du Congrès de Strasbourg (9–18 juin 1946): “Il est décidé d’admettre Eilenberg au sein de Bourbaki, s’il accepte et si le group brésilien n’y met pas son veto. Eilenberg sera charge de faire une rédaction de Topologie algébrique.” (Archives Bourbaki 1946, p. 7). The “Brazilian group” included, of course, André Weil, who taught at the Universidade de São Paulo from 1945 to 1947. The invitation letter—quite puzzlingly dated on Juin 21, 1948—is reproduced in (Krömer 2006, p. 128). It contains a proposal somewhat different from the decision taken at the Strasbourg congress (what could explain the considerable time lapse between the two dates): “Nous vous proposons aussi, si cela vous convient, de vous charger, en collaboration avec notre bien-aimé disciple André Weil, d’un rapport concernant les propriétés élementaires de l’homotopie et des espaces filtrés [sic, but it must be emended to “espaces fibrés”].” The report in question was actually produced by Eilenberg and Weil, and should correspond to the text titled Rapport SEAW sur la topologie préhomologique (Archives Bourbaki 1948–1949) (“SEAW” is an obvious acronym). This is confirmed, in the obituary note (Bass et al. 1998) (p. 1345), by Cartan, whose collaboration with Eilenberg had began in 1947. Before Cartan, another member of Bourbaki, Claude Chavalley, had started collaborating with Eilenberg: their first joint paper (Chevalley & Eilenberg 1948) was presented to the American Mathematical Society in 1946 and published in 1948. 49 The unnumbered issue of “La Tribu” reporting on the congress held in Paris from March 15 to 18, 1947 relates what follows: “Lecture est faite de l’Appendice 1 (Applications universelles) du Chap. 3 d’Algèbre. L’auditoire est vivement intéressé, et une partie de l’auditoire est séduite. Il se manifeste néanmoins une assez vive opposition à l’insertion du dudit appendice au Chap. III d’Algèbre. Ne sont pas opposants: Charles [Ehresmann], parce qu’on y parle du groupe libre (!); Cartan, qui aime mieux le voir au Chap. III d’Algèbre que dans le livre de théorie des ensembles, où l’on n’aura aucun exemple à montrer. Les opposants n’ont pas de suggestion précise à faire quant à la place la plus opportune de cet Appendice.” (Archives Bourbaki 1947b, p. 5) A section titled Applications universelles—clearly taking inspiration from (Archives Bourbaki 1947a)—will be included in Chap. 4 of Théorie des ensembles (Bourbaki 1957, Sect. 3). 50 The author of the Rapport says that A , A and A are “les 3 axiomes de Weil.”. 1 2 3

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σ , σ  are on the sets E, E  ⊂ E, respectively, σ  is said to be induced by σ whenever: I1 the injection of E  into E is a T -mapping; I2 if f : F → E is a T -mapping such that f (F) ⊂ E  , then f , regarded as a map from F to E  , is a T -mapping (Archives Bourbaki 1947a, p. 1). It is manifest that the notion of induced structure is more general and more manageable than that of transport of structures defined in the Fascicule de résultats. In addition to mappings between sets endowed with the same kind of structure, the author of Rapport sur les applications universelles defines a class of mappings between S-sets and T -sets, called (S − T )-mappings ((S-T)-applications), as those satisfying the property: any composition f ◦ ϕ of an (S − T )-mapping ϕ with a T mapping f is an (S − T )-mapping. A “universal problem” (in the original, problème (U)) consists in associating, to any S-set E, a T -set F0 and an (S − T )-mapping ϕ0 such that, given any T -set F and any (S − T )-mapping ϕ : E → F, there exists a T -mapping f : F0 → F making the diagram E ϕ

ϕ0

F0 f

F commutative (Archives Bourbaki 1947a, p. 3). It is then shown that, under suitable additional assumptions, a universal problem has a unique solution, which satisfies, moreover, the following condition: if two T -mappings from F0 to F coincide on the subset ϕ0 (E), they are the same. As an application, the completion of a uniform space is obtained as solution of a universal problem (Archives Bourbaki 1947a, pp. 4–5). It is not hard to guess who is the anonymous author of the Rapport sur les applications universelles: he is to be identified with Pierre Samuel, who participated as cobaye in the congress held in Paris from June 22 to July 4, 1945 and was admitted to the Bourbaki group shortly afterward.51 Indeed, Samuel’s 1948 paper On universal mappings and free topological groups (Samuel 1948) has a substantial overlap with the Bourbaki report and discusses the same examples (characterisation of uniformisable spaces, “Alexandroff’s T2 -space”, free topological groups). It seems fair to argue that, since at least as early as 1947, some members of the Bourbaki group (including Cartan, Chevalley, Samuel, and possibly Ehresmann and Weil) were favourably disposed (or not hostile) to embracing ideas that one could broadly define as “functorial”, such as those of natural isomorphism, universal construction, invariant, duality. If it is true that originally, around the mid 1930s, “Bourbaki’s treatise was planned as a tool kit for the working mathematician” (Dieudonné 1982, p. 620), then it has to be recognised, I believe, that about ten years later the basic set-theoretic tools provided by the Fascicule de résultats were a bit 51 The report of the Paris congress contains the rather famous sonnet Le filtre written by Samuel in imitation of Mallarmé’s sonnet whose first line is Le vierge, le vivace et le bel aujourd’hui (Archives Bourbaki 1945a, p. 3). It should be reminded that Samuel received his doctorate from Princeton University in 1947; his advisor was Oscar Zariski.

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rusted, or at any rate, inadequate to deal effectively with the new concepts and the new problems arising in such areas of mathematics as homology algebra, differential topology, or algebraic geometry. A rather cumbersome definition of the notion of “structure” in terms of échelles d’ensembles, no general definition of the notion of homomorphism, no way to have relations between structures of different kinds, no way to characterise natural isomorphisms: all these drawbacks were in need of urgent fixing. But no serious effort in this direction was made by the Bourbaki group. Quite the contrary, in 1948 there appeared, under the name of Nicolas Bourbaki, a paper that attempted to build a philosophy (or should I rather say an ideology) of mathematics based on the very concept of “structure.” The short essay L’architecture des mathématiques (Bourbaki 1948c), addressed not to fellow mathematicians but to a general public, was included in the miscellaneous volume Les grands courants de la pensée mathématique, edited by the polymath François Le Lionnais, whose planning had started before the war, in 1942 or even a few years earlier.52 After an introductory section (La mathématique ou les Mathématiques?) dealing with the classical question of the unity of mathematics, the author—who we know is Dieudonné—presents his quite peculiar conception of the axiomatic method.53 This is not to be confused with the “logical formalism”: Ce que se propose pour but essentiel l’axiomatique, c’est précisément ce que le formalisme logique, à lui seul, est incapable de fournir, l’intelligence profonde des mathématiques. De même que la méthode expérimentale part de la croyance a priori en la permanence des lois naturelles, la méthode axiomatique trouve son point d’appui dans la conviction que, si les mathématiques ne sont pas un enchaînement de syllogismes se déroulant au hasard, elles ne sont pas davantage une collection d’artifices plus ou moins “astucieux”, faits de rapprochements fortuits où triomphe la pure habilité technique. (Bourbaki 1948c, pp. 37– 38)

The axiomatic method shows how “trouver les idées communes enfouies sous l’appareil extérieur des détails propres à chacune des théories considérées” (Bourbaki 1948c, p. 38): to achieve this goal, its main and essential conceptual tool— according to Dieudonné—is that of “structure”. Since the paper is conceived to be of non-technical nature, the notion of “structure” is introduced in an elementary and informal way by considering the example of groups, and it is given essentially the same definition that is stated in the Fascicule de résultats (see Bourbaki 1948c, p. 40, fn (3)). One the major advantages of this approach—it is maintained—is that of providing an “ordering principle” (principe ordonnateur), that could be aptly defined

52 See

(Salon 2016), pp. 203–206. Le Lionnais’s Les grands courants de la pensée mathématique included articles by three members of Bourbaki, who signed them with their own names: Dieudonné (David Hilbert (1862–1943), pp. 291–297), Weil (L’avenir des mathématiques, pp. 307–320), Godement (Les méthodes modernes et l’avenir des mathématiques concrètes, pp. 321–326). 53 Dieudonné had already addressed this question in his 1939 paper Les méthodes axiomatiques modernes et les fondements des mathématiques (Dieudonné 1939). The paper was positively reviewed by Alonzo Church, who made the only criticism “that the account is disproportionately dominated by the great name of Hilbert”, whilst “Frege receives but the briefest passing mention [...]” (Church 1939). In L’architecture des mathématiques Frege’s name is not even mentioned.

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as “Cartesian”,54 consisting in the idea of a “hierarchy of structures” (hiérarchie de structures) (Bourbaki 1948c, p. 43). The simplest ones are, of course, the structuresmères (namely, algebraic, order, and topological structures), besides which there are […] les structures qu’on pourrait appeler multiples, où interviennient à la fois deux ou plusieurs des grandes structures-mères, non pas simplement juxtaposées (ce qui n’apporterait rien de nouveau), mais combinées organiquement par un ou plusieurs axiomes qui les relient. (Bourbaki 1948c, p. 44)

So, structures—and the associated notion of “isomorphism”, which allows “to give the same name to different things”—provide also a classification system for mathematical disciplines.55 The author does not deny that such a view of mathematics is too schematic, highly idealised and, above all, “rigid” (figée), “for nothing is more foreign to the axiomatic method than a static conception of science” (Bourbaki 1948c, p. 45). Taking these correctives into account, and bearing in mind that the hierarchy of structures is a circular one—from the centre to the periphery—, mathematics can be likened to a city that gradually evolves from a simple to a more complex, if not chaotic, form: telle une grande cité, dont les faubourgs ne cessent de progresser, de façon quelque peu chaotique, sur le terrain environnant tandis que le centre se reconstruit périodiquement, chaque fois suivant un plan plus clair et une ordonnance plus majesteuse, jetant à bas les vieux quartiers et leurs dédales de ruelles, pour lancer vers la péripherie des avenues plus directes, plus larges et plus commodes.56 (Bourbaki 1948c, p. 45) 54 Dieudonné quotes twice from Descartes’s Discours de la méthode, and in particular he makes explicit reference to the second of the famous four Cartesian precepts: “[l’axiomatique] ‘divisera les difficultés por les mieux résoudre”’ (Bourbaki 1948c, p. 38); the exact quotation (in modernised spelling) is: “Le second, de diviser chacune difficulté que j’examinerais, en autant de parcelles qu’il se pourrait, et qu’il serait requis pour les mieux résoudre.” The third Cartesian precept seems appropriate to describe Diuedonné’s idea of hierarchy of structures: “Le troisième, de conduire par ordre mes pensées, en commençant par les objets les plus simples et les plus aisés à connaître, pour monter peu à peu, comme par degrés, jusques à la connaissance des plus composés, et supposant même de l’ordre entre ceux qui ne se précèdent point naturellement les uns les autres.” (Descartes 1982, p. 18). 55 In his 1980 paper Nicolas Bourbaki and contemporary mathematics Cartan writes: “It proved impossible to retain the classical division into analysis, differential calculus, geometry, algebra, number theory, etc. Its place was taken by the concept of structure, which allowed definition of the concept of isomorphism and with it classification of the fundamental disciplines within mathematics.” (Cartan 1980, p. 177). 56 It may be interesting to compare this simile to the famous one made by Wittgenstein in his Philosophische Untersuchungen (18): “Unsere Sprache kann mann ansehen als ein alte Stadt; Ein Gewinkel von wieviel Gäßchen und Plätzen, alten und neuen Häusern, und Häusern mit Zubauten aus verschiedenen Zeiten; und dies umgeben von einer Menge neuer Vororte mit geraden und regelmäßigen Straßen und mit einförmigen Haüsern.” [“Our language can be seen as an ancient city; a maze of little streets and squares, of old and new houses with additions from various periods; and this surrounded by a multitude of new boroughs with straight regular houses.”] (Wittgenstein 1958, pp. 8–9). Notice that the two similes convey opposite views: Wittgenstein’s “maze of little streets and squares” continues to exist in the center of his language-city, while the “vieux quartiers et […] dédales de ruelles” of Dieudonné’s mathematics-city are demolished and rebuilt.

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Even though not addressed to specialists (or perhaps just because of that), L’architecture des mathématiques, translated into English in 1950 (Bourbaki 1950), became Bourbaki’s programmatic manifesto. It presented, however, a view of mathematics quite remote from the new ideas that were emerging at that time. The notion of principal fibre bundle, for example, first introduced by Ehresmann and Jacques Feldbau in 1941 (Ehresmann & Feldbau 1941), “is not really just that of a few sets with some added structure, but it is built up in a complex way out of previous structural concepts (space, group, and group action)” (Mac Lane 1980, p. 364) Analogously, the notion of sheaf, originally devised by Jean Leray and developed, among others, by Cartan in his Séminaire held at the École Normale Supérieure in 1948–1949 and 1950–1951, is too convoluted to be described by means of the definition of “structure” given in the Fascicule de résultats.57

5 Structural Stubbornness (1950–1958) It is not my purpose here to give a detailed account of the debate on category theory that arose within the Bourbaki group in the first half of 1950s.58 It will be enough to recall a few key facts. After the invitation letter he received in 1948 (if the date is correct), Eilenberg attended several Bourbaki’s meetings. In 1951 he was charged with the task of writing a report on functors.59 More precisely, at the “Congrès œcuménique de Pelvoux-le-Poët” (June 25–July 8, 1951) the following decision was adopted: Chapitre IV (Structures). Sammy fera un rapport sur ce qu’on pourra y dire des foncteurs, homomorphismes, variances, structures induites, etc. On y ajoutera l’appendice des applications universelles. (Archives Bourbaki 1951b, p. 9)

Jointly with Cartan, Eilenberg wrote the book Homological Algebra (Cartan & Eilenberg 1956), a milestone of 20th century mathematics, which went to print in 1956, but was completed already in 1953. In this work they used profusely the language of functors: in particular, through the fundamental concept of derived functor (defined by means of projective and injective resolutions), they were able to “show how the cohomology theories of groups, Lie algebras and associative algebras fit into a uniform pattern” (Cartan & Eilenberg 1956, p. viii). The volume contains an appendix by David Buchsbaum introducing the notion of “exact category”, which provides an

57 For a detailed and illuminating exposition of the development of sheaf theory see (Houzel 1990).

Cartan’s sheaf theory is mentioned in the report of Bourbaki’s “Congrès œcuménique” held in June 1948 (Archives Bourbaki 1948, p. 35). 58 See e.g. Corry (2004), Krömer (2006). 59 Eilenberg was nicknamed “Sammy” and given the honorific titles of “grand Distributeur” and “grand Foncteur” at the “Congrès de l’horizon”, Royaumont, October 8–15, 1950; see (Archives Bourbaki 1950, p. 1).

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abstract natural framework suitable to develop the machinery of homological algebra and duality.60 Mac Lane too participated in one Bourbaki’s meeting, but this contact did not end in any kind of collaboration. Mac Lane himself wrote about his experience with a grain of irony: Early in the 1950s some members of Bourbaki, seeing the promise of category theory, may have considered the possibility of using it as a context for the description of mathematical structure. It was about this time (1954) that I was invited to attend one of Bourbaki’s private meetings — not the Bourbaki seminar, but a meeting where draft volumes are torn apart and redesigned. Bourbaki did not then or later admit categories to their volumes; perhaps my command of the French language was inadequate to the task of persuasion. (Mac Lane 1988, p. 337)

It should be emphasised that the banning of functors from the volumes of Éléments de mathématique was a deliberate choice, not the result of inertia. For example, at the “Congrès nilpotent” (Celles-sur-Plaine, March 1–8, 1953) it was agreed to suppress any allusion to functors in Algèbre II.61 During approximately the same years, the work on the chapter dealing with “structures” was progressing quite slowly. After having decided in 1951 to insert this chapter after that on ordered sets, ordinals, cardinals and integers,62 at the “Congrès de la motorisation de l’âne qui trotte” (Pelvoux-le-Poët, June 25–July 8, 1952) the still unsolved issue of the definition of the notion of homomorphism (also called représentation) of structures was addressed: Chapter IV (Structures): le Congrès a travaillé pour dégager la notion d’homomorphisme. Une nouvelle rédaction sera polie cet automne par un Caucus Américain, puis envoyée au Congrès de Février. (Archives Bourbaki 1952, p. 3)

The crucial suggestion for tackling the problem was made by Eilenberg: […] Sammy a proposé de définir une structure comme partie d’un ensemble de l’échelle (et non plus comme élément), ce qui a l’air de donner une bonne définition des représentations (modulo certaines petites complications de “variance” co ou contra); si ça marche, on a alors facilement les notions de structure induite, de produit et quotient (cf. papier Samuel des “Universal mappings”). On décide d’essayer le système Sammy. (Archives Bourbaki 1952, p. 8)

60 See

also (Buchsbaum 1955). Exact categories were to be called “abelian categories” by Alexandre Grothendieck in his groundbreaking paper Sur quelques points d’algèbre homologique (Grothendieck 1957), published in 1957. 61 “Supprimer les allusions aux foncteurs” (Archives Bourbaki 1953a, p. 9). As late as 1982—then after his long-lasting collaboration with Grothendieck on preparing the eight parts of the Éléments de géométrie algébrique, published from 1960 through 1967—, Dieudonné declared his aversion to categories and functors: “For many other parts of mathematics, it is certainly possible to use the language of categories and functors, but they do not bring any simplification to the proofs, and even in homological algebra (treated, for modules, in a recent Bourbaki chapter), one can entirely do without their use, which would only amount to introducing extra terminology.” (Dieudonné 1982, p. 622). 62 See (Archives Bourbaki 1951a, p. 3).

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Dieudonné, growing more and more exasperated, produced three successive versions of the chapter on “structures” between April 1953 and March 1954.63 The last version (Archives Bourbaki 1954) is essentially identical to the printed text (Bourbaki 1957), which appeared one year after Grothendieck’s celebrated Tôhoku paper (Grothendieck 1957).64 There are many significant differences between this book and the Fascicule de résultats. First, the notion of échelle d’ensemble is replaced by a much more complex notion. A schéma de construction d’échelon (SCE for short) is defined as a sequence c1 = (a1 , b1 ), c2 = (a2 , b2 ), . . . , cm = (am , bm ) of pairs of integers such that (1) if bi = 0, then 1 ≤ ai ≤ i − 1; (2) if ai = 0 and bi = 0, then 1 ≤ ai ≤ i − 1 and 1 ≤ bi ≤ i − 1. An SCE is said to be sur n terms (n-SCE for short), if n is the biggest integer occurring in pairs of the form (0, bi ) with bi > 0. Let S = (c1 , c2 , . . . , cm ) be an n-SCE. Given n terms E 1 , . . . , E n in a theory T stronger that set theory (i.e. a theory of types), a construction d’échelon, de schéma S, sur E 1 , . . . , E n is a sequence A1 , A2 , . . . , Am of m termes in T determined by the following conditions: (1) if ci = (0, bi ), then Ai = E i ; (2) if ci = (ai , 0), then Ai = P(Aai ); (3) if ci = (ai , bi ) with ai = 0 and bi = 0, then Ai = Aai × Abi . The last term Am of the previous construction is called échelon de schéma S sur les ensembles de base E 1 , . . . , E n and denoted by S(E 1 , . . . , E n ). Second, the notion of transport of “structures” is introduced in a rigorous way. Let x1 , . . . , xn , s1 , . . . , s p be distinct letters (lettres) which are not constants of the theory T , and A1 , . . . , Am terms of T in which no letter xi , s j occurs. Given (n + m)-SCEs Si , . . . , S p , the relation T {x1 , . . . , xn , s1 , . . . , s p }: “s1 ∈ S1 (x1 , . . . ,xn , A1 , . . . , Am )} and s2 ∈ S2 (x1 , . . . , xn , A1 , . . . , Am )} and · · · and s p ∈ S p (x1 , . . . , xn , A1 , . . . , Am )}” is called a typification of the letters s1 , . . . , s p . The definition of transportable relation is quite involved, and I reproduce it in the original in order to give the flavour of Dieudonné’s quite distinctive style:

63 Dieudonné’s exasperation is evident, for example, in the comment he inserted in the last preliminary version (“état 8”, 1954): “Contrairement à La Tribu, on a conservé la distinction entre ‘caractérisation typique’ et ‘axiome’ d’une espèce de structure, parce que c’est ainsi qu’on procède dans la pratique. Le rédacteur n’a pas compris pour quelle raison bizarre la ‘structure d’ensemble’ devrait être repoussée au § 2, où elle n’a évidemment que faire, et il a laissé les choses en l’état. […] Le rédacteur se déclare incapable de rédiger quelque chose de sensé et de non vaseux sur les “identifications en cercle’ et acceptera toute rédaction que le prochain Congrès voudra bien faire sur ce sujet (palpitant!).” (Archives Bourbaki 1953b). 64 The copyright is dated 1957, but the printer’s statement at the end of the book indicates that it was printed on March 7, 1958.

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Soit R{x1 , . . . , xn , s1 , . . . , s p } une relation de T , contenant certaines des lettres xi , s j (et éventuellement d’autres lettres). Dire que R est transportable (dans T ) pour la typification T, les xi (1 ≤ i ≤ n) étant considérés comme ensembles de base principaux et les Ah (1 ≤ h ≤ m) comme ensembles de base auxiliaires, c’est dire que la condition suivante est satisfaite: Soient y1 , . . . , yn , f 1 , . . . , f n des lettres distinctes entre elles, distinctes des xi (1 ≤ i ≤ n), des s j (1 ≤ j ≤ p), des constantes de T et de toutes lettres figurant dans R ou dans les Ah (1 ≤ h ≤ m). Soit d’autre part Ih (1 ≤ h ≤ m) l’application de Ah sur lui-même. Alors, la relation “T {x1 , . . . , xn , s1 , . . . , s p } et ( f 1 est une bijection de x1 sur y1 ) et · · · et ( f n est une bijection de xn sur yn )”

(1)

entraîne, dans T , la relation

où on a posé

R{x1 , . . . , xn , s1 , . . . , s p } ⇐⇒ R{y1 , . . . , yn , s1 , . . . , s p }

(2)

s j =  f 1 , . . . , f n , I1 , . . . , Im  S (s j ) (1 ≤ j ≤ p).

(3)

(Bourbaki 1957, pp. 11–12)

In Eq. (3) the expression  f 1 , . . . , f n , I1 , . . . , Im  S denotes the canonical extension, through the SCE S, of the mappings f 1 , . . . , f n , I1 , . . . , Im (see (Bourbaki 1957, p. 9) for a precise definition). The key point is that relations may fail to be transportable.65 So, the notion of transport of structures introduced in the Fascicule de résultats of set theory appears to be not only heuristic (since a heavy machinery of definitions is needed in order to make it rigorous), but also incorrect (in that it may fail to work when there are also auxiliary sets). An 18 pages long appendix on trasportability criteria is inserted at the end of Dieudonné’s exposition (Bourbaki 1957, pp. 51–69). Surprisingly (or maybe not), this appendix is entirely dropped from the second edition of Chapitre 4. Structures (Bourbaki 1966), as well as from all subsequent editions of Théorie des ensembles, and replaced by a short, quite user-unfriendly statement: On reconnait aisément, dans les cas usuels, si une relation est transportable (pour une certaine typification). (Bourbaki 1966, p. 12)

The remaining of Sect. 1 of (Bourbaki 1957) is devoted to give a precise definition of isomorphism (again in terms of transport of structures) and introduces in a rigorous way the notions of induced structure and of equivalence between structures. On the other hand, Sect. 2 presents entirely new concepts, and the titles of some of its subsections are telling in themselves: Morphismes, Structures initiales, Structures finales. Section 3 is devoted to universal mappings. The members of the Bourbaki group were fairly obstinate in adhering to their structural credo, even at a time when it clearly appeared that the formalism of “structures” was inadequate to tackle effectively some of the major problems arising in mathematics. But the very language they were eventually forced to adopt reveals that they had already lost their war. = p = 2 and if T is the typification “s1 ∈ x1 and s2 ∈ x1 ”, the relation x1 = x2 is not transportable. 65 For example, if n

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Fréchet, M. (1928). Les espaces abstraits et leur théorie considérée comme introduction à l’analyse générale. Paris: Gauthier-Villars. Frege, G. (1976). Wissenschaftlicher Briefwechsel. G. Gabriel, et al., (Eds.), T. Hamburg: Felix Meiner (English edition abridged by B. McGuinness and translated by H. Kaal, Philosophical and mathematical correspondence. Oxford: Basil Blackwell, 1980). Grothendieck, A. (1957). Sur quelques points d’algèbre homologique. Tôhoku Mathematical Journal, 9, 119–221. Houzel, C. (1990). A short history: les débuts de la théorie des faisceaux. In M. Kashiwara & P. Schapira (Eds.), Sheaves on manifolds (pp. 7–22). Berlin, Heidelberg: Springer. Krömer, R. (2006). La “machine de Grothendieck” se fonde-t-elle seulement sur des vocables métamathématiques? Bourbaki et les catégories au cours des années cinquante. Revue d’histoire des mathématiques, 12, 119–162. Krömer, R. (2007). Tool and object. A history and philosophy of category theory. Basel, Boston, Berlin: Birkäuser. Lefschetz, S. (1942). Algebraic Topology. New York City: American Mathematical Society. Mac Lane, S. (1938). The uniqueness of the power series representation of certain fields with valuations,. Annals of Mathematics, 39, 370–382. Mac Lane, S. (1939). Some recent advances in algebra. American Mathematical Monthly, 46, 3–19. Mac Lane, S. (1950). Duality for groups. Bulletin of the American Mathematical Society, 56, 485– 516. Mac Lane, S. (1980). The genesis of mathematical structures, as exemplified in the work of Charles Ehresmann. Cahiers de topologie et géométrie différentielle catégoriques, 21, 353–365. Mac Lane, S. (1988). Concepts and categories in perspective. In P. Duren (Ed.), A century of mathematics in America, Part I, History of mathematics (Vol. 1, pp. 323–365). Providence (RI): American Mathematical Society. Mac Lane, S. (1997). Van der Waerden’s modern algebra. Notices of the American Mathematical Society, 44, 321–322. Mac Lane, S. (2005). A mathematical autobiography. Wellesley (Mass.): A.K. Peters. Marquis, J.-P. (2009). From a geometrical point of view. A study of the history and philosophy of category theory. Springer, s.l. Mayer, W. (1929). Über abstrakte Topologie. Monatshefte für Mathematik und Physik, 36, 1–42. Mehrtens, H. (1979). Die Entstehung der Verbandstheorie. Hildesheim: Gerstenberg. Noether, E. (1921). Idealtheorie in Ringbereichen. Mathematische Annalen, 83, 24–66. Noether, E. (1927). Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern. Mathematische Annalen, 96, 26–61. Ore, Ø. (1932). Van der Waerden on algebra [review of van der Waerden’s Moderne Algebra]. Bulletin of the American Mathematical Society, 38, 327–329. Ore, Ø. (1935). On the foundations of abstract algebra, I. Annals of Mathematics, 36, 406–437. Ore, Ø. (1936a). On the foundations of abstract algebra, II. Annals of Mathematics, 37, 265–292. Ore, Ø. (1936b). L’algèbre abstraite, "Actualités scientifiques et industrielles" 362. Paris: Hermann. Poincaré, H. (1908). Science et méthode. Paris: Flammarion. Salon, O. (2016). Le disparate: François Le Lionnais. Tentative de recollement d’un puzzle biographique. Paris: Othello. Samuel, P. (1948). On universal mappings and free topological groups. Bulletin of the American Mathematical Society, 54, 591–598. Schlimm, D. (2011). On the creative role of axiomatics. The discovery of lattices by Schröder, Dedekind, Birkhoff, and others. Synthese, 183, 47–68. Steinitz, E. (1910). Algebraische Theorie der Körper. Journal für die reine und angewandte Mathematik, 137, 167–309. van der Waerden, B. L. (1930–1931). Moderne algebra (2 vols). Berlin: Springer. van der Waerden, B. L. (1975). On the sources of my book moderne algebra. Historia Mathematica, 2, 31–40.

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The Wrapped Dimension of Bourbaki’s Structures Mères Alberto Peruzzi

Abstract The term “structures mères” (mother structures) first appeared in a paper by Bourbaki, published in 1948, under the title “L’architecture des mathématiques”. It was introduced to name a kind of elementary building block for the “architecture” of mathematics, but in fact Bourbaki’s set-theoretic presumptions prevented the concept of structure mère from playing anything beyond a token role in Bourbaki’s Éléments de mathématique and so its potential remained undeveloped. A general, principled, concept of structure and the existence of structures mères in mathematics together form a characteristic feature of Bourbaki’s species within the genus called “structuralism”, which, after its emergence early in the 20th century, flourished in many areas of research in mid-century, before undergoing a swift decline. The renaissance of structuralism in the philosophy of mathematics which has taken place over the past two decades is different from Bourbaki’s species for two main reasons: it primarily pivots on category theory to capture an even more general, but no less principled, concept of structure and does away with structures mères altogether—a notion which from the outset is inconsistent with a purely relational view of mathematics. Usually, discussions about Bourbaki’s species abstract from its genus. Here it is argued that this approach is not profitable, for the new structuralism resurrects problems which the earlier structuralism without structures mères failed to solve. So, as not to run the risk of repeating that failure, further arguments are in need. Refined arguments have been provided in support of the new structuralism, but they do not avoid that risk, since they fail to identify the meaning-roots of the notions used to understand structures, failing in particular to address the structure grounding problem. To avoid the risk, we shall employ a method of addressing the problem which calls for the existence of structures mères, defined in a suitable fashion. What follows is a continuation of a research program which, begun within a phenomenological perspective on the foundations of mathematics, is intended to take advantage of the pluralistic overview suggested by John Bell, and to meet Bill Lawvere’s philosophical demands. Indeed, Bourbaki’s claim that structures mères exist was from the start accompanied A. Peruzzi (B) Department of Humanities, University of Florence, Via Della Pergola, 60, Florence, Italy e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 A. Peruzzi and S. Zipoli Caiani (eds.), Structures Mères: Semantics, Mathematics, and Cognitive Science, Studies in Applied Philosophy, Epistemology and Rational Ethics 57, https://doi.org/10.1007/978-3-030-51821-9_5

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by an irreducible plurality of kinds-of-structure, which at first blush is at odds with Bourbaki’s set-theoretic assumptions. Such a plurality was open to either a gestaltist or a merely taxonomical reading but neither could suffice, as structures mères call for an inherent generativity. Thus arose a grammar of universal “schemes”, the term used in the present paper, as prime patterns of structuration, which, made precisely explicit in mathematics, play an active role in every science. Section 1 introduces the motivations for Bourbaki’s structuralism and points out the issue raised by a set-theoretic definition of structure. Section 2 deals with Bourbaki’s attitude towards foundational problems: from the lack of attention to logic to the decision to leave category theory aside, while identifying some of its basic notions and employing it in the solution of problems (e.g. in algebraic geometry, by Alexandre Grothendieck). Section 3 relates Bourbaki’s relationism to historical approaches contrasting substantialism, conceptual atomism and compositionality. Section 4 offers a quick look at the structure-centred relational approach to general linguistics since Saussure’s Cours in 1916 and then the spread of structuralism to many other fields. Section 5 reports on the “golden age” of structuralism in continental Europe and some of the general objections this view provoked. Section 6 examines the connection between structuralism and the view of axiom systems as implicit definitions of the concepts axioms express, with reference to classical debates such as those which opposed Poincaré to Russell and Hilbert to Frege. Here it is argued that, when the concept of structure is intended to be implicitly defined, the view lacks explanatory power. Section 7 addresses the “structure grounding problem”, suggesting a perspective quite different both from formalism and realism, as it rests on kinesthetic schemes whose action is expressed in category-theoretic terms. Finally, Sect. 8 focuses on the semantic roots of the word “structure” to emphasise the primary role of spatial interaction patterns as generators of every variety of abstract structure.

1 Bourbaki’s Perspective Any kind of mathematical structuralism of course hinges on the notion of structure. The ubiquity of this notion in mathematical discourse is a product of Bourbaki’s influence. But it should be noted that even outside mathematics “structure” became an all-encompassing word in the mid 20th century,1 in the twofold conviction that structures are everywhere and anything we talk about can only be identified relative to a structure.2 Nevertheless, the common usage of the term “structure” in ordinary, scientific or purely mathematical language does not entail a commitment to the existence of structures mères.3

1 For

a discussion of the factors which determined this outcome, see Peruzzi (1996). Lawvere (1969), Bell (1986). 3 Or schemata, to stay at the Greek. Of course, “scheme” is not used here in the specific Grothendieck’s sense. 2 See

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“L’architecture des mathématiques”4 was intended to be a manifesto of the group of Bourbakists.5 It succeeded in promoting the novelty of their perspective, particularly insofar as it focused not so much on foundations, but on the architecture of a system of interrelated mathematical “buildings”. The principal motivation had two facets: deployment of formal rigour and attention to the mathematical practice in each “building”. The initial aim was not to focus on a list of prime notions and axioms relating these notions to one another, starting from which (a) any other mathematical notion would be definable and (b) any specific axiom system concerning specific areas of mathematics would be justified and provided with explicit formulation. But they soon realised that beyond a Fascicule de résultats (subtitle of the 1939 volume by Bourbaki) concerning the set-theoretic background, precise axiomatic presentation could not be avoided, while the truly important thing remained the architecture (and city planning), to be described in terms of structures. As a formalised relational framework was not at hand, the architectural design employed the language of sets, with emphasis on salient building-shapes, corresponding to three basic kinds-of-structure, namely, algebraic, order and topological. These, the structures mères, would, in suitable combinations, serve to cover the immense variety of mathematical structures. The very definition of structures mères embodies a compromise: these three basic kinds-of-structure were not mothers at all, but daughters of a zero degree of structure, namely sets, as defined in terms of ∈, whereas in a purely relational architecture the hierarchy of sets ought to be described as a special instance of notions pertaining to the structures mères. In this case it does not matter which particular form of set theory is chosen; in fact Bourbaki selected a version of ZFC– (where the minus sign indicates that the axiom of foundation is lacking).6 But, taken as mere collections, kinds-of-structure have no selective generative power by themselves: they simply identify three classes of sets, like Linnaeus’ three reigns—the vegetal, the animal and the mineral—or the ten “categories” after Aristotle, also misunderstood as maxima genera. At best, they can be put in correspondence with different principles, and patterns, of formation/construction, and, since the goal was not taxonomy but functional design (to retain the architectural metaphor), the inherent potential of structures mères duly failed to materialise. Thus the exclusive attention paid to the kind of set theory Bourbaki adopted also prevented the achievement of an exhaustive, formal, explicit theory of structures which are truly basic, and so not daughters of the same mother. Disregarding the fact that the structures mères provide the ingredients for any combination of structures, whereas Bourbaki has a decidedly selective attitude toward 4 Bourbaki (1948). The paper was actually written by Jean Dieudonné. Its English translation, “The

Architecture of Mathematics”, appeared on The American Mathematical Monthly, Vol. 57, No. 4 (Apr., 1950), pp. 221–232. 5 We make occasional use of this anglicisation of a noun used in French and Italian to refer to the members of the group of mathematicians who assumed “Bourbaki” as their collective name—the origins of which are described by Maschaal (2000). 6 As to whether the resulting setting was as rigorous as it was intended to be, or not, see Mathias (2015) and Lolli (2019) (this volume).

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which (composed) structures are relevant, the problem remains that, for the architectural approach to be consistent, there are two options. Option A: sets have to be taken as a fourth source. Option B: the universe of sets has to be characterised in a purely relational way in terms of algebras, spaces and orders. Neither option was pursued, though already the 1948 manifesto displays an awareness of the provisional character (and the inconvenience) of the resulting classification of structures. Options A and B suggest, with hindsight, that the notion of category needs to be made explicit and that it is necessary to investigate the universal maps which connect, for instance, spaces to sets and vice versa, possibly together with (if option B is chosen) an algebraic formulation of set theory.7 Both options stand in the spirit of a radically structuralist view, which set theory, as presented in the Éléments, does not realise, since the three structures mères, far from acting as autonomous embodiments of mathematical form, function only as highlands within the sea of sets—so to speak, because there was no universal set. Doubtless, pragmatic requirements served to justify the compromise. At the time, each algebra, order type, or topology was introduced by means of the ∈-language, in which sets are identified by their global elements (in view of the strong ε-axiom adopted by Bourbaki), thus “bottom-up”, whereas a fully structural view requires us to identify elements through and only through their participatory role in a structure. This structure has to be specified in advance of, or at least simultaneously with, such elements. It cannot be given after the introduction of elements. There are many aspects of the general definition of structure which should be taken into account.8 In the Fascicule de résultats structure was confined to what can be inductively obtained by combining products and powers of sets. Suffice it to recall here that after Tarski the notion of structure enters the scene with formal semantics, in which the notion of model of any syntactic presentations of a theory T is defined by considering all possible interpretations of the language of T by the inductive definition of T-terms and T-formulae. Within this setting a structure is defined as a set D (the “domain” of the interpretation) together with a set of selected individuals and/or subsets in D, a selected family of relations between D-elements, and finally a family of operations, on D-elements, with respect to which D is closed, in a manner adequate for the associated T-language (a many-sorted language calls for further details). Algebras, orders and spaces are just particular examples of this general, set-theoretic notion of structure. But model theory was not part of Bourbaki’s program and neither was universal algebra, so these essentially metamathematical notions remained unexplored.

7 The

renaissance of structuralism in the philosophy of mathematics over the past two decades primarily pivots on category theory, but not exclusively, as witnessed by Resnik (1982), Shapiro (1997) and Hellman (2001). 8 Bourbaki’s notion of structure is widely discussed by Corry (1992).

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2 Bourbaki’s Look on Logic, Sets and Categories Along with set theory, logic—and not just propositional but at least first-order logic— had to be presupposed as an additional source, and in both cases other problems arose, associated, respectively, with the supposedly minimal commitment to infinitary assumptions and the elimination of quantifiers through Hilbert’s ε/τ operators.9 Nevertheless, it would be unfair to ignore the fact that the subsequent extensive use of set-theoretic language in many fields of mathematics is mainly due to Bourbaki and that use was far from obvious, even though it may appear to be so to those later educated in analysis, algebra or geometry through that very use. Bourbaki was also instrumental in fixing set-theoretic notation, and introduced current terms for specific kinds of maps and particular set-theoretic construction methods (such as filters) which have become standard. As a result, there is a feeling of natural continuity when we pass from set theory to any specific mathematical area. A language and a style provided the unifying glue which turned, in Bourbaki’s hands, a plural into a singular: les mathématiques became la mathématique. As if the cunning of reason were at work, Bourbaki’s nonchalant attitude towards logic and foundational issues, even if deserving of rebuke, paved the way for dealing with sheaves in algebraic geometry and for Grothendieck’s idea of a “universe”, tailored to the requirements of “the working mathematician”10 rather than furnishing a universal realm for the whole of mathematics. We must be careful, however, when making appeal to these sorts of considerations, in order to avoid too easy a backward “mitigating” case. A consistently and fully structural view focused on the architecture of mathematics had to allow for logic and sets, even though at the time there appeared to be no way of achieving this; thus, even had Chevalley’s book dealing with logic and categories not been lost,11 it could at best offer hints at the future task of showing how logic and abstract sets could be introduced in purely structural terms, while leaving the gap unfilled. Had Option A and Option B been examined, a different scenario from what was to become the standard one (classical logic plus an axiom system for sets, after Zermelo) could have been discerned, one also different from a commitment to constructivity, on the face of it at odds with structuralism, as well as from the priority of metamathematical warrants, at least as they passed from Hilbert to Gödel. These warrants, of a recursion-theoretic nature, were on the whole avoided by Bourbaki’s structure-oriented formalism, a formalism which did not arise from the requirements of finitism or computability. To be fair also on this matter, we should add that the relevance of such gaps in Bourbaki’s work is ambiguous, depending on whether the gaps are intended to 9 Such

problems feed criticisms focusing on the Bourbaki’s lack of rigour, as Mathias emphasised, but see also Anacona et al. (2014) for a defence from some of these criticisms. 10 The term comes from Bourbaki (1949). After Saunders Mac Lane used the term in the title of Mac (1971), it rapidly came to be associated with an acknowledgment of the importance of the actual practice of doing mathematics within any account of its foundations. 11 Reference is of course to its preliminary version, as any volume in the Éléments de Mathématique had to pass a severe check by the community of affiliates.

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be taken as fundamental or as contingent matters of fact. Considered as a work in progress, open to future completion (and revision), the gap-filling device had already been prepared by 1945, with the definition of categories and functors by Saunders Mac Lane and Samuel Eilenberg (a member of the Bourbaki group five years later). Category theory acquired foundational meaning when Bill Lawvere succeeded in showing that (1) sets admit a purely categorical axiomatisation in ETCS,12 (2) the intuitionistic behaviour of connectives and quantifiers is inherent in topos structure through suitable adjunctions, (3) logical self-reference and topological fix point theorems are specific versions of a general, categorial, property. But the impact of (1), (2) and (3), rather than establishing Bourbaki’s structuralism as an adequate view of mathematics, in fact changed the overall picture. While algebras, orders and spaces were conceived as being housed in a single category (of sets), they proceeded to migrate to different ambient categories from which they inherit features: a basic algebraic structure such as that of a group shows up as a topological group in the category of spaces, and the duality of Stone spaces, originally relative to Boolean algebras and power sets, acquires other meanings in different ambient-categories. This might suggest looking at the three structures mères in an even more abstract way than did Bourbaki, and we do not have to look far, given Grothendieck’s move from a space to a site in his definition of the base of a topos. But a price has to be paid for this increase in abstraction: form seems to drift apart from content and the intuitive grounding of notions involved in the three structures mères is relegated to a hazy background. “Intuition” is a noun foreign to the structuralist lexicon, so that the roots of our access to the very meaning of “form” become obscured. In transforming mother-structures into daughter-structures, one automatically satisfies the grounding requirement for the “universe” of abstract sets as to be composed of lauter Einsen, i.e., pure “ones” in Cantor’s sense,13 and if reasons can be found for the autonomy and salience of the three basic kinds-of-structure, these cannot be “daughters”. That being the case, how can they be formally presented? Any answer will commit us to conceiving space-structure as a general pattern presented synthetically rather than analytically, order-structure likewise as a general pattern independent of any specific collection of entities which are (totally or partially) “ordered”, and algebra-structure as a general pattern of operation, combination, action (when an algebra acts on a structure, see Cayley’s representation theorem for groups). Filling gaps with other gaps is no recommendable option, of course. Once again, the answer came with category theory: think, respectively, of synthetic differential geometry and pointless topology, of categories of functors from Cop to E, where C is a category with at most one morphism between any two objects and E is a topos, and the monad construction. But is a category-theoretic phenomenology of structures the end of the story?

12 ETCS

stands for the Elementary Theory of the Category of Sets, see Lawvere (1964). Valuable adjustments to ETCS were later accomplished by Colin McLarty, see McLarty (1991). 13 See Lawvere (1994).

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If the list of structures mères is not taken as fixed (to recover the “openness” of the 1948 manifesto), it is reasonable to expect that, whatever kinds-of-structure are added as basic, their list remains finite and as short as possible. Otherwise, with the increase of their number, structuralism with structures mères would lose what distinguishes it from other types of structuralism (to which we shall come next). Simply insisting that the list is finite does not suffice and adding that it is irreducible to a singleton, in principle or simply as a matter of fact, i.e. relative to the state of the art, is a task in itself: that of ensuring the mutual independence of a (possibly extended) collection of kinds-of-structure, and since de facto adequacy does not explain what makes this possible, a metamathematical proof would be needed. In view of this additional gap, the framework envisaged by Bourbaki retains more than historical interest: the gap’s presence risks projecting its own shadow on the “architectural” view in the cleansed and re-enacted form it has come to assume within the structuralism of recent years, mainly set up through category theory. How much of Lawvere’s “materialistic” view survives the transition to such an arrangement is not clear.14 The way to fill Bourbaki’s gaps calls for making the concrete roots of abstract structures explicit, unless we are ready to explain away one characteristic feature of Bourbaki’s species of structuralism and to apply to mathematics the structuralism which, outside of mathematics, we must admit has failed.

3 Structure as Substance-Erasing Since the Presocratics, among the questions defining the goals of philosophy, one of the most important was: What are the prime, or ultimate, elements, constituents, “substances”, of which reality is composed? This question led immediately to other two questions: What is the status of such elements? How do they combine to produce what we experience? The contrast between “prime” and “ultimate” constituents correspond to the direction in which they are sought, by the choice, in attempting to answer these fundamental questions, of proceeding forward or backward, and adopting a bottom-up or a top-down order. Aristotle drew a sharp distinction between what is first for us and what is first per se, but in such a way that each direction is not prevented from matching with the other. Moreover, the search for what is first per se was linked to the search for first principles, pertaining to the very structure of Being, thus to the “seeds” of what there is as a synolon of Form and Matter. The link between “seeds” and principles had to do with the notion of category and specifically with the ten categories he proposed. This link was preserved by philosophers in disagreement with Aristotle, even by those who were inclined to a relationist ontology. The debate

14 Lawvere’s

ideas on the matter are summarised in Peruzzi (2000). McLarty (2004) and Landry and Marquis (2005) provide excellent replies to arguments against the independence of category theory from notions which are supposed to be essential properties of ∈-based set theory.

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about which system of categories is the true one lasted for many centuries.15 In the 20th century a change occurred, so influential to become the received view, which is by now close to common sense. This change was announced as undermining the very meaningfulness of the starting question. It was no sudden burst, being rather the endpoint of previous research-paths leading from individuals, as individual entities endowed with individual essences, to relational nets, within which any individual acquires its identity. The primacy of relations had been already argued at the dawn of modern science and shaped in manifold ways since the 17th century: Locke’s objections against the Aristotelean notion of substance, the view of space and time advocated by Leibniz against Newton, the equational form of laws in which different quantities are mutually related, the shift of geometry from figures to the background out of which figures emerge, the emergence of the concept of field in physics, the idea of “system” at the birth of social sciences (Emile Durkheim: “individu écarté, il ne reste que la société”), the progressive abstraction leading from algebraic operations on numbers to algebras which admit a plurality of different kinds of instances, the idea of biological evolution as concerning the pairing of species and their ecological niches. It was an overall trend which in 1905 Ernst Cassirer summarised as the achievement of a view centered on Funktion-begriffe over one centered on Substanzbegriffe, and thirty years later he finally talked of “structure” as standing above the specific, material, nature of its constituents. Even the Knowing Subject of the Kantian tradition had become a “lattice” of structures. When Cassirer made explicit reference to “structuralism” as the resulting paradigm of thought,16 the noun was already in use, having been introduced by Roman Jakobson to name the approach which found expression in Ferdinand de Saussure’s project of “general linguistics”. Such a paradigm had the virtue of setting idealistic and materialistic ontology, as well as positivism, aside: fons et origo was Structure, and Structure can be investigated in its own terms, independently of the particular, concrete, manifestations of any instance of Structure. Was this turn irreconcilable with the bottom-up, compositional approach, which also endured in exact science and science-oriented philosophy? Or could the two perspectives be linked with one another? If so, how? Bourbaki’s notion of structures mères is relevant here as it might allow the contrast to be viewed as a superficial appearance of a principled dialectics in action also within mathematics. But it only might. Now, consider a fictitious triangle ABC, with “purely” syntactic structure of the language-of-structures as vertex A, semantic structure of an empirically interpreted language as vertex B, and cognitive structure as vertex C. The side AB can be drawn under the guide of logic, the side BC by means of information processing systems (whether based on classical programming or neural networks or possibly quantum computing). So, we could claim (as many did), AC is obtained by composition of AB and BC. What if AC can also be traced directly, in a manner which possibly admits a factorisation through AB and BC? That is exactly the change in perspective 15 See 16 See

Peruzzi (2018). Cassirer (1945).

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cognitive grammar is concerned with, and its focus on basic patterns of meaning provides support for a hypothesis: that the place such patterns have in the architecture of semantic competence corresponds to the place structures mères have (actually, might have), according to Bourbaki, in the architecture of mathematics. This hypothesis suggests a different end to the story which started with the search of prime components of Being and arrived at structuralism. It is not just that Aristotle’s warning “Being is said in many ways” has to be updated as “Structure is said in many ways”, for the variety of “ways” is constrained by principles, which in turn correspond to basic structure-patterns which in their turn correspond to building blocks of meaning construction, and finally the existence of such building blocks corresponds to the existence of structures mères.

4 The Eraser’s Past The Cours de linguistique générale by Saussure appeared (posthumously) in 1916 and soon became a standard work, whose influence spread in continental Europe from Paris to Moscow. A key role in this regard was played by the Prague Linguistic Circle (1929–1939) which, centred around its two leading figures, Nicolaj Trubeckoj and Roman Jakobson, produced the famous series of Traveaux. This series started with the manifesto containing the Circle’s Theses, which appeared in 1929, that is, the same year as the Wissenschaftliche Weltauffasung which boldly presented the tenets of the Vienna Circle, a link between the two passing through Copenhagen, with Luis Hjemslev’s project of a “glossematics”. In fact, Saussure’s legacy was also decisive in the spread of semiology to disciplines well beyond linguistics. As mentioned above, Jacobson used the term “structuralism” to refer to Saussure’s approach, though in the Cours Saussure talks of a language as a système, rather than as a structure. Apart from terminology, the main thesis was clearly expressed in Sect. 3 of the Cours: “La langue est un système dont toutes les parties peuvent et doivent être considérées dans leur solidarité synchronique”,17 and already in the notes for the (second) course of general linguistics in 1908–1909 Saussure stated that “tout est syntactique dans la langue, tout est un système”, to support a purely internal investigation of the structure of any language, shielded from socio-historical and psychological factors. In strict correspondence with such autonomy of structure, “systemic” synchrony was sharply contrasted in the Cours with diachrony. This distinction played an indispensable role in Saussure’s conception of a science of language, in view of the fact that, if linguistic changes lack fully systemic character, there is no room for clear-cut general laws: whatever is meant by “genesis” of structure, the analysis of structure can and must be pursued on its own. The primacy of a synchronic approach marked a turn with respect to the 19th century’s evolutionary view of language. Later versions (such as Jakobson’s) of

17 The

quotation is from the new edition, Saussure (1971) p. 124.

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linguistic structuralism opened up a class of diachronic phenomena as no less systematic, and curiously one of the first attempts aimed at a proper “structural” semantics was proposed by Jost Trier, in 1934, in terms of the notion of Wortfeld (word-field); it concerned right the variation of semantic fields in the history of a natural language— something which might have been of relevance for a philosophy of “language games” which pays attention to how a “way of life” actually evolves. Direct application of the synchronic view of language as a system led Saussure to define the meaning of an expression as its role in the system (structure), this role being intended as the oppositional value of an expression with respect to others of the same class (Cours, Chap. 4). He illustrated this by means of an example which was to become paradigmatic, namely chess, where any piece is identified by its relations, within the game, with any other piece, while its material instantiation is irrelevant to the (oppositional) value defining each piece. Thus, meaning is no absolute, point to point, correspondence, but only relative, and internal, to the given system, and “abstraction faite de son expression par les mots, notre pensée n’est qu’une masse amorphe et indistincte”.18 If the ambient structure changes, meaningas-value changes too, and, as Jacques Derrida later remarked, if there is any structure extractable from texts in a given language, it’s only due to other texts in another language (“il n’y a pas de hors-texte”), implicitly suggesting a language-centred holistic view. The guiding ideas of linguistic structuralism found their most effective application in phonology thanks to the Prague School and, beyond phonology, were used by Hjemslev to fashion an algebra of language structure, with the traditional notions of Form and Content now essentially correlative. The fertility of this approach was acknowledged by Claude Lévi-Strauss when he credited Saussure for being “the Copernican Revolution” in the human sciences, claiming that it is not so much that language belongs to man as man belongs to language, to point out the antisubjectivistic orientation of structuralism. Once system-relativity is complemented with (pragmatic) context-relativity, the outcome is only partially similar to what ordinary language philosophers identified as the dawn of a study of natural/historical languages free of the logic-induced guilt feelings toward formal languages. Yet there was no sign of an account of that “generative” power of a structure producing strings of lexical items with hierarchical noun-phrases and verb-phrases, which makes language different from chess. Hence questions such as: What makes such structure possible? Why is the range of structures for human languages not as wide as it could formally be? To what extent does the structure of a language mould an image of the world? The answers to these questions given by some structuralists came up against obstacles through the evidence accumulated by linguistic anthropology. Other structuralists’ refusal to provide answers shifted the burden to still others who could not, for the sake of consistency, subscribe to structuralism. In fact, explanatory debts by structuralism in linguistics were never repaid, but if a theory of language is authorised to deny they are debts, how can it be part of science? 18 Saussure

(1971) p. 156.

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When a mathematical theory of syntactic structure for natural languages was at hand (through Noam Chomsky’s use of recursion theory) structuralism could still withdraw to semantics, resulting in a methodological gap between the analysis of syntactic competence and the analysis of semantic competence. This gap began to shrink only when cognitive grammar approached the syntax-semantics boundary in a new way, which so far is lacking a proper, precise theory, while the anti-atomistic character of structuralism survived in both the hermeneutic and the analytic variants of the “linguistic turn”, bouncing to holism all the way down. But what if a principled, structure-centered relationism were to enter the scene? This is what is suggested by a structuralist philosophy of mathematics with category theory as its lynchpin. Even today, objections to the resulting picture follow Solomon Feferman’s line, that is: category theory is not capable of a fully autonomous foundation, as it presupposes the notions of collection and operation. Replies to this and similar objections have been argued for.19 One idea underlying such replies is that we can continue to uphold Hilbert’s distinction between the genetic and the axiomatic method, a distinction reinforced, rather than overcome, even by Husserl, i.e., by the philosopher who intended to recover the original experiential Sinngebung (sensebestowal) behind mathematical models of the physical world, as Husserl himself legitimised the distinction between a descriptive and a genetic phenomenology. Indeed, both distinctions are a profitable start, but not the end, provided that we disavow Derrida’s idea that the external of a language is another language. Together with making the contentual roots of formal notions explicit, the notions used in the genetic method can be recognised as being susceptible of axiomatisation, so that the two methods meet and the way they meet is no less explicit and precisely identified. Such a perspective embodies a dialectical view of foundations which realises Lawvere’s intent, i.e., not to leave the “constitutive” ingredients of the axiomatisation of structures to an informal background which departs from the scientific image of the world as it is shaped by mathematics. The exploration of such a dialectical view can follow different paths. One attempt at melding the formal, possibly axiomatic, view (à la Bourbaki), of structures as autonomous end-products with a genetic method, instead of ratifying their separation, was “genetic epistemology” as conceived by Jean Piaget.20 Previously, a member of the Prague Circle, Jan Mukarowský, had suggested the similarity between structuralism (in linguistics) and the view of the set of axioms of a theory T as an implicit definition of the meaning of each primitive term of T.21 In 1949 André Weil’s appendix to part I of Lévi-Strauss’ Les Structures élémentaires de la

19 See

the excellent analysis provided by Landry (2013) of different formulations of mathematical structuralism in connection with category-theoretic foundations. 20 See Piaget (1950) and Piaget (1955). 21 See the papers collected in Mukarowský (1978). Mukarowský also noted in each case a holistic philosophy at work.

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parenté appeared22 and in 1968 other traits d’union were made accessible to even a non-academic audience through a successful survey book by Piaget.23 Piaget’s commitment to overcome the divide between mathematical and nonmathematical structuralism met specific obstacles. The way he approaches logic (actually, only classical propositional logic) as a symbolic end-product, in algebraic form, of non-symbolic interaction patterns could hardly be accepted by Bourbaki. Even less acceptable would be Piaget’s dialectical interpretation of the growth of structure, with its focus on cognitive development, which Frege would have censured as “psychologism”. On the other hand, according to the structuralist Piaget, such growth is no return to a form of subjectivism: it is rather a structural dynamics to be embedded into the “biology of cognition”, far from the ontological neutrality of Bourbaki’s view. Besides, Piaget established an analogy between the historical development of mathematics and cognitive development, through what he named as abstraction réfléchissante, as an ascent to increasing structure completion (closure under more and more symmetries) and an ascent to increasing abstraction by self-reference, so that what comes first in conceptual order comes last in consciousness. The analogy rested upon a simplification on both sides as if stages of structuration, be they ontogenetic (in psychology) or historical (in mathematics), were governed by one global meta-structure, covering any sort of problem against “modal”,24 domainspecific, structure. In either case we have counter-evidence.25 Nonetheless, the idea of common developmental patterns deserves further attention if we wish to make “architecture” relevant for mathematical education as was in fact Bourbaki’s intention.

5 Back to the Golden Age? The footprint of structuralism in mid 20th century continental Europe was broad enough to entitle us to identify the period from 1945 to the early seventies as its “golden age”, with applications to almost any philosophical topic and an intertwining with major trends in European philosophy. There was an extended debate on which sort of structure was appropriate for each specific subject, and on the nature of a general framework capable of embracing such a variety of domain-dependent structures.26 Even when of doubtful solidity, that debate was a mark of the Zeitgeist and it 22 See

Lévi-Strauss (1949). (1968). Only later, Piaget realised the need of category-theoretic notions to deal with evolving structures, but in the lack of the notion of adjunction he could not account for the universality of “free” structures, see Peruzzi (1980). 24 Here, “modal” is intended not in sense of logic but in the sense of perceptology. 25 For a list of experimental outcomes, see Karmiloff-Smith (1992), with also the proposal of how Piagetian tenets can be revised. 26 See Dosse (1992) for a detailed chronicle, with prevailing (understandably so) attention for debates in France. 23 Piaget

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affected not only linguistics, anthropology and developmental psychology but also a wide range of other areas, from semiotics (Roland Barthes, Umberto Eco, Algirdas Julius Greimas, Jurij Lotman), to psychoanalysis (Jean Lacan, Julia Kristeva), from sociology and political sciences (Luis Althusser, Michel Foucault27 ), to literary criticism (Gerard Génette, Marcello Pagnini), while gestalt psychology as well as the field-theoretic model of contemporary physics were taken as further confirmation instances. During the seventies the golden age came to an end and it was not long before the new generation of European graduate students in core areas of philosophy as epistemology, philosophy of language and philosophy of science, could conduct their research in total ignorance of structuralism, apart from the obvious exception of semiotics-oriented research, whose lexicon, at any rate, was no longer paradigmatic but secondary with respect to the analytic and the hermeneutic lexicon. In fact, some ideas of structuralism not only survived by shedding their skin but also reached a much more precise articulation, as in the line of research pursued by Jean Petitot.28 Today, structuralism has been revived as a general view of mathematics, with relevant feedbacks on philosophy of science, within which the holistic trait of structuralism was already mainstream, from Quine’s global “web of belief” to Kuhn’s “paradigms”. It is plain that the revival of structuralism in the philosophy of mathematics has specific motivations and is based on a precise notion of structure (far from the cloud-concept recurrent in common usage). The question is: to what extent does it bypass the difficulties met by previous structuralism of non-Bourbakist species? If we compare the conceptual frameworks of structuralism and logic-driven analytic philosophy, they appear to be historically, conceptually and methodologically alien to one another, as the mere comparison of references at the end of research papers in philosophy shows over an extended period of time. Was the spread of analytic philosophy stimulated by targeting European structuralism? Did the growth of the sets/categories debate specifically concern Bourbaki’s legacy? The answers to both questions are negative. The track of past structuralism was erased exactly as most structuralists intended to erase any “genetic” issue, and in this, structuralism was just the other side of the “linguistic turn” coin. If it was not credited with possessing equal novelty, the reason lies in a globally successful advertising campaign. But if we end up restoring, in a much more sophisticated way, at least part of what was erased, how are we to deal with the debts of the original structuralism? Formal sophistication does not repay them. How are we to decide what is the import, within a purely relational setting, of a commitment to universals which are presumed to underlie any manifest, concrete, instance of structure? If this commitment is what makes category theory relevant for foundational issues, how can the existence of specific universals (say, in the forms of adjoint functors) be accounted for from a 27 Foucault’s L’archéologie du savoir, published in 1969, was originally titled L’archéologie du structuralisme. 28 See Petitot (1985). Originally influenced by Greimas, Petitot developed an original perspective by approaching the process of structure-formation in terms of differential topology and linking such approach to epistemological considerations, set aside by the customary, non-genetic, structuralism of the golden age.

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holistic point of view? It is to be hoped that the answers will not be merely pragmatic, for, were the state of the art beyond the reach of the language of structures, mind would be cast in the role of the man in the old joke in search of his keys only under the streetlight; and were it not, appeal to universals would start a regress in explanation. How, then are we to avoid this regress? Nobly abstaining from responding to this question would give rise to an objection which, mutatis mutandis, is in the same spirit as the famous reply given in 1966 by Jean Paul Sartre when, with reference to Foucault’s Les mots et le choses, he took a position on structuralism. In an interview appeared that year on L’Arc (issue 30), he bitterly insinuated that structuralism, with its formalistic and static tenets, is a regressive ideology: “il remplace le cinéma par la lanterne magique, le mouvement par une succession d’immobilités. […] Derrière l’histoire, bien entendu, c’est le marxisme qui est visé. Il s’agit de constituer une idéologie nouvelle, le dernier barrage que la bourgeoisie puisse encore dresser contre Marx.”

6 Implicitly Defined Structures There is a sense in which the loss (of what has been forgotten) can be seen as being largely compensated by what the logic-driven, analytic approach and the recursive, “generative”, bottom-up, view of structures had to offer, but in a specific, technical, sense which has a well-established history independently of the tag of “structuralism”. Tags count, but ideas are cunning, move under different tags, and we cannot be guided by tags only. Not only can the notion of mathematical structure already be seen at work in van der Waerden’s Moderne Algebra (1930–1931) but the view of axioms to which Mukarowský referred was more than a century old, making its entrance in 1818 with the Essai sur la théorie des definitions by Joseph Diaz Gergonne, and its philosophical meaning was thrust into the foreground in the dispute about geometry between Henri Poincaré and Bertrand Russell at the end of the 19th century. As that dispute was and remains instructive for one of the “debts” alluded above, it is suitable to recall that the young Russell, beyond his previous ascription of synthetic a priori status to the axioms of projective geometry, claimed29 that previous access to the meaning of primitive terms is needed in order to understand the axioms, as these are expressed by means of such terms.30 Poincaré replied that “Si on veut isoler un terme et faire abstraction des ses relations aves d’autres terms, il ne reste plus rien; ce term ne

29 See

Russell (1899). Actually, Russell replied to Poincare’s review of Russell’s Essay on the foundations of geometry, published two years before. 30 Such tenet lends itself to both an epistemic and a realist reading. Russell’s Kantian attitude at the time could allow him to overlap the two readings. This was no longer the case when Russell worked out a foundational view expectedly free from any synthetic a priori element.

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devient pas seulement indéfinissable, il deviant vide de sens”31 —a reply to be paired with Poincaré’s conventionalism about geometry. In general, early and mid 20th century structuralists did not show much familiarity with the axiomatic method but, according to Mukarowský, they could and should have regarded the axiomatic formulation of what a structure is as its implicit definition: if, as Saussure argued, meaning is role in a structure, the same applies to the meaning of “structure”. Now, neo-structuralists perfectly manage the axiomatic machinery and have at hand a formal language, that of categories and functors, to express the updating of that relational idea. The updated version, however, is not in automatic agreement with the “categorial turn” insofar as this is taken to be (according to Saunders Mac Lane) the completion of Klein’s Program,32 or taken to come closer to the inherent dialectics of reality (as “reflected” in thought, according to Lawvere): in either case, universals (and invariants) are more than a conventional device serving economy of thought; and the fact that this contrast was already a problem for Poincaré, as he admitted synthetic a priori elements rooted in mathematical intuition,33 is a clue to the questions set aside. An almost simultaneous controversy similar to the one opposing Hilbert to Russell, and one equally on the status of the axioms of a mathematical theory, occurred between Frege and Hilbert.34 Their opposite standpoints, as witnessed in their correspondence, do not exhaust the range of configurations realism and formalism can take, e.g., there is room for a structural realism no longer committed to categoricity demands as well as for a formalism with a Kantian face.35 The relevance of structuralism’s history to its present day manifestation in the philosophy of mathematics does not lie in the many forms revealed by historical analysis a philosophical standpoint can take. The relevance arises from a theoretical reason, to wit, it shows that what was missing is not achieved by entrenched formulation. It can only be achieved by enriching the framework with “morphogenetic” considerations irreducible to the two argument lines of realism and formalism, especially when they result in assigning epistemic priority to semantics over syntax, or vice versa. If this enrichment is called for to deal with questions about the meaning of primitive notions of geometry, as was in the case in both debates, Poincaré versus Russell and Frege versus Hilbert, it is also called for to resolve questions about the meaning of any notion in terms of which a foundational project is formulated, thus also about the notion of structure and the sense in which a list of axioms (with possibly axiom-schemes) can be said to define what “structure” means. 31 Poincaré

(1900) p. 78. Marquis (2009). 33 Poincaré assigned synthetic a priori status to the notion of group, as defined by the group axioms, and to the schema of arithmetic induction. 34 For the place arguments addressed by Frege and Hilbert have in the history of the theory of definition, see Peruzzi (1983). 35 Just consider the shift from the intended model of a theory to the collection (or the category) of its models), when the use of the definite articles is justified up to elementary equivalence, or isomorphism or at face value. 32 See

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In the following sections we shall see what form such enrichment can take. For the moment, suffice it to note that if the framework is not so enriched, we are left with the contemplation of a two-sided coin. Side one. Suppose, for any set of sentences, its truth-conditions determine the meaning of the notions—such as closed terms and predicates—used to express each sentence, and suppose also that for each notion there is a finite set of sentences sufficient to determine its meaning. Axiomatic systems concerning one kind-ofstructure or all possible structures are a case in point. What can be implicitly defined saves us from having to commit ourselves to any sort of intuitive acquaintance with meaning. But then the stability of notions across different axiom systems for different kinds-of-structure seems just an effect of homonymy. Side two. If truth calls for an interpretation of the language and this interpretation is inductively defined starting from the notions used by our axioms, we need to specify what the notions refer to in a suitable universe-of-discourse, which is expected to be characterised by means of other notions the meaning of which is defined by other axioms. This applies to the notion of “structure” too. But if a universe-of-discourse is considered as a structure the characterisation of which is rightly the purpose of axioms, there is a problem (and one prior to categoricity demands): we are left with the choice between being trapped in a vicious circle or in an infinite regress (the universe-of-discourse is the output of another theory’s syntax). Invoking category theory in order to deal with the most general, mathematically relevant and finally precise, notion of structure is part of this problem, not its solution. If more than one model of a theory T is possible, the set of relations which the axioms posit between the primitive notions is unable to determine the interpretation. When the same applies to a general theory of structure, the result can be tagged with a “minus”, to mark it as a failure (as Frege would mark it), but the axioms can be taken to uniquely characterise the reference of the notion in another sense, which calls for a “plus” sign, to mark it as a success, if the wider the class K of T-models is, the more valuable T-axioms are (were it not that K has to be uniquely characterised in turn). In the case of the notion of structure these two opposite perspectives coexist as the plurality of models becomes a structure on its own. Assigning both a “plus” and a “minus” is disputable (at least), thus relativisation is needed, and since plurality also concerns the characterisation of what an interpretation is, it also concerns reference and truth … relative to a structure, hence the “plus” and the “minus” are far from being absolute. Rather, (*) the plus and the minus turn out to be essentially relative to a context of reference, and finally relativisation applies to claim (*). Yet those who assert this essential relativity are supposed to understand the meaning of their assertion. Joined together, Side one and Side two condition any hope of success of the effort to understand (or, to justify our understanding in a non-Pickwickian way) what we mean by “structure”. Since the steps performed on Side one and Side two are recursively repeatable by a back-and-forth strategy, also the claim (*) receives a “plus” and a “minus”. Full-fledged relationism, once formalised and consistently applied to the notion of structure from top to bottom in language and metalanguage, has the same effect as positing a sequence of “homunculi”, each conscious of what

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the previous one means, to explain consciousness: the end of this sequence cannot be seen; at the same time their nesting is supposed to be absolutely understood and, if there were a fixpoint and we were able to avoid self-reference paradoxes, this fixpoint ought to be determined relative to the internal structure we are not able to determine. As is often the case in philosophy, such an outcome appears as skepticism-withfaith, a conjunction which however embroidered with subtleties can be expressed by saying that we cannot know what we mean but, after all, we are not speaking but spoken, say by Language, as the Structure of all structures, which is unattainable by limited instances of “speakers” such as us. Indeed, if there is no other way to approach the notion of structure apart from joining Side one and Side two in one coin as described above—so that the globally relational view is constitutive of meaningfulness—, a declaration of faith is at hand: the Whole secretes structures and this secretion is all-pervasive. We can only understand fragments of structuration in actu, as the Whole we need to posit to achieve structural closure transcends rationality. Those who find skepticism-with-faith unpalatable would presumably revert to more manageable questions. How can a set of axioms uniquely determine a specific kind-of -structure? How can a set of notions mutually linked by axioms determine the meaning of each notion? An answer was advanced by Alessandro Padoa in his talk at the International Congress of Philosophy, held in Paris in 1900: lack of implicit definability implies the lack of explicit definability. This is “Padoa’s Criterion” and its converse is the “Beth Property”. Both are provable in standard model theory for first order languages endowed with classical logic and their generalisations to other languages has been widely investigated. Here, a key issue concerns which kind of equivalence relation is considered between models. The notion of isomorphism being not first-order within set-theoretic semantics, if structure is intended set-theoretically, and the set theory we use is first-order, the questions above find no satisfactory answer (as already suggested by Skolem’s Paradox). At best, we could try to sandwich the notion of structure between a theory of structures axiomatically presented and the phenomenology of structures in mathematical practice. So doing brings us back to the problematic coin (Side one - Side two). More refined answers to definability questions are obtained by passing to the category Mod(T) of models of a theory T, rather than just the class K of its models. Once we describe a theory as a category, T-models become functors and morphisms in Mod(T) are T-preserving maps. However, when T is the intended theory of structures, refinement is not sufficient to resolve the Side one—Side two conundrum: the structure grounding problem remains unsolved.

7 Structure Schemes Meta-structural issues like those referred to in Sect. 3 were not available to the Bourbakists. Also, if two foundational frameworks were available and they were equivalent with respect to the body of mathematical knowledge to be accounted for, but different in the structures mères they posit, the choice would be merely pragmatic,

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in the same sense argued by Poincaré for geometry—then, what has to be taken as a “basic” kind of structure depends of the overall usability of the framework in which it is basic. A minimal elaboration of these two remarks invites us to be generous: Bourbakists cannot be blamed for the unfaced risk of circularity, or regress. Meanwhile, as far as concerns philosophy of science, apart from an idea advanced by Carnap which remained undeveloped,36 logical empiricism had set aside model-theoretic issues as those hinted at the end of the last section, in favour of only one kind of equivalence relation between models, i.e., empirical equivalence, accompanied by a sudden demand for consistency with not only a given body of evidence but with any possible body of evidence, thus by reference to a totality which is beyond the reach of empiricism. In the light of a “structural” approach to scientific theories as proposed by Joseph Sneed and Wolfgang Stegmüller, the effect of the circularity mentioned above was even more extensive, since the notion of a network of models calls for a theory of its own, and, under holistic assumptions, the same network view would apply to mathematical theories, thus to the notion of set as well, whereas set-theoretic notions were taken at face value. Substantial progress on the matter might have been made by exploiting the refinements indicated in the last paragraph of Sect. 5. This did not happen, but even if it had, the questions alluded to would have remained unanswered, and if we suppose that the boundary conditions of a refined architecture of mathematics are only of pragmatic, postmodern, holistic, deconstructionist nature, the structure grounding problem is not merely unsolved, but unsolvable (our rational warrants in the land of structures are surrounded by skepticism-with-faith), since the claim that structuralism is suitable for mathematics only requires a justification prevented by epistemic holism; otherwise (if holism is rejected) a reason should be provided for the new form of structuralism in mathematics to be extensible to the whole of knowledge. Then we would be committed to solve the structure grounding problem, but no solution can be achieved within the skepticism-with-faith scenario. By setting aside such quandaries, what became the mainstream view was one aspect of early structuralism, to wit, the relational characterisation of any notion (vs. bottom-up construction) married to praxis-oriented conventionalism (vs. the old idea of mathematical truth ultimately founded on intuition or pure rationality): the verités de raison became conventions to be selected by indispensability arguments together with criteria of economy and empirical usability. This marriage was supported throughout the 19th century by the ascent of mathematics towards “abstract” structure and to formalism. The outcome was that philosophy (of mathematics) turned out to be what Jean Dieudonné ironically characterised as a metatheoretical extravaganza to be indulged in only on Sundays, having no impact whatsoever on the weekly activity of the working mathematician. It was a “two-job” marriage. We were issued a sedative for our ontological and epistemological turmoils, for it was in the name of freedom (we are free to adopt any conventions we choose) and 36 See

Awodey and Carus (2001).

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pragmatic wisdom (we are only expected to choose what we take as useful), against the presumption of mirroring reality or relying on a mandatory epistemic footprint, as if there were no intrinsic constraint on the range of structures accessible to us. Moreover, the notion of structure was regarded as the proper tool for taking on the task of shielding any talk of what mathematics is about from “brute” materialism, from the threat of “psychologism” or any other external impurities arising from social and historical factors, as well as from the need to assume that essences are grasped by some sort of intuition and, finally, from the threat of being trapped in interpretative issues such as those concerning Kant’s legacy in epistemology, e.g. in which way “forms-of-intuition” (space, time and motion), “categories” and “schemes” are related to each other to make knowledge possible. In so doing an inner tension came equally to the surface, for, though Bourbaki’s structures mères were three, algebra came first in practice and as Vladimir Arnold ironically remarked, in contrast to the importance Isaac Barrow assigned to figures, “Bourbaki writes with some scorn of Barrow that in his book in a hundred pages of text there are about 180 drawings. Concerning Bourbaki’s books it can be said that in a thousand pages there is not one drawing, and it is not at all clear which is worse”.37 To prevent the impression that we speak of “unpaid debts” of structuralism while the previous reference to “intuitive” aspects of understanding the notion of structure is unjustifiably taken to be transparent, we have to admit that (A) reference to intuition looks old-fashioned indeed in the age of (neuro-)cognitive sciences; (B) certain previous remarks seem to suggest that we have implicitly superimposed a Kantian view on a perspective which, as far as it is structuralist, is presumed to be epistemologically, and ontologically, neutral; (C) if there are structures mères which convey patterns of form and content, no way-out à la Cassirer is at hand. “Intuitive” notions which may count as constituents of both an updated Aesthetics and an updated Analytics (with reference to the Critique of Pure Reason) are here intended as projections of schemes, so that the schemes Kant designated as mediating between mutually independent “pure” intuitions and “pure” concepts constitute the primary toolbox. Thus there is no commitment to intuition of concepts, as already argued by Kant, but also no commitment to intuition of space as such. Rather, schemes are associated with different basic patterns of kinesthetic configurations, and such patterns, inherent in bodily experience, provide the grounding of structures mères. In this way the window of possible formal set-ups shrinks to the window made possible by the contentful presuppositions required to understand form. This all conforms with Bourbaki’s view of mathematics as disconnected from physical reality. Two motivations for that view can be found in André Weil’s idea of the dharma, with the related injunction to be free of (inessential) individuality—mirrored in the choice of a collective name—, and the influence the philosophy of Émile Meyerson had on Chevalley: pure mathematics as shelter, highland of supreme stability, from the chaos of the external world, as if the material roots of abstraction could be entirely ripped off and the “order of the world” which made them possible could leave no trace in any accessible kind-of-structure. 37 Arnold

(1990) p. 20.

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If such schemes exist, structures mères can also exist, and if there are structures mères, they can be nothing more than kinds-of-structures. They are what they do and they can do it only as specific structures of different kinds endowed with the power to generate other structures through no less specific patterns of mutual integration. We have a meter stick to test this generative power, namely, universality in the sense of category theory, thus by means of adjunctions, possibly iterated and generalised. With hindsight, one can see that Bourbaki’s oeuvre suffered from the lack of the concept of category, which in fact remained outside the scope of Bourbaki’s “paquet abstract”.38 But a still more severe lack was that the concept of universality was not dealt with in its full generality. Similarly, while Bourbaki failed to provide a proper treatment of logic, the lack of a “structural” presentation of logical notions was even more sorely felt. The freedom involved in an infinite list of arbitrary choices to be collected together allows for the direct formation of specific sets, but free structures, such as free groups, are principled: they come with an associated universality condition to be satisfied and are identified, up to isomorphism, by suitable adjoint functors; thus, even though the generic model of a theory can be thought of as an “ideal element” in Hilbert’s sense, it is uniquely determined (up to isomorphism)—starting with the category of models of a geometric theory—by such conditions and thus it is anything but arbitrary. On the other hand, it would be unfair not to acknowledge that Bourbaki does essentially define the concept of freely generated algebra in terms of universal mappings. What is not recognised and made fruitful is that the concept is an example of the general concept of adjunction, as defined by Daniel Kan, and yet Grothendieck was one of the first to make profitable use of the notion. “Free objects” are byproducts of the existence of an adjoint to a forgetful functor. In 1956 Pierre Cartier had proposed the introduction of categories as an alternative foundation, but the proposal was rejected, as stated on the June issue of La Tribu, n. 39, of the same year, in favour of an extension—of the system already envisaged— which could host categories and functors, and, to this aim, NBG seemed (“à premier vue”) to them a better option than ZFC. On the same n. 39, after saying they are by now convinced that category theory is important, a concrete plan is made explicit, together with the typically Bourbakist assignment of internal tasks. The plan concerns precisely a further Chap. 5 of the set theory volume, after Chap. 4 on structures: “Chap. 5 (Catégories et foncteurs)—Pour commencer Grothendieck rédigera une espèce de Fascicule de Résultats en style naïf, afin que Bourbaki se rende compte de ce qu’il est utile de pouvoir faire. On formalisera ensuite.”.39 Outside of mathematics, as already noted, only certain versions of the XXth century’s European structuralism made reference to universals (in phonology, developmental psychology and anthropology) and the existence of such universals was not 38 See

Anacona et al. (2014) for the motivations which blocked the Bourbakists to enrich their framework with category-theoretic tools, even after Grothendieck made massive and essential use of them in algebraic geometry. 39 As rightly noted by Corry, the work of Charles Ehresmann drew an original connection between structures and categories, but it remained aside of the Éléments.

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and could not be justified by relational considerations alone, as long as one adhered to Saussure’s original approach, which had been popularised through the interpretation of ethnolinguistic data as later suggested by Edward Sapir and Benjamin Lee Whorf. The majority of those who around the mid 20th century promoted structuralism either explicitly denied universals or disregarded the task of formalising the notion (so much so that the so-called “post-structuralist” movement had no need to target the existence of structure-crossing universals). The inner tension of structuralism, as universals-oriented or universals-free, was not resolved. Provided we are not guided solely by tags, we can see that other paths of 20th century philosophy show the same unresolved tension between stances of structuralist orientation which admit cross-domain concepts—which, being trans-categorial, had been named “transcendental” in the Middle Ages—and stances which deny or abstain from committing to them. In this connection one could mention Wittgenstein’s idea of language games, Quine’s belief holism, Gadamer’s concept of the hermeneutical circle, Nelson Goodman’s world-versions. Their common emphasis is on a particular point: the set (system, network) RS of mutual links in a whole S identifies the content of each item (state, node) X, with the role X has in S (or in S plus context-relative parameters which are left to be freely variable) so that, as Saussure anticipated, meaning is drained from the system. Much less or null emphasis is placed on the fact that the words used to express this “role” can only exploit notions which cross the boundaries of each language game, paradigmatic setting of knowledge, tradition’s horizon, and world-version, while arguments in support of the resulting cloud-concept of structure rely on a supposedly absolute divide between (I) a notion of compositionality according to which the generators are pointlike, tailored to only one of the possible implementations of the notion, i.e. that which takes set theory as the only provider of a rigorous, extensional semantics, and (II) a free-floating notion of relationality, the features of which are essentially context-dependent, with the exception that a relation is… a relation, and the context is… the context, thus structure-independent, contrary to the assumptions. Forms of relational ontology had already been explored in antiquity, for instance by the Stoics. It was only modern times, however, that saw the emergence of the view of individuals as sets of properties, and of each property as identifiable only in relation with a collection of other properties. The idea of axioms as implicit definitions was, in a way, the culmination of this line of thought and it is revealing that the 20th century philosophy of mathematics, language and knowledge start with the assertion of this idea, but the novelty was that finally a logic of relations (together with relation algebras) was reached: an outstanding progress, the recognition of which does not prevent the acknowledgment that it was achieved in a non-relational formal setting. Yet structuralists of the golden age made scarce and inefficient use of such progress, and Bourbaki’s concept of structure mères did not provide a way of unifying compositionality and relationality. As a consequence, the opposition between (I) and (II) persisted. Such unification only became possible once adjoint functors proved to be the key to “universals”. On the other hand, since category theory by itself prescribes no structure mère, in the end it seems we can only say that

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being “mother” or not is a contingent, pragmatic issue. If indeed that is the case, we come back to the picture of Sect. 3. If more can be said, it is because of the schemes as laid out above.

8 The Wrapped Dimension To gain an idea of what this substantive notion of structure mère can mean, let us take a side road and consider the etymology of “structure”, at the same time remaining fully aware that in so doing we risk introducing an off-topic gadget. Latin has the verb struere, meaning to build, pile up, with structum as past participle. Also English has the verb strew, curiously suggesting a converse action of deploying: to spread, spill, disperse, scatter, disseminate. Both come from a common proto-indoeuropean root *ster[e]—(its tracks are also in the Greek sternon, breast, Latin sternere, to stretch, Russian stroji, order, German Stern, and in many words of common use, as strategy, (sub-)stratum, stereophonic. In Latin there are various nominal forms associated with the past participle structum. Besides in-structio and de-structio, two forms matter here. One is costructio, meaning the action of building something by assembling various pieces (cum), but also meaning the outcome of that action, thus referring both to a kind of process and a kind of object arising from that process. The other form is structura, nominalisation of the future participle of struere, meaning something which is going to be constructed and expected to be completed in the next future,40 thus a sort of anticipated object, the shape of which we are able to envisage independently of actual completion. Both the resulting notions, of construction and structure, proved to be the source of innumerable metaphorizations and acquired an increasingly abstract sense, which also caused the two to diverge, e.g., in the philosophy of mathematics, and if it is worthwhile to recall that they were born linked to each other, the reason is a philosophical one. Mac Lane’s remark that topos theory allows Brouwer to reconcile his mathematical work (in topology) with his philosophy (intuitionism) is a case in point: here the original link is recovered, revealing yet another gap in Bourbaki’s tendency to eschew a properly structuralist view of logic, since different logical structures are associated with different truth-value objects, whereas the primacy of place assigned to Boolean algebra41 blocks the reconciliation to which Mac Lane refers. In a similar but less precise way, Piaget tried to recover the original link between structure and construction: “There is no structure without a construction, abstract or generic”.42

40 The

very noun “future”, from Latin futurus, is a perfect fossil of future participle!

41 On this issue, see Anacona et al. (2014) Sect. 4. As is well known, categorial logic also deals with

logical systems which do not require the topos structure to provide their semantics. Peruzzi (1980).

42 See

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Now, what-is-going-to-be-structured makes tacit reference to the notion of path, metaphorically lifted from physical space to the phase space of a system (change as motion from a state to another), and to the notion of action, metaphorically lifted from what can be performed on physical bodies to what can be performed on abstract objects to be structured. Likewise, in positing structures mères, tacit appeal is made to basic patterns of action, out of which any con-struction is accessible to understanding. The algebra of actions on a structure forms a structure on its own, one which can be identified with an agent; and there can be many agents which can be collected together to form yet another and another structure… This has nothing to do with idealism, as the agents (and in particular, the “con-structors”) can be human beings or other sorts of organisms or things. Any philosophy which relies, exclusively or primarily, on the analysis of language, be it ordinary or formalised, which accepts its lexicon as a history-free gift, is of no help in identifying patterns of conceptual structure because these are all mixed up, in current usage, with their byproducts, so that the variety of historical sediments of path/action transfers appear conjoined with their sources. We are aided in carrying out the task of identifying such patterns by objectifying the generating structureschemes in mathematical language, so that the primitive notions used in identifying the basic kind-of-structures and how they combine mirror their own roots. Rather than considering these roots as inaccessible to mathematics, mathematics is required to identify them and to make the dialectics of structure and construction explicit: this means taking compositional and structural constraints into account, both of which are two-faced, bottom-up and top-down. Recursion itself, in its various forms, presupposes a back-ground structure which can support both the base and the inductive steps. The features of different recursive hierarchies depend on which back-ground is considered, but the stability of this background is in turn the outcome of processes amenable to mathematical description. If we take static, formal axiomatics as endproducts and simply draw a line there, we are left with a frozen ontology and its counterpart, a cryologic, and if each of them is now plural, the conditions for this plurality cannot be found in the widest phenomenology of ontological systems and formal logics. From the 17th century up to the present day, science has progressively shed all static views of nature. A similar change is at hand with the category-theoretic approach to mathematical structures: it makes manifest the dialectics described above and enriches it by taking into account local and global, as well as internal and external, aspects of any given structure. Abstraction is needed here as well to formalise the principles of this twofold dialectics, but, as anticipated, it has to be fine-tuned. For example, the “logical structure” of a theory T formulated in a language L rests on the L-atomic formulae, and is generated by the primitive notions of T. The standard formalisation of logic collapses basic patterns of thought in parallel with the “flattening” induced by the general notion of predicate, of the differences between states and processes in moulding units of meaning, leading to the assimilation of actions, kinds, and static or dynamic, qualitative or relational states. In the same vein, abstract sets continue to support the structures mères instead of constituting a special case of these. It is no surprise, accordingly, that through this overlapping of

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linguistic and ontological flattenings, formal semantics appears to be independent of space and motion, as these are alien to “pure” logic. Remarks such as these are often made to serve the cause of a philosophy which insists that all formal languages have inherent limits and shuns the use of mathematical tools. In the present case, they rather serve the opposite cause, starting with the elimination of flattenings of conceptual structure and with making its basic construction patterns explicit. Consonant with this aim are ideas already present in Charles Fillmore’s contribution to “structural semantics”,43 and even before this, the restriction to a specific domain found in Vladimir Propp’s morphology of fairy tales, as elaborated by the Russian Formalists. In both cases there are structures mères in the above sense, but, since they concern narrative scripts, can be the source of confusion. The point is that Fillmore reenacted case grammar, with a focus on “frames” as basic units of sentence structure, and his work was seminal for the project of cognitive linguistics (as pioneered by George Lakoff, Ronald Langacker and Leonard Talmy), which brought to the fore the vast phenomenology of spatial patterns in language. Analysis of logical form was not, in principle, beyond the scope of this project, though in fact it was dealt with only marginally, echoing Bourbaki’s indifference to issues of logical form which proved to be central to analytic philosophy. The link between meaning-gestalts and structures mères still awaits systematic investigation, notwithstanding the fact that the very lexicon of grammar and logical syntax provides a rich source of lifted spatial frames, as witnessed by terms such as subject, object, adective, preposition, inference, deduction, as well as notions such as individual, variable, connective not to mention that of a map from … to… Emphasising this pervasive “spatiality” often elicits the reaction that, with regard to logic and mathematics, this “spatiality” is nothing more than a manner of speaking and has no specific, and indispensable, place in foundational studies. The situation is similar to the 18th century’s debate (Buffon, Hutton) on fossils of an extinct aquatic species found on mountains: in that case the theory of evolution and geology provided an explanation, whereas in the present case it seems no explanation is needed or that it is devoid of mathematical relevance. But any formal language retains a trace of the intuitive, dynamic, “spatiality”, and this track is indispensable in understanding the extraction of meaning in abstraction.44 As this extraction is constantly presupposed while remaining unanalysed, all talk of structures contains what we shall call a “wrapped” or “concealed” dimension. This dimension can be teased out by joint examination of the morphogenesis of structure and the structure of morphogenesis, against the (opposite) sense Hilbert and Husserl gave to the separation of the axiomatic and genetic methods. The analysis of this wrapped dimension affects the questions posed by structuralism and its debts. Here the figure/ground pattern, associated with position and motion, change and stability (under a group of actions, say) below our “abs-tract” concepts of object and map, enters the picture, and this and other patterns affect our choice of the sort of formal language we use to deal with variable structures. 43 See 44 See

Fillmore (1968). Peruzzi (1994).

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No matter how “neutral” our discourse is intended to be, from the very start we resort to patterns of kinesthetic nature in conjunction with procedures by which something of macro-size is identified as stable, although its micro-components undergo variations, and used to fill the slots of each pattern. It is in connection with the success of such procedures we use notions such as in/out, from/to, up/down, but it’s the dynamics of the components which allows for the stability of that entity we ascribe a noun or a proper name. Such patterns are so space-laden and motion-laden that, if we were to attempt a global suspension (as a supercharged Husserl’s epoché) of such an alphabet of meaning, no syntactic structure would remain to support the process of abstraction. If structures mères exist, they can be found at work right in this dimension as soon as it is “unwrapped”. In recovering original space and motion patterns, it is not that topological structure ought to come first, for topological knowledge can emerge only through notions concerning order and composition of operations. Structures mères arise simultaneously (as Bourbaki suggested) and this also applies to their principal relationships, as well as their mutual integration, both of which are expressible in functorial language. Clearly, our work remains unfinished. Research continues on the way universal patterns of structure-and-construction can be precisely correlated with schemes of kinesthetic intuition—which ground, and format, the meaning of any sentence by integrating structures mères with each other. Indeed, this genetic-axiomatic perspective raises further questions which remain unanswered here, of concern for the working mathematician and for the working philosopher, as such schemes, rather than pointing to a disembodied “spatiality”, reveal the most concrete, and direct, material sources of structure for us as they act in structuring thought. As for Bourbaki’s structures mères, let me make two concluding remarks. First, the existence of structures mères is supported by the above perspective only if they are constructed independently of ∈-based set theory. Second, such support is at odds with the formalistic spirit of structuralism. The first remark leads us out of Bourbaki’s actual axiomatic framework and is in perfect agreement with category-theoretic foundations; the second remark makes the present genetic-axiomatic perspective different from most versions of structuralism, in particular when its neutrality takes the form of a type-theoretical translation. But what is the import of category theory for the structure grounding problem when structures mères are conceived within such a perspective? Were it not for an opinion about category theory that is still widespread, it should go without saying that no theory is a language, and no language, by itself, is a specific theory. Of course, any choice of language constrains the range of theories expressible in it and any theory exploits the resources offered by the language in which it is expressed. This applies, in particular, to a language and a theory of structures in which objects (structures) and maps (between structures) are taken at par, in which objects-ofmaps (by exponentials) can be formed, and in which objects can be taken to index or parametrise other objects; thus this applies to category theory as well. Category theory is not enjoined to identify the structures mères, nor is it required to specify their exact status, whether epistemological or ontological; nevertheless,

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category theory allows one to make the notions of genericity and ideality (à la Hilbert) precise in terms of “universal mapping properties”. Category theory does not identify the actual “generators”, so to speak, of meaningunderstanding, nor does it endorse the commitments of previous foundational projects such as logicism or intuitionism; but, category theory is necessary in that it constrains the range of such “generators” by means of universality conditions, and, in the present perspective, these formal constraints have to conform with content constraints provided by basic patterns of meaning. This conformity lies within the architecture of mathematics, and in particular within an architecture based on structures, insofar as that architecture is made possible by the reflexive objectification of kinesthetic schemes. This being the case, what is necessary also determines what needs to be added to solve the structure grounding problem.

References Anacona, M., Arboleda, L. C., & Pérez-Fernandez, F. J. (2014). On Bourbaki axiomatic system for set theory. Synthese, 191, 4069–4098. Arnold, V. (1990). Huygens and Barrow. Birkhäuser, Basel: Newton and Hooke. Awodey, S., & Carus, A. W. (2001). Carnap, completeness and categoricity. The Gabelbarkeitssatz of 1928. Erkenntnis, 54, 145–172. Bell, J. (1986). From absolute to local mathematics. Synthese, 69, 409–426. Bourbaki, N. (1948). L’architecture des Mathématiques. In F. Le Lionnais (Ed.), Les grands courants de la pensée mathématique (pp. 35–47). Marseille: Cahiers du Sud. Bourbaki, N. (1949). Foundations of mathematics for the working mathematician. Journal of Symbolic Logic, 14, 1–8. Cassirer, E. (1945). Structuralism in modern linguistics. Word, 1, 99–120. Corry, L. (1992). Nicolas Bourbaki and the concept of mathematical structure. Synthese, 92, 315– 348. de Saussure, F. (1971). Cours de linguistique générale, edited by F. Bailly, A. Sechehaye and A. Riedlinger. Paris: Payot. Dosse, F. (1992). Histoire du structuralisme: Le champ du signe, 1945–1966. Paris: La Découverte. Fillmore, C. (1968). The case for case. In E. Bach, R. T. Harms (Eds.), Universals in Linguistic Theory (pp. 1–25). Holt, Rineheart and Winston (Part two). Hellman, G. (2001). Three varieties of mathematical structuralism. Philosophia Mathematica, 9, 184–211. Karmiloff-Smith, A. (1992). Beyond modularity: A developmental perspective on cognitive science. Boston: MIT Press. Landry, E. (2013). The genetic versus the axiomatic method: responding to Feferman 1977. The Review of Symbolic Logic, 6, 24–50. Landry, E., & Marquis, J. P. (2005). Categories in context, historical, foundational and philosophical. Philosophia Mathematica, 13, 1–43. Lawvere, F. W. (1964). An elementary theory of the category of sets. In Proceedings of the National Academy of Science of the U.S.A (Vol. 52, pp. 1506–1511). For an extended reprint, with commentary by C. McLarty, see Reprints in Theory and Applications of Categories (Vol. 11, pp. 1–35) (2005). Lawvere, F. W. (1969). Adjointness in foundations. Dialectica, 23, 281–296.

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Lawvere, F. W. (1994). Cohesive toposes and Cantor’s “lauter Einsen”. Philosophia Mathematica, 3, 5–15. Lévi-Strauss, C. (1949). Les structures élémentaires de la parenté. Paris: PUF. Lolli, G. (2019) THIS VOLUME. Mac Lane, S. (1971). Categories for the working mathematician. Berlin: Springer. Marquis, J.-P. (2009). From a geometrical point of view: A study in the history and philosophy of category theory. New York: Springer. Maschaal, M. (2000). Bourbaki, une société secrète de mathématiciens. Paris: Belin -Pour la science. Mathias, A. (2015). In lode della logica. La Matematica nella Società e nella Cultura, Rivista dell’Unione Matematica Italiana, 8, 43–74. McLarty, C. (1991). Axiomatizing a category of categories. Journal of Symbolic Logic, 56, 1243– 1260. McLarty, C. (2004). Exploring categorical structuralism. Philosophia Mathematica, 12, 37–53. Mukarowský, J. (1978). Structure, sign and function. New Haven: Yale University Press. Peruzzi, A. (1980). Jean Piaget e l’epistemologia. Antologia Vieusseux, 58, 21–32. Peruzzi, A. (1983). Definizioni. La cartografia dei concetti, Franco Angeli, Milano (Errata corrige on www.academia.edu.AlbertoPeruzzi). Peruzzi, A. (1994). Constraints on universals. In R. Casati, B. Smith, & G. White (Eds.), Philosophy and the cognitive sciences (pp. 357–370). Vienna: Hölder-Pichler-Tempsky. Peruzzi, A. (1996). Debiti dello strutturalismo. Teoria, 16, 21–63. Peruzzi, A. (2000). Anterior future. Rendiconti del Circolo Matematico di Palermo, 64, 227–248. Peruzzi, A. (2018). Delle categorie. Firenze: Edizioni Via Laura. Petitot, J. (1985). Morphogenèse du sens. Paris: PUF. Piaget, J. (1950). Introduction à l’épistémologie génétique. Tome I: La pensée mathématique. Paris: PUF. Piaget, J. (1955). Les structures mathématiques et les structures opératoires de l’intelligence. L’enseignement des mathématiques. Nouvelles Perspectives (pp. 11–33). Paris: Delachaux Niestlé, Neuchâtel et. Piaget, J. (1968). Le structuralisme. Paris: PUF. Poincaré, J. (1900). Sur les principes de la géométrie. Réponse à M. Russell. Revue de Métaphysique et de Morale, 8, 73–86. Resnik, M. (1982). Mathematics as a science of patterns: epistemology. Nous, 16, 95–105. Russell, B. (1899). Sur les axioms de la géométrie. Revue de Métaphysique et de Morale, 7, 684–707. Shapiro, S. (1997). Philosophy of mathematics: structure and ontology. Oxford: Oxford University Press.

The Basic Structures of Motor Cognition Silvano Zipoli Caiani

Abstract One of the main issues in the philosophy of cognitive science is to understand how natural systems represent and manipulate informational content. This chapter focuses on the representational formats that are involved in action cognition, that is, the structures that informational contents can take to guide our actions. Contrary to a common view, I argue that there are different formats by which information can take part in cognition and that this difference in the structure of contents allows relevant cognitive abilities, such as the planning and execution of practical skills, to be accounted for.

1 Introduction One of the main issues in the philosophy of cognitive science is to understand how natural systems represent and manipulate informational content. This issue can be addressed from several points of view. For example, many are interested in clarifying the notion of informational content to find a place in nature for it (e.g., Dretske 1995; Fodor 1990; Millikan 2009), whereas others are concerned with the computational architecture linking sensory input to behavioral output (e.g., Marr 2010; Horst 2011; Jeannerod 2006). This chapter focuses on the representational formats that are involved in cognition, that is, the structures that informational contents can take to deliver information. Contrary to a common view, I argue that there are different formats by which information can take part in cognition and that this difference in the structure of contents allows relevant cognitive abilities, such as the planning and execution of practical skills, to be accounted for. To defend this claim, I proceed as follows. In Sect. 3, I present the common view according to which cognition involves only propositional representations, that is, S. Zipoli Caiani (B) Department of Humanities, Università degli Studi di Firenze, Firenze, Italy e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 A. Peruzzi and S. Zipoli Caiani (eds.), Structures Mères: Semantics, Mathematics, and Cognitive Science, Studies in Applied Philosophy, Epistemology and Rational Ethics 57, https://doi.org/10.1007/978-3-030-51821-9_6

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intentional mental states that convey informational content endowed with a propositional format (Davidson 1970; Searle 1983; Williamson 2017). Let call this conception “propositionalism”. The main reason to endorse propositionalism is that the propositional format of mental contents allows for the rationality of thought to be conceived as a logical calculus on mental representations, with the relevant consequence of making actions as rational consequences of intentions. Accordingly, if propositionalism is true, the propositional format of our intentions would be suitable for providing the information needed to select and execute the intended action. Then, in Sect. 4, I show that a propositional format of representation can be rendered as a true or false predicate-argument structure. This view allows the structure of an intention to be represented as a relation between type-free, independent variables, so that the intention to perform an action in the environment can be conceived as a link among the agent, an object and a motor goal. However, despite the perfect fit of conceiving action intentions as predicate-argument structures with propositionalism, such an account suffers from issues that are not easy to solve. I show that there are at least two problems whose solutions are not within the reach of the classical view. The first issue concerns the prescriptive role of action intentions. Notably, the predicate-argument structure alone does not explain how an intention to execute an action determines the series of motor acts that constitute the action. More precisely, the classical view is not able to account for how a propositional description of a motor skill hangs up with the right set of bodily movements that allows the execution of that skill. The second issue concerns the empirical evidence concerning the functioning of the agent’s motor system. Evidence from cognitive neuroscience has indeed revealed that the motor and pre-motor cortexes contain a system of functional states that are causally involved in planning and executing already structured and goal-oriented sequences of motor acts instead of broadly conceived action concepts (e.g., Rizzolatti et al. 1998). According to this evidence, it has become common to consider the cortical motor system as a sort of ‘vocabulary’ where each word conveys the representation of an already structured chain of movements that fits with the agent’s intention to execute an action. Now, inasmuch as we consider the states of the motor system as conveying information about already structured motor acts, they should be conceived as mental states with a prescriptive format rather than a descriptive one. Based on these considerations, in Sect. 4, I provide an account of action intentions using a non-predicative interpretation of the action concepts in terms of motor acts. My strategy is to avoid trouble with propositionalism by suggesting that the action concepts that constitute an intention to act can be interpreted as a function linking an initial bodily state to a goal state. Such a structure resembles that of a morphism, which configures a dynamical change in the agent’s body (or a part of it), moving it from an initial shape to a final shape. Such a morphism represents the execution of a motor act, which in turn may be suitable or unsuitable to make true a propositional description of the action intention. I conclude by arguing for a hierarchy of mental representations in motor cognition, according to which a propositional intention is reliant on the structure of the

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embedded action concept. This involves that the truth value of the propositional content of an intention depends on the specific motor act denoted by the action concept. Now, since a motor act can be conceived as a morphism defined on an object, and since the same motor act can be realized by different types of morphisms, the truth value of a propositional intention changes because of the specific morphism denoted by the embedded action concept. It will be clear, at the end, that action concepts have a non-propositional structure, which however can be part of a propositional structure.

2 Propositionalism According to a common assumption in theoretical cognitive science, cognitive systems represent the environment by means of representational states that function as a medium between a sensory stimulus and the behavioral response (e.g., Haugeland 1978; Pylyshyn 1986). Following this representational theory of mind, the mental states of an agent can be seen as symbolic structures that convey information in a propositional format (Davidson 1970; Fodor 1980; Searle 1983; Williamson 2017). Propositions are commonly treated as informational vehicles of, to use ordinary terminology, the semantic contents of mental states and thus are taken to be central to the understanding of our cognitive abilities. Thus, it is possible to say that the agent s believes or perceives or desires that p, by ascribing s with the relevant mental state, whose content is precisely the proposition “p”. Now, since states such as believing, perceiving and desiring are paired with propositions and since they are commonly considered to be paradigmatic examples of mental states, it seems to be an intuitive step to hypothesize that all mental states should be construed as relations to propositional contents. Such a view, also known as “propositionalism”, is so widely accepted among cognitive scientists and philosophers of mind that the expressions “mental content” and “propositional content” are often used interchangeably. Indeed, it is quite common that, after having adhered to the representational theory of mind, scientists and philosophers take for granted the propositional format of mental representations, this without any consideration of the question as to whether there are non-propositional mental states (Campbell 2018; Grzankowski 2016; Montague 2007; Sinhababu 2015). Scholars are attracted to propositionalism for a variety of reasons, including mainly the desire to provide an adequate connection between reasons and actions. Indeed, one of the main advantages of conceiving cognitive systems in terms of mental states with a propositional content is that this move allows for modeling the thought of an agent as a propositional calculus. As reasoners, indeed, we make rational thoughts by inferring conclusions from premises according to rules, which are plausibly characterized as formal schemes, such as “if p then q; p, therefore q”, “p or q; non-q, therefore p”, etc., where p and q are symbolic propositional structures. Accordingly, if rationality is characterized by patterns of formal relations between propositions, and humans are reasoners, it seems prima facie that we do a better

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job capturing the rational thought of humans by attributing them propositional states (Fodor 1980; Searle 2001; Sinhababu 2015; Stanley and Williamson 2001). Remarkably, propositionalism has relevant consequences on the attempt to provide an explanation for the agent’s motor behavior. According to this view, the rational explanation of an agent motor behavior can be provided only by means of patterns of propositional representations, inasmuch as propositions are considered the only type of mental contents involved in the modeling of rational thought (Fodor 1980; Searle 2001; Stanley and Williamson 2001). Following this line, a certain agent’s practical ability to execute a skillful action can be explained by attributing the agent a propositional intention to execute precisely that action in a certain way. Thus, for an agent s to have the practical ability to execute action F in a certain way w, she must possess the propositional representation that w is a way to execute F, and she must be able to link such propositional representation to her intention to execute F (Pavese 2017; Stanley 2011; Stanley and Williamson 2001).1 For example, the practical ability of an agent to grasp a mug to drink can be explained by attributing the agent with the intention to grasp the mug to drink, which involves the propositional content “I grasp the mug to drink”. It is because the agent has the intention with the right propositional content that she can grasp the mug to drink. In other words, according to this classical view, knowing how to grasp a mug to drink is entirely defined in terms of propositional representations that rationally determine the plethora of motor acts required to execute this action.

3 The Structure of Propositional Representation Propositional contents are generally thought of as interpreted linguistic items, which own the property of being the bearers of truth and falsity. Notably, propositional contents have been conceived of as sentences that affirm or deny a predicate concerning a subject. Now, since sentences consist of predicates, arguments and adjuncts, the format of propositional contents reflects the predicate-argument structure. The predicate-argument structure depends on the very nature of the predicate and the number of arguments that can be related to it. Such a structure may be adopted to either assign a property to a single variable (the subject) or relate two or more variables to each other, so that a given predicate can be represented as a relation with only one variable, between two variables or even more. For instance, the proposition “Sally plays” is classically modeled as a unary relation with the structure P(s), where P denotes the predicate and s denotes the argument. Differently, the proposition “Sally plays chess” is a binary relation with the structure P(s, c), where P is the predicate and (s, c) is the argument composed 1 It should be noted that the thesis according to which practical knowledge is a case of propositional

knowledge, also known as “intellectualism” (Stanley and Williamson 2001), is a consequence of the more general view according to which all intentional states are propositional states, also known as “propositionalism” (Davidson 1970).

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of an ordered pair of variables, namely, the subject and complement. This format of representation can be intended to denote the existence of a specific relation P between the ordered list of components s and c, so that, under semantic interpretation, such a structure can be conceived of as a true or false description concerning a factual condition. Notably, according to the common view, P(s, c) is true if the relation P holds between s and c, whereas it is false otherwise. For the sake of the present argument, it is challenging to understand whether and how propositions describing skillful actions can be represented using the predicateargument structure. Accordingly, we may focus on a particular class of predicates involving transitive action verbs, such as “to push”, “to pull”, “to kick” and “to grasp”, to cite only few of them. Transitive action verbs are the essential conceptual components of the propositional descriptions concerning our interactions with the environment, so that it is usually assumed that a propositional mental representation of the actions we intend to execute must involve such a type of component. Take for example the intention to grasp the mug, whose propositional content is “I grasp the mug”. In this case, the action concept “to grasp” works as a predicate and can be represented as the relation G holding between the relevant argument variables. On the basis of these considerations, predicates involving transitive action verbs can be represented by means of a binary structure of the form G(s, o), where G denotes the action concept expressed by the action verb, s is the subject who executes the action, and o is the target object in the environment with which the subject s interacts by means of the action represented by G. Importantly, in this view, any transitive action verb (G) is equated to a particular relation between the subject (s) and the environmental object (o), so that under semantic interpretation the structure G(s, o) can be regarded as a true or false description of a factual condition. Thus, G(s, o) will be true if the subject s executes the action represented by G on the target o, differently it will be false. For example, take the binary structure G(i, m) involving the predicate G and the ordered pair of variables i and m. Such a structure can be intended as a formal representation of a sentence asserting that the relation G holds between the variables i and m. Thus, under semantic interpretation, the structure G(i, m) can be used as a format for propositional contents describing transitive actions such as “I grasp the mug”. It should be noted that this structure can be enriched by adding further arguments (as parameters), such as to obtain a format like G(i, m, d). Such structure type may be employed to provide a formal model for predicates involving transitive verbs, which need to be characterized according to specific parameters, such as the goal of the action. Indeed, the representation of a motor instruction elicited by an action concept is not fixed once and for all, but rather depends on the “sense” or “intension” with which the action concept is used. For example, consider the case of the transitive action verb “to grasp”. This verb can be used to capture a wide range of different

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motor interactions with the same object2 such that, to specify the actual chain of movements denoted by this concept, it is essential to clarify the goal of the action in addition to the subject and the target. Thus, for example, “I grasp the mug to drink” and “I grasp the mug to wash it” are propositional contents based on the same action concept; however, they describe different motor goals, which in turn involve the execution of different sets of movements. Importantly, it is also because of this difference in the motor meaning of the action concept “to grasp” that the propositional content “I grasp a mug to drink” and the propositional content “I grasp the mug to wash it” may present different conditions of truth. It is a matter of fact that, when grasping the mug to drink, the agent performs movements with fewer degrees of freedom than when she grasps the cup to wash it. Such a difference can be formalized by means of the adoption of conventional symbols for different argument roles. Accordingly, the first proposition can be represented by means of the structure G(i, m, d), and the second proposition can be represented as G(i, m, w). This means that, after semantic interpretation, the structure G(i, m, d) will be true if the relation G holds between i and m according to d, and the structure G(i, m, w) will be true if the relation G holds between the i and m according to w.

4 Why Propositionalism Does Not Work: The Case of Motor Cognition According to propositionalism, cognition is a matter of manipulating only propositional contents, that is, mental representations characterized by a predicate-argument structure (Sects. 2 and 3). As a consequence of this view, an agent’s practical ability to execute a motor action must be accounted for in terms of propositional knowledge concerning the way to execute that action. This allows for explaining the underlying rationality of motor cognition through the powerful tools of propositional logic (Pavese 2017; Stanley 2011; Stanley and Williamson 2001). That said, though the possibility of providing a propositional account of action planning and execution may look prima facie appealing, it must comply with two requirements: first, it must account for the prescriptive role of action intentions; second, it must be consistent with the available empirical evidence.

2 For example, a mug can be grasped with a power grip or with a precision grip depending of the aim

of the action. Interestingly, in both cases, the verb “to grasp” is used without further discrimination (Bratman 1999; Pacherie 2008; Searle 1983; Moneglia et al. 2014).

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4.1 The Prescriptive Role of Motor Representations Philosophers and cognitive scientists widely agree that an agent intending to perform a skilled action needs to transform her intention into the execution of the related bodily movements (Bratman 1999; Mele and Moser 1994; Pacherie 2008). This, in turn, requires the prescription of an ordered sequence of mental states, which implies a hierarchical organization of different elements (e.g., motor acts, movements, and muscular contractions) linked to one another to generate a successful motor behavior. Accordingly, depending on the level of analysis we select, an ordinary action can involve different components and thus can be represented by means of different structures. This poses an issue for the propositional account of motor cognition. Propositional representations indeed do not have the right structure to prescribe instructions on how to execute an action. Notably, it remains unclear why an intention endowed with a certain predicate-argument structure should give to the agent the ability to successfully execute an action while another intention with a different predicate-argument structure may not give rise to the same ability. The problem here is to understand the way a predicate-argument structure hangs up on the bodily movements that constitute an action. Thus, on the pain of being unable to explain why propositional representations allow action execution, a criterion is needed to translate the propositional content delivered by propositional representations into the actual execution of the related action (Fridland 2013, 2016). As we saw in the previous section 3, a propositionally structured intention to execute an action can be rendered as a predicate-argument structure with the form G(i, m, d), where G is the action concept, i is the subject, m the target object and d the aim of the action. According to a common view, G(i, m, d) is true if G holds between i and m according to d. Now, since action intentions should have an executive function to give rise to actions execution, the question is this: in what way may this representational structure prescribe the right motor instructions for G(i, m, d) to be true? The format G(i, m, d) indeed represents a relation between three elements but says nothing regarding how to obtain such a relation. In other words, what the propositional format G(i, m, d) lacks is information about how to motorically arrange the three variables of the argument to make the proposition true. More precisely, the structure G(i, m, d) does not provide information concerning the chain of motor movements that agent i should perform to instantiate relation G with the object m to obtain the goal d. Consider again the case of the proposition “I grasp the mug to drink”. We saw that such a proposition can function as the content of an intention to act, namely, the intention to grasp a mug with the aim to drink (Sect. 2). Such an intention should guide the agent to execute the related action, prescribing the chain of movements required to make it true. Moreover, we saw that such a propositional content can be rendered according to the predicate-argument structure G(i, m, d) (Sect. 3). Remarkably, it should be noted that G(i, m, d) represents a relation between the components of the action denoted by G, but does not have the resources to provide information

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concerning how to execute that action. This means that, if the content of the intention to grasp the mug to drink has the propositional content “I grasp the mug to drink” with the predicate-argument structure G(i, m, d), it does not have enough resources to provide guidance on how to grasp the mug to drink. In sum, the predicate-argument structure alone does not explain how an intention to execute an action determines the motor acts that constitute the intended action. Thus, accounting for how a propositional description of a motor skill hangs up with the right set of bodily movements that allows the execution of that skill remains to be determined. Remarkably, this consideration has consequences on the explanatory function of the classical propositional view. Additionally, it should be noted that a propositional description of an action cannot function as an explanation of the actual execution of the action, inasmuch as the meaning of the former is dependent on the motor acts that constitute the latter. It is because the agent is able to perform the set of movements w that constitute the way to execute action F that she can intend to do F by means of w (Bermúdez 2007; Pacherie 2011). Indeed, the prescriptive function of the executable action concepts that constitute an intention to act depends on the possession of the practical abilities to which such an action concept refer. For example, the meaning of the action concept, say “to grasp”, can be related to the class of motor acts that characterize the specific way in which the agent carries on this action. Remarkably, without such a relation, it seems hard to attribute an executive function to such an intention. Take again the case of the propositional content “I grasp the mug to drink” and its related predicate-argument structure G(i, m, d). We saw that G(i, m, d) is true if G holds between i and m according to d, but for this to be possible, it is necessary for i to know how to execute the action G on m according to d. Therefore, for an intention to prescribe the execution of a grasping action, such an intention must already possess a relation to the representation of the related motor acts that characterize the intended action to grasp (Pacherie 2011). Thus, if circularity is to be avoided, this representation of movements cannot in turn involve propositional descriptions but rather must be conceived in a basic format that allows a more direct interaction with the agent’s motor system. Contrary to the propositional view, therefore, having a predicate-argument structure is not enough to account for the ability to execute a skilled action, being the executable action concepts that constitute the predicate dependent on the motor acts that the agent already knows how to perform.

4.2 The Mental Representation of Action To develop an account of the representational structures underlying skilled action planning and execution, we need also to feature empirical facts regarding the structure of motor cognition. Interestingly, evidence from cognitive neuroscience reveals that the motor and pre-motor cortexes contain a system of functional states involved in the execution of already structured and goal-oriented sequences of motor acts, instead of a system of states functionally involved in mere series of single movements

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or muscular displacements (Rizzolatti et al. 1998). Indeed, motor neurons in these regions show activation patterns that are correlated to the goal of the action sequence in which the act is embedded (Andersen and Buneo 2003; Fogassi and Luppino 2005; Iacoboni 2006; Umiltà et al. 2008). According to this evidence, the cortical motor system can be viewed as a sort of “vocabulary” in which the words are the physical states of the motor cortex and the related meanings are expressed in terms of already structured motor acts (Rizzolatti et al. 1988). Such a vocabulary can be conceived of as a system of rules linking the agent’s intentions to act to the related bodily effectors and patterns of movements (Fadiga et al. 2000). Each word of the vocabulary can be seen as delivering the instruction to guide the activity of the body to execute the chain of acts that fits with the agent’s intention to execute a skilled action. Accordingly, a certain agent’s intention, say that of grasping the mug to drink, can be satisfied by means of the execution of a series of movements and interactions, such as “extending the arm to reach the cup”, “open the hand to grab the handle,” “retract the arm to bring the cup to the mouth”, etc., each of which is encoded by an already structured representation within the agent’s motor system. Importantly, this empirical evidence has relevance for the issue of establishing the structure that information should have to successfully execute a skilled action. Indeed, inasmuch as the representational states that form the vocabulary of motor acts convey already structured instructions, they cannot be considered as the bearers of truth or falsity: since instructions do not describe but rather prescribe events, they cannot be true or false. Rather, they can be suitable or not suitable according to an intended motor goal (e.g., to grasp to drink). As a result, the contents of each representation pertaining to the vocabulary of a motor act cannot be easily formatted by means of propositional structures, being a basic feature of propositions the bearing of true or false values. Then, since the functional states of the motor system cannot be seen as conveying true or false informational contents, they should be conceived of as mental states with already structured content and a non-propositional format.

5 Descriptive Versus Prescriptive Structures It is now possible to show how propositional representations differ from representations that are structured in motor format, and then, how referring to such a difference in format may help to avoid troubles with propositionalism (Sects. 2 and 3). To do that, it is important to be clear about the structure of propositional contents that is relevant here. We previously saw that a propositional representation of an action intention can be rendered by means of a predicate-argument format whose structure is G(i, m, d). However, we also saw that such a structure suffers from a series of flaws that cannot be easily solved. First, the predicate-argument structure G(i, m, d) does not have enough resources to describe how to select and implement the series of motor acts that are needed to execute an action (Sect. 3). Accordingly, since it is usually

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assumed that an intention to act can be transformed in an actual motor plan subserving action execution, it seems that the predicate-argument structure G(i, m, d) is not suitable for this aim. Second, over the last decade empirical research has shown that, at the level of the motor and pre-motor cortexes, neural activity does not codify single movements or muscular displacements, but rather already structured and goaloriented sequences of motor acts. According to this evidence, it has become common to consider the cortical motor system as a sort of “vocabulary” where each word delivers the representation of an already chain of structured motor acts that fits with the agent’s intention to execute an action. Now, inasmuch as we consider the states of the motor system as conveying motor instructions rather than goal descriptions, they should be viewed as mental states with a non-propositional format. So far so good, but what is the structure that characterizes such a type of nonpropositional motor representation?

5.1 The Structure of Action Concepts The first thing to do to avoid trouble with the classical approach to motor cognition (Sects. 2 and 3) is to provide a satisfactory account of the prescriptive function of the intention to act. In the previous Sects. 4 and 5, we saw that conceiving action intentions as propositional predicates is not a viable solution to this issue. Indeed, propositional representations are usually characterized as true or false predicateargument structures, such as descriptions, rather than instructions. However, though it is a common view to see action intentions as delivering propositional contents, this common view is not the only game in town. Introducing an alternative account, special attention should be focused on the action concepts that constitute our intentions to act, such as the action verb “to grasp” within the intention to grasp the mug to drink. According to the classical view, action concepts are predicates and are conceived as relations between the variables of the argument. Such a view, however, does not account for the prescriptive function that the action concepts play within our action intentions (Sects. 4 and 5). In a different vein, action concepts can be conceived as practical representations concerning the execution of the motor acts that constitute the intended action. According to this view, the action concept that constitutes an intention should prescribe instructions on how to execute the intended action by means of the appropriate motor acts. Notably, a motor act can be conceived of as a function linking two bodily states: an initial condition and a goal condition. Motor acts are events that involve the transformation of the agent’s body, such that the execution of an action can be described as the passage from an initial bodily state to a final bodily state. Accordingly, an action concept can be represented as a function mapping of different conditions of the body over time in a phase space, such that its structure will be that of a functor rather than that of a predicate. Thus, the action concept G can be represented as the functor G ci →cg , where G symbolizes the action concept that

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constitute an intention to act and ci → cg is the functor linking the initial condition of the agent’s body to a related goal state. Take for example the case of the intention to grasp the mug to drink. According to the common propositional conception, it can be rendered as a predicate-argument structure G(i, o, d), which however does not have enough resources to explain how the action concept G relates to the execution of the related action (Sect. 4). However, it is possible to think of the action concept G as a practical, non-propositional representation that can be rendered by means of the structure G ci →cg . According to this interpretation, ci is the initial condition of the agent’s body before starting the action to grasp, and cg is the goal state that the body should reach at the end of the grasping. Now, the structure G ci →cg can be read as prescribing a motor act that is defined by a previously structured series of movements that allow for passing from ci to cg according to the functor ci → cg . Interestingly, such an interpretation allows for linking the action concept G to the actual execution of the related action, that is, to the motor act defined by the function ci → cg . Importantly, the functor G ci →cg does not have the same structure as that of the predicate G(i, o, d). Indeed, the structure G(i, o, d) is usually intended to describe a relation holding between the variables (i, o, d), whereas the structure G ci →cg denotes a function linking two topological conditions characterizing the agent’s body (i.e., bodily states).3 Now, while the predicate-argument structure can be true or false depending on whether the relation G holds between the variables of the argument, the structure G ci →cg does not describe, but rather prescribes, a variation on the conditions characterizing the agent’s body, that is, a series of movements allowing transformation of the body, or a functional part of it, from the condition ci to the condition cg . Accordingly, instead of providing a true or false description of the factual execution of an action, the functor G ci →cg prescribes how to execute an action by means of the pattern of previously structured bodily movements required to pass from condition ci to condition cg . All this suggests that action verbs can be described not as a priori relations of a static character, but rather as dynamical schemas of bodily variation. The resulting structure grounds the meaning of action concepts in topological-dynamical patterns concerning the transformation of bodies in space with a privileged fineness of grain (Peruzzi 1994, 2000).4 We saw that the motor cortex represents already structured motor acts according to their motoric goal instead of representing arbitrary chains of movements. Notably, the motor cortex does not convey information about the kinematics of single muscles and joints; rather, it prescribes already structured chains of motor acts learned by 3 This

view allows for conceiving actions as special types of morphisms, that is, transformations of bodily states in different bodily states. Once it is realized that action concepts cannot be merely interpreted as predicates, and that the constructive constituents of intentions are bodily grounded, the related representational structure is strictly linked to the categorical analysis of logic (see Peruzzi 2000). 4 Linguistic investigations such as those of Talmy (2005), Jackendoff (1987), Lakoff and Johnson (1999) provided evidence of the pervasive presence of bodily-related structures in the semantics of natural language.

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conditional visuomotor association (Ferretti 2016). Therefore, although it is possible to account for a skilled action at several levels of description, motor processing always represents skilled actions by prescribing chains of motor acts whose structure has been fixed by the agent’s motoric expertise given by previous visuomotor exercise (Rizzolatti et al. 1988; Rizzolatti and Luppino 2001; Ferretti 2017). It should be noted that conceiving action concepts as a functors of the form G ci →cg allows for conveying all the relevant information required to prescribe how to execute the related action (Ferretti and Zipoli Caiani 2018, 2019). Notably, among all the possible motor patterns of transformations that a body can perform to obtain the condition cg from the condition ci , many of them can be excluded because incompatible with the physical constraints of the agent’s body, while only a small part of the remaining patterns have acquired motoric value on the basis of the previous association between the same initial condition and the goal state.5 Remarkably, although G ci →cg has the structure of a prescription and therefore cannot be the subject of truth or falsity, it can be assessed as suitable or unsuitable in respect to a certain intention to act. For example, take the action intention associated with the predicate-argument structure G(i, o, d). Such a structure can be interpreted as a true description if the relation denoted by the action concept G holds among the variables i, o and d. However, for the structure G(i, o, d) to be true, it is necessary that the agent knows the way to link the action concept G that constitutes her intention to the execution of the suitable motor movements that allows for her to perform G. Indeed, it could be the case that an agent intends to execute the action G but does not have the representation of the right instructions concerning how to link the action concept G to the movements suitable for making true the predicate-argument structure G(i, o, d). This may happen either because the action concept G that constitutes the intention is not associated with any motor instruction or because the action concept G is associated with the wrong motor instruction. In the second case, the action concept G can be represented as a functor G ci →cg prescribing a motor act that is not suitable for making G(i, o, d) true.

6 Conclusion: Hierarchy of Structures According to a widely shared view among philosophers and cognitive scientists, intentional mental states are relations to propositional contents, that is, relations to true or false predicate-argument structures of the form P(a), where P is the predicated concept and (a) is the argument of the predicate (Sects. 2 and 3). One of the main 5 There

is evidence of the fact that long training in executing a skilled action changes the cortical organization of the motor cortex. For example, Monda et al. (2017) have recently shown significant differences in the motor cortex excitability between trained athletes and non-athletes, supporting the hypothesis that training determines a specific organization of the motor cortex. Further evidence of the existence of differences in the functional organization of the motor cortex between trained and non-trained agents can be revealed by experiments on “motor imagery” (e.g., Lacourse et al. 2005; Wei and Luo 2010).

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consequences of this classical view is that, since cognitive processing involves only propositional contents, it should be possible to account for any relevant cognitive task by relying only on representations with a predicate-argument structure. Contrary to this view, I argued that there is at least a relevant cognitive task that cannot be addressed by means of propositional contents only. This task concerns the planning and the execution of motor actions. The main problem of a propositional approach to motor cognition is that intentional representations characterized by a predicate-argument structure have not enough resources to account for how our action intentions can induce the right chain of motor acts required to execute the action. Indeed, inasmuch as propositional representations describe relations, but not prescribe instructions to instantiate relations, it is hard to understand how an intentional mental state can have a functional role in determining the movement of a body. Indeed, the predicate-argument structure G(i, m, d), by means of which propositional intentions to act can be characterized, ascribes the action concept G on the argument variables i, m, d, but says nothing about how to prescribe such a condition. More precisely, what the predicateargument structure G(i, m, d) lacks is the relevant information about how to arrange the three variables of the argument in order to make true the propositional content of the intention. This means that the structure G(i, m, d) does not provide information concerning the chain of motor movements that the agent should perform in order to execute the action denoted by the concept G that constitute the related intention to act (Sect. 4). On a different vein, action concepts can be conceived as representations characterized by a practical, non-propositional format, which provides information concerning how to execute the intended action. According to this view, the action concept that constitutes an intention should prescribe motor instructions, rather than deliver a mere description of the action. Thus, an action concept can be represented as a function G ci →cg mapping from the bodily condition ci to the bodily condition cg , where G symbolizes the action concept that constitute an intention to act, and ci → cg the functor linking the initial condition of the body to a related goal condition. In should be noted that this way of modeling action concept has the merit to avoid the problem of prescription which afflict the classical view. Notably, the structure G ci →cg does not merely describe, but rather prescribes a variation on the conditions characterizing the agent’s body, that is, a series of movements allowing to transform the body, or a functional part of it, from the condition ci to the condition cg. Rather than providing a true or false description, the functional structure G ci →cg prescribes how to execute an action by means of the series of previously structured bodily movements required to pass from the bodily condition ci to the condition cg . This because the motor cortex represents already structured motor acts, whose structure has been fixed by the agent’s motoric expertise given by previous visuomotor experience. As a result, contrary to the classical view, we saw that a suitable account of motor cognition involves at least two different representational formats. The propositional format based on the predicate-argument structure G(i, m, d), and the pragmatic, nonpropositional format of representation based on the functorial structure G ci →cg . While the former allows to convey the informational contents that characterize intentions,

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the latter allows to convey the information required to execute the propositional intended action. Interestingly, this view configures a hierarchy of representational formats according to which the propositional description finds its meaning in the pragmatic representation of action concept. Notably, once an agent intends to execute an action, she represents that action by recruiting the same motoric structure that is involved in the representation of the motor instruction. This amounts to saying that the action concepts that constitute a propositional intention share the same format of a pragmatic representation, which is the non-predicative, functorial structure used both when we think about actions through an intention and during covert or overt action performance. As a result, this explains why and how a propositional intention hangs up with the bodily movements that constitute the appropriate outcome of an action.

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Topological Aspects of Epistemology and Metaphysics Thomas Mormann

Abstract The aim of this paper is to show that (elementary) topology may be useful for dealing with problems of epistemology and metaphysics. More precisely, I want to show that the introduction of topological structures may elucidate the role of the spatial structures (in a broad sense) that underly logic and cognition. In some detail I’ll deal with “Cassirer’s problem” that may be characterized as an early for runner of Goodman’s “grue-bleen” problem. On a larger scale, topology turns out to be useful in elaborating the approach of conceptual spaces that in the last twenty years or so has found quite a few applications in cognitive science, psychology, and linguistics. In particular, topology may help distinguish “natural” from “not-so-natural” concepts. This classical problem that up to now has withstood all efforts to solve (or dissolve) it by purely logical methods. Finally, in order to show that a topological perspective may also offer a fresh look on classical metaphysical problems, it is shown that Leibniz’s famous principle of the identity of indiscernibles is closely related to some wellknown topological separation axioms. More precisely, the topological perspective gives rise in a natural way to some novel variations of Leibniz’s principle.

1 Introduction Traditionally, logic is considered as one of the essential pillars of philosophy. Both epistemology and metaphysics are considered as closely related to logic: Epistemology can be conceived of as a sort of applied logic that is engaged in describing the basic laws of (correct scientific) thought, and metaphysics as investigating the most basic structures of the world thereby providing a kind of logic of reality. In modern times, the traditional basic role of logic for epistemology and metaphysics has become more and more doubtful. Logic in the traditional sense has lost its fundamental role for both areas. Even if Feyerabend’s notorious “Anything goes” T. Mormann (B) University of the Basque Country UPV/EHU, Donostia-San Sebastian, Spain e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 A. Peruzzi and S. Zipoli Caiani (eds.), Structures Mères: Semantics, Mathematics, and Cognitive Science, Studies in Applied Philosophy, Epistemology and Rational Ethics 57, https://doi.org/10.1007/978-3-030-51821-9_7

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is an untenable exaggeration, it has turned out that traditional logic covers only a limited aspect of the multi-facetted cognitive endevours of modern epistemology and metaphysics. Logic no longer can be considered as a fundamental and unproblematic ingredient of human cognition that can be taken simply for granted. With respect to the areas of cognitive science and epistemology Alberto Peruzzi described this new situation as follows: The structure of thought is classically described in logical terms. But the roots of logical structure lie deeper than mere symbolic manipulation, for logic inherits meaning from patterns of bodily interaction that involve essential correlations of space and quantity (hence, of geometry and algebra). (Peruzzi 2000, 169/170).

It goes without saying that the investigation of the “origins” and the “meaning” of logic are questions that cannot be treated exhaustively in a short paper. In this paper I’d like to address some aspects of these questions in a very elementary manner. 1. What are the roots of logical structure and cognition, which lie “deeper than mere symbolic manipulation? 2. What does it mean that logic “inherits meaning” from patterns of bodily interaction that involve space and quantity? My answer to these questions, then, in a nutshell, is the following (cf. Lakoff 1987): The roots of logic and cognition lie in our activities as living beings that have to cope with the various challenges of a spatio-temporally structured material world. The spatialtemporal structure of the world is described by some kind of generalized geometry, geometry understood here in a broad as topology as theory of spatial structures in general.

For any classical account of geometry such a project of a geometrical formulation of logic, cognition, and metaphysics would be a hopeless endevour. What is needed is a “geometry” that goes far beyond the confines of traditional (Euclidean) geometry. Fortunately, in the last 150 years or so modern mathematics has provided the conceptual means for such a comprehensive theory of generalized geometry aka topology.1 While Euclidean geometry has served as source of inspiration for traditional philosophy over the centuries, in contrast, topology as its modern success or can hardly claim to play an analogous role for contemporary philosophy. However successful and fruitful topology may have been in the realm of mathematics and other sciences (empirical science and computer sciences), according to most philosophers, topology is to be considered just as highly sophisticated mathematical theory without philosophical relevance. This state of affairs can hardly be considered as optimal. Fortunately, there is evidence that it may be overcome in the future. In cognitive science, linguistics, and related disciplines modern mathematics has established itself in a way that might help philosophers understand that the ideas of modern 1 Among

the protagonists who played a leading role in this endeavor one may mention Poincaré, Hausdorff, and Kuratowski.

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mathematics can indeed be philosophically relevant and fruitful. The present paper may be considered as a modest contribution to this endevour. More precisely, the aim of this paper is to show that topology may play a role in elucidating the conceptual roots of the logics and its role in epistemology and metaphysics.2 The outline of the paper is as follows. In the next Sect. 2 the philosopher Ernst Cassirer and the psychologist Kurt Lewin are briefly presented as two precursors of a kind of topological epistemology avant la lettre. This is interesting not only in itself but also as evidence that analytic philosophy cannot claim to have a monopoly on an affinity to modern mathematics and science—Cassirer and Lewin have been scholars with strong roots in “continental” philosophy. In Sect. 3 we deal with the first highly non-trivial relation of logic and topology that was presented in the 1930s by the mathematician Marshall H. Stone, namely, Stone’s Representation Theorem (cf. Stone 1936, 1937). Stone’s achievement, which according to many mathematicians is to be considered as a trail-blazing result of 20th century mathematics, shows that the relation between logic and topology is much more than just a nice metaphorical relation but touches the very essence of these disciplines. In Sect. 4 the modern theory of conceptual spaces, inaugurated in the 1990s by Peter Gärdenfors and other scholars is reconsidered from a topological point of view. As an elementary example that topological considerations may be fruitfully applied in the realm of metaphysics, in Sect. 5 topological separation axioms are used to define a variety of novel versions of Leibniz’s classical Principle of the Identity of Indiscernibles (LPI).

2 Cassirer’s Problem: Begriffsbildung Beyond Traditional Logic The necessity of going beyond the traditional logic as the art of correctly manipulating logical symbols was not a problem perceived solely in the context of analytical philosophy. This problem arose well before analytical philosophy proper came into being. Evidence for this is Ernst Cassirer’s Neo-kantian philosophy of science. Cassirer’s philosophy of science was concept-oriented par excellence: The theory of the concept becomes a cardinal problem of systematic philosophy. It becomes the nub around which logic, epistemology, philosophy of language and cognitive psychology are rotating (Cassirer 1928, 163).

Already in his first systematic opus magnum Substance and Function (1910) Cassirer had conceived philosophy of science as a theory of the formation of scientific concepts. His theory was empirical in the sense that philosophy should not decree 2 In

this endeavor I want to be as accessible as possible. That is to say, often things could be formulated in a mathematically or logically more elegant and more general way (cf. Peruzzi 2000 or Petitot 1995). I prefer to express myself in a much more pedestrian way in order to show that the topological way of doing philosophy (at least in its beginnings) does not depend on a heavy formal apparatus.

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what scientific concepts were and how they worked from the philosophical armchair, so to speak. Rather, since scientific concepts evolved in the historical evolution of science, it was the task of philosophy of science to study this conceptual development of science and to make philosophical sense of it, not to legislate it according to some preconceived philosophical ideas. He considered traditional logic, as a part of traditional philosophy, as epistemologically deficient: Traditional logic comprises, so to speak, with the same love “valid” and “unvalid”, “useful” and “non-useful” concepts. Herein resides, … seen from an epistemological point of view … its decisive fault. Hence now it becomes evident that a concept may possess its complete formal “correctness” in the sense of classical logical theory, that it may be construed quite “correctly” without telling us anything about its specific epistemological usefulness (Cassirer 1928, 132).

Already 20 years earlier in the introduction to Substance and Function he mentions the following drastic example that can be traced back to the 19th century logician and philosopher Hermann Lotze3 : If we group cherries and meat together under the attributes red, juicy and edible, we do not thereby attain a valid logical concept but a meaningless combinations of words, quite useless for the comprehension of the particular cases. Thus it becomes clear that the general formal rule in itself does not suffice; that on the contrary, there is always tacit reference to another intellectual criterion to supplement it. (Cassirer 1910(1923), 7).

As Cassirer points out, in Aristotle’s philosophy this other supplementary criterion is that the gaps, which are left in logic, are filled in and made good by metaphysics. For Aristotle a concept is not an arbitrary group of things. The selection of what is common remains an empty play of ideas if it is not assumed that what is thus gained is, at the same time, the real Form which guarantees the causal and teleological connection of particular things. For Aristotle, according to Cassirer: The process of comparing things and of grouping them together accodring to similar properties … does not lead to what is indefinite, but if rightly conducted, ends in the discovery of the real essences of things. (Cassirer 1910(1923), ibid.)

This close Aristotelian connection between logic and metaphysics has disappeared in modern times. We have to rely on other, not always well-understood means to distinguish honest concepts from meaningless, but formally correct conceptual chimeras. In our daily practice of life and science, we are hardly ever confronted with Cassirer’s 3 In

Anglo-Saxon analytic philosophy, Nelson Goodman and Carl Gustav Hempel are said to have been the first philosophers who put the issue of distinguishing between “valid” and “non-valid” concepts on the agenda. As is well-known, this has lead to a whole “industry” that is engaged in finding criteria for distinguishing between “non-valid” predicates such as “grue” and “bleen” on the on side and entrenched or projectible ones as “blue” and “green” on the other side. Hempelian paradoxes of confirmation have found hardly less attention. Formulated in a very simplistic way they tackle the problem why a green frog cannot count as a confirmation of the “law” that “All ravens are black” although a green frog may considered as a confirmation of the logically equivalent “law” that “All non-black things are no-ravens.

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(or Goodman’s or Hempel’s) problem. We seem to have an innate intuition of which concepts are natural and which are not. Philosophy cannot remain content with this intuitive state of affairs. It remains an important task of epistemology and philosophy of science to explicate how this is achieved. The aim of this paper is to tackle this problem by using geometrical and, more generally, topological methods that may be conceived as a kind of relativized or historized a priori. As a first step in the task of coming to terms with Cassirer’s problem it is expedient to formulate Cassirer’s problem in a more precise manner by interpreting concepts extensionally. This means that we identify a concept A with its extension (conceived as a subset of a universal set U) such that the concept A is represented by the set of objects that instantiate it. Then, Cassirer’s problem can be informally expressed in the following way: There are many more subsets A ⊆ U than natural concepts (predicates, properties). Set theory does not distinguish between “good” sets, which could represent “natural” concepts, and “bad” sets that do not. It does not offer means to overcome this shortcoming by itself. Nevertheless, it gives some useful hints. The deficiencies of a purely set-theoretical modeling of concepts can be overcome if the unstructured base sets U of traditional extensional logics are replaced by structured conceptual spaces: Then the basic thesis of a geometrical or topological epistemology can be formulated as follows: Natural concepts correspond to well-formed, subsets of a conceptual spaces. Non-natural concepts are represented by geometrically non-well-formed sets.

Such a geometrical epistemology is partially an empirical theory: we do not know a priori which geometric structures of conceptual spaces are appropriate for the characterization of natural concepts of a domain. We have to find out this through the empirical investigation of examples. Let us consider the following well-known example: The simplest model for a conceptual space of color predicates (“red”, “blue”, “yellow”, …) is the well-known “color circle” or, a bit more advanced, the “color spindle” that represents basic color predicates such as “red” or “blue” by certain areas or regions of these spaces (cf. Gärdenfors 2000, 2014). Due to the geometrical and topological structure of the circle, this representation renders plausible that disjunctive color predicates such as “red or green” fail to be natural predicates. This is confirmed by empirical investigations according to which there are no natural languages that possess simple color terms for expressing disjunctive color properties. In other words, there is a correspondence between “intuitive naturalness” and “structural wellformedness”, since connected areas of the circle are geometrically better-behaved than scattered and disconnected one. From this example one may conjecture that a necessary condition for color predicates to be natural is that they can be represented by connected subsets of the color circle. As will be argued in more detail later, being represented by connected areas of the representing space is only one requirement among others that “good” concepts have to satisfy. It is a non-trivial problem to find other useful criteria.

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Regardless of the details of how this can be achieved, already here it is expedient to warn against a possible fundamental misunderstanding concerning the epistemological status of the spatial structure of conceptual spaces. It would be quite misleading to conceive them as “mere metaphorical talk” or just as “pictorial illustrations” (“Veran-schau-lichungen”) without cognitive significance. As an early witness for this thesis, I’d like to mention the psychologist and philosopher Kurt Lewin, who may be considered as the first scientist who ever envisaged a deep non-trivial relation between topology and cognitive psychology.4 Since Lewin may not be so well-known among philosophers, the following succinct biographical remarks may be in order. Lewin was a former student of Cassirer, later he belonged to the Berlin group of philosophers and scientists that gathered around Reichenbach and may be considered as an ally of the Vienna Circle in their common aim to render philosophy a scientifically respectable entreprise. In the Introduction to his book The Principles of Topological Psychology (Lewin 1936) he wrote: The Principles of Topological Psychology is the result of a very slow growth. I remember the moment when – more than ten years ago – it occurred to me that the figures on the blackboard which were to illustrate some problems for a group in psychology might after all be not merely illustrations but representations of real concepts. … Knowing something of the general theory of point sets, I felt vaguely that the young mathematial discipline of “topology” might be of some help in making psychology a real science. (Lewin (1936, vii))

Lewin’s The Principles of Topological Psychology may be seen as the historically first attempt to apply topological concepts to problems of cognitive psychology. I don’t want to enter into a detailed discussion of Lewin’s book. The important point is to take serious Lewin’s thesis according to which the concepts of psychology may be usefully represented spatially: … that the figures on the blackboard might be not merely illustrations but representations of real concepts. I had already … in 1912 defined the thesis … that psychology, dealing with manifolds of coexisting fact, would be finally forced to use not only the concept of time but that of space too. (ibidem)

That is to say, for Lewin—and this renders him a precursor of modern geometrical or topological accounts of cognitive science and epistemology—the phenomena of logic and cognition can be fruitfully represented by geometrical and topological concepts. Sure, Lewin’s “topological psychology” uses only quite elementary and intuitive topological concepts. A stubborn despiser of a topological perspective may still dismiss it as merely metaphorical. In order to definitively refute the prejudice that topology has nothing more to offer than mere metaphorical illustrations for logical phenomena. In order to refute this claim, in the next section I’d like to briefly discuss Stone’s representation theorem in some more detail. Stone’s theorem is to be considered as the first deep connection between logic and topology. It establishes a

4 For

some interesting remarks on Lewin’s “topological field semantics” see Wildgen (2000, 216– 217) and Wildgen (2001).

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profound relation between logic and topology that up to now has only insufficiently been exploited philosophically, or so I want to argue.5

3 Topological Aspects of Logic: Stone’s Representation Theorem Since the advent of Tarskian semantics the set-theoretical representation of predicates and relations has become a standard method in logic. An innate weakness of set-theoretical semantics is that it ignores “Cassirer’s problem”, i.e., predicates may have any extensions whatsoever, a distinction between “good” and “not-sogood” predicates according to the well-formedness of their extensions cannot be not drawn—all subsets A of a semantic universe U are equally admissible. Since some time quite a few authors are engaged in the task of refining the set-theoretical approach by amending set-theory by topological methods. For instance, if one is interested in distinguishing vague concepts from precise concepts, topology may come to the rescue in a quite natural way. Instead on a set-theoretical universe one relies on an appropriate topological space. Then vague predicates A may be characterized by the property that all objects x that instantiate A do this in a stable way, i.e., any sufficiently small variation x’ of an x that instantiates A still instantiates A. To make this precise a topological structure is required. A recent, very interesting elaboration of this topological refinement of classical set-theoretical semantics can be found in Ian Rumfitt’s The Boundary Stones of Logic (Rumfitt 2015). In this work, Rumfitt offers (among many other things) a topological account of conceptual spaces that sheds interesting new light on the role of prototypes for the topological structure of conceptual spaces.6 This shows that topological structures (and possibly other structures as well) may be used to refine traditional logic in such a way that Cassirer’s problem can be tackled. In a nutshell, this procedure can be characterized as a structural amendment of the classical approach. It is based on the idea that appropriate relational systems such as topological spaces provide information that can be exploited in some way or other to distinguish between natural and not-so-natural concepts. Topology, however, may not only be used to refine classical logic by distinguishing, for instance, between vague and precise concepts. Topology is also helpful for investigating and elucidating the topic of Boolean7 algebra itself. This is evidence that topology is a kind of general logic of space, is more fundamental than logic as 5A

modern presentation of its mathematical content may be found in Mac Lane and Moerdijk (1992). 6 More precisely, relying on a prototype account of concepts he shows that prototypes can be used to define a topological structure on conceptual spaces that can distinguish between “good” (“natural”) concepts and “not-so-good” (“non-natural”) concepts. This prototype topology is very different from familiar topologies such as the topology of Euclidean spaces. 7 Actually, this is not true only for Boolean logic but for other logics as well. Incidentally, the case of Boolean logic is only the historically first one. Stone’s theorem may be considered as a source of a

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understood traditionally as a purely symbolic discipline quite unrelated to any spatial considerations. It is the achievement of Marshall Stone to have set the foundations for a topological reformulation of logic that turned to be extremely fruitful in the following decades of the 20th century till today (cf. Mac Lane and Moerdijk 1992). The modest aim of this paper is only to provide a superficial look onthe most elementary aspects of this new perspective onto logic. In a nutshell Stone’s great achievement is to have constructed for each Boolean algebra B a topological space St(B) in an essentially 1-1-way such that the conceptual (algebraic) devices of the algebra B could be used to shed new light on the topology of St(B) and viceversa, i.e., the conceptual (topological) devices of the topological space St(B) offered to insights into the algebraic (logical) structure of B. More precisely, Stone showed that for each Boolean algebra B one can construct a topological space St(B): (1) The points of the Stone space St(B) of B can be constructed as the maximal filters (ultrafilters) of the Boolean algebra B. As has been shown, this construction has may applications in lattice theory, order theory, and algebra in general. As is well-known, without the assumption of the axiom of choice or some similar principle the existence cannot be ensured. Hence the construction of a Stone space St(B) may be characterized as a conceptually very demanding endevour. (2) The set St(B) of ultrafilters of B is topologized in a canonical way, such that in the topological space St(B) the the elements of B can be identified with very special genuine topological structures of St(B), namely, with clopen sets, i.e., sets that are open and closed with respect to the canonical topology of the space St(B).

The spaces that are treated in traditional topology such as the Euclidean spaces E only have very few clopen regions, namely, the empty set Ø and the entire space E itself. Thus, the topologization of Boolean logic carried out by Stone is very far away from any intuitive and merely “metaphorical” geometrical reformulation of logic. There is no need to go into the details here. Be it sufficient to say that the Stone space St(B) of a non-trivial Boolean algebra B is a rather strange space in the sense that its topology has many clopen elements. In a nutshell, then, Stone’s representation theorem of Boolean logic may be encapsulated in the formula B = Clopen(St(B))

(1)

Here B is any Boolean algebra, St(B) the Stone space of B, and Clopen(St(B)) the set of clopen sets of the topological space St(B). In the decades after Stone had formulated his theorem, it found many applications and generalizations in a variety of mathematical disciplines (cf. Johnstone 1982). This is not the place to deal with this issue in any detail. Just one remark. At first view, one might have doubted the topological relevance of Stone’s theorem, since Stone spaces are very different from the topological spaces that traditional topology has dealt with. This impression is quite misleading. A generalization of Stone’s theorem establishes that for many “ordinary” topological spaces an analogous theorem can be proved: If flourishing quite general discipline of “spatial logics” documented, for instance, in the monumental Handbook of Spatial Logics (2007) edited by Aiello, Pratt-Hartmann and van Benthem.

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X is a reasonably well-behaved topological space, for instance, a Euclidean space, then the lattice OX of open sets is a complete Heyting algebra. Moreover, any wellbehaved Heyting algebra H uniquely defines a topological space pt(H) such that the lattice of open sets O(pt(H)) is isomorphic to H. That is, one obtains the analogous correlation for Heyting algebras and general topological spaces as the one that Stone proved for Boolean algebras and Stone spaces. Stone showed that topology was not just a mathematical discipline among others, but rather a method or a conceptual organon to be applied fruitfully in many areas of mathematics that at first view seemed to be quite unrelated to topology proper. This led him to formulate what became famous as “Stone’s maxim”, namely, the dictum that, being confronted with any mathematical problem whatsoever, “one must always topologize”, i.e., one must always attempt to conceive it as a topological problem such that its solution could be achieved by topological means. Taking into account that logic, as one of the corner stones of philosophy, has its roots in topology, topology understood as general theory of spatial structures, it does not seem unreasonable to generalize Stone’s maxim to at least some areas of philosophy as well: Given a philosophical problem, say, of logic or metaphysics, one should try to exhibit its topological (i.e. spatial) aspects in order to apply topologically inspired methods to solve it. If this general strategy should turn out to be feasible, formal philosophy, which has been traditionally dominated by logical problems and logical methods for more than one century, would give way to a kind of topological philosophy. In certain area this shift is already under way. As an example one may mention epistemology for which the theory of conceptual spaces plays an ever more important role. To be sure, in order that a geometrical or topological epistemology can get off the ground, a sufficiently general notion of space is necessary. We should be prepared to accept a topology of logic, and, more generally, a topology of thought that uses conceptual spaces whose topology differs considerably from the familiar topology of Euclidean space we are accustomed to.

4 Conceptual Spaces as a Framework for Topological Epistemology Over the last two decades or so conceptual spaces as a framework for the geometrical or topological representation of concepts and knowledge has been highly influential. It has contributed a lot to elucidate and articulate the spatial roots of our knowledge and cognition. Many researchers in cognitive psychology and cognitive science have contributed to the development of the conceptual spaces approach (even Kurt Lewin may be considered as an early precursor). Nevertheless, this approach gained real momentum only in the last decade of the 20th century with the work of Peter Gärdenfors (cf. Gärdenfors 2000, 2014).

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The basic technical idea of this approach is that concepts can be usefully represented geometrically or topologically as regions of well-defined geometrical structures, so-called conceptual spaces. More precisely, according to Gärdenfors, concepts can be represented by convex regions of conceptual spaces that result from Voronoi tessellations that are defined with the help of the prototypes of these concepts. More generally, meanings can be represented by geometrical structures of conceptual spaces (cf. Gärdenfors 2014). This geometrical representability is no coincidence. Rather, our conceptual systems are constructed by us, who are creatures that live in a spatio-temporally structured material world. Thus, [T]hese conceptual systems grow out of our bodily experience and make sense in terms it; moreover, the core of our conceptual systems is directly grounded in perception, body movement, and experience of a physical and social character. (Lakoff 1987, xiv)

For all experiences whatsoever (perhaps apart from mystical experiences) their spatial character is essential. That means in order to be meaningful and communicable they have to be conceived as being located in a conceptual space that is metaphorically related to physical space in some way or other. This is—in very general terms—the content of Lakoff’s Spatialization of Form Hypothesis” (Lakoff 1987, p. 283) that lies at the heart of his Cognitive Semantics approach whose explicit aim is to elaborate a theory of embodied cognition that conceives all our cognitive endeavours—even the most abstract ones such as mathematics—as determined by the contingencies of our existence as natural beings in a material spatio-temporal world. According to cognitive semantics, conceptual structures depend on the perceptions and actions of the cognizing subjects. This entails that these structures reflect in some way or other the geometrical (=topological) structure the spatio-temporal world those subjects live. As Lakoff convincingly argued, this entails that cognitive structures are not propositional, but image-schematic. This means, ultimately, that they depend on the topological structures of the spatio-temporal material world we belong to. The regions of conceptual spaces that correspond to concepts are constituted by the sets of points that represent those objects, processes, or events which exhibit the key perceptual properties associated to each considered region. Every conceptual space is endowed with a metric that measures the similarity between objects. For any two objects, events, processes, or whatever the elements of the conceptual space are assumed to represent, x is the more similar to y the closer it is to y according to the metric. This notion of similarity is a similarity-in-agiven-respect, that is encapsulated in the specific metrical structure of a space—not an overall similarity. It may well be the case that some aspects of the objects to be compared are ignored by that metrical structure. One of the best known and most familiar example of a conceptual space is the three-dimensional Euclidean space which serves to represent proximity relations between physical objects. Other examples of conceptual spaces are the spaces of temporal, auditory, olfactory and taste experiences. For the latter ones it may be not clear that they can be endowed with a fully-fledged Euclidean structure. Rather, they are endowed only with a simpler topological structure of some kind.

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In Gärdenfors’s conceptual spaces the categorization of an object g under a particular concept is the result of a process that (i) evaluates the distances of g from the prototypes of all the relevant concepts within the considered context, and (ii) classifies an object g under the concept C whose prototype pC is the most similar to g. If it happens that there are several different prototypes p1 , …, pn maximally close to g, then g is located on the boundary of the cells defined by the prototypes p1 , …, pn (for details cf. Rumfitt 2015). Cassirer’s problem of how natural concepts can be distinguished from gerrymandered “Goodmanian” concepts shows up in the context of conceptual spaces in a quite natural manner: As said, concepts are identified with specific well-formed spatial regions. For instance, the concepts “blue” and “green” correspond to wellformed regions of the color space. In contrast, the Goodmanian counterparts “grue” and “bleen” turn out to be as represented by non-well-formed or at least less wellformed regions of the color space (cf. Gärdenfors 2000, Chap. 3, Mormann 1993, 231). This is indeed the case. Thus, due to the geometrical structure, not all subsets of a conceptual space correspond to natural concepts. Due to the selective force of the topological structure not all subsets of a conceptual space C correspond to natural concepts. However, Gärdenfors does not give necessary and sufficient conditions that a region represents a natural concept. Instead, he only proposes a necessary condition for naturalness, to wit, convexity. A subset A of an appropriate (metrical) space C is convex iff for any pair of points x, y of A the line segment connecting x and y lies, in its entirety, in A as well. Then, according to Gärdenfors, a “good” concept is to be represented by a convex region of a conceptual space. For the following it is sufficient to consider some elementary examples. Let us assume that the conceptual space C has the structure of an Euclidean space (or some derivative of it, e.g., a locally Euclidean manifold such as the well-known color spindle, (cf. Gärdenfors 2014, p. 23, Gärdenforst 2000)). Then, convex regions as representatives of natural properties in conceptual spaces may be defined as follows: Definition 4.1 (“Criterion C”) A region R is convex iff for any two points x and y in R, all points between x and y are also in R. A natural property is represented by convex region of a conceptual space D.  Gärdenfors’ motivation for the convexity requirement invokes the idea of learnability: if some objects located at x and z are both instantiations of a “convex” concept C, then any object y that is located “between” x and z will also instantiate the concept C. This suggests that for any object y that is located in this way, it is easily learned that it also instantiates C—provided one already knows that x and z are instantiations of C. As Gärdenfors pointed out more than once, a plausible theory of concepts should explain somehow that a conceptual system (i.e., a system of concepts) is learnable by finite creatures possessing only a limited capacity of memory. “Criterion C” is intended to cope with this challenge. “Learnability” is not the only adequacy condition that conceptual systems have to satisfy if they strive to be considered as adequate. Recently, Douven and Gärdenfors

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have proposed a list of “design principles (for conceptual systems based on conceptual spaces with prototypes)” that good conceptual systems should satisfy (Douven and Gärdenfors 2018): Definition 4.2 (“Design Principles for Conceptual Systems”) (1) Parsimony: The conceptual structure must not overload the system’s memory. (2) Informativeness: The concepts should be informative, meaning that they should jointly offer a good and roughly equal coverage of the domain of classification cases. (3) Representation: The conceptual structure should be such that it allows the system to choose for each concept a prototype that is a good representative of all items falling under the concept. (4) Contrast: The conceptual structure should be such that prototypes of different concepts can be so chosen that they are easy to tell apart. (5) Learnability: The conceptual structure should be learnable, ideally from a small number of instances.  To be sure, the principles (1)–(5) may not always single out a unique system among several rival ones, but at least they help narrow down the number of competitors. This is the best one can hope for. After all, the strict connection between metaphysics and epistemology seems to be once and for all lost, conceptual systems remain forever more or less corroborated hypotheses that have to be assessed by appropriate pragmatic criteria. The general design criteria proposed by Douven and Gärdenfors may be taken as general guidelines for coming to terms with this task, not as fool-proof recipes that guarantee fully satisfying results.

5 Topological Metaphysics: Separation Axioms as Leibnizian Principles of The Identity The aim of this section is to show that topology may be a useful tool to elucidate a classical metaphysical principle in a new way, namely, Leibniz’s Principle of the Identity of Indiscernibles. More precisely, it will be shown that from a topological perspective Leibniz’s original principle is only one member of a large family of similar principles that all can be conceived as topological separation axioms. Leibniz’s Principle of the Identity of Indiscernibles (hereafter called LPI) is usually formulated as follows: If, for every property F, object x has F if and only if object y has F, then x is identical to y. In symbolic notation: 5.1 Leibniz’s Principle of the Identity of Indiscernibles Let F be a property, and x, y be objects. Then Leibniz’s Principle of the Identity of Indiscernibles holds if and only if the following implication holds:

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(LPI)∀x, y∀F(F x ↔ F y) → x = y. (Stanford Encyclopedia, Forrest 2016).



The logically equivalent transposition of (LPI) is sometimes called the Principle of Dissimilarity of the Diverse (PDD), namely: If x and y are distinct objects, then there is at least one property F that x has and y does not, or that y has and x does not have. In symbolic notation: 5.2 The Principle of the Dissimilarity of the Diverse (PDD)∀x, y(x = y → ∃F ((Fx and not Fy) or (Fy and not Fx))) (Stanford Encyclopedia, Forrest 2016).  The status of LPI is controversial, to put it mildly. According to some, it is simply false, to others, it is trivially true. Still others claim that there are various versions of LPI, some of them true, others false. These diverging contentions indicate that not all people understand the same as “Leibniz’s principle of the identity of indiscernibles”. This is indeed the case. For instance, if one accepts quantification over all properties whatsoever, then no Leibniz principle is needed at all. As Whitehead and Russell put it: It should be observed that by indiscernibles“[Leibniz] cannot have meant two objects which agree as to all their properties, for one of the properties of x is to be identical with x, and therefore this property would necessarily belong to y if x and y agreed in all their properties. Some limitation of the common properties necessary to make things indiscernible is therefore implied by the necessity of an axiom. … [W]e may suppose the common properties required for indiscernibility to be limited to predicates. … (Whitehead and Russell, Principia Mathematica (Introduction, p. 57))

In other words, Whitehead and Russell exclude “identity properties” such as “being identical with x” from the domain over which the quantifier of Leibniz’s principle runs since such properties as not defined by predicates but predicate functions. Dealing with Leibniz’s principle of the identity of indiscernible we are confronted with what may be called the „limitation problem“, i.e., theproblemofgivingcriteriahowtorestrictthedomainofpropertiesthathavetobetakenintoaccountforLeibniz’sprinciple. However the limitation problem is solved, not all properties should be admitted: As just said, iff or all objects x one admits the property „being identical with x “Leibniz’s principle holds for trivial reasons. Thus, „being identical with x “should not be considered as a „good“ property that has to be taken into account for a reasonable Leibniz’s principle. So, in dealing with Leibniz’s principle we are confronted again with a version of Cassirer’s problem, namely, the distinction between „good“ and „not so good“ properties, i.e., properties that are admitted to occur in the range of the principle’s quantifier and those that are not. As we shall see, topology may help cope with this problem.

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The very key observation here is that the validity of Leibniz’s principle is nothing but the assertion that the topological separation axiom T0 holds (cf. Steen and Seebach, Jr. (1978, p.11)). In a nutshell, LPI is just T0 in disguise. More precisely, the following holds: (1) LPI is equivalent to T0 for appropriately defined topological properties. (2) T0 is the simplest member of a large family of topological separation axioms Ti , 0 ≤ i. Thus, it is plausible to conjecture that at least some of the higher separation axioms Ti (i > 0) may give rise to various novel Leibnizian principles that deserve to be studied in metaphysics. The details are as follows. From an extensional point of view we may (pace Wittgenstein) identify a world X with a set of objects existing in X. Then a property may be identified with a subset F ⊆ X. In this setting LPI boils down to the assertion that two objects x, y ∈ X are identical iff they are elements of the same (admissible) properties F ⊆ X. Clearly, if any subsets F ⊆ X whatsoever are allowed to serve as properties, Leibniz’s principle is trivially true. If one wants to avoid this unwelcome result, one has to exclude some unnatural properties in some way or other. In other words, one has to solve Whitehead and Russell’s limitation problem, or, to express it in still another way, one has to cope with Cassirer’s problem of distinguishing between good and not-so-good properties. Here topology may come to the rescue. The very definition of a topological structure can be interpreted as a response to the limitation problem in the sense that a topological structure on a set X can be used to distinguish topologically well-behaved subsets from non-well-behaved ones. The details are as follows: 5.3 Definition Let X be a set and PX its power set. A topological structure on X space is a relational structure (X, OX) with OX ⊆ PX satisfying the requirements that Ø, X ∈ OX, finite intersections and arbitrary unions of elements of OX are elements of OX. The elements of OX are called open sets, their set-theoretical complements are closed elements of the topological structure (X, OX).  On every set X there are two extreme topological structures (X, O0 X) and (X, O1 X) defined by O0 X: = {Ø, X} and O1 X: = PX. With respect to set-theoretical inclusion ⊆ all topological structures (X, OX) on X lie between these two extreme topologies: O0 X ⊆ OX ⊆ O1 X. Obviously, for a non-trivial set X there are many different topological structures OX. Thus, the point of topologizing a world X is not simply to note that a topology on X exist, but to choose an interesting one. 5.4 Definition (Topological Properties) Let (X, OX) be a topological space, A topo-logically admissible property of x ∈ X is an open set a ∈ OX such that x ∈ a. The sys-tem of admissible properties of x is denoted by O(x): = {a; x ∈ a and a ∈ OX}. The Heyting algebra OX is called the topological property system of (X, OX).  After these clarifications we can tackle the problem of determining those topological structures that give rise to interesting Leibnizian principles LPI and those which do not:

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5.5 Examples (Leibniz’s Principle LPI of the Identity of Indiscernibles for topological property systems.) Let (X, OX) be a topological space. (1) For the indiscrete topology O0 X = {Ø, X} Leibniz’s principle of the identity of indiscernibles is clearly false: According to the property system of O0 X, there is only one non-empty property, namely the trivial property X that is instantiated by all elements of X. Thus, Leibniz’ principle is false for all worlds X that have more than one element. (2) For the discrete topology O1 X = PX, Leibniz’s principle of the identity of indiscernibles is trivially true since it contains for every object x ∈ X the identity property “being identical with x”, namely, the singleton {x}. Hence, distinct object x = y have different properties {x} and {y}, and LPI is true.  Admittedly, this is not very exciting. But at least it suggests how the undesired “identity properties” “being identical with x”, incriminated already by Russell and Whitehead in their discussion of the Leibniz principle, can be excluded by topological means in a natural way: A necessary condition a „good“ topology OX should satisfy is that it has no isolated points, i.e.,{x} ∈ / OX for all x ∈ X. What about intermediate topologies OX strictly between O0 X and O1 X? At first look, the problem whether a topological space (X, OX) satisfies the topological Leibniz principle LPI for its canonical property system OX may appear to be a somewhat contrived hybrid of a philosophical a topological problem. Actually, it is not. Rather, this question has attracted considerable interest among topologists since the beginnings of topology more than one hundred years ago. More precisely, we will show in a moment that the validity of LPI is equivalent to the weakest topological separation axiom T0 : 5.6 Definition Let (X, OX) be a topological space, x, y ∈ X, and a ∈ OX. The topological space (X, OX) is a T0 -space iff the for all distinct x, y there exists an open set a ∈ OX such that either x ∈ a and y ∈ / a, or x ∈ / a and y ∈ a.  The axiom T0 has often been considerated as a minimal requirement that “good” topological spaces have to satisfy. To obtain interesting specific results, however, one has to assume the validity of certain stronger separation axioms Ti, i > 0. As the starting point for the endeavor of interpreting topological separation axioms as Leibnizian principles of the identity of indiscernibles we begin with the following easily proved proposition: 5.7 Proposition A topological space (X, OX) satisfies the separation axiom T0 if and only if the topological property system OX satisfies Leibniz’s law LPI.  In order to emphasize the intimate relation between T0 and LPI, in the following let us denote LPI by LPT0. Proposition (5.7) opens the gate for constructing a close connection between topological separation axioms and Leibnizian principles of the identity of indiscernibles: T0 is just the first member of a large family of separation axioms Ti , i ≥ 0. The fact that Leibniz’s principle and T0 are equivalent suggests that not only T0 but also

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other members Ti of the family of separation axioms can be interpreted in terms of (generalized) Leibnizian principles of identity of indiscernibles. This is indeed the case as is confirmed by the following few examples (that could be easily multiplied): 5.8 Definition A topological space (X, OX) is a T1 -space iff for all distinct points x, y there exist open sets a, b ∈ OX such that x ∈ a, and y ∈ b, such that x ∈ / b and y ∈ / a.  For the corresponding Leibnizian principle one obtains: 5.9 Theorem (LPT1 -priniciple of the identity of indiscernibles) A system of properties P: = {F; F ∈ P} satisfies the T1 -Leibnizian principle of the identity of indiscernibles iff for all (LPT1 ) ∀x, y(x = y → ∃F ((Fx and not Fy)or ∃G(Gy and not Gx))) A topological space (X, OX) is a T1 -space iff the topological property system OX  satisfies (LPT1 ). There is no reason to stop here. One may go on to interpret further separation axioms as principles of the identity of indiscernibles analogously: 5.10 Definition The topological space (X, OX) is a T2 -space iff for all distinct points x, y there are disjoint open sets a, b ∈ OX such that x ∈ a and y ∈ b.  5.11 Theorem A property system P satisfies the Leibnizian principle (LPT2 ) iff for all distinct objects x and y there are properties F and G such that x has property F and all elements that have property F lack property G, and y has property G and all elements that have g lack property F.A topological space (X, OX) is a T2 -space iff  the topological property system {O(x); x ∈ X} satisfies (LPT2 ). The axioms T0 , T1 , and T2 are not equivalent, but satisfy the strict chain of implicationsT2 ⇒ T1 ⇒ T0 . Hence the principles (LPT0 ), (LPT1 ) and (LPT2 ) are not equivalent but satisfy the analogous chain of implication LPT2 ⇒ LPT1 ⇒ LPT0 . Separation axioms Ti determine important features of topological spaces. It seems plausible to conjecture that the corresponding principles of identity (LPTi ) determine important metaphysical features of the worlds for which they hold. Thus, topology turns out to provide useful means for distinguishing metaphysically different worlds in a more fine-grained way than the usual conceptual apparatus based on set theory. These examples may suffice to argue that topology is a conceptual useful device to define interesting and well-behaved property systems that give rise to non-trivial topological identity principles LPTi . To be sure, there are other spatial structures that may can also be used for this purpose, for instance, convexity structures of conceptual spaces (see Sect. 3). It should be noted, however, that convex structures and topological structures are structurally rather similar: Both can be defined by closure operators (cf. van de Vel 1993).

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6 Concluding Remarks Not just any topological structure is appropriate as a means for distinguishing between good and not-so-good, i.e., natural and not-so-natural concepts. The point is to find illuminating topologies for specific purposes. This may be difficult: It may turn out that a certain (topological or convex) structure imposes formal conditions on naturalness that are unacceptable for intuitive reasons. If this is the case, the nit has be replaced by another, more appropriate structure. Hopefully, thereby an equilibrium between our intuitions concerning naturalness and the formal conditions imposed by the spatial structures may be achieved. Since the end of 19th century a general topological concept of space has been developed that comprises the traditional Euclidean concept only as a very special case. Philosophy should attempt to exploit this newly gained general perspective. Then, topology as a general theory of spatial structures may serve as a kind of conceptual toolkit at least as versatile and fruitful as traditional logic. In terms of traditional philosophy of science the task of finding a “good” topology may be described as the task of formulating a kind of relativized or historized a priori element that is expressed in topological (=geometrical) terms. This thesis may be elucidated by briefly recalling an essential ingredient of Kant’s original account that still plays an essential role in the contemporary discussion, namely, the a priori and its role in the constitution of the objects of science. For Kant, Euclidean geometry of space functioned as a constitutive framework for physics. This framework was not empirical rather, it first rendered possible properly empirical discoveries (cf. Friedman 2001, 62). For example, Euclidean geometry enables us to represent physical phenomena such as the paths of particles, their velocities, and the forces that act on them geometrically or topologically, i.e., by certain spatial structures—for instance, by vectors, tensors, differential forms, and many other geometrical constructs. Since Einstein’s relativity theories one knows, however, that the framework of Euclidean geometry is not the only possible one. There is more than one constitutive geometrical framework possible, and these different frameworks compete with each other with respect to simplicity, fruitfulness, precision, and other theoretical and practical virtues. The different frameworks compete with each other. To take into account the dynamic aspects of the evolution of scientific knowledge, a particular structuralization of a conceptual space is to be conceived as only one phase in a process of the evolution of ever more sophisticated conceptual structures. For the philosophical discussion concerning the role of a priori aspects of scientific knowledge, this means that pragmatic aspects of the a priori become increasingly important in post-Kantian philosophy of science (see Sects. 4 and 5). The presence of spatial structures of some kind or other everywhere in the representation of empirical knowledge provides a good argument for the thesis that topology may be a fruitful source for the formulation of a relativized or historized Kantian a priori. A detailed elaboration of this thesis is a task for future work.

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References Aiello, M., Pratt-Hartmann, I. E., van Benthem, J. F. A. K. (Eds.) (2007). Handbook of spatial logics. Springer. Cassirer, E. (1910). Substanzbegriff und Funktionsbegriff . English edition: Curtis, W. C., Curtis M. C. (1923). Substance and function. Chicago, London: The Open Court Publishing Company. Cassirer, E. (1928). Zur Theorie des Begriffs. Bemerkungen zu dem Aufsatz von Georg Heymans. Kant-Studien, 33(1–2), 129–136. Douven, I., Gärdenfors, P. (2018). What are natural concepts? A design perspective. Mind and Language. Forrest, P. (2016). The identity of indiscernibles. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy, https://plato.stanford.edu/archi–ves/win2016/entries/identity-indiscernible/. Friedman, M. (2001). Dynamics of reason. Stanford/California: CSLI Publications. Gärdenfors, P. (2000). Conceptual spaces. The geometry of thought. Cambridge-Massachusetts: The MIT Press. Gärdenfors, P. (2014). The geometry of meaning, semantics based on conceptual spaces. Cambridge/Massachusetts: The MIT Press. Johnstone, P. T. (1982). Stone spaces. Cambridge: Cambridge University Press. Lakoff, G. (1987). Women, fire, and dangerous things. What categories reveal about the mind. Chicago: University of Chicago Press. Lewin, K. (1936). The principles of topological psychology. McGraw Hill, New York and London. Mac Lane, S., & Moerdijk, I. (1992). Sheaves in geometry and logic. A first introduction to topos theory. Springer. Mormann, T. (1993). Natural predicates and topological structures of conceptual spaces. Synthese, 95(2), 219–240. Peruzzi, A. (2000). The geometric roots of semantics. In L. Albertazzi (Ed.), Meaning and cognition (pp. 169–202). Amsterdam: John Benjamins. Petitot, J. (1995). Morphodynamics and attractor syntax: constituency in visual pattern perception and cognitive grammar. In R. F. Port & T. Van Geldern (Eds.), Mind as motion (pp. 224–281). Cambridge/Massachusetts: MIT Press. Rumfitt, I. (2015). The boundary stones of logic. An essay in the philosophy of logic. Oxford: Oxford University Press. Steen, L. A., Seebach Jr., J. A. (1978). Counterexamples in topology. Springer. Stone, M. H. (1936). The theory of representations for boolean algebras. Transactions of the American Mathematical Society, 40, 36–111. Stone, M. H. (1937). Applications of the theory of boolean rings to general topology. Transactions of the American Mathematical Society, 41, 345–481. Wildgen, W. (2000). The history and future of field semantics: from Giordano Bruno to dynamic semantics. In L. Albertazzi (Ed.), Meaning and cognition (pp. 203–226). Amsterdam: John Benjamins. Wildgen, W. (2001). Kurt Lewin and the Rise of “Cognitive Science” in Germany: Cassirer, Bühler, Reichenbach. In L. Albertazzi (Ed.), The dawn of cognitive science, early European Contributions, series synthese (pp. 299–332). Dordrecht: Kluwer. Van de Vel, M. L. J. (1993). Theory of convex structures. Amsterdam, North-Holland: Elsevier.

The Difficulty of Neutrality A Graph-Theoretical Solution Caterina Del Sordo

Abstract This paper deals with the philosophical movement of neutral monism. The main thesis of neutral monism (NM) is that the fundamental entities of empirical reality are neutral, that is, neither mental nor physical. In particular, this paper focuses on the concept of neutral entities. The question arises as to whether and to what extent it is possible to give a definition of a neutral entity without contradicting NM. First, the paper provides an overview of the definitions of neutral entities that the literature currently offers. Thereafter, the proposal of Carnap (1928) and Dipert (1997) which defines neutral entities by using graph-theoretical structures is discussed in detail.

1 Introduction This investigation aims to provide a contribution to the philosophical area of ontology. Broadly speaking, ontology seeks a general theory of existing and non-existing entities. Clearly, an appealing ontological theory should not resemble a world catalogue. Rather, it should try to obtain much via very few assumptions. One of the cheapest ontological theories that the philosophical scene currently offers is neutral monism. The original authors of neutral monism were Ernst Mach, William James and Bertrand Russell. In general, neutral monism maintains that only one type of entity exists and that the entities of this type are neutral, that is, neither mental nor physical. All the other empirical entities, from mental contents to ordinary and scientific entities, arise as appropriate configurations of the neutral ones. Regrettably, the literature on neutral monism lacks a clear definition of “neutral entities”. Abriefhistorico-philosophical overview shows that to make neutral monism a viable ontological theory an appropriate definition of neutral entities is urgently needed. C. Del Sordo (B) University of Urbino, Urbino, Italy e-mail: [email protected] University of the Basque Country UPV/EHU, Leioa, Spain © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 A. Peruzzi and S. Zipoli Caiani (eds.), Structures Mères: Semantics, Mathematics, and Cognitive Science, Studies in Applied Philosophy, Epistemology and Rational Ethics 57, https://doi.org/10.1007/978-3-030-51821-9_8

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Ernst Mach called his neutral entities “elements” (see, for example, Mach 1886, p. 16) without ever settling on a stable definition of them. This has made the literature on Mach’s neutral monism oscillate between two tendencies. On the one hand, one finds the classical, idealistic interpretation of Mach’s elements, which interprets elements as sensations and Mach’s neutral monism as a form of phenomenalism (see, for example, Schlick 1918, Sect. 25). On the other hand, one finds a recent, realistic interpretation of Mach’s elements, which interprets them as physical fundamental forces and Mach’s neutral monism as an enhanced form of physicalism (see Banks 2014, Chap. 6 and passim). The literature on Russell’s neutral monism features the same oscillation. Russell outlines his neutral entities as an “intrinsic character of matter” (see Russell 1927, p. 384) without giving a clear definition of it. This has also made the literature on Russell’s neutral monism oscillate between two tendencies. On the one hand, one finds the interpretation of Stoljar (2001) and Montero (2015), who conceive Russell’s intrinsic character of matter as something physical. Therefore, they hold that Russell’s neutral monism is a form of physicalism. On the other hand, one finds the interpretation of Chalmers (2002, 2015), who conceives Russell’s intrinsic character of matter as something psychical, or proto-phenomenal. Accordingly, one may say that Russell’s neutral monism is a form of phenomenalism or proto-phenomenalism. Finally, William James called his neutral entities “pure experiences” (see, for example, James 1977, p. 170) without providing a definition that separates them sharply from other psychical and physical entities. Accordingly, James’s neutral monism has also received various interpretations, ranging from naturalism to phenomenology and panpsychism (see Cooper 2002, Chap. 2). This paper has two main objectives: first, to systematize the problem of defining neutral entities and, second, to discuss a solution to it. The second paragraph of this paper (Sect. 2) divides neutral monism into three main currents and provides the problem of defining neutral entities with an analytical systematization. I give this problem the name of the “difficulty of neutrality”. Three solutions to the difficulty have been proposed to date: the solution via graphs (SG), the solution via elements (SE) and the solution via quiddities (SQ). As the graph-theoretical solution appears to be the basic one, the third and fourth paragraphs undertake its explanation and discussion. Therefore, in Sect. 3 a graph-theoretical definition of neutral entities is introduced. The question then (Sect. 4) arises as whether the definition via graphs is actually free from the difficulty of neutrality or not. As will become clear, none of the SG supporters offer a fully-fledged solution to this problem. In conclusion, in Sect. 5 the paper briefly sketches the definitions of neutral entities provided by the solution via elements and the solution via quiddities. Clearly, they deserve their own extensive discussions. Here, I only mention some of their fundamental aspects to finally construct a preliminary framework for a proper definition of neutral entities.

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2 The Difficulty of Neutrality The literature on neutral monism is divided into three main currents: (1) structural neutral monism (SNM); (2) classical neutral monism (CNM); (3) Russellian neutral monism (RNM). All these currents share the following thesis: (NM), the fundamental entities of empirical reality are neutral, i.e., neither mental nor physical. Structural neutral monism has its roots in Carnap’s early idea of relational reconstruction of empirical reality that Carnap developed in two early manuscripts, Vom Chaos zurWirklichkeit (1922) and Die Quasizerlegung (1923), and in Der logische Aufbau der Welt (1928). Classical neutral monism originates from the ontological theories one finds in many writings of Ernst Mach, for example in Analysis of Sensations (1886) and in Knowledge and Error (1905), and in William James’s Essays in Radical Empiricism (1912). Russellian neutral monism stems from the ontological and epistemological theses of Bertrand Russell’s Analysis of Mind (1921) and Analysis of Matter (1927).1 The main problem that plagues the literature on neutral monism stems from the purported psycho-physical neutrality of the fundamental entities and arises as soon as one tries to provide a definition of them without contradicting (NM). The problem of defining neutral entities arises as a difficulty, which I refer to as the difficulty of neutrality. As one takes empirical reality to be the touchstone of our judgements on ordinary or scientific objects, the fundamental entities can be either (i) transcendentally constituted, to wit, produced by an impersonal subjectivity of some kind, or (ii) mind-independent. Well, if via (i) one holds that neutral entities are transcendentally constituted, then this contradicts (NM), for neutral entities would somehow end up being mental. Via (ii), if neutral entities are mind-independent, then they must be either (ii.1) physical or (ii.2) not physical. Clearly, when following (ii.1), one blatantly contradicts NM. When following (ii.2), one finds two alternatives. Indeed, if neutral entities are not physical, then they should be either (ii.2.1) mental or (ii.2.2) abstract. On the one hand, via (ii.2.1) one directly contradicts NM. On the other hand, via (ii.2.2) one does not directly contradict (NM), but the position will be at least strongly at odds with the empirical constraints of (NM). One may illustrate the difficulty of neutrality by means of the following tree-like diagram (Fig. 1).2

1 According

to some authors (see for example Banks 2003, 2014), Russell forms with Mach and James the triad of CNM. However, as the divergences in the theory of knowledge between Russell’s neutral monism and Mach’s and James’s are crucial for further investigations, I will keep Russell separate from CNM (along the same lines, see, for example, Bostock 2012, Chap. 11). 2 “n.e.” stands for “neutral entities”.

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Fig. 1 Tree-like diagram illustration of the difficulty of neutrality

As one sees, all the left-side branches of the difficulty are closed as they directly contradict (NM). Branch (ii.2.2) does not directly contradict (NM), and one may consider its respective branch as partially open or closed.3 The literature on neutral monism basically has three ways of finding a solution: SNM: the solution via graphs (SG), CNM: the solution via elements (SE), RNM: the solution via quiddities (SQ). The solution via graphs is cheaper than the others and is somehow entailed by them. It deserves an appropriate examination first to better understand the other two. For this reason, this paper explains and examines the solution via graphs. The discussion of the solutions via elements and via quiddities will be addressed by later contributions.

3 The Solution via Graphs The solution via graphs has been proposed by structural neutral monism. It was developed in the works of Carnap (1922, 1923, 1928) and Dipert (1997). According to this solution, neutral entities are conceived as vertices of an asymmetric graph whose appropriate subgraphs constitute all the ordinary and scientific entities (a world graph, as it were). A graph G is a structure that consists of a set V(G) of

3 This is a basic version of the difficulty of neutrality. It can be enriched according to the definitions

of mental, physical and abstract entities one introduces. Enriched versions can be introduced and discussed in the light of the various kinds of monism arising from the Russellian neutral one (see for example Stoljar 2013 and Coleman 2017). The basic version of the difficulty suffices for the purposes of this paper.

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Fig. 2 Three examples of (“small”) graphs

objects called vertices together with a set E(G) of unordered pairs of vertices called edges. Some examples of (“small”) graphs are given below (Fig. 2). A vertex v is describable according to its degree, to wit, the number of edges that have v as an endpoint, and according to the degrees of its adjacent vertices. A graph is asymmetric if each vertex is univocally identifiable by its degree and the degrees of its adjacent vertices. For example, among graphs (A), (B) and (C), only graph (C) is asymmetric. Taking the example of graph (A), with the aid of labels, one may see that only vertex c is univocally identifiable (Fig. 3). Indeed, a and e are both of degree 1 and are adjacent to vertices of degree 2. Vertices b and d are both of degree 2 and adjacent to one vertex of degree 1 and to one of degree 2. Vertex c is the only vertex of degree 2, which is adjacent to only vertices of degree 2. Taking the example of graph (B), one may see that none of the vertices is univocally identifiable, as they are all of degree 2 and adjacent to vertices of degree 2. Instead, one sees that graph (C) is asymmetric (Fig. 4). Indeed, f is of degree 1 and is adjacent to a vertex of degree 3. Vertex mis of degree 1, as well as f, but mis adjacent to a vertex of degree 2. Vertex gis of degree Fig. 3 Example of a graph containing only one univocally identifiable element

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Fig. 4 Example of asymmetric graph

3, as well as i, but g is adjacent to a vertex of degree 1, while i is not. Vertex h is of degree 2 and is adjacent to vertices of degree 3. Vertex l is also of degree 2 but is adjacent to a vertex of degree 1. The attractiveness of SG lies in its identification of neutral entities with pure structural entities, viz. the vertices of an asymmetric world graph. This choice avoids both referring to physical substrata and resorting to some cognitive faculties to distinguish otherwise undetectable entities. Therefore, SG purportedly avoids facets (i), (ii.1) and (ii.2.1) of the difficulty of neutrality, that is, all the closed branches of the diagram. However, it has to cope with the drawbacks of (ii.2.2). Indeed, to ensure that SG is a solution to the difficulty, it must serve the purpose of making up empirical reality.

4 Weak Neutral Monism and Strong Neutral Monism Both Carnap (1928) and Dipert (1997) consider empirical reality to be composed of two parts: a purely experiential and subjective one and a purely structural and objective one. Together with this alignment, their respective proposals of SG exhibit a paradigmatic divergence. Carnap aims to reconstruct or constitute via SG only the structure of empirical objects and considers their experiential content as already given from the outset by means of unanalysable Elementarerlebnisse. Dipert, instead, aims to construct via SG both the experiential and structural components of empirical objects. This divergence brings different scopes to the drawbacks of (ii.2.2). Following Carnap, SG serves to constitute a structural description of empirical reality. The drawbacks of (ii.2.2) arise as ambiguities that one may encounter in ascribing a structural definite description to empirical objects. To address this problem, a concrete example will be useful. It can be seen how structures arise in empirical reality when one learns, for example, how to recognize objects in the starry sky. Initially, the starry sky looks like a chaos of dots. To give the dots a stable label, one starts distinguishing some asterisms, namely, patterns of dots. The starry sky can be considered a huge graph such that its vertices represent the stars and its edges represent the patterns of stars, the asterisms and the constellations. If one has to

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Fig. 5 Picture of a starry night

Fig. 6 Finding Polaris by drawing asterisms (I)

identify the Polaris’s dot, for example, then one has to find a structural description for it, otherwise according to its experiential content, it would still look like many other dots (Fig. 5). Therefore, one needs to embed Polaris’s dot in a super-pattern, or hyperedge,4 by imagining a new connection line to the Pointers, alpha and beta, of the Big Dipper (Fig. 6). Either because of a very starry night or because of poor visibility of the Pointers, it might happen that two dots have connection lines similar to those of Polaris. Hence, 4 The

hyperedge generalizes the concept of edge in the definition of the hypergraph. In particular, a hypergraph is a generalization of a graph G in which E(G) is considered to be a family of unordered tuples of vertices called edges, or hyperedges.

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Fig. 7 Finding Polaris by drawing asterisms (II)

one needs to enlarge the super-pattern, to wit, the hyperedge composed by Polaris and alpha and beta as vertices, by imagining another new connection line with one of Cassiopeia’s stars. If other ambiguities still occur, then one may detect the asterism of the Little Dipper and single out Polaris’s dot as its endpoint (Fig. 7). According to Carnap (1928), whenever structural ambiguities are obtained, two alternatives occur: Where such a definite description is not univocally possible, the object domain must be enlarged or one must have recourse to other relations. If all relations available to science have been used, and no difference between two given objects of an object domain has been discovered, then, as far as science is concerned, these objects are completely alike, even if they appear subjectively different.5

On the one hand, two empirical objects may share the same structural description because of a lack of information. In the case of Polaris, for example, one fills in the lacuna by enlarging the first super-pattern, composed of Polaris and the two Pointers of the Big Dipper, to Cassiopeia and then, if this is still not sufficient, to the Little Dipper. On the other hand, if later on the two objects still share the same structural description, then it means that they are structurally the same and differ only in some subjective aspects, such as, for example, appearing in two different moments of one’s sequence of mental representations. Accordingly, one may say that Carnap’s SG overcomes the drawbacks of (ii.2.2) by relying on the dynamical increase and enrichment of structural information of both scientific and ordinary knowledge. Following Dipert instead, SG serves to construct both the structural and experiential components of empirical reality. On the one hand, as far as the structural component is concerned, the drawbacks of (ii.2.2) along with their settlement turn 5 Carnap

(1928), p. 27.

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out to be the same as in Carnap’s SG (see Dipert 1997, pp. 350–6). On the other hand, as far as the experiential component is concerned, the question arises as follows: how is it possible to construct the experiential content of material and psychical objects from scratch out of pure abstract entities? Dipert offers no consistent solution to this problem. Rather, to cope with it, he ends up undertaking panpsychism, as the quote below shows: The entities closest to a common-sense notion of an entity will be connected subgraphs […] Physical objects, even the finest subatomic particles, certainly do not correspond to vertices. Instead, they themselves are composite entities, subgraphs of the world graph […] Thoughts are also subgraphs of a certain sort […] There might first seem to be no place in these cold graphs for minds, consciousness, and other mental phenomena – unless, that is, everything is mental. […] we should perhaps consider seriously the possibility that something like the panpsychism of Spinoza, Leibniz and Peirce is true, and that vertices are pure feelings […] constituting distinct thoughts and objects only when connected to other such entities.6

To summarize, Carnap’s SG relies on a weak version of neutral monism, as it were, where abstract neutral entities constitute the structural components of empirical reality. Dipert’s SG instead relies on a strong version of neutral monism, where both the structural and experiential components are constructed out of abstract neutral entities. Carnap overcomes the drawbacks of (ii.2.2) at the price of a weak neutral monism. In addition, if one tries to develop a strong neutral monism via Carnap’s SG, then the experiential component of empirical objects could not, in any case, be constituted, as other difficulties would then arise (see, for example, Goodman’s difficulty of companionship7 ). Dipert proposes a strong neutral monism, but to avoid the drawbacks of (ii.2.2), he finds himself embracing panpsychism. This way, he ends up undertaking an option that SG initially succeeded to avoid, namely, that of the facet (ii.2.1), which directly contradicts NM.

5 Conclusions A brief outline of the solutions via elements and via quiddities is in orderhere. The solution via elements has been proposed by classical neutral monism. It consists of an event-based ontology in which neutral entities are identified with not-compounded and not-repeatable events, called “elements” (see Mach 1872, 1886, 1905 and Banks 2003, 2014). Here, the difficulty arises as soon as one is asked to specify what type of events elements are. The solution offers two different answers thereto, an historicoscientific one and a graph-theoretical one. The first answer defines elements by using an historico-scientific approach and identifies them with the fundamental entities of contemporary physical inquiry. The second answer basically follows SG and identifies elements with vertices of an asymmetric world graph. Following the first answer, the solution via elements ends up undertaking the facet (ii.1) of the difficulty 6 Dipert

(1997), pp. 352–8 (emphasis mine). (1951), pp. 160–1.

7 Goodman

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of neutrality, which directly contradicts (NM). Following the second answer, it must cope with the drawbacks of (ii.2.2), as well as SG. None of the two answers fully overcomes the difficulty of neutrality. The solution via quiddities has been developed by Russellian neutral monism. According to SQ, neutral entities can be considered noumenic entities, otherwise known as “quiddities” (Langton 1998; Lewis 2009; Chalmers 2015). Quiddities underlie empirical reality but cannot be known either by ordinary or scientific knowledge. On the one hand, one may take SQ as a promising proposal, as it does not overtly undertake any facet of difficulty of neutrality. However, on the other hand, one may also take it as an idle proposal, as it offers no positive definition of neutral entities either. In conclusion, to define neutral entities without running into the difficulty of neutrality, two preliminary tasks must be carried out. First, one has to dismiss those solutions that lead to contradiction with (NM), such as Dipert’s strong neutral monism and the historico-scientific approach of SE. Thereafter, one shall reconsider only those solutions that develop either an open branch of the difficulty, such as the graphtheoretical approaches of SG and SE, or other theoretical alternatives. Accordingly, three options will deserve to be examined and revised: the weak neutral monism of SG, the graph-theoretical approach of SE and, finally, SQ. Indeed, the weak neutral monism of SG and the graph-theoretical approach of SE succeed in providing a positive definition of neutral entities without contradicting (NM). Additionally, SQ, although without providing a positive definition of neutral entities, unfolds epistemological considerations that might be useful to prevent sceptical scenarios. Clearly, none of the three options has provided a fully satisfactory solution to the problem, but they all contain useful theoretical hints that deserve to be developed.

References Banks, E. (2003). Ernst Mach’s World Elements: A Study in Natural Philosophy, Dordrecht: Kluwer Academic Publisher. Banks, E. (2014). The realistic empiricism of Mach, James, and Russell. Neutral Monism reconceived: Cambridge University Press, Cambridge. Bostock, D. (2012). Russell’s Logical Atomism, Oxford: Oxford University Press. Carnap, R. (1922). Vom Chaos zurWirklichkeit, Unpublished Ms., Archive of Scientific Philosophy, Special Collections Department, Hillman Library, University of Pittsburgh, RC-081-05-01. Carnap, R. (1923). Die Quasizerlegung—Ein VerfahrenzurOrdnungnichthomogener Mengen mit den Mitteln der Beziehungslehre, Unpublished Manuscript, Archive of Scientific Philosophy, Special Collections Department, Hillman Library, University of Pittsburgh, RC-081-04-01. Carnap, R. (1928). Der logische Aufbau der Welt. Berlin: Weltkreis Verlag. Cooper, W. (2002). The unity of William James’s Thought. Nashville: Vanderbilt University Press. Chalmers, D. (2015). Panpsychism and panprotopsychism. In T. Atler & Y. Nagasawa (Eds.), Consciousness in the physical world. perspectives on Russellian monism. New York: OUP. Chalmers, D. (2002). Consciousness and its place in nature. In D. Chalmers (Eds.), The blackwell guide to the philosophy of mind. Oxford: Blackwell reprinted in Philosophy of mind: classical and contemporary readings (pp. 247–272). New York: OUP. Coleman, S. (2017). Panpsychism and neutral monism. How too make up one’s mind. In G. Brüntrup & L. Jaskolla (Eds.), Panpsychism. contemporary perspectives. New York: OUP.

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Dipert, R. (1997). The mathematical structure of the world: the world as graph. The Journal of Philosophy, 94(7), 329–358. Goodman, N. (1951). The structure of appearance. Harvard: Harvard University Press. James, W. (1977). The writings of William James. J. J. McDermott (Ed.), Chicago: Random House. James, W. (1912). Essays in radical empiricism, Reprinted: University of Nebraska Press, Lincoln, 1996. Langton, R. (1998). Kantian humility. our ignorance of things in themselves. New York: OUP. Lewis, D. (2009). Ramseyan humility. In D. Braddon-Mitchell & R. Nola (Eds.), Conceptual analysis and philosophical naturalism. Cambridge: MIT. Mach, E. (1905/1976). Knowledge and error—sketches on the psychology of enquiry (trans. T. J. McCormack & P. Foulkes). Dordrecht: D. Reidel. Mach, E. (1886/1959). The analysis of sensations (trans. C. M. Williams & S. Waterlow) New York: Dover. Mach, E. (1872/1910). The History and Root of the Principle of the Conservation of Energy (P.E.B. Jourdain trans.). Chicago: Open Court. Montero, B. (2015). Russellian physicalism. In T. Atler & Y. Nagasawa (Eds.), Consciousness in the physical world. Perspectives on Russellian monism. New York: OUP. Russell, B. (1921). The analysis of mind. London: George Allen & Unwin. Russell, B. (1927). The analysis of matter. London: Kegan Paul, Trench, Trubner. Schlick, M. (1918/1974). General theory of knowledge (trans. A. E. Blumberg). Wien/New York: Springer. Stoljar, D. (2001). Two conception of the physical. Philosophy and Phenomenological Research, LXII, 22. Stoljar, D. (2013). Four kinds of Russellian monism. In U. Kriegel (Ed.), Current controversies In philosophy of mind. New York: Routledge.

Structures, Archetypes, and Symbolic Forms. Applied Mathematics in Linguistics and Semiotics Wolfgang Wildgen

Abstract A basic question in the philosophy of mathematics concerns the analytic or the synthetic character of mathematics (cf. Kant). Since the mathematics of Greece, a relevant mapping of geometry to phenomena of space (architecture, astronomy, etc.) and musical scales had been accepted. In his philosophy of symbolic forms, Ernst Cassirer considered a gradient that links mathematics with language and myth. They have in common the symbolic mapping of phenomena in human apperception to systems of signs. This chapter concerns the relation between mathematics and linguistic, visual, and musical signs. In a first step, we reconsider Plato’s geometrical explanation of the universe and the human mind in his dialogue “Timaeus” and the use of a similar strategy in René Thom’s topological semantics that applies major results of catastrophe theory. Thom’s derivation of a list of semantic archetypes and their linguistic applications are summarized regarding results in Wildgen (1982). Beyond the semiotics of language, a further application shows the relevance of geometrical archetypes in classical paintings (e.g. Leonardo da Vinci) and tonal music (e.g. the blues schema). The final remarks consider evolutionary aspects of archetypal structures in semiosis.

1 With the increase of distance, new and important values are added. These are mainly: the greater independence from actual use and its contexts, the capacity of generalization, and thus of further insights useful in later uses, and possibly a global understanding.

W. Wildgen (B) Universität Bremen, Waibliner Weg 16, 28215 Bremen, Germany e-mail: [email protected] Institut für Allgemeine und Angewandte Sprachwissenschaft, Bremen, Germany

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 A. Peruzzi and S. Zipoli Caiani (eds.), Structures Mères: Semantics, Mathematics, and Cognitive Science, Studies in Applied Philosophy, Epistemology and Rational Ethics 57, https://doi.org/10.1007/978-3-030-51821-9_9

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1 Introduction The philosophy of symbolic forms conceived and developed by Ernst Cassirer (1874– 1945) between 1920 and 1945 establishes epistemological parallelism between different types of cognitive grip humans have on reality. A scale of more and more rational apprehension of reality links the symbolic forms: myth > language > science (mathematics). In myth, the distance between the cognitive capture and reality is minimal. In language, the distance is larger and most users become aware of the arbitrariness of words concerning their correlates (referents). In the mathematical sciences (e.g. theoretical physics) individual references seem to dissipate, i.e. the distance is maximal.1 What is left is a schematic or diagrammatic seizure of major aspects of reality (e.g. physical laws). Mathematics since the program of Erlangen (Felix Klein) and the program of axiomatization by Hilbert are further abstracted to find a way to basic invariants and concepts of order and structure. If Cassirer considered the scale leading to mathematics (applied in the sciences) starting from language (with a look back to myth), René Thom (1923–2002) started from new results of stability theory and differential topology (mid-fifties until the seventies of the last century) and looked for deep regularities in biology and language. His semantic archetypes are not an abstraction from mathematics towards logics but in the opposite direction instantiations, interpretations of mathematically sophisticated theorems in fields beyond the classical applications of mathematics (physics and more recently chemistry), i.e. in biology (morphogenesis), linguistics (valence patterns, narratives) and semiotics. My major question will be: Is such an applicative elaboration of mathematics into the domain of human behavior (such as speaking and creativity in visual art and music) a possible and rewarding strategy? I will discuss this topic with Thom’s catastrophe theoretical model of verbal valence and consider fundamental structures in visual art, specifically in some paintings of Leonardo. Eventually, I ask if tonal music exhibits archetypes comparable to those characteristic for language.

2 Mathematics and the Problem of Modeling in the Humanities Since the rise of science and reflections on science in antique Greece, the enigma of the intellectual and empirical relevance of mathematics was a basic concern of philosophy. Mathematics is neither the direct consequence of psychic realities, i.e. individual, spontaneous activities of the mind nor is it the effect of external, empirically accessible conditions. It rather unfolds its effectiveness in a kind of world between a subjectivity based on the senses and objectivity linked to the surrounding nature. The “ideas” of Plato try to catch this intermediate and seemingly independent reality. Geometry and its principles which were later the content of Euclid’s demonstrations seemed to be the hidden kernel of reality, whereas arithmetic with the law

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of repetition seemed to be more practical, linked to human action. In Kant’s philosophical development the same problem led to the concept of synthetic a priori, i.e. principles which are necessary for empirical insight and discovery but are themselves not derived from empirical knowledge. Cassirer has described Kant’s itinerary on behalf of this problem in the second volume of his treatise “Das Erkenntnisproblem in der Philosophie und Wissenschaft der neueren Zeit” (Cassirer 1922b: book 8). Since Kant’s time, the world of mathematics has exploded and the seemingly monolithic field has been at the same time diversified and (partially) unified, e.g. in Felix Klein’s Erlangen Program, in Hilbert’s program, in the set-theoretical axiomatization of the Bourbaki group, and category theory. The disciplines beyond physics, such as biology, psychology, and linguistics came into the reach of mathematical methods and the discussion in Kant (1971) had to be enlarged: How is knowledge in biology, psychology, linguistics beyond empirical description possible and what kind of mathematics, which province or subfield of mathematics should be the intellectual instrument of such knowledge. In the forties and fifties, Jean Piaget tried to show that the Bourbaki program and set-theoretical models may be the key to general insights in psychology (genetic psychology); Chomsky and in the sequel Artificial Intelligence (A.I.) saw logical calculus and computer programs as the key to advanced knowledge in linguistics and psychology. Cassirer considered in the last years of his life Klein’s Erlangen Program and its relation to the psychology of perception as a possible application. He had, however, major objections against such a direct comparison or even mirroring of a mathematical in an empirical field (cf. Cassirer 1945; Wildgen 2009a). In his philosophy of symbolic forms, Cassirer broadened the epistemological view by considering a larger field of basic knowledge accesses ranging from myth to language and at the end to mathematically specified (natural) science. Later he added art, technology, and ethics (law) to the set of symbolic forms. The question of the kind of mathematical tools adequate for each of these symbolic forms will be my major topic in the following.2

3 The Diversity of Symbolic Forms and the Search for Fundamental Structures (“Structures Mères”) The philosophy of symbolic forms was first sketched by the philosopher Ernst Cassirer in a conference at the Warburg Institute in Hamburg in 1921. This philosophy aimed to consider the expression of something in the mind (“Geistiges”) by

2 In

the fourth chapter of his “Phänomenologie der Erkenntnis”, Cassirer says (with Leibniz) that mathematics should have a proper “fundamentum in re”, i.e. only in the context of scientific theories do they have the status of a symbolic form, become operative as conceptual tools for the enhancement of scientific knowledge. Cf. Cassirer, [1929] 1982: 418f. I thank the reviewer for his question on this behalf.

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signs accessible to the senses (“sinnliche Zeichen”). In the twenties, Cassirer (1923– 1929) developed this philosophy in three volumes: (1) Language, (2) Myth and (3) the Phenomenology of Knowledge (science). This sequence implies a developmental hierarchy: myth could have preceded full-fletched linguistic forms, such as narratives, and science presupposes language. Later (in the forties) Cassirer wrote two articles on art as a symbolic form. In the present paper, I shall just consider some basic questions of this issue. 1. If we consider a plurality of symbolic forms, is there a set of basic principles (archetypes) governing the different genres of symbolic forms? Our focus will be on language, visual art, and music. Is there a set of archetypal structures governing these symbolic forms? For biological reasons, they are surely not inborn ideas. As the symbolic forms depend on different types of perception via different organs, they should be constantly emergent (self-organized) patterns, whose epistemic nature has to be further elucidated. René Thom‘s theory of “saillance” and “prégnance” could be the key for such an elaboration; cf. Thom (1988), Wildgen (2010b), and for the underlying epistemology, Wildgen (2020) forthcoming. 2. In mathematics, different programs tried to find such archetypes for mathematically/logically organized sciences, e.g. Felix Klein’s Erlangen program which puts group theory into the center, the structuralism of the Bourbaki group which starts from the axioms of set theory or in more recent time category theory. In the context of the structuralism of the Bourbaki group, the term “structures mères” had been coined. The major concern was the unification of mathematics based on specific far-reaching subfields like set-theory. Category theory has been introduced by S. Eilenberg and S. MacLane in 1942. It tries to create an overarching theory which reorganizes the three major fields: sets, groups and topological spaces, using analogies between these three domains and generalizing their basic notions to objects, morphisms, isomorphisms and products (cf. Mac Lane 1998). In the spirit of Cassirer’s critic of the application of group theory to psychology (Cassirer 1945), it is not evident that the archetypes of mathematical architecture (structures mères) must also be the archetypes of the basic symbolic forms, such as language or art. Implicitly this is also a criticism of the program of Jean Piaget who tried to mirror the architecture of mathematics in Bourbaki’s program to levels of development in the psychology of cognition and language.3 We will argue that the archetypes of the genres of symbolic forms can take profit of some mathematical tools but that they are not specifically related to 3 In the epistemology of Analytic Philosophy (e.g. in the work of Wolfgang Stegmüller) one tries to

reorganize different empirical theories based on set theory and logic (which are thus assumed to be universal tools for the organization of empirical knowledge). In the continuity of this program, Mormann proposed to use category theory for the same aim (cf. Balzer and Moulines 1996: Chap. 13). Applications to linguistics and more specifically to semantics are rare if not inexistent. Insofar as all other formalizations can be reanalyzed using category theory, most algebraic grammars and logical semantics should allow for a reformulation in the context of category theory. This will however not bring new insights for linguists.

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unification programs inside mathematics and logics. They must rather be evaluated concerning a deeper understanding of these genres of symbolic forms. The fact that logical and mathematical tools like predicate logic, formal languages, computer languages or free algebras can be used for modeling in linguistics is for the main part due to the analogy between language and the logical tools which since Aristotle’s logics took natural language as a kind of intuitive starting point. 3. Mathematics is a necessary and advanced tool if we want to understand the basic genres of human access to reality (Cassirer’s symbolic forms). The dismissal of mathematical tools by Lakoff, Talmy, and Langacker is an over-reaction against the dominant trend of Chomskyan linguistics and in the sequel of Logical Grammar (it’s rival since the work of the logicians Montague (1974), Barwise and Perry (1983), and others).4 One should consider the problem of applying mathematics rather in the perspective of other empirical sciences, such as biology, psychology, sociology, etc. and their use of mathematics. A set of relevant insights is given empirically; their mathematical analysis concerns the proper organization of pertinent empirical descriptions and their generalization. This allows us to formulate new and far-reaching questions and programs. I will not dive into the difficulties in the interaction of empirical science and mathematics, but as Kant observed already before 1800, there is a touch of synthesis in mathematics which makes it relevant for the growth of scientific knowledge. 4. There are two major questions left: (1) Does the human capacity to use and invent symbolic forms which has led to different genres have a common source, a general denominator? In an evolutionary perspective, the roots may even lie beyond the human species and may have emerged later given the specific conditions created by human evolution (e.g. with the rise of large-scale cultures). (2) How did these genres of symbolic forms which may have existed and coevolved during long periods interact? Could this co-evolution explain the easiness of blends between the different genres? Simple technologies and a human protolanguage may have co-existed since the Homo erectus (2 million years B.P.), The use of colors and body adornment may have evolved since 1 million years B.P. Art, painting, sculpture and music could have emerged since the era of the second Out-of-Africa migration (100.000 to 50.000 years B.P.). In this period, the protolanguage of Homo erectus had already found the form characteristic for all living humans today (probably with the rise of species Homo sapiens 300.000 years B.P.). The use of color and adornments are witnessed since a million years, visual art and musical instruments are attested since 50.000 years. I assume that the archetypal structures of language, art, and music which I will discuss further coexisted in the period since speciation and interacted producing various blends. Paleolithic images and the use of abstract graphical signs coexisted in the Paleolithic caves (cf. Wildgen 2004b, 80–83) and first writing systems 4

The critique of George Lakoff is directed against Chomskyan and logical models in linguistics. Lakoff (1987): Part II is a critique if the Objectivist Paradigm “reality is structured in a way that can be modeled by set-theoretical models” (ibid.: 159) and of the Formalist Enterprise “natural language syntax and semantics is just a special case of formal syntax and semantics” (ibid.: 219).

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emerged ca. 8.000 years ago. Early forms of music certainly used the human voice and thus created songs (a blend of music and language). The blend of technology and first mathematical concepts came up in the context of astronomy, geography, and architecture in antique civilizations. Early forms of geometry and arithmetic must be assumed for the first civilizations (5000–3000 years BC). All historically documented symbolic forms are therefore at least in their complex mode influenced by the blending of symbolic forms in the civilizations which followed the Paleolithic societies, i.e. in the Neolithic societies and later. We assume that the fundamental principles of the genres of semiosis are older than historical cultures and point to the original autonomy of the specific symbolic forms. The search for separate archetypes or “structures mères” remains therefore significant.

4 Archetypes of Language and Thom’s Conjecture One direction of search for universal features of human language concerns the sound shape, i.e. phonetics. This was the major aim of “sound laws” established by historical and comparative linguistics in the 19th century (cf. the neo-grammarian movement with Brugmann, Osthoff, and Schleicher). The other pathway considers meaningful entities like words, syntactic constructions, and larger utterances like text and discourse. In the following I shall only consider construction on the sentence level, specifically the valence patterns described by Tesnière (1959), Fillmore and many others and the far-reaching conjecture of René Thom in catastrophe theoretical semantics (cf. Wildgen 1982, 1985, 1994) which established a systematic parallel between the architecture of elementary catastrophes and the hierarchy of syntactic constructions. “Archetypes” in Thom’s usage refer to Plato and his dialogue Timaeus (Greek: T´ιμαιoς), where he pleads for a geometric foundation of natural laws and even laws of the human mind. The favorite geometrical building blocks in Plato’s treatise are triangles, regular surfaces and regular solids (the five Platonic solids). In modern differential topology, which is a generalization and dynamical elaboration of Greek and premodern geometry, the elementary catastrophes (cuspoïds and umbilics) and the three symbolic genres correspond to regular surfaces (polygons ~ cuspoïds), double-faced surfaces (dihedra ~ umbilics), and regular (Platonic) solids (polyhedra ~ symbolics); cf. Slodowy (1988) and Wildgen (1994: 49–56). Basic contributions to this new version of Plato’s conjecture were the development of “analysis situs” since Gottfried W. Leibniz (1646–1716), the stability theory of Henri Poincaré (1854–1912), the Erlangen program based on group theory by Felix Klein (1849–1925) and modern dynamical systems theory elaborated by Hassler Whitney (1907–1989), René Thom (1923–2002) and others and applied by Christopher Zeeman, René Thom, Ilya Prigogine and Hermann Haken in the second half of the 20th century. This astonishing history leading from Pythagoras (ca. 600 BC) to Thom (1972) is not yet a guarantee that such conjecture is relevant or even valid. Before I consider arguments for the validity of Thom’s conjecture I want to point

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to the fact that such a hypothesis is not at the same level as Chomsky’s conjecture in 1957 that languages are a kind of free monoïds or of Montague’s conjecture that natural languages are a kind of formal language (e.g. intensional logics; cf. his article: “English as a formal language”). The difference is obvious if we consider the scope of applications. Thus catastrophe theory and at the same level Prigogine’s systems far from equilibrium, Haken’s synergetics or Mandelbrot’s fractal geometry have their major applications in physics, chemistry, and biology, i.e. in the field traditionally called natural science. The second realm of Plato’s conjecture, the architecture of the mind is a domain only assessed with caution by these authors (cf. Zeeman’s models of the brain and of behavior in animals and humans; synergetic models of the brain and symbolic behavior, Prigogine’s models of the cooperative behavior of animals etc.). Quantitatively precise and elaborate applications in mathematical terms are mostly beyond the scope of the humanities. The models in the humanities mentioned above assume a fundamental continuity between the natural and the life sciences and therefore the non-autonomy of systems surfacing in the humanities; this is at least a very controversial issue. The surplus-value of the “dynamic systems” approach is the generality of its hypotheses or proposed “laws”. The possible shadow of such advantages is a loss of specificity to the field, the difficulty to match the conjecture of a structural mapping against the enormous variability of human languages and human behavior in general. But, even if the conjecture had a poor correspondence to documented facts, it could be significant for further research and the growth of knowledge. In Peirce’s terms, it is neither deductive nor inductive, but abductive, i.e. it uses deductive techniques and considers inductive methods to evaluate the results of deduction. The abstract character of Thom’s topological conjecture may be brought down to experimental research looking at results in neuropsychology. In the psychology of vision, the relevance of spatial and “imagistic” analyses of cognition, memory, and language becomes evident (cf. for an overview Petitot 2008). The basic problem in the transition between perception—cognition—motor controls is the proper mapping from one internal representation (in a realistic sense) to the other. The mapping must conserve basic topological and dynamic characteristics and can forget metrical details, variations of a type of object or event. Therefore, the problem of a structurally stable mapping lies at the heart of every theory of representation and semantics. The crucial result in this field is the theorem of Whitney. Whitney’s theorem (for mappings from plane to plane) says that locally (in the environment of a point) we can only find three types of points (all other types become identical to these if perturbed): a. regular points (Morse-points); they do not qualitatively change under perturbation; we may say that they have a static identity (of self-regulation), b. fold-points (a frontier line between a stable and an unstable domain appears), c. cusp-points (two stable attractors conflict and one of them may appear or disappear). Thom’s classification expands this list in the domain of real analysis and Arnol’d (1984) presents a list for the more general case of complex analysis. In the present

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context, it is important to note the basic difference between static stability and process stability. a. Static stability: The prototypical (local) systems are the potential functions: V = x2 (one can add a function Q which contains more quadratic terms and constants, e.g. V = x2 +Q; Q = c1 .y21 + c2 .y22 + … + cn.yn2 . The singularity of the unfolding is given by the gradient: V =2x = 0. The stable system V = x2 has a minimum (V = 2 > 0) as its singularity. The dual of this function is V = −x2 ; it is the prototype of an unstable singularity, V = −2x = 0; V = −2 < 0; it is a maximum. Figure 1 shows the two dynamical systems and as physical analogues the motion of pendulums with damping. The minimum of a dynamical system is called an attractor, the maximum a repellor (Fig. 2). b. Process stability. Most dynamical systems are not structurally stable, they degenerate under small perturbations. Nevertheless, they can, under specific conditions, have a stable evolution called “unfolding”. These special cases can be called highly ordered instabilities or catastrophes. The measure of degeneracy is given by the minimum number of unfolding parameters, it is called the co-dimension. As the theorem of Thom shows, we have three basic types of dynamical systems: A (cuspoïds), D (umbilics) and E (symbolics). The list of elementary catastrophes is a list of ideal, regular dynamic forms or archetypes of processes. Table 1 Normal

Fig. 1 A set of regular polygons (above), the di-hedron (middle; square with rotation on the axis) and the regular polyhedrons (Platonic solids)

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Fig. 2 Basic dynamical systems

Fig. 3 Configurations of conflict

forms of some stable unfolding shows the list of ideal process forms (cf. Arnol’d 1972: 254 and Gilmore 1980: 11). In this contribution, I must dramatically simplify the mathematical aspects of the theory. This can be done if we consider just the configurations of conflicts in behavior-space. Figure 3 shows the conflict lines for three (compact) unfoldings of the Ak -family5 called: the cusp (A3), the butterfly (A5) and the star (A7). Every regime Ri has, locally, the form of a stable attractor (V = x2 ), the lines are transitions or conflict lines. An even simpler (linear) picture is given by a diagrammatic representation of the stable attractors in the unfolding ( = minimum, = maximum, = vector field). cusp (A3): butterfly (A5): star (A7):

5 The

family Ak of cuspoïds is the real resolution of the (complex) cyclic group already described by Felix Klein in 1874; see for more details Slodowy (1988: 77–80).

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In the family Dk (umbilics) the notion of a saddle ( ) must be introduced (if we add a quadratic function, e.g. y2 to the members of the Ak -family, maxima become saddles; cf. Gilmore (1980: 119f). . cusp (A3) + y2 : If the elliptic umbilic is made compact, i.e. if attractors ⊕ close the saddle connections, we obtain the maximal substructure with four minima: This is the basic type of a configuration with two internal variables (quick dynamics) and four attractors (cf. Wildgen 1985: 204–212).The configuration with 1, 2, and 3 linearly arranged attractors (on x) and the configuration of 4 attractors in a two-dimensional (x-y) plane are the fundamental concepts in catastrophe theoretical semantics (cf. Wildgen 1982 for an introduction in English and Thom (1983) for the translation into English of his major articles treating semiotics and linguistics). A further notion must be informally introduced: the linear path in an elementary unfolding (this aspect has been elaborated in Wildgen (1982) and in more detail in Wildgen 1985). If we consider linear paths in an unfolding, we can classify types of processes. In this contribution only the most basic types will be used; the schemata are abbreviations of explicit dynamical descriptions. In Fig. 6 the schemata called EMISSION, CAPTURE and (bimodal) CHANGE are derived from the catastrophe set (set of extrema) of the cusp. The diagrammatic simplification eliminates the lines of (unstable) maxima, the circles symbolize the bifurcation points (type ‘fold’: V = x3 ).

Fig. 4 The bifurcation outline of the (three) unfolding variables. Aazami, Keeton and Petters (2019: 2)

Fig. 5 Diagram of the compactified elliptic umbilic

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Fig. 6 The derivation of archetypal diagrams from the “cusp”

Thom considered only catastrophes with a co-dimension equal to or lower than the dimensionality of space-time (≤4). With this assumption and if the symmetrical catastrophes (compact, i.e. closed by minima on both sides of the Dynkin diagram; cf. Fig. 5) are preferred, the basic scenarios of change and process are processes in the cusp (A3), in the butterfly (A5) and in the elliptic umbilic (D-4; the elliptic umbilic compacted in the double cusp X9). On this basis linear pathways in the bifurcation-plane can be classified and we obtain a set of process-types characterized by the catastrophic jumps (changes of attractor).6 This mathematical structure can have different models depending on the interpretation of the variables and the type of representational space. In Wildgen (1994: 126) four stratified domains of interpretation are distinguished: (1) locomotion in space, (2) change in a quality space, (3) external action/interaction, and (4) internal (mental) action. The analogy between process-types derived in catastrophe theory and basic schemes of processes denoted by verbs in human languages led René Thom to formulate his conjecture (first in 1968 and with more details in 1972).

4.1 Thom’s Conjecture Given a dynamic situation the analysis of structural stability cuts out pieces of the continuous process: 6 The

original set depicted by Thom in his book of (1972) was further specified and enlarged in Wildgen (1985).

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a. in the neighborhood of singularities (catastrophes), b. these segments have a maximum complexity of 3 (with one component) or 4 (with two components). The restrictions explain the fact that the valence of sentence construction has a maximum of three (under specific conditions a maximum of four). The schemata of processes derived from catastrophe theory map into the sentence types: intransitive (valence 1), transitive (valence 2) and bi-transitive (valence 3). In causative constructions, a further elaboration to valence 4 can occur. The empirical evaluation of this conjecture is complicated given the diversity of thousands of human languages and the difficulties of a coherent classification which reflects the semantic (functional) values of the structures found and their comparison. In the tradition of grammar writing in India, Greece, and Rome and during the middleages case theory was a basic element. The classification of cases as specific dynamic relations between a verb and its nominal/pronominal relata is obvious in the grammar of Sanskrit, Greek, and Latin. In contemporary Romanic and Germanic languages inflectionally marked cases are partially replaced by prepositions. In languages of the agglutinative type which use sets of affixes (pre- and suffixes), a larger set including rather differentiated spatial relations appears (e.g. case-suffixes in Finnish). Synthetic languages (e.g. Chinese) which lack inflection and affixes use word order and stress to mark these relations. If no marking occurs, the meaning of the verb and its nominal adjuncts and common knowledge are sufficient to infer the implicit use of case-roles. Hjelmslev (1935) proposed a configuration of different factors which define caseroles and speaks of a “sub-logic” underlying these classifications. Thom’s conjecture specifies the relational logic Hjelmslev (and even before him Peirce) proposed and adds spatial and temporal (dynamic) features to this concept, thus completing a tradition of grammatical thought since antiquity. I must refer to Wildgen (1982, 1985, 1994) and a more recent overview of the case-theoretical discussion in Wildgen (2017). The structures and processes introduced in this section are more general than language. This is immediately obvious if we consider other applications of catastrophe theoretical concepts in optics (caustics), fluid dynamics (waves) and biological rhythms (e.g. heartbeat). In the realm of symbolic forms sketched by Cassirer, the archetypes introduced by Thom refer to basic types of construction of complexes with a small number of governing principles (forces). They allow for stability even under the effect of change and motion.

5 Universals of Shape and Figural Archetypes in Leonardo’s Paintings In the visual domain, i.e. in visual perception and memory and as a sequel in visual art, constructions similar to those found in language appear. They allow the mind to construct complex scenes from local visual input. If we see a visual object, e.g. a

Structures, Archetypes, and Symbolic Forms … Table 1 Normal forms of some stable unfolding

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Ak : V = ±xk+1 ±y2 + Q; Q = −z21 − … −z2s +z2s+1 + … +z2n Example: A2 : V = + x3 (fold); co-dimension = k − 1 = 1; unfolding: V = x3 + vx Dk : V = x2 y ± yk − 1 + Q (Q as above) Example: D4 : V = x2 y − y3 (elliptic umbilic); co-dimension = k − 1 = 3; unfolding: x2 y − y3 + wy2 + uy + vx E6 : V = x3 ± y4 + Q; co-dimension = 5 (≈ tetrahedron) E7 : V= x3 + xy3 +Q; co-dimension = 6 (≈ octahedron/cube) E8 : V =x3 + y5 + Q; co-dimension = 7 (≈ dodecahedron/icosahedron)

square, we must first focus on the corners and then superimpose these focalizations in order to recognize the object (cf. Petitot 2008: 386, Fig. 1). In a complex image this scanning is much richer, the eye jumps from one relevant point to another and produces a temporal sketch of the image along the line of the scanning process. As Rudolf Arnheim (1904–2007) has shown in his psychology of visual art, forces are directing such a neural scan: the shape of the frame (rectangular or circular, etc.), the diagonals in a rectangular frame, the horizontal and vertical middle lines and the center established by these vectors.7 Arnheim (1988) speaks of the principle of centrality. The symmetry of this force-field is in most paintings broken. Thus the horizontal line dominates; i.e. the line of locomotion; in the vertical dimension, the area below the center dominates, prototypically creating the opposition of land (earth) versus sky (air). Intermediate zones may be water (e.g. in pictures of a seaside) or towns with towers. The left-right opposition is also broken; insofar as the left dominates (at least in cultures where writing is from left to right). Symmetries, their instability, and broken symmetries exhibit the process type called cusp in the theory of catastrophes. The Last Supper of Leonardo in Milan demonstrates these vectors of forces and the principle of centrality. In this example, we have one central attractor (Jesus) and four decentered attractors. The elementary catastrophe with a maximum of five attractors (in a line) has the germ V = x10 and codimension 8 (10 = k + 1; k = 9; codimension = k − 1 = 8; cf. Table 1). The group of twelve apostles is first ordered along a straight line (abandoning the natural order around a table) and then separated into four groups; each group has again a center, although all of them show an attraction towards the major center (Jesus). The whole motion pattern implicit in the static image is driven by an utterance of Jesus in the center: One of you will betray me. When Leonardo was still a master working in the “bottega” of Andrea del Verrocchio (1435–1488), he authored (or co-authored) the painting “Annunziazione” (1470–1473). It has at its center the implicit wording of the message of her pregnancy; visually this center is void. The communication between the archangel and Mary was a difficult topic for theological reasons. Petitot (2004: 75–80) showed, how 7 Cf.

Verstegen (2005) for an overview of Arnheim’s contributions to the psychology of vision and visual art.

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Fig. 7 Diagonal und straight force-fields and four groups of apostles in the periphery of Jesus (the center)

Fig. 8 Leonardo da Vinci, “Annunciation”, Florence, The Uffici (1470–1473)

Piero Della Francesca in the “Annunziazione” of Perugia painted in (1465) chose a non-generic perspective to avoid a direct line of impact between the archangel and Mary. In an earlier fresco of Piero in Arezzo painted in (1455), a column is separating both figures and a third element, God Father above the archangel, is the sender of the message. In the early fifteenth century, Fra Angelico (1400 ca—1455) painted textlines between both persons, and in a tympanum of the chapel of Mary in Würzburg (early 15th century), a letter band is placed between them, additionally a tube links Mary’s ear with the mouth of God Father above. Leonardo shows an eye-link going from Mary to the archangel (or behind and above him a and the eye-link of the angel goes to her womb (yellow). Mary’s right hand points to a word or sentence in the bible. Thus, God’s word, the prophecy of her pregnancy, is part of the message. The construction is triangular with the central vertical axis touching the lectern (Fig. 8).8

8 This

short analysis is my response to a suggestion by the reviewer. The source of the message, God, can be associated with the white surface between two Italian cypresses. As in other paintings by Leonardo, e.g. “The virgin of the rocks”, 1483–1485, the theological content may be heretical.

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Basic triangle of figural composition The main line of sight parallel to Mary’s grip on Jesus Static field of Mary’s body on the knees of Anne Mary’s grasping at Jesus; Jesus’ grasping at the lamb, (2 vectors) The lamb resists (1 vector)

Position of the figures on earth (cf. the position of feet)

Fig. 9 Leonardo da Vinci: St. Anne with Mary, Jesus and the lamb (1509/10; Paris, Louvre)

A figural grouping can also unite a set of utterances, thus realizing a kind of visual narrative. The single sentences have the underlying dynamics we have described in the chapter above. I shall give a short analysis of the thematic composition in Leonardo’s paintings of St. Anne (Fig. 9). The painting contains a rich geometric and dynamic structure (weights, bar centers, force-vectors,9 gaze-directions, etc.). A purely static representation would be insufficient for both the pictorial and the narrative aims of the painting; it is given by three positional fields with a barycenter (two triangles and a semi-circle). The sight vectors (looking at something) and the grasp vectors exhibit the forces, which constitute the kernel of the visual narrative. Furthermore, this piece is typical for Leonardo’s art which consistently exemplifies the concept of dynamic valence. In the case of this painting, we have on the surface a quaternary constellation: Anne—Mary—Jesus—lamb. If one considers the force fields and actions, one notices that basic interaction links: Mary—Jesus—the lamb. Mary pulls on Jesus Jesus pulls on the lamb The lamb resists Jesus resists being pulled away from the lamb. 9A

specific consequence of the mathematical conceptualization of dynamics is the calculus of vectors introduced by William Rowan Hamilton (1805–1865). In Fig. 10 three force-vectors and four sight vectors are given. In a catastrophe theoretical frame, one would consider two vector-fields, one sight field and one grasp-field and their attractors (minima) and repellors (maxima).

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Mary

main attractor 1

transient attractor

Jesus

lamb

main attractor 2

Fig. 10 The dynamical archetype of transfer (giving) and a fiber on it (with attributed contents at the right)

There is a conflict between Mary who tries to prevent Jesus from seizing the lamb and Jesus who notices this (he looks back to her) but resists against her action. This triad constitutes a force field that dominates the message of the painting. A first schematic representation introduces two vectors with one repellor and two attractors: Mary

Jesus

lamb

The constellation of forces between Mary—Jesus—the lamb corresponds to the basic archetype of transfer. The archetype is derived by considering a path in the catastrophe set of the butterfly: germ V = x6 ; cf. Fig. 10). In the center of the catastrophe set three attractors (=minima) coexist and the change occurs along a path in this zone; for technical details, cf. Wildgen (1982, 1985). As the archetype does not describe all the interactions in the pictorial composition, one has to add further features which ask for an elaboration of the model: • Anne supports/anchors the whole event (physically and genealogically), she is a fourth attractor which sustains the event happening on her knees. This anchoring is visible in the position of the feet and the central triangle of gravitational stability in Fig. 10. • The manner of “transfer” is further elaborated in the painting and could be described in a sentence like Mary tries to prevent Jesus from seizing the lamb. This complex sentence goes beyond the elementary schema shown in Fig. 11. The innovation by Leonardo does not break with the tradition of his Renaissanceprecursors (cf. the development of the topic of the Last Supper before and after Leonardo in Wildgen 2004a, 2010a, 2013), it is rather the climax of this development, which will be imitated in the following centuries. It is followed by the mannerist period and baroque art. Such a climax is a singularity in the sense of catastrophe theory (cf. Wildgen 2016). As such it has a very strong impact on later developments

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although it is itself unstable and induces rapid changes which show up already in the late work of Michelangelo, and more radically on that of Tintoretto and Arcimboldo (cf. Wildgen 2013: 107–111).

6 Geometrical Archetypes and Semiotic Construction in Music Musical scales have a mathematical nature which was already focused on by Greek mathematics (the Pythagorean School); in the eighteenth century, it was further developed by the mathematician Leonard Euler in his Euler-space based on the dimensions: octave, fifth and third. Modern models of music were introduced by Mazzola (1990); his torus of thirds shows the basic intervals organized in three-dimensional space and relations/motions on this torus. The twelve notes of the classical tempered scale (0, …, 11) are ordered by the interval of a major third (meridian circle), e.g. (2), (6), (10), and a minor third (equatorial circle), e.g. (2), (5), (8), (11). The spiral movement on the torus (see the arrows) corresponds to the intervals of seconds, i.e. the sequence of 12 halftones. The smallest distances on the torus are minor and major thirds, which is why the Torus is called torus of thirds. The construction of a melody (or a musical motive in a larger piece of music) has such a basic space as its background, e.g. the Euler-space based on the octave, the fifth and the third, or the Mazzola-space based on the major and minor thirds. In this case, the fifth and the fourth are composite intervals. As in the case of linguistic systems, many different scale-systems (even scales which use smaller intervals and smooth transitions or variations of intervals) are possible. The tonal system which appeared

Fig. 11 The torus of thirds (cf. Mazzola 1990: 246, 251)

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after the modal scales of the Middle Ages had its climax around 1700 and began to be questioned at the end of the 19th century. It is still the dominating tonality in Western and Western-dominated public music. In the section on archetypes in language, I have sketched the results of catastrophe theoretical semantics and I will look for similar structures in music. Within the framework of Western tonality, one can find examples of comparable constructional principles. The scale-step theory in musicology (theoretically described in the 19th century by Hugo Riemann and others based on the music since Renaissance times) can be compared in its architecture with the valence patterns in language. The tonic (I) corresponds to the nominative case in the nominative-accusative languages, which is often unmarked and is used in the intransitive sentence (valence 1). The dominant (V) or second most important scalestep corresponds to the accusative appearing in the transitive sentence (valence 2), the subdominant (IV), or the third most important level corresponds to the dative, etc. In a melody, the tonic corresponds to the general tune, e.g. C-major, the dominant is realized by the tune G-major, the subdominant by the tune F-major. Many popular songs move just between these three chords, eventually changing to the correspondent minor tunes (C-major – A minor, etc.). Dominant seventh chords (with four components) are used for transitions between the three elementary steps. In arrangements and Jazz-compositions many further chords based on the sixth, ninth, etc. are added. These dependencies or dominance relationships have an anchor in the natural scale of music which is physically (and psycho-acoustically) based on the overtone series; thus the dominant is one fifth above the tonic and this interval is the acoustically basic one, the subdominant is one fourth below the tonic, etc. As an example for the systematic and almost exhaustive use of this scheme of melodic unfolding one can take the classical blues-scheme (also many European folk songs; in the AfroAmerican blues, “blue notes” which have their origin in African pentatonic scales are added). The tonal hierarchies may be considered as syntactic, insofar as they organize the inherent topological structure of the tonal scale in time (in the syntagmatic dimension). A difficult question in musical semiotics concerns the semantics of music. Does music have a descriptive/denotative function or is it only appealing to feelings? If this is the case, how do musical signs refer to feelings, to emotion and what are the emotive categories which make up the meaning of musical signs? In Wildgen (2018, 2019), an answer to this question is given, but it would be too long to unfold this issue here. Concerning our topic “structures mères” it is just necessary to point out that impressionistic classifications of emotions which have a long tradition (from Aristotle to Descartes and later) are not sufficient. One must rather look to archaic and evolutionary basic emotions to find a proper ground for the analysis of archetypes of emotional meaning in music. Of specific importance for music is the neuro-evolutionary role of the basic emotion called “Play” by Panksepp and Biven (2012). In the case of language, the valence has its natural foundation in event and action archetypes or basic scenarios such as the movement of a subject (valence 1), acting of a subject on an object (valence 2), transaction in giving/taking (valence 3). The

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basis is indeed different in language and music, although motion patterns are relevant in both cases. The functions and modes of action of the two symbolic forms are different. However, the order of steps in the central relational patterns is comparable. This anchoring of symbolic systems in principles of nature (the phenomenal environment) can be called archetypical (see Wildgen 1985). Such archetypes (literally original sketches) must remain hypothetical and can only gain their justification from a global analysis of the system. The large field of contingent, context-dependent, and ultimately random phenomena in symbolic systems is caused by a multitude of acting forces. They form the area of arbitrariness which Saussure has placed at the center of his concept of a linguistic system. This “arbitrariness” is, however, restricted by natural conditions which govern the space of alternatives and the criteria of selection which establish habits.10 Archetypes or “structures mères” specific for a genre of symbolic forms are the stable background which is responsible for the reproduction and stability of a given symbolic form. In René Thom’s terms, they form the “morphogenetic germs”, on which quasi-Darwinian mechanisms of variation and selection may operate. As such, they are a proper field of mathematical model-building. In Cassirer’s terms, they are the substratum “in re”, which can be conceptualized mathematically.

7 Final Remarks The genres of symbolic forms advocated by Cassirer (1922a) show a tremendous variety of forms and they exploit different physical, energetic fields and sensual (neural) resources. Therefore it seems to be extremely difficult to corroborate Cassirer’s intuition with empirical data. The possible evolutionary source, the symbolic capacity, as a major characteristic of the species Homo sapiens emerged probably with our species ca. 300.000 ago. Almost one century after Cassirer had formulated the conjecture (around 1921) and after the consolidation of evolutionary biology (after 1930) and its further elaboration in the 20th century, one has to accept a major change in the theoretical and philosophical context of his proposal. This could lead to a more continuous view, in which basic communicative faculties in the animal kingdom are at the source of human symbolic capacity; cf. Wildgen (2009b). In the present contribution, the existence of basic principles (archetypes or “structures mères”) has been shown for the symbolic forms: language, visual art, and music. 10 The area between the archetypal level and arbitrary rules in semiotic systems may be called semiotic stylistics. In the act of creation or creative use, highly motivated sign gestures and content structures beyond conventionalized patterns may show up. In relation to saliency (“saillance”), these may be highly coordinated simultaneous features. Thus, in an orchestral work, musical gestures are supported by different instrumental groups and cooperate to create a rich fabric of sounds. In the realm of content (“prégnance”), utterances may have a multiplicity of relevant readings, i.e. be enriched, or compress a field of meanings into a very short, allusive expression, e.g. in poetry. This intermediate area asks for a more specific treatment, which lies beyond the scope of the present contribution.

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Amazingly, mathematical structures which stand in the tradition of antique geometry and the Pythagorean conjectures surfacing in Plato’s Timaeus are again candidates for the fundamentals of symbolic forms and the cognitive faculties which they exhibit.

References Aazami, A. B., Keeton, Ch.R., & Petters, A. O. (2019). Magnification cross sections for the elliptic umbilic caustic surface. Universe, 161. Arnheim, R. (1988). The power of the center: a study of composition in the visual arts. Berkeley: University of California Press. Arnol’d, V. I. (1972). Normal forms near degenerate critical forms in the Weyl Groups of Ak, Dk, and Ek, and lagrangian singulaities. Functional Analysis and its Applications, 6, 254–272. Arnol’d, V. I. (1984). Catastrophe theory. Berlin: Springer. Balzer, W., & Moulines, C. U. (Eds.) (1996). Structuralist theory of science. focal issues, new results. Berlin: Springer (Chapter 13: Mormann, Thomas, Categorical Structuralism). Barwise, J., & Perry, J. (1983). Situations and attitudes. Cambridge (Mass.): MIT-Press. Cassirer, E. (1922a). Der Begriff der symbolischen Form im Aufbau der Geisteswissenschaften, again in: Idem, 1956. Wesen und Wirkung des Symbolbegriffs, Wiss. Buchgesellschaft, Darmstadt. Cassirer, E. (1922b). Das Erkenntnisproblem in der Philosophie und Wissenschaft der neueren Zeit, Vol. 2, Reprint 1991. Wiss. Buchgesellschaft, Darmstadt. Cassirer, E. (1923–1929). Die Philosophie der symbolischen Formen, vol. 1 (1923): DieSprache, vol. 2 (1925): Das mythische Denken, vol. 3 (1929): Phänomenologie der Erkenntnis, Gesammelte Werke. Reprint 1982. Wiss. Buchgesellschaft, Darmstadt (cf. also Hamburger Ausgabe, vol. 11–13, Meiner, Hamburg). Cassirer, E. (1945). Reflections on the concept of group and the theory of perception, again in: Idem, 1979. D. P. Verene (Ed.), Symbol, myth and culture. Essays and lectures of Cassirer 1935–1945 (271–291). London: New Haven. de Saussure, F. (1916). Cours de linguistique générale. Paris: Éditions Payot, new edition 1995. Gilmore, R. (1980). Catastrophe theory for scientists and engineers. New York: Wiley. Haken, H. (1988) Information and self-organization. A macroscopic approach to complex systems. Berlin: Springer. Hjelmslev, L. (1935) La catégorie du cas. Etude de grammaire générale I, II (reprinted by Fink, Munich, 1972). Kant, I. (1971) Kritik der reinen Vernunft. Republished after the 1st and 2nd edn. R. Schmidt (Ed.), Hamburg: Meiner. Lakoff, G. (1987). Women, fire, and dangerous things. What categories reveal about the mind. Chicago: The Chicago University Press. Mac Lane, S. (1998). Categories for the working mathematician. New York: Springer. Mazzola, G. (1990). Geometrie der Töne. Elemente der mathematischen Musiktheorie. Basel: Birkhäuser Verlag. Montague, R. (1974). Formal philosophy. Selected papers of Richard Montague. New Haven: Yale University Press. Panksepp, J., & Biven, L. (2012). The archaeology of mind. Neuroevolutionary origins of human emotion. New York: Norton. Petitot, J. (2004). Morphologie et esthétique. Paris: Maisonneuve & Larose. Petitot, J. (2008). Neurogéométrie de la vision. Modèles mathématiques et physiques des architectures fonctionnelles, Les Éditions de l’Ecole Polytechnique, Paris. Prigogine, I. (1980). From being to becoming. Time and complexity in physical science. San Francisco: Freeman.

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