Philosophical Approaches to the Foundations of Logic and Mathematics In Honor of Stanisław Krajewski 9004445943, 9789004445949

Table of contents :
Contents
Introduction. Philosophy Asking about the Foundations of Logic and Mathematics
Chapter 1. From Speculative to Practical Foundations of Mathematics: A Communication-Centered Account
Chapter 2. Is There an Absolute Mathematical Reality?
Chapter 3. Mathematical Modalities and Mathematical Explanations
Chapter 4. Apriorism, Aposteriorism and the Genesis of Logic
Chapter 5. On Some Problems with Truth and Satisfaction
Chapter 6. Inductive Plausibility and Certainty: A Multimodal Paraconsisent and Nonmonotonic Logic
Chapter 7. On the Reception of Cantor's Theory of Infinity (Mathematicians vs. Theologians)
Chapter 8. How Is the World Mathematical?
Chapter 9. An Analysis of Paradoxes
Chapter 10. Computation and Visualization Thought Experiments after Lakatos's Heuristic Guessing Method (Semantics of Thought Experiments – Pt. Mathematical Thought Experiments)
Closing Words. Are Dogs Logical Animals?
Index of Persons
Index of Subjects

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Philosophical Approaches to the Foundations of Logic and Mathematics

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Poznań Studies in the Philosophy of the Sciences and the Humanities Founding Editor Leszek Nowak (1943–2009)

Editor-in-Chief Katarzyna Paprzycka-Hausman (University of Warsaw)

Editors Tomasz Bigaj (University of Warsaw) – Krzysztof Brzechczyn (Adam Mickiewicz University) – Jerzy Brzeziński (Adam Mickiewicz University) – Krzysztof Łastowski (Adam Mickiewicz University) – Joanna Odrowąż-Sypniewska (University of Warsaw) – Piotr Przybysz (Adam Mickiewicz University) – Mieszko Tałasiewicz (University of Warsaw) – Krzysztof Wójtowicz (University of Warsaw)

Advisory Committee Joseph Agassi (Tel-Aviv) – Wolfgang Balzer (München) – Mario Bunge (Montreal) – Robert S. Cohen† (Boston) – Francesco Coniglione (Catania) – Dagfinn Føllesdal (Oslo, Stanford) – Jacek J. Jadacki (Warszawa) – Andrzej Klawiter (Poznań) – Theo A.F. Kuipers (Groningen) – Witold Marciszewski (Warszawa) – Thomas Müller (Konstanz) – Ilkka Niiniluoto (Helsinki) – Jacek Paśniczek (Lublin) – David Pearce (Madrid) – Jan Such (Poznań) – Max Urchs (Wiesbaden) – Jan Woleński (Kraków) – Ryszard Wójcicki (Warszawa)

Volume 114 The titles published in this series are listed at brill.com/ps

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Philosophical Approaches to the Foundations of Logic and Mathematics In Honor of Professor Stanisław Krajewski

Edited by

Marcin Trepczyński

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Cover illustration: View of the Tatra Mountains from the observation tower on Lubań, photograph by Jakub Hałun. This image has been shared under the CC BY-SA 4.0 license. Copy-editor: Barbara Przybylska The Library of Congress Cataloging-in-Publication Data is available online at http://catalog.loc.gov LC record available at http://lccn.loc.gov/2020054337

Typeface for the Latin, Greek, and Cyrillic scripts: “Brill”. See and download: brill.com/brill-typeface. ISSN 0303-8157 ISBN 978-90-04-44594-9 (hardback) ISBN 978-90-04-44595-6 (e-book) Copyright 2021 by Marcin Trepczyński. Published by Koninklijke Brill NV, Leiden, The Netherlands. Koninklijke Brill NV incorporates the imprints Brill, Brill Hes & De Graaf, Brill Nijhoff, Brill Rodopi, Brill Sense, Hotei Publishing, mentis Verlag, Verlag Ferdinand Schöningh and Wilhelm Fink Verlag. Koninklijke Brill NV reserves the right to protect this publication against unauthorized use. Requests for re-use and/or translations must be addressed to Koninklijke Brill NV via brill.com or copyright.com. This book is printed on acid-free paper and produced in a sustainable manner.

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Contents Introduction: Philosophy Asking about the Foundations of Logic and Mathematics 1 Marcin Trepczyński 1

From Speculative to Practical Foundations of Mathematics: A Communication-Centered Account 12 Vladislav Shaposhnikov

2

Is There an Absolute Mathematical Reality? Zbigniew Król

3

Mathematical Modalities and Mathematical Explanations Krzysztof Wójtowicz

4

Apriorism, Aposteriorism and the Genesis of Logic Jan Woleński

5

On Some Problems with Truth and Satisfaction Cezary Cieśliński

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Inductive Plausibility and Certainty: A Multimodal Paraconsisent and Nonmonotonic Logic 193 Ricardo Silvestre

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On the Reception of Cantor’s Theory of Infinity (Mathematicians vs. Theologians) 211 Roman Murawski

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How Is the World Mathematical? Michael Heller

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An Analysis of Paradoxes Anna Wójtowicz

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123

150

175

238

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Computation and Visualization Thought Experiments after Lakatos’s Heuristic Guessing Method (Semantics of Thought Experiments – Pt. Mathematical Thought Experiments) 271 C. Peter Hertogh Closing Words: Are Dogs Logical Animals? Jean-Yves Béziau

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Index of Persons 307 Index of Subjects 309

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Introduction

Philosophy Asking about the Foundations of Logic and Mathematics Marcin Trepczyński

Abstract In this introduction to the volume Philosophical Approaches to the Foundations of Logic and Mathematics dedicated to Professor Stanisław Krajewski, I argue that it is worth asking about such foundations and putting together logic and mathematics when we conduct such considerations. There are various problems and questions which apply to both of them. But what is more important, putting them together allows us to make new observations, as in the case of the phenomenon of emergence in mathematics described by Stanisław Krajewski. I also show that asking about the foundations is constitutive for logic and mathematics, and essential for philosophy as such; furthermore, it also plays a significant role for the humanities.

Keywords foundations of logic and mathematics – philosophy of logic – philosophy of mathematics – mathematical logic – Stanisław Krajewski

1

Stanisław Krajewski and the Foundations

Is it worth asking about both the foundations of logic and mathematics? Are logic and mathematics so closely related that we can benefit from putting them together? Is it still important to ask philosophical questions concerning logic and mathematics, including questions about their foundations? This volume is published in honor of Professor Stanisław Krajewski to celebrate his 70th birthday. Through this initiative, the authors want to pay tribute and express their affinity to this outstanding scholar, author of numerous works, organizer of scientific events, social activist, colleague and friend. His scientific activity and works seem to support a positive answer to the questions opening this volume. In his early career he was a mathematician.

© Marcin Trepczyński 2021 | DOI:10.1163/9789004445956_002

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However, he gradually began to concentrate on logic. Finally he adopted a philosophical perspective, within which he studied problems concerning both logic and mathematics. Moreover, thanks to his religious interests, he also started to examine the relationships between logic, mathematics and theology. From his writings we can see how rich the emerging issues are when we juxtapose these various perspectives. His analyses reveal that the special entanglement existing between mathematics and logic is not limited to the simple fact that more than a century ago, mathematical logic was invented. This fact was only a cornerstone for the inquiries that showed much more. The first of them (and perhaps most spectacular) were projects focused on the logical approach to mathematics and the mathematical approach to logic, represented by two schools, which confronted their views in 1900 at the International Congress of Philosophy in Paris (Mancosu, Zach and Badesa 2009, 319). This “dualism” was well expressed a few years later by Zermelo: The word “mathematical logic” can be used with two different meanings. On the one hand one can treat logic mathematically, as it was done for instance by Schröder in his Algebra of Logic; on the other hand, one can also investigate scientifically the logical components of mathematics. (Zermelo 1908, after: Mancosu, Zach and Badesa 2009, 320)1 These attempts were followed by projects of searching for the logical foundations of mathematics and the mathematical foundations of logic. So this common offspring of logic and mathematics called “mathematical logic” brought to existence an idea that one of them may deliver foundations to the other. One of the aspects of such approaches were the projects of axiomatizing the theory of natural numbers by Giuseppe Peano (Epstein 2011, xxi), and geometry, taken up independently by the Italian school of logic and David Hilbert (Mancosu, Zach and Badesa 2009, 324), using a modern understanding of axiom, which were followed by Alfred North Whitehead’s and Bertrand Russel’s axiomatization of logic (Epstein 2011, xxii). What is more, we should recall that at the turn of the 19th and 20th centuries, new questions arose concerning reasoning in mathematics. As Richard L. Epstein put it, “[t]he reasoning appropriate to mathematics became itself a subject of study for mathematicians” (Epstein 2011, xvi). This means that reflection on mathematics entered

1 As Mancosu, Zach and Badesa point out, “The first approach is tied to the names of Boole and Schröder, the second was represented by Frege, Peano, and Russell” (Mancosu, Zach and Badesa 2009, 324).

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the area of interest of logic, which was traditionally focused on the ways of reasoning. But Stanisław Krajewski showed that there are other interesting problems that can be observed when we put mathematics and logic together. For instance, when analyzing the phenomenon of emergence in mathematics, the situation in which essentially new (unexpected) properties or regularities appear, he points out examples that can be viewed from the perspective of the logical foundations of mathematics. As such examples, he first recalls the Skolem paradox, and then an opposite fact that all the theories of natural numbers (in first order logic) admit uncountable models. However, he emphasizes that in these cases, it is doubtful that we are really facing emergence. As a very serious example of emergence, he provides Gödel’s discovery from 1931 and “the advent of undecidability as a consequence of one simple step consisting of piecing together multiplication and addition”, or vice versa (Krajewski 2012, 102; Krajewski 2011, 17). He argues that in this case, even if logicians and mathematicians had become used to Gödel’s theorem, the initial surprise connected with this discovery is not eliminated, because it is still difficult to explain why this undecidability occurs (which is connected with another deep problem – the inadequacy and insufficiency of our definitions of numbers, supplemented with our implicit resources, or in other words: our tacit knowledge) (ibid., 103). Moreover, Stanisław Krajewski notes that we can address very similar philosophical questions to mathematics and logic. This concerns, for instance, the above-mentioned idea of axiomatization and the foundations of mathematical and logical axiomatic systems. The question about the origin of their axioms refers to both mathematics and logic. He points out that we can answer this by saying that their origin is intuition. But this reply is still not sufficient, and in fact we have a broad range of possible answers, from widely understood formalism, positivism or conventionalism (claiming that we can choose axioms arbitrarily) to the quite opposite, often called realism or Platonism (referring to certain existing objects and their properties) (ibid., 37). In the reflection on mathematics and logic, there is the very similar and still open question concerning the ontological status of, respectively, numbers (cf. Gillies 2013, 31; Dejnozka 1996, 165–166; Mayberry 2000, 29–32) as well as propositions (see: Gochet 2012, 51–54; Epstein 2011, 2; Dejnozka 1996, 147). This is another point where mathematics and logic meet, and another reason to consider them together when asking about their foundations. Finally, the topic of such foundations is connected with several key concepts, which, along with the development of set theories and generally mathematical logic, became common to them. At the very least, we can enumer-

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ate the concepts of set, relation, infinity and so on (Epstein 2011, xxi). This community of concepts and problems enables the reflections concerning such concepts to be applicable both to considerations on logic and on mathematics. To conclude this section, we can refer to some intuitive similarities or closeness between logic and mathematics, which seem to lie under all the abovementioned entanglements and to some extent explain them. One of them is their property, which we can call formalism. Putting it very simply and very generally, mathematics is about formal objects (either real or just arbitrarily assumed) or a formal approach to such objects, and logic is about formal aspects of thinking (in some natural or unnatural language). However, logic is also about some formal objects of thinking (or of language, like propositions). This formal approach is, of course, connected with abstracting from what is real or from particulars; however, it does not exclude empirical investigation. Second, even though the first of them is about some objects and the latter mainly about thinking, there is another strong link: thinking must have its object. When we conduct a logical analysis of our thinking, it is always thinking about (formal) objects, which can also be described by mathematics. And finally, when we describe objects through mathematics, it is an act of thinking, and we do it logically, so we can formalize such reasoning. At the same time, we also can mathematically describe our models of thinking. To quote Richard Epstein on this aspect, who to some extent limited Zermelo’s definition cited above: The word ‘mathematical’ in ‘classical mathematical logic’, then, had two meanings: the mathematization of models of reasoning, and the use of formal logic to formalize reasoning in mathematics. (Epstein 2011, xxii) According to all the reasons presented above, it seems worthwhile to follow Stanisław Krajewski’s paths and consider – within reflections concerning the foundations – logic and mathematics together.

2

The Significance of Asking about Foundations

In the previous section, I tried to argue that it was worth putting logic and mathematics together, especially in the context of the question about foundations. However, there still remains the problem of the importance of such questions. There is, of course, an obvious answer, namely that it can always surprise us with new and interesting results. But it seems that there are other important reasons.

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Philosophical questions referring to logic and mathematics have been present from the beginnings of philosophy. Mathematics suddenly jumped into the heart of the first philosophical inquiries when Pythagoreans gave their spectacular answer to the question of the foundations (gr. arché or archai) of reality. In their view, the ultimate elements of reality were numbers; after their master Pythagoras, they claimed that numbers were eternal, indestructible and that to some extent, all the things of the world depended on them (Solomon and Higgins 2016, 114; Reale 2001, 29). Their theory explained what the world was built of and how it worked. But at the same time, it opened the question about the status of mathematical objects. If numbers are the bricks or even the building blocks of reality, and if there is harmony (understood as a combination of proportions) everywhere, numbers and proportions must be real, objective or even absolute. In subsequent centuries, the problem of mathematical objects was not neglected, and raised by Plato, Aristotle and their followers. One of the consequences of this was Plato’s theory of objective mathematical objects, accessible only to human reason within its act called dianoia.2 Another one was Aristotle’s criticism of Pythagorean and Platonic theories, supplemented by his theory of science, including the theory of mathematics, the subject of which were forms, which in the aspect of existence or being were considered as existing in the material world, whereas in the aspect of notions and definitions – were independent from matter.3 Moreover, these early reflections concerning mathematics could have inspired the development of logic. Although logic was independently developed by Greek philosophers, starting with Socrates, Plato and the sophists, and by successive generations who created impressive logical theories and identified many logical rules, it is probable (although still debatable) that the idea of axiomatization embodied in 4th century BC geometry influenced Aristotle4 and was the basis for his theory of sciences, understood as axiomatized deductive systems, and perhaps for his theory of deductive reasoning. And logic was being developed, on the one hand, along with philosophical considerations about language (the sophists, Plato, Aristotle), the objectivity of truth and the possibility of objective knowledge (Socrates and Plato vs. the sophists), and on the other hand, together with numerous paradoxes (e.g. Eleatic School), very often connected with reflections on reality and movement, but finally leading to logical ones. They produced questions about the nature and role of logic and 2 See: Plato, Republic, VI, 510C and 511E. 3 See: Aristotle, Physics, II, 2, 193b–194a; Aristotle, Metaphysics, VI (E), 1, 1026a; Aristotle, XIV (N), 2, 3, 5. 4 See possible scenarios and an in-depth discussion in: McKirahan 2017, 135–138.

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finally provided mature answers and theories formulated by Aristotle and the stoics. I tend to show that questions about the foundations of logic and mathematics accompany philosophy from the beginning, and that they are also very inspiring, fruitful and help to open new fields of problems. Philosophy simply needs them. What is more, it seems that such questions come from the natural inclination of philosophy to ask about the foundations, conditions, beginnings of anything. Whatever exists, either as something real, or as a phenomenon, can be of interest to philosophy. Especially if it is considered important. And for the spirit of philosophy, such questions as: why, how, from where and from what, etc., are essential. Getting rid of them would probably lead to the annihilation of philosophy. So as long as logic and mathematics exist as an important part of human activity, and we are doing it, it is both natural and important to ask about their foundations. Another reason is that to some extent, these questions are constitutive for the sciences, including logic and mathematics. We can learn from the history of science: mathematics and logic were practiced before formulating the mature theories describing them and their methods. However, such activity cannot prove that its results are necessarily true and cannot point out what it is about and how it is related to the world. It can just argue that it works. Today we know that we cannot be sure about the foundations and these questions are still open (cf. Krajewski 2011, 130–137; Nievergelt 2012, xiv). However, this continuous asking, this movement of thought which undermines subsequent theories provides awareness of their limitations and helps to improve it, and on this basis it allows logic and mathematics to be treated as sciences. Mathematicians and logicians should be aware or should think about what they are doing, what they are standing on, what they are talking about, even if they are working “within” one logic, or one of the logics or above them… Being aware that the foundations are fuzzy, flickering or doubtful, and continuously examining them is completely different from disregarding them. Finally, it seems that these questions are important not only for the philosophy of science, but also generally for the humanities. In one of his books, Stanisław Krajewski attempted to answer the question “Is mathematics a humanistic science?” Among many (possible) humanistic aspects of mathematics, he analyzed how the results of mathematics or logic may be inspiring for the humanities. The most appealing example is the inspiration for the humanities provided by Gödel’s theorem when placed in a wrong context. An example of the broader use of this theorem given by Stanisław Krajewski is its popular formulation: the truth is elusive, there are unprovable truths. He points out that if we try to generalize such a theorem, applying it to any the-

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ory or discourse which cannot be formalized as a mathematical theory, such a generalization is not justified and may be incorrect (Krajewski 2011, 24–25). However, he admits that such an interpretation may express philosophically important beliefs, showing that a broader interpretation of Gödel’s theorem is close to the postmodern attitude (ibid., 140), and somehow expresses its general convictions. Similarly, he also notes that the erosion of certainty, reported by some philosophers of mathematics, connected with the pluralism of theories in mathematics (which also can be referred to as pluralism in logic, or in this context: logics), may support not only postmodernism in mathematics, but postmodernism in general (ibid., 137). Finally, he emphasizes that although strict results concerning the foundations of logic and mathematics, such as Gödel’s theorem, neither settle philosophical disputes nor help in discovering new facts or regularities, they can be truly inspiring for reflections in many areas of the humanities. To sum up, inquiries concerning the foundations of logic and mathematics are important not only for logicians and mathematicians. They can be (and they are) great inspiration for the whole of philosophy and even the humanities, at least as a trigger producing or awakening some important intuitions.

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Various Paths of Inquiries Concerning the Foundations

For the reasons provided above, it seems very important to continually ask questions about the foundations of logic and mathematics. Therefore, we believe discussion of these issues, raised in the 11 articles of this volume, to be worthwhile. The articles can be ordered in three groups based on the diverse philosophical approaches to the concept of foundations. First of all, we ask about foundations understood as a real basis for logic and mathematics, combining the ontological and epistemological perspectives. This is connected with the traditional questions “an sit?” and “quid sit?”, so: are there any foundations at all, and what are they? Among the foundations understood in this way, we refer to such topics as the reality of mathematical objects according to mathematical Platonism, as well as to apriorism and searching for a real basis of logic. We begin with an article that refers to the debate concerning foundationalism in mathematics and the understanding of the notion of foundations. In the paper From Speculative to Practical Foundations of Mathematics: A Communication-Centered Account, Vladislav Shaposhnikov discusses various positions presented in the 20th century to show an evolution in thinking about the foundations “in the post-Gödelian intellectual landscape”, which – starting

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from the classical approach, followed by anti-foundationalism – finally leads to two recent trends: the anthropological and practical understanding of foundations. As he points out, both of them can be interpreted in terms of communication with both human and non-human agents. The next two articles refer to the problem of mathematical Platonism. In the first – Is There an Absolute Mathematical Reality? – Zbigniew Król argues that the “extreme” position presented by Gödel can be successfully replaced with several versions of phenomenological Platonism. As he shows, absolute Platonism and “phenomenological Platonism” are different with regard to their ontological presuppositions, but in epistemological terms (phenomenologically construed), they are indistinguishable. Analyzing particular possible positions, he also refers to the issue of the possibility of mathematical alter-theories and the idea of the plurality of “absolute worlds”. In the article Mathematical Modalities and Mathematical Explanations, Krzysztof Wójtowicz addresses the problem of the applicability of mathematics in the natural sciences and its connection with the debate between mathematical realists and anti-realists. The author discusses Geoffrey Hellman’s position, which states that the explanatory power of mathematics does not entail realism and the anti-realist can freely use modal notions without being committed to possible worlds or abstract structures. He shows that Hellman’s approach is useful to better understand mathematical modality and reconstruct or reinterpret (in line with this idea) mathematical tools needed in science in an anti-realistic manner. Finally, the article Apriorism, Aposteriorism and the Genesis of Logic by Jan Woleński is focused on the “roots” of logic in a genetic perspective. The author sets up an opposition between apriorism and aposteriorism, tries to answer what is logic and discusses the issue of pluralism, to finally defend moderate aposteriorism in the philosophy of logic. At the end, he argues that evolutionary biology and genetics provide data for conceiving logic as rooted in the properties of DNA. The second group is devoted to the foundations understood as the basic concepts for logic and mathematics, such as truth and satisfaction, certainty and plausibility, as well as set and infinity. In the article On Some Problems with Truth and Satisfaction, Cezary Cieśliński presents two strategies or two approaches to semantics. One of them adopts as a basis the notion of satisfaction and treating the concept of truth as defined in terms of satisfaction, and the opposite one places the notion of truth as primary. As he points out, the definability relation between the notions of truth and satisfaction is sometimes problematic and the decision to ascribe primacy to one or the other has important consequences. Taking

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into account both the model-theoretic approach and the axiomatic one, he analyses both strategies and proposes a minimally adequate semantic theory, guaranteeing a mutual definability of these notions. Ricardo Silvestre refers to the issue of the unity and plurality of logical systems. In the paper Inductive Plausibility and Certainty: A Multimodal Paraconsisent and Nonmonotonic Logic, he shows that in addition to the strategy of seeking common assumptions for many logics, it is also possible to follow the opposite one and combine different logics into coherent systems. Following the idea of combining modal logic and paraconsistent logic, he presents the concept of paranormal modal logic and explains how it can be extended to use the notions of plausibility along with an inductive reasoning mechanism by introducing a non-classical multimodal logic of plausibility and certainty. In turn, Roman Murawski refers to the common notions of logic and mathematics, which are set and infinity. In the paper On the Reception of Cantor’s Theory of Infinity (Mathematicians vs. Theologians), he describes mathematicians’ objections to Cantor’s set theory and his concept of actual infinity, and the much more positive reactions of Catholic theologians of his time, with whom he was in contact, being convinced of the great significance of this theory for metaphysics and theology. The author focuses on Cantor’s correspondence with Cardinal Franzelin, within which the German mathematician refined his theory of infinity in the context of metaphysics and theology. The third group is centered on the foundations understood as the basic mechanisms that work in the practice of using logic and mathematics. Discussed are: scientific practice, in the context of the relationship between theories and to what they refer, the role and status of paradoxes and thought experiments in philosophical practice, as well as the essence of reasoning. Michael Heller, in his paper How Is the World Mathematical?, starts with the problem of the interaction between mathematical theories and the empirical data to which they can be applied, resulting in new information. The main question he focuses on is: “How can a formal language (such as that of mathematicians) be effective and, in particular, how can it be effective in the physical world?” Using categorical semantics, he explains the idea of mutual causal influences between the mathematical structure of a given theory and its natural semantics (going both ways), and the notion of “adjointness”. In the paper An Analysis of Paradoxes, Anna Wójtowicz points out the special role or power of paradoxes. After presenting how paradoxes are defined and what distinguishes them, she classifies them in three groups to indicate how they relate to fundamental ontological and epistemological problems. Those from the first group reveal the weaknesses of our intuitions. The second group consists of paradoxes indicating the weaknesses of the theories describ-

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ing a too complicated reality. The last group are those which finally leave us with a question about the possibility of establishing correct meanings. In the article Computation and Visualization Thought Experiments after Lakatos’s Heuristic Guessing Method (Semantics of Thought Experiments – Pt. Mathematical Thought Experiments), C. Peter Hertogh refers to thought experiments in two ways. He presents them as a kind of foundation: as constituting a basic philosophical methodology in the sciences and philosophical disciplines, including logic and mathematics. At the same time, he points out that thought experiment analyses themselves are part of logic, semantics. The author presents examples of such analyses, using various logical approaches. We conclude the volume with a short article written in a lighter form (with a pinch of salt): Are Dogs Logical Animals? by Jean-Yves Béziau. The author analyzes several examples, supplemented with pictures, to consider the question of whether animals use logic.

References Balaguer, Mark, Platonism and Anti-Platonism in Mathematics, Oxford University Press 2001. Dejnozka, Jan, The Ontology of the Analytic Tradition and Its Origins: Realism and Identity in Frege, Russell, Wittgenstein, and Quine, Rowman & Littlefield 1996. Epstein, Richard L., Classical Mathematical Logic: The Semantic Foundations of Logic, Princeton University Press 2011. Gillies, Donald, Frege, Dedekind, and Peano on the Foundations of Arithmetic, Routledge 2013. Gochet, Paul, Outline of a Nominalist Theory of Propositions: An Essay in the Theory of Meaning and in the Philosophy of Logic, Reidel Publishing Company, DordrechtBoston-London 2012. Krajewski, Stanisław, Czy matematyka jest nauką humanistyczną, Kraków 2011. Krajewski, Stanisław, Emergence in Mathematics, “Studies in Logic, Grammar and Rhetoric” 2012, no. 27(40), 95–105. Mancosu, P., Zach R., Badesa C., The Development of Mathematical Logic from Russell to Tarski: 1900–1935, in: ed. Leila Haaparanta, The History of Modern Logic, Oxford University Press 2009, pp. 318–470. Mayberry, John P., The Foundations of Mathematics in the Theory of Sets, Cambridge University Press 2000. McKirahan, Richard D., Principles and Proofs: Aristotle’s Theory of Demonstrative Science, Princeton University Press 2017.

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Nievergelt, Yves, Foundations of Logic and Mathematics: Applications to Computer Science and Cryptography, Springer Science & Business Media, New York 2012. Reale, Giovanni, Il pensiero antico, Milano 2001. Solomon, Robert C., Higgins, Kathleen M., The Big Questions: A Short Introduction to Philosophy, 10th ed., Cengage Learning, Boston 2016. Zermelo, Ernst, Mathematische Logik. Vorlesunggehalten von Prof. Dr E. Zermelo zu Göttingenim S.S. 1908. Lecture notes by Kurt Grelling. Nachlaß Zermelo, Kapsel 4, Universitätsbibliothek Freiburg im Breisgau, 1908.

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Chapter 1

From Speculative to Practical Foundations of Mathematics: A Communication-Centered Account Vladislav Shaposhnikov

Abstract The present paper is devoted to several major changes in how the foundations of mathematics are understood – changes that have been taking place in the postGödelian intellectual landscape. The foundationalist and speculative approaches of the previous historical period first gave way to the “no foundations” approach and, next, to two concurrent trends of “anthropological” and “practical” approaches to the foundational problem. An attempt is made to demonstrate that each of the two recent trends can be interpreted in terms of communication with both human and nonhuman agents.

Keywords philosophy of mathematical practice – foundational crisis – anti-foundationalism – conceptual metaphors – life-world (Lebenswelt) – Gödel – Wittgenstein – Voevodsky – univalent foundations

… I thought that certainty is more likely to be found in mathematics than elsewhere. But I discovered that many mathematical demonstrations, which my teachers expected me to accept, were full of fallacies, and that, if certainty were indeed discoverable in mathematics, it would be in a new kind of mathematics, with more solid foundations than those that had hitherto been thought secure. But as the work proceeded, I was continually reminded of the fable about the elephant and the tortoise. Having constructed an elephant upon which the mathematical world could rest, I found the elephant tottering, and proceeded to construct a tortoise to keep the elephant from falling. But the tortoise was no more se-

© Vladislav Shaposhnikov 2021 | DOI:10.1163/9789004445956_003 Marcin

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cure than the elephant, and after some twenty years of very arduous toil, I came to the conclusion that there was nothing more that I could do in the way of making mathematical knowledge indubitable. (Russell 1956 [1952], 54–55)

… The mathematical problems of what is called foundations are no more the foundation of mathematics for us than the painted rock is the support of a painted tower. (Wittgenstein 1978, 378)

… A foundation makes explicit the essential general features, ingredients, and operations of a science as well as its origins and general laws of development. The purpose of making these explicit is to provide a guide to the learning, use, and further development of the science. A “pure” foundation that forgets this purpose and pursues a speculative “foundation” for its own sake is clearly a nonfoundation. (Lawvere and Rosebrugh 2003, 235)

∵ For a very long time – at least from Galen (Grant 1989) to Descartes (1985, Vol. I, 10–13) and from Descartes to Hilbert1 – mathematics was considered to be the paragon of certitude. Though there is a great variety of “foundational work” or “foundational jobs” (see Feferman 1985, 229; Maddy 2019), the main target of foundational research, in my opinion, has been to explain that fascinating certitude and, perhaps, also to provide support to it, if some kind of support is needed. For some cultural environments, the certitude of mathematics goes without saying; for others, it looks rather dubious. Through most of the 19th century and into the early 20th century, the mathematical foundations were recognized as being deeply problematic. The situation is often described as a “foun-

1 Hilbert called mathematics “[the] paragon of reliability and truth ([das] Muster von Sicherheit und Wahrheit).” See (Hilbert 1926, 170); (Hilbert 1967 [1925], 375).

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dational crisis (Grundlagenkrise).”2 It does not necessarily mean that something was seriously wrong with the mathematical practice of the time, but there certainly was an obvious incongruity between the unprecedented explosive growth of new and highly abstract mathematics and the far more conservative and slowly changing philosophy of mathematics. In this paper, I am concerned with a later story; it begins in the 1930s and goes up to some of today’s controversies. On the way, I pay close attention to the conceptual metaphors that are used while talking about the foundations of knowledge, especially mathematical knowledge. I begin with a preliminary discussion and dismissal of the popular idea of “the three crises” in the history of mathematics, which I treat as a kind of fable with some didactic intention (section 1). Then, I turn to a general epistemological consideration of the foundational problem and its foundationalist and anti-foundationalist solutions (section 2). Next, I examine how the anti-foundationalist epistemological tendencies showed their worth in the philosophy of mathematics in the form of the popular “no foundations” trend from the 1930s to the 1980s (section 3). The main hero of section 3 is Ludwig Wittgenstein, whose philosophy of mathematics by no means just exemplifies the “no foundations” approach, but also prepares the ground for a more positive discussion of the “foundations without foundationalism” (section 4). In section 4, I study some typical approaches to the foundations of mathematics in the philosophy of mathematical practice. My examples are taken mainly from the 1980s and the 1990s, with Saunders Mac Lane’s and Eric Livingston’s approaches as model ones to the problem of foundations. I finish this section with the distinction between “anthropological” and “practical” foundations of mathematics. Section 5 is devoted to the “practical” foundations of mathematics. The idea to introduce “practical” foundations in contrast to “theoretical” or, better, “speculative” ones comes from the 1990s and became popular in the 2000s. It is closely associated with the computer revolution in mathematics and contemporary complexity challenges. The main hero of this section is Vladimir Voevodsky with his “univalent foundations of mathematics” – the most widely discussed foundational project of the last decade. In the conclusion, I reflect on the foundational meaning of communication. The theme of communication runs through the whole paper, though only in the end it is favored with separate consideration. This is because only recently have mathematicians seemed to become fully aware of the crucial role of effective communication for maintaining the celebrated certainty of mathematics.

2 Hermann Weyl seems to be the one who coined the term. See (Weyl 1921); (Weyl 1998).

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Prelude: The Myth of the Three Crises

The locus classicus of the story, as far as I know, is paragraph five of chapter one, entitled “The Three Crises,” of Fraenkel and Bar-Hillel’s influential book Foundations of Set Theory (Fraenkel and Bar-Hillel 1958, 14–15). The authors are anxious to sketch a historical perspective of the early twentieth century foundational crisis, i.e., to convince their readers that it is in no way unique for the history of mathematics. To achieve that end, they add to the picture “the first crisis in foundations of mathematics” (ibid., 198), the one provoked by the discovery of the incommensurability of magnitudes and Zeno’s paradoxes in the 5th century B.C.E., and “the second crisis” (ibid., 15, 198), caused by the problems with the foundations of mathematical analysis in the modern period. This simple story turned out to have considerable mythmaking potential. One of the standard symbolic meanings of number three is “completeness” or “totality.” “The three crises” were nearly uniformly allocated among all main historical periods: one to ancient history, one to the modern era, and one to our time. Here the medieval period is omitted, but it is still too often considered to be a dead one in the history of science, so it makes no difference. Anyway, the message is quite clear: foundational crises regularly emerge in the history of mathematics; our crisis is not the first and, apparently, not the last one. In my opinion, the authors of the first half of the 20th century, who felt that in the 1950s mathematics was still undergoing a foundational crisis (ibid., 15), were inclined to unwarrantedly project their peculiar way of seeing the current situation in mathematics both on the past and the future of their discipline. The idea of the foundational crisis in pre-Euclidean Greek mathematics has been widely popular at least since Paul Tannery, who famously called the discovery of incommensurable magnitudes “a true logical scandal (un véritable scandale logique)” (Tannery 1887, 98).3 It is still the case, even though already in the 1960s and 1970s the idea of the first mathematical crisis met valid criticism both from historians of Pythagorean philosophy and from historians of Greek mathematics: available ancient sources seem to give us no firm basis for such a belief (See Burkert 1972 [1962], 455–465; Knorr 1975, 307–312). Talking of the “second” foundational crisis, we should go into detail a bit more. The foundational issues in the 18th century presupposed first and foremost ontological questions: do infinitely small quantities, imaginary quantities, negative quantities, or some other mathematical entities really exist? The 3 According to Struik, the words of Tannery indicate “a crisis in Greek mathematics” (Struik 1948, 50).

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philosophy of mathematics of the time oscillated between mathematical realism and mathematical fictionalism. For the first position, mathematical objects in some way existed both in the real world and in God’s mind. For the second one, they were useful human-made fiction, just like characters of great literature and art. When people asked whether mathematical theories were true or reliable, they had in mind that very dilemma. In 1734, George Berkeley famously called Newton’s fluxions “the Ghosts of departed Quantities” (Berkeley 1992, 199). His words should be considered against that kind of background.4 The correctness of mathematical methods was also among the basic issues, but in that aspect as well, one has to choose some particular position from the spectrum with two opposite ends: loyalty to the rigorous geometrical methods of the Ancients plus the standards of Aristotelian logic, at one pole, and fascination with practically effective, but badly grounded, new analytic heuristics at one’s own risk at the other pole. Indeed, the basic concepts and principles of mathematics were widely discussed among mathematicians throughout the 18th century.5 At the same time, we find no serious anxiety about the lack of a generally accepted resolution of the foundational issues in their milieu. There are no signs that the mathematicians of the first half and middle of the 18th century took their discipline as being in crisis. They were quite aware of certain ambiguities that invaded the fundamental mathematical concepts of the day. Nevertheless, it never affected their professional self-consciousness to any considerable degree. They were quite sure that a fuller and more detailed “Account of the Grounds”6 would dissolve the foundational problems completely.

4 Cf. with the following remark: “The problem of foundations did not exist in the eighteenth century as we understand it nowadays. Mathematicians were more occupied with defining the ‘principles’ of the calculus. They were concerned with the ontological status of the objects of the calculus and with the correctness of the methods of the calculus according to the standards of Aristotelian logic. […] British mathematicians had an empiricist philosophical background. […] Even though nobody tried to develop an empiricist methodology of mathematics, it was somehow implied that mathematics too had to possess a certain empirical foundation.” (Guicciardini 1989, 38–39). 5 See (Boyer 1959, ch. VI: The Period of Indecision). Gert Schubring insists: “It is a widespread opinion to think of eighteenth-century mathematics as unconcerned with the foundations and as interested only in the further development of analysis. As we have seen with the concepts of negative numbers as well as with the infinitely small quantities, the mathematicians were, in contrast, very anxious to clarify basic concepts.” (Schubring 2005, 285). In my opinion, Schubring overestimates the level of anxiety among the mathematicians of the 18th century giving no convincing evidence for it being really high. 6 The words by Colin Maclaurin from his Treatise of Fluxions (1742). Cited in (Guicciardini 1989, 47).

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Apparent obscurity of the metaphysics of contemporary mathematics was partly compensated at the time by its successful applicability to the physical world. Even more so: there seemed to be no considerable gap between physical and mathematical entities (Daston 1988, 221–224; Shaposhnikov 2014, 189–193). Such a gap, if it existed at all, was well veiled by the widespread popularity of “naïve abstractionism.”7 Mathematics was generally considered to be a study of quantities, which made it so easy to blur the transition between physical and mathematical matters.8 Newton was convinced that his fluents and fluxions had “an existence in nature.”9 Colin Maclaurin based his presentation of calculus on a sort of kinematic geometry, taking space, time, motion, and velocity as his primitive notions.10 When Leonhard Euler illustrated the difference between “constant quantities” and “variable quantities” (mathematical distinction) with an example of “a shot fired from a cannon with a charge of gunpowder” (physical event), he experienced no difficulties in transitioning from physical quantities – the amount of gunpowder, the angle of elevation of the cannon above the horizon, the distance traveled by the shot, the length of time the shot is in the air, the length of the barrel, the weight of the shot and the force of the explosion – to mathematical quantities, such as constant, 7

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By this name, Jeremy Gray designated “the idea that mathematics deals with idealizations of familiar objects”. See (Gray 1992, 228). The classic example of naïve abstractionism is given by d’Alembert in his “Discourse préliminaire” (1751). See (d’Alembert 1995, 16–25). I should mention that D’Alembert constantly talked about “abstractions” not “idealizations” contrary to Gray’s definition of naïve abstractionism. Both notions are closely connected though they are not identical. To illustrate his definition, Gray (1992, 228–229) gives a quote from a later work by D’Alembert: “In nature for example there is certainly no perfect circle, but the more it approaches that state the more nearly it has exactly and rigorously the properties of a perfect circle that geometry establishes” (“Géométrie,” 1785). It argues to the same end: not only mere abstraction, i.e., separation of properties, but even idealization, i.e., a shift from imperfect to perfect, creates no gap between physical and mathematical realms for Nature can provide us with all intermediate degrees of perfection. D’Alembert divides “Science of Nature” into “physics” and “mathematics.” “Physics” deals with some general properties of bodies (corporeal individuals), such as extension, movement, impenetrability and their varieties. He continues: “Quantity, another more general property of bodies, which supposes all the others, has formed the object of mathematics. Quantity, the object of mathematics, could be considered either alone and independent of real and abstract individual things from which one gained knowledge of it, or it could be considered in these real and abstract beings, or it could be considered in their effects investigated according to real or supposed causes; this reflection leads to the division of mathematics into pure mathematics, mixed mathematics and physico-mathematics.” “Detailed Explanation of the System of Human Knowledge,” in (d’Alembert 1995, 151–153). Tractatus de quadratura curvarum (1704). Cited in (Guicciardini 1989, 51). In Treatise of Fluxions (1742). See (Guicciardini 1989, 49–51).

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variable, function (“how the variable quantities depend on each other”), increment, ratio of increments, vanishing increment or differential, etc. up to the full conceptual apparatus of differential and integral calculus.11 A much higher awareness of possible deep-lying errors and the corresponding anxiety level or even dread (Gray 2004, 25–27), which resulted in the “new attitude towards the foundations of mathematics” (Bottazzini 1986 [1981], 90)12 that we find among the leading mathematicians of the 19th century make an obvious contrast with the previous frame of mind. Not only mathematics but the whole scientific enterprise, in a way, had a new start (Gray 2004, 24) at the time. The new situation has been perfectly captured in the term “the second scientific revolution” (Kuhn 1977, 218, 220, 147). First of all, mathematicians changed their social status. They were no longer independent scholars or academicians but university professors in their majority. So, they were inclined to pursue a more rigorous approach “for didactic purposes” (Bottazzini 1986, 91). Second, it was the period of “the growing separation of mathematics from physics” (Gray 2004, 37), as well as of abstract or pure mathematics, i.e., of analysis, including arithmetic and algebra – from concrete mathematics, i.e., geometry and mechanics (Bottazzini 1989, 91). Consequently, ontological issues, as far as mathematical objects are concerned, became rather confused, for the standard approaches, such as naïve abstractionism or Kantian transcendentalism, failed to cope with the new situation, at least without major modification (cf. Gray 1992). Third, mathematics, together with other sciences, was separated from Christian theology, which ceased to be considered genuine science (Shaposhnikov 2016, 33–35). Pure mathematics, as a result, obtained a reputation of a completely autonomous newly-crowned queen of sciences. Noblesse oblige. Mathematics, to be as perfect as it should be, needed a new kind of foundation, intrinsic and absolute. This is why the foundations of infinitesimal calculus turned into a crucial mathematical problem only in the 19th century. We should do justice to Fraenkel and Bar-Hillel: they situate the second crisis “at the beginning of the 19th century” (Fraenkel and Bar-Hillel 1958, 15), not in the 17th or 18th centuries. It may seem that the “second” crisis is quite distinct from the “third” one. They are separated in time by almost a century. The former was provoked by obscurity and the shakiness of the foundations of the calculus while the latter – by the set-theoretic and logical paradoxes. Nevertheless, in my opinion, they should be treated as successive stages of

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See Euler’s preface to Institutiones Calculi Differentialis (1755), in: (Euler 2000, v–vii). Bottazzini’s central example is Niels Henrik Abel’s letters dated 1826.

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the same crisis.13 At a deeper level, that crisis was caused by the quest for an absolute and autonomous foundation for mathematics and its inevitable failure.14 What kind of moral may be drawn from the preceding discussion of the story of the three crises? To begin with, it is crucial to be more sensitive both to historical ruptures and historical continuities, even in the case of logic and mathematics. And above all, one should keep in mind that logical and mathematical discourses are always “situated culturally and historically” as any other variety of discourse, to be sure (Crease 1997, 3). We are still naturally inclined to see “the foundations of mathematics” exclusively through the prism of the early-twentieth-century stories and myths, but it is surely not the only option available. Tracking the changes in attitude towards the foundations of mathematics in the course of history may be rather instructive and even have a liberating effect as far as our way of doing the philosophy of mathematics is concerned (Cf. Ferreirós 2005). What is more, it may also help to introduce new insights for improving our comprehension of the foundational debate in the mathematics of the first decades of the twentieth century, though I am not going to dwell upon the subject in this paper. Here, I start with the origins of the foundational issues in ancient Greek philosophy, but only to get a convenient reference point for my discussion of some post-Gödelian trends in thinking and talking about the foundations.

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Turtles All the Way Down

The foundational problem, so brilliantly emphasized by Bertrand Russell (see the first epigraph to this paper), seems to be a Greek one. It was launched, 13

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Cf. an approach to the “third” crisis declared by Ferreirós: “The foundational crisis is usually understood as a relatively localized event in the 1920s, a heated debate between the partisans of ‘classical’ (meaning late-nineteenth-century) mathematics, led by Hilbert, and their critics, led by Brouwer, who advocated strong revision of the received doctrines. There is, however, a second, and in my opinion very important, sense in which the ‘crisis’ was a long and global process, indistinguishable from the rise of modern mathematics and the philosophical and methodological issues it created.” (Ferreirós 2008, 142). In my view, the period of “the rise of modern mathematics” may safely be prolonged into the past to include not only the last three decades of the 19th century, which Ferreirós apparently had in mind, but almost the whole century. One can find a similar position in Gray, who wrote: “This nineteenth-century revolution resembles the scientific revolution identified by Butterfield in being long, slow, and without an organized leadership having a specific programme, more than it does Kuhn’s examples. […] It is my contention that by then [by the period of Brouwer, Hilbert and Gödel] the revolution had occurred, and that it were the implications that caused the fuss.” (Gray 1992, 245–246). For a more detailed account see (Shaposhnikov 2016). Marcin Trepczyński - 978-90-04-44595-6 Downloaded from Brill.com 03/03/2024 09:20:15PM via University of Wisconsin-Madison

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as far as we know, in the Aristotelian methodology of science (Prior and Posterior Analytics) and Pyrrhonian skepticism (Sextus Empiricus). It is closely connected with the issues of giving explicit justification for our beliefs and producing or refuting proofs. In Republic (books VI–VII), Plato famously criticized mathematicians for being reconciled with the hypothetical character of their theories and “not travelling up to a first principle” (511a). He contrasted mathematics with dialectics, which “makes its way to a first principle that is not a hypothesis (ἀρχὴ ἀνυπόθετος)” (510b).15 Moreover, mathematics presupposes dialectics as “placed at the top of the other subjects like a coping stone (θριγκός)” (534e).16 Plato’s “unhypothetical first principle of everything” (511b)17 or “the form of the good (ἡ τοῦ ἀγαθοῦ ἰδέα)” (508d–509b),18 which is being discovered by dialectics and forms the ultimate basis for all mathematical disciplines, apparently leads us into the divine realm, joining mathematics and religion together. His mathematician-philosopher should possess or be granted divine powers. It is no wonder that Plato hints at such superhuman abilities making his Glaucon, in response to Socrates’ words “the good is not being, but superior to it in rank and power,” suddenly exclaim “By Apollo, what a daemonic superiority!” (509c).19 Aristotle, who usually preferred to take a more mundane stance than his mentor, just observed: “it is impossible that there should be demonstration of absolutely everything; there would be an infinite regress, so that there would still be no demonstration” (Metaphysics, 1006a).20 The epistemic regress problem was stated explicitly as the following trilemma for a non-skeptic or dogmatist:21 Horn A: An infinite regress of foundations Horn B: A circle in the process of grounding Horn C: Hypothetical foundations This classic approach takes for granted that the process of justification is linear or at least can be easily and naturally linearized. I will call this presuppo15 16 17 18 19 20 21

Plato 1997, 1131. Ibid., 1150. Ibid., 1132. Ibid., 1129–30. Ibid., 1130. Aristotle 1995, vol. II, 1588. It is often called nowadays “the Münchhausen Trilemma.” The term was coined by Albert. See (Albert 1985, 16–21). Nevertheless, the Trilemma was already well known to the ancients. It is also called “the Agrippan Trilemma” for it constitutes a part of the so-called five Agrippan skeptical modes. See, e.g., (Sienkiewicz 2019).

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sition the linearizability hypothesis. Propositions are imagined to be like beads on a string or links of a chain along which one can move in a step-by-step manner in both directions.22 In a slightly more complicated case, we can allow our progressive movement to be ramified while our regressive movement should still be linear. In both cases, if we follow our deductions in the opposite direction asking for justification of some accepted propositions, we inevitably meet the aforestated Münchhausen Trilemma. It looks as if we are supposed to imagine a sort of dialogue between two persons, say U and V : U : – I know P1 is true. V : – Why are you sure that P1 is true? U : – Because I know that P2 is true and P2 ⊢ P1 . V : – Why are you sure that P2 is true? U : – Because I know that P3 is true and P3 ⊢ P2 . V : – Why are you sure that P3 is true? Etc. Here Pi may designate a simple proposition or a composite one, being the conjunction of a finite group of propositions: the linearizability hypothesis states that any ramifying regress can be reduced to a linear or serial regress (Berker 2015, 322–323). Horns A and B of the Münchhausen Trilemma are usually considered to be completely unacceptable, so we are left with horn C as the only way out.23 Aristotle, Descartes, and numerous others agree that this is the case. So we inevitably face the issue of first principles. Epistemological foundationalism is a standard name for the line of thought pointed out by horn C. The main foundationalist metaphors for the system of knowledge are that of “a tree” and of “a building.”24 The first principles are “the roots of a tree” and “the foundations of a building,” respectively. No plant can thrive without 22

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Cf. the use of this image as an illustration of the nature of deduction in Descartes: “This is similar to the way in which we know that the last link in a long chain is connected to the first: even if we cannot take in at one glance all the intermediate links on which the connection depends, we can have knowledge of the connection provided we survey the links one after the other, and keep in mind that each link from first to last is attached to its neighbour.” René Descartes, Rules for the Direction of the Mind (Descartes 1985, vol. I, 15). Aristotle, Posterior Analytics, 72b5–73a20. In (Aristotle 1995, vol. I, 117–18). Both metaphors were used by Descartes. For the “tree of philosophy” metaphor, see his “Preface to the French edition” of his Principles of Philosophy (1647), in (Descartes 1985, vol. I, 186). The metaphor of a building can be found in his Discourse on the Method (1637) and Meditations on First Philosophy (1641). See (Descartes 1985, vol. I, 114–117, 122, 125; Descartes 1984, vol. II, 12, 366–383, 407). Descartes especially stressed the last metaphor: “my method imitates that of the architect.” See (Descartes 1984, vol. II, 366).

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long roots going deep into the soil, and no building can be secure without firm foundations. Both metaphors were highly popular with mathematicians for describing the structure of their science in the 19th and most of the 20th centuries.25 Conversely, epistemology and cognitive psychology drastically changed our attitude towards classical foundationalism and foundationalist metaphors during the second half of the 19th century and the 20th century. Let us then pose a question: “Why should a deduction or a justification process be seen as a linear one?” The burden of proof or final justification, in the classic scheme, lies completely with an individual epistemic subject U . It cannot be shared with anyone. That is why it tends to be linear as a soliloquy. Another epistemic subject V in that scheme is not a real interlocutor and partner but is reduced to a formal interiorized interrogator. The picture looks like an epistemic “Robinsonade.” Such a solitary epistemic subject seems to be no more than a replica of the One God of philosophers. From Charles Peirce26 to Donald Davidson27 and on, many authors have convincingly argued that taking an individual epistemic subject as a natural epistemological model is an undesirable oversimplification. Leaving out the regular communication between subjects as the true basis of any epistemic activity, we put the whole epistemological enterprise at risk of fatal distortion. From Pierre Duhem to W.V.O. Quine28 and on, we are gradually persuaded to look at the justification process in science (mathematics and logic included) as a highly complicated and holistic one. The holistic thesis states that no

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These metaphors can be traced back to ancient Platonism. Compare, e.g., the metaphors of a gardener (ϕυτουργός) and an architect (τεκταινόμενος) for God in Plato (Republic, 597d; Timaeus, 28c), and the metaphor of an architect (ἀνὴρ ἀρχιτεκτονικός) for God in Philo (De Opificio Mundi, IV, 17–19). By the way, the metaphor of God the Gardener we also find in Genesis 2: 8, where He is said to have planted a pleasure-garden (παράδεισος). See (Schlimm 2016). Schlimm tries to oppose the building and the plant metaphors, though his examples rather show their close connection. His examples of their use include texts by Moritz Pasch, Richard Dedekind, Felix Klein, Gottlob Frege, Henry Poincaré, David Hilbert and Hermann Weyl. I would add Bertrand Russell to that collection. See, e.g., (Russell 1917 [1902]). Fighting “the spirit of Cartesianism,” Peirce firmly insisted on the shift from individual to community in epistemology. See (Peirce 1868). Davidson says: “[R]ationality is a social trait. Only communicators have it.” To visualize the crucial role of the second person in cognition, he introduced a handy “triangulation” metaphor. See Donald Davidson, “Rational Animals [1982],” in (Davidson 2001, 105); Donald Davidson, “The Second Person [1992],” in (ibid., 119). I am talking about the so-called Duhem–Quine thesis. See, e.g., (Gillies 1993, 98–116); (Verhaegh 2017).

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proposition can be justified or refuted in isolation; only a group of propositions, i.e., a theory, or even a group of theories, can be justified or refuted. Holistic justification is in no way a mere logical procedure, but rather a pragmatic one. The interconnection of propositions, as well as that of theories, is nonlinear; it can hardly be linearized, if at all. It is rather a web-like structure (cf. Quine and Ullian 1978). The introduction of a collective epistemic subject and nonlinear or holistic justification processes may change our attitude towards horns A and B, and, as a result, to horn C as well. Epistemological infinitism takes horn A as a true way out: the chain of reasons should be considered as infinite, i.e., “there are no foundational reasons” and hence no “basic beliefs” (Turri and Klein 2014, 1).29 In my opinion, the most serious objection to infinitism is the so-called “finite mind objection” (ibid., 12–13). It can be traced back to Aristotle, who claimed that “it is impossible to go through infinitely many things (τὰ ἄπειρα).”30 Impossible for whom? Obviously, for someone whose mind is finite. And yet, even a finite mind can (at least in principle) deal with the infinite if it is the potential infinite, i.e., always actually something finite.31 Still another doubt may be raised. Can one cope with a very big, though finite, chain of arguments? “One” means here a mortal individual, not only whose mind is finite, but whose lifespan is severely restricted, that is also finite.32 In my view, there is a lack of fit between the infinitist’s claim that a potentially infinite chain of reasons or arguments is being dealt with and the finite time of human life. Nevertheless, it can be easily reconciled as soon as we shift from an individual to a collective epistemic subject, even if we do not challenge the linearity of the process. In this case, the building of a chain of reasons may be thought of as something passing from one individual epistemic subject to another in potentially infinite succession.33

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The same text is also available as (Klein and Turri 2014). Aristotle, Posterior Analytics, 72b10–11. (Aristotle 1995, vol. I, 117). We again owe this idea to Aristotle. See Aristotle, Physics, 206a9–208a25. (Aristotle 1995, vol. I, 351–354). Infinitists Peter D. Klein and John Turri talk about potentially infinite chains of reasons. Infinitists Klein and Turri agree that “we have finite lives and finite minds.” See (Turri and Klein 2014, 12). It is worth noting that though Descartes envisioned collective epistemic efforts, he restricted them to the progress of knowledge on the basis of the foundations already laid by him alone, which he considered to be final. He wrote in his Discourse on the Method: “by building upon the work of our predecessors and combining the lives and labours of many, we might make much greater progress working together than anyone could make on his own.” See (Descartes 1985, vol. I, 143). He never thought of the reworking of the

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Epistemological coherentism is a position that dispels the unacceptability of horn B: “beliefs are justified in virtue of forming a circle” (Murphy 2006). But to make coherentism inviting, a strictly linear conception of justification should be abandoned. Selim Berker states this idea with the utmost clarity: For once the non-skeptical alternatives have been portrayed in this [linearized] manner, the coherentist option looks, to put it bluntly, stupid – a wholly unattractive position that only a truly desperate philosopher would endorse. How can a mere circle of support, of whatever length, make a belief appropriate to hold? It is no coincidence that coherentists tend to prefer the metaphor of a web to that of a circle. For most coherentists, it is crucial to keep in the structure that gets washed out when we assimilate a ramifying regress into a serial regress. (Berker 2015, 323–324) Peter Murphy conveys the coherentists’ position also by appealing to the metaphors of something web-like: Coherentists are fond of metaphors like rafts, webs, and bricks in an arch. These things stay together because their parts support one another. Each part both supports, and is supported by, other specific parts. So too with justified beliefs: each is both supported by, and supports, other beliefs. This means that among support relations, there are symmetrical support relations: one belief can support a second (perhaps mediately through other beliefs), while the second also supports the first (again, perhaps, mediately). Beliefs that stand in sufficiently strong support relations to one another are coherent, and therefore justified. (Murphy 2006) One of the most celebrated examples of a coherentist metaphor for our justification enterprise is “Neurath’s Raft” (or “Ship,” or, more often, “Boat”): That we always have to do with a whole network of concepts and not with concepts that can be isolated, puts any thinker into the difficult position of having unceasing regard for the whole mass of concepts that

foundations as an everlasting collective enterprise. The position of Kant on the issue was quite similar. In any true science – and mathematics was one of his examples – the revolution takes place only once. It consists in laying the only possible foundation and may be “brought about by the happy inspiration of a single man.” After it, “the road to be taken onward could no longer be missed, and the secure course of a science was entered on and prescribed for all time and to an infinite extent.” See (Kant 1998, 107–108).

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he cannot even survey all at once, and to let the new grow out of the old. Duhem has shown with special emphasis that every statement about any happening is saturated with hypotheses of all sorts and that these in the end are derived from our whole world-view. We are like sailors who on the open sea must reconstruct their ship but are never able to start afresh from the bottom.34 Where a beam is taken away a new one must at once be put there, and for this the rest of the ship is used as support. In this way, by using the old beams and driftwood, the ship can be shaped entirely anew, but only by gradual reconstruction.35 Otto Neurath emphasized that “we cannot start from a tabula rasa as Descartes thought we could” (Neurath 1973, 198), but still believed it is possible in mathematics, though not beyond. He wrote later on, having in mind the concrete sciences, both natural and social, but excluding the formal sciences, i.e., logic and mathematics: We possess no fixed point which may be made the fulcrum for moving the earth; and in like manner we have no absolutely firm ground upon which to establish the sciences. Our actual situation is as if we were on board ship on an open sea and were required to change various parts of the ship during the voyage. We cannot find an absolute immutable basis for science; and our various discussions can only determine whether scientific statements are accepted by a more or less determinate number of scientists and other men. New ideas may be compared with those histor-

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In one of his later works, Neurath put his metaphor in even clearer words: “We are like sailors who have to rebuild their ship on the open sea, without ever being able to dismantle it in dry dock and reconstruct it from the best components.” Otto Neurath, “Protocol Statements [1932/33],” in (Neurath 1983, 92). Otto Neurath, “Anti-Spengler [1921],” in (Neurath 1973, 198–99). Neurath’s boat has obviously merged Plutarch’s metaphor of the Ship of Theseus (Theseus, 23.1), which is constantly rebuilt, and Plato’s metaphor of the Raft, which compares a man, who is using the best of human theories for the lack of some divine doctrine, to the one, who uses a raft (σχεδία) to sail across the dangerous sea for the lack of a firmer vessel (Phaedo, 85c–d). See (Plato 1997, 74). Neurath’s boat metaphor is used in his texts five times over a span of more than thirty years (from 1913 to 1944), so, actually, there are five Neurath boats. Thomas Uebel summarizes: “One central theme – perhaps the central one – of Neurath’s philosophy is the absence of epistemic foundations and their reducible contextuality of knowledge and justification. The continuity of this theme is illustrated by Neurath’s frequent employment of the simile which subsequently Quine made common coin: we are like sailors, who have to repair their boat on the open sea, without ever being able to pull into dry dock.” (Uebel 2004, 4). For a detailed discussion see (Uebel 1996).

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ically accepted by the sciences, but not with an unalterable standard of truth.36 He also echoed the same metaphor when he noted: “Finally, we find ourselves all together in the same ship and are co-operating even when we think we are fighting one another.”37 Thomas Uebel comments on Neurath’s epistemology: Neurath’s Boats teach that in place of ultimate justification there can only be spot checks, and that these legitimating procedures themselves needed legitimation which in turn cannot be foundational either. Scientific knowledge is a communal project that has to hold itself in place. (Uebel 1996, 93) I think we can safely apply Neurath’s metaphor to mathematics as well, though it means to blur the demarcation line between formal and concrete sciences and to overcome the idea of the unique status of mathematics within the system of human knowledge (cf. Shaposhnikov 2014, 193–195). A circle in justification turns into circulus vitiosus, or a vicious circle, i.e., something unacceptable, only when interpreted as something static. On the contrary, when it is taken to be a dynamic pattern, it is quite common and widely, almost ubiquitously, used in different thought practices. The so-called “hermeneutic circle” methodology may serve here as a paradigmatic example.38 Hermeneutic techniques are no more considered as restricted exclusively to the sphere of humanities as opposed to the natural science, but rather as universally applied, including the variety of scientific practices,39 as well as

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Otto Neurath, “Unified Science and Its Encyclopedia [1937],” in (Neurath 1983, 180–181). Otto Neurath, “Universal Jargon and Terminology [1941],” in (Neurath 1983, 229). It seems that, for the first time, we meet the circle as an epistemological pattern in Plato’s theory of anamnesis (Meno, 86a–c). See (Plato 1997, 886). Shklar associates the hermeneutic circle with Neo-Platonism: “Hermes carried the messages of the gods, and hermeneutics is the art of reading them. The circle with a message, the hermeneutical circle, as a Neo-Platonic image, designed to intimate the relation of an infinite, eternal and omnipresent God to his creation, and it makes its most significant appearance in the late Middle Ages, never to leave our imaginative literature thereafter.” (Shklar 1986, 450). Cf., e.g., Ginev’s hermeneutic philosophy of science. “Scientific practices are capable, solely in their interrelatedness, of articulating reality in meaningful entities and structures. […] The mutual dependence of whole and particular units forms a hermeneutic circle of the reality’s meaningful articulation. In a manner similar to the process of reading a text, this articulation moves in each context of configured practices from particular

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mathematics education and mathematics research (Brown 1991; Brown 1994; Brown 2002). Any epistemic justification cannot be absolute or universally applicable; it presupposes some target audience, some real or potential dialogue, and some particular communicative situation. Only understanding makes an argument sound and a conclusion valid. It is understanding that makes justification occur. “Human understanding functions by interpretation and its product is meaning” (Heelan 1998, 278). Together with the last two, hermeneutics inevitably enters the stage. Suggesting hermeneutics as an innovative approach to the epistemic regress problem means more than just turning from static to dynamic aspects. In addition to the shifts from an individual to a collective epistemic subject and from a linear to a holistic justification process, we should also introduce a historical dimension of epistemic acts. The latter means that we stop looking at justification sub specie aeternitatis and begin to treat it as characterized by historical situatedness or historicality, i.e., as taking place within a particular community and as gradually changing within a particular historical time. Justification processes are seen then as irretrievably immersed in the flow of history and into the so-called “life-world (die Lebenswelt),” which is “characterized by the action of embodied human inquirers in communication with one another and with their environment against a background of active cultural networks” (ibid., 277).40 One of the implications of this “historical turn” is that the quest for the coherence of our beliefs (even in mathematics) has no absolute beginning or end: if a belief system is complicated enough, it is usually coherent only to some degree at any given moment (cf. Olsson 2003, subst. rev. 2017). Such an understanding of justification naturally leads to the rise of interest in inconsistent theories (logical consistency is normally considered to be the basic level of coherence) and even to the so-called “inconsistency tolerance.”41 By now the image of knowledge – not only mathematical knowledge but human knowledge in general – has gotten rather far from the classics, as-

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meanings to the contextual whole and back again.” (Ginev 2016, 15). See also (ed. Crease 1997). In this paper, I refer to the “life-world” as a general name for pre-epistemic existential conditions for acquiring and testing knowledge. I am not going to distinguish between or discuss the varieties of its use in Edmund Husserl, Alfred Schütz, or outside of the phenomenological tradition. See (ed. Meheus 2002); (Vickers 2013); (eds. Bueno and Vickers 2014); (eds. MartínezOrdaz and Estrada-González 2017); (Berte, 2007); (Weber 2009); (Mortensen 1996, subst. rev. 2017).

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sociated with the names of Aristotle, Descartes, and Kant. Cognitive science accustoms us to embodied and enactive cognition, emotional cognition, and distributed and social cognition.42 In addition to everything else, these new trends by and by irreversibly change our attitude towards justification and the foundations of knowledge. To summarize the discussion in this section, I may say that for a collective and historically embedded epistemic subject, who is practicing holistic justification in a hermeneutic vein, the horns of the Münchhausen Trilemma look no more as if they are mutually exclusive alternatives. Horn A may be easily reconciled with horn B because the hermeneutic kind of circularity is actually a spiral.43 Therefore, it can go on ad infinitum, being free from identical repetition. We can add horn C to the picture by examining the issue of the so-called “prejudices (Vorurteile) as conditions of understanding.”44 We cannot start our hermeneutic process without such “prejudices” or, better, pre-understandings. They may be conscious, half-conscious or even unconscious. We may try to become aware of them and state them explicitly as a kind of “foundations,” then doubt and correct them within the correlative processes of understanding and justification. Nevertheless, we are unable to get rid of any unjustified background knowledge. It makes all our epistemic constructions inevitably hypothetical. So the old joke about the turtles (or rocks) that inevitably are “all the way down”45 has proved to be no joke after all. It is how things stand, at least as regards the issue of background knowledge. Does such a picture of our epistemic activities still leave some room for any straightforward use of the concept of foundations?

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See, e.g., (Varela, Thompson and Rosch 1991); (Thompson 2007); (Lakoff and Johnson 1999); (Damasio 1995); (Hutchins 1995). Cf. (Ricoeur 1986, 312, 314); (Adams 2017); (García Landa 2004). “What appears to be a limiting prejudice from the viewpoint of the absolute selfconstruction of reason in fact belongs to historical reality itself. If we want to do justice to man’s finite, historical mode of being, it is necessary to fundamentally rehabilitate the concept of prejudice and acknowledge the fact that there are legitimate prejudices.” (Gadamer 2004, 289). Charles Peirce also spoke in support of “prejudices” and tried to vindicate them. See (Peirce 1968, 140–141). Cf. an attempt at the history of this joke “Turtles All the Way Down [March 27, 2020],” Wikipedia, https://en.wikipedia.org/wiki/Turtles_all_the_way_down (access: 29.05.2020).

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Mathematics without Foundations? Philosophers and logicians have been so busy trying to provide mathematics with a “foundation” in the past half-century that only rarely have a few timid voices dared to voice the suggestion that it does not need one. I wish here to urge with some seriousness the view of the timid voices. I don’t think mathematics is unclear; I don’t think mathematics has a crisis in its foundations; indeed, I do not believe mathematics either has or needs “foundations.” The much touted problems in the philosophy of mathematics seem to me, without exception, to be problems internal to the thought of various system builders. The systems are doubtless interesting as intellectual exercises; debate between the systems and research within the systems doubtless will and should continue; but I would like to convince you (of course I won’t, but one can always hope) that the various systems of mathematical philosophy, without exception, need not be taken seriously. (Putnam 1967, 5)

This is the famous passage that opens Hilary Putnam’s 1967 paper “Mathematics without Foundations.” And it indicates a gradually increasing tendency towards revisiting the early-twentieth-century foundational situation as being a true “crisis.” This tendency can be tracked in the changing environment of post-Gödelian mathematics,46 that is, approximately since the year 1930 when Kurt Gödel announced his famous incompleteness theorems at the Second Conference on the Epistemology of the Exact Sciences in Königsberg. Hermann Weyl, the one who bluntly declared the new foundational crisis of mathematics in 1921, drastically changed his mind afterward, mainly because of the results of Gödel. He wrote on the subject in the 1940s: Since then the prevailing attitude has been one of resignation. The ultimate foundations and the ultimate meaning of mathematics remain an open problem; we do not know in what direction it will find its solution, nor even whether a final objective answer can be expected at all. “Mathematizing” may well be a creative activity of man, like music, the products of which not only in form but also in substance are conditioned by the

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Cf. (Körner 1967). Körner warns his readers: “The philosophy of mathematics in so far as it analyses the structure of mathematical thought, may conflict with mathematics not only by being mistaken, but also by, for example, having lost touch with the subject or having been left behind by its actual development” (ibid., 118).

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decisions of history and therefore defy complete objective rationalization. (Weyl 1949, 219) Moreover, Weyl now tended to the assimilation of mathematics back into physics: A truly realistic mathematics should be conceived, in line with physics, as a branch of the theoretical construction of the one real world, and should adopt the same sober and cautious attitude toward hypothetic extensions of its foundations as is exhibited by physics. (Ibid., 235) He was apparently inspired by Gödel’s realistic stance on logic and mathematics.47 For example, Gödel wrote that “logic and mathematics (just as physics) are built up on axioms with a real content which cannot be ‘explained away’” (Gödel 1990 [1944], 132).48 The true meaning of Gödel’s realism remains a moot point (see, e.g., Parsons 1995). Did he have any clear understanding of the correlation between the domains of mathematical and physical objects? Was his position closer to Platonism, which separates the domains of physics and mathematics while retaining certain similarities between them, or Aristotelian realism, which denies that separation while stressing the considerable difference between the two? In his 1951 Gibbs lecture, Gödel is quite certain that “mathematics describes an objective world just like physics and is not our free creation. He advocates a sort of “conceptual realism (Platonism)” (Gödel 1995 [1951], 314). He claims that “mathematics describes a non-sensual reality, which exists independently both of the acts and [of] the dispositions of the human mind and is only perceived, and probably perceived very incompletely, by the human mind” (ibid., 323). Gödel readily dismisses Aristotelian realism: “I have purposely spoken of two separate worlds (the world of things and of concepts), because I do not think that Aristotelian realism (according to which concepts are parts or 47

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Cf. “What allowed Putnam to be so nonchalant about foundations is his belief in the reality of mathematics. Mathematical entities are so strongly entwined with the scientific project that it makes no sense, according to Putnam, to think of them as less real than such entities as electrons or water.” (Wagner 2019, 384–385). It looks as if we are getting back to the 18th-century belief that the ultimate justification of mathematical theories may be found in the fundamental unity (though without identity) between mathematics and natural science. For example, Putnam wrote: “The point is that the real justification of the calculus is its success – its success in mathematics, and its success in physical science.” (Putnam 1979 [1975], 66). These words are among those cited by Weyl (1949, 235).

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aspects of things) is tenable” (ibid., 321). In fact, it is not so easy to tell Platonism from Aristotelianism when considering the point of view of a realisticallyoriented mathematician. When Gödel quotes French mathematician Charles Hermite to give an example of mathematical realism, he has an obvious problem in classifying him as Platonist or Aristotelian realist. He finally cuts Hermite’s words into two parts to treat the first part as a manifestation of mathematical Platonism and the second one as a turn towards Aristotelian realism. The discussion of Hermite’s position clearly shows that Gödel rejects Aristotelianism only because he fails to understand how concepts can be “parts or aspects of things” (ibid., 323). What is worth noting is that Gödel’s conceptual realism frees him from anxiety about the insecure state of set-theoretic foundations of mathematics. His position turns out to be somewhat close to that of Hilary Putnam in this respect. [O]ur knowledge of the world of concepts may be as limited and incomplete as that of [the] world of things. It is certainly undeniable that this knowledge, in certain cases, not only is incomplete, but even indistinct. This occurs in the paradoxes of set theory, which are frequently alleged as a disproof of Platonism, but, I think, quite unjustly. Our visual perceptions sometimes contradict our tactile perceptions, for example, in the case of a rod immersed in water, but nobody in his right mind will conclude from this fact that the outer world does not exist. (Ibid., 321) Such a realistic position turns a disastrous “bankruptcy” into a mere “misunderstanding” that can and should be routinely corrected. It is the way Hao Wang commented on Gödel’s views on set-theoretic paradoxes (Wang 1974, 187–188).49 Gödel banishes the paradoxes from mathematics into logic and epistemology, i.e., philosophy. [T]here exists, I believe, a satisfactory foundation of Cantor’s set theory in its whole original extent and meaning, namely axiomatics of set theory interpreted in the way sketched below.50 It might seem at first sight that the set-theoretical paradoxes would doom to failure such an undertaking, but closer examination shows that they cause no trouble at all. They are a

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Please note that the presentation of Gödel’s views in Wang’s 1974 book was checked and approved by Gödel himself. See (Parsons 1998, 17). Gödel is talking here of the so-called “iterative concept of set”.

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very serious problem, not for mathematics, however, but rather for logic and epistemology.51 Since 1959, Gödel became involved in studying Husserl’s works52 and was inclined to clarify his own philosophy of mathematics with the help of phenomenology. Those works gave Gödel a fair chance to specify his realism and to find out how concepts can be “parts or aspects of things” after all. Dagfinn Føllesdal comments on the situation in the following way: Gödel had expressed views on the philosophy of mathematics similar to those of Husserl long before he started to study him. What he found in Husserl was not radically different from his own view; what impressed him seems to have been Husserl’s general philosophy, which provided a systematic framework for a number of his own earlier ideas on the foundations of mathematics. (Føllesdal 1995, 427–428; Føllesdal 2014, 325) For Husserl, mathematical intuition is not only, to some extent, similar to the perception of physical objects, but the two are inextricably intertwined though definitely not identical. Mathematical objects and physical bodies (and objects) inhabit one and the same world, not different ones (Føllesdal 1995, 439). And what about Gödel? Gödel’s mathematical realism was already evident in his 1944 paper on Russell’s logic. Still, his fully-fledged theory of mathematical intuition was revealed only in his supplement to the 1964 edition of “What is Cantor’s Continuum Problem?” In this supplement, he highlights the analogy between mathematical intuition and sense-perception, though – as Charles Parsons points out – his “claims about this analogy are strong but not very much developed” (Parsons 1995, 61). First of all, Gödel dwells on the similarities and differences between the undecidability (from some accepted axiomatic system) of Euclid’s fifth postulate in geometry, on the one hand, and Cantor’s continuum hypothesis in set

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Kurt Gödel, “What is Cantor’s Continuum Problem? [1964]” in (Gödel 1990, 258). In the first variant of this paper (1947), the passage looks differently: “[T]here exists a satisfactory foundation of Cantor’s set theory in its whole original extent […]. It might at first seem that the set-theoretical paradoxes would stand in the way of such an undertaking, but closer examination shows that they cause no trouble at all. They are a very serious problem, but not for Cantor’s set theory”. See (Gödel 1990 [1947], 180). Wang 1987, 12, 28, 98, 219–222; Wang 1996, 164–172; Føllesdal 1995, 427–428; Føllesdal 2014, 325–326.

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theory, on the other. Does obtaining proof of undecidability make a particular postulate or hypothesis neither true nor false? Yes, but only on a formal level, “i.e., if the meanings of the primitive terms are left undetermined,” answers Gödel (1990 [1964], 267). The choice and use of formal systems are usually underpinned and backed up by some informal vision, considerations, and needs, as well as interconnections with other branches of science. In geometry […] the question as to whether Euclid’s fifth postulate is true retains its meaning if the primitive terms are taken in a definite sense, i.e., as referring to the behavior of rigid bodies, rays of light, etc. The situation in set theory is similar; the difference is only that, in geometry, the meaning usually adopted today refers to physics rather than to mathematical intuition and that, therefore, a decision falls outside the range of mathematics. (Ibid.) Unlike the objects of Euclidean geometry, the ones of transfinite set theory – he continues – “clearly do not belong to the physical world, and even their indirect connection with physical experience is very loose (owing primarily to the fact that set-theoretical concepts play only a minor role in the physical theories of today)” (ibid.). Nevertheless, this difference should not be overestimated. But, despite their remoteness from sense experience, we do have something like a perception also of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true. I don’t see any reason why we should have less confidence in this kind of perception, i.e., in mathematical intuition, than in sense perception, which induces us to build up physical theories and to expect that future sense perceptions will agree with them, and, moreover, to believe that a question not decidable now has meaning and may be decided in the future. The set-theoretical paradoxes are hardly any more troublesome for mathematics than deceptions of the senses are for physics. (Ibid., 268) It is worth mentioning that Cantor’s transfinite set theory was at least partially motivated by its potential applications in physics and especially life sciences (a letter from G. Cantor to M. G. Mittag-Leffler, 22 September 1884) (see Ferreirós 2004). Gödel hoped very much that the emergence of some new mathematical intuitions would finally lead to new axioms, making Cantor’s continuum hypothesis decidable. Then he proceeds to some explanations on

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the nature of mathematical intuition, which strongly remind one of Husserl’s ideas. It should be noted that mathematical intuition need not be conceived of as a faculty giving an immediate knowledge of the objects concerned. Rather it seems that, as in the case of physical experience, we form our ideas also of those objects on the basis of something else which is immediately given. Only this something else here is not, or not primarily, the sensations. That something besides the sensations actually is immediately given follows (independently of mathematics) from the fact that even our ideas referring to physical objects contain constituents qualitatively different from sensations or mere combinations of sensations, e.g., the idea of object itself, whereas, on the other hand, by our thinking we cannot create any qualitatively new elements, but only reproduce and combine those that are given. Evidently the “given” underlying mathematics is closely related to the abstract elements contained in our empirical ideas. It by no means follows, however, that the data of this second kind, because they cannot be associated with actions of certain things upon our sense organs, are something purely subjective, as Kant asserted. Rather they, too, may represent an aspect of objective reality, but, as opposed to the sensations, their presence in us may be due to another kind of relationship between ourselves and reality. (Gödel 1990 [1964], 268) “This passage expresses in a condensed and cryptic way a key to Husserl’s theory of perception, which is the basis for his and Gödel’s philosophy of mathematics,” says Føllesdal (2014, 326). Here, Gödel bridges the gap between mathematical intuition and reality with the help of the “immediately given.” There are two closely connected types of “immediately given”: sensations and “the abstract elements contained in our empirical ideas.” Though the text is “cryptic” (Føllesdal) indeed and the author’s position is “not very much developed” (Parsons), it is certainly suggestive of Husserl’s ideas of the connection between “perception (Wahrnehmung)” and “eidetic intuition” or “essential insight (Wesensschau)”.53 Føllesdal proposes to decipher Gödel with the help of Husserl in the following way: We perceive physical objects, but the notion of a physical objects involves a comprehensive structure that includes arithmetic, geometry and 53

Cf. “Perhaps Husserl’s considerations of Wesenschau can be borrowed to support G[ödel]’s belief in the objective existence of mathematical objects” (Wang 1987, 304).

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other branches of mathematics. The notion of a physical object contains, for example, the idea of individuation: features are grouped together as belonging to the same individual. Individuals are distinct from one another although they may be similar, and they may keep their identities through changes. This structuring into objects, their properties and relations between them is crucial to perception. These abstract elements are incorporated into what we perceive; there is no perception without such elements. They are features of the world we experience, they are not objects of perception, but objects of eidetic intuition. We anticipate these features, but this does not make them any more subjective than the physical objects, which are themselves part of this structure. […] So, we experience these “abstract elements contained in our empirical ideas” not because they are physical objects that affect us causally, but because they are elements of that same objective reality. They are anticipated in our experience of the world, and the further course of our exploration of the world may lead us to give up or revise some of these anticipations. So even “the abstract elements contained in our empirical ideas” are subject to constraints and revision. (Føllesdal 2014, 337) According to this interpretation, “abstract elements are incorporated into what we perceive,” hence, physical and mathematical objects “are elements of the same objective reality.” And what is more, our picture of the mathematical world, as well as that of the physical world, is just a web of anticipations, which is always incomplete and open to revision. Unlike Kant, Husserl was not a foundationalist after all, as Føllesdal clearly demonstrated (Føllesdal 1988; Føllesdal 1995, 437–438; Føllesdal 2010, 44–45). Tackling the problem of ultimate justification – even in the case of mathematics – inevitably leads to consideration of background knowledge and the life-world. In Crisis (1936), Husserl makes it quite obvious: What is needed, then, would be a systematic division of the universal structures – universal life-world a priori and universal “objective” a priori – and then also a division among the universal inquiries according to the way in which the “objective” a priori is grounded in the “subjectiverelative” a priori of the life-world or how, for example, mathematical selfevidence has its source of meaning and source of legitimacy in the selfevidence of the life-world. (Husserl 1970, 140) In Føllesdal’s interpretation, Husserl’s “a priori” does not refer to any absolutely secure ground:

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For Husserl, the a priori is that which we anticipate, that which we expect to find, given the noema54 we have. Phenomenology studies and attempts to chart these anticipations, but as we know, our anticipations often go wrong, our experiences turn out differently from what we expected, and again and again we have to revise our views and our expectations. (Føllesdal 1988, 115; cf. Føllesdal 2014, 322) Talking about mathematics, Husserl discerns two levels: “mathematical selfevidence” or “‘objective’ a priori” and “self-evidence of the life-world” or “lifeworld a priori.” The first level “is grounded” and “has its source of meaning and source of legitimacy” in the second one. Føllesdal uses an iceberg metaphor to illustrate this two-level approach to ultimate justification. To quote him again: Husserl’s key observation, which I regard as an intriguing contribution to our contemporary discussion of ultimate justification, is that the “beliefs,” “expectations,” or “acceptances” that we ultimately fall back on, are unthematized, and in most cases have never been thematized. Every claim to validity and truth rests upon this “iceberg” of unthematized prejudgmental acceptances […]. One should think that this would make things even worse. Not only do we fall back on something that is uncertain, but on something that we have not even thought about, and have therefore never subjected to conscious testing. Husserl argues, however, that it is just the unthematized nature of the life-world that makes it the ultimate ground of justification. “Acceptance” and “belief” are not attitudes that we decide to have through any act of judicative decision. What we accept, and the phenomenon of acceptance itself, are integral to our life-world, and there is no way of starting from scratch, or “to evade the issue here through a preoccupation with aporia and argumentation nourished by Kant or Hegel, Aristotle or Thomas”.55 Only the life-world can be an ultimate court of appeal: “Thus alone can that ultimate understanding of the world be attained, behind which, since it is ultimate, there is nothing more that can be sensefully inquired for, nothing more to understand.”56 (Føllesdal 1995, 438; cf. Føllesdal 2010, 45)

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55 56

Føllesdal uses here one of the central concepts of Husserl’s phenomenology. The structuring activity of our consciousness Husserl calls noesis, while noema is the name for a corresponding structure. Husserl 1970, 132. Husserl 1969 [1929], 242.

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If ultimate justification is not explicit, does it mean that our beliefs have no foundations at all? Or should we radically change the very meaning of the term “foundations”? It is quite possible to interpret Husserl the way Føllesdal did, but how close was Gödel’s position to that of Husserl’s? Gödel never referred to Husserl in the works published during his lifetime, but in his Nachlass, one finds strong evidence of his sympathy with Husserl’s phenomenology. In a draft of a lecture from the 1960s, Gödel criticizes Hilbert’s approach to the foundations of mathematics while retaining his “rationalistic optimism”:57 […] the certainty of mathematics is to be secured not by proving certain properties by a projection onto material systems – namely, the manipulation of physical symbols – but rather by cultivating (deepening) knowledge of the abstract concepts themselves (die Erkenntnis der abstrakten Begriffe selbst) which lead to the setting up of these mechanical systems, and further by seeking, according to the same procedures, to gain insights (Einsichten) into the solvability, and the actual methods for the solution, of all meaningful mathematical problems.58 Talking about the “projection” of mathematical properties “onto material (mechanical) systems” and an attempt to secure mathematical certainty by “the manipulation of physical symbols,” Gödel had in mind the central idea of Hilbert’s metamathematics. Hilbert famously referred to “certain extralogical concrete objects that are intuitively present as immediate experience prior to all thought (gewisse, außer-logische konkrete Objekte, die anschaulich als unmittelbares Erlebnis vor allem Denken da sind)” (Hilbert 1967, 376; Hilbert 1926, 171). He wrote: This is the basic philosophical position that I consider requisite for mathematics and, in general, for all scientific thinking, understanding, and communication. And in mathematics, in particular, what we consider is the concrete signs themselves (die konkreten Zeichen selbst), whose shape, according to the conception we have adopted, is immediately

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On the origin of the expression see (Wang 1974, 325). Gödel shared Hilbert’s belief that in mathematics, there was no ignorabimus. See, e.g., Kurt Gödel, “Undecidable Diophantine Propositions [the late 1930s],” in (Gödel 1995), 164–165. Kurt Gödel, “The Modern Development of the Foundations of Mathematics in the Light of Philosophy [1961 or later],” in (Gödel, 1995), 383.

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clear and recognizable (unmittelbar deutlieh und wiedererkennbar ist). (Ibid.) Instead, Gödel suggests “cultivating (deepening)” and “extending” our knowledge of abstract concepts as a basis for future mathematical “insights.” This better knowledge of abstract concepts, “a clarification of [their] meaning,” he hoped to gain with the help of Husserl’s phenomenological method, as the further reading of the draft reveals. He describes this method in the following way: Here clarification of meaning consists in focusing more sharply on the concepts concerned by directing our attention in a certain way, namely, onto our own acts in the use of these concepts, onto our powers in carrying out our acts, etc. But one must keep clearly in mind that this phenomenology is not a science in the same sense as the other sciences. Rather it is (or in any case should be) a procedure or technique that should produce in us a new state of consciousness in which we describe in detail the basic concepts we use in our thought, or grasp other basic concepts hitherto unknown to us. (Gödel 1995 [1961 or later], 383) Further on, Gödel proposes to view the correlative evolution of physics and mathematics as “a systematic and conscious extension” of the development of a child that proceeds in two mutually related directions. The first direction is experimenting with the objects of the external world, while the second one is with the use of language and “the basic concepts on which it rests,” including logical inference. In fact, one has examples where, even without the application of a systematic and conscious procedure, but entirely by itself, a considerable further development takes place in the second direction, one that transcends “common sense”. Namely, it turns out that in the systematic establishment of the axioms of mathematics, new axioms, which do not follow by formal logic from those previously established, again and again become evident. It is not at all excluded by the negative results mentioned earlier59 that nevertheless every clearly posed mathematical yes-or-no question is solvable in this way. For it is just this becoming evident of

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Gödel is talking here about his own famous incompleteness theorems announced in 1930 and published in 1931. See (ibid., 381).

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more and more new axioms on the basis of the meaning of the primitive notions that a machine cannot imitate. (Ibid., 385)60 In Gödel’s interpretation, Husserl’s phenomenology is “the application of a systematic and conscious procedure” in the second direction. He saw phenomenological research as a potential source of new mathematical axioms and hoped to revitalize Hilbert’s rationalistic optimism on this ground. But, according to the later Husserl, the “deepening” of our knowledge of abstract concepts that was sought by Gödel leads us straight into analyzing how the “‘objective’ a priori” is grounded in the “life-world a priori” and emerges from it. It seems that Gödel never used the term “Lebenswelt,” but some scholars tend to think that he should have been sympathetic to the idea. For instance, Richard Tieszen says: Gödel was apparently […] impressed by the phenomenological claim that philosophy calls for a different method from science, and that it can provide a deeper foundation for science by reflecting on everyday concepts. Gödel was here presumably influenced by the role of what Husserl later came to call the Lebenswelt in our scientific thinking. (Tieszen 1992, 182–183) Contrariwise, the evidence collected by Hao Wang seems to testify against this hypothesis. According to that evidence, Gödel voiced (in private conversations) serious dissatisfaction with Husserl’s Crisis and especially with Wittgenstein’s later philosophy (Wang 1987, 58–67, 219; Wang 1996, 164, 177–182, esp. 178–179). Possibly, Gödel was of two minds about the new understanding of the foundations of mathematics; so it is no accident that he published nothing on the subject during his lifetime. Nevertheless, apparently unaware of this, Gödel – in some of his later unpublished works – got rather close, not only to the later Husserl but to the later Wittgenstein as well. Despite all the differences between them, both of the two great philosophers demonstrate remarkable affinity as regards the foundational problems (Føllesdal 2005, 127–142).61 Ludwig Wittgenstein was a leading exponent of the “no foundations” approach, who was not taking mathematical philosophy “seriously” (as Hilary Putnam put it). His attitude towards the foundational issues was firmly estab60 61

Cf. “What Turing disregards completely is the fact that mind, in its use, is not static, but constantly developing”. Gödel as quoted in (Wang 1974, 325). On Husserl and Wittgenstein, in their convergences and divergences, see also (Taylor 1978); (González-Castán 2015).

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lished already in the early 1930s. In his lectures on philosophy for mathematicians at Cambridge in the academic year of 1932–33 he was saying, according to the notes taken by Alice Ambrose: Is there a substratum on which mathematics rests? Is logic the foundation of mathematics? In my view mathematical logic is simply part of mathematics. Russell’s calculus is not fundamental; it is just another calculus. There is nothing wrong with a science before the foundations are laid. (Ambrose 2001, 205) In his so-called The Big Typescript (about 1933), Wittgenstein discussed the possibility of treating mathematics as no more than a “calculus,” that is, a sort of “game” with strictly fixed “rules,” which can be compared to chess62 or patience. Mathematics, thus, is not a “theory,” is not a system of “propositions” that can be considered true or false as saying something about the physical realm or, at least, some independent mathematical realm. He wrote: Since mathematics is a calculus and hence isn’t really about anything, there isn’t any metamathematics. […] No calculus can decide a philosophical problem. A calculus cannot give us information about the foundations of mathematics. […] Number is not at all a “fundamental mathematical concept”. There are so many calculations in which numbers aren’t mentioned. So far as concerns arithmetic, what we are willing to call numbers is more or less arbitrary. For the rest, what we have to do is to describe the calculus – say of cardinal numbers – that is, we must give its rules and by doing so we lay the foundations of arithmetic. Teach it to us, and then you have laid its foundations. (Hilbert sets up rules of a particular calculus as rules of metamathematics.)63

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The comparison of mathematics with chess can be traced back to the second volume of Frege’s Grundgesetze der Arithmetik (1903), sections 90–91. See (Frege 1960, 185–187). However, Frege emphasized differences – arithmetic can be applied, chess cannot – while Wittgenstein emphasized similarities. Cf. “The system of calculating with letters is a new calculus; but it does not relate to ordinary calculation with numbers as a metacalculus does to a calculus. Calculation with letters is not a theory. This is the essential point. In so far as the ‘theory’ of chess studies the impossibility of certain positions it resembles algebra in its relation to calculation with numbers. Similarly, Hilbert’s “metamathematics” must turn out to be mathematics

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A system’s being based on first principles is not the same as its being developed from them. It makes a difference whether it is like a house resting on its lowest walls or like a celestial body floating free in space which we have begun to build beneath although we might have built anywhere else. Logic and mathematics are not based on axioms, any more than a group is based on the elements and operations that define it. The idea that they are involves the error of treating the intuitiveness, the selfevidence, of the fundamental propositions as a criterion for correctness in logic. A foundation that stands on nothing is a bad foundation. (Wittgenstein 1974, 290, 296–297) Wittgenstein’s peculiar understanding of the nature of mathematics and its “foundations,” made him feel its practices to be rather robust. This is why he was so far from sharing the anxiety of mathematical philosophers, such as Russell and Hilbert, about the so-called “hidden contradictions” in mathematics. Something tells me that a contradiction in the axioms of a system can’t really do any harm until it is revealed. We think of a hidden contradiction as like a hidden illness which does harm even though (and perhaps precisely because) it doesn’t show itself in an obvious way. But two rules in a game which in a particular instance contradict each other are perfectly in order until the case turns up, and it’s only then that it becomes necessary to make a decision between them by a further rule. Mathematicians nowadays make so much fuss about proofs of the consistency of axioms. I have the feeling that if there were a contradiction in the axioms of a system it wouldn’t be such a great misfortune. Nothing easier than to remove it. (Ibid., 303) His approach to the foundations of mathematics looked rather “eccentric”64 at the time. In my view, the shortest way to get to the core of his position towards

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in disguise.” From shorthand notes of Waismann (December 1930). (ed. McGuinness 1979, 136). Cf. (Wright 1980, vii). Ray Monk wrote about “Wittgenstein’s Quixotic assault on the status of pure mathematics”; in his philosophy of mathematics Wittgenstein was “tilting at windmills” for “he was working against the stream of modern civilization.” (Monk 1991, 326–328).

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logic and mathematics is to characterize it by the adjective “anthropological,” following Wittgenstein himself who once said (spring 1944): For mathematics is after all an anthropological phenomenon. (Wittgenstein 1978, 399) To treat mathematics as an anthropological phenomenon means to look at it as a group of cultural practices paying special attention to the ways of the acquisition and transition of such practices within society. Wittgenstein’s approach was, by all means, groundbreaking. Nevertheless, it does not presuppose that it was unique, as some scholars have stated (cf. Hintikka 2000, 49–50; Sokuler 1994, 72). First of all, there was Oswald Spengler, who denied the universality of mathematics and insisted that mathematical practices are culturally dependent and culturally diverse. His book Der Untergang des Abendlandes (the first volume published in 1918) was well known to Wittgenstein and he acknowledged its influence.65 Second, there was Piero Sraffa, an Italian economist and Marxist, who was held in high esteem by Wittgenstein. They had continual discussions from 1929 to 1946. Wittgenstein especially thanked him in his 1945 Preface to Philosophical Investigations. An anecdote exists that Wittgenstein gained his “anthropological” way of looking at philosophical problems primarily from those talks with Sraffa.66 Wittgenstein may also have been influenced, through Sraffa, by some ideas of the leading Italian Marxist Antonio Gramsci, for whom Marxism was “the philosophy of praxis,” who advocated absolute historicism67 and introduced the concept of “political and cultural hegemony” (see Sen 2003, 1240–1255). Later on, Marxism and Spengler were recognized as the classic sources of the sociological approach to mathematics (Restivo 1983, sections VI and VII). Some scholars established a trend of reading Wittgenstein’s philosophy

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“Wittgenstein’s Remarks on the Foundations of Mathematics must be one of the few attempts that have ever been made to put Spengler’s bold but diffuse ideas on a firm footing.” (Bloor 1983, 164–165). On Spengler and Wittgenstein see also (DeAngelis 2007). See (McGuinness 1982, 36–39); (Monk 1991, 261). For a detailed discussion of the true meaning of that anthropological shift in Wittgenstein’s thought and the exact impact of Sraffa’s criticism on its course, see (Engelmann 2013, 148–166). Cf. the following words by Gramsci: “If the philosophy of praxis affirms theoretically that every ‘truth’ believed to be eternal and absolute has had practical origins and has represented a ‘provisional’ value (historicity of every conception of the world and of life), it is still very difficult to make people grasp ‘practically’ that such an interpretation is valid also for the philosophy of praxis itself, without in so doing shaking the convictions that are necessary for action.” (Gramsci 1971, 406).

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of mathematics in the same sociological vein.68 Moreover, there emerged the sociological tradition that pretended to “touch the very heart of mathematical knowledge” (see Bloor 1991 [1976], 84). Wittgenstein tries to show that mathematics is neither an empirically contingent phenomenon (as customs and beliefs, studied by ethnographers and cultural anthropologists, can be) nor something ideal, necessary and absolute, i.e., a collection of eternal truths (as Frege and early Russell thought it to be). Mathematics is normative but in the sense of being a set of socially established rules. This means, above all, that treating any other set of rules of the same kind as a “foundation” for the first set just leaves us within the same social game. The only true foundation for such rules is the network of social communication itself.69 That is why mathematics is “an anthropological phenomenon” and can be compared with the game of chess. Among Wittgenstein’s remarks, which date from October 1939 to April 1940, one finds the following: Are the propositions of mathematics anthropological propositions saying how we men infer and calculate? – Is a statute book a work of anthropology telling how the people of this nation deal with a thief etc.? – Could it be said: “The judge looks up a book about anthropology and thereupon sentences the thief to a term of imprisonment”? Well, the judge does not USE the statute book as a manual of anthropology. The prophecy does not run, that a man will get this result when he follows this rule in making a transformation – but that he will get this result, when we say that he is following the rule. What if we said that mathematical propositions were prophecies in this sense: they predict what result members of a society who have learnt this technique will get in agreement with other members of the society? ‘25 × 25 = 625’ would thus mean that men, if we judge them to obey the rules of multiplication, will reach the result 625 when they multiply

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For instance, David Bloor claimed that “Wittgenstein solves Mannheim’s problem,” i.e., the problem of thinking sociologically about the universally accepted theories of logic and mathematics. (Bloor 1973, 173). See also (Bloor 1983); (Bloor 1997). Within the aforementioned trend, Wittgenstein is treated as “a key figure in philosophy who initiated a ‘sociological turn’ by showing that the compulsive force of logical and mathematical rules is inseparable from a communal consensus on how such rules are to be applied in particular circumstances of action.” See (Lynch 1993, 161). Cf. “Much of the Remarks focuses on elementary arithmetic. Its concept of ‘the foundations of mathematics’ is the robust one of the teacher who speaks of elementary schooling as laying the foundations for later learning. So from the beginning Wittgenstein takes a social rather than a logical definition of his subject matter.” (Bloor 1973, 183).

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25 × 25. – That this is a correct prediction is beyond doubt; and also that calculating is in essence founded on such predictions. That is to say, we should not call something ‘calculating’ if we could not make such a prophecy with certainty. This really means: calculating is a technique. And what we have said pertains to the essence of a technique. This consensus belongs to the essence of calculation, so much is certain. I.e.: this consensus is part of the phenomenon of our calculating. In a technique of calculating prophecies must be possible. And that makes the technique of calculating similar to the technique of a game, like chess. But what about this consensus – doesn’t it mean that one human being by himself could not calculate? Well, one human being could at any rate not calculate just once in his life. […] It is clear that we can make use of a mathematical work for a study in anthropology. But then one thing is not clear: – whether we ought to say: “This writing shews us how operating with signs was done among these people”, or: “This writing shews us what parts of mathematics these people had mastered”. (Wittgenstein 1978, 192–194, 197–198) Wittgenstein also remarks a few pages later: I have not yet made the role of miscalculating clear. The role of the proposition: “I must have miscalculated”. It is really the key to an understanding of the ‘foundations’ of mathematics. (Ibid., 221) In his remarks on the same subject from spring 1944, he succeeded in making his point much clearer: This might be expressed: if calculation reveals a causal connexion to you, then you are not calculating. Our children are not only given practice in calculation but are also trained to adopt a particular attitude towards a mistake in calculating [= a departure from the norm]. What I am saying comes to this, that mathematics is normative. But “norm” does not mean the same thing as “ideal”. […] The mathematical Must is only another expression of the fact that mathematics forms concepts. And concepts help us to comprehend things. They correspond to a particular way of dealing with situations. Mathematics forms a network of norms. (Ibid., 425, 430–431)

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Moreover, Wittgenstein calls mathematics “a family” (spring 1944) (ibid., 399). He seems to mean that its “language-games (Sprachspiele)” evolve according to the mechanism of “family resemblances (Familienähnlichkeiten)” (Wittgenstein 2009, 36) (in Remarks on the Foundations of Mathematics, Wittgenstein does not use the expression; he simply says “family”). By the way, the metaphor of “family resemblances” is a biological one. It makes it quite natural to compare Wittgenstein’s remarks on mathematics with that of Konrad Lorenz (biological inheritance)70 and Leslie White (cultural inheritance), also from the 1940s.71 It is already existing mechanisms of transition (reproduction of grammar rules, rule-following and the overall effect of multiple proofs72) that make mathematics secure, not some special foundation-building activity. What we actually need is just to take a fresh look at the things we have had at our disposal from time immemorial: The difficult thing here is not, to dig down to the ground; no, it is to recognize the ground that lies before us as the ground. (Wittgenstein 1978, 333, circa 1943/1944) Such considerations help us to understand Wittgenstein’s words (spring 1944) used as an epigraph to this paper: What does mathematics need a foundation for? It no more needs one, I believe, than propositions about physical objects – or about sense impressions, need an analysis. What mathematical propositions do stand in need of is a clarification of their grammar, just as do those other propositions.

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Cf. “Here too we cannot give any foundation (except a biological or historical one or something of the kind); all we can do is to establish the agreement, or disagreement between the rules for certain words, and say that these words are used with these rules” (about 1933) (Wittgenstein 1974, 304). See (Lorenz 1941/1942, 104–106); (White 1947). As Bloor put it: “So the question is whether mathematical rule following is to be located amongst the instincts or the institutions?” (Bloor 1973, 186). He attempts to show that Wittgenstein’s answer was surely “amongst the institutions.” Rav, who discussed the biosocial nature of mathematics, suggested telling between the “logico-operational” and the “thematic” components of mathematics, i.e., between the biological and the socio-cultural strata within it. (Rav 1989, 58). The issue of biological and neurophysiological basis for mathematics, as a vast array of social practices, is still a big and vital one. “Every mathematical proof gives the mathematical edifice a new leg to stand on. (I was thinking of the legs of a table.)” (Wittgenstein 1978, 399, spring 1944).

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The mathematical problems of what is called foundations are no more the foundation of mathematics for us than the painted rock is the support of a painted tower. (Ibid., 378) What really supports the “painted tower” and connects it with the “painted rock”? It is the sheet of paper or other material they are painted on. Within the paper, both the “rock” and the “tower” are interconnected and interdependent. This “paper” (or other material for painting) may serve as a good metaphor for a human community tied up with communication, both verbal and nonverbal. Mathematics as a “network of norms” presupposes particular people that are involved in numerous interwoven language-games, as their “forms of life (Lebensformen)” (Wittgenstein 2009, 11, 15, 94), and thus share the same Lebenswelt.73 And we may say, following David Bloor, who apparently stands on Wittgenstein’s shoulders, that if we have some problems at the level of formal norms (contradictions, etc.), which are just a “surface structure,” they always can and should be sorted out through an interpretative or hermeneutic process and informal negotiations. Bloor calls this principle “the priority of the informal over the formal”: How does the priority of the informal over the formal express itself? The answer is two-fold. First, informal thought may use formal thought. It may seek to strengthen and justify its predetermined conclusions by casting them in a deductive mould. Second, informal thought may seek to criticise, evade, outwit or circumvent formal principles. In other words the application of formal principles is always a potential subject for informal negotiation. (Bloor 1991, 133) Wittgenstein was a philosopher, not a working mathematician. Is it possible to find similar tendencies towards the “no foundations” approach among the mathematicians of the same historical period? In the post-Gödelian mathematical landscape, the Bourbaki project was one of the most notable phenomena, especially from the 1940s to the 1960s. No wonder their attitude towards the foundations of mathematics constitutes a matter of particular interest. Below, I try to emphasize some telling changes in their presentation of the issue in comparison with that of David Hilbert and his school.

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On the problem of whether there is one shared life-world or we all live in different lifeworlds, according to Husserl, see (Føllesdal 2010, 41–43).

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The Nicolas Bourbaki group exploited an architectural metaphor to the fullest when they began their famous manifesto “The Architecture of Mathematics” with asking whether it is appropriate to liken contemporary mathematics to the tower of Babel, and culminated with likening modern mathematics to a megalopolis. It is like a big city, whose outlying districts and suburbs encroach incessantly, and in a somewhat chaotic manner, on the surrounding country, while the center is rebuilt from time to time, each time in accordance with a more clearly conceived plan and a more majestic order, tearing down the old sections with their labyrinths of alleys, and projecting towards the periphery new avenues, more direct, broader and more commodious. (Bourbaki 1950, 230) Note the idea of occasionally rebuilding the city center, which apparently means a revision of the foundations. Bourbaki’s architectural metaphor substantially differs from that of Kant. For Kant, the foundations are laid once and forever; they presuppose no future revision or change. The next quote confirms that Bourbaki take a recurring revision of the foundations to be a normal part of the history of mathematics. Historically speaking, it is of course quite untrue that mathematics is free from contradiction; non-contradiction appears as a goal to be achieved, not as a God-given quality that has been granted us once for all. Since the earliest times, all critical revisions of the principles of mathematics as a whole, or of any branch in it, have almost invariably followed periods of uncertainty, where contradictions did appear and had to be resolved. (Bourbaki 1949, 2) Finally, we see that the Bourbaki group projected onto the future their historical insight into the way mathematics evolves. To sum up, we believe that mathematics is destined to survive, and that the essential parts of this majestic edifice will never collapse as a result of the sudden appearance of a contradiction; but we cannot pretend that this opinion rests on anything more than experience. Some will say that this is small comfort; but already for two thousand five hundred years mathematicians have been correcting their errors to the consequent enrichment and not impoverishment of their science; and this gives them the right to face the future with serenity. (Bourbaki 1968, 13)

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The revision or even drastic change of the foundations does not suppose a “crisis”; it is just a matter of routine, a normal part of the progress of mathematics. One of the leaders of the Bourbaki group, Jean Dieudonné, wrote in the 1980s: Young mathematicians […] no longer even know that there was once a “crisis of foundations”. This is in any case a completely misleading name, when we compare what happened in mathematics with the real crises in physics, when relativity and quantum mechanics forced the physicists radically to modify their conception of natural phenomena. In mathematics, we can at most speak of a certain uneasiness caused by the “paradoxes”; but apart from Brouwer and his disciples, there has hardly been a mathematician who was led to make the slightest change in his way of presenting proofs. Furthermore, the period of the supposed “crisis”, 1895–1930, was one of the most fertile in history […]. It would be even more nonsensical to speak of “crisis” at the present day, given the never-equalled abundance of solutions to ancient problems and of discoveries of new methods which we are witnessing. […] Mathematicians working in these fields hardly know what the logicians are doing, and if they sometimes hear about it, they pay no more attention than they would to sciences quite remote from mathematics, such as biology or geology. If logicians sometimes show signs of surprise at this separation, it is because they have failed to take into account the evolution of mathematics over the last fifty years. Nothing that they do can help any of the mathematicians in question to solve their problems, the only ones in which they are really interested; for if they have by chance heard of the “paradoxes”, as far as they are concerned they are nothing but pseudoproblems. (Dieudonné 1992 [1987], 234–235)74 In the title of his chapter six, from which this lengthy passage was taken, Dieudonné puts the word “foundations” in scare quotes (ibid., 203) as if to say that though the word is widely used, he doubts the reality or correct representation of the phenomenon. He also characterizes the period between 1800 and 1930 as an epoch “in which doubts and controversies about the nature of mathematics were at their height” (ibid.). Those doubts and controversies

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He also describes the “crisis of foundations” with an adjective “prétendue” (in the French original), which was quite accurately translated as “alleged” (Dieudonné 1992, 224; Dieudonné 1987, 234).

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“left a positive gain in the increased precision of mathematical language, and in particular they made way for enormous advances in mathematical logic in the twentieth century” (ibid.) Nevertheless, in his opinion, the debates which arose from them “are scarcely of interest now except to historians and philosophers” (ibid.), i.e., they are of no interest to mathematicians any more. While philosophers were articulating the “no foundations” approach, mathematicians were ascertaining the widening gap between the working mathematicians and the mathematical logicians (cf. Halmos 1985, 202–206; for discussion see Lolli 2008). In the early 1980s, Solomon Feferman described mathematicians’ common attitude towards the foundations of mathematics at the time as follows: There is currently a general malaise about the logical approach to the foundations of mathematics. […] Though these various critical views diverge from each other in many significant respects, they share two common themes: (i) mathematics is more reliable than any of the foundational schemes which have been proffered by the logicians to “secure” it, and (ii) the logical analysis of mathematics bears little or no relation to actual mathematical practice. (Feferman 1985, 229)

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Foundations without Foundationalism75

Wittgenstein’s mature philosophy, later on, was recognized as one of the major sources of the so-called “practice turn” in philosophy and science studies, and among other fields, in the field of philosophy of mathematics (Soler et al. 2014, 4–5, 10). The core idea of that turn has been aptly summarized by Valeria Giardino in the following way: [S]cience is not conceived anymore as true and justified belief that has to be examined, so to speak, in vitro, but as a mingle of different practices that should be considered in vivo, looking at the behaviors and the habits characterizing the people involved in them. (Giardino 2017, 19) 75

This expression is surely taken from the title of the following book: (Shapiro 1991). Shapiro defines foundationalism as “the view that it is possible and desirable to reconstruct mathematics on a secure basis, one maximally immune to rational doubt” (Ibid., v). In his book, he attempts to discuss some foundational issues – related to the use of higher-order logic – from an “anti-foundationalist” point of view. Nevertheless, Shapiro is far from accepting the later Wittgenstein’s “no foundations” approach; he prefers to look for a compromise with traditional Platonism (Ibid., xv).

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From the 1970s on, among those, who systematically applied the new approach to mathematics were David Bloor, Eric Livingston,76 Donald MacKenzie,77 and many others.78 Nevertheless, within the philosophy of mathematics sensu stricto this approach, known now under the name of the “philosophy of mathematical practice,”79 still has to fight for its place in the sun.80 The foundational issues have been mostly associated with “condemned” foundationalism and lack popularity with the philosophers of mathematical practice, and yet, not all of the adherents discarded these issues altogether. They have rather tended to change the very meaning of foundational research. I will take my first example from the 1980s. It is the views of Saunders Mac Lane, “the last mathematician from Hilbert’s Göttingen” (McLarty 2007). He put forth a “network” conception of mathematics: Our central observation is this: the development of Mathematics provides a tightly connected network of formal rules, concepts, and systems. Nodes of this network are closely bound to procedures useful in human activities and to questions arising in science. The transition from activities to the formal Mathematical systems is guided by a variety of general insights and ideas. Within the formal network, new developments are stimulated and guided by conjectures, problems, abstractions, and the constant desire to understand more. (Mac Lane 1986b, 409) Mac Lane visualized his conception with the help of the following diagram (Mac Lane 1986b, 409; Mac Lane 1986a, 7):

76 77 78

79 80

Livingston 1986; Livingston 1987, ch. 14–19; Livingston 1999; Livingston 2006; Livingston 2008; Livingston 2015. MacKenzie 2001; MacKenzie 2004; MacKenzie 2005; Barany and MacKenzie 2014. I purposely name here only some sociologists and sociologically-oriented philosophers that have paid close attention to mathematics along the lines specifically discussed in this paper. David Bloor and Donald MacKenzie represent the so-called “Edinburgh school” in Science and Technology Studies (STS) while Eric Livingston belongs to the school of ethnomethodology, pioneered by Harold Garfinkel at the University of California, Los Angeles. Both schools have been obviously influenced by later Wittgenstein, and the latter one also by the “Lebenswelt” concept of later Husserl and other phenomenological philosophers. See (Lynch 1993). In 2009, the Association for the Philosophy of Mathematical Practice (APMP) was founded. See their website http://www.philmathpractice.org/about/ (access: 29.05.2020). For an up-to-date discussion see also (Carter 2019).

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The key characteristic of mathematics for Mac Lane is its formality. As a result of this formality, mathematics is absolutely precise and absolutely rigorous. It means that mathematical rules may be carried out, say, by a computer; they are completely “independent of persons.” Moreover, that is why mathematics can serve as a perfectly firm background for human communication: “The formal can be communicated well, without ambiguity” (Mac Lane 1986b, 410). Mathematical forms “are suggested but in no way determined” (ibid., 412) by some everyday human activities and scientific conundrums; mathematics enjoys considerable freedom.81 Mac Lane’s position in the philosophy of mathematics is in no way empiricism; no empirical evidence or experiment can disprove a single mathematical proposition. Mathematics “is about form and not about fact” (ibid., 454); mathematical forms “can be many steps removed from the original facts” (ibid., 456) He names his position in the philosophy of mathematics “formal functionalism” (ibid., 456; Mac Lane 1986a, 12) Mathematical formalizations normally presuppose something that is formalized in them. Mac Lane calls that something “underlying ideas,” (Mac Lane 1986b, 414–415) “suggestive ideas” (Mac Lane 1986b, 155; Mac Lane 1986a, 7) or “guiding ideas” (Mac Lane 1986b, 289, 447; Mac Lane 1986a, 7). The mathematical ideas arise from some exact procedures and problems that emerge within ordinary “human cultural activities” (Mac Lane 1986b, 34–35)82 and scientific practices or the body of mathematics itself (Mac Lane 1986b, 416).83 These ideas are usually vague and nebulous and can be formalized in many different ways (Mac Lane 1986b, 414–415; Mac Lane 1986a, 7). Formalization 81 82 83

“The axiomatic method is a declaration of independence for Mathematics.” (Ibid., 418). At p. 35, the author gives a table of examples, which connects some particular “activities,” corresponding “ideas” and their mathematical “formulations.” On this page, see the table of some ideas that have arisen within mathematics and their formal versions.

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in mathematics takes a number of steps and levels (Mac Lane 1986b, 414) to arrive eventually at a vast array of “full-fledged axiomatic theories, connected with each other in an elaborate network” (Mac Lane 1986a, 6). Finally, mathematics has not only origins but applications as well (Mac Lane 1986b, 444– 446). I assert that subjects of Mathematics are extracted from the environment; that is, from activities, phenomena, or science – and that they are then later applied to that – or other – environments. Thus number theory is “extracted” from the activity of counting, and geometry is extracted from motion and shaping. The exact mechanism of this “extraction” has not been described in detail here; it will clearly vary considerably from case to case. I have deliberately chosen this word “extraction” to be close to the more familiar word “abstraction” – and with the intent that the Mathematical subject resulting from an extraction is indeed abstract. Mathematics is not “about” human activity, phenomena, or science. It is about the extractions and formalization of ideas – and their manifold consequences. (Ibid., 418) Mac Lane especially stresses the roots, as well as applications, of mathematical formalisms both in everyday human activities and specific scientific practices. The “development of the formal from the factual is a long historical process” (ibid., 6) he admits. What is more, it is a circular process: The world has many underlying regularities which, once extracted, can be analyzed and understood by Mathematical form. Because it is formal, the same Mathematical notions can apply to widely different phenomena. (Ibid., 446) Why are we looking for the foundations of mathematics, according to Mac Lane? He answers that we are craving for “the assurance of correctness” of our mathematical results, for “a security blanket” (ibid., 406). There are two kinds of assurance of correctness – the “local” and the “global” – and, hence, two types of foundational questions. The local assurance is concerned with establishing the correctness of mathematical proofs within a chosen formalism. Mac Lane specifies: Then a proof is correct provided it can be restated in terms of that particular formalism; for example, the standard formalism consisting of the Zermelo–Fraenkel axioms for set theory, with the axiom of choice, as

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formulated in a suitable language and using the first order predicate calculus with its rules of inference. Such a formalism does not as yet assure absolute freedom from contradiction, since the Hilbert program to achieve this end has not yet succeeded. However, it does assure us that the developments as made follow accepted conventions and therefore as secure as various past development of mathematics. There is of course no uniqueness about this particular standard formalization. Others may employ intuitionist set theory with an appropriate intuitionist logic, or may even follow still other versions of set theory. One such is the recent development of topos theory, which provides the notion of a well-pointed topos as one alternative foundation for (almost) all of mathematics. This – and other – alternatives have not been extensively studied. (Mac Lane 1986a, 4–5) On the contrary, the global assurance is all about how different parts of mathematics fit each other and the rest of human practices and activities. He continues: There are also difficult questions of a ‘global’ character about foundations. One of these questions has the form: ‘Does this particular piece of mathematics really belong to mathematics; that is, is it consonant with the nature of mathematics?’ The reason this is a question of foundation is that mathematics is ultimately correct, not because it follows the formal rules of the Zermelo–Fraenkel axioms for sets, but because it depends on an analysis of various mathematical forms and their realization in different scientific applications. In other words, mathematics is right, not primarily because it obeys the local correctness involved in checking out formal logic, but primarily because it exhibits global correctness; in that the statements are tightly connected with the rest of mathematics, science, and human activities. In other words, again, the forms under study in mathematics are those which are closely connected, in many different and complicated ways, to the ultimate business of scientific knowledge or to basic human activities. For this reason the question of foundations of mathematics must be based on a study of what mathematics actually is: an examination of the multiple connectivities of the various disciplines and their relations to other parts of human knowledge. Study of these matters is not a question of techniques in logic but a question of examination of the nature and inter-connections of different parts of mathematics. (Ibid., 5–6)

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Mac Lane apparently contrasts here global and local foundations of mathematics. It seems to be a polemical exaggeration. Rather, the point is that local foundational questions qua foundational questions make sense only against a global foundational background. It gives any local foundations their flexibility, making them changeable and historical. As Mac Lane puts it, “there is no single and absolute foundation for Mathematics,” because “any such fixed foundation would preclude the novelty, which might result from the discovery of new form” (Mac Lane 1986b, 454–455). The so-called “foundations” turn out to be no more than historically changing “proposals for the organization of Mathematics” (ibid., 406). There was quite a number of such local “foundations” in the history of mathematics. And any of them has its benefits as well as pitfalls. Neither can be called completely successful. Being one of the creators of category theory, Mac Lane was far from admiring set-theoretic foundations for mathematics, giving preference to categorical ones.84 But what is far more important from his point of view, “this variety of proposals for organization reflects the diversity and richness of Mathematics” (ibid., 407). Mac Lane’s network conception of mathematics and his distinct shift in the understanding of the correctness and security from “foundation” to “organization” (cf. McLarty 2013) were signs of the time. He was in no way the only one. For example, in his 1999 paper on the role of proofs in mathematics, Yehuda Rav pinpointed the change of the focus with crystal clarity: Consider the construction of a skyscraper. In order to secure the stability of the edifice, it has to be seated on solid foundations and erected stagewise, level after level, from the bottom to the top. Call the model of such a structural stability and reliability skyscraper grounding. The construction of a spaceship, on the other hand, is rather different. It is fabricated out of numerous components, each component being separately manufactured and tested for correct functioning and reliability. The spaceship as a whole is an assemblage of the individual components, constructed under the guidance of system engineers to insure that all the components function coherently in unison. Call the model of the resulting structural stability and reliability systemic cohesiveness. (Rav 1999, 27–28)

84

As Mac Lane claims, “the purported foundation of all of Mathematics upon set theory totters.” “Set theory is a handy vehicle, but its constructions are sometimes artificial. Moreover, it is clearly far too general; as Hermann Weyl once remarked, it contains far too much sand.” (Ibid., 359, 407).

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A spaceship metaphor – which echoes both Neurath’s raft and Wittgenstein’s “celestial body floating free in space” (Wittgenstein 1974, 297)85 – is introduced here to describe a new vision of mathematics. As far as mathematics is concerned, Rav claims, “skyscraper grounding […] has lost its credentials” while “systemic cohesiveness is a more viable model for the reliability of mathematical knowledge” (Rav 1999, 28–29). What is more, Rav strongly emphasizes the social nature of this cohesiveness: Mathematics is a collective art: the social process of reciprocal crosschecks seems to be the only way to weed out errors and guarantee the overall coherence and stability of mathematical knowledge. (Ibid., 36) The stability and secure flight of the spaceship of mathematics are sociallygained and gained primarily by giving, checking and correcting mathematical proofs. This is the point of Rav’s paper. The most outspoken advocacy of the social nature of mathematical proof, as far as I know, was made by the American mathematician and computer scientist Joseph Goguen in his paper “What is a Proof?” which was created in 1997, repeatedly edited until 2001, but never published.86 First of all, he introduces the concept of “proof event”: Mathematicians talk of “proofs” as real things. But the only things that can actually happen in the real world are proof events, or provings, which are actual experiences, each occurring at a particular time and place, and involving particular people, who have particular skills as members of an appropriate mathematical community. (Goguen 1997, par. 1) He also introduces the concept of “proof participant”:

85 86

Cf. Hersh’s eagle metaphor (Hersh 2005, 155). Goguen, “What is a proof? [1997, last revision 2001].” Available at https://cseweb.ucsd .edu/~goguen/papers/proof.html. Goguen acknowledges that his paper was “in part inspired by remarks of Eric Livingston”. He also comments at the end of it: “A more formal version of the above material would of course include citations to relevant work in ethnomethodology and the sociology of mathematics, especially work of Harvey Sacks, Eric Livingston, and Donald MacKenzie.” In his turn, Eric Livingston thanks Joseph Goguen for help and the time they spent together. Livingston visited the University of California, San Diego on Study Leave (where Goguen was a professor since 1996 until he passed away in 2006). See (Livingston 2008, ix).

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A proof event can have many different outcomes. For a mathematician engaged in proving, the most satisfactory outcome is that all participants agree that “a proof has been given.” Other possible outcomes are that most are more or less convinced, but want to see some further details; or they may agree that the result is probably true, but that there are significant gaps in the proof event; or they may agree the result is false; and of course, some participants may be lost or confused. In real provings, outcomes are not always just “true” or “false”. Moreover, members of a group need not agree among themselves, in which case there may not be any definite socially negotiated “outcome” at all! Each proof event is unique, a particular negotiation within a particular group, with no guarantee of any particular outcome. Going a little further, the traditional distinction between a proof giver and a proof observer is often artificial or problematic; for example, a group of mathematicians working collaboratively on a proof may argue among themselves about whether or not a given person has made substantive contributions. Hence we should use general terms like provers and proof participants, whatever their role and their distribution over space and time, and we should speak of “proof givers” and “proof observers” only to the extent that participants do so. (Ibid., par. 5–6) Goguen stresses “the rich social life of mathematicians.” Such a sociallyoriented optics urges him, on the one hand, to look at the cases of a single prover and a purely mental proof event as derivative or degenerate ones, and on the other, to bring interpretation and communication issues to the forefront. Though any mathematical proof event presupposes some physical objects mediating social communication, such as spoken, written or printed words, gestures, diagrams, formulae, 3D models, etc., only proper interpretation in the due communicative context turns them into “proof objects.” Moreover, it is often crucial to take into consideration the temporal order in which those proof objects appear within a particular proof event. This means that proof objects are actually “proof processes”: “diagrams being drawn, movies being shown, and Java applets being executed.” Goguen also calls upon his readers to switch attention from “the unusual achievements of particular ‘heroic’ individuals” to specific mathematical practices (“counting” is one of the most primitive examples) and “the communities that embody those practices.” Mathematics is seen in the same vein by American sociologist and ethnomethodologist Eric Livingston, who is working at the University of New England, Australia:

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[…] I have come to see the “doing of mathematics” in terms of the concrete details of collaborative proving at the blackboard, the ways that provers work out and write mathematical argumentation, the dynamic features of seminar presentations, the specifics of collegial discussions, the recognition of mistakes, and the pursuit and utility of ultimately (or seemingly) futile lines of inquiry. I am interested in notes written on paper napkins, blackboard erasures and repairs, and the material that ends up in the wastebasket. Mathematicians are engaged in the work of mathematics, often preoccupied by it, and it is this work that, for me, makes up mathematical activity and practice. (Livingston 2015, 200) In the 1980s, Livingston developed his conception of “the living foundations of mathematics”. He was awarded his Ph.D. in sociology for a thesis on the subject at the University of California, Los Angeles in 1983 (Livingston 1983). It was supervised by the creator of ethnomethodology, Harold Garfinkel. In 1986, this Ph.D. thesis was published as The Ethnomethodological Foundations of Mathematics (Livingston 1986). Livingston contrasted his approach to foundations with the traditional one. In his next book, he emphasized: When the ethnomethodology of mathematics was first undertaken, it was intended as an investigation of the “foundations” of mathematics. “Foundations,” however, was not understood in terms of a disengaged accounting procedure such as those of mathematical logic; instead, it was understood in terms of the situated work practices of professional mathematicians – that is, as the living foundations or “genetic origins” of the discipline. (Livingston 1987, 94–95) Here, we apparently have an allusion to Husserl’s understanding of the “origin (der Ursprung),”87 and an adjective “living,” applied to the foundations, corresponds with the “life-world (die Lebenswelt)” of phenomenology. Harold Garfinkel and his followers tried to turn those phenomenological insights into a working method of sociological inquiry. Garfinkel calls such a program of inquiry “the Lebenswelt origins of the sciences” (Garfinkel and Liberman 2007). In the preface to his first book, Livingston announced:

87

See (Husserl 1989 [1936]); (Hacking 2010). Husserl’s Ursprung does not mean simply a starting point or beginning in history, a historical origin, it also means something extra, a transcendental origin, that is, the source, the ground, the evidence of objectivity that manifests itself any time someone practices geometry.

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This book is a study of the foundations of mathematics, but in the original sense. It is a study of the genetic origins of mathematical rigor, examining the proofs of ordinary mathematics and investigating how the adequacy of such proofs, for the purposes of everyday mathematical inquiry, is practically obtained. The book formulates and, in a certain sense, solves the problem of the foundations of mathematics as a problem in the local production of social order. (Livingston 1986, x) In one aspect, Livingston’s living foundations are quite similar to Mac Lane’s global foundations, while in the other, they may be opposed. Both approaches pose the following question as a foundational one: what identifies an exact practice as a piece of mathematics? In Livingston’s variant, it runs: By the question of “genetic origins” we must understand the problem of finding what, in situ, for a local production cohort of provers, makes their work identifiably the work of mathematics, of finding what sustains their enterprise together as the work of the discovering science “mathematics”. The genetic origins of mathematics do not lie in the potentially problematic character of certain proofs, but in the fact that provers can demonstrate the truth of a proposition, for all provers, as their situated accomplishment. It is this daily achievement of professional mathematicians that permits and makes intelligible the attempt to demonstrate the existence of proof-independent methods of substantiating that achievement. The problem of genetic origins is, in this way, the primordial foundations problem. (Livingston 1987, 130) Both for Mac Lane and Livingston, this foundational question leads us, first, to a consideration of the inner organization of mathematical discourse and, second, to the outer connections of mathematics. Nevertheless, Mac Lane treats it as a “global” one, in contradistinction to the “locality” of the traditional understanding of the foundations of mathematics, while Livingston stresses “locality” of his understanding as opposed to traditional foundations, aimed to establish “a priori, globally assured, situationally transcendent rigor” (ibid.). The difference between the two points of view may be reconciled by treating them as macro- and micro-level approaches, respectively. For Mac Lane, mathematics is a single and (more or less) unified network of formal systems, relatively autonomous from both nature and culture, despite all the interplay between mathematics and its background. Livingston – as an ethnomethodologist – in principle, deals exclusively with the local practices of proof or discovery “that arise from within the situated, lived-work of doing professional mathematics –

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that is, as a work-site phenomenon” (Livingston 1986, 7). He is aimed at making the inner sociality and materiality of those practices discernible. Livingston’s solution to the foundation problem consists in the demonstration that any mathematical proof is a “Lebenswelt pair,” as Garfinkel calls it (Garfinkel 2002, 187–190), that is, the pair “the-material-proof/the-practicesof-proving-to-which-that-proof-is-irremediably-tied” (Livingston 1986, 171). The first segment of the pair (a “material detail”) is a theorem formulation and its proof published in a paper book or existing in some digital form. It is treated by Livingston as no more than “an account of the work of its production” or “a proof account” for short, a “template” of some course of action. The second segment of the pair (“practice”) is a corresponding lived-work of proving (ibid.). The living foundations of mathematics are the actual way in which the segments of a mathematical Lebenswelt pair are “irremediably” tied together. The only way to study them is by finding out how particular material proofs are understood and actualized as provings, and how particular acts of proving are materialized with the help of some symbolic notation. According to Livingston, the transcendental character of mathematics means “ideal,” “objective,” “anonymous as to authorship” or “evident” character of this kind of reasoning (ibid., x, 10, 23, 138, 150). It is explicated as “the disengagability […] from the actual, temporally-situated, circumstantial, at-theboard or with-pencil-and-paper work of doing mathematics” (ibid., 34). At the same time, he claims this transcendental character to be no more than “the witnessed, local achievement” (ibid., xi) of the living mathematical demonstrations themselves. This character is created in the local production of proofs and exists only in their local reproduction. Thus Livingston’s understanding of the transcendental character of mathematics contrasts markedly with any “platonic” (ibid., 6, 32, 44, 97) interpretation. In his largely negative review of Livingston’s first book, David Bloor insists that Livingston failed to be convincing enough in showing how localized work gives rise to transcendental rigor in mathematical practice (Bloor 1987). Perhaps Bloor is right; still, Livingston’s reconceptualization of the foundational issues looks promising to me. The new understanding of the foundations of mathematics that emerges within the philosophy of mathematical practice (as the examples of Saunders Mac Lane and Eric Livingston have shown) suggests distinguishing between the two coexisting levels of foundational studies. The first one deals with grasping how mathematical practices are rooted in everyday human activities and non-mathematical practices. Let me call it anthropological foundations. The second one is about the ways of the intrinsic organization and transformation of some mathematical practices or even all of them. It is mainly driven by mathematical, not anthropological or philosophical concerns, though it also

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may rest upon particular philosophical underpinnings, not to mention anthropological ones. This second level may be called practical foundations.

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Practical Foundations

The main change that drastically affected the everyday activities of the working mathematicians in the last half a century has been the advent of computers. It is even defensible to designate the transformation of mathematical practices as the computer revolution in mathematics (Fillion 2019, 200–202; Shaposhnikov 2019b). Moreover, digital computers may be treated, at least from some points of view, as new “members” of the mathematical community (Shaposhnikov 2018; Shaposhnikov 2019a). Nicolaas Govert de Bruijn – one of the patriarchs of automated theorem checking and the development of formal mathematical proofs by human-machine collaboration – wrote: Some people feel that mathematical thinking depends on the existence of a real world of mathematical objects, an idea called Platonism. For communication between mathematicians this idea is irrelevant: in a mathematical discussion between a believer and a non-believer none of the two notices their different backgrounds. And of course it is irrelevant in mathematical communication with a machine. The machine does not store the mathematical objects we are talking about. The only thing it may have to store is what has been said thus far. […] The language of mathematics cannot be verified on the objects: there are too many of them. The only thing we can do is apply the rule that things are correct if they have been correctly said. The notion of correctness is not formulated in terms of a mathematical reality, but involves rules about how a statement should be related to material that has been said before. […] I believe that every Platonist would be converted at once when having to explain his mathematics to a machine. (de Bruijn 1991, 10–11) De Bruijn insists that when computers enter the network of mathematical communication, it profoundly changes our ways of doing mathematics and thinking about mathematics. These changes affect our view of foundations as well. For instance, de Bruijn, while developing a formal language of his own and the associated automated proof checker Automath in the late 1960s, was forced to question the status of Zermelo–Fraenkel set theory (ZF) as a natural foundation for mathematics (ibid., 11–14).

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Discussing the problems preventing the so-called QED project – the project of the total formalization of the main body of contemporary mathematics (Boyer et al 1994) – from successful realization, Freek Wiedijk (a leading expert in the field) emphasized, among other things, two closely connected issues. First, we need a user-friendly formalization of mathematics: An aspect where “QED-like systems” currently are notoriously bad at is communication. A formalization is completely useless for communicating the mathematics that is formalized in it. Here is the most to be gained for formal mathematics. (Wiedijk 2007, 122) Second, there is still no consensus on the choice of a practical foundation for such a total formalization (different proof assistants have different foundations): Ideally, the system should be set up in such a way, that the mathematics that is formalized in it can survive a “redesign of the foundations”. Suppose that one builds a QED-like library on top of Coq,88 and then changes one’s mind and switches to a HOL-like system.89 Basically, this will mean that one will have to start from scratch. It seems very arrogant to think that we already know what the best foundations for our formal library should be, and one would therefore prefer not to be “locked in” to a specific version of the foundations. […] I do not believe that the QED system will consist of many different systems living peacefully together. One of the systems – hopefully the best one – will kill the others. That is how evolution works. It is this system that should be built. (Ibid., 131) The term “practical foundations of mathematics” was popularized by Paul Taylor, who published a book with such a title (Taylor 1999). Already his reviewers wondered about the true meaning of the title. One of them suggested: The somewhat mysterious word “practical” in the title simply insinuates that the author doesn’t want “foundations” to be understood in any “foundational” sense. The aim rather is to exhibit and study the math-

88 89

One of the most popular proof assistants (interactive theorem provers). This group includes, according to Wiedijk, the following provers: HOL Light, Isabelle/HOL and ProofPover.

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ematical principles behind logic and induction as needed and used for the formalisation of (the main parts of) Mathematics and Computer Science. […] the book provides a coherent picture of “modern” categorical foundations as opposed to the somewhat “old-fashioned” set-theoretic ones. The former are as strong as the latter as long as classical logic is assumed. However, the advantage of categorical foundations, besides being closer to mathematical practice, is that they aren’t biased towards classical logic like traditional set theory. Therefore, the book can be strongly recommended to everybody interested in foundations, typically rather a mathematically inclined computer scientist than a “working mathematician” who in general has less need for and, accordingly, less interest in foundations. (Streicher 2000, 155–157) A short description on the back cover of Taylor’s book says: Practical Foundations collects the methods of construction of the objects of twentieth century mathematics, teasing out the logical structure that underpins the informal way in which mathematicians actually argue. Although it is mainly concerned with a framework essentially equivalent to intuitionistic ZF, the book looks forward to more subtle bases in categorical type theory and the machine representation of mathematics. These words can be read as an attempt to answer what “practical foundations” are. First, they are claimed to be an articulation of “the logical structure that underpins the informal way in which mathematicians actually argue.” The words “informal” and “actually” are crucial here. Such foundations pretend to meet the everyday needs of contemporary and, perhaps, future working mathematicians. Second, the working mathematicians in question are computer-oriented if not computer scientists. Third, the two previous points make “practical” foundations to be seen as an alternative to traditional settheoretic foundations. The “more subtle basis” is developed along categorical and constructivist lines. Taylor is far from being explicit on the unifying idea behind his understanding of the foundations of mathematics. He formulates his intention as a number of hints. The book opens with one of them: Foundations have acquired a bad name amongst mathematicians, because of the reductionist claim analogous to saying that the atomic chemistry of carbon, hydrogen, oxygen and nitrogen is enough to understand biology. Worse than this: whereas these elements are known with

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no question to be fundamental to life, the membership relation and the Sheffer stroke have no similar status in mathematics. (Taylor 1999, viii) So “practical” foundations are not intended to be reductionist, we are not expected to rewrite existing mathematics in some new and artificial manner. We are rather supposed to unveil, to make explicit the sort of “natural” foundations that are already at work in the living organism of modern mathematics. Taylor continues: Our subject should be concerned with the basic idioms of argument and construction in mathematics and programming, and seek to explain these (as fundamental physics does) in terms of more general and basic phenomena. This is “discrete math for grown-ups”. […] You should not regard this book as yet another foundational prescription. I have deliberately not given any special status to any particular formal system, whether ancient or modern, because I regard them as the vehicles of meaning, not its cargo. I actually believe the (moderate) logicist thesis less than most mathematicians do. This book is not intended to be synthetic, but analytic – to ask what mathematics requires of logic, with a view to starting again from scratch. (Ibid., viii–ix) Thus we are advised not to confuse the “vehicles” with the “cargo.” Informal mathematical theories are highly valuable “cargo” while an underlying logic and basic formalisms are neither more (nor less!) than “the vehicles of meaning.” These “vehicles” are not absolute; they must be just practically effective, that is, they must maintain the correctness of mathematics on the highest possible level and facilitate communication both between humans and in a human-computer variant. Thomas Callister Hales – an American mathematician, who is widely known for settling the honeycomb (Hales 2001) and the Kepler conjectures, also one of the leading authorities in the field of computer-assisted formal proofs – stated the contrast between the theoretical and the practical foundations of mathematics in the following way. According to Hales, “Bourbaki’s Theory of Sets was designed as a purely theoretical edifice that was never intended to be used in the proof of actual theorems” (Hales 2008, 1371). In the cases of both Hilbert and Bourbaki, formalization of mathematical proofs was a separate project in its own right. It peacefully coexisted with informal or semi-formal mathematical practices of working mathematicians, who may happily ignore the artificial constructions of the formalists. (Perhaps, it would be better to use the word “speculative,” instead of “theoretical” to characterize

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the type of foundations criticized by Hales. Please revisit the last epigraph to this paper.) The ever-increasing involvement of working mathematicians in a constructive dialogue with the computer in the hope of coping with the notorious problem of complexity (Davies 2005) changes their attitude towards formalization and the underlying foundational issues. Now, the foundational problem turns out to be practical, for it directly affects the obtaining and verification of ordinary mathematical results. In Hales’s own words: In Bourbaki’s view, the foundations of mathematics are roped-off museum pieces to be silently appreciated, but not handled directly. There is an opposing view that regards the foundational enterprise as unfinished until it is realized in practice and written down in full. This article sketches the current state of this endeavor. It has been necessary to commence afresh, and to retool the foundations of mathematics for practical efficiency, while preserving its reliability and austere beauty. (Hales 2008, 1371) Hales is in no way a casual observer of the newer tendencies in mathematical practice. He authors a 1998 computer-assisted proof of the famous Kepler conjecture on dense sphere packings, which was impossible for humans to completely verify (Hales and Ferguson 2006). This is why he attests: My interest in formal proofs grows out of a practical desire for a thorough verification of my own research that goes beyond what the traditional peer review process has been able to provide. (Hales 2008, 1378) He launched the Flyspeck project and headed the team, which, in 2014, eventually obtained a formal proof of the Kepler conjecture, being checked with the help of some HOL-like proof assistants (Hales et al 2017). Hales favors the QED-project, metaphorically naming it “the sequencing of a mathematical genome” (Hales 2008, 1379). He finishes his 2008 survey article with the words, obviously emblematic of the ongoing shift in the meaning of the foundations: Outsourcing is the brute force solution to the Q.E.D. manifesto. Most researchers, however, prefer beauty over brute force; we may hope for advances in our understanding that will permit us someday to convert a printed page of textbook mathematics into machine-verified formal text in a matter of hours, rather than after a full week’s labor. As long as transcription from traditional proof into formal proof is based on human

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labor rather than automation, formalization remains an art rather than a science. Until that day of automation, we fall short of a practical understanding of the foundations of mathematics. (Ibid.) Now let us turn to what seems to be the latest substantial advance along the line under discussion. A Russian-American mathematician Vladimir Voevodsky, a 2002 Fields Medalist, was one of those who dared to face the challenge of gaining a truly practical understanding of the foundations of mathematics. For about fifteen years – since approximately 2002 and until his untimely death in 2017 – he was developing a new approach that came to be known as “Homotopy Type Theory or Univalent Foundations of Mathematics (HoTT/UF).” In 2012–13, Voevodsky, with the help of Steve Awodey and Thierry Coquand,90 organized a special HoTT/UF program in the School of Mathematics at the Institute for Advanced Study in Princeton, where he was a professor (Awodey and Coquand 2013). More than fifty scholars took part in the program with the twenty-seven official participants as its core. One of the results of their collaboration during that academic year was a six-hundred-page book,91 which was intended to serve as a textbook on the new subject for humans.92 The introduction to this book states: “we […] believe that univalent foundations will eventually become a viable alternative to set theory as the ‘implicit foundation’ for the unformalized mathematics done by most mathematicians.”93 Awodey and Coquand emphasize both continuity and novelty, introducing HoTT/UF: This research program was centered on developing new foundations of mathematics that are well suited to the use of computerized proof assistants as an aid in formalizing mathematics. Such proof systems can be

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Thierry Coquand has been a leading figure in the Coq proof assistant development team. The Univalent Foundations Program at the Institute for Advanced Study, Homotopy Type Theory: Univalent Foundations of Mathematics (Princeton, 2013). This book is freely available at https://homotopytypetheory.org/book/ (access: 29.05.2020). See also (Bauer 2013). From the Preface to the 2013 HoTT/UF book: “Univalent foundations is closely tied to the idea of a foundation of mathematics that can be implemented in a computer proof assistant. Although such a formalization is not part of this book, much of the material presented here was actually done first in the fully formalized setting inside a proof assistant, and only later ‘unformalized’ to arrive at the presentation you find before you – a remarkable inversion of the usual state of affairs in formalized mathematics.” (The Univalent Foundations Program 2013, iv). Ibid., 2.

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used to verify the correctness of individual mathematical proofs and can also allow a community of mathematicians to build shared, searchable libraries of formalized definitions, theorems, and proofs, facilitating the large-scale formalization of mathematics. The possibility of such computational tools is based ultimately on the idea of logical foundations of mathematics, a philosophically fascinating development that, since its beginnings in the nineteenth century, has, however, had little practical influence on everyday mathematics. But advances in computer formalizations in the last decade have increased the practical utility of logical foundations of mathematics. Univalent foundations is the next step in this development: a new foundation based on a logical system called type theory that is well suited both to human mathematical practice and to computer formalization. (Awodey and Coquand 2013, 1) Here we, once again, meet many already familiar ideas. HoTT/UF are introduced as “implicit” or “natural” foundations for the working mathematician of today. In contrast to ZFC (the Zermelo–Fraenkel axioms plus the axiom of choice), which are theoretical – i.e., useless – foundations, HoTT/UF are hoped to be practical – i.e., useful – foundations. Their practicality is twofold. They are simultaneously oriented towards unformalized or highly intuitive human mathematics and completely formalized machine mathematics, being both human- and machine-friendly. The leaders of the new approach are fully aware of the radical novelty and the proper place of HoTT/UF against the historical background: By the 1950s, a consensus had settled in mathematics that the program of logical foundations, while perhaps interesting in principle or as its own branch of mathematics, was going to be of little use to the general practice of mathematics as a whole. […] But with recent advances in the speed and capacity of modern computers and theoretical advances in their programming has come a remarkable and somewhat ironic possibility: the use of computers to aid in the nearly forgotten program of formalization of mathematics. For what was once too complicated or tedious to be done by a human could now become the job of a computer. With this advance comes the very real potential that logical foundations, in the form of computer formalizations, could finally become a practical aid to the mathematician’s everyday work, through verifying the correctness of definitions and proofs, organizing large-scale theories, making use of libraries of formalized results,

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and facilitating collaboration on the development of a unified system of formalized mathematics. Gödel may have shown that mathematics cannot be entirely formalized in principle, but in practice there are still great benefits to be had from formalization when sophisticated computer systems can be brought to bear. This new conception of foundations of mathematics, so closely tied to the use of computers to provide the guarantee of formal rigor and to aid in handling the explosion of complexity, seems so natural that future historians of mathematics may well wonder how Frege and Russell could have invented the idea of formal systems of foundations before there were any computers to run them on. Nor is it a coincidence that foundations work so well in combination with computers; as already stated, the modern computer was essentially born from the early research in logic, and its modern programs and operating systems are still closely related to the logical systems from which they evolved. In a sense, modern computers can be said to “run on logic.” This is the starting point of the univalent foundations program: the idea that the time is now ripe for the development of computer proof assistants based on new foundations of mathematics. (Ibid., 6) The two motivating factors behind the HoTT/UF project as practical foundations for mathematics are quite obvious. The first one is to avoid mistakes and to be sure about the correctness of our mathematical results. In the next step, it leads to the idea of relying on machine verification of mathematical proofs for human verification fails to be secure enough. Voevodsky’s reminiscences testify to this (Voevodsky 2014). Since the 1980s, he had been engaged in developing new highly abstract and extremely complicated mathematical theories, such as motivic cohomology. It is very easy to make a mistake when discussing such matters and extremely hard to detect it. Voevodsky was to find this out the hard way himself. He traced a mistake in the proof of a key lemma in one of his papers (published in 1993) while lecturing on the subject and checking every step in his arguments with the help of one of his colleagues in the 1999–2000 academic year. He wrote: This story got me scared. Starting from 1993, multiple groups of mathematicians studied my paper at seminars and used it in their work and none of them noticed the mistake. And it clearly was not an accident. A technical argument by a trusted author, which is hard to check and looks similar to arguments known to be correct, is hardly ever checked in detail. (Ibid., 8)

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Another, even more regrettable, incident happened with the main result of his 1989 paper (coauthored with Michael Kapranov). In 1998, Carlos Simpson constructed a counterexample, but the reputation of Voevodsky and Kapranov in mathematical society was too strong to doubt their result. Simpson’s counterexample was ignored until 2013 when Voevodsky located a mistake in the 1989 paper himself. “Mathematical research currently relies on a complex system of mutual trust based on reputations,” he commented (ibid.). In the early 2000s, Voevodsky was working on even more complicated higher-dimensional mathematics than before, but he was now extremely anxious about the correctness of his own and similar results. He recollected: But to do the work at the level of rigor and precision I felt was necessary would take an enormous amount of effort and would produce a text that would be very hard to read. And who would ensure that I did not forget something and did not make a mistake, if even the mistakes in much more simple arguments take years to uncover? I think it was at this moment that I largely stopped doing what is called “curiosity-driven research” and started to think seriously about the future. I didn’t have the tools to explore the areas where curiosity was leading me and the areas that I considered to be of value and of interest and of beauty. So I started to look into what I could do to create such tools. And it soon became clear that the only long-term solution was somehow to make it possible for me to use computers to verify my abstract, logical, and mathematical constructions. (Ibid.) Unfortunately, the state of the art in the development of “practical” proof assistants at the time was still too far from what he needed to secure his work in algebraic geometry. What is more, Voevodsky fully understood that the key problems lay as deep as the foundations of mathematics: to create a truly “practical” proof assistant one had first to create adequate “practical” foundations, which were still lacking. The primary challenge that needed to be addressed was that the foundations of mathematics were unprepared for the requirements of the task. Formulating mathematical reasoning in a language precise enough for a computer to follow meant using a foundational system of mathematics not as a standard of consistency to establish a few fundamental theorems, but as a tool that can be employed in everyday mathematical work. (Ibid.)

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Thus, Voevodsky embarked on his foundational venture that finally resulted in the introduction of HoTT/UF. He combined CIC (the calculus of inductive constructions, which was used for the development of the formal deduction system of the Coq prover), the homotopy type theory and a particular way to encode mathematical notions in terms of homotopy types to get a cohesive alternative both to ZFC and categorical foundations.94 He also introduced the so-called “univalence axiom.” Daniel Grayson, one of the official participants of the 2012–13 HoTT/UF program at Princeton, comments on its meaning: The formal mathematical language, together with the univalence axiom, fulfills the mathematicians’ dream: a language for mathematics invariant under equivalence and thus freed from irrelevant details and able to merge the results of mathematicians taking different but equivalent approaches. (Grayson 2018, 428)95 The second motivating factor behind the HoTT/UF project, along with the quest for correctness and security, seems to be the one for more convenient communication among mathematicians. Voevodsky was consciously looking for foundations that would be natural for a working mathematician, that is, allowing “to directly express” (Voevodsky 2014, 9) the statements about mathematical objects of the kind he was interested in – the objects of higherdimensional algebraic geometry. Voevodsky hoped that in a few decades, the use of handy proof assistants for doing mathematics would become ubiquitous. According to Julie Rehmeyer, who interviewed Voevodsky in 2013: He also predicts that this will lead to a blossoming of collaboration, pointing out that right now, collaboration requires an enormous trust, because it’s too much work to carefully check your collaborator’s work. With computer verification, the computer does all that for you, so you can collaborate with anyone and know that what they produce is solid. That creates the possibility of mathematicians doing large-scale collaborative projects that have been impractical until now. (Rehmeyer 2013) 94

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Voevodsky was surely not the only one who labored in this direction. In his own words: “I have been working on the ideas that led to the discovery of univalent models since 2005 and gave the first public presentation on this subject at Ludwig-Maximilians-Universität München in November 2009. While I have constructed my models independently, advances in this direction started to appear as early as 1995 and are associated with Martin Hofmann, Thomas Streicher, Steve Awodey, and Michael Warren.” (Ibid., 9). Grayson gives a formulation of the univalence axiom on pages 442–443.

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As Michael Harris, a mathematician at Columbia University, put it, in contemporary mathematics “foundations” should be understood “not as the bedrock of certainty but as a common language”; and this common language is regularly reworked to be kept up to date (Harris 2015, 65). We may add that there is no need to oppose “certainty” to “a common language,” that is, to a vehicle of communication. To create and maintain a dense network of communication – which includes both humans and computers – seems to be the only way to guarantee mathematical certainty. The successful cross-checks Yehuda Rav was talking about presuppose free and unhampered communication within the mathematical community. Hence, a steady interest in any improvements that facilitate it and make it more convenient. This means that communication is obviously obtaining a foundational meaning nowadays. The arrival of the HoTT/UF project was in no way a bolt out of the blue. It ingeniously and successfully combined several ideas and approaches that were already in the air. The first strand can be traced back to Bertrand Russell’s theory of types, which was introduced at the very beginning of the twentieth century. Then followed the idea of typing as used by de Bruijn in his proof checker Automath in the 1960s, Per Martin-Löf’s intuitionistic type theory in the 1970s, and the Calculus of Constructions and the Coq proof assistant on its base, created by Thierry Coquand and his colleagues in the 1980s. The second strand got its start in the idea of geometry-based foundations of mathematics or “the geometrical source of knowledge” as stated by the later Gottlob Frege.96 It obtained a new life in the creation of the category theory in the 1940s and later on in the ideas of Alexander Grothendieck up to the 1980s. Finally, both strands were united during the 1990s and the 2000s, and homotopy type theory emerged. Voevodsky shares the honor of creating it with Martin Hofmann, Thomas Streicher, Steve Awodey and Michael Warren, with Michael Makkai as their forerunner. The QED project has been widely known since its appearance in the first half of the 1990s; mathematical structuralism, as a general methodology, has been extremely popular since at least the 1980s, and the categorical foundations for mathematics as an alternative to the set-theoretic ones have been repeatedly discussed since the 1960s.97 It would be wise not to make any predictions as to the future of the HoTT/UF project. In an interview, Thomas Hales said about Voevodsky:

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I mean here some of Frege’s drafts that were made about 1924/25. See (Frege 1979, 267–281). See, e.g., (Lawvere 1966); (Feferman 1977); (Linnebo and Pettigrew 2011); (Marquis 2013).

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His ideas gave a new way for all mathematicians to do what they do, a new foundation. The foundations of math are like a constitutional document that spells out the governing rules all mathematicians agree to play by. He has given us a new constitution. (Rehmeyer 2017) Hales is among the enthusiasts. There are skeptics as well. To give but a few instances, I can name the mathematician Michael Harris (see Harris 2015) and the philosopher of mathematics Penelope Maddy (see Maddy 2019).

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Conclusion: On the Foundational Meaning of Communication

In the year 2014, a young Israeli mathematician, Ittay Weiss,98 suggested switching to Version 2.0 of the QED Manifesto.99 He proposes “a different ideal” as far as the formalization of mathematics is concerned. The 1994 QED Manifesto (QED 1.0) was aimed at a complete formalization for the sake of precision, rigor and, finally, absolute correctness of mathematical results. It said: “The QED system will provide a beautiful and compelling monument to the fundamental reality of truth” (Boyer et al 1994, 240). Surely, it was seen as a communal project with open participation and communal benefits, but free and unhampered communication was not its primary concern. Weiss, who keeps in mind Freek Wiedijk’s and Thomas Hales’s expert judgments (see the earlier discussion), as well as his own experience as a mathematician, makes the facilitation of communication the focus of his QED 2.0. While QED 1.0 seeks to maximize the formality of mathematical proofs, QED 2.0 is concerned with the maximization of readability and the cognitive comprehensibility of them. In the case of long proofs, there is an unresolved tension between the two aspirations. Available ways of formalization required mathematicians to choose between completely formal proofs and easily understandable proofs. To achieve both ends at the same time seems to be impossible by now. The point of Weiss’s presentation is that to concentrate all our energy exclusively

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Ittay Weiss obtained his B.Sc. (2001) and M.Sc. (2003) in mathematics from the Hebrew University of Jerusalem, and his Ph.D. (2007) from Utrecht University (Netherlands). He is currently a senior lecturer in mathematics at the University of Portsmouth (UK). His research interests lie in the areas of topology, homotopy theory, metric geometry and category theory, as well as the interface between computers and mathematicians. First presented at the Asia-Pacific World Congress on Computer Science and Engineering (November 4–5, 2014, Nadi, Fiji). See (Weiss 2014). A full-length article was published later: (Weiss 2016).

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on achieving a complete machine-readable formalization of the main body of contemporary mathematics (QED 1.0) has no prospects. It should be complemented with parallel work on building human-friendly computer systems that will help make the sharing of mathematical ideas and texts far more convenient, and hence to maintain intense communication between mathematicians and with all kinds of users of mathematics (QED 2.0). It is crucial to state that QED Version 2.0 is neither meant to replace Version 1.0 nor to indicate that it is in any sense better. By analogy to programming languages, the difference between QED 1.0 and QED 2.0 is akin to the differences between low level programming paradigms and higher level ones, respectively. If formalizability and cognitive readability can be unified by some currently unknown clever mechanism […], or if the process of turning a ‘higher level’ proof into a formal ‘lower level’ one can be automated, then QED 1.0 and QED 2.0 will merge into a single product. Otherwise, each paradigm will have its own merits and its own applications, and products will coevolve utilizing various mixtures of the two versions. (Weiss 2016, 804) Weiss formulates eight goals for QED 2.0. (1) Independence from Convention. The envisaged QED 2.0 system will be able to “decouple content from presentation.” The goal is to set mathematicians free from a rigid choice of notation by making all sorts of conventional notations automatically convertible. Then a mathematician can easily convert any mathematical text into his favorite convention. (2) Independence of Content. The reader will be able to automatically adapt any mathematical text to the chosen level of her mathematical training. (3) Dissemination of New Results. Easy and open access to the whole base of mathematical results. Automatic search for auxiliary results for a chosen new result plus automatic adaptation to the needed level of mathematical training. (4) Modularity and Reusability. Reusability of any piece of mathematical writing, which will be presented as a universal module that can be easily combined with other modules with automated coherence check. (5) Learning and Teaching Management. The development of convenient tools both for lecturers and students of mathematics. (6) Language Flexibility. The automated translation of mathematical texts into any ordinary language (English, German, French, Chinese, Russian, etc.) (7) Organization of Knowledge. Navigation in the ocean of mathematical results. Special tools for detailed search and comparison of mathematical items. Tools for assessing the impact and significance of particular mathematical results or theories. The clarification of the inner connections and the overall structure of mathematics. (8) Testing

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Ground for Other Domains of Knowledge. The QED 2.0 system may serve as “a testing ground” for developing similar systems in other sciences and beyond (ibid., 805–806). Surely Weiss is not the first and the only one to discuss these matters.100 For instance, Michael Kohlhase’s “The Flexiformalist Manifesto” (Kohlhase 2012)101 comes to mind (Weiss refers to some of his works). By now, Kohlhase is one of those who talk earnestly about “Big Math” and its challenges: It seems clear that the traditional method of doing mathematics, which consists of well-trained, highly creative individuals deriving insights in the small, reporting on them in community meetings, and publishing them in academic journals or monographs is reaching the natural limits posed by the amount of mathematical knowledge that can be held in a single human brain – we call this the one-brain barrier. (Carette et al 2019, 4) Well, one may ask what these ideas have to do with the foundations of mathematics. Weiss compares his own initiative with the introduction of TeX/LaTeX into the everyday work of mathematicians: “TeX has nothing to do with mathematics. It is a typesetting system. Similarly, QED 2.0 has nothing to do with formal mathematics, logic, or semantics. It is a typesetting ideology” (Weiss 2016, 813). Nevertheless, I believe that there is a strong connection. Increased sensitivity of contemporary mathematicians to communication problems seems to have a hidden foundational meaning, even when there is no open talk about foundations. It seems that the maintenance of stability in the local communication flow within mathematical communities is taking on the role of traditional foundations of mathematics before our very eyes.

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A similar approach to that of Ittay Weiss was manifested by another young mathematician, Felix Breuer, who graduated from Freie Universität Berlin in 2006 and obtained his Ph.D. from the same institution in 2009. His communication-centered “Open Mathematics” project was presented in his personal blog (2010–14), http://blog.felixbreuer.net/ blog/archives/ (access: 29.05.2020). See especially his posts “Not Only Beyond Journals, Not Only Beyond Papers. Beyond Theorems” (February 27, 2012), “Towards a Standard File Format for Formal Sketches of Mathematical Articles” (June 18, 2012), “From Open Science to Open Mathematics” (July 14, 2013) and “Thoughts on QED+20” (July 21, 2014). Breuer’s texts can serve as a useful introduction to the current frame of mind. Michael Kohlhase is a German computer scientist who obtained his Ph.D. from Universität des Saarlandes in 1994. He is president of the OpenMath Society and a trustee of the Mathematical Knowledge Management Interest Group.

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The most appropriate metaphor for this new vision of foundations is that of a “thick thread,” “rope” or “cable”. We meet it in Charles Peirce and Ludwig Wittgenstein. The passage in question from Peirce runs: Philosophy ought to imitate the successful sciences in its methods, so far as to proceed only from tangible premises which can be subjected to careful scrutiny, and to trust rather to the multitude and variety of its arguments than to the conclusiveness of any one. Its reasoning should not form a chain102 which is no stronger than its weakest link, but a cable whose fibres may be ever so slender, provided they are sufficiently numerous and intimately connected. (Peirce 1868, 141) Wittgenstein elaborated on the metaphor, adding a length limitation for each fiber: Why do we call something a “number”? Well, perhaps because it has a – direct – affinity with several things that have hitherto been called “number”; and this can be said to give it an indirect affinity with other things that we also call “numbers”. And we extend our concept of number, as in spinning a thread (ein Faden) we twist fibre on fibre. And the strength of the thread resides not in the fact that some one fibre (eine Faser) runs through its whole length, but in the overlapping of many fibres. (Wittgenstein 2009, 36) Wittgenstein spoke about continuity and changes in the use of words, while Peirce discussed scientific justification through arguments. I think the “thread” metaphor is also good enough to visualize how mutual understanding, security, and reliability are achieved in mathematics. We can further elaborate on the “thread” metaphor by merging it with the “web” metaphor. Finally, we get the whole “fabric” instead of many isolated “threads”. Not surprisingly, the “fabric” metaphor is an old and well-known one.103 In the first scene of Goethe’s Faust, the Earth Spirit says: “So schaff ich am sausenden Webstuhl der Zeit/ Und wirke der Gottheit lebendiges Kleid” (“I work at the whirring loom of time/ and fashion the living garment of God” 102 103

Peirce obviously hints at Descartes’ metaphor of “chain” for deductive inference and mathematical proof. Cf. Quine’s use of the metaphor: “the fabric of science”, “a man-made fabric”, and “a fabric of sentences”. (Quine 1961 [1951], 42, 46); (Quine 1960 [1954], 374); (Quine Ullian 1978, 82); (Verhaegh 2017, 15).

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508–9).104 Faust has failed to compete with the Earth Spirit, and he states later on, while talking to Mephistopheles, that “Der große Geist hat mich verschmäht,/ Vor mir verschließt sich die Natur./ Des Denkens Faden ist zerrissen,/ Mir ekelt lange vor allem Wissen” (“The Great Spirit rejected me with scorn,/ and Nature’s doors are closed against me./ The thread of thought is torn asunder,/ and I am surfeited with knowledge still” 1746–9).105 Still, later on, Goethe’s Mephistopheles – appearing in the disguise of Faust in front of a student – mocks traditional logic as a way of linearizing “the fabric of thought (der Gedankenfabrik)” at the expense of the loss of its spirit and hence its true meaning: Zwar ists mit der Gedankenfabrik Wie mit einem Webermeisterstück, Wo Ein Tritt tausend Fäden regt, Die Schifflein herüber-hinüberschießen, Die Fäden ungesehen fließen, Ein Schlag tausend Verbindungen schlägt. Der Philosoph, der tritt herein Und beweist Euch, es müßt so sein: Das Erst wär so, das Zweite so Und drum das Dritt und Vierte so, Und wenn das Erst und Zweit nicht wär, Das Dritt und Viert wär nimmermehr. Das preisen die Schüler aller Orten, Sind aber keine Weber geworden. Wer will was Lebendigs erkennen und beschreiben, Sucht erst den Geist herauszutreiben, Dann hat er die Teile in seiner Hand, Fehlt, leider! nur das geistige Band. Encheiresin naturae nennts die Chemie, Spottet ihrer selbst und weiß nicht wie.

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Although in fact the fabric of thought is like a masterpiece of weaving, for which one treadle moves a thousand threads as back and forth the shuttles fly and threads move quicker than the eye and a single stroke makes a thousand ties, nonetheless the philosopher comes and proves to you it had to be thus: the first was so, the second so, and hence the third and fourth are so; but if there were no first and second the third and fourth could never exist. Students applaud this everywhere, but fail to master the weaver’s art. To understand some living thing and to describe it, the student starts by ridding it of its spirit; he then holds all its parts within his hand except, alas! for the spirit that bound them together – which chemists, unaware they’re being ridiculous, denominate encheiresin naturae. (1922–41)106

Goethe 2014, 16. Ibid., 45. Ibid., 49.

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There is an apparent correspondence between Nature as “the living garment of God” and knowledge of Nature as “the fabric of thought” in Goethe.107 Mathematical knowledge can also be seen as a “fabric of thought” that is created, mended and maintained by constant combined efforts of numerous mathematicians who join their forces within the social fabric of the mathematical community. Earlier in this paper, I distinguished between the anthropological and practical foundations of mathematics. Both can be readily interpreted in terms of communication. The contemporary quest for practical foundations turns out to be a quest for a new common mathematical language that is specially designed to facilitate communication (pari passu in humanto-human, human-to-computer, and computer-to-computer variants) to the highest practically possible degree. Anthropological foundations, in their part, bring to light the Lebenswelt underpinnings of various mathematical practices, investigate the character and role of social contacts and cognitive environment for doing mathematics. All these, as we have seen, is also possible to discuss in terms of communicative actions, which consider both human and non-human agents. Foundations in such a context are understood as cognitive, cultural, and social mechanisms that provide local stability and reliability of communication flows.

… One may object to the perspective adopted in this paper, claiming that after all, it is not about “the foundations of mathematics” in the proper sense of the phrase. For example, Costas Drossos – while contrasting analytic and holistic approaches to foundations – says about the latter approach: “The other version of foundations, actually, is not a foundation at all” (Drossos 2006, 102) His second version of foundations is very close to what I call “practical foundations.” He illustrates the version by quotes from such distinguished mathematicians as William Lawvere (see the third epigraph to this paper) and Yuri Manin. The second quote is genuinely representative. Here it is: I will mean “foundations” neither as the paraphilosophical preoccupation with the nature, accessibility, and reliability of mathematical truth, nor as a set of normative prescriptions like those advocated by finitists

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These metaphors of Goethe are closely connected with those of the lifting of Isis’s πέπλος (veil or mantle) for uncovering the secrets of Nature, which dates back to Plutarch’s On Isis and Osiris (354b, 10). See (Plutarch 1970, 130–131). For discussion, see (Hadot 2006).

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or formalists. I will use this word in a loose sense as a general term for the historically variable conglomerate of rules and principles used to organize the already existing and always being created anew body of mathematical knowledge of the relevant epoch. […] In all cases, foundations in this wide sense is something which is of relevance to a working mathematician, which refers to some basic principles of his/her trade, but which does not constitute the essence of his/her work. (Manin 2007 [2004], 49) First of all, I would like to note that there is no such thing as a generally accepted or standard understanding of what the foundations of mathematics are or should be. Nevertheless, we have some paradigmatic or classical examples. “Classical foundations” (Wagner 2019, 381) refer – first and foremost – to the three “isms”: logicism, intuitionism, and formalism. While foundational matters appeared to be central to the philosophy of mathematics in the first decades of the 20th century, they were – as is widely known – marginalized later on. It is especially so within the philosophy of mathematical practice (Azzouni 2005, 9–13; Hersh 2005, 155–158). The concurrent tendency – to change the meaning of the very phrase “foundations of mathematics” while preserving the discussion of foundational concerns among the key topics – is much less visible and much less talked about. Nevertheless, it cannot be dismissed as a mere misuse of the phrase in question. The new meanings of the old words have emerged as a natural reaction to the changing cultural and philosophical background and have developed in accordance with that background. What is more, there are substantial precedents for using the phrase “foundations of mathematics” in a new way, as the words of Yuri Manin cited above illustrate. To add to the examples already given in this paper, I will refer here to a 2005 article by Reuben Hersh, entitled “Wings, Not Foundations!” In this article, the two new meanings I am talking of (“practical” and “anthropological” foundations) are convincingly discerned. The first meaning is captured when Hersh notes that “in the common talk of ‘ordinary mathematicians’, the expression ‘foundations of mathematics’ has degenerated to mean ‘universal language for mathematics’” (Hersh 2005, 157–158). He does not approve of this usage and sees it as a misnomer. I obviously can hardly agree with such a critical evaluation of it. But he rightly stresses a possible pluralism of common mathematical languages. Hersh introduces the second meaning in the following passage: If we do want to talk about “foundations of mathematics”, we have to recognize that there can be many different kinds of foundations. It has become automatic to interpret “foundations” as “logical” or “axiomatic”

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foundations. This notion of foundations is based on seeing mathematics as a collection of statements or formulas, a library of inscriptions. The foundation then is the inscription from which all the other inscriptions can be derived. But it’s equally valid to regard mathematics as a historical process, part of the intellectual and cultural history of humanity. Then one could ask for historical foundations. It is equally legitimate to see mathematics as embedded in society, as a part of the socio-economicpolitical life of our times. Then one could ask for the socio-economicpolitical foundations of mathematics. And certainly one can think of mathematics as an activity of the individual mind/brain, a function of the nervous system. Then one could seek the psychological/neurological foundations of mathematics. In fact, all three kinds of activity are going on today. I have become interested in still a different kind of foundation, what is sometimes called “phenomenological foundation.” (Ibid., 158–159) To designate Hersh’s “many different kinds of foundations,” I use one umbrella term – “anthropological foundations” – because all of them account for one thing: how mathematics manages to thrive as a complex array of practices carried out by human beings. Now classical foundations of mathematics are often discarded as “speculative foundations.” They are found guilty of being “wholly detached from the discipline itself” (Rodin 2014, 74, 99, 113–114, 204),108 i.e., having almost nothing to do with the keen interests and real worries of the so-called working mathematicians. They are also accused of a vain attempt at introducing some external foundation for mathematics to provide its certainty. Classical foundations tried to secure the grand mathematical edifice with the help of logic, philosophy, or some other extramathematical means. These foundations tended to be prescriptive rather than descriptive to actual mathematical practices. Their further development is seen to be safely recognized as a highly specialized field of expertise within mathematics or logic, provided that they free themselves from “paraphilosophical” pretensions. The alternative vision of the foundations of mathematics craves to be free of the faults of the classical ones. It splits foundational concerns into two strata: internal (“practical foundations”) and external (“anthropological foundations”). However, the split does not presuppose that there is no permanent interplay between the strata. Practical foundations are supposed to be flex-

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Rodin manifestly follows in the footsteps of William Lawvere (cf. Lawvere 2003, 213).

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ible enough to readily meet the historically changing and diverse concerns of the working mathematicians. The unification tendency ought to be balanced against the pluralistic one. Anthropological foundations – though their study in philosophy and social or other sciences is predominantly descriptive – not only makes mathematical activity possible but may also be and indeed is rather suggestive as far as new mathematical ideas are concerned. Is the time ripe to announce the arrival of a new paradigm?

Acknowledgments I would like to thank an anonymous reviewer for his/her stimulating comments and constructive suggestions that helped me to improve this paper substantially.

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Livingston, Eric. “The Disciplinarity of Mathematical Practice.” Journal of Humanistic Mathematics 5, no. 1 (January 2015): 198–222. https://doi.org/10.5642/jhummath .201501.12. Lolli, Gabriele. “Why Mathematicians Do Not Love Logic.” Workshop “Linguaggio, verità e storia in matematica,” Mussomeli (CL), 9 febbraio 2008. http://homepage.sns .it/lolli/articoli/Lolli.pdf (access: 29.05.2020). Lorenz, Konrad. “Kants Lehre vom Apriorischen im Lichte gegenwärtiger Biologie.” Blätter für Deutsche Philosophie: Zeitschrift der Deutschen Philosophischen Gesellschaft 15 (1941/42): 94–125. Lynch, Michael. Scientific Practice and Ordinary Action: Ethnomethodology and Social Studies of Science. New York: Cambridge University Press, 1993. MacKenzie, Donald. Mechanizing Proof: Computing, Risk, and Trust. Cambridge, MA: The MIT Press, 2001. MacKenzie, Donald. “Computers and the Sociology of Mathematical Proof.” In New Trends in the History and Philosophy of Mathematics, edited by Tinne Hoff Kjeldsen, StigAndur Pedersen, and Lisa Mariane Sonne-Hansen, 67–86. Odense: University Press of Southern Denmark, 2004. MacKenzie, Donald. “Computing and the Cultures of Proving.” Philosophical Transactions of the Royal Society, Series A 363 (2005): 2335–347. https://doi.org/10.1098/rsta .2005.1649. Mac Lane, Saunders. “Mathematical Logic is Neither Foundation Nor Philosophy.” Philosophia Mathematica (Series II) 1, no. 1/2 (1986): 3–13. https://doi.org/10.1093/ philmat/s2-1.1-2.3. Mac Lane, Saunders. Mathematics, Form and Function. New York: Springer-Verlag, 1986. Maddy, Penelope. “What Do We Want a Foundation to Do? Comparing Set-theoretic, Category-theoretic, and Univalent Approaches.” In Reflections on Foundations: Univalent Foundations, Set Theory and General Thoughts, edited by Stefania Centrone, Deborah Kant, and Deniz Sarikaya, 293–311. Cham: Springer International Publishing, 2019. https://doi.org/10.1007/978-3-030-15655-8_13. Manin, Yuri I. “Georg Cantor and His Heritage [2004].” In Yuri I. Manin, Mathematics as Metaphor: Selected Essays, 45–54. Providence, RI: American Mathematical Society, 2007. Marquis, Jean-Pierre. “Categorical Foundations of Mathematics or How to Provide Foundations for Abstract Mathematics.” The Review of Symbolic Logic 6, no. 1 (March 2013): 51–75. https://doi.org/10.1017/S1755020312000147. Martínez-Ordaz, María del Rosario, and Luis Estrada-González, eds. Beyond Toleration? Inconsistency and Pluralism in the Empirical Sciences. HUMANA.MENTE Journal of Philosophical Studies 10, no. 32 (August 2017): iii–vi, 1–245 (a special issue).

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McGuinness, Brian, ed. Ludwig Wittgenstein and the Vienna Circle: Conversations recorded by Friedrich Waismann, transl. Joachim Schulte and Brian McGuinness. Oxford: Blackwell Publishing, 1979. McGuinness, Brian. “Freud and Wittgenstein.” In Wittgenstein and His Times, edited by Brian McGuinness, 27–43. Chicago: University of Chicago Press, 1982. McLarty, Colin. “The Last Mathematician from Hilbert’s Göttingen: Saunders Mac Lane as Philosopher of Mathematics.” The British Journal for the Philosophy of Science 58, no. 1 (March 2007): 77–112. https://doi.org/10.1093/bjps/axl030. McLarty, Colin. “Foundations as Truths which Organize Mathematics.” The Review of Symbolic Logic 6, no. 1 (March 2013): 76–86. https://doi.org/10.1017/ S1755020312000159. Meheus, Joke, ed. Inconsistency in Science. Dordrecht: Kluwer Academic Publishers, 2002. Monk, Ray. Ludwig Wittgenstein: the Duty of Genius. New York: Penguin Books, 1991. Mortensen, Chris. “Inconsistent Mathematics [1996, substantive revision 2017].” Stanford Encyclopedia of Philosophy, principal editor Edward N. Zalta. https://plato .stanford.edu/entries/mathematics-inconsistent/. Murphy, Peter. “Coherentism in Epistemology [2006].” In Internet Encyclopedia of Philosophy, edited by James Fieser and Bradley Dowden. https://www.iep.utm.edu/ coherent/. Neurath, Otto. “Anti-Spengler [1921].” In Otto Neurath, Empiricism and Sociology, edited by Marie Neurath and Robert S. Cohen, 158–213. Dordrecht: D. Reidel Publishing Company, 1973. Neurath, Otto. Philosophical Papers 1913–1946, edited and transl. by Robert S. Cohen and Marie Neurath. Dordrecht: D. Reidel Publishing Company, 1983. Olsson, Erik. “Coherentist Theories of Epistemic Justification [2003, substantive revision 2017].” Stanford Encyclopedia of Philosophy, principal editor Edward N. Zalta. https://plato.stanford.edu/entries/justep-coherence/#JusCohScrLewBonCon (access: 29.05.2020). Parsons, Charles. “Platonism and Mathematical Intuition in Kurt Gödel’s Thought.” The Bulletin of Symbolic Logic 1, no. 1 (March 1995): 44–74. https://doi.org/10.2307/ 420946. Parsons, Charles. “Hao Wang as Philosopher and Interpreter of Gödel.” Philosophia Mathematica (Series III) 6, no. 1 (February 1998): 3–24. https://doi.org/10.1093/ philmat/6.1.3. Peirce, Charles Sanders. “Some Consequences of Four Incapacities.” Journal of Speculative Philosophy 2 (1868): 140–57. Plato. Complete Works, edited by John M. Cooper. Indianapolis: Hackett Publishing Company, 1997.

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Schubring, Gert. Conflicts between Generalization, Rigor, and Intuition: Number Concepts Underlying the Development of Analysis in 17–19th Century France and Germany. New York: Springer, 2005. Sen, Amartya. “Sraffa, Wittgenstein, and Gramsci.” Journal of Economic Literature 41, no. 4 (December 2003): 1240–255. https://doi.org/10.1257/002205103771800022. Shapiro, Stewart. Foundations without Foundationalism: A Case for Second-order Logic. Oxford: Clarendon Press, 1991. Shaposhnikov, Vladislav A. “The Applicability Problem and a Naturalistic Perspective on Mathematics.” In Philosophy, Mathematics, Linguistics: Aspects of Interaction. Proceedings of the International Scientific Conference: St. Petersburg, April 21–25, 2014, edited by Grigori Mints and Oleg B. Prosorov, 185–97. Saint Petersburg: Euler International Mathematical Institute, 2014. Shaposhnikov, Vladislav A. “Theological Underpinnings of the Modern Philosophy of Mathematics. Part I: Mathematics Absolutized.” Studies in Logic, Grammar and Rhetoric 44(57) (2016): 31–54. https://doi.org/10.1515/slgr-2016-0003. Shaposhnikov, Vladislav A. “Distributed Cognition and Mathematical Practice in the Digital Society: From Formalized Proofs to Revisited Foundations.” Epistemology and Philosophy of Science 55, no. 4, 2018: 160–73 (in Russian). https://doi.org/10 .5840/eps201855474. Shaposhnikov, Vladislav A. “Towards Open Mathematics: The Transformation of the Practice of Mathematical Proof from Individual to Socio-digital.” Bulletin of Moscow University, Series 7 ‘Philosophy’, no. 1 (February 2019): 79–94 (in Russian). Shaposhnikov, Vladislav A. “To Outdo Kuhn: On Some Prerequisites for Treating the Computer Revolution as a Revolution in Mathematics.” Epistemology and Philosophy of Science 56, no. 3, 2019: 169–85 (in Russian). https://doi.org/10.5840/ eps201956357. Shklar, Judith N. “Squaring the Hermeneutic Circle.” Social Research 53, no. 3 (Autumn 1986): 449–73. Sienkiewicz, Stefan. Five Modes of Scepticism: Sextus Empiricus and the Agrippan Modes. Oxford: Oxford University Press, 2019. Sokuler, Zinaida A. Ludwig Wittgenstein and His Place in 20th Century Philosophy. Dolgoprudny: Allegro-Press, 1994 (in Russian). Soler, Léna, Sjoerd Zwart, Vincent Israel-Jost, and Michael Lynch. “Introduction.” In Science After the Practice Turn in the Philosophy, History, and Social Studies of Science, edited by Léna Soler, Sjoerd Zwart, Vincent Israel-Jost, and Michael Lynch, 1–43. New York: Routledge, 2014. Streicher, Thomas. “[Book Review:] Practical Foundations of Mathematics, Paul Taylor, Cambridge University Press, 1999.” Science of Computer Programming 38, no. 1–3 (August 2000): 155–57. https://doi.org/10.1016/S0167-6423(00)00009-5.

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Chapter 2

Is There an Absolute Mathematical Reality? Zbigniew Król

Abstract This paper analyses mathematical Platonism in a dualistic version, reconstructible from the Gibbs lecture of Kurt Gödel, that assumes the existence of an absolute, ideal mathematical reality together with its basic object, the absolute set, construed as expressible in a single consistent theory – namely, that of proper mathematics. It argues that this position is contradicted by phenomenological data, current mathematical knowledge, and mathematical practice. Several plausible versions of phenomenological Platonism are given as alternatives to Gödel’s extreme position. From a pragmatic and phenomenological point of view, embracing the notion of absolute mathematical reality obliges one to accept a version of mathematics that is impoverished, both conceptually and in respect of its content.

Keywords mathematical Platonism – ideal mathematical reality – absolute mathematical worlds – Gödel – phenomenology – insight – mathematical intuition – universal language of mathematics – mathematical semantics – alter-theories

1

Preliminary Remarks

In this paper, I consider the problem of the possibility of the existence of an absolute mathematical reality, whose mode of existence would be different from that of the real world in that it would be non-temporal, ideal and unchanging, and, along with this, the idea of its basic component, i.e. the absolute set. Acknowledging the existence of these means accepting some kind of dualistic mathematical Platonism.1 I shall initally seek here to explain the

1 This kind of mathematical Platonism will be different from monistic Platonism, i.e. the kind of Platonism in which one accepts the real existence of mathematical objects, but holds

© Zbigniew Król 2021 | DOI:10.1163/9789004445956_004

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concepts of “absolute reality” and “absolute set” with reference to the description of Kurt Gödel, before further specifying them in Sections II and III below. The reference to Gödel seems justified, given that he derives clear mathematical consequences from this view, as well as inquiring into the matter of its conformity with mathematical knowledge. Assuming Gödel’s conceptual and philosophical apparatus as a starting premise, my intention is to try to show that traditional dualistic mathematical Platonism, which is chiefly defined by the concepts of absolute mathematical reality and an absolute set, is unsustainable – for both philosophicophenomenological and mathematical reasons. In particular, it seems implausible and inconsistent when one takes into consideration certain more recent mathematical achievements. Thus, the present analysis will be undertaken in a way that begins from Gödel’s own postulated inner philosophical and mathematical world, at least as presented in his Gibbs lecture (see: Gödel 1951). Of course, the most frequently studied topic, also in Polish literature, was the philosophical consequences of Gödel’s famous theorems (cf. e.g. Krajewski 2003). There are, however, some other variants of this position that can be adhered to, which I shall also point to in this paper. In particular, I shall also consider the possibility of the existence of a plurality of absolute worlds, pointing out the phenomenological and practical limitations of this conception.2 In the course of pursuing these goals, I hope to bring a new perspective to bear on some longstanding issues. The argument here will be developed on two levels, which at some stages, at least, are not easily separated out in any completely unambiguous way: one philosophical, the other mathematics-related. Because of this, the reporting

that they exist in the same way as other objects postulated by some scientific theory. For instance, Quine’s Platonism accepts the existence of mathematical objects indispensable to the description of physical reality. Thus numbers, algebras, sets, etc., exist in the same way as electrons, atoms, forces, or energy. Such objects posited as existing in line with a scientific theory are typically indicated by means of his quantificational criteria of existence. In the context of such Platonism, the problem of epistemic access usually reduces to the problem of epistemic access to sensory reality (cf. also Maddy 1990). 2 The issues of Platonism in the context of mathematical practice, Gödel’s position, and abstract objects in mathematics, have been extensively discussed in the literature. Compare, for instance: the special issue of Philosophia Mathematica, volume 14, issue 2, June 2006, “Kurt Gödel (1906–1978) on Logic and Mathematics”; Kennedy 2015; Atten 2015; Linnebo, Pettigrew 2011; McLarty 2012; Panza, Sereni 2013; Baaz et al. 2011; McEvoy 2012; Parsons 2010; Mancosu 2008; Schlimm 2013; Pettigrew 2008; Parsons 2008; Kerkhove 2008; Callard 2007; Tieszen 2006; Davies 2005; Carter 2004; Restall 2003; Beall 1999; Tieszen 1994.

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of research here, and hence also the correlative structure of the text itself, do not fit readily into a strictly linear sequence: to properly grasp the argument, one must therefore have read the whole text. More specifically, in Section III, I shall appeal in general terms to data that draws on phenomenological approaches. Section IV, meanwhile, is mainly devoted to formal-ontological considerations, which involve certain mathematical findings, including those concerning mathematical practice (pragmatics). In that section, we shall also have recourse to phenomenological data, but in the specific context of the mathematical findings presented there. An analysis of the problem from the point of view of phenomenology is, in my opinion, both interesting and necessary, and gains further motivation from Gödel’s own deep interest in phenomenology, his references to the role of insight, and his search to uncover new axioms of set theory through phenomenology. In addition, the phenomenology of mathematics provides answers to difficulties that arise in the context of naturalist and causal epistemologies – difficulties associated with an alleged lack of cognitive access to (ideal) “platonic mathematical reality”.3 In this paper I seek to show that phenomenology can explain how one gains intellectual and instant access to a non-sensory reality (taking for granted, of course, some rudimentary familiarity with the main ideas of phenomenology, such as ought to be obligatory for all philosophers), as well as why the subject matter of mathematics is given to the subject of mathematical knowledge as possessing some properties – even infinitely many ones – independently of the will of the subject. For many years now, a renaissance of sorts has been underway, reflected in the growing presence of phenomenology in the philosophy of mathematics.4 Therefore, I shall presume to think that even conclusions touching merely on the plausibility of this kind of Platonism as seen from the viewpoint of phenomenology will be of potential interest to readers – even those not well acquainted with phenomenological methods or, in the worst case, hostile to them. For the purposes of the present analysis, I shall therefore deploy a (hypothetical) interpretation of Gödel’s view regarding the essence of mathematics, which, in my opinion, is to be found (at least) in (Gödel 1951). My intention is not to pursue matters of interpretative exegesis: rather, I shall confine my pre-

3 Cf. the debate prompted by the work of Benacerraf (1965) and Benacerraf (1975), and, for example, Chihara (1982), Pataut (2017), Brown (2012) and Calderon (1996). 4 Regarding the presence of phenomenology in the philosophy of mathematics, see the exemplary work found in: Tragesser 1972; Føllesdal 2016; Hopkins 2016; Hartimo 2015; Hill, Silva 2013; Hartimo 2012; Atten 2006; Leng 2002, as well as the works of R. L. Tieszen, H. Wang, etc., listed in the References section at the end of this article.

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sentation of Gödel’s views to the points explicitly formulated by him in that lecture, and to reminding the reader of the basic topography of Gödel’s philosophical world. Even so, I should add that I think that the concepts of “absolute set” and “absolute mathematical reality” are worth considering for their own sake, not only for reasons that are strictly a priori and philosophical, but also because they possess importance from a historical point of view inasmuch as they pertain to the possibility of the existence of some forms of mathematical Platonism that are, and certainly have been, quite frequently endorsed in the philosophy of mathematics.5

2

Formulation of the Problem

The title of the present article undoubtedly requires some additional clarification before we can proceed further. What, after all, we may once again ask, is “absolute mathematical reality”, and what is the “absolute concept of set” that, as we shall in due course see, turns out to be associated with it? The first thing to say is that in using this term I have in mind the idea of Gödel himself, present in several of his texts, but in particular in the Gibbs lecture itself (see: Gödel 1951). This concerns the view holding that what mathematicians are engaged in is investigating a world of absolute non-sensory objects (i.e. ones that exist objectively, independently of our consciousness and knowledge, and which exhibit a different kind of existence from sensory reality).6 According to Gödel, the basic mathematical concept is the concept of a set.7 “Proper mathematics” (Gödel’s term as defined in: Gödel 1951) is made up of just all those propositions that hold true always of absolute mathematical reality. The truth of proper mathematical propositions consists in the veracity of those propositions, and not in some model or other – coming instead as it does

5 See, for instance, the well-known work of Hardy (1940). Cf. also Król (2006), in which I provide a list of sources and of supporters of this type of Platonism. It should be added that several dozen types of mathematical Platonism are distinguished there. 6 Cf., for instance, the following quotation from Gödel (1951): “Thereby I mean the view that mathematics describes a non-sensual [sic] reality, which exists independently both of the acts and [of] the dispositions of the human mind and is only perceived very incompletely by the human mind” (ibid., 323). 7 In the classical mathematics embraced by Gödel, “(…) all of mathematics is reducible to abstract set theory” (ibid., 305–6). “… [T]he consistency of the axioms of set theory (…) is self-evident (and therefore can be dropped as a hypothesis) if set theory is considered to be mathematics proper” (ibid., 307, note 7). Such a view is frequently espoused as the basis of so-called “set-theoretical Platonism”.

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from correspondence with absolute mathematical reality. Gödel, of course, realizes that there are many different models, and knows about the relativization of concepts in formalized mathematics, and so on. However, he himself is concerned with true propositions in the absolute sense – as tools for describing absolute mathematical reality.8 The multiplicity of different mathematical theories and models available does not prevent him from adhering to the idea of absolute mathematical truth, because – as with physical theories – that multiplicity need not entail a plurality of corresponding worlds, or that reality itself is somehow contradictory or undefined; rather, it indicates only that our descriptions and knowledge are limited. Gödel clearly states that there are, indeed, propositions that are absolutely true (i.e. true always and everywhere) – and, moreover, he gives examples of these. The “absolute concept of set” is a concept that refers to a “set in itself” or, rather, to a universe composed of such objects: objects that exist and exhibit properties and mutual interrelations independently of our knowledge, sensory experience and cognitive acts, and which are the basic elements (or material) of absolute mathematical reality. To sum up: Gödel’s postulated “inexhaustibility” of proper mathematics results from the limitations of its linguistic description. Mathematical reality surpasses any such verbal or linguistic description. The truth of the propositions of proper mathematics comes from insight: i.e. direct intellectual observation or seeing, construed as apprehension of or contact with an inexhaustible (linguistically and axiomatically) mathematical reality.9 Phenomenology thus emerges as a natural candidate for explaining how this is possible. Gödel constructed this state of affairs on the basis of his investigations into natural-number arithmetic, where the set of all true propositions in the stan8 The propositions of mathematics in formalised hypothetico-deductive systems have only conditional, hypothetical veracity. The propositions of “proper mathematics”, though, have absolute veracity: i.e. independently of any concrete formalization or theory and on the basis of insights given directly. Therefore, the propositions are independent of each other, or can be reduced to a subset of such independent propositions (see alter-theories as discussed in the final section here). Such propositions must always exist and are exemplified by propositions of the type “If such and such axioms are assumed, then such and such a theorem holds” (Gödel 1951, 305), or by individual propositions belonging to a finitist theory of natural numbers (for example “2 + 2 = 4”). The same proposition can be a part of “proper mathematics” and, say, Peano arithmetic (PA). However, in the latter instance it will only possess hypothetical veracity. 9 “For if he perceives the axioms under consideration to be correct, he also perceives (with the same certainty) that they are consistent. Hence he has a mathematical insight not derivable from the axioms” (ibid., 309). “The truth, I believe, is that these concepts form an objective reality of their own, which we cannot create or change, but only perceive and describe” (ibid., 320).

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dard model (complete arithmetic) is inexhaustible, as evidenced by any of the possible first-order axiomatic systems containing Peano arithmetic (PA). We are now in a position to give a more precise formulation of the issue under consideration, and to present the basic mathematical implications of this ontological standpoint as identified chiefly by Gödel. At the outset, let us ask about two matters: 1. Is it plausible to accept, for philosophical and mathematical reasons, the existence of an absolute (unchangeable, non-temporal, ideal) mathematical reality, that is “platonic” in the sense that it is independent of any acts of the subject practising mathematics, and has a different way of existing from the sensorily-perceived reality of everything that we call the “external world”? 2. Is it possible and plausible to consider the (absolute) concept of a set as a basic ontic element, and as a basic concept describing this reality? It seems that where proper mathematics is concerned, the object of our investigation should, in addition to the above, exhibit “extra-linguistic consistency” as one of its features. As we have seen, for Gödel the positing of consistency need not take on a linguistic form: this is evidenced by formalised systems, in that insight into the truth of axioms is at the same time a guarantee that they are not contradictory. As a feature pertaining to the “inexhaustibility” of mathematics, there remains – from the point of view of the linguistic aspect – the incompleteness of the axioms of proper mathematics. There are two ways to extend the initial set of axioms. In the case of the first one, the axiomata are simply independent, so it is possible to add both a given axiom (“independent proposition”), as well as its negation. This is the case, for example, with the continuum hypothesis. Due to the different conceptions of sethood and the lack of a standard model for ZF(C), this proposition is true only in some models. According to Gödel, such a situation indicates that the initial system of axioms does not describe proper mathematics, and “proper” axioms are still waiting to be discovered, so that we are, in effect, still engaged in looking for more general axioms or some new axioms.10 Gödel therefore excludes the so-called splits: i.e. the possibility of building proper mathematics in two or more alternative and mutually contradictory directions. The second possibility concerns propositions that, admittedly, are formally independent of the axioms of the initial theory (in the sense that they are “undemonstrable”, unprovable), but where some given independent sentence is at the same time true of absolute mathematical reality (in the standard model), and is

10

Cf. e.g. Gödel 1947/1987, Gödel 1944/87.

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so on the basis of an insight. If the propositions of the finitary arithmetic of natural numbers are to be regarded as propositions of proper mathematics, then this is the case, for example, with Gödel’s independent proposition for a Peano arithmetic of natural numbers (PA), formulated in the PA-language. Such an independent sentence, but only it, and not its negation, is a true proposition on the basis of insight and is true in the standard model because it has an arithmetical interpretation. This proposition, although independent of other demonstrable propositions of PA, does not allow for the inclusion of its negation without changing the PA model at the same time. The model therefore contains more information than a merely syntactically defined and consistent theory would. The same holds true for the proposition stating the existence of a model for PA, which is equivalent (under Gödel’s theorem of completeness and Gödel’s second theorem) to asserting the consistency of PA. This proposition (i.e. Gödel’s independent statement), however, is not directly formulable in the language of PA, and counts as true on the basis of an insight that leads us to consider this proposition true and not its negation. The proposition also has an arithmetical interpretation (cf. the encoding method used by Gödel – so-called Gödel numbering) and, for this reason, furnishes a true proposition in the standard PA model on the basis of an insight into the truth of a proposition equivalent to it stating the consistency of PA. This sentence cannot, however, be added consistently to PA (cf. Gödel’s second theorem). Propositions of this type, i.e. true without proof, are propositions of the complete theory of arithmetic, meaning that they belong to the set of all true propositions in the standard model of arithmetic. Absolute mathematical reality therefore has the features of a “super-model” for all mathematics. This indicates that proper mathematics should have a standard model. On the other hand, as I mentioned, ZF(C) and other standard set theories do not have such a single standard model. Any currently known theories cannot, therefore, as a whole be partial theories contained in proper mathematics; but, on the other hand, Gödel claims that some of the propositions or axioms of such theories (and thus, probably, some of their sub-theories or fragments) are absolutely true propositions of proper mathematics. It seems, therefore, that the only fragments of current mathematics that can belong to proper mathematics are those fragments of theories that have standard models. Finitistic number theory is a good example here, with the fragment corresponding to it in ZF + (V = L). Gödel is concerned by the fact that the truths of finitistic arithmetic, which he considers truths of proper mathematics, and the truths of set theory – albeit most probably only some of those currently known (cf. his remarks about the continuum hypothesis in other texts (e.g. Gödel 1947/87) – which he also

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counts among proper mathematics, in relation to the possibility of interpreting the theory of natural numbers within set theory, are only superficially related. An expression of this (current) superficiality is the fact of the uselessness of the set-theoretical properties of natural numbers (i.e. natural numbers treated as sets) when it comes to the creation of number theory. Is it possible to answer questions (1) and (2) above affirmatively? Can Gödel’s position be plausibly maintained? What are the difficulties facing this position?

3

Selected Problems and New Theoretical Possibilities Considered from a Philosophical Point of View

Before embarking upon the analyses below, I should like to draw attention to the advantages of the Platonist conception discussed here, which recognizes the existence of absolute reality – and thus the object of investigation of proper mathematics. It explains the objectivity of mathematical knowledge, in the sense of the independence of properties of mathematical objects from the will of the subject who creates mathematical knowledge, and the possibility of intersubjective recognition and communication of results, along with the possibility of surprising mathematicians by stating unexpected properties, and the fact of the same mathematical truths being discovered independently by different mathematicians. Amongst other things, it also offers a rational semantics for mathematical theorizing and explains what, from a phenomenological point of view, is given in acts of consciousness (e.g. insights) on the part of the subject that creates mathematics, and so on. The same advantages are, however, also offered by other Platonist conceptions of mathematics. In particular, it is not necessary to accept the existence of an absolute mathematical reality in order to be counted a mathematical Platonist. For reasons of space, what I present below falls short of a fully elaborated description of a phenomenological inquiry. Instead, my focus will be on just conveying certain ideas and general results that emerge from such an investigation. I shall assume some general knowledge of the methods and conceptual apparatus of phenomenology within the ambit of, for example, Husserl’s Logical Investigations, (Husserl 1900–1), but at the same time will often employ my own intuitive terms, with the aim of simplifying and helping to capture certain phenomenological concepts.11 From the point of view of phe11

Husserl avoids pursuing phenomenological inquiries that pertain specifically to the real or ideal existence of something, or related ontological questions about different ways

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nomenology, it is undoubtedly the case that objects of thoughts are directly given, and that they present themselves to the subject engaged in practising science or mathematics in the context of some conscious acts. For instance, neither quantum mechanics nor linear algebra are sensory objects composed of atoms, electrons, chemical substances, etc., and neither of them has sensory properties as such, yet both are given to the subject of cognition via certain mental or intellectual acts. There are many round things in the real world, but geometric circles or spheres are not (the same as) sensory objects. 3.1 Phenomenological Platonism One could, for example, assume that mathematical reality is given as an intentional correlate (or intentional correlates) of acts of consciousness, but that it does not exist completely independently of those acts. Despite this, these correlates can be (and usually are) given as ideally (i.e. perfectly) identical in numerically distinct acts of a cognizing consciousness. For example, one can grasp the same object via sensory perception or through a recollection of it, or one can just imagine it. The acts are different, yet they correspond to the same object. Mathematicians can therefore know exactly the same thing in an independent way, and in numerically separate acts or experiences of consciousness.12 Then it is impossible to determine whether such objects are plural, or if we are dealing with a single and absolute such object. Hence, such objects are also referred to as “universals”, in that they do not exist individually. In particular, this also applies across multiple acts of the same subject: the mathematician can return to the course of his/her thoughts after some time has passed, recognizing them as having just “the same” object as his/her previous considerations. Moreover, from a phenomenological point of view, such a connection to already discovered properties, grasped intellectually, occurs not in the form of a cognitive act relating to an object completely independent of us, but as an act of recognition of the content and object of this act as be-

12

of existing or modes of existence. However, in classical phenomenology it is possible to consider such questions from the point of view of a priori formal ontology – without any assumption of, say, a Meinongian-style metaphysics. Therefore, I take it upon myself to consider the consistency of the phenomenological data with the hypothesis of there being an absolute mathematical reality. Cf. also the difference between ontology and metaphysics in Ingarden’s phenomenology (Ingarden 2013–16). Obviously, the concept of identity with regard to acts calls for a detailed and non-trivial explanation. When, and how, if at all, can we conclude that something can be given identically in certain acts that are of different quality, or even only numerically different? In this respect, I would like to refer the interested reader to Husserl’s Logical Investigations, passim: e.g. “Investigation V”, Chapter III.

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ing themselves a result of some previous instance of one’s being engaged with them: i.e. as something significantly connected with some other act(s) or conscious activity. The correlates of such objectifying acts of consciousness may therefore be the same at an ideal level for each subject and act. These features make possible the reduction of each individual consciousness to a so-called “transcendental consciousness”. Nevertheless, the internal consistency of such a grasp requires the fulfilment of certain cognitive conditions pertaining to what I call the “reconstruction of the hermeneutical horizon”,13 not to mention others, about which Husserl writes in the Logical Investigations.14 In epistemological terms (phenomenologically construed), absolute Platonism and “phenomenological Platonism” are indistinguishable both at the level of pure phenomenology and with regard to certain other correlative features, with full phenomenological analysis and (as we will see in the next section) formal-ontological analysis both preferring a phenomenological position. The two positions are, however, different with regard to their ontological presuppositions. It should be added that the basic problem, whose analysis Gödel omits, is that absolute (ideal) objects are not individual objects but general ones (universals or essences bereft of individual properties), so their manner of existence must be different from that in which individual sensory (physical, material) reality exists. 3.2 Moderate Phenomenological Platonism Apart from phenomenological Platonism, another alternative is to be found in a complementary variant of absolute Platonism: namely, moderate phenomenological Platonism. This view posits that mathematical reality does exist, admitting that it does so partly independently of acts of consciousness, but only to the extent that these acts discern it. Mathematical reality arises (and reveals itself) together with the act(s) of consciousness of which it is a correlate. Two variants are possible here. In the first (III.2A.), mathematical reality continues its existence independently of the act of consciousness that initiates it, while in the second (III.2B), it exists only when “brought to life” in the context of some act by virtue of being an intentional correlate of that act, and “perishes” when not the object of any act. The second position (III.2B) is unsustainable, if only because each product of consciousness is intrinsically available for subsequent acts of consciousness: e.g. for higher-order acts supervening or otherwise depending upon a given act, such as recalling, imagining,

13 14

See Król 2015. Cf. footnote 7 above.

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or judging. Such a product only exists by virtue of doing so in the stream of consciousness: the act ceases to exist, but the correlate is supposed to somehow remain present and accessible within the stream of consciousness for other acts and intellectual operations – e.g. in a recollection. Moreover, the record of the results of individual instances of engaging with something in the context of some act are also available to other subjects of consciousness, if only as a product of culture, and so continue to exist. For example, the cultural (and intentional) product entitled “Chopin’s Mazurka in C sharp minor Op. 63 No. 3” continues to exist, even though there are no acts anymore – not even the subject himself who called this intentional object into existence. However, how and in what way such “culturally autonomized” intentional objects exist is a matter for clarification. This can be accomplished by referring to the structure of the phenomenological consciousness of the act. Undoubtedly, the existence of phenomenological objects (correlates of the acts) is, in cultural terms, a phenomenological fact; hence, the problem itself, of whether and by virtue of what such objects exist completely independently of any act of consciousness and any subject (i.e. stream of consciousness), becomes an issue to be addressed in the context of the analysis of the acts of consciousness of a given community of subjects. We may ask, for instance, whether such an object would continue to exist if it were the case that some record of information about such an intentional object contained in a certain “physical representation” of it (e.g. musical notation, as in a score) were not able to be “brought to life” in the context of the act of at least one subject. It seems that it would not exist, because the essence of the intentional object is that it is a correlate of some act of consciousness, and the physical representations themselves are, as a rule, completely different real objects, which do not have any properties identical to the objects that are represented by the intentional objects. Mathematical and scientific objects are also cultural products. If, on the other hand, it were thought that the essence of intentional objects lay in the supposed fact that when the consciousness of acts appears they may become correlates of those acts, then, from the phenomenological perspective, an absolute mathematical reality could be said to exist. This would be similar to the situation with sensory reality: the fact that of that reality’s being seen, heard, etc., is not itself the basis for asserting its existence, yet its essence does lie in the possibility that it possesses of revealing itself in sensory experience. So, to sum up, I would say here that the fact that the object of mathematical investigations is given to us in related and consecutive acts of consciousness, but only as having been previously disclosed in just those (antecedent) acts, makes position III.2A preferable – the position stating that, in phenomeno-

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logical terms, mathematical reality is not completely independent of acts of consciousness, and continues to exist after our first intentional engagement in the context of some act of consciousness. On the other hand, the necessary a priori fact that each act of intentional engagement construes an object (correlate) that already at the moment of this engagement possesses more features (cf. the explanation below) than are grasped and realized explicitly indicates that mathematical reality, from the moment of the initial encounter in some act or other, also possesses some features in an absolute way. The two variants just discussed – both III.2A and the initially eliminated III.2B – can, as is already evident from the abovementioned considerations, be combined with other ones (III.2A.1, III.2A.2, III.2B.1 and III.2B.2), in which, additionally, we state either that a correlate is only that which is actively given, or that it, together with features the subject realizes explicitly, as an a priori necessary fact (always or sometimes) exhibits more features related to the initial features – ones whose subject is not (yet) actively, i.e. explicitly, conscious. An example of this situation could be a statement of a definition or description of a concept or object – say, a well-founded relation, and then a subsequent investigation of the properties of this concept and its relations to other concepts or objects. It is precisely the statement of this definition (i.e. the “assignment”) of certain objects, together with certain other conditions (both realized and unrealized by the subject), that causes certain other relations and dependencies, not yet explicitly assigned by any conscious (fully aware) operations which exist objectively (i.e. in a manner that can be disclosed phenomenologically, logically, etc.) between these objects: i.e. those still to be discovered and consciously grasped. In this sense, each intellectual opening up of a new mathematical situation simultaneously opens up an entire, undiscovered and unrealized world of new (a priori) mathematical dependencies. As an example, let us take some statement offering a definition of a square: i.e. an intellectual grasp of this object as a flat polygon that has four identical angles and four identical sides. This allows for further investigation and for the discovery (i.e. arrival at a conscious realization by the subject) of its additional properties: i.e. that it has certain properties of symmetry, that the sum of its internal angles is equal to the sum of four right angles, that it can be divided in several ways into triangles, that it can be inscribed in a circle, that a circle can be inscribed in it, etc. This phenomenon explains why the mathematician feels so strongly that he or she is only the discoverer of a hitherto unknown world, which he or she comes to know and investigates. The question of the manner of mathematical reality’s existence is therefore reduced to the question of the mode of existence of each and every new opening up of a research problem in mathematics, and each such opening up engenders the appearance of an infi-

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nite number of further, already defined and objectively existing dependencies holding between the multifarious “worlds of mathematics” – i.e. the various objects, theories and even practitioners of mathematics located in different historical epochs. It looks, moreover, as if the creation of a correlate in the context of a certain act of consciousness is connected with the excessive assigning of certain objective properties, which each time exceed the intention of a given act. The act discloses only a part of the field of objectively existing a priori relations. Here, too, Gödel’s position and those discussed above converge almost completely, and indeed, they would be indistinguishable were it not for Gödel’s commitment to the global consistency of absolute mathematical reality. From the phenomenological point of view, the grasping of an object with its excessive properties need not be noncontradictory relative to other previously grasped objects. In each such opening and “assignment” of the object conceived (i.e. grasped in the act of consciousness), there will be areas of underdetermination corresponding to alternative and mutually contradictory but absolutely possible a priori variants of its defining features: i.e. to the previously mentioned “splits”, excluded by Gödel.15 These observations once again serve to eliminate the variants from group III.2.B, and to confirm or render preferable variant III.2A.2. The fact that the mathematical reality available to a human being in this excess sphere is always connected with some act of consciousness that “assigns it” also speaks against absolute mathematical reality as a basis for phenomenological experience. Both historically and substantively, such an excess pertains to every independent act, no matter whether it concerns what Gödel himself designated as the object of investigation of proper mathematics or any other form of human mathematics. Such a universal excess in respect of features of the correlates of acts indicates that to each object grasped in an act there would have to correspond a property of absoluteness. That, in turn, would run contrary to Gödel’s own conception and intentions. While the issue of absolute reality certainly requires further investigation, the phenomenological data do seem to point in the direction of a falsification of the conception of absolute mathematical reality. Let us therefore try to consider whether it is possible to resolve the issue through formal-ontological means, using, instead, data provided by mathematics itself in the context of the phenomenological findings just outlined.

15

It seems that pure essences (universals) can be embellished with some less general or lower-level properties. However, these secondary and less general properties are not individual properties, so the object still remains a universal.

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The Absolute Concept of Set as Reflected in Mathematics

Gödel’s fundamental intention was to discuss the ontology of mathematics in the light of mathematical facts and theorems. At this point, I shall confine myself to those remarks which indicate that mathematics itself to a certain significant extent falsifies absolute Platonism: i.e. those concerned with the absolute concept of set and mathematical practice. This has the consequence that if one wishes to remain in line with current mathematical knowledge, then one cannot give an affirmative answer to the two questions raised in the second section. What mathematics is is revealed not only in its concrete results, theories and theorems, but also in how we practise this discipline and in the process of its historical development, as these also serve to disclose what mathematics can be. Let us recall that absolute mathematical reality exists independently of our knowledge of it, just like the real world. Mathematics itself is a group of theories whose role is to describe this reality, while proper mathematics is a grouping together of just those theorems that have succeeded in capturing absolute mathematical reality, so that they adequately describe it, stating its absolute properties in the form of unconditional, temporally transcendent truths. But is the absolute truth of a theory, or more broadly of knowledge, guaranteed by the absolute mode of existence of the object with which it is concerned? One may reasonably ask whether the fact of the objective existence of the world ensures, or at least enables, the existence of absolute knowledge about it. Has man created at least one scientific theory to date, which is absolutely true? It seems not. Newtonian Mechanics, Quantum Mechanics, the General Theory of Relativity, and all the other scientific theories offered by naturalists are, at best, only partially true: i.e. they may contain some propositions that adequately capture properties of the world. We know that none of the currently existing scientific theories in physics can be true in their totality, because they are mutually contradictory: i.e. they postulate contradictory features of the world. For example, quantum mechanics postulates the discontinuity of time and space (something like “space-time foam”), and the General Theory of Relativity their continuity. Yet how could we state any propositions that we take to be absolutely true independently of such theories? What means do we have for this purpose? Perhaps the situation is different (and less hopeless) in the case of mathematics and the absolute world of mathematics, just because it is unchangeable, timeless and ideal, so that it maintains its identity over the course of multiple investigations.

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Gödel appears to hold that after two-and-a-half thousand years of mathematical development, we have reached a point where we can be sure that some propositions of set theory and finitist number theory adequately describe some part of absolute mathematical reality. But on what basis do we state that these propositions – mainly axioms and propositions stating the simplest of properties – are propositions of proper mathematics? Gödel gives his answer not on the basis of evidence, but on that of insight. Yet could it really be the case that all other propositions, including the axioms of “human mathematics”, were accepted without any insight or intuitive basis, in that they, too, were usually accepted without evidence? Analysis of the development of mathematics shows that this was not the case, since every axiom in historical mathematics was established on some basis, and only recently have we departed from intuitive evidence and obviousness by accepting axioms on the basis of certain effects and consequences which cause their adoption.16 Axioms do not have to be obvious, but there must be obvious reasons why we formulate and accept them. Gödel’s conception, therefore, leads to a divergence with respect to matters of insight: some human insights capture absolute reality, and some do not. Does this mean that the latter insights must be false? It seems not, because mathematics, apart from the propositions that Gödel ranked as belonging to proper mathematics, also contains an inexhaustible number of true propositions that say nothing about sets and are not statements about the finite properties of finite (standard) natural numbers. So how can we distinguish, in practice, insights that do present absolute reality directly in an originary kind of way from those that, though correct and counting as also presenting something in an originary manner, nevertheless present an object different from the only absolute object of mathematical investigations? Let us call this latter object, construed as the ordinary object of mathematical investigations, “the object of historical mathematics” – i.e. of “human mathematics”. It should be noted, and it follows from Gödel’s theses, that among the propositions of proper mathematics, all those propositions of human mathematics should also be ranked which, admittedly, do not explicitly contain any terms referring to sets and finitary properties of natural numbers, but which can be reduced to or written in an equivalent form with the use of only these terms: i.e. equivalently translatable or expressible (interpretable) propositions in absolute mathematics. Thus, an ancient mathematician might not have been aware that in formulating certain claims pertaining to the arithmetic of

16

Cf. e.g.: Maddy 1988; Maddy 1990; Maddy 1992; Maddy 2011; Schlimm 2013.

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natural numbers on the basis of human insights, (s)he was speaking about certain sets, although now we can show that most of his/her arithmetical claims are indeed related to the properties of certain sets. But what does that mean? Well, the insights of human mathematics can lead to the same (i.e. equivalent) results as insights into absolute mathematical reality. Thus, absolute truths can also be formulated without any insight into absolute mathematical reality, and what we establish in human mathematics and its insights can be just as true as what proper mathematics brings us to. Owing to the fact that this was the case in historical forms of mathematics, insights in human mathematics may pertain not to sets or natural numbers, but rather to other objects. An example is the treatment of geometrical objects as sets of points. For the first time, such an approach appeared in the work of Bolzano (1804).17 Earlier, it was possible to formulate Euclidean geometry without basing it on the intuition of a set, but nevertheless insights based on these other intuitions are interpretable in set theory and strictly speaking, lead to identical (i.e. equivalent) results. Until the 19th century, no one thought of geometrical formations as being composed of points. Moreover, in the Elements of Euclid, this view does not appear anywhere, and is even ruled out (cf. Król 2015, Part I). No one, however, can deny that the geometry of the Elements is based primarily on a series of intuitive insights. What, then, is given in the insights of human mathematics, and how can we distinguish these human insights from absolute ones? It seems that the only reasonable answer to this question is one provided not by insights and their analysis, but by the state and historical advancement of our human mathematical knowledge. Gödel, it would appear, came to the conclusion that the whole of mathematics is a form of set theory for reasons not based on absolute insights. This is a claim made in, and justified in terms of, human mathematics. Asserting that “everything is a set”, like claims to the effect that “everything is a natural number” or the later one18 asserting that “natural numbers are created and everything else is the work of man”, were formulated as a result of large-scale accomplishments in mathematical knowledge with respect to reduction to set theory and number theory (Dedekind sections, etc.). It was only the development of the foundations of human mathematics, and of its reduction, that made it possible to formulate the view that the concept of a set is a fundamental one, and to give reasons

17 18

For more information, see (Król 2015, 78). I.e. Feferman’s, in which he paraphrases a well-known saying by Kronecker formulated in the wake of the analysis of predicative mathematical systems (cf. Feferman 1969, 95).

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and justification for the view that the truths of set theory belong to proper mathematics. The sheer insight in and of itself, even if absolute, would not have convinced anyone. Another thesis, which seems to stem from Gödel’s views, asserts the following: a) that the whole of mathematics, i.e. all its theories and truths, can be expressed as set-theoretical claims, and b) that there is no theory other than set theory, and no concept other than that of sethood, which possess this property. Moreover, if there were to be two different basic theories in the sense of a) and b), then even if they were equivalent, in the sense that each proposition of the (absolute) set theory could be translated into some proposition of the other theory and vice versa, they would amount to a pair of equivalent theories of absolute mathematics, while being based on different insights. Moreover, it cannot be said that there is no theory different from set theory – even in its absolute version – that is richer than it, in the sense that certain truths of this richer theory are inexpressible in set theory, while all the theses of set theory are expressible in this richer theory. Today’s (human) mathematics can point to category theory as a good candidate for being the sort of theory just referred to in the previous paragraph (cf. Goldblatt 1979, chapter 12.6). From the phenomenological point of view, these two theories are based on different insights. From this perspective, distinguishing set theory as proper mathematics is only an accidental historical circumstance, and not an unquestionable, absolute truth. The development of mathematics and the study of its historical forms corroborate the thesis that there is only human mathematics with its human insights, and that this is a sine qua non for the (hypothetical) existence of anything distinguishable as proper mathematics. Moreover, the conviction that, for example, the mathematics of ancient Greece can be expressed using the theory of sets is in no way justified.19 The fact that certain fragments of ancient mathematics may be interpreted using the theory of sets – e.g. the theories contained in the Elements of Euclid – does not justify the thesis that this is set theory, because it is certainly not the case that all the features of objects considered in ancient mathematics have set-theoretical equivalents. This holds, for instance, for certain intensional properties that are indistinguishable when extensional conceptions of sethood are employed. Similarly, some properties that are very straightforwardly formulable in language, and by means of ancient mathematics, start to look extremely complicated when we try to write them down in contemporary algebraic notation. This applies, for example, to the theory of

19

I offer a precisely worked-out defense of this claim in (Król 2015).

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the substitution of baselines in Book X of Euclid’s Elements.20 Thus, in effect certain insights are only possible if one moves beyond absolute set-theoretical reality. This brings us to my second question. Is there an absolute concept of set? This issue, as it turns out, is related to the question of the possibility (or impossibility) of accepting the idea of an absolute mathematical reality. The absolute concept of set is – at least according to Gödel – a good candidate for “absolute reality”. As we have stated above, proper mathematics, as a set of true propositions about absolute reality, cannot exhibit consistency as a feature in any form that could be written in a formalized language, since every such notation would itself have to be inconsistent, while amongst the propositions of absolute mathematics, we also encounter a proposition stating that there can be no contradiction between propositions (theories) obtained on the basis of a grasp of absolute reality. It seems, however, that a theory containing only absolute protocol statements – i.e. sentences corresponding directly to absolute reality (and not to the absolute properties of proper mathematics) – must be inconsistent (since it is up to proper mathematics to state this contradiction). This also applies to all partial formulations of the theory of proper mathematics, in part because they take into account the linguistic inexhaustibility of that mathematics. Locally, therefore, any subset of the protocol statements of absolute mathematics and the corresponding theory must be consistent. Due to the reference of protocol statements of proper mathematics to absolute mathematical reality, a key part of it will be those protocol statements that are independent of other propositions. We may consider them to be axioms of proper mathematics, and state that they form an independent set. Moreover, due to the aforementioned lack of splits, it should be assumed that absolute mathematical reality is a unitary and non-contradictory affair, albeit linguistically inexhaustible. Accepting the above claims entails consequences that are hard to accept and unrealistic from the point of view of mathematical practice. It is wellknown that there are many different and mutually contradictory conceptions of sethood and the corresponding set theories. There are, for example, extensional and non-extensional conceptions of sethood. The latter use some form of negation or limitation of the axiom of extensionality accepted by extensional conceptions. We are considering and investigating conceptions and theories of well-founded and non-well-founded sets, as well as many other rel-

20

Cf. the preceding footnote above.

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atively non-consistent ones. Gödel’s conception requires either the selection of at least one of these conceptions (or such a subset of them as is consistent) as reflecting the absolute concept of set (limited to at least some extent by inexhaustibility), or the creation of a new set theory of the utmost generality, in which all of the conceptions considered so far are interpretable. Gödel seems to recognise and prefer extensional and founded conceptions of sets. But is it therefore necessary to reject conceptions of cyclic and other non-wellfounded sets? After all, Gödel wanted to arrive at axioms of set theory that would be such that the continuum hypothesis would be either true or false, and not independent, as in the current human version of ZFC – i.e. ZermeloFraenkel set theory with the axiom of choice. (Such a property is exhibited by Gödel’s conception of a universe of sets with the axiom of constructability V = L, because in this theory and model, the continuum hypothesis is true.) It seems impossible, however, to construct such a theory as would contain all of the rich abundance of different possible set theories known to us, where this would often mean taking into account sets displaying contradictory properties. We would always have to reject something as not belonging to proper mathematics, and as not relating to absolute mathematical reality. What, then, is the object of the insights and theorems of these rejected theories of human mathematics? Are they mere linguistic-syntactic formations, and as such, lack their own semantics? There are various ways to deal with the problems just raised. Firstly, such non-standard set theories may have models in standard set theory: e.g. nonwell-founded and other “nonclassical” theories have models that are objects describable in ZFC, in the sense that certain cyclic sets are isomorphic with some well-founded sets.21 Second, non-classical theories can be interpreted in terms of classical set theories (e.g. ZF(C), NBG – i.e. that of von NeumannBernays-Gödel). Third, it may be possible to build a (currently unknown) set theory of sorts that is so general and new that all other theories are interpretable or somehow expressible in it, so that in choosing something as the absolute set theory we lose little from the repository of human mathematical knowledge. Fourth and finally, one can give up the thesis that “everything is an absolute set” and simply acknowledge that each theory and conception creates and investigates its own world of concepts, and that it is impossible to attribute global consistency to the totality of these worlds. There may, therefore, be many absolute worlds, not one. The latter conception corresponds better to accepted mathematical practice, and does not lead to a schizophrenic

21

See: Aczel 1988; Król 2018.

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dichotomy between human and “proper” mathematics. It is also in accordance with moderate phenomenological Platonism, and especially with variant III.2A.2 above, as it is more indicative of the relationship between mathematics and the acts of consciousness that create it. I would, moreover, like to remind the reader that the latter conception also explains the fact that despite the dependence of mathematics on certain acts of consciousness and, inasmuch as not everything in the cultivation and creation of mathematics has the form of acts of consciousness, on certain intellectual activities, our products, or rather correlates of acts, always have more objectively-given properties than we realize at the starting point – properties still waiting to be discovered and brought to conscious realization. Such objects appear as existing in part independently of the acts of intentional encountering that found them. But the conception of a multitude of absolute worlds is also impossible to maintain on the basis of the previously indicated phenomenological data and analogous arguments – as with the case of one absolute world. Even so, I would like to offer some other arguments below aimed at trying to save the conception of a multitude of absolute worlds. At this point, I wish to devote a little more time to the third of the possibilities just considered, which seems to support the idea of a single absolute world. First of all, it is possible to give an example to illustrate the direction in which one must go, and the intuitive insights (i.e. intuitions) which would need to be invoked as the basis, when seeking to construct a known set theory of the utmost generality. Such a theory should, for example, combine extensional theories with non-extensional and intensional ones. In such a theory, it may be possible to prove the continuum hypothesis not as an absolute truth or axiom, but as a theorem – i.e. a proposition that holds true under certain conditions about, for example, form: “sets having such and such properties, which can be expressed in this theory, have a property which is stated by the (general) continuum ((G)CH) hypothesis”, with this not only reflecting one’s acceptance of (G)CH as an additional axiom, but the property in question also resulting – as in the case of Gödel’s axiom of constructibility – from other propositions. Having considered elsewhere the conception of such a set, together with the proposal for an axiomatic notation of its key properties,22 I shall only highlight here some quite basic, informal intuitions. According to extensional conceptions (i.e. those based on the axiom of extensionality), a set is understood as a formation that is completely determined by its “content”: i.e. its elements.

22

Cf.: Król 2010; Król 2015, 297–331.

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Sets that have the same elements are identical and, vice versa, only one set can be created from the given elements. In particular, intensionally distinct sets are extensionally indistinguishable: for example, the intensionally different “set of persons in city Y” and “set of women, children and men in city Y” are extensionally identical, because they have the same elements, despite the fact that the reasons for which these elements were included in a given set are intensionally different. If we consider a set as a formation defined not only by its elements, but also by its “envelope” – e.g. the concept under which the elements fall – then the sets will be considered identical not only if they have the same elements, but also if they have the same envelope. The formal notation for these intuitions allows us to consider extensional conceptions as only one of the possibilities, and not an apodictic a priori situation of which it could be said that tertium non datur. In addition, in my view a distinction should be made between sets of something – i.e. sets of “full-blooded objects” – and sets of unstructured points. For example, a set of two Lie algebras is not identical to a set whose elements are two ants. The second issue worth considering is the attempt to show that one can create a language in which the whole of classical mathematics can be formally and consistently expressed – i.e. all of its known theories, including those contradictory to each other – contrary to the common belief that the impossibility of constructing a single universal language and theory capable of expressing all of mathematics follows from Gödel’s famous theorems. From these theorems, it only follows that this cannot be done in a certain class of languages. I would like to give an example of an algorithm for creating such a universal language, restricting myself only to the truths of mathematics and theories that can be expressed in propositional formulae of finite length. Formally speaking, such a language would be the appropriate language for writing down not only absolute mathematics but, above all, human mathematics. I shall use the example of two different first-order theories, ZF(C) and PA, which we shall try to write down in a single formal first-order theory.23 Let us first list the axioms of both theories without alteration, so that they form one theory. Such a theory will not be consistent. In order to make it so, assuming that the initial theories are also each consistent, it is necessary to distinguish all symbols denoting relations, so that there are no identical relations in both theories, including the relation of identity. Likewise, we need to establish the symbols that denote constants. However, the theory will still be contradictory, and it will be necessary to distinguish between the quantifiers of the two

23

This example comes from an as yet unpublished paper (Król, Lubacz 2019).

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theories: for instance, let “∃a ” be an existential quantifier in ZF(C), and “∃c ” one in PA. The same should be done for universal quantifiers. The theory thus obtained will be “formally” consistent, but in order for all this to have mathematical sense, it will also be necessary to impart that sense to the new symbols that denote quantifiers, so that they can work in a single unified theory. To this end, further new axioms may be added; for example, the following: (1) (2) (3) (4) (5)

∃x. F (x) ≡ ∃c x. F (x) ∨ ∃a x. F (x); ∃c x. F (x) → ~∃a x. F (x); ∃c x. F (x) → ∃x. F (x); ∃a x. F (x) → ∃x. F (x); F(a) → ∃c x. F (x) ∨ ∃a x. F (x).

In the general case, if we allow that part of the objects of both theories is common, or that there are more than two types of object, then condition (2) can be abandoned or weakened; e.g.: (2’) ∃c x. F (x) ∨ ∃a x. F (x) → ∃x. F (x) It is also possible to introduce other additional existential and universal quantifiers. Conditions (1)–(5) indicate that the universe of our theory is the sum (disconnected or not, in the case of (2’)) of two classes of objects (here: sets (C) and numbers (A)). Thus, our theory is interpretable in a Tarskian sense, in terms of a theory with limited quantifiers: (6) ∃c x. F (x) ≡ ∃x ∈ C. F (x); (7) ∃a x. F (x) ≡ ∃x ∈ A. F (x). We could now also think of further specific axioms, subject to the needs and purposes of the research being conducted. To conclude, I would like to give an example of a situation which clearly shows the limitations of Gödel’s conception of absolute mathematics and the existence of one absolute world. This situation indicates the need to accept at least two absolute worlds of proper mathematics. For each consistent theory (and hence also for set theory), it is possible to construct a theory whose axioms are, in the intuitive sense, contradictions (or, more precisely, negations) of the initial theory. To illustrate this, I shall show how such an alter-theory for ZFC and PA may be constructed. Let us first select a subset of independent axioms from those of ZFC. (The separation axiom can be proved using the replacement, pairing,

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power set and union axioms. The existence of a set, including the empty set, results from the axiom of infinity.) For each independent ZFC-specific axiom, let us list either its negation or a proposition stating (in intuitive terms) something that is opposite to the initial theory axioms, and thus a partial negation of the initial axiom. (By “partial” I mean here including only one, or some portion, of the conditions defining full negation.) The alter-axiom to the pairing axiom will state the following: that for some a and b, there is no set whose elements are only a and b, etc. If the axioms of the initial theory are independent, their negations must also be independent of each other. It is therefore also possible to consider partial alter-theories, where some axioms may be identical to the initial theory axioms. If the initial theory is consistent and has a model, then the alter-theory must also be consistent and possess a model.24 An alter-theory will undoubtedly constitute an interesting “shadow” of the initial theory, but need not be mathematically trivial. It follows from Gödel’s first theorem that in any given language there cannot exist only the theory and its alter-theory, if the initial theory happens to be incomplete. If, in a given language, only that theory and its alter-theory exist, then the theory must be complete. In the case of alter-ZFC, this theory cannot have models describable in ZFC: i.e. ones that are ZFC objects. Thus, we also arrive at a negative answer to the question of whether everything is a set. Analogically, the alter-theory of proper mathematics will describe precisely those objects that differ from sets or finite natural numbers. An interesting example to consider is the alter-theory of PA. Let us list the negations of all of the PA axioms. Alter-PA cannot possess a finite model, because it would be categorical, and PA itself would then have to be categorical. We may well then wonder what causes the alter-PA not to have a finite model. It is the negation of each proposition, being part of the induction scheme. A structure where 0 has two different antecedents, a and b, ~(a = b = 0), i.e. Sa = Sb = 0, may serve as an exemplary model in this regard. In order to ensure the veracity of the negation of particular PA axioms, i.e. alter-PA axioms, 24

I confess that I am simplifying the matter somewhat here, in that we really ought to take a few other circumstances into account, too. The procedure for constructing an altertheory is based on the possibility of adding to a given set of consistent axioms both a proposition independent of these axioms and its negation, without generating contradiction. If the initial axioms are independent of each other, then to each of them can be added, consistently, the negation of some other (independent) axiom. Analogously, the initial axiom can be “replaced”. By iterating this operation, an alter-theory to a consistent theory that is itself consistent can be constructed. Obviously, the logical axioms remain the same in both theories.

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let us assume the following additional dependencies: a + 0 = b, S(a + b) = 0, b × 0 = a, b × a = a. This finite structure of several elements is enough to constitute a model for all six first alter-PA axioms. However, in order to satisfy the alter-induction axioms, it is necessary to include an infinite, at least countable, set of consecutive different individuals, because there must be an exception to each formula of the form A(n) = [n = a or n = b or n = 0 or … n = ck ]. This means that for each finite number of constants ck , there must exist a constant ck+1 that is different from them in the alter-PA model. Alter-theories can undoubtedly be employed to investigate the properties of initial theories as known to us from human mathematics. In particular, they may be used to investigate the independence or demonstrability of certain propositions, e.g. in PA. Returning to the issue of absolute mathematical reality, we can assert that an appropriate alter-theory should exist for each version of the theory responsible for locally axiomatising proper mathematics – and, especially, for an independent set of protocol-truths obtained on the basis of insights into absolute mathematical reality. The difficulty in sustaining such a conception of proper mathematics rather arises from the analysis of problems pertaining to the relationship between theory and alter-theory. What does the alter-theory of some locally noncontradictory version of proper mathematics say about it? And on the basis of what insights do we create such a theory? Moreover, it would seem to be an entirely historically contingent matter that we ourselves have developed, and are now working with, some given theory and not its alter-theory. It seems that the insights that lead to the alter-theory are of a completely different kind to those involved in creating the theory. Proper mathematics therefore has a “shadow” in the form of proper alter-mathematics, and it is impossible to maintain the uniqueness of the object of proper mathematical investigations. In this sense, proper mathematics is completely indistinguishable from some part of human mathematics that has turned out to be richer in terms of both content and model. The objection could be raised that the construction of instances of altertheory is being carried out here in a purely mechanical way, without the use of insight. Yet that cannot be the case, in that the analysis of alter-theory usually functions to show that from an intuitive point of view, it lacks certain axioms. For example, in alter-ZFC, in order to be able to somehow intuitively imagine what we are talking about and try to answer some intuitively posed questions, it is natural to add another axiom: namely, ∃y∀x. ~(y ∈ x). (This proposition is a negation of one of the theses of ZFC, but in general this need not mean that the negation of a thesis is an alter-ZFC thesis.) Thus, the analysis and understanding of alter-theory is carried out on the basis of certain “alterna-

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tive insights”. It is therefore necessary to assume the existence of at least two absolute worlds, if there exists at least one theory expressing proper mathematics. The final thing to consider is a certain awkward consequence of Gödel’s views. If there is an absolute mathematical reality, then proper mathematics amounts to a privileged mathematical theory. The absence of “splits” also makes it the only mathematical theory capable of adequately describing absolute mathematical reality. However, each basic theory of human mathematics known to us – PA, set theory (e.g. ZFC), CT (category theory), and so on – describes the same structures, but usually in a non-equivalent in generalitate way. For example, apart from the usual properties referred to by PA, the natural numbers in ZFC have some other excessive or unexpected properties. In ZFC, we can ask “Does the number 2 belong to the number 4?”, or “What is the power set of number 10?”, etc. These questions do not make sense in PA. The situation in CT is even odder. Natural numbers, or rather NNO (natural number objects), often have significantly different properties in different categories, e.g. in different topoi. These descriptions are different simply by virtue of not being equivalent, and it is difficult to claim that the standard model as we know it from PA is picked out in them. The question then arises: which of these descriptions is the sole appropriate one? Can we really talk about one absolute instance of the natural numbers? The assertion that behind all these theories lies one absolute reality, part of which consists in the natural numbers, is not based on any mathematical facts, but rather stands in contradiction to them. We have as many natural numbers as there are possible theories. Each such theory is based on certain intuitions and insights, and these in turn can be compared with the others, and investigated with respect to the relations in which they stand. PA comes closest to our intuitions and the way we tend to picture things, because it is more in accordance with historical forms of arithmetic than set theory or CT. Absolute mathematics is more of a postulate and a prospect than a really existing theory. Accepting the existence of absolute reality requires an acknowledgement that all mathematical theories – if they are based on authentic intuitive insights – should furnish at least some sort of non-trivial equivalent sub-scope-level (e.g. finitary) description of the same objects. In the wake of the preceding remarks, as a last non-contradictory possibility, there remains the fourth of the abovementioned ways of dealing with the difficulties explored here: namely, acceptance of moderate phenomenological Platonism in version III.2A.2, and with this a plurality of “absolute worlds” of mathematics. Phenomenologically, however, the concept of a plurality of absolute worlds has its defects. For example, the number of possibilities and

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underdeterminations pertaining to mathematical objects given to us indicates that we create the worlds of mathematics in our acts rather than uncovering them. To be sure, our discovering of other worlds can be based on more than just alternative insights, but the example of alter-theory still demonstrates that each of our constructions employing a certain “deliberate model” is also an involuntary creation (rather than such a discovery) of an alternative world. Hence, such a conception requires that there be infinitely many worlds of absolute mathematics, with the proper mathematics of these worlds possessing completely different properties from those postulated or anticipated by Gödel. This point applies not only to formal properties, but also to the role of absolute insights themselves. We are thus brought to a point in our considerations where it appears justifiable to assert that human mathematics, with its multiplicity of systems, theories and conceptions (and not only set theory), seems more interesting and more intellectually fascinating than the sublime world of absolute mathematics as sifted through the fine sieve of absolute insights and truths. Absolute mathematics requires us to abandon most of the mathematics that we deal with on an everyday basis. Most importantly, proper mathematics does not exist without human mathematics, and could never be known and discovered by us as humans. The objects of inquiry in human mathematics can be platonic or ideal just as in proper mathematics, without there being any requirement to accept some kind of unitary global consistency holding between them – or, rather, as obtaining across the theories that describe them. Instead of practising proper mathematics, we can achieve the same results by practising human mathematics. But what the latter itself really amounts to, and which conception of it is to be preferred, is something I would like to leave to the reader to decide.

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Benacerraf, P., “What numbers could not be”, in: [Benacerraf, Putnam 1987], pp. 272– 294. Benacerraf, P., “Mathematical truth”, Journal of Philosophy 70(1973), pp. 661–680; (cf. in: [Benacerraf, Putnam 1987], pp. 403–420). Benacerraf, P. Putnam, H., Philosophy of mathematics. Selected readings, reprint of II. ed. 1983, Cambridge University Press, Cambridge 1987. Bolzano, B., Betrachtungen über einige Gegenstände der Elemetargeometrie, Prague 1804. Boolos, G., “The iterative conception of set”, in: Benacerraf, Putnam [1987], p. 486–502. Brown, J. R., Platonism, Naturalism, and Mathematical Knowledge, Routledge, New York and London 2012. Cf. Philosophia Mathematica, Paseau, C., Volume 20, Issue 3, 1 October 2012, pp. 359–364. Calderon, M. E., “What Numbers Could Be (And, Hence, Necessarily Are)”, Philosophia Mathematica, Volume 4, Issue 3, 1 September 1996, pp. 238–255. Callard, B., “The Conceivability of Platonism”, Philosophia Mathematica, Volume 15, Issue 3, 1 October 2007, pp. 347–356. Carter, J., “Ontology and Mathematical Practice”, Philosophia Mathematica, Volume 12, Issue 3, 1 October 2004, pp. 244–267. Chihara, C., “A Gödelian thesis regarding mathematical objects: do they exist? and can we perceive them?” Philosophical Review 91, (1982), pp. 211–227. Davies, E. B., “A Defence of Mathematical Pluralism”, Philosophia Mathematica, Volume 13, Issue 3, 1 October 2005, pp. 252–276. Feferman, S., “Systems of predicative analysis”, in: Hintikka [1969], p. 95–127. Føllesdal, D., “Richard Tieszen. After Gödel. Platonism and Rationalism in Mathematics and Logic”, Philosophia Mathematica, Vol. 24, Issue 3, 1 October 2016, pp. 405–421. Gödel, K., “Russell’s Mathematical Logic”, in: The Philosophy of Bertrand Russell, ed. Schilpp, P. A., Northwestern University, Evanston 1944, pp. 123–153. Gödel, K., “What is Cantor’s continuum problem?”, in: Benacerraf, Putnam [1987], p. 470–485; (first published in: The American Mathematical Monthly, 1947, 54, pp. 515–525). Gödel, K., “Some basic theorems on the foundations of mathematics and their implications”; in: Kurt Gödel. Collected Works. Volume III. Unpublished Essays and Lectures, ed. Feferman, S., Oxford University Press, New York and Oxford 1951, pp. 304–323. Goldblatt, R., Topoi. The Categorial Analysis of Logic, North-Holland Publishing Company, Amsterdam, New York and Oxford 1979. Hardy, G. H., A Mathematician’s Apology, Cambridge University Press, Cambridge 1940. Hartimo, M., ed., Phenomenology and Mathematics. Phaenomenologia; 195, Dordrecht, Springer, 2010. Cf. Centrone, S., Philosophia Mathematica, Volume 22, Issue 1, 1 February 2014, pp. 126–129 and Volume 19, Issue 3, 1 October 2011, pp. 369–370.

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Tieszen, R. L., “Mathematical Realism and Gödel’s Incompleteness Theorems”, Philosophia Mathematica, Volume 2, Issue 3, 1 September 1994, pp. 177–201. Tieszen, R. L., “Gödel’s path from incompleteness theorems (1931) to phenomenology (1961)”, Bulletin of Symbolic Logic 2(1998), pp. 181–203. Tieszen, R. L., Phenomenology, Logic, and the Philosophy of Mathematics, Cambridge, Cambridge University Press, 2005. Cf. Philosophia Mathematica, Ronzitti, G., Volume 16, Issue 2, 1 June 2008, pp. 264–276. Tieszen [2006] Tieszen, R. L., “After Gödel: Mechanism, Reason, and Realism in the Philosophy of Mathematics”, Philosophia Mathematica, Volume 14, Issue 2, 1 June 2006, pp. 229–254. Tragesser, R., “Eidetic analysis, informal rigour, and a phenomenological critique of Carnap’s notion of explication”, Philosophy and Phenomenological Research 33(1972), pp. 48–61. van Atten, M., Essays on Gödel’s Reception of Leibniz, Husserl, and Brouwer, Logic, Epistemology, and the Unity of Science; 35, Springer, Dordrecht 2015. Cf. Philosophia Mathematica, Volume 23, Issue 3, 1 October 2015, pp. 438–439. van Atten, M., Brouwer meets Husserl: On the Phenomenology of Choice Sequences, Synthese Library, vol. 335, Springer, Dordrecht 2006. Cf. Philosophia Mathematica, Franchella, M., Vol. 16, Issue 2, 1 June 2008, pp. 276–281. Van Kerkhove, B., Van Bendegem, J. P., eds., Perspectives on Mathematical Practices: Bringing Together Philosophy of Mathematics, Sociology of Mathematics, and Mathematics Education, Logic, Epistemology, and the Unity of Science, 5, Springer, Dordrecht 2007. Cf. Philosophia Mathematica, Volume 16, Issue 1, 1 February 2008, p. 146. Wang, H., “The concept of set”. In: Benacerraf, Putnam 1987, pp. 530–570. Wang, H., “Reflections on Kurt Gödel”, The MIT Press, Cambridge Mass., London 1987. Wang, H., “To and from philosophy – discussions with Gödel and Wittgenstein”. Synthese 88(1991), pp. 229–277. Wang, H., “Skolem and Gödel”, Nordic Journal of Philosophical Logic 1(1996), pp. 119– 132.

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Chapter 3

Mathematical Modalities and Mathematical Explanations Krzysztof Wójtowicz

Abstract In the philosophy of mathematics, modal concepts appear in the realism-antirealism debate in different contexts and roles. (1) They are used by advocates of the antirealistic position as a tool for giving anti-realistic (re)interpretations of mathematics. Geoffrey Hellman’s modal structuralism (discussed in this text) is a prominent example. (2) The notion of a mathematical modality is also used by supporters of realism in the context of contemporary versions of the indispensability argument (which is based on the claim that mathematics per se has some explanatory contribution to science). An interesting tension appears as both sides of the debate refer to modal concepts, albeit in a different way (and obviously drawing different conclusions). The aim of the article is to discuss this matter. I formulate several hypotheses concerning the possibility of giving an account of the mathematical explanation based on Hellman’s anti-realist account of mathematics. Hellman considers the indispensability argument to be a serious challenge to the anti-realist camp, so he tries to give a non-realistic account of the applicability of mathematics using modal notions. However, he does not discuss the problem of mathematical explanation, which has become popular some years after the publication of Hellman’s main work. I claim that Hellman’s approach could be used to provide a better understanding of the mathematical modality in question – and also to undermine the realistic thesis. It allows one to reconstruct (or: reinterpret) the mathematical tools needed in science in an anti-realistic manner, making it coherent with accounts of explanation formulated in terms of a mathematical modality. In this paper, I discuss the possibility of pursuing such program, and formulate some tentative hypotheses hoping to initiate a discussion. The results might be a challenge for the mathematical realist.

© Krzysztof Wójtowicz 2021 | DOI:10.1163/9789004445956_005 Marcin

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Keywords mathematical explanation – mathematical realism – Enhanced Indispensability Argument – modal structuralism – Hellman’s structuralism – program explanations – abstract explanations – explanations by constraints

One of the fundamental problems in the philosophy of mathematics is the applicability of mathematics in the natural sciences: why is it possible to apply mathematical techniques in the description of the non-mathematical world? And what philosophical conclusions concerning the metaphysical and epistemological status of mathematics might be drawn from this fact? Quine’s account has been very influential in the discussion. His concept is based on his holistic view of science and on his doctrine of ontological commitments. Together with the thesis that mathematics is indispensable in science, they lead to mathematical realism. Willard Van Orman Quine’s indispensability argument has become the central pro-realistic argument in the debate, and a strong counter-reaction from the antirealist camp was inevitable. Hartry Field’s Science Without Numbers (Field 1980) opened a new phase in the debate. The original indispensability argument has been given a modified form in the last few years and now it is mostly discussed in the form of the Enhanced Indispensability Argument (EIA). It has emerged in the context of the discussion concerning non-causal (in particular: mathematical) explanations in science as a reaction to the anti-realist claims that the indispensability of mathematics is not sufficient to justify mathematical realism. According to these claims, indispensability – in order to play its (alleged) role in the pro-realistic argumentation, should have a very special form, i.e. it should be explanatorily indispensable. So the main question to address is whether mathematics per se offers an explanatory contribution to physics (or – more generally – to science). So the analysis of the explanatory role of mathematics becomes central to the debate. Some of the important accounts of mathematical explanation make use of the concept of mathematical modality, which is stronger than natural modality. So the problem of its semantics, metaphysics and epistemology arises. As it is natural to perceive the truths of mathematics as characterizing the class of possible worlds, there is a certain type of modal realism involved in the Enhanced Indispensability Argument. Geoffrey Hellman’s modal structuralism, presented in (Hellman 1989), and discussed and refined in a number of his publications (Hellman 1990, 1996, 2001, 2005), is a very important contribution to the debate. Hellman devel-

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oped a nominalistic (anti-realistic) account of mathematics based on a primitive modality. His theory can be used to construct a “modally self-contained” account of mathematical explanation that does not lead either to mathematical realism or to modal realism. In the paper I present the general idea of such an account. I am aware of the fact that some of the ideas presented here have the character of a preliminary sketch – but nevertheless I hope to initiate a discussion. The structure of the article is as follows: In (1) The Indispensability Argument I briefly recount the main features of the classic indispensability argument, as it is crucial for all the discussed problems. A comment on some of the anti-realist responses (Field’s classic strategy, and more recent theories of Azzouni, Melia, Balaguer and Yablo) is also given. In (2) Hellman’s structuralism, Hellman’s modal structuralism is presented. In (3) The Enhanced Indispensability Argument (EIA), the general problem of mathematical explanations is shortly commented on in the context of the ontological discussion. In (4) Modal notions and mathematical explanations, the focus is on the accounts which stress the modal character of mathematical explanations (i.e. program explanations or abstract explanations). In (5) An antirealistic account of mathematical modality, it is claimed that Hellman’s approach naturally fits into the modal approach: it does not provide an interpretation for modal notions in terms of possible worlds, but treats them as primitive. This poses a challenge for the realist whose argumentation is based on EIA – and who uses modal notions in the discussion. (6) Concluding remarks presents a short summary.

1

The Indispensability Argument

The contemporary realism-antirealism debate has been dominated by Quine’s indispensability argument, which is (together with its descendants) the most widely-discussed argument in favor of mathematical realism.1 In succinct form, it rests on two premises: 1. Mathematics is indispensable in science.

1 Gödel’s version of mathematical realism (Platonism) is also widely discussed, especially since his Collected Works were edited. But the indispensability argument (including its modified forms) is still the most popular pro-realistic argument in the contemporary debate.

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2.

The ontology of a (scientific) theory should be identified in a uniform way – in particular, the objects over which we quantify in the theory, should be recognized as existent. According to the second premise, ontological commitments of scientific theories should be taken at face value. But this means that we need a precise method of identifying the underlying ontology, as it is not always given in an explicit form.2 The solution is provided by Quine’s famous criterion of existence, according to which “to be is to be a value if a variable”. We can therefore identify the ontological commitments of a scientific theory by reconstructing (“translating”) it as a first-order theory (i.e. in first-order logic) and identifying the existential sentences, which are logical consequences of the theory.3 According to Quine’s criterion, we are committed to the existence of all the entities we (indispensably) quantify in our scientific theories.4 This is widely believed to be the case of mathematical objects as mathematics is an essential component of physics, and it seems obvious that mathematical terms and notions cannot be “paraphrased away”. So, from these two premises, the conclusion that mathematical objects exist follows.5 Quine’s doctrine has therefore a holistic character: The totality of our so-called knowledge or beliefs, from the most casual matters of geography and history to the profoundest laws of atomic physics or even of pure mathematics and logic, is a man-made fabric 2 “The common man’s ontology is vague and untidy in two ways. It takes in many purported objects that are vaguely or inadequately defined. But also, what is more significant, it is vague in its scope; we cannot even tell in general which of these vague things to ascribe to a man’s ontology at all, which things to count him as assuming” (Quine 1981, 9). This remark applies to scientific theories as well, and they need a precise existence criterion, too. 3 Quine’s doctrine rest on the assumption that first-order logic has a distinguished character, i.e. on the first-order thesis. Here I do not discuss the first-order thesis – in particular the problem of logical pluralism, the status of logical systems, the notion of logical constants etc. (for discussion see for instance: Sher 1991; Shapiro 2014; Russell 2019). 4 “A theory is committed to those and only those entities to which the bound variables of the theory must be capable of referring in order that the affirmations made in the theory be true.” (Quine 1948, 33). 5 Quine’s doctrine does not have an “essentialist” character: the postulated objects may be considered to be positions in a structure, and their character, role and status is determined by this overall structure. An interesting analysis of objects in terms of “eigen-solutions metaphor” is given in (Stern 2011, 2015): “The fundamental insight of this epistemological framework is summarized by the celebrated aphorism of Heinz von Foerster – Objects are tokens for eigen-solutions. In other words, objects, and the names we use to call them, stand for and point at such invariant entities” (Stern 2015, 56). I am grateful to the anonymous reviewer for the inspiring remarks on this topic.

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which impinges on experience only along the edges. Or, to change the figure, total science is like a field of force whose boundary conditions are experience. (Quine 1951, 42)6 It is important to observe that the indispensability argument applies only to those fragments (theories) of mathematics which are applied in empirical theories. So identifying the ontological basis for applied mathematics becomes an essential problem.7 Quine’s stance has been often described as quasi-empiricism, as Quine obviously is an empiricist (and naturalist). But the term “quasi-empiricism” also refers to the views of Imre Lakatos – and it is interesting to compare these two uses of the term. Quine’s account focuses on the ontological issues and on the justification of mathematics as a fragment (tool) of science (which results from his holistic views (combined with the doctrine of ontological commitments). Lakatos is mainly concerned with the internal mechanism of the development of mathematics, and stresses its similarities to empirical science (hypothesis verification, fallacies, counterexamples and their role in both theory and concept formation, etc.). So this is a different kind of quasi empiricism than Quine’s.8 However, Lakatos’ quasi-empiricism can be combined in a natural way with Quine’s account as the development of mathematics is often dri-

6 (Resnik 2005) provides a presentation and discussion of Quine’s holism in the context of the philosophy of mathematics. 7 Quine admitted that theories of pure mathematics (for instance fragments of set theory) are on par with uninterpreted systems, as the indispensability argument cannot be applied to them. The identification of the “mathematical ingredient” which is necessary in science is an intriguing problem, as applied mathematics makes use of a limited fragment of available mathematical notions, theories and techniques. In particular, in some cases mathematical techniques are created to meet the needs of science. Calculus is a clear example of a situation where mathematical tools and notions were introduced together with (or even: as a fragment of) a physical theory. (I want to thank the anonymous reviewer for inspiring observations on this topic). 8 “Whether a deductive system is Euclidean or quasi-empirical is decided by the pattern of truth value flow in the system. The system is Euclidean if the characteristic flow is the transmission of truth from the set of axioms ‘downwards’ to the rest of the system – logic here is an organon of proof ; it is quasi-empirical if the characteristic flow is retransmission of falsity from the false basic statements ‘upwards’ towards the ‘hypothesis’ – logic here is an organon of criticism. But this demarcation between patterns of truth value flow is independent of the particular conventions that regulate the original truth value injection into the basic statements. For instance a theory which is quasi-empirical in my sense may be either empirical or non-empirical in the usual sense: it is empirical only if its basic theorems are spatiotemporally singular basic statements whose truth values are decided by the time-honoured but unwritten code of the experimental scientist” (Lakatos 1976, 206).

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ven by the needs of science.9 An important support for this argument comes from Hilary Putnam, who forcefully claimed that it is impossible to reconcile scientific realism with the view that mathematical objects are mere fictions, accusing the adherents of this way of thinking of intellectual dishonesty: It is like trying to maintain that God does not exist and angels do not exist while maintaining at the very same time that it is an objective fact that God has put an angel in charge of each star and the angels in charge of each of a pair of binary stars were always created at the same time! (Putnam 1975, 74) This argument for mathematical realism is therefore often called the “QuinePutnam indispensability argument”. The indispensability argument has been subjected to fierce debate, and there are several strategies within the anti-realist camp. Broadly speaking, they can be divided into two groups: (i) those which try to undermine the indispensability claim; (ii) those which do not accept the inferential step from the indispensability of mathematics to accepting a realistic mathematical ontology. The most influential in the first group is Field’s strategy (Field 1980). Field claims that mathematical notions are not indispensable for formulating our best scientific theories. Given the omnipresence of mathematics in contemporary physics, this claim sounds quite peculiar at first. However, Field tried to reconstruct the necessary tools in a synthetic way, i.e. without invoking any mathematical vocabulary. His strategy is often called “geometric”, as it is based on providing a reconstruction of mathematical notions using spatio-temporal predicates.10 Broadly speaking, Field provides a reconstruction of elementary calculus claiming it to be the first step in the nominalistic enterprise. This might be viewed as a realization of Quine’s description of nominalism: As a thesis in the philosophy of science nominalism can be formulated thus: it is possible to set up a nominalistic language in which all of natural science can be expressed. The nominalist, so interpreted, claims that a language adequate to all scientific purposes can be framed in such a way 9 10

I am indebted to an anonymous reviewer for inspiring remarks concerning Lakatos’ views in this context. In some sense, it is the opposite of Hilbert’s strategy from Grundlagen der Geometrie: instead of producing models for geometric notions, Field provides geometric interpretations for notions like continuity, derivative, etc.

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that its variables admit only concrete objects, individuals, as values – hence only proper names of concrete objects as substituends. Abstract terms will retain the status of syncategorematic expressions, designating nothing, so long as no corresponding variables are used. (Quine 1939, 708) But this involves quite strong assumptions concerning the metaphysical status of space-time points and regions (Field assumes them to be physical objects), and the strength of the mathematical tools reconstructed in this way is limited. An important step in Field’s program is proving the conservativeness of theories using mathematical vocabulary over their nominalistic counterparts. Conservativeness – broadly speaking – means that no new nominalistically statable conclusions can be derived after enriching the language with mathematical terms and after adding mathematical assumptions.11 Informally, conservativeness principle can be stated as: (C):

Let S be a mathematical theory, N – a synthetic (nominalistic) theory, and α – a nominalistic sentence. If α is not a consequence of N , then α is not a consequence of N + S. (Here N + S means the “mixed” theory, i.e. the scientific theory augmented with mathematical notions.)12

Field also considers the general case of adding set-theoretic vocabulary to physical theories and treating physical objects as urelements. He formulates and justifies a general thesis concerning the conservativeness of set theory with respect to synthetic theories. Informally speaking, if we treat physical objects as urelements and formulate a nominalistic theory T using some (nominalistic) vocabulary τ, then adding the axioms of set theory will not lead to new (nominalistic) conclusions.13

11

12 13

A theory S is a conservative extension of a theory T with respect to the class of sentences Φ if any φ ∈ Φ which is a consequence of S is also a consequence of T. Loosely speaking, theory S does not allow the justification of more consequences φ ∈ Φ than theory T. In the case of non-conservative extensions, new corollaries (from class Φ) arise. It is important to stress that the notion of conservativeness has two versions: semantic and syntactic (roughly speaking: model-theoretic versus proof-theoretic). In the case of first-order theories they coincide, but this need not be the case for logics without completeness. By transposition: if α is a consequence of N + S then it is also a consequence of N . Take ZFU to be the standard set theory ZFC with urelements and let τ be any signature (i.e. a set of extralogical symbols). We can then consider a theory ZFU(τ), which is set

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The conservativeness of mathematized theories over their (purely) nominalistic version implies that set theory can be used as an ontologically neutral tool, as it does not imply new (nominalistic) consequences. The reason for enriching the nominalistic (synthetic) vocabulary with mathematical terms is to make the inferences easier – but the detour into the realm of alleged abstracta does not change the class of the (nominalistic) conclusions, as the mathematical versions of the theories are conservative over their nominalistic counterparts. It makes our lives (as scientists) easier, but without extra metaphysical costs. The use of abstract, mathematical methods in science is justified, as they only play an auxiliary role. They simplify the inferences but do not provide any novel inferences within the class of nominalistic assertions.14 So, if we believe the nominalistic scientific theories, we are free to use mathematics as a tool: even if mathematics is indispensable for practical reasons, this is not an inherent indispensability. Moreover, according to Field the nominalistic (or nominalized) versions of scientific theories are better than their mathematical versions, as they do not refer to abstract objects which are not causally connected with the explanandum. This allows intrinsic explanations to be given of physical phenomena as, according to Field, “underlying every good extrinsic explanation there is an intrinsic explanation” (Field 1980, 44). Scientific laws are grounded in the purely physical properties of objects even if their formulations refer to mathematical notions – and the nominalistic version reveals the structure. This means that there are also methodological advantages in giving a synthetic reconstruction. There has been much discussion concerning Field’s program – virtually all later accounts (for instance: Balaguer, Chihara, Hellman, Maddy, Resnik, Shapiro – the list is far from being exhaustive) comment on it. If mathematical notions could indeed be interpreted in a nominalistically acceptable way, it would undermine the indispensability argument – as irreducible mathematical notions would cease to be indispensable. However, it is far from ob-

14

theory with all the comprehension axioms for formulas from the enriched language, i.e. for τ ∪ {∈}. Let T be a theory with vocabulary τ. The following principle holds: (C0 ) If T is consistent, then ZFU(τ) + T ⁎ is also consistent (Field 1980, 17). Here by T ⁎ we denote a version of theory T, which is – loosely speaking – restricted to non-mathematical objects. Remember Hempel’s famous claim that mathematics plays the role of a theoretical juice extractor: “The techniques of mathematical and logical theory can produce no more juice of factual information than is contained in the assumptions to which they are applied; but they may produce a great deal more juice of this kind than might have been anticipated upon a first intuitive inspection of those assumptions which form the raw material for the extractor.” (Hempel 1945, 391).

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vious that it could ever be done, and the consensus seems to be that Field’s nominalization strategy failed, as it faces formidable technical and conceptual problems. For instance (Shapiro 1983) shows that – in the case of first-order theories – there is no syntactic conservativeness of mathematical theories over their nominalistic versions.15 Moreover, the crucial representation theorem cannot be proven as any first-order theory of spacetime has non-standard models.16 It is also far from clear how far Field’s strategy can be extended. There are formidable difficulties with presenting a synthetic paraphrase of scientific theories. Indeed, if such a (theoretically attractive) paraphrase was possible, it would be an important argument in favor of nominalism. But for instance, Malament (1982) argues against the possibility of extending Field’s strategy to Klein-Gordon’s field theory; he also discusses Hamiltonian mechanics and quantum mechanics as examples of theories where nominalization seems to be very hard (if possible at all) to achieve. It should be mentioned that some partial results have been obtained – for instance (Tallant 2013) gives a nominalistic paraphrase of the fragments of number theory, which are important for Baker’s cicada argument. But this is a rather piecemeal result, concerning a very elementary application of mathematics: very sparse resources are sufficient to formalize the tools needed to formulate the cicada argument. Even if we accept such proposals, they give – at best – only very limited paraphrases of scientific theories. And the burden of proof is still on the side of these nominalists who claim that a nonmathematical paraphrase of scientific theories is possible. Technical problems are obviously an obstacle, but there are also important philosophical doubts to address. One on them is concerned with the metaphysical status of space-time points and space-time regions. According to Field, they should be treated as physical objects. But (Malament 1982) points out that they do not have mass, they do not move, and are not subject to any changes. It is therefore not justified to consider them to be physical objects: they are rather locations for physical objects, and it is the objects that act in a causal way, not their locations. (Resnik 1985) argues that the role of space-time points in physics is similar to that of mathematical objects rather than to that of standard physical objects. Unlike electrons, forces and planets, there are no observational data that would lead to the discovery of space-time points. Ac15

16

This conclusion is achieved by a version of Gödelian argument: it is possible to perform an appropriate encoding of the syntax to retrieve a nominalistic sentence, which is provable in mathematical theory, but unprovable in the nominalistic version. For details see (Shapiro 1983). Field addresses these problems in (Field 1989).

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cording to Resnik, Field’s arguments on the causal impact of points and areas of space-time are unconvincing – we might equally well claim that numbers have causal effects because the temperature rises when the number of people in a room rises (Resnik 1985, 166). But it is necessary to assume the space-time points and regions have physical character in order to give the synthetic reconstruction of even such a simple theory as Newton’s theory of gravitation. So even if the technical problems with giving the reconstructions could be overcome, deep philosophical problems remain unsolved. The problems with putting Field’s ideas to work has led many adherents of nominalism to look for an easier road. The term “easy road” was used in (Colyvan 2010) to describe a group of approaches, in which the indispensability of mathematics is not denied. It is rather the second premise – according to which indispensability gives us good reason to believe the existence of mathematical objects – which is questioned.17 I shortly comment on some of the “easy road” contributions: (1) (Azzouni 2004) claims that quantifier commitments (i.e. being in the range of quantification) does not imply existence, as some of the objects might be mere posits (with an “ultra-thin” epistemic access). Azzouni mentions four conditions (robustness, monitoring, refinement, and grounding (Azzouni 2004, 129–136)) which are fulfilled by direct observation – but also by procedures which give us epistemic access to some unobservable objects (like elementary particles). The epistemic access fulfilling these conditions is thick epistemic access. Thin epistemic access is different: it is based on the role entities play in scientific theories, so it is theoretically-motivated, rather than direct access. In particular, accepting the thinly accessed entities involves explaining why they cannot be accessed in a thick way. Azzouni also accepts the existence of objects of this kind: they have to “pay their Quinean rent”. But mathematical objects are different – we only have ultra-thin epistemic access to them, which is a typical feature of, for instance, fictional objects. We can posit them without taking their reality into account. While a thinly accessed posit can be considered real, ultra-thinly accessed posits cannot.18 And this is exactly the case of mathematical objects. 17

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By contrast, the adherents of the “hard road” (Field being the most prominent example) try to attack the indispensability premise by giving suitable reconstructions of mathematical tools. “Should [a thin posit] fail to pay its Quinean rent when due, should an alternative theory with different posits do better at simplicity, familiarity, fecundity, and success under testing, then we have a reason to deny that the thin posits, which are wedded to the earlier theory, exist – thus, the eviction of centaurs, caloric fluid, ether, and their ilk from the universe.” (Azzouni 2004, 129).

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(2) (Melia 2000) suggests to “weasel out” the reference to mathematical objects. In his opinion, it is possible to quantify over objects while denying their existence. The negative existential claims are a way of taking back some details that appeared in the narrative. “[J]ust as in telling a story about the world, we are allowed to add details that we omitted earlier in our narrative, so we should also be allowed to go on to take back details that we included earlier in our narrative.” (Melia 2000, p. 470). In Melia’s opinion, the claim that the given object we have been referring to does not really exist is adding such a missing detail. So it is perfectly justified to say, for instance, that there is a prime number bigger than 100 – but nevertheless mathematical objects do not exist. (3) (Balaguer 1998) advocates a kind of fictionalism about mathematics trying to provide a semantics for scientific theories without commitments to abstract objects. The general idea is expressed as “The nominalistic content of a theory T is just that the physical world holds up its end of the ‘T bargain’, that is, does its part in making T true”. (Balaguer 1998, 135). So, the non-nominalistic parts of the theory have a non-factual character. (4) (Yablo 2002) proposes to treat mathematical claims as metaphorical so as not to interpret the existential claims of mathematics literally.19 In Yablo’s opinion there is no clear boundary between the portions of scientific discourse which should be treated literally and those which are a part of the metaphor. And in this situation Quine’s criterion cannot be applied as it leads to a wrong ontology. A necessary condition to apply it is “to ferret out all traces of nonliterality in our assertions. If there is no feasible project of doing that, then there is no feasible project of Quinean ontology.” (Yablo 1998, 233). All these strategies fall into the class of “easy road strategies” and they have been analyzed, for instance, in (Colyvan 2010, 2012) (and discussed in many other papers). Field’s program falls into the “hard road to nominalism” – i.e. a technical project of providing suitable reconstructions of scientific theories so as to get rid of the ontological commitments of mathematics. And there is also Hellman’s modal-structuralist approach, which has some features of both these strategies. On one hand, it is similar to the easy road strategies, as it does not try to eliminate the mathematical from science, but it gives a direct reinterpretation of mathematics in modal terms. However, this might be an argument for considering Hellman’s strategy as falling into the “hard road camp” – as it

19

Yablo’s theory is based on Walton’s account of metaphors (Walton 1993): we often invoke metaphorical talk (and generally: non-literal talk) to describe some real situations, without claiming that all the objects that have been mentioned really exist.

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rests on a rather complex technical program, and its aim is to give a suitable reformulation of mathematics.20 But regardless of the “philosophical affiliation” of Hellman’s program, I consider it to be the most impressive (both from the technical and philosophical points of view) among the anti-realist strategies. It might be very promising for the anti-realist as a strategy against certain forms of pro-realistic argumentation.21

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Hellman’s Structuralism22

Hellman accepts the claim that mathematics is indispensable in natural sciences and stresses the importance of this fact for philosophical discussion. However, the conclusions he draws from this fact are fundamentally different from those of the realists. He does not accept the claim that the indispensability of mathematics justifies interpreting the notion of truth in mathematics in terms of a correspondence between mathematical sentences (and theories) and an objectively existing realm of mathematical objects. Although Hellman claims that mathematical claims have a logical value, this premise does not lead to the conclusion that abstract objects exist. So he advocates realism in truth value combined with anti-realism in ontology. But there is a price to pay for the elimination of the ontology of abstracta: it is necessary to accept a “modal ideology” – in particular, modality has to be considered a primitive notion.23 According to Hellman, the problem of mathematical knowledge should be discussed in the context of scientific knowledge – and he considers Quine’s position (according to which the status of mathematical knowledge is characterized by its role in science) to be promising in this respect. In particular, he stresses the need of taking into account the role of mathematics in science – in particular, the problem of applicability. But he rejects the realistic interpretation of mathematics suggesting that we can interpret (or perhaps

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Leng argues that Hellman’s program is more similar to the easy road strategies, but not without hesitation; the phrase “both Hellman and the easyroaders” (Leng 2017, 414) suggests that Hellman’s strategy might be considered to be a kind of a “third road”. (I would like to thank the anonymous reviewer for pressing me to be more precise on this matter). This is not a welcomed conclusion for me as I have strong realistic sympathies. Nevertheless I think that Hellman’s approach can be problematic for the mathematical realist – and inspiring for the discussion. The presentation will be brief, and it will skip many (indeed, most of…) the technical and philosophical subtleties of Hellman’s program. In this respect, Hellman’s account is similar to Chihara’s theory presented in (Chihara 1990), but the technical details are quite different.

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even formulate) our scientific physical theories in such a way that “rather than commitment to certain abstract objects receiving justification via their role in scientific practice, it is the claims of possibility of certain types of structures that are so justified” (Hellman 1989, 96–97). Hellman’s strategy is very different from Field’s. Indeed, he expresses reservation concerning Field’s program. It is perfectly reasonable to claim that if Field’s program succeeded it would probably be the most elegant way of eliminating the ontological commitments of mathematics. But Hellman points out that Field’s strategy fails in the context of quantum mechanics (Hellman 1989, 112–115), as the mathematical tools needed there are too strong to allow for a nominalistic reconstruction. In particular, even second-order real arithmetic RA2 (which is one of theories reconstructed by Hellman) does not allow Gleason’s theorem to be proven, which is relevant for quantum mechanics (in particular for the discussion of the theories based on the concept of hidden parameters).24 Hellman’s inspirations are two-fold: (i) the general structuralist view on mathematics; and (ii) the idea of using modal notions instead of realistic semantics. In short: “mathematics is the free exploration of structural possibilities, pursued by (more or less) rigorous deductive means” (Hellman 1989, 6). Mathematical objects do not exist, and mathematics concerns only the possibility of the existence of certain structures.25 Thus, loosely speaking, one can get rid of the ontology of abstract objects by accepting postulates expressing the possibility of the existence of certain objects. We might say that the truthmakers are not mathematical structures, but rather the possibility of their existence. Modal structuralism rests on two pillars: (i) the hypothetical component; and (ii) the categorical component. (ad i) The hypothetical component is concerned with claims like “α would be the case if some objects/structures existed”. For example, arithmetical sentences are paraphrased as conditional modal sentences. Informally speaking,

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The case of RA2 is important, as it is – in a sense – the strongest theory that might perhaps be interpreted in a nominalistically acceptable way. One of Hellman’s inspirations is the strategy outlined in (Putnam 1967), where Putnam justifies the idea of eliminating ontological commitments of mathematics through the use of modal concepts. Putnam discusses Fermat’s hypothesis as an example (today, after Wiles’ proof we know that it is a theorem). According to Putnam, claiming that it is true means only that the axioms of PA necessarily imply Fermat’s hypothesis so that assuming the existence of natural numbers is not necessary. In general, statements concerning natural numbers are reducible to the following two claims: (i) ω-sequences are possible; (ii) sentences of the form: “if α is an ω-sequence, then…” are necessary truths (Putnam 1967, 301).

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instead of saying that the sentence φ is true in the standard model for arithmetic, we will say that if an ω-sequence existed then sentence φ would be true there. So, in the case of number-theoretic sentences, the hypothetical component could be formulated in the following informal way: (*)

If X was an ω-sequence, then sentence φ would be true there.

This intuitive formulation needs to be made precise. After a very thorough analysis, Hellman proposes the paraphrase φMSI , which – in informal terms – states that sentence φ would be necessarily true in any domain X with respect to a relation f behaving like the successor.26 But obviously, it does not claim that such a structure exists! (ad ii) The second crucial component – i.e. the categorical component – expresses the fact that the existence of a certain structure is possible. In the case of arithmetic, it is a claim concerning PA2 and might be formulated as: ⬨∃X ∃f (∧PA2 )X (s, f ) Informally, it amounts to the thesis that it is possible that a certain structure X exists, where the axioms of second-order arithmetic are fulfilled. The justification of this principle is quite complex, and will not be discussed here. Second-order Peano arithmetic PA2 is the weakest of the theories considered by Hellman – and obviously not strong enough to serve the purpose of physics. Hellman provides a similar analysis for second-order real arithmetic RA2 i.e. real analysis: the first-order variables range over real numbers, and the second-order variables range over sets of real numbers. RA2 is important for two reasons: (i) a substantial part of mathematics, including large parts of applied mathematics can be reconstructed within it; (ii) RA2 can be interpreted nominalistically and can be considered to be the strongest theory that is subject to such an interpretation. Hellman also discusses the extension of the modal-structuralist interpretation to set theory (with physical objects as urelements). Assuming the full strength of ZFU would make the reconstruction of mathematical tools needed in physics very easy – but the philosophical price to pay is too high. Therefore 26

To be more precise, the paraphrase has the following form: (φMSI ) ◻∀X ∀f [∧PA2 ⇒ φ]X (s, f ) where: (a) ∧PA2 is (the conjunction of) the axioms of second order arithmetic; (b) [α]X is the formula φ with its quantifiers relativized to class X ; (c) (s, f ) means that the successor symbol s has been replaced by a two-place relation variable f .

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Hellman considers ways of reconstructing these tools within simpler theories, like RA2 , as this is a way to minimize the strength of the modal assumptions.27 Analyzing the technical details is outside the scope of this paper. It should be mentioned that Hellman stresses the importance of giving an account of the mathematical proof as the source of mathematical knowledge, so he provides a translation scheme – i.e. a method of translating ordinary mathematical proofs into proofs (arguments), which are acceptable for modal structuralism.

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The Enhanced Indispensability Argument

An important class of anti-realist accounts is based on the claim that the indispensability of mathematics in science is not sufficient to justify its realistic interpretation: mathematical notions have to be indispensable in the right way in order to be given a realistic interpretation. This has motivated a new version of Quine’s indispensability argument: it has taken on the form of the Enhanced Indispensability Argument (EIA), where the explanatory virtues of mathematics are of primary importance. The received view on scientific explanation(s) seems to be that they have to be causal.28 But the thesis that the classic causal accounts of scientific explanation are not satisfactory has received strong support in the last years – and the notion of non-causal explanations is discussed extensively.29 In particular it is claimed that mathematics per se can play an explanatory role. The EIA might be viewed as a kind of inference to the best explanation account, according to which explanatory indispensability is a sufficient condition for

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In (Hellman 1996) an interpretation of RA3 is given but here some additional assumptions concerning plural quantifiers are needed. They might be considered an additional “ideological cost”. The thesis that scientific explanation should have a causal character enjoyed a privileged status in the discussion for a rather long time; here are some relevant opinions: “To give scientific explanations is to show how events and statistical regularities fit into the causal structure of the world.” (Salmon 1977, 162); “Causal processes, causal interactions, and causal laws provide the mechanisms by which the world works; to understand why certain things happen, we need to see how they are produced by these mechanisms.” (Salmon 1984, 132); “Causal explanation is the unique mode of explanation in physics.” (Elster 1983, 18). For an introductory presentation, see for instance (Reutlinger, Andersen 2016) and the monograph (Reutlinger, Saatsi 2018). (Pincock 2015) discusses concrete examples; (Saatsi 2018a) gives an interesting overview; the monograph (Lange 2017) is devoted to particular explanations of this kind – to mention just a few references.

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existential claims.30 The crucial premise of EIA is that mathematics plays a genuine explanatory role in science, i.e. that the explanatory job is done (at least partially) by mathematics. So in order to put this argument to work we have to provide examples of genuinely mathematical explanations in science – not just of physical explanations, where mathematical terms are used (this would be very easy, as virtually all physical claims are formulated with the use of mathematics). The topic of what constitutes the special character of mathematical explanations in science (i.e. what makes them “genuinely mathematical explanations” – not just explanations where mathematical notions appear) is much discussed; the monograph (Lange 2017) contains a wealth of examples with an extensive discussion.31 It is not possible to discuss the problem in detail – I will therefore only give some examples of phenomena with explanations which might be considered to be mathematical. One of the simplest, but instructive and much discussed is the famous system of seven bridges in Königsberg. It is well known that it is not possible to take a walk crossing all the bridges in such a way that each bridge is crossed exactly once. And the explanation is mathematical (by a simple fact in graph theory), not physical – as this impossibility does not depend on any physical properties of the bridges. Another nice example is the Borsuk-Ulam theorem offering a mathematical explanation of the fact that there are always two antipodal points on the surface of the Earth, where two physical parameters (take – for instance – pressure and temperature) are equal (Baker 2005, 2009; Baker, Colyvan 2011).32 A very famous example is the periodical life-cycle of a species of cicadas (13 and 17 years): here a fact in number theory is claimed to do the explanatory job rather than just biological facts concerning this particular species (Baker 2005).33 Another biological example concerns bees, and 30

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Saatsi gives the following brief characteristic: “The advocates of the explanatory indispensability argument have argued that we should extend our realist commitments from typical unobservable realist posits, such as electrons, to mathematical and other abstracta. This holistic application of explanatory reasoning in defence of realism has become another point of contention in the scientific realism debate.” (Saatsi 2018, 203). (Lange 2013) analyzes the notion of a distinctively mathematical explanation. The Borsuk-Ulam theorem states that if f is a continuous function from the sphere into R2 , two antipodal points x, y exist, such that f(x) = f(y). This is perhaps the most popular of the elementary examples discussed in the last years – it is very easy to understand and makes use of very elementary mathematical notions. Of course, there is also a biological ingredient in the explanation: there are bounds on the length of the periods and there is a natural hypothesis that in the evolutionary history of the cicadas, there were predators exhibiting periodical behavior. Of course, it would have been evolutionarily advantageous for the cicadas not to meet them and prime periods – such as 13 and 17 – minimize the number of such meetings.

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the particular structure of honeycombs. It is explained by the (mathematical) fact that hexagonal tiling minimizes the total perimeter length – which means that the amount of wax is minimal, and this gives the bees an evolutionary advantage.34 Many examples of this kind are discussed in the literature.35 The interplay between the problem of mathematical explanations in science and the problem of explanations within mathematics should be mentioned here as well. Explanations within mathematics are a debated topic (see Mancosu 2018, for discussion and comprehensive references). One of the problems to mention is the problem of the explanatory character of mathematical proofs – and their possible explanatory contribution in scientific explanation. Consider theorem A which appears in the explanans for a physical phenomenon. Has its proof any explanatory relevance? On one possible view, the proof of A indeed has some contribution to the explanation: intra-mathematical explanations may spill over into the empirical realm. The idea is that if, say, the Borsuk-Ulam theorem is explained by its proof and the antipodal weather patterns are explained by the Borsuk-Ulam theorem, it would seem that the proof of the theorem is at least part of the explanation of the antipodal weather patterns. (Baker, Colyvan 2011, 327) So, there is some kind of transitivity: the theorem explains the scientific facts, and the proof explains the theorem (so – indirectly – also the facts). In the second view, the proof does not matter: it is only theorem A that has explanatory input. Given that the proof justifies the theorem, we are then entitled to make use of the theorem, e.g., in applications to physical facts. (…) The role of the proof of that theorem is to justify the acceptance of that theorem. (Daly, Langford 2009, 648)36 Of course, the thesis that mathematics makes a genuine explanatory contribution to science is strongly disputed, especially by anti-realists. The detailed

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The optimality of hexagonal tiling has been proved by (Hales 2000, 2001). However, (Räz 2013, 2017) argues that this example rests on a misunderstanding, as real-live honeycombs are not flat, so that Hales’ results cannot be applied to them (at least not directly). See for instance (Lyon, Colyvan 2008), (Baron 2014) – and the extensive bibliography in (Lange 2017). (Wójtowicz 2019) discusses this problem in more detail.

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discussion of this topic is outside the scope of this paper. Here I assume it as a working hypothesis and discuss it in the context of Hellman’s theory.

4

Modal Notions and Mathematical Explanations

An important group of approaches to the problem of mathematical explanations is based on the claim that – broadly speaking – mathematics provides an abstract characterization of physical systems, and that this characterization has a modal character. The seminal paper for these approaches is (Jackson, Pettit 1990). The authors claim that explanations of physical facts (events, processes) should be based not only on identifying particular scenarios and examining the causal nexus of events – but also on identifying the abstract property of the system in question. This abstract property of the system is – in a sense – responsible for the particular course of events, but is not a causal property: “not efficacious itself, the abstract property was such that its realization ensured that there was an efficacious property in the offing” (Jackson, Pettit 1990, 114). One of the examples of phenomena to explain is a glass cracking after pouring hot water into it. Of course, we know the molecular mechanism: a group of molecules moving with a high momentum hits a particular place and causes changes in the structure of the glass at the micro-level – which means that the glass cracks. But we rather want to know what general features of the system are responsible for this course of events – i.e. for the inevitable cracking of glass in certain circumstances. This inevitability is of a mathematical rather than a (purely) causal character. We are interested not in the particular detailed scenario (for instance: what was the exact momentum of the particular particles?) – but rather an explanation of the inevitability of cracking, which is a quite general phenomenon. Take another simple example from Jackson’s and Pettit’s paper: a square peg does not fit into a hole with a diameter equal to the side of the peg. We can explain the impossibility of putting the peg into the hole by mathematics alone, we do not need to know anything more (apart from assuming, e.g. the impenetrability of the peg, its constant shape, etc.). This is a much better explanation of the impossibility of fulfilling the task than by describing the detailed location, rotation, speed, etc. of the peg bumping into the particular piece of the board. According to Jackson and Pettit, identifying the mathematical structure of the problem, rather than identifying the particular causal nexus explains the phenomenon.37 37

We might give very complicated examples of such phenomena, but also very elementary illustrations are available: if we have 5 apples in the basket, and 2 apples are taken away,

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Another instructive example are soap bubbles, which form special structures – in particular, they meet at certain, fixed angles, obeying Plateau laws. This example was studied by (Lyon 2012), who discusses the idea of considering the mathematical explanations as program explanations (in the sense of Jackson and Pettit). Lyon gives the following, general characteristic of program explanations: They cite properties and/or entities which are not causally efficacious but nevertheless program the instantiation of causally efficacious properties and/or entities that causally produce the explanandum. And, importantly, they cite mathematical properties and/or entities that are doing (at least part of) this programming work. (Lyon 2012, 567) He goes on to discuss the mechanism, indicating that even a very detailed knowledge of the behavior of the soap particles would lack an important ingredient: it would not give us the knowledge that forming the Plateau’s soap film is necessary and that it is the result of a certain optimization process (minimizing the surface area) taking place there.38 Plateau’s laws are in a sense programming the soap film to behave in a certain way. Lyon stresses the modal character of the program explanations: a certain scenario proves to be inevitable. And importantly: it is mathematics which provides this constraint, not physics.39 The same example is discussed in a detailed way by (Pincock 2015), who develops a slightly different account of the explanation. His characterization is as follows: Abstract explanations involve an appeal to a more abstract entity than the state of affairs being explained. I show that the abstract entity need

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the inevitability of 3 apples remaining in the basket stems from (very elementary) mathematics, not from laws of physics (of course, assuming some basic facts – for instance the stable behavior of the apples – they do not duplicate or disappear). “Changing the exact locations and movements of the soap molecules would not change their final macroscopic form. The mathematical explanation involving the mathematical fact from the theory of minimal surfaces gives us this modal information.” (Lyon 2012, 567). We might use the analogy with computer programs: the functioning of the hardware is subject to mathematical limitations. Even if the laws of physics were different, the (classical) computer would not solve any problem, which we know (from mathematics) to be unsolvable. It is not physics which makes solving the halting problem impossible, the limitations are of a different kind.

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not be causally relevant to the explanandum for its features to be explanatorily relevant. (Pincock 2015, 857)40 But regardless of the subtle differences between the accounts, there is always some modality in play. It has a mathematical character and it is stronger than the physical (natural) modalities: the inevitability of some facts is claimed to be stronger than just inevitability by the laws of nature. They have to take place regardless of the particular causal laws, and of the particular form of laws of nature. Modality might be naturally considered to define a set of constraints which any system has to obey.41 An important class are what Lange calls “distinctively mathematical explanations”, which have a modal character as well: “The mathematical facts figuring in distinctively mathematical explanations possess one such stronger variety of necessity: mathematical necessity” (Lange 2017, 10). These types of explanations (program, abstract, explanations by constraints, distinctively mathematical explanations) are therefore all based on the notion of the mathematical modality being stronger than the natural one (but not as strong, as the logical modality). But what do we mean by saying “It is mathematically necessary that…”? The standard explication is given in terms of possible worlds: mathematical truths hold in all possible worlds. In other words: they determine the borderline of the possible, as a world where mathematical truths do not hold is impossible.42 An interpretation in terms of a version of modal realism seems therefore to be most appropriate. But what do we mean when we say that some modal truth holds? And obviously this interpretation is not welcomed by the nominalists who reject the existence of abstract objects. So they need to give an account of a mathematical explanation in anti-realistic terms. In my opinion, Hellman’s theory is well suited to provide a smooth epistemology and semantics for modalities involved in the explanatory enterprise without making commitments to any form of modal realism.43 40

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“We think we have an explanation when we have found a (1) classification of systems using (2) a more abstract entity that is (3) appropriately linked to the phenomenon being explained. Whenever an explanation has these three features I will say that we have an abstract explanation.” (Pincock 2015, 867). (Lange 2017) studies this class in great detail: “Explanations by constraint work not by describing the world’s causal relations, but rather by describing how the explanandum arises from certain facts (“constraints”) possessing some variety of necessity stronger than ordinary laws of nature possess.” (Lange 2017, 10). I do not discuss the problem of impossible worlds. Some authors take this notion as sound, I am skeptical. Consider Baker’s cicada example: the realistic account is straightforward (there are numbers, there are mathematical truths – and we appeal to them when explaining biological

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An Antirealistic Account of Mathematical Modality

The explanatory (and explicatory) roles of mathematics and modal metaphysics are intermingled. (Some of) the realists argue for mathematical realism on the basis of the explanatory virtues of mathematics. These virtues are connected with the modal status of mathematics, which sets constraints (programs) on the physical world so a notion of a mathematical modality is invoked. But the standard explication of the notion of modality is via possible worlds semantics. This realistic possible-worlds semantics in the background (defining the mathematical modality) leads to a realistic semantics for mathematics (via the EIA). On the other hand, the class of possible worlds is characterized by mathematics, as any possible world has to obey the laws of mathematics. A kind of circularity lurks in the background: the realistic interpretation of mathematics is based on its explanatory virtues, which are provided by postulating mathematical modalities – however, modal claims (underlying these modalities) are justified by appealing to mathematical truths.44 This might be a source of possible problems for the realistic approach. The anti-realist might make use of this weakness by arguing that the notion of mathematical modality (used by the adherents of the explanatory role of mathematics) is not sound. The discussion is then shifted to the problem of the status of modal notions – and the burden of proof is transferred to the realist who is willing to use them in the pro-realistic argumentation.45 But Hellman’s proposal might also be used by the anti-realist who does not reject the notion of mathematical modality, and even grants that this modality (and – more generally – mathematics) can have indispensable explanatory contributions to science. Obviously, the anti-realist

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facts), so the challenge for the anti-realist is to give a satisfactory account in modal terms. The general idea might be to reformulate number-theoretic claims in modal terms, like “If natural numbers existed, then the cicada life cycle would obey some number-theoretic constraints – anyway, natural numbers do not exist, but the life cycle obeys these rules”. But we face a very general problem: is it justified to appeal to non-existent entities (phenomena, forces, etc.) when formulating scientific explanations? The adherents of EIA (including of course Baker) would surely disagree. (I am indebted to the anonymous reviewer for inspiring comments). The answer might be in terms of the inference to the best explanation account: we explain scientific facts by using the notion of modality, and the best explanation of this modality is via mathematical realism. One of the referees suggested that this circularity might be an important argument for introducing modalities. Indeed, if we think of them in primitive terms – the circularity disappears, as we do not have to justify the reduction of modalities to possible worlds, or – more generally – to any class of truths.

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has to reject the explication on the modal concepts in realistic terms. Hellman’s theory is promising in this respect: a primitive modality is the source of the truth of mathematical claims. The notion of truth of mathematical sentences need not involve the claim that mathematical objects exist. There are no mathematical structures – the only assumption we are obliged (and entitled) to make is the possibility of their existence. Importantly, Hellman does not assume the existence of possible structures – on the contrary, he assumes only the possibility of their existence, which is a much weaker claim.46 Modality is a primitive notion, we do not reconstruct it in (familiar) set theoretic terms: in particular, we do not interpret the possibility of α as the existence of a structure (model) M, where α is true. Similarly, we do not interpret the necessity of α by appealing to a realistic notion of a class of structures where α invariably holds.47 A subtle problem arises here: Hellman makes use of a primitive logical modality, which need not have a specific mathematical character. But the modality I mentioned in the previous paragraph might have a specific mathematical character – which relies on stronger assumptions than purely logical. So an anti-realist relying on the notion of a mathematical modality has to make stronger assumptions than Hellman (purely logical modality might not be sufficient – just as purely logical principles were not sufficient to provide a reconstruction of mathematics, as the failure of the logicist program – at least in its original form – shows clearly). But in this case, the antirealists might make use of this primitive modality in a direct way – without relying on Hellman’s complex strategy – in order to account for the role of mathematics.48 So, in particular, the objectivity of mathematics does not have its source in any act of faith of the realists. But nevertheless, mathematics retains its objectivity: Hellman is an advocate of realism in truth value (from which realism in ontology does not follow). He is able to retain the objective character of mathematics by invoking modal notions treated as primitive. This fits well into the view that there is some kind of mathematical modality, which has impact on

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If we write it in a semi-formal way, Hellman’s claim is rather ⬨∃SM(S) than ∃S⬨M(S): he does not postulate the existence of possibilia. For a discussion of a primitive notion of modality see (Hale 2013). Field also made use of a primitive modality in (Field 1991) (see: Shapiro 1993; Wójtowicz 2001, for discussion); Balaguer’s fictionalist account from (Balaguer 1998) might also be interpreted in this way. (Leng 2007) discusses Field’s ideas and argues for a primitive modality as a basis for a fictionalist account of mathematics. (I am indebted to the anonymous reviewer for indicating the problem of the relationship between the logical and the mathematical modalities and for the helpful suggestions).

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our world (it sets some non-causal constraints), and which has explanatory potential. Hellman’s modal account can be applied to explain the epistemic source and the metaphysical status of these constraints. It is not necessary to interpret them as truths in the class of possible worlds: they have a “modally self-contained” character, as the modality is primitive. We are freed of any ontological commitments which might arise when giving an interpretation of the modality in question. Ontological assumptions are eliminated and replaced by a kind of modal ideology. The advantage of this approach is that it uniformly treats the modal notions involved in the discussion concerning both the ontology and the explanatory value of mathematics. The price to pay are the ideological assumptions which have to be made: it is necessary to assume a primitive mathematical modality, which applies to our world, setting constraints on the possible courses of events (and describing the realm of structural possibilities49). In Hellman’s account, the anti-realist can freely use modal notions without being committed to possible worlds or abstract structures. And even the explanatory force of mathematics (couched in modal terms) does not support the realistic interpretation.

6

Concluding Remarks

The classic indispensability argument is based on the role mathematics plays in science. In its more recent form, it has taken the form of the EIA, where the explanatory role of mathematics plays a crucial role. Of course, this is not a causal kind of explanation, and it might be explicated in terms of a mathematical modality, setting constraints or programming the world via its abstract properties (here I neglect the subtle differences between the accounts). From the realistic point of view, this modality can be interpreted in terms of mathematical truth, which presupposes the existence of mathematical structures. This account is coherent – but obviously not acceptable for the antirealist. This means that a natural problem arises for these antirealists, who accept the notion of non-causal explanation, and moreover accept the idea of a mathematical modality setting constraints/ programming the world. They need to explain the status of modality without presupposing the realm of abstract structures. Hellman’s account is very promising here: it gives a coherent

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“Mathematics is the free exploration of structural possibilities, pursued by (more or less) rigorous deductive means” (Hellman 1989, 6).

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account of mathematics in modal terms (also of applications of mathematics, which is crucial for the debate), and this account can be extended to the anti-realistic interpretation of mathematical modality entering explanations of diverse kinds. The notion of mathematical truth is preserved (but interpreted differently). This poses a challenge to mathematical realism based on the indispensability argument, also in its recently discussed form.

Acknowledgments I would like to thank both anonymous reviewers whose detailed and insightful comments helped us to improve presentation of the paper. Of course, I am responsible for any mistakes. The preparation of this paper was supported by NCN grant 2016/21/B/HS1/01955.

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Elster, J. (1983) Explaining Technical Change. Cambridge: Cambridge University Press. Field H. (1980) Science Without Numbers: A Defence of Nominalism. Oxford: Blackwell. Field H. (1989) Realism, mathematics and modality. Blackwell, Oxford, Cambridge. Field H. (1991) “Metalogic and Modality”, Philosophical Studies, 62, 1–22. Hale B. (2013) Necessary beings. An Essay on Ontology, Modality, and the Relations Between Them. Oxford University Press, Oxford. Hales, T. C. (2000) “Cannonballs and honeycombs”, Notices of the American Mathematical Society, 47 (4), 440–449. Hales, T. C. (2001) “The Honeycomb Conjecture.” Discrete and Computational Geometry, 25, 1–221. Hellman G. (1989) Mathematics without Numbers, Oxford: Clarendon Press. Hellman G. (1996) “Structuralism without structures”, Philosophia Mathematica, 3, vol. 4, 100–123. Hellman G. (2001) “Three varieties of mathematical structuralism”, Philosophia Mathematica, 9(2), 184–211. Hellman G. (2005) “Structuralism”, in: Shapiro, S. (ed.) The Oxford Handbook of Philosophy of Mathematics and Logic, 536–562. Oxford University Press, Oxford. Hempel C. G. (1945) “On the Nature of Mathematical Truth,” American Mathematical Monthly 52, 543–546. Reprinted in: Benacerraf P. Putnam H. (eds.), Philosophy of Mathematics, second edition, 1983, CambridgeUniversity Press, Cambridge, 377–393. Jackson F., Pettit P. (1990) “Program explanations: A general perspective”, Analysis, 50(2), 107–117. Lakatos I. (1978) “A renaissance of empiricism in the recent philosophy of mathematics?” British Journal for the Philosophy of Science, 27 (3), 201–223 (reprinted in: Lakatos I. Philosophical Papers, t. 2: Mathematics, Science and Epistemology, Worall J., Currie G. (red.), Cambridge University Press, 24–42). Lange M. (2013) “What Makes a Scientific Explanation Distinctively Mathematical?”, British Journal for the Philosophy of Science 64 (3), 485–511. Lange M. (2017) Because Without Cause, Oxford University Press, Oxford. Leng M. (2007) “What’s there to know? A Fictionalist Approach to Mathematical Knowledge”, in: Leng M., Paseau A., Potter M., (eds.), Mathematical Knowledge. Oxford: Oxford University Press, 84–108. Leng M. (2017) “Mathematical Realism And Naturalism”, in: The Routledge Handbook of Scientifc Realism, Routledge, New York, Saatsi J. (ed.), 407–418. Lyon A. (2012) “Mathematical Explanations of Empirical Facts, and Mathematical Realism”, Australasian Journal of Philosophy 90 (3), 559–578. Lyon A., Colyvan M. (2008) “The Explanatory Power of Phase Spaces”, Philosophia Mathematica 16 (2), 227–243.

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Malament D. (1982) “Review of Science Without Numbers”, Journal of Philosophy, 79, 523–534. Mancosu, P. (2018) “Explanation in Mathematics”, The Stanford Encyclopedia of Philosophy (Summer 2018 Edition), Edward N. Zalta (ed.), URL = https://plato.stanford .edu/archives/sum2018/entries/mathematics-explanation/. Melia, Joseph (2000) “Weaseling Away the Indispensability Argument”. Mind, 109, 455–79. Melia, Joseph (2002) “Reply to Colyvan”, Mind, 111, 75–79. Pincock C. (2015) “Abstract Explanations in Science”, British Journal for the Philosophy of Science, 66, 857–882. Putnam H. (1967) “Mathematics without foundations”, Journal of Philosophy, 64, 5–22, reprinted in: Benacerraf P. Putnam H. (eds.), Philosophy of Mathematics, second edition, 1983, CambridgeUniversity Press, Cambridge, 295–311. Putnam H. (1975) “What is mathematical truth?”, reprinted in: Mathematics, matter and method: philosophical papers, Vol. 1, 2nd. Ed. Cambridge University Press (1979), 60–78. Quine W.V.O. (1939) “Designation and Existence”, Journal of Philosophy, 36, 701–709. Quine W.V.O. (1948) “On What There Is”, The Review of Metaphysics, 2(1), 21–38. Reprinted in From a Logical Point of View, Cambridge, Harvard University Press, 1953, 1–19. Quine W.V.O. (1951) “Two dogmas of empiricism”, Philosophical Studies, 2, 20–43, reprinted in: From a Logical Point of View, Cambridge, Harvard University Press, 20–46. Quine W.V.O. (1981) “Things and Their Place in Theories”, w: Theories and Things, Cambridge, Mass: The Belknap Press of Harvard University Press, 1–23. Räz T. (2013) “On an Application of the Honeycomb Conjecture to the Bee’s Honeycomb”, Philosophia Mathematica, 10 (5), 322–333. Räz T. (2017) “The Silent Hexagon: Explaining Comb Structures”, Synthese, 194, 1703–1724. Resnik M.D. (1985) “How nominalist is Hartry Field’s nominalism?”, Philosophical Studies, 47, 163–181. Resnik M.D. (2005) “Quine and the web of belief”, in: The Oxford Handbook of Philosophy of Mathematics and Logic, Shapiro S. (ed.), Oxford University Press, Oxford, 412–436. Reutlinger A., Andersen H., (2016) “Abstract versus Causal Explanations?”. International Studies in the Philosophy of Science, 30 (2), 129–146. Reutlinger A., Saatsi J. (2018) Explanation Beyond Causation. Philosophical Perspectives on Non-Causal Explanation, Oxford University Press, Oxford.

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Russell G. (2019) “Logical Pluralism”, The Stanford Encyclopedia of Philosophy (Summer 2019 Edition), Edward N. Zalta (ed.), URL = https://plato.stanford.edu/archives/ sum2019/entries/logical-pluralism/. Saatsi J. (2018) “The Limits of explanatory reasoning”, in: The Routledge Handbook of Scientifc Realism, Routledge, New York, Saatsi J. (ed.), 200–211. Salmon W. (1977) “A Third Dogma of Empiricism.” In: Basic Problems in Methodology and Linguistics, edited by Robert Butts and Jaakko Hintikka, 149–166. Dordrecht: Reidel. Salmon W. (1984) Scientific Explanation and the Causal Structure of the World, Princeton: Princeton University Press. Shapiro S. (1983) “Conservativeness and incompleteness”, Journal of Philosophy 60, 524–531. Shapiro S. (1993) “Modality and Ontology”, Mind, 102, 455–481. Shapiro S. (2014) Varieties of Logic, Oxford: Oxford University Press. Sher G. (1991) The bounds of logic. A generalized viewpoint. MIT Press, Cambridge, Massachusetts and London. Stern J.M. (2011) “Constructive Verification, Empirical Induction, and Falibilist Deduction: A Threefold Contrast”, Information, 2, 635–650, 2011. Stern J.M. (2015) “Cognitive-Constructivism, Quine, Dogmas of Empiricism, and Muenchhausen’sTrilemma”. In: Springer Proceedings in Mathematics &Statistics, 118, 55–68. Tallant J. (2013) “Optimus prime: paraphrasing prime number talk”, Synthese 190, 2065–2083. Walton K. (1993) “Metaphor and Prop Oriented Make-Believe”, European Journal of Philosophy, 1, 39–57. Wójtowicz K. (2001) “Some remarks on Hartry Field’s notion of logical consistency”, Logic and Logical Philosophy, (9), 199–212. Wójtowicz K. (2020) “The status of mathematical proofs and the Enhanced Indispensability Argument”, in: Formal and Informal Methods in Philosophy, Będkowski M., Brożek A., Chybińska A., Ivanyk S., Traczykowski D., (eds.), Brill, 180–194. Yablo, S. (1998) “Does Ontology Rest on a Mistake?” Aristotelian Society, Supplementary Volume, 72, 229–61. Yablo, S. (2002) “Go Figure: A Path Through Fictionalism,” Midwest Studies in Philosophy, 25, 72–102.

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Chapter 4

Apriorism, Aposteriorism and the Genesis of Logic Jan Woleński

Abstract This paper defends moderate aposteriorism in the philosophy of logic. Yet logic is conceived as a set of tautologies, that is, truths in all models (possible worlds). Aposteriorism assumes genetic empiricism, that is, the view that every piece of knowledge is rooted in experience. Since according to present trends in the philosophy of logic, experience cannot generate universal truths, empiricism (aposteriorism) cannot explain the nature of logic. The paper shows that evolutionary biology and genetics provide data for conceiving logic as rooted in the properties of DNA.

Keywords consequence operation – semantics – topology – DNA – evolution – genetic code

1

Introduction

This paper is closely related to (Woleński 2012), in which I outlined a naturalistic account of the genesis of logic. Now I will try to enlarge this philosophical perspective by locating the problem in a more general setting, namely in the context of the apriorism/aposteriorism controversy – this means that I consider the issue of the genesis of logic as epistemological. As a matter of fact, standard textbooks of the philosophy of logic (Quine 1970; Putnam 1991; Haack 1978; Fisher 2007; Cohnitz, Estrada-Gonzáles 2019), as well as collections, like (Jacquette 2001; Shapiro 2005; Jacquette 2007), although they, more or less focus on apriorism and aposteriorism, do not directly address to the question “How did logic arise?” Perhaps the problem is left to cognitive science or evolutionary psychology, but I will not enter into the causes of this situation. To repeat, my position consists in considering the genesis of logic as a legitimate epistemological issue, independently of its actual connections with investigations in other disciplines. This does not mean that philosophers working in the philosophy of logic should ignore what psychologists or cogni-

© Jan Woleński 2021 | DOI:10.1163/9789004445956_006

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tive scientists say about the genesis of our (as well as other animals’) mental faculties. A convenient starting point consists in considering three questions about the epistemology of logic listed in (Cohnitz, Estrada-Gonzáles 2019, 140). They are as follows: (A) Is logic revisable? (B) Is logic a priori or a posterori? (C) Can we justify logic? Clearly, the answers depend on a view concerning the epistemological status of the laws of logic. Take for granted that logic is universal in the sense that its laws are true in all circumstances (this account will be more closely analyzed below). If aposteriorism (or empiricism, if you like) claims that all knowledge is based on experience, it has a serious problem with answering the questions (A)–(C). In particular, the revision of logic seems to be at odds with its universality. However, the apriorist (or rationalist, if you like) also encounters serious trouble, because he or she must assume that the possessors (humans or other species) of logic are equipped with a special disposition to grasp logic. Even if this ability explains the universality property, it seems at odds with logic as revisable. Inspecting (A)–(C), we easily observe that (B) appears fundamental. And if so, any analysis of whether logic is a priori or a posteriori presupposes a view on the genesis of logic. My further analysis proceeds as follows. Section 2 is devoted to an analysis of apriorism and aposteriorism, section 3 offers a formal definition of logic, section 4 contains various philosophical comments on logic as already defined, and section 5 – a proposal, which tries to reconcile aposteriorism, the evolutionary account of the genesis of logic and the universality property. Still one preliminary remark is in order. A customary view consists in including logic, together with mathematics, in the so-called formal sciences. Doubtlessly there are important and convincing reasons for such an account. On the other hand, I guess that similarities between logic and mathematics do not exclude looking at logic and its philosophical problems as somehow autonomous with respect to mathematics. Even if we admit that we have a considerable evidence for answering the question “Where does mathematics comes from?” to use the nice title of Lakoff, Núñez 2000, but to a much lesser extent, to handle “Where does logic comes from?”, the second question is legitimate. Although the empiricist in the philosophy of logic assumes, like his or her “compatriot” in the philosophy of mathematics does, that the embodied mind “invented” logic, we have no bones with cuts imitating reasoning, although we have various traces of primitive calculations.1 Yet I will employ deliberations on the genesis of mathematics only parenthetically. 1 I guess that logical tools and philosophical distinctions used in this book are standard. On the other hand, suggestions offered in section 6 try to interpret biological facts in order to

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I follow the approach presented in Ajdukiewicz 1947 and Ajdukiewicz 1949, Chapter 3 (all page or chapter references are to translations or first editions, if cited in the bibliography at the end of the paper) with some comments from the present point of view. He considered the problem of the sources of knowledge as one of the three principal questions of epistemology – the others are truth and the limits of knowledge. The basic distinction is that of the psychological (genetic) origins of knowledge vs. the epistemological (methodological) sources of knowledge. Here is the explanation (Ajdukiewicz 1949, 22–25; italic in the original): There was controversy as to whether among the concepts which we came across in the mind of an adult human being there are innate thoughts and concepts, or whether the concepts and thoughts which we possess are entirely formed by experience. Those who believe in the existence of innate ideas are called genetic rationalists or nativists.; those who hold the opposite opinion are called genetic empiricists. […]. The problem of genetic rationalism and empiricism as dealing with the origin of our ideas and beliefs, which we have briefly discussed, is a problem of a distinctly psychological character. It is concerned in factual fact with the way of in which thoughts come to be in the human mind. With this psychological problem there has been connected, and somehow confused, another problem, not psychological, but methodological or methodological in character. This is the problem of how we can arrive at the fully justified knowledge of reality, that is by what method we can arrive at knowledge which is true. This problem belongs to the theory of knowledge that is to the discipline which is concerned with not with the factual occurrence of cognition, but with its truth and justification. […] in the previous discussion the terms ‘rationalism’ and ‘empiricism’ were prefaced with the word ‘genetic’; now, in contrast, we should talk about methodological rationalism and empiricism. […] the position opposed to methodological empiricism […] we shall call it apriorism. explain the genesis of logic. I would like to stress that I am not attempting to enrich biology, but to use it philosophically. This remark is intended as a reply to some reservations made by the reviewers who expressed their doubts concerning the status of the considerations contained in this paper. Of course, I am fully aware that many of my assertions could be the starting point for extensive papers or even monographs. However, this is a typical situation in philosophy. On this occasion, I would like to express my indebtedness to the reviewers for their valuable comments.

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Note that Ajdukiewicz himself did not use the term ‘aposteriorism’, but this label seems equally legitimate. Ajdukiewicz’s explanations lead to some doubts concerning the character of genetic empiricism (its famous slogan is nihil est in intellectu quod non prius fuerit in sensu) and rationalism (its dictum is nihil est in intellectu nisi intellectus ipse). One reading of this distinction, suggested by the contrast genetic/methodological, is that both positions belong to psychology (today, perhaps prefaced by cognitive), not to epistemology, but the second interpretation, supported to some extent (this qualification is fairly relevant) by the intuition that if we have a contrast of two accounts, in the discussed context, genetic and methodological rationalism (empiricism), and the latter has a philosophical (epistemological) import, its opposition shares the same feature. I am not insisting that this is just the case (hence, I employed the phrase ‘to some extent’ above), but I consider treating genetic empiricism and rationalism as an admissible position in the analysis. Moreover, Aristotle, Locke, Hume or John Stuart Mill represented genetic empiricists, but Plato, Descartes, Leibniz or Kant represented genetic rationalists (nativism). Thus, let us investigate the relations between empiricism (aposteriorism) and rationalism (apriorism) from a philosophical point of view. I repeat once again that this circumstance does not exclude appealing to empirical data. By the way, the present situation in epistemology and cognitive science is also somehow ambiguous as far as the issue concerns the distribution of philosophical and empirical aspects in the analysis of knowledge (cognition). At first glance, if a philosopher accepts genetic empiricism, he or she represents aposteriorism and reversely; a similar correlation is expected in the case of nativism and apriorism. The historical examples are straightforward, because Aristotle, Locke, Hume and Mill were genetic empiricists and aposteriorists, but Plato, Descartes and Leibniz – nativists and apriorists. However, Kant does not fit this scheme – for him, experience is the source of cognition, but a priori elements of knowledge decide its truth and justification. Kant famously divided propositions according to two criteria. For him, every proposition can be reduced to the form ‘S is P’. Take into account the relation between the content of S and P. If the content of P is contained (that is, smaller or at most equal) to the content of S, the entire proposition is analytic – it is synthetic in the opposite case. Second, a proposition is a posteriori, if it is justified on the basis of (sense) experience – it is a priori in the opposite case. Combining both criteria, we obtain four kinds of propositions, namely analytic a posteriori, analytic a priori, synthetic a posteriori and synthetic a priori. Since, according to Kant, all analyticals are a priori, we finally have three classes of propositions: (i) analytic; (ii) synthetic a priori; (iii) synthetic a pos-

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teriori. Looking at (i)–(iii) from a different axis, first we divide propositions into a priori and a posteriori, and, second into analytic and synthetic. Now, assuming that the set of analyticals a posteriori is empty, we also obtain the list (i)–(iii). Kant used the above classification of propositions in characterizing the sciences. Logic consists of analytic propositions. Kant identified them with tautologies, not extending our knowledge. Logic for him was trivial and completed by Aristotle; he maintained that nothing important happened in logic after the Stagirite. Historical records illustrate synthetic a posteriori assertions and provide new factual knowledge (later called idiographic). For Kant, syntheticals a priori are the most important propositions – they are present in arithmetic, geometry and theoretical physics. Kant’s famous question was as follows ‘How are synthetic a priori propositions possible?’ More specifically, how are propositions extending our knowledge and a priori possible? According to a simplified account of aposteriorism and apriorism, the concept of syntheticals a priori is apparently inconsistent, for no true proposition can be simultaneously synthetic and a priori. Without entering into details about Kant’s Copernican revolution in philosophy, as he called, not quite modestly, his well-known response to the question concerning the possibility of synthetic a priori propositions, they are generated by the structure of our mind (reason), in particular, by the concepts of time (arithmetic) and space (geometry), and categories, especially causality, which is relevant for physics. Kant’s solution tried to overcome the opposition of apriorism and aposteriorism in the traditional setting of these views. He argued that all kinds of propositions, that is, (i), (ii) and (iii), are admissible in our knowledge. Admitting (ii) was at odds with traditional apriorism, but admitting (i) and (iii) contradicts aposteriorism. Ajdukiewicz (see Ajdukiewicz 1947) proposed a refined scheme for aposteriorism and apriorism. His ingenious idea consisted in coordinating the types of propositions with the species of both competing views concerning the methodological sources of knowledge. This proposal can be illustrated by the following table: (I) Radical apriorism – only propositions a priori (analytic and synthetic); (II) Moderate apriorism – all kinds of propositions according to the list (i)–(iii); (III) Radical aposteriorism – only synthetic a posteriori propositions; (IV) Moderate aposteriorism – analyticals and synthetic a posteriori propositions. Plato, Descartes and Leibniz represented (I), Kant – (II), Mill, Comte, Mach, Avenarius – (III), Hume – (IV) (later this combination was adopted by logical

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empiricism). Ajdukiewicz’s map also works for more specific historical points. For instance, one can point out that Plato’s distinction of intuitive and discursive knowledge can by captured by saying that the former is synthetic a priori, but the latter – analytic, in most popular versions of moderate aposteriorism, for instance, that of logical empiricism, mathematics and logic are analytic, the rest of the sciences – synthetic a priori. Ajdukiewicz’s approach via (I)–(IV) essentially helps in discriminating the advantages and disadvantages of particular views. Leaving radical apriorism as having exclusively historical significance, moderate apriorism has the resources to explain the genesis of all kinds of propositions expressing knowledge via appealing to the sense-experience as well as innate factors, but, on the opposite edge, radical aposteriorism ascribes the same role to empirical data. Yet (II) has difficulties with a closer accounting of the nature of what is innate. Various and conflicting interpretations of a priori intuition (for instance in the Kantian or Husserlian sense) very well document the issue in question. Radical aposteriorism must accept the empirical status of logic and mathematics, and this view does not cohere with the universality property of logic and the independence of mathematical proof from empirical data (I quite intentionally omit the problem of the universality of mathematics). Finally, moderate aposteriorism deals well with empirical problems, but fails in explaining how analytic propositions appeared as the result of cognition. Using another perspective, that is, combining the genetic and methodological perspective, radical apriorism is also nativism – all genuine knowledge is episteme and dependent on innate ideas, empirical knowledge is at most doxa in Plato’s sense, moderate apriorism – is partial nativism and partial genetic empiricism, but with the distinguished role of innate elements. Radical aposteriorism is also genetic empiricism, and moderate aposteriorism – genetic empiricism. Now although radical aposteriorism is ready to pay the cost of rejecting the universality property of logic, moderate aposteriorism keeps the view that logic is universal and does not abandon genetic empiricism. Thus, we have a tension between the main epistemological postulates of the position (IV). Let me add (see the remarks on the relation between the present paper and Woleński 2012 in the Introduction above) that naturalism can be incorporated into both kinds of aposteriorism. Anticipating the results of my analysis, I will defend moderate aposteriorism cum naturalism. Any project of defending this or that philosophical view, in this case apriorism or aposteriorism, requires some preliminary (or parallel) explanations, in particular, relating to the status of the sources of cognition. I observed above that the concept of intuition in apriorism requires additional explanation. On

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the other hand, the same concerns the notion of experience or empirical data. Sensualism (Berkeley, Condillac) offers the most restrictive approach – only senses provide empirical information. Hume and Kant admitted psychological experience. What about the knowledge of language? Further possible extensions concern logical experience, mathematical experience, axiological experience or religious experience. I must confess that I have no idea how to define experience in a satisfactory manner. My own preferences are consistent with Hume and Kant – they consist in accepting sense experience and psychological experience as basic forms. Agreeing that this characterization is not complete, I reject all extensions proposed by apriorists consisting in admitting various forms of intuition.

3

What Is Logic?

The word ‘logic’ has various applications. Polish tradition, having earlier anticipations, distinguishes logic in a narrow sense and broad sense. Whereas the latter (sensu largo) consists of semantics (or semiotics), formal logic and the methodology of science, the former (sensu stricto) is reduced to formal logic. Another distinction differentiates logica docens (theoretical logic, logic as a theory) and logica utens (applied logic, practical logic). Still another approach (see Bedürftig, Murawski 2018, 94) lists logic as a scientific discipline, as a language and as a system of calculus. I will focus on formal logic understood as defined by the concept of logical consequence. More particularly, I will concentrate on the classical first-order logic with identity (see Woleński 2004 for a justification of this choice) with some remarks on other systems. To complete the preliminary remarks in this section, let me add that the study of logic cannot be separated from metalogic. Consequently, I include metalogical themes into logic as such. In particular, any definition of logic has to employ the tools defined in metalogic. The concept of logical consequence (symbolically, Cn) appears as basic for defining logic (I follow Woleński 2012 and incorporate some fragments from this paper here). Assume that the operation Cn satisfies Tarski’s well-known general axioms, i.e. (a) denumerability of the language (a set of sentences) L in which logical calculus is formulated; (b) X ⊆ CnX (the inclusion axiom; X, Y are sets of sentences of L); (c) if X ⊆ Y , then CnX ⊆ CnY (monotonicity of Cn); (d) CnCnX = Cn (idempotence of Cn) and; (e) if A ∈ CnX, then there is a finite set Y ⊆ X such that A ∈ CnY (Cn is finitary). Cn is a mapping of the type 2L → 2L , that is, transforming subsets of L into its subsets. If we add the deduction theorem to the above axioms, that is, the formula (I employ the

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provability operator ⊢ and use a simplified formulation, more accurate for my discussion (1)

If A ⊢ B, then ⊢ A ⇒ B,

(the version with Cn: if B ∈ Cn{A}, then (A ⇒ B) ∈ Cn{A}) logic can be defined as the set of consequences of the empty set, symbolically (2) LOG = Cn∅. Furthermore, if we define a deductive system (theory) S by the condition S = CnS, LOG becomes the only common part on the consequences of all sets of sentences or the only deductive system included in non-logical systems. Otherwise speaking, logic is the only non-empty intersection of the family of all subsets of L. This fact implies that logic is included in the consequences of each set of sentences, which underlines its universal character. If CnX ⊆ X, then X = CnX due to the inclusion axiom. Moreover, if CnX ⊆ X, we say that that X is closed by the consequence operation of logical consequence. Thus, in the case of deductive systems, Cn does not extend X beyond itself. The concept of logical consequence belongs to the syntax of language. On the other hand, the notion of logical following (entailment) is a semantic counterpart of Cn, defined as the relation holding between A and B, if it is impossible that A is true and B is false. Consequently, if A ∈ CnX and X consists of true sentences, the sentence A also must be true as well. How to compare that A ∈ Cn and A is true? The answer is given by the completeness theorem, one of the most important metalogical results. Assume that the symbol MX refers to a model of the set X, that is, a structure in which all sentences belonging to X are true (I skip the definition of M – we write VER(M) for denoting the class truths in M. The (semantic) completeness theorem says (3) A ∈ CnX iff (if and only if) for any model MX , A ∈ VER(MX ). The above statement is a strong completeness theorem. Its weak version has the following form (4) A ∈ Cn∅ iff for any M, A ∈ VER(M). The last assertion, although it looks like a trivial instantiation of (2), has a considerable philosophical content. It allows us to define logic by

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(5) A ∈ LOG iff for any M, A ∈ VER(M). In words, A is a logical theorem if and only if A is true in every model (every state of affairs, every situation, every possible world, domain, etc.) The definition (2) can be qualified as artificial, because the derivability from the empty set of premises arguably appears as strange and even false – Cn functions as an abbreviation for a definite set of rules of inference. Thus, the expression Cn∅ should be read as follows “the set of consequences of the empty set” obtainable by rules of inference encoded by Cn’. On the other hand, the phrase for any M, A ∈ VER(M) possesses a clear meaning. Moreover, it fits well with the traditional intuition (see above about Kant) that logical laws are tautologies. More importantly, the right part of (5) precisely indicated why logic is universal.2 It is so because theorems of logic are universally truths. Due to the fact that various definitions of logic derived from (2) (logic as the only common part of the consequences of all sets of sentences, as the only nonempty intersection of the family of all subsets of L or as included in the consequences of each set of sentences) are equivalent, we have various, syntactic and semantic, features of the universality property of logic. Still another mark of the universality in question follows the next theorem (see Grzegorczyk 1974, p. 227–236): (6) Logic does not distinguish any extralogical content. The theorem of Löwenheim-Skolem-Tarski is related to (6). This theorem says (7) A first-order theory S has an infinitely denumerable model iff it has a model of arbitrary infinite cardinality, is frequently considered paradoxical, because, for example, it implies that the arithmetic of real numbers has a model in the domain of natural numbers. However, if we apply (7) to first-order logic (it is a fortiori a first-order theory), it is plausible. Obviously, the cardinality of models is an extralogical property.

2 Non-logical true statements are not universal in this sense. Consequently, if we introduce the concept of empirical universality, it must be otherwise defined than via truth in all models. The problem of mathematical universality provides a special problem due to the existence of non-standard models.

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Thus, purely logical means do not suffice in order to discriminate models of different cardinality.3 However, there are some doubts concerning (6) as applied to the first-order logical system. In particular, the concept of identity leads to difficulties. Identity, commonly considered as a logical constant, is not definable in first-order logic. Leibniz’s definition that, roughly speaking, two objects are identical, provided that they share the same properties, is obtainable in second-order logic. Hence, identity must be characterized in first-order logic by suitable axioms (reflexivity, symmetry, transitivity). Having identity, we can define so-called numerical quantifiers, represented by the scheme ‘there exists exactly n objects’ and they seem to express extralogical facts. Since the identity symbol allows us to define such quantifiers, (6) might be questioned as applied to first-order logic in its customary understanding. On the other hand, first-order logic with identity satisfies the completeness theorem, the identity predicate should be included in the list of logical constants. We have two ways out, namely (a) consider identity as an extralogical concept and restrict logic to first-order calculus without identity, or (b) to say that since every first-order theory is strongly complete, the status of identity can be settled by a convention. Both strategies are consistent with the universality property of logic. Anyway, we can conclude that the universality property of (first-order) logic is well-defined by various metalogical attributes of this system generated by metalogical theorems. The statement that Cn closes sets of sentences as long as CnX ⊆ X, suggests some analogies with topology, since certain properties of this operation satisfy Kuratowski’s axioms for topological spaces. Let Cl denote the closure operation of a topological space, and X, Y – any subspaces (subsets); I intentionally use the same letters for denoting the set of sentences and the sets investigated by topology. Then (see Duda 1986, p. 115): (a) Cl∅ = ∅; (b) ClClX = ClX; (c) Cl(X ∪ Y) = ClX ∪ ClY. Operations Cn and Cl differ from each other as far as the matter consists of axioms (a) and (c), because, in the case of logic, set Cn∅ is non-empty and CnX ∪ CnY ⊆ CnY) but the reverse inclusion does not hold. The first difference is founded on the specific definition of logic, which does not possess a clear topological sense, while the other one indicates a partial analogy between closed sets in the topological sense and deductive

3 I do not touch other aspects of (7), except (see below) its role in the Lindström theorem. Thus, the Löwenheim-Skolem-Tarski theorem is a tool for explaining the status of first-order logic and its universality. See Bays 2014 for a concise report on the problems relating to the theorem in question.

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systems in the logical sense, because (c) does not hold for arbitrary sets of sentences. Thus, ‘logical’ closure is weaker than the topological one. The set of theses of logic is with certainty non-empty and it is a system. It can be treated as a specifically closed topological space, with individual theorems as its points. The topology {∅ = Cl∅, X} is minimal (see Wereński 2007, p. 124) in the sense that the smallest one cannot be examined. Furthermore, we have Cl∅ ⊆ ClX since for each X, ∅ ⊆ X. Let us agree (this is a convention) that Cl∅ is a topological equivalent of logic. The motivation for this convention consists in taking into consideration t that a proof of logical theorems does not require any assumption. The evident artificiality of this convention can be essentially weakened by the acknowledgment that closing an empty set produces any theorem of logic. It can be shown that if A and B are theses of logic, then Cn{A} = Cn{B}, which means that any two logical truths are deductively equivalent. Let us assume that X (this time as a set of sentences) is consistent and consists of the set X′ of logical tautologies and a set X′′ of theorems outside logic. Thus, X′ = Cn∅ and X′′ ⊆ J − X′ . Sets X′ and X′′ are disjoint and constitute mutual complements in the set (space) X. Since set X′ is closed, its complement, i.e. X′′ is an open set. The introduced convention about Cl∅ allows one to “topologize” the properties of sets of theses, in particular, it makes it possible to treat the set X (of theses) as a clopen set. From the intuitive point of view, the operation of logical consequence encodes inference rules of deriving some sentences from other sentences, that is, the deduction of conclusions from defined sets of premises. Deduction, at the same time, is infallible, that is, it never leads from truth to falsity – this statement is a simple consequence of the definition of Cn in the semantic setting.

4

Logic or Logics

What about second (higher)-order logic, many-valued logic, intutionistic logic, modal logic, paraconsistent logic, non-monotonic logic, fuzzy logic, etc. (see Priest 2008 for a survey of various logics, although not all)? Is logic local or global? Should we admit the pluralism of logic (see Beall, Restall 2006)? Should we speak about deviant logic (see Haack 1996)? Terminology varies from one to another author – this fact indicates that particular labels express various intuitions. Since even a fragmentary discussion of the issue exceeds the scope of this paper, I limit myself to a few very general remarks. If someone accepts (2), agrees that Cn is monotonic, and adding special Cn-axioms, for instance, the excluded middle, we obtain bivalence. Let us agree that that monotonicity and bivalence define classical logic. Hence, non-monotonic logic

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and many-valued logic are deviant. A more complicated situation concerns intuitionistic or paraconsistent logic, because both might be viewed as bivalent under special truth-definitions. Leaving paraconsistency aside (its concept of truth is very peculiar – in particular, it is unclear why some dialetheias, that is, pairs of contradictory statements are true, but others false), intuitionistic logic can be identified as a subsystem of classical. This is the point of view of ‘classicism’. On the other hand, the intuitionist would say that his or her logic is non-classical (deviant) from the start and, eventually, adds that the characteristic (adequate) matrix for intuitionistic logic is infinite. The next complication relates to an ambiguity of the label ‘non-classical logic’. In one sense, it means deviancy, but in another – refers, for instance, to modal logic, also to systems which are two-valued and employ the standard axiomatization of Cn (some additions must be implemented for (1) as working in modal logic). In general, we can distinguish deviant non-classical logics and non-classical extensions of the classical system, including modal logic, higher-order logic, logic with generalized quantifiers, logic with infinite formulas – all lists of logics in the present section are incomplete). Yet the situation is still more complex, because if we take, for instance, intuitionistic logic or three-valued logic as a basic system, it has non-classical (deviant) modal extensions. Due to this variety of logics, there appears a small chance to arrive at a clear and transparent map of logics and the relations between them. Moreover, (2) allows us to define every system of logic as the set of logical consequences of the empty set, provided that (1) is available in our metalogic. Even more, if we assume that S is an arbitrary axiomatic theory, ω is one of its axioms, and ω ⊢ B, then by (1) we obtain ⊢ ω ⇒ B, and this means that ω ⇒ B ∈ Cn∅. So-called if-thenism was one of Russell’s interpretations of logicism in the foundations of mathematics (it was Russell’s case) – it consists in considering mathematics as a collection of conditional propositions in which consequents were derivable from antecedents. Russell did not claim that ω and B are logical truths. However, one can go a step further and use the old idea that axioms do not require proofs at all, we can say that every axiom is a logical truth. A plausible principle that if A is a logical theorem and if A ⊢ B, B also belongs to logic, every theorem of any axiomatic theorem is a logical truth. Perhaps it is close to some forms of radical apriorism, in particular, to Platonism with its concept of episteme. Such an extension of logic as was done in the previous paragraph is very problematic. As Tarski observed (see Tarski 1936), the list of logical constants has an importance. According to Tarski (see also Tarski 1986), the division of words into logical and non-logical is vague and conventional to some extent. If we decide that every word is a logical constant, each implication becomes

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a logical theorem. Clearly, this view is not acceptable, but its rejection leaves considerable space for possibilities. If we suppose that propositional connectives, first-order quantifiers and identity are the only logical constants, then first-order logic is the only logic. Of course, other choices are possible as well and can be executed, for instance, taking membership as a logical concept leads to including set theory in logic. The universal property as defined by (3) justifies the choice of first-order logic as the logic. The completeness property can be attributed to many other systems termed ‘logical’. Take, for example, modal logic. Particular modal systems are complete with respect to special frames – deontic logic requires a non-reflexive accessibility relation, the system S5 – the accessibility relation which is reflexive, symmetric and transitive, etc. My main thesis states that first-order logic is the only one having the unrestricted universality property. This circumstance relates to the Lindström theorem stating that the strongest first-order logic has the Löwenheim-Skolem property (if a theory has an infinite model, it also has a countable infinite model) and the compactness property – another version of this theorem states that first-order logic results from substituting compactness with completeness. One might say that even if we restrict our choice to first-order logic, the classical system has rivals, because we can define intuitionistic logic or manyvalued logic as universal. Eventually, one might say that we reject monism (or globalism), saying that there is only universal logic, and adopt pluralism – the view that logics are local and connected with specific domains, for example, the scope of constructive proofs is well-defined and requires intuitionistic (effective) proofs. Furthermore, reasoning about vagueness employs fuzzy logic, operating inconsistencies – paraconsistent logic or dealing with partial or approximate truth – many valued logic. Appealing to metalogic can help here. Hitherto, I only indicated that it is a part of logic. However, the status of metalogical investigations requires additional remarks. Take, for example, the theory of Cn and the completeness theorem. Even if LOG is formalized, its metatheory MLOG is at least partially informal. It uses logical concepts, for instance, connectives and quantifiers, special concepts, like logical consequence, provability, consistency or completeness, traditionally associated with the properties of logical systems, notions from set theory and other branches of mathematics, for example, algebra. If we formalize metalogic, we must do so in an informal meta-meta-theory. Typically, classical logic is employed in metatheory, even in the case of non-classical or deviant logics. Now we encounter the question of whether non-classical logic suffices for developing metalogical investigations of non-classical logic. Intuitionism provides a good example. The question is, for example, whether the completeness theorem for intuitionistic logic can be proved by purely intuitionistic methods. Although

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the issue is controversial, we can take for granted that the answer is negative. If so, the classical system should be considered as prior to metalogic. This view does not exclude applying non-classical tools, for instance, constructive ones, in classical metalogic, but the latter is a general frame in metalogical research. We can also combine globalism (the distinguished position of classical logic) and localism (the role of non-classical and deviant systems in particular investigations). Anyway, classical logic has a privileged status, due to its formal properties and role in metalogic.

5

Some Philosophical Comments on the Definition of Logic

Let me return to Ajdukiewicz’s table. Defining analyticals as tautologies, that is, propositions (I will also use the name ‘sentence’ as coextensive with the term ‘proposition’) true in all models, radical apriorism, moderate apriorism and moderate aposteriorism agree that logical laws are tautological. Moreover, all views qualify logic as a priori. However, this does not imply that being a priori and being analytic are the same. The equivalence in question holds for moderate aposteriorism, but not for both forms of apriorism. In general, apriorism distinguishes analytic sentences (a priori by definition) and syntheticals a priori (I skip the question of the nature of the synthetic a priori). Consequently, the logical a priori is marked by analyticity for apriorism as well as moderate aposteriorism. Necessity appears as another property of logical laws. Clearly, a negation of tautology is inconsistent, because if A is true in every model M, then ¬A is (must be) false in every model (according to apriorism, negations of syntheticals a priori are also inconsistent, but I do not enter this difficult problem). Radical aposteriorism must either deny that logical laws are analytic, tautological and necessary or define these attributes in a special way, for example, by saying that they are fully confirmed by empirical data. Apriorism in both its species and moderate aposteriorism agree that logical laws are analytic, tautological and a priori, but all properties are equivalent for the last position, but not for former ones – more specifically, only ‘A is necessary’ and ‘A is a priori’ are coextensive for apriorism. Consequently, excluding radical aposteriorism, being tautological is the most specific property attributed to logic by all not radically empiricist philosophical views. The concept of analyticity requires further comment. The issue is very extensive (see Woleński 2004a for a systematic presentation; I follow the settings in this paper). In fact, at least 62 definitions of what is an analytic proposition were proposed in the 19th and 20th centuries; many of them are variants of the most popular account, consisting in defining analyticity as truth in virtue

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of meaning. I restrict myself to proposals closely relating to (4) and (5). We define: (8) A ∈ Ansyn iff A ∈ Cn∅; (9) A ∈ Ansem iff for any model M, A is true in M. The indexes in formulas (8) and (9) express that the former definition is syntactic, but the latter – semantic. Thus, we have analytic sentences in the syntactic sense and analytic sentences in the semantic sense. The completeness theorem establishes the equivalence of (8) and (9). Moreover, the analytic propositions in both senses can be termed as absolute. This property is perhaps intuitively accounted by (8) – the derivation of absolute analytic sentences does not require any premises. Consequently, first-order logic is absolutely analytic. One could observe that (8) and (9) depend on fixing some metalogical issues, namely relating to Cn and M. That is true, but one might reply that if the metalogical matters are established, the absoluteness inside the logic is guaranteed. Anyway, and to anticipate further remarks, the path from metalogic to logic is relevant. What about relative analytic sentences? Let the symbol CS refers to socalled conceptual schemes. I take this notion as primitive – it can be illustrated by theories, sets of linguistic stipulations, etc. We have two statements (see Gochet 1986, see also Hintikka 2003 – both works are directed against Quine’s well-known skepticism about analyticity, but I omit this problem) (10) (11)

There is a sentence, which is analytic in all CS; For any CS, there is a sentence analytic in CS.

Clearly, if A is a sentence analytic in the sense of (11), A does not need to be analytic in the sense of (10), but the reverse implication holds. Postulates of empirical theories or definitions in the legal sense illustrate case (11). We can consider instances of (11) as relative analytic propositions. Clearly, absolute analytic propositions are eo ipso relative but not reversely – this means that absolute relative propositions are a special case of relative. Let CS be axiomatized by the set Ω. We define (12) (13)

A ∈ Analsyn (CS) iff A ∈ CnΩ A ∈ Analsem (CS) iff A is true in every model of Ω.

Accordingly, we have relative syntactic analyticals and relative semantical analyticals. If CS is a first-order theory, it is complete – therefore, (12) and

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(13) are equivalent. However, this statement cannot be generalized to all CS, because, for instance, legal stipulations do not generate theories in the exact metalogical sense. Anyway, (12) and (13) entail that analytic sentences exist that are not tautologies. The philosophical importance of the distinction of absolute and relative analyticals consists in providing a device for analysing the problem of whether arithmetic or modal logic are analytic or not. We can say that the latter can be considered as consisting of relative analytic sentences. Thus, Kant’s solution via synthetic a priori sentences is not the only one capturing the intuition that logic and mathematics differ. A further extension of the concept of analyticity consists in taking pragmatic aspects of language into considerations. We can define (14)

A ∈ Analprag (CS) iff A is true in every intended model of Ω.

This means that pragmatic analytic sentences are truths in the intended models. The first-order arithmetic AR of natural numbers provides an example for (14). According to the Löwenheim-Skolem theorem, it possesses models of different infinite cardinalities. However, its intended interpretation qualifies countable infinite models as “good”, “proper”, etc., because the set of natural numbers is accounted as infinite but denumerable. This is a fact relating to intentions, not to syntactic or semantic settings. Absolute analyticals, syntactic and semantic, remain pragmatic in the sense of (14), but relative analytic propositions require special treatment. From the purely formal point of view, standard and non-standard models of AR are equally good, that is, truths in both categories of models are perfectly legitimate. The pragmatic decision about what is standard qualifies some true sentences of AR as correct from the intuitive point of view. Clearly, they still satisfy conditions (12) and (13). We have inclusions (I change some points present in Woleński 2004a) ABSOLUTE ANALYTICALS ⊂ RELATIVE ANALYTICALS, ABSOLUTE ANALYTICALS ⊂ PRAGMATIC ANALYTICALS, but not all RELATIVE ANALYTICALS can be regarded as pragmatic analytic sentences. The concept of pragmatic analytic sentences is usable in the case of conceptual schemes not being theories; this gives an additional reason for rejecting the view that the terms ‘tautology’ and ‘analytic sentence’ are coextensive. Moreover, that if a set X of sentences is a conceptual scheme, it must contain analytic sentences, because their lack would annihilate the logical and conceptual relations inside X (see Hintikka 2003). One of the most popular conceptions of apriority identifies being tautology (analytic), being and being necessary and being a priori. All these properties are localized in language. Consequently, they are attributed to propositions.

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Concentrating on the last, we have the linguistic approach to apriority. It was one of the most characteristic views of logical empiricism. On this proposal, the a priori is entirely independent of experience. This concurs with Kant’s related thesis, but restricted to logic – nevertheless, one could say about the absolute a priori in this case (for Kant, absoluteness is an attribute of every a priori, analytic or synthetic). Now, if we speak about the absolute a priori, a natural question concerns the relative apriora. A straightforward answer runs: well, let us identify this kind of apriority with relative analyticity. However, I prefer another account. First of all, the qualification that something (a sentence) is a priori, traditionally refers not to having a specific logical form (being a tautology), but to the way the propositions are justified (see Casullo 2003). Historically speaking, we have the following proposals of how to justify the a priori (see Moser 1987; the list in this paper is longer; the names of the representatives are exemplifications; see also Casulo, Thurow 2013 for a historical and systematic survey of various conceptions of apriority): (a) transcendentalism (Kant – space, time, categories); (b) psychologism (Fries – human mental constitution); (c) pragmatism (James – schemes organizing cognition oriented toward utility); (d) linguisticism (the Vienna Circle – rules of language); (e) conventionalism (Poincaré – conventions); (f) anthropologism (later Wittgenstein – forms of life); (g) intuitionism (Brentano, Chisholm – intuition). Omitting (a) and (d) having an absolutist interpretation, all the others admit a relative reading. One hint in this respect comes from the theory of probability, particularly from Bayes’ theorem, that is, the formula (15)

P(e/f ) = P(f /e)P(e) : P(f ),

where P(e/f ) – a conditional probability of the occurrence of a event e with respect to the evidence e, P(f /e) – a conditional probability of f with respect to e, P(e) – the initial probability of e, P(f ) – the initial probability of f , P(f ) ≠ 0. The initial probability is sometimes characterized as a priori, the conditional probability – as a posteriori. Thus, (15) shows how to use a priori probability in assessing a posterior one. In specific applications, for example in statistics, we formulate a hypothesis H with certain a priori probability – this means that P(H) is independent of a performed empirical investigation, although it can be guided by former ones. I do not claim that (15) provides a general form of justification, but I use Bayes’ formula in order to show the possibility of saying that the label a priori can refer to something relative to something else. Unfortunately, (15) does not help with the problem of justifying logic, if we adopt its definition expressed by (4)–(5). Since logic is true in every model, any

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sample of data confirms logical laws. This means that the a priori probability of H being a logical theorem is always equal to 1. Even if we say that every justification of a sentence, independently of its apriority or aposteriority, is relative to some data (I am inclined to this view), we obtain only that logical laws are a special case in this respect, because they have a stable (= 1) initial probability confirmed by arbitrary empirical evidence. The relativity of a priori can also be interpreted in accordance with Hintikka’s remark on the role of the analytic elements of a conceptual scheme. Let CS be a conceptual scheme and Anal – its analytic element. Thus, we can say the latter is a priori with respect to the former. Now, remembering that logic is a part of every deductive system, we obtain that logic is a priori relatively to every theory and other conceptual schemes. It appears a matter of convention whether we say that this understanding of a priori is absolute or relative.

6

On the Genesis of Logic

However, the end of the preceding section does not explain the genesis of logic.4 Eventually, (15) can be used by the radical aposteriorist in order to show that logical laws arose through the long accumulation of evidence by humans to the extent that they (laws) are insensitive to further data. This was Mill’s position, shared by Tarski (see Tarski 1987) – neither Mill nor Tarski used Bayes’ theorem in their formulations of aposteriorism. However, I am interested in combining the universality of logic with moderate aposteriorism. Once again, this view shares genetic empiricism, that is, the thesis that nihil est in intellectu quod non prius fuerit in sensu. On the other hand, one of the principal views of the contemporary philosophy of science is that empirical procedures generate hypotheses, which are tentative and subject to revision by new evidence. Thus, we have a conflict, because, on one hand, logic, in accordance with genetic empiricism, should be viewed as empirical or a posteriori, but, on the other hand, it is defined as a priori (true in all models or belonging to every conceptual scheme). Is there any way out?

4 One of the reviewers maintains that the question of how logic evolved is irrelevant for logic as such. I do not agree with this view, because the answer to this question is very important for the controversy between apriorism and aposteriorism. The referee is right, provided that logic is discovered, (Frege’s view), not created. My view is different – I prefer to say that logic is a cognitive tool which appeared in the long process of human epistemic enterprise – I think that the word ‘created’ is misleading in the context being considered.

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I will outline the approach presented in (Woleński 2012). The first step is to assume that logic as defined by (4)–(5) is a product of logical competence, analogical to linguistic competence in the sense of Chomsky. Leaving aside an interpretation of Chomskian nativism, logical competence refers to a determined disposition of biological organisms which are capable of performing special mental functions called inferences. This view can be called naturalistic nativism and says that logical competence is not eternal; it appeared in the Cosmos at some point in time and is rooted in the biological structure of organisms. To return to one methodological point, already mentioned, I do not treat my comments as belonging to biology. My analytical interest is of a philosophical nature and remains within evolutionary epistemology. Logical competence is the ability to use operation Cn. A logical theory is not just logical competence, but its articulation or, to employ another conceptual scheme, logical competence is a disposition to some actions, but logic is a product of such “logical” actions. The path from logical competence to abstract logical systems had to be very long and its full reconstruction meets with serious difficulties. Anyway, logica utens was before logica docens. As it typically happens in the case of theories, their content goes beyond their cognitive generators and practical applications. In fact, the development of logical theories was and is strongly dependent on nature and the needs of communicative interactions within human society, in particular, by the nature of language. Consequently, the story of language and the story of logic go together. In particular, the nature of logic has affinities in the creative character of language. The latter manifests itself by the fact that the output of linguistic performances is via linguistic competence richer than their input. Analogically, the scope of inferential actions increases due to the mastery of logical competence. However, specific acts of reasoning were not the first word in the development of logic. If we like to be faithful to genetic empiricism, we must show that logical competence is rooted in the biological structure of the subjects becoming the users of logic. The evolutionistic classical approach to the development of human mental abilities to use a language or reasoning is – in an outline – as follows (see: Lieberman 2005; Tomasello 2010). The Universe exists since about 15 billion years ago (all the dates here are given in approximation), the age of Earth – 4.5 billion years, the first cells appeared a billion years later, multi-cellular organisms after the next 2.5 billion years, plants are aged from around 500 million years ago, reptiles – 340 million years, birds – 150 million, and apes – 7 million years. The genus of Homo appeared two million years ago, Homo erectus – from 1 million to 700,000, and Homo sapiens – 200,000 years ago. The cultural-civilizational evolution marked by language (in the understanding of

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our modern times), the alphabet and writing began 8,500 years ago. If this temporal assessment is correct, 3.5 billion years from the appearance of the first cells to the appearance of civilization and culture was completely sufficient to form a mind capable of performing typical intellectual activities, in particular, to perform logical operations. It cannot be ruled out that those rudiments of logical competence occurred in Homo erectus. Establishing the exact date of the start of logical competence in the course of evolution seems not particularly important for the main topic of the present analysis. Perhaps a plausible cognitive-evolutionist view in this respect might claim, for instance, that the inferential ability appeared by way of randomly acting mutations, and, as an effective adaptive tool, it was developed by homo sapiens, also thanks to available and ever more perfect linguistic devices. Logical theory appeared as the final product of a long evolutionary process. This suggestion can be considered as an adaptation of the classical concept of language evolution.5 Neo-Darwinian evolutionism connects the appearance of life and its further evolution with entropic phenomena (see: Brooks, Wiley 1986; Küppers 1990). This perspective leads to the need for indicating anti-entropic phenomena, i.e. mechanisms which regulate stability of organisms and their internal order, and thereby determine the continuation their existence (see Kauffman 1993). The decisive event to force a serious revision of evolution theory was the discovery of the structure of DNA by Crick and Watson in 1953 (the model of the double helix), as well as further research into genetic encoding. These results demonstrated the need to more profoundly link evolution with genetics. The notion of genetic information and ways of its transference became the key instruments of a new biological synthesis, that of genetics and the classical theory of evolution (frequently called – the synthetic theory of evolution). Notice that formal analogies between information and entropy caused biologists and philosophers of biology to take a much closer interest in the relations between the first notion and the course of biological processes since as early as the 1920s (see Yockey 2005). Several facts established by molecular biology are significant from the point of view of this paper. First, passing genetic information is directed from DNA through RNA (more precisely: mRNA – the letter ‘m’ denotes that RNA is a messenger, that is, an agent passing information) to proteins. This observation makes the so-called main dogma of molecular biology (I omit certain exceptions, e.g. the case of viruses), but at least in eukaryotic

5 Applications of evolutionary biology supplemented by information theory to problems of epistemology are quite common in contemporary philosophy (see Flodidi 2010). My analysis attempts to extend this approach to logic.

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organisms (including humans), the transmission of information is coherent with this dogma. Second, genetic information is passed in an ordered, linear, discrete and sequential manner. Third, DNA particles are subject to replication (copying) and recombination (regrouping). Fourth, the intracellular information system encodes and processes information encoded as an interpreted and computational system. Fifth, passing on genetic information is not deterministic but probabilistic, and thanks to this, genetic novelties may appear. Although a temptation to consider genetic information as consisting of alphabets, ordered by syntax and having some (proto) semantic features is natural, its nature is physical. Consequently, the relation of physical information as something quantitative to semantic information as qualitative is still mysterious and perhaps will remain as such forever.6 Perhaps it is so that the properties of genetic information determine the semiotization of mental processes, but it is a fairly speculative assumption, yet philosophically plausible. If genetic information were a language, even in an approximate sense, we could look for the genesis of logical competence directly on the microbiological level. Unfortunately, this is not the case. On the other hand, there are some prospects for a more complex analysis. One of the axioms of probability calculus states that there exists an event which in probability obtains the value 1 (this event can be identified as the entire space in which the probability measure is defined). We can interpret this axiom as preventing the dispersion of probabilities ascribed to particular occurrences, that is, subsets of the whole space. In other words, this axiom saves the differences in the amount of information – it is a condition of its flow. Thus, it performs an anti-entropic function, i.e. blocks the dispersion of information, it protects it in this way. Operation Cn can also be understood as an instrument protecting information from its dispersion, since it prevents the formation of false information on the basis of true information. However, saying ‘true’ or ‘false’ operates on the level of logic as a system. In order to connect Cn with genetic facts, we need to find a microbiological counterpart of the consequence operation. Since saving genetic information appears as a vital function of all organisms, we can assume that this function has its source in genetic code. This assumption is justified by the properties of such codes (see Klug, Cummings, Spencer 2006, p. 307), for example, linearity, the regularity of elements (the stable of ‘letters’), near universality,

6 The problem of how physical information is transformed into semantic information appears as crucial in philosophical discussions on the concept of information. See Floridi 2013 for an extensive analysis of this question.

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etc. (in some explanations), distinctions between initial and terminal states, semi-conservativeness (each replicated particle of DNA has one old string and one new one), conservativeness (the parent string is conserved as a result of synthesis in two new strings), or dispersed (old strings are dispersed in new ones). Anyway, the genetic information existing earlier is inherited by helixes formed by way of replication. The transmission of genetic information has two aspects – first, it must be conservative to protect the old information, and, second, it has to be open in order to be able to produce and accumulate information. Consequently, the instruments responsible for protecting information as well as enabling its increase must be physically embodied in the processes of genetic information. Thus, the nature of genetic codes (more precisely, a part of their equipment), syntactically very regular, can be considered the embodied mechanism resulting with Cn as a mental product determined by the semiosis of genetic information – this is, to say once again, the most mysterious element of the entire construction proposed in this paper. The second aspect of the transmission of genetic information, responsible for its openness, generates strategies for increasing empirical data – perhaps it is a source of so-called inductive logic or, considering using Bayesian strategies, innate. Once again (see note 1), I consider these remarks as philosophical, not as ones increasing biological knowledge. At the end of the section 3, I pointed out some analogies between Cn and the closure operation in topology. Going further in this direction, deduction within closed sets ‘leads’ to accumulation points in the topological sense, but inferring new non-logical (non-tautological) propositions in a non-deductive way corresponds to the definition of an open set, that is the set including all its neighbourhoods. To put it in a different way, the transition to neighborhoods of sentences as points in spaces in the set X′′ , that is, the extension of this set can be non-deductive. Consequently, the entire space of reasoning, including deductive and non-deductive inference, is a clopen set, that is, closed and open. Consider the double helix. It has a topological interpretation in Bates, Maxwell 2005, the book entitled DNA Topology. It can be understood in a dual way, first, it suggests that DNA has a topological structure, second – that the structure of the genetic space can be modelled by topology. This ambiguity is nothing new – for instance, we say that the geometry of physical space has definite geometrical properties and, on the other hand, is an abstract mathematical model of related physical reality. Assume that we use the label ‘topology of DNA’ in the second way. Yet we can argue that the real properties of DNA, particularly those associated with the transmission of genetic information, result in the topological modelling of this part of microbiological reality. Thus, DNA has a structure displayed by the concept of the clopen set and the

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natural properties of what is modelled by topology constitutes the ontological background of Cn and relates to the mechanisms of the protection/dispersion of information. Let me put the problem of the biological-topological evolution of logic in this way. Assume that A is a tautology. Hence, if B ∈ Cn{A}, is a tautology as well; A and B are not (or minimally) informative. On the other hand, every sentence belongs to Cn{A ∧ ¬A} – hence the contradictions are maximally informative. This is a paradox (see Bar-Hillel, Carnap 1952, Floridi 2013, p. 111–114)), because we are interested in safe (protected) and rich information. If we associate C with clopen sets, we have an explanation of why logic blocks the dispersion of information. On the other hand, logic as such does not contribute to increasing information. Consequently, the mechanism of obtaining new information has to be inductive (non-deductive). Anyway, logic started to work on the basis of already accessed information. Since humans invented theories and other conceptual schemes, this evolutionary novelty forced the invention of apriora and tautologies as devices of the stabilization of cognitive units. Logic became a by-product of this process – and it did not fall from Heaven – to paraphrase the title of von Ditfurth’s book (Der Geist fiel nicht vom Himmel in the German original).

References Ajdukiewicz, K. (1947), Logika i doświadczenie Przegląd Filozoficzny 40, 3–22; Eng. tr. (J. Giedymin), Logic and Experience. In K. Ajdukiewicz, The Scientific WorldPerspective and Other Essays 1931–1963. D. Reidel Publishing Company: Dordrecdt. Ajdukiewicz, K. (1949), Zagadnienia i kierunki filozofii. Teoria poznania. Metafizyka, Czytelnik: Warszawa; Eng. tr. (by H. Skolimowski, A. Quinton), Problems & Theories of Philosophy. Cambridge University Press: Cambridge. Bar-Hillel, Y., Carnap, R. (1952), An Outline of a Theory of Semantic Information, Technical Report no. 247 of the Research Laboratory of Electronics, Massachusetts Institute of Technologic, Cambridge, Mass; repr. in Y. Bar-Hillel, Language and Information. Selected Essays on their Theory and Application. Addison-Wesley Publishing Company, Reading, Mas. 1964, 221–274. Bates, A., Maxwell, A. (2005), DNA Topology. Oxford University Press: Oxford. Bays, Th. (2014), Skolem’s Paradox, The Stanford Encyclopedia of Philosophy (Winter 2012 Edition), Edward N. Zalta (ed.), URL = https://plato.stanford.edu/entries/ paradox-skolem/. Beall, J. C., Restall, G. (2006), Logical Pluralism. Clarendon Press: Oxford.

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Bedürftig, Th., Murawski, R. (2018). Philosophy of Mathematics. Walter de Gruyter: Berin. Bradbury, J., Vehrencamp, S. (1998), Principles of Animal Communication. Sinauer Associates, Inc.: Sunderland, Mass. Brooks, D., Wiley, E. (1986). Evolution as Entropy. Toward a Unified Theory of Biology, University of Chicago Press: Chicago. Casullo, A. (2003), A Priori Justification. Oxford: University Press: Oxford. Casullo, A., Thurow, Y. C. (eds.) (2013), The A Priori in Philosophy. Oxford University Press: Oxford. Conitz, D., Estrada-Gonzáles, L. (2019), An Introduction to the Philosophy of Logic. Cambridge University Press: Cambridge. Duda, R. (1986), Wprowadzenie do topologii, t. 1: Topologia ogólna (Introduction to Topology, v. 1: General Topology). Państwowe Wydawnictwo Naukowe: Warszawa. Fisher, J. (2007), On the Philosophy of Logic. Wadsworth Publishing Company, Inc.: Belmont, CA. Floridi, L. (2010), Information: A very Short Introduction, Oxford University Press, Oxford. Floridi, L. (2013), The Philosophy of Information. Oxford University Press: Oxford. Gochet, P. (1986), Ascent to Truth. A Critical Examination of Quine’s Philosophy. Philosophia Verlag: München. Grzegorczyk, A. (1974). An Outline of Mathematical Logic. Fundamental Results and Notions Explained in All Details. D. Reidel Publishing Company: Dordrecht. Haack, S. (1978), Philosophy of Logics. Cambridge University Press: Cambridge. Haack, S. (1996), Deviant Logic. Fuzzy Logic. Beyond the Formalism. University of Chicago Press: Chicago. Hintikka, J. (2003), A Distinction Too Few or Too Many? A Vindication of Analytic vs. Synthetic Distinction. In Constructivism and Practice. Ed by C. Gould, Rowman & Littlefield: Lanham, Maryland, 41–74. Ilachinski, A. (2011), Cellular Automata. A Discrete Universe. World Scientific: Singapore. Jacquette, D. (ed.) (2001), Philosophy of Logic. An Anthology. Blackwell: Oxford. Jacquette, D. (ed.) (2007), Philosophy of Logic. Elsevier: Amsterdam. Kauffman, S. (1993), The Origin of Order. Self-Organization and Selection in Evolution. Oxford University Press: Oxford. Klug, W., Cummings, M., Spencer, Ch. (2006), The Concepts of Genetics. Prentice-Hall, Inc.: Upper Saddle River, NJ. Küppers B.-O., Information and the Origin of Life. The MIT Press: Cambridge, Mass. Lakoff, G., Núñez, R. E. (2000), Where Mathematics Comes Form. How the Embodied Mind Brings Mathematics into Being. Basic Books: New York.

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Lieberman Ph. (2005), Toward an Evolutionary Biology of Language. Harvard University Press: Cambridge, Mass. Moser, P. (1987), Introduction. In A Priori Knowledge. Ed by P. Moser, Oxford University Press: Oxford 1–14. Priest, G. (2008), Non-Classical Logic. From If to Is. Cambridge University Press: Cambridge. Putnam, H. (1971), Philosophy of Logic. George Allen and Unwin: London. Quine, W. van O. (1970), Philosophy of Logic. Prentice-Hall, Inc.: Englewood-Cliffs. Shapiro, S. (ed.) (2005), The Oxford Handbook of Philosophy of Logic and Mathematics. Oxford University Press: Oxford. Tarski. A. (1936), Über den Begriff den logischen Folgerung. In Actes du Congrès International de Philosophie Scientifique, v. 7. Hermann: Paris, 1–11. Eng. tr. (by J. H. Woodger), On the Concept pf Logical Consequence. In A. Tarski, Logic, Semantics, Metamathematics. Papers from 1931 to 1938. Clarendon Press: Oxford 1956, 408–420. Tarski, A. (1986), What are Logical Notions? History and Philosophy of Logic 7, 143–154. Tarski, A. (1987), [A Letter of Alfred Tarski [to M. White], September, 23, 1944. The Journal of Philosophy 84, 29–32. Tomasello, M. (2010), Origins of Human Communication. The MIT Press: Cambridge, Mass. Wereński, S. (2007), Topologia (Topology). Wydawnictwo Politechniki Radomskiej: Radom. Woleński, J. (2004), First-Order Logic: (Philosophical) Pro and Contra, w: First-Order Logic Revisited, ed. by V. Hendricks, F. Neuhaus, S. A. Pedersen, U. Scheffler, H. Wansing, λογος, Berlin 2004, 369–399; repr. in J. Woleński, Essays on Logic and Its Applications in Philosophy, Peter Lang, Frankfurt am Main 2011, 61–80. Woleński, J. (2004a), Analytic vs. Synthetic and Apriori vs. A Posteriori. In Handbook of Epistemology. Ed. by I. Niiniluoto, M. Sintonen and J. Woleński, Kluwer Academic Publishers, Dodrecht 2004, 781–839. Woleński, J. (2012), Naturalism and the Genesis of Logic, in Papers on Logic and Rationality. Festschrift in Honor of Andrzej Grzegorczyk (Studies in Logic, Grammar and Rhetoric 27(40)). Ed. by K. Trzęsicki, S. Krajewski, University of Bialystok, Bialystok 2012, 223–240; repr. in J. Woleński, Logic and Its Philosophy, Peter Lang: Berlin 2018, 57–70. Yockey, H. (2005), Information Theory, Evolution and the Origin of Life. Cambridge University Press: Cambridge.

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Chapter 5

On Some Problems with Truth and Satisfaction Cezary Cieśliński

Abstract We compare two approaches to semantics: one of them takes the notion of satisfaction as basic, treating the concept of truth as defined in terms of satisfaction; the second approach treats the notion of truth as primary. We search for a minimally adequate semantic theory, guaranteeing a mutual definability of truth and satisfaction.

Keywords truth – satisfaction – definability

In this paper we compare two approaches to semantics, clearly visible in contemporary semantic theorizing. With the first approach, one takes the notion of satisfaction as basic, treating the concept of truth as defined in terms of satisfaction. Adopting the second approach, one treats the notion of truth as primary. Both strategies are intertwined with another major division in contemporary formal semantics, namely the division into the model-theoretic and axiomatic approach to semantic notions. We start by providing a few introductory remarks concerning the model-theoretic/axiomatic distinction; then in the next section we sketch the background and provide examples of theories taking satisfaction/truth as the primary notion. Model-theoretic versus axiomatic approach. In the model-theoretic approach, given a formal language L, we start by defining a general notion of a model of this language. Informally, the idea is that a model of L is an arbitrary structure with a non-empty universe, containing the interpretations of all extra-logical symbols of L.1 Given a model M of L, we then build a defini-

1 For example, if L has exactly two relational symbols R and Q, the first one unary and the second forming atomic formulas of the form ‘Q(x, y)’, then the model of L will contain exactly

© Cezary Cieśliński 2021 | DOI:10.1163/9789004445956_007

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tion of ‘a formula of L being satisfied in M’ or ‘a sentence of L being true in M’ (depending on whether we take satisfaction or truth as primary). In the last step, we can choose a class of models of L and declare that models belonging to this class are to be called ‘intended’ or ‘standard’. The idea is then that satisfaction/truth in one of these models corresponds to satisfaction/truth for formulas of L as we normally understand them. In the axiomatic approach, our starting point is also a fixed, concrete language L. However, instead of defining satisfaction/truth in a model, we then extend L with a new predicate symbol. Thus, depending on one’s choice of satisfaction or truth as basic, one can consider the language LS obtained from L by introducing the binary satisfaction predicate ‘S(x, y)’, or the language LT , obtained by introducing the unary truth predicate ‘T(x)’. In the next move, an axiomatic system in LS or LT is defined. The guiding idea is that the axioms and rules of this system are chosen in such a way as to characterize the meaning of the satisfaction or truth predicate.2

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The Primacy of Satisfaction or Truth: Background

1.1 Satisfaction as the Primary Notion The history of the approach emphasizing satisfaction as primary goes back to Tarski’s classic paper (Tarski, 1933), which describes the foundations of the model-theoretic approach to semantics.3 Tarski’s construction is well known and I will not present it in detail; let me only summarize its main features: – Given a model M of L, we define an assignment in M as an arbitrary function assigning the elements of the universe of M to the variables of L. – The notion of a value of a term t of L in a model M under an assignment v is defined by the usual recursive clauses. – A satisfaction relation between M, a formula ϕ of L and an assignment v (‘ϕ is satisfied in M under v’) is defined by the usual recursive clauses. In particular, the notion of a value of a term is used to define satisfaction for atomic formulas of L. – Truth is defined in terms of satisfaction: a sentence ϕ of L is true in M iff it is satisfied under some assignment (equivalently, iff it is satisfied under all assignments). two relations on its universe, namely, a unary relation for interpreting ‘R’ and a binary one for interpreting ‘Q’. 2 For more on the differences between the model-theoretic and the axiomatic approach, see Chapter 2 of (Cieśliński, 2017). 3 Here I describe Tarski’s construction as model-theoretic, even though the notion of a model was introduced only later in (Tarski and Vaught, 1957). Tarski (1933) does not employ it.

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As already mentioned earlier, in the final move the concept of a standard or intended model of L can be introduced, with the satisfaction/truth of a formula in such a model corresponding to the satisfaction/truth under our usual way of interpreting the formula in question. For example, working in ZFC we can construct the set ω and define the model N = (ω, S, +, ×, 0) of the arithmetical language with the operations of successor, addition and multiplication defined on ω. We can then declare as ‘standard’ an arbitrary model of the arithmetical language isomorphic to N . The idea is that truth in the standard model corresponds to truth of arithmetical sentences as we understand them. Starting from the 1960s, the axiomatic approach began to gain ground. The initial motivation had to do with the fact that first-order arithmetical theories have non-standard models and these models contain non-standard formulas (objects seen by models as formulas even though they are not formulas in the real world).4 In view of this, the question naturally arises whether it is possible to develop semantics for all the formulas in the sense of the model, including the non-standard ones. Pioneering work in this direction has been done by Robinson (1963) and Krajewski (1976); in particular, in (Krajewski, 1976) the important notion of a satisfaction class was introduced, which since then has played a crucial role in the axiomatic approach to semantics. Roughly, the idea is to characterize a satisfaction class S in a model M as a set of pairs (ϕ, v), such that ϕ is a (possibly non-standard) formula, v is an assignment and S satisfies the compositional clauses characterizing the satisfaction predicate. These clauses, in turn, are treated as axioms of a theory of satisfaction. We provide now a concrete illustration of the approach. Let LPA be the language of first-order arithmetic. We assume that LPA has the usual logical vocabulary (variables, quantifiers, connectives, the identity symbol and brackets) and that the set of its primitive extralogical symbols is defined as {‘ + ’, ‘ × ’, ‘0’, ‘S’} (symbols for addition, multiplication, zero, and the successor function). In addition, we adopt the following notational abbreviations: – ‘x ∈ Var’ is an arithmetical formula with a natural reading ‘x is a variable’, – ‘x ∈ Tm’ is an arithmetical formula with a natural reading ‘x is an arithmetical term’, – ‘x ∈ Tmc ’ is an arithmetical formula with a natural reading ‘x is a constant arithmetical term’,

4 Given a non-standard model M of arithmetic, choose a non-standard element c and consider an arbitrary arithmetical formula ϕ. Then every expression of the form ‘ϕ ∧ ϕ ⋯ ∧ ϕ’ is also a formula, no matter how many times ϕ is iterated. Now, a nonstandard formula will be obtained by constructing a conjunction in which the component ϕ is repeated c times. Then the model M will recognize such a conjunction as a formula, but in the real world it is not a formula at all, because c is not a natural number.

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– ‘x ∈ FmLPA ’ is an arithmetical formula with a natural reading ‘x is a formula of LPA ’, – ‘x ∈ SentLPA ’ is an arithmetical formula with a natural reading ‘x is a sentence of LPA ’, – ‘x ∈ Asn(y)’ is an arithmetical formula with a natural reading ‘x is an assignment for free variables in formula y’,5 – ‘v′ = v[x/a]’ is an arithmetical formula with a natural reading ‘v′ is an assignment differing from v at most in that it assigns a to the variable x’, – ‘x = val(t, v)’ is an arithmetical formula with a natural reading ‘x is the value of the term t under an assignment v’. Let LS be the result of extending LPA with the binary predicate ‘S(x, y)’. We are ready now to define a satisfaction theory built over Peano arithmetic (PA), treated here as the base theory of syntax. Definition 1 Let CS− be a theory in LS axiomatized by the usual axioms of PA together with the following axioms for the satisfaction predicate: – ∀s, t ∈ Tm ∀v ∈ Asn(t = s) (S(t = s, v) ≡ val(s, v) = val(t, v)) – ∀φ ∈ FmPA ∀v ∈ Asn(φ) (S(¬φ, v) ≡ ¬S(φ, v)) – ∀φ, ψ ∈ FmPA ∀v ∈ Asn(φ ∨ ψ)(S(φ ∨ ψ, v) ≡ (S(φ, v) ∨ S(ψ, v))) – ∀a ∈ Var ∀φ(x) ∈ FmPA ∀v ∈ Asn(∀aφ(a))(S(∀aφ(a), v) ≡ ∀x S(φ, v[x/ a])) For example, the intuitive meaning of the first axiom is that given arbitrary terms t and s and an assignment v of numbers to variables occurring in these terms, the formula ‘t = s’ is satisfied under v if and only if the values of these terms under v are the same. The other three axioms state the compositional properties of satisfaction for negation, disjunction and general quantifier (e.g., the second axiom states that the negation of ϕ is satisfied under v iff ϕ itself is not satisfied under v). In effect, CS− is obtained by turning the usual Tarskian recursive clauses defining satisfaction into axioms. Note that the axioms of CS− comprise all the arithmetical axioms of induction but no induction for formulas with the satisfaction predicate. In other words, in the induction schema (Ind)

[φ(0) ∧ ∀x(φ(x) → φ(S(x)))] → ∀xφ(x),

we are allowed to substitute an arbitrary arithmetical formula (without the satisfaction predicate) for the schematic letter ϕ, since the resulting substitution of (Ind) is an arithmetical axiom of PA. However, when working in CS− , we are 5 Intuitively, this means that x is a finite function assigning numbers to all variables which are free in y.

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not allowed to substitute formulas of LS containing the satisfaction predicate for the schematic letter ϕ. Nevertheless, the extended induction for all formulas of LS can also be introduced axiomatically. Definition 2 We define CS as a theory in LS axiomatized by all the axioms of − CS together with all the substitutions of (Ind) by formulas of LS . A satisfaction class in M is a subset of M which makes CS− true, that is, a subset S of M such that (M, S) ⊧ CS− . If (M, S) ⊧ CS, then we say that S is an inductive satisfaction class in M. 1.2 Truth as the Primary Notion Horwich (1999) famously proposed that truth should be taken as a primitive notion of a semantic theory. In his view, all the truth axioms should take the form of the so-called Tarski biconditionals, that is, they should have the form ‘T(ϕ) ≡ ϕ’ for all propositions ϕ belonging to a chosen (possibly quite comprehensive) class.6 Axiomatic truth theories taking Tarski biconditionals as the only axioms are often called ‘disquotational’ in the literature. According to Horwich, truth is conceptually simple and this simplicity is revealed by the disquotational axiomatization. In effect, in his opinion truth should not be characterized in terms of reference or satisfaction lest ‘the extreme conceptual purity and simplicity of minimalism is thrown away for nothing’ (see Horwich 1999, 111). It should be emphasized that even if we adopt a compositional (not purely disquotational) approach, it is still possible to treat truth as basic.7 A classical example is the theory CT− defined below, whose axioms are designed to characterize the notion of arithmetical truth, that is, the notion of truth as applied to sentences of the language of first-order arithmetic. Definition 3 Let CT− be a theory in LT axiomatized by the usual axioms of PA together with the following axioms for the truth predicate: – ∀s, t ∈ Tmc (T(t = s) ≡ val(y) = val(s)) – ∀φ(SentPA (φ) → (T¬φ ≡ ¬Tφ)) – ∀φ∀ψ(SentPA (φ ∨ ψ) → (T(φ ∨ ψ) ≡ (Tφ ∨ Tψ))) 6 Due to the liar paradox it cannot be the class of all proposition, hence some restrictions are necessary. 7 In fact, both truth and satisfaction have been used in the literature in the role of the basic concept of a compositional semantic theory. Thus, e.g., in (Krajewski, 1976), (Kaye, 1991) and (Enayat and Visser, 2015) satisfaction is the primary notion, with a satisfaction class defined as an interpretation of the satisfaction predicate. On the other hand, in (Kotlarski et al., 1981), (Engström, 2002) and (Leigh, 2015) the discussion is conducted in terms of truth, not satisfaction.

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˙ – ∀v∀φ(x)(SentPA (∀vφ(v)) → (T(∀vφ(v)) ≡ ∀xT(φ(x)))) For example, the intuitive meaning of the first axiom is that given arbitrary constant terms (no variables occurring in them) t and s, the sentence ‘t = s’ is true if and only if the values of these terms are the same. The other three axioms state the compositional properties of truth for negation, disjunction and general quantifier (e.g., the second axiom states that the negation of an arithmetical sentence ϕ is true iff ϕ itself is true). Note that the last axiom of CT− involves numerals, that is, arithmetical constant terms of the form S…S(o), with arbitrarily many iterations of the successor symbol. Roughly, the last axiom states that a generally quantified sentence ∀vϕ(v) is true iff the result of every substitution of a numeral for the variable in ϕ(v) is true (indeed, this is how the dot symbol over ‘x’ should be interpreted). A slightly different quantifier axiom would employ arbitrary constant terms instead of numerals: – ∀v∀φ(x)(SentPA (∀vφ(v)) → (T(∀vφ(v)) ≡ ∀t ∈ Tmc T(φ(t)))) In this modified version, the axiom states that a generally quantified sentence ∀vϕ(v) is true iff the result of every substitution of a constant term (not necessarily a numeral) for the variable in ϕ(v) is true. Sometimes we will refer to the theory from Definition 3 as the numeral variant of CT− . A corresponding theory with the quantifier axiom employing arbitrary constant terms will be called a term variant of CT− . Similarly to the case of CS− , the theory CT− contains arithmetical induction only, that is, only arithmetical substitutions of (Ind) are allowed. Again, the extended induction can be introduced axiomatically. Definition 4 We define CT as a theory in LT axiomatized by all the axioms of CT− together with all the substitutions of (Ind) by formulas of LT . Observe that CS− and CT− share the following trait. The axioms of CS− correspond to the recursive clauses of Tarski’s model-theoretic truth definition (indeed, they are obtained from these very clauses by replacing ‘satisfaction in a model’ by a formula with the primitive satisfaction predicate). The same holds for CT− : in our model-theoretic construction we could define arithmetical truth directly, without going through satisfaction first (just replace the axioms of CT− by the strictly analogous recursive clauses). In this paper I will not be discussing other model-theoretic and axiomatic characterizations of truth. Let me only emphasize that many such characterizations have been proposed in the literature. Prominent examples include Kripke’s model-theoretic characterization of the truth predicate and the axiomatic systems KF (Kripke-Feferman) and FS (Friedman-Sheard).8 The book 8 For Kripke’s model-theoretic proposal, see (Kripke, 1975).

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(Halbach, 2011) contains a detailed presentation of these and other axiomatic theories of truth.

2

Comparing Truth with Satisfaction: The Basic Framework

Does it really matter whether we choose truth or satisfaction as our primary notion? I take it for granted that the choice is not particularly important provided that the two notions are mutually definable. Thus, assume that my clauses defining/axiomatizing satisfaction permit me to define the truth predicate and to derive useful properties of truth; assume also that my favorite clauses defining/axiomatizing truth permit me to adequately characterize the notion of satisfaction. In such a situation the decision to make satisfaction or truth primary would boil down to an aesthetic choice, supported perhaps by simplicity considerations. For example, if our final objective is characterizing the notion of truth, then introducing satisfaction seems like a detour and a direct definition/axiomatization of truth might look more elegant. However, as we are going to see, the matter is not always this simple and the definability relation between the two notions is sometimes problematic. It is in such situations that the choice between truth and satisfaction has real consequences. Before discussing the problematic cases, let us consider another question first. If we want to characterize the notion of truth, is the detour via satisfaction always avoidable? Perhaps our earlier example of arithmetical truth is quite special in this respect? In the case of arithmetic, compositional truth axioms of CT− can be formulated directly for a very simple reason: every natural number has a name, that is, an arithmetical term denoting it (some of these names are typically selected as canonical under the label ‘numerals’). Indeed, this observation can be formalized in our arithmetical theory (say, Peano arithmetic), yielding a corresponding arithmetical theorem.9 It is exactly this fact which permits us to formulate the quantifier axiom of CT− , which states that a universally quantified sentence is true if and only if every relevant numeral substitution produces a true sentence. After all, since every number is named by a numeral, nothing is omitted. However, when we move beyond arithmetic, the nameability assumption might no longer be valid. A typical example is set theory. We will not be able to prove in ZFC that every set has a name, nor (so one could claim) will such an assumption be true under any reasonable purported interpreta9 Namely, the sentence ‘∀x∃t[Tmc (t) ∧ val(t) = x]’ (‘For every x there is a constant term t taking x as its value’) is provable in PA.

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tion of set-theoretical language. In such cases it is still possible to characterize truth axiomatically by means of Tarski biconditionals (i.e., Horwich style), but a direct formulation of compositional truth axioms is problematic. Introducing the satisfaction predicate is easier because in CS− we do not rely on the nameability assumption. Nevertheless, the above reservation does not settle the issue between truth and satisfaction. In fact, compositional theories of set-theoretic truth have been investigated in the literature. A prominent example is (Fujimoto, 2012), where the base set-theoretical language is assumed to contain a constant symbol cx for every x belonging to the universe of all sets. While it should be admitted that this is not the usual language of set theory,10 the trick permits us to build the counterparts of the well-known axiomatic systems of truth and the detour via satisfaction can still be avoided. In what follows, when comparing truth and satisfaction, we will take into account the following desiderata for theories Th of truth or satisfaction.11 (1) Th should be a theory of satisfaction/truth and not of something else. (2) Th should be strong enough to prove the compositional clauses for satisfaction/truth. (3) If Th is a theory of satisfaction, a truth predicate should be definable in Th. (4) If Th is a theory of truth, a satisfaction predicate should be definable in Th. Condition (1) might seem very obvious: after all, if it is a theory of truth or satisfaction that we are after, it would be plainly redundant to incorporate in Th the body of knowledge concerning, e.g., stars or elephants. However, as we will see, things are not always that simple. In general, condition (1) leads to various questions about the admissibility of theoretical tools in our semantic theory. Can Th contain the means for discussing natural numbers? Would it be possible to use set theory freely in Th? Can Th contain higher order quantification? Questions of this sort do not have answers as immediate as the corresponding question about the elephants. Adopting condition (2) eliminates typical Horwich-style or disquotational axiomatizations from our inquiry, as it is known that proving compositional

10 11

Indeed, after adding the constants for all sets, the language ‘inevitably becomes of proper class size’, see (Fujimoto, 2012, p. 1486). Various desiderata for truth theories have been formulated in the literature; see (Leitgeb, 2007) for a useful review. Conditions (1) and (2) on the list below are pretty standard; on the other hand, conditions (3) and (4) have been rarely (if ever) discussed in the literature.

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clauses is usually beyond their reach.12 While some authors claimed that this deficiency in itself makes disquotational axiomatizations inadequate, I am not taking this route.13 Accordingly, condition (2) should be read not as a condemnation of disquotational theories, but as a decision to focus on compositional (non-disquotational) axiomatizations. In short, in the context of the present paper we are not concerned with deflationary views on truth and satisfaction. Indeed, I think that the most interesting comparisons to be made between truth and satisfaction are those between general claims derivable in truth and satisfaction theories – claims, let us add, whose proofs require some compositional principles. Conditions (3) and (4) require that Th permits us to recognize the connection between two basic semantic notions: satisfaction and truth. I think that in fact characterizing truth has been typically one of the objectives to be achieved by the introduction of the satisfaction predicate (cf. (Tarski, 1933)), so (3) accords well with the actual practice. Condition (4) introduces a symmetry in this respect and guarantees that none of the two approaches (truth or satisfaction as primary) enjoys an initial advantage over the other one.

3

The Assessment of Truth and Satisfaction Theories

As noted earlier, applying condition (1) (‘Th should be a theory of satisfaction/truth and not of something else’) is not an easy and straightforward matter. Here I adopt the minimal strategy: roughly, the idea is that the theory should contain as little as possible about other matters than truth or satisfaction. I take it for granted that truth/satisfaction will apply to syntactic objects, hence our theory is bound to contain some theory of syntax. In the discussion below I simply assume that Th contains satisfactory arithmetical resources to reproduce all syntactic constructions needed for the characterization of the notion of truth or satisfaction. After adopting the minimal strategy, axiomatic truth/satisfaction theories move to the foreground. As for the model-theoretic approach, the main worry is that it engages a too heavy set-theoretical artillery, thus violating condition (1) of the previous section. For illustration, consider again the notion of arithmetical truth. Clearly in model-theoretic constructions some form of the axiom of infinity is needed in order to construct models of the arithmetical lan12 13

See, for example, (Cieśliński, 2017, p. 51 and p. 55). Indeed, in (Cieśliński, 2017) I defended the disquotational theories against this sort of criticism.

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guage; moreover, the notion of a standard or an intended model is eventually employed in order to explain the notion of truth of arithmetical sentences as we normally understand them.14 Axiomatic theories do not carry this baggage and as such, they are more attractive. In the discussion that follows, I focus on the axiomatic theories of arithmetical truth of the CS/CT variety. The purpose of the next definition is to make conditions (3) and (4) precise. Definition 5 Let ThS and ThT be theories in LS and LT , respectively. – We say that ThS defines the predicate T(x) of ThT iff there is a formula τ(x) of LS such that for every ψ ∈ LT , if ThT ⊢ ψ, then ThS ⊢ ψτ(x) , where ψτ(x) is the result of replacing every occurrence of T(t) in ψ ( for an arbitrary term t) with τ(t). – We say that ThT defines the predicate S(x, y) of ThS iff there is a formula σ(x, y) of LT such that for every ψ ∈ LS , if ThS ⊢ ψ, then ThT ⊢ ψσ(x,y) , where ψσ(x,y) is the result of replacing every occurrence of S(t, r) in ψ ( for arbitrary terms t and r) with σ(t, r). Definition 5 gives the precise sense to the definability conditions (3) and (4). Thus, for example, when we ask whether CS is able to define the notion of arithmetical truth, the idea is that we work with some theory of truth ThT (e.g., CT) in the background and we inquire whether it is possible to interpret the truth predicate in terms of satisfaction in such a way that all the theorems of ThT become provable in CS.15 Now, it is easy to observe that CS and CT are indeed definitionally equivalent in our sense. The following fact is a folklore observation. Fact 6 CS defines the truth predicate of CT and CT defines the satisfaction predicate of CS. Proof. In order to define the truth predicate of CT in CS, apply any of the usual, familiar methods: define τ(x) as x ∈ Sent LPA ∧ ∀yS(x, y) (‘x is an arith-

14

15

The last move, that is, invoking the notion of the standard model, is quite demanding indeed. See (Gaifman, 2004), where the question is considered ‘Which, if any, of some given models is the standard one?’ Gaifman notes that we can answer this question provided that we adopt the notion of a power set of an infinite set. Then on p. 15–16 he comments: ‘But this is highly unsatisfactory, for it bases the concept of natural numbers on the much more problematic shaky concept of the full power set. It is, to use a metaphor of Edward Nelson [1986], like establishing the credibility of a person through the evidence of a much less credible character witness’. This notion of definability is stricter than the usual notion of interpretability between theories in that we require that only one predicate (truth or satisfaction) is reinterpreted. In particular, in the case of theories of arithmetical truth and satisfaction, the arithmetical part of the language is kept constant, i.e., it does not receive a new interpretation.

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metical sentence satisfied under all assignments’) or x ∈ SentLPA ∧ ∃yS(x, y) (‘x is an arithmetical sentence satisfied under some assignment’) or x ∈ SentLPA ∧ S(x, ε) (‘x is an arithmetical sentence satisfied under the empty assignment’). The availability of full induction in CS permits us to demonstrate the provable equivalence of all these versions of the truth predicate; it permits us also to prove every axiom of CT with ‘T(x)’ replaced by our chosen formula with the satisfaction predicate. Defining the satisfaction predicate of CS in CT is also quite straightforward: just define σ(x, y) as the formula x ∈ FmLPA ∧ y ∈ Asn(x) ∧ T(xy ), where xy denotes the result of substituting numerals for variables free in a formula x in accordance with the assignment y.16 Again, the availability of full induction in CT permits us to demonstrate that all the axioms of CS become provable in CT once we replace the satisfaction predicate with σ(x, y). In effect, Fact 6 shows that CS and CT satisfy conditions (3) and (4). Clearly both theories prove the compositional clauses for satisfaction or truth, thus satisfying also condition (2). How about condition (1)? Are CS and CT theories of satisfaction/truth only, or perhaps some external subject matter is smuggled into their axioms? Indeed, there is one worry: in both theories we have full extended induction at our disposal, for all formulas of LS or LT , respectively. It should also be emphasized that extended induction plays a crucial role in the justification of Fact 6. So, how innocent is extended induction? It has been argued (notably, by Field (1999)) that the axioms of extended induction should not be treated as forming the core of a theory of truth. The reservations apply just as well to theories of satisfaction. As Hartry Field puts it: [the axioms of induction] would hold for any other predicate, and what they depend on is a fact about the natural numbers, namely, that they are linearly ordered with each element having only finitely many predecessors. (Field, 1999, p. 538) Further he adds: It is something about our idea of natural numbers that makes it absurd to suppose that induction on the natural numbers might fail in a language 16

For example, let x be the arithmetical formula ‘∃v(v + r = z)’ and let y be the assignment ((r, 5), (z, 13)). Then the expression xy refers to the arithmetical sentence ‘∃v(v + 5 = 13)’, with numerals for 5 and 13 substituted for the free variables r and z.

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expanded to include new predicates (whether truth predicates or predicates of any other kind): nothing about truth is involved. (Field, 1999, p. 539) Using the formulation from condition (1), the upshot is that the axioms of extended induction are about ‘something else’ than truth or satisfaction. We do not accept them because of our beliefs concerning specifically truth or satisfaction. Indeed, if we accept them, then we would be just as ready to accept induction for an arbitrary new expression predicated of natural numbers. In short: the principle of induction expresses our way of understanding the notion of a natural number, and not the notion of truth or satisfaction.17 In effect, CS and CT share the fate of the model-theoretic approach: their machinery goes beyond the minimal approach, which apart from the basic theory of syntax, permits us to employ only the principles characterizing the notion of truth or satisfaction. This leaves us with CS− and CT− as decent candidates for the role of minimal theories of truth and satisfaction. They satisfy condition (1) – apart from the base theory of syntax, they contain just the compositional satisfaction/truth axioms, hence trivially satisfying also condition (2). Are these theories perfect, then? Well, the trouble is that without the extended induction conditions (3) and (4) become problematic. First, can we recover the truth predicate in CS− ? The problem is that none of the usual methods seems to work. For illustration, let us consider two typical strategies: (a) truth is defined as satisfaction under some assignment, (b) truth is defined as satisfaction under the empty assignment. An example of a problem concerning (a) is the question of how to prove that truth commutes with negation. Using (a) as the truth definition, our claim becomes: ∀ψ ∈ SentLPA [∃yS(¬ψ, y) ≡ ¬∃yS(ψ, y)]. Working in CS− , fix ψ ∈ SentLPA . By the compositional axiom of CS− for negation, we observe that ∃yS(¬ψ, y) is equivalent to ∃y¬S(ψ, y). However, this is not enough: our task is to obtain the information that ψ is not satisfied under any assignment. But how can we reach this? 17

As I take it, this diagnosis is pretty independent of the philosophical views on the nature of natural numbers. Whatever your philosophical conception of a number is, the point is only that the principle of induction is number-theoretic and not truth/satisfactiontheoretic.

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At this point, the usual strategy is to prove the lemma about the behavior of satisfaction under assignments which are indistinguishable for a given formula. The lemma runs as follows: Lemma 7 For an arbitrary ψ ∈ FmLPA , for all assignments v and w such that for every variable x free in ψ, v(x) = w(x), S(ψ, v) ≡ S(ψ, w). Since sentences contain no free variables, it immediately follows that for an arbitrary sentence ψ the condition ∃y¬S(ψ, y) is equivalent to ¬∃yS(ψ, y), which ends the proof of the compositional truth clause for negation. However, the trouble is that the proof of Lemma 7 proceeds by induction on the syntactic complexity of ψ, with a formula containing the satisfaction predicate playing the role of the inductive hypothesis. In other words, the proof employs extended induction (for formulas of LS ), which is simply not present in CS− . In effect, this road to proving the compositional truth clause is blocked. Choosing option (b) – defining truth as satisfaction under the empty assignment – solves the problem with negation. Trivially, CS− proves that: S(¬ψ, ε) ≡ ¬S(ψ, ε), which under option (b) is simply the compositional truth clause for negation. However, now a serious difficulty appears elsewhere. Namely, there is a problem with proving the compositional truth clause for a quantifier, which in the present setting has the following form: ˙ ε)). ∀v∀φ(x) ∈ FmPA (S(∀vφ(v), ε) ≡ ∀xS(φ(x), In order to appreciate the difficulty, observe first that the left side of the above biconditional, that is S(∀vϕ(v), ε), is (provably in CS− ) equivalent to ∀x S(ϕ(v), ε[v/x]), informally: for every number x, ϕ(v) is satisfied under an assignment of x to variable v. However, this falls short of the mark, as our task is to deliver the right side of the biconditional, which states that for every number x, the sentence obtained from ϕ(v) by substituting a numeral denoting x for a free variable v, is satisfied under the empty assignment. But how to obtain the right side of the biconditional? Again, the natural method to proceed would be by induction on the complexity of ϕ: just assume the property in question for simpler formulas and derive it for more complex ones. Unfortunately, this would require again the availability of induction for LS (not just for the arithmetical language), which is simply not available in CS− . Hence, this road is blocked as well. Of course, the difficulties just mentioned do not demonstrate that there is no way for CS− to define a truth predicate. In fact, I am not aware of any proof,

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published or not, of such a general negative result. Nevertheless, I find the following hypothesis plausible. Hypothesis 8 CS− does not define the truth predicate of CT− . If the hypothesis is true, then condition (3) is violated and CS− does not fare well as a decent minimal theory of satisfaction. Something is still missing. How about CT− and condition (4)? Can we accept CT− and still recover the notion of satisfaction? I am afraid that also here we need to proceed cautiously. The answer might depend on the shape of the quantifier axiom of CT− , namely, it might depend on whether we employ the numeral variant or the term variant of the quantifier axiom (see the explaining remarks below Definition 3). If the first option is chosen, there is the following positive result. Observation 9 CT− in its numeral variant defines the satisfaction predicate of − CS . Proof. We are working in CT− . In what follows, the notation ‘ϕy ’ will be used in the same sense as in the proof of Fact 6 (that is, the expression refers to the result of substituting numerals for variables free in a formula ϕ in accordance with the assignment y). We define the satisfaction predicate in the following manner: S(ϕ, y) := ϕ ∈ FmLPA ∧ y ∈ Asn ∧ T(ϕy ). It remains to be demonstrated that under such a rendering of the satisfaction predicate, every axiom of CS− is provable in CT− . I will not check all the axioms, restricting myself to verifying that the compositional axiom of CS− for quantified formulas is provable in CT− . To this aim, it is enough to observe that the following conditions are equivalent (provably in CT− ) for an arbitrary formula ϕ and an arbitrary assignment y: 1. S(∀vϕ(v), y), 2. T(∀vϕ(v)y ), 3. ∀xT(ϕ(x)y ), 4. ∀xT(ϕ(v)y[v/x] ), 5. ∀x S(ϕ(v), y[v/x]) The equivalence between 1 and 2 is purely notational (by the definition of S); the same concerns the equivalence between 4 and 5. 2 and 3 are equivalent by the compositional axiom of the numeral variant of CT− for quantified sentences. The equivalence of 3 and 4 follows from the fact provable already in PA that ϕ(x)y = ϕ(v)y[v/x] .18 18

Keep in mind that ϕ(x) is the result of substituting a numeral denoting x for a variable v in ϕ(v). In effect, it is an arithmetical fact that substituting numerals in accordance with

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However, note that the above proof becomes incorrect if the quantifier axiom of CT− is adopted in its term variant. For the term variant, we would only obtain the following equivalences: 1. S(∀vϕ(v), y), 2. T(∀vϕ(v)y ), 3’. ∀t ∈ Tmc T(ϕ(t)y ) The problem is that now the last condition 3’ is not equivalent to 4. Indeed, 4 could be true with 3’ being false, since 4 demands truth under every substitution of a numeral for the variable v, while 3’ demands truth under every substitution of a constant term for the variable v. Changing the truth definition in such a way as to accommodate constant terms (not just numerals) does not help. Thus, one could consider the following: S(ϕ, y) iff every result of substituting constant terms denoting elements of y for appropriate variables in ϕ is true.19 Unfortunately, with this definition of satisfaction, the problem for negation reappears. We need to obtain the clause: S(¬ϕ, y) ≡ ¬S(ϕ, y). However, in the present proposal, the right side of the biconditional is ‘It is not the case that every term substitution of ϕ is true’ and there seems to be no way to obtain from this in CT− the left side, that is ‘every term substitution of ¬ϕ is true’. Again, the natural strategy permitting us to obtain the desired conclusion would be to argue by extended induction, which is not available in CT− . Exactly as in the case of the definition of truth in CS− , this does not demonstrate that one cannot define satisfaction in the term variant of CT− . However, in view of the fact that overcoming the difficulties described above seems to require extended induction for formulas of LT , I find the following hypothesis plausible. Hypothesis 10 CT− in its term variant does not define the truth predicate of − CS .

19

the assignment y in ϕ(x) produces exactly the same result as substituting numerals in ϕ(v) in accordance with y[v/x] (intuitive reason: the numeral denoting x is substituted for the variable v in both cases). Employing our previous example, let again x be the arithmetical formula ‘∃v(v + r = z)’ and let y be the assignment ((r, 5), (z, 13)). Under the present definition, S(x, y) requires not only the truth of ‘∃v(v + 5 = 13)’ but also the truth of analogously obtained sentences with arbitrary constant terms denoting 5 and 13 (for example, the definition requires that the sentence ‘∃v(v + (2 × 2) + 1 = 9 + 4)’ be also true).

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All in all, the effect is that the numeral variant of CT− satisfies condition (4). On the other hand, if Hypothesis 10 is true, then condition (4) is not satisfied by the term variant of CT− .

4

Conclusions

The assessment of satisfaction and truth theories in light of conditions (1)–(4) brought the following results. – Model-theoretic characterizations of satisfaction and truth violate condition (1), introducing a heavy theoretical apparatus which goes beyond the subject matter of satisfaction or truth. – Axiomatic theories CS and CT guarantee mutual definability of truth and satisfaction, but they also violate condition (1) introducing extended induction, which goes beyond the subject matter of satisfaction or truth. – CS− satisfies conditions (1) and (2), but possibly violates condition (3) (see Hypothesis 8). – The numeral version of CT− satisfies all the conditions including (4). – The term version of CT− satisfies conditions (1) and (2), but possibly violates condition (4) (see Hypothesis 10). In effect, all the theories have been deemed unsatisfactory with the only exception being the numeral version of CT− . In these concluding remarks I suggest a remedy against the weaknesses of satisfaction theories. Since I am not entirely satisfied even with the numeral version of CT− , a remedy will also be suggested here. In general, the remedies are new axioms, which can be added to truth and satisfaction theories. In the case of CS− , one possible solution is to introduce Lemma 7 as a new axiom. Then it is easy to observe that in CS− supplemented with this axiom, the truth predicate of CT− can be defined by one of the usual methods. An alternative approach, adopted by Enayat and Visser (2015), is the introduction of the axiom of extensionality for the notion of satisfaction. In order to formulate the axiom, let us introduce an auxiliary notion first. Let ϕ, ψ be arithmetical formulas and let x and y be assignments for ϕ and ψ respectively. We define the relation (ϕ, x) ~ (ψ, y) as holding if and only if substituting numerals in ϕ in accordance with x produces exactly the same sentence as substituting numerals in ψ in accordance with y – that is, both operations lead us to exactly one and the same syntactic object. The axiom of extensionality can now be formulated as follows: ∀ϕ, ψ∀x, y[(ϕ, x) ~ (ψ, y) → S(ϕ, x) ≡ S(ψ, y)]. Marcin Trepczyński - 978-90-04-44595-6 Downloaded from Brill.com 03/03/2024 09:20:15PM via University of Wisconsin-Madison

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It transpires that adding such an axiom to CS− permits us to define the notion of truth of CT− .20 Coming back to CT− , I find the discrepancy between the numeral variant and the term variant of this theory somewhat disquieting and indicating a weakness of both theories. The point is that opting for a numeral variant on the basis that it alone can define satisfaction (assuming that Hypotesis 10 is true) seems rather artificial. After all, what is so special about numerals? Why should any priority be given to them? Since I do not have a good answer to this question, I would opt instead to extend CT− with the additional regularity axiom, guaranteeing that co-referring constant terms can always be freely substituted in the scope of the truth predicate. Adding such an axiom would obliterate the difference between the numeral and the term variant of CT− and, ultimately, such a theory would be my candidate for a ‘minimally decent’ theory of arithmetical truth.

Acknowledgments The research presented in this paper was supported by the National Science Centre, Poland (NCN), grant number 2017/27/B/HS1/01830.

References Cieśliński, Cezary. The Epistemic Lightness of Truth. Deflationism and its Logic. Cambridge University Press, 2017. Enayat, Ali and Visser, Albert. New constructions of satisfaction classes. In: T. Achourioti, H. Galinon, K. Fujimoto, and J. Martínez-Fernández, editors, Unifying the Philosophy of Truth, pages 321–335. Springer, 2015. Engström, Fredrik. Satisfaction Classes in Nonstandard Models of First Order Arithmetic. Chalmers University of Technology and Göteborg University, 2002. Hartry Field. Deflating the conservativeness argument. Journal of Philosophy, 96: 533– 540, 1999. Kentaro Fujimoto. Classes and truths in set theory. Annals of Pure and Applied Logic, 163: 1484–1523, 2012. Haim Gaifman. Non-standard models in a broader perspective. In A. Enayat and R. Kossak, editors, Non-standard Models of Arithmetic and Set Theory, The Contemporary Mathematics Series, pages 1–22. American Mathematical Society, 2004. 20

For details, I refer the reader to (Enayat and Visser, 2015), see also (Cieśliński, 2017, pp. 112–114).

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Volker Halbach. Axiomatic Theories of Truth. Cambridge University Press, 2011. Paul Horwich. Truth. Clarendon Press, second edition, 1999. Richard Kaye. Models of Peano Arithmetic. Clarendon Press, 1991. Henryk Kotlarski, Stanisław Krajewski, and Alistair Lachlan. Construction of satisfaction classes for nonstandard models. Canadian Mathematical Bulletin, 24(3): 283– 293, 1981. Stanisław Krajewski. Non-standard satisfaction classes. In W. Marek et al., editor, Set Theory and Hierarchy Theory. A Memorial Tribute to Andrzej Mostowski, pages 121–144. Springer, 1976. Saul Kripke. Outline of a theory of truth. The Journal of Philosophy, 72 (19): 690–716, 1975. Graham Leigh. Conservativity for theories of compositional truth via cut elimination. Journal of Symbolic Logic, 80(3): 845–865, 2015. Hannes Leitgeb. What theories of truth should be like (but cannot be). Philosophy Compass, 2: 276–290, 2007. Abraham Robinson. On languages which are based on non-standard arithmetic. Nagoya Mathematical Journal, 22: 83–117, 1963. Alfred Tarski. Pojęcie prawdy w językach nauk dedukcyjnych. Prace Towarzystwa Naukowego Warszawskiego. Wydział III, 1933. Translated as ‘The Concept of Truth in Formalized Languages’ in (Tarski, 1956, 152–278). All page references are for the translation. Alfred Tarski. Logic, Semantics, Metamathematics. Clarendon Press, Oxford, 1956. Alfred Tarski and Robert L Vaught. Arithmetical extensions of relational systems. Compositio Mathematica, 13: 81–102, 1957.

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Chapter 6

Inductive Plausibility and Certainty: A Multimodal Paraconsisent and Nonmonotonic Logic Ricardo Silvestre

Abstract Is it possible to combine different logics into a coherent system with the goal of applying it to specific problems so that it sheds some light on foundational aspects of those logics? These are two of the most basic issues of combining logics. Paranormal modal logic is a combination of paraconsistent logic and modal logic. In this paper, I propose two further combinatory developments, focusing on each one of these two issues. On the foundational side, I combine paranormal modal logic with normal modal logic, resulting into a paraconsistent and paracomplete multimodal logic dealing with the notions of plausibility and certainty. On the application side, I combine this logic with Reiter’s default logic, resulting into an inductive and consequently nonmonotonic paraconsistent and paracomplete logic able to represent some key inductive principles.

Keywords paranormal modal logic – multimodal logic – Reiter default logic – Carnap’s pragmatical probability

1

Introduction1

Richard Epstein has famously opened his book on the semantic foundations of logic (to which Stanisław Krajewski has collaborated) asking: “Why are there so many logics?” (Epstein 1995, p. xix). The apparent consequence of this logi-

1 This paper is an extended version of the paper entitled “An Inductive Modal Approach for the Logic of Epistemic Inconsistency”, which appeared in the Brazilian journal Abstracta: linguagem, mente e ação, v. 6, p. 136–155, 2010.

© Ricardo Silvestre 2021 | DOI:10.1163/9789004445956_008

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cal pluralism, that “there is no one right way to reason, no one notion of necessity” (Epstein 1995, p. xix), seems disturbing, to say the least. Epstein’s purpose is to show that there is a structural unity based on some assumptions common to many logics, despite the plurality of logical systems. One can follow a different and in some sense opposite strategy. Would it be possible, for instance, to combine different logics into a coherent system with the goal of applying it to specific problems so that it sheds some light on foundational aspects of those logics? (Carnielli and Coniglio 2016). These practical and foundational purposes of the enterprise of combining logics appear very clearly in the combination of modal logic and paraconsistent logic (da Costa and Carnielli 1986) (Fuhrmann 1990) (Goble 2006) (Silvestre 2006). First of all, paraconsistent logics are often advertised for application in knowledge representation and data-base management, since it is plausible that a data-base could contain contradictory information. They are also touted as necessary in deontic contexts, since it is plausible that a body of law or other normative system could both require and prohibit something. Thus, it is natural to extend basic paraconsistent logics with modalities to represent knowledge or belief or to represent obligation, prohibition permission, etc. (Goble 2006). Second, modal logic and paraconsistent logic seem to be already foundationally connected. For example, some have defended the idea that normal modal logic already embodies a kind of paraconsistency.2 If we define in S5 an operator ~ as ¬◻ (~α =def ¬◻α), we get a unary operator that does not satisfy the principle of explosion (it is not true that from α and ~α we conclude any formula β) and has enough properties to be called a negation, which might entitle us to classify S5 as a paraconsistent logic (Béziau 2002). Some years ago, I proposed a combination of modal logic and paraconsistent logic which instantiates both the practical and foundational goals. I have called it paranormal modal logic (Silvestre 2006, 2012, 2013). From the practical viewpoint, the motivation behind paranormal modal logic lies in the concept of inductive plausibility, understood here as Carnap’s concept of pragmatical probability (Carnap 1946), and the problem of inductive inconsistencies (Hempel 1960) (Perlis 1987). When one seriously takes into consideration the contradictions that are sure to arise from the use of inductive inferences (Pequeno and Buchsbaum 1991) (Gabbay and Hunter 1991), it

2 A normal modal logic is modal logic which has K (◻(α → β) → (◻α → ◻β)) as a valid principle and modus ponens, generalization (from α conclude ◻α) and the rule of uniform substitution as valid inference rules. See (Hughes and Cresswell 1996).

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can be shown that there is not one, but two authentic approaches to deal with the problem (Silvestre 2007). These skeptical and credulous approaches to induction, as I have called them, give rise to two different plausibility notions which bear important relations to the field of paraconsistent and paracomplete logics (Loparíc and da Costa 1984). While the skeptical plausibility is a paracomplete notion, the credulous plausibility is a paraconsistent one. The idea of paranormal modal logic then is to analyze these two notions inside a modal framework. First of all, we have a modal operator ? (used in a post-fixed notation) meant to represent the notion of credulous plausibility. Alike to ⬨, α? is true in (plausible) world w iff α is true in at least one world w′ such that wRw′ , where R is the accessibility relation between plausible worlds. In addition to ?, there is an ◻-like operator ! meant to represent the notion of skeptical plausibility or acceptance: α! is true in w iff α is true in all plausible worlds w′ such that wRw′ . While the primitive negation ¬ is, in connection with ?, paraconsistent – we might have both α? and ¬(α?) – in connection with ! it is paracomplete – it might be that neither α! nor ¬(α!) are true. I say thus that ¬ is a modalitydependent paranormal negation. From the foundational point of view, in the same way that normal modal system K can be extended into D, T, B, S4, S5, etc., the most basic paranormal modal logic K? can be extended into corresponding paranormal modal systems: add α! → α? (axiom D? ) to K? and you have system D? ; add α! → α (axiom T? ) to K? and you have system T? ; add α → α?! (axiom B? ) to T? and you have system B? ; add α! → α!! (axiom 4? ) to T? and you have system S4? ; add axiom B? to S4? and you have S5? . Second, while the accessibility relation of K? has no restrictions, D? ’s models are serial, T? ’s are reflexive, B? ’s are reflexive and symmetric, S4? ’s are reflexive and transitive and S5? ’s are reflexive, transitive and symmetric. Third and more important, it is possible to show that these paranormal modal systems are equivalent to their normal counterpart: K? is equivalent to K, D? to D, and so on and so forth (Silvestre 2013). From a conceptual point of view, plausible worlds are maximized rational views on how the actual world is, obtained, for example, with the help of some inductive reasoning mechanism. That is why whatever is true in at least one plausible world w can be taken as plausible; but only weakly or credulously plausible, for there might be other plausible worlds which contradict w in the pertinent sense. On the other hand, if α is true in all plausible worlds, then α is plausible in a skeptical and much stronger sense: α is part of what we might take as a consensus among all those maximized rational views on how things are. An ideal rational agent will therefore be forced to tentatively accept α as true. The use of the word “tentatively” is important because this plausible

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acceptance is not the same as certainty: as a plausible statement, α is still subject to revision. Giving these considerations, as well as some key aspects of the philosophical framework behind paranormal modal logic, two further combinatory developments can be thought of. First, following the original motivation of the first versions of paranormal modal logic (Pequeno and Buchsbaum 1991), we might think of using the notions of plausibility along with an inductive reasoning mechanism, therefore giving rise to an inductive and consequently nonmonotonic paraconsistent and paracomplete logic. Second, since ? and ! represent epistemic notions, it might be useful to investigate the relation between these notions and other epistemic notions, notoriously the notion of certainty. This is significant, for when we look deep at the epistemic nature of inductive inferences, we see that in the same way that the conclusions of such inferences must be marked with a plausibility operator, their premises should also be referred to with the help of some epistemic notion. My purpose in this paper is to advance in these two directions. For the sake of simplicity, I shall consider only the propositional case. In the next section I briefly present paranormal modal logic K? . In Section 3, I introduce a multimodal logic meant to function as a logic of plausibility and certainty. In Section 4, I use this multimodal logic along with a nonmonotonic inferential mechanism to obtain a sort of inductive logic of plausibility and certainty. I then show, in the same section, how some basic principles of inductive reasoning might be formalized inside this framework. Finally, in the last section, I lay down some conclusive remarks.

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Paranormal Modal Logic

As I have said, the intended meaning for the modal operators ? and ! of paranormal modal logic are the notions of credulous plausibility and skeptical plausibility (or acceptability). If α is a formula, then the α? and α! mean, respectively, “α is credulously plausible” and “α is skeptically plausible or accepted”. Both ? and ! are introduced as primitive symbols of the language, as are ¬, →, ∧ and ∨. ¬ is our modality dependent paranormal negation: while in connection with ? it behaves paraconsistently, in connection with ! it behaves paracompletely. →, ∧ and ∨ are used according to their usual meanings. There is an extra monadic symbol meant to represent classical negation: ~. Along with ↔, it is defined as follows: α ↔ β =def (α → β) ∧ (β → α) and ~α =def α → p ∧ ¬p, where p is an arbitrary propositional symbol. Below is the axiomatics of K? , the most basic paranormal modal logic:

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Positive Classical Axioms P1: α → (β → α) P2: (α → (β → φ)) → ((α → β) → (α → φ)) P3: α ∧ β → α P4: α ∧ β → β P5: α → (β → α ∧ β) P6: α → α ∨ β P7: β → α ∨ β P8: (α → β) → ((φ → β) → (α ∨ φ → β)) P9: ((α → β) → α) → α Paranormal Classical Axioms A1: (α → β) → ((α → ¬β) → ¬α) wherein β is ?-free and α is !-free A2: ¬α → (α → β) wherein α is ?-free A3: α ∨ ¬α wherein α is !-free Non-Positive Additional Classical Axioms N1: ¬(α → β) ↔ (α ∧ ¬β) N2: ¬(α ∧ β) ↔ (¬α ∨ ¬β) N3: ¬(α ∨ β) ↔ (¬α ∧ ¬β) N4: ¬¬α ↔ α Paranormal Modal Axioms K1: α? ↔ ~((~α)!) K2: (¬α)! ↔ ¬(α!) K3: (¬α)? ↔ ¬(α?) Modal Axioms K? : (α → β)! → (α! → β!) Rules of Inference MP: α, α → β/β N! : α/α! The calculus of paranormal modal logic is a modal extension of positive classical logic (axioms P1–P9). Schemas of formula A1–A3 correspond to the negative axioms of classical logic, restricted in such a way as to take into account the paraconsistent and paracomplete behavior of ? and !, respectively. Trivially enough, since these restrictions apply only to non-modal formulas, the non-modal fragment of paranormal modal logic behaves exactly like in classical logic. Axiom schemas N1–N4 are meant to restore the deductive power of paranormal modal logic weakened by the restrictions of A1–A3. The paranormal modal axioms K1–K3 set the basic properties of the modal operators ! and ?. K1 states that in connection with classical negation ~, ? and ! are the dual operators of each other. K2 and K3 state that the skeptical plau-

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sibility of ¬α is equivalent to the skeptical implausibility of α, and that the credulous plausibility of ¬α is equivalent to the credulous implausibility of α, respectively. K? is the paranormal equivalent to normal modal logic’s axiom K and, finally, MP is the rule of modus ponens and N! is the paranormal rule of necessitation.3 The definition of the relations of deduction and logical consequence of paranormal modal logic makes use of the distinction between global and local premises. From an axiomatic point of view, the difference between them is that the rule of necessitation (in our case N! ) can be applied only to those formulas derived exclusively from the set of global premises.4 Let A and B be two sets of formulas, A representing the set of global premises and B the set of local premises, φ a formula and L a paranormal modal logic. I define φ’s being L-deducted from A and B (in symbols: A ⊹ B ⊢L φ) in the standard way. If B = ∅ I write A ⊢L φ; and if A = B = ∅, I write ⊢L α, in which case I say that φ is an L-theorem. In the case of K? for example, I say that φ is K? -deducted from A and B (in symbols: A ⊹ B ⊢K? φ) and that φ is a K? -theorem (in symbols: ⊢K? φ). A frame in paranormal modal logic is a pair ⟨W, R⟩ where W is a non-empty set of entities called worlds (or plausible worlds) and R is a binary relation on W called accessibility relation. A model is a triple ⟨W, R, ν⟩ where F = ⟨W, R⟩ is a frame and ν is a function mapping elements of P and W to truth-values 0 and 1. The max-min modal valuation functions Ω and ℧ which, given a model M = ⟨W, R, ν⟩ and a world w ∈ W, map formulas to truth-values 0 and 1, are defined as follows: ΩM,w (p) = ℧M,w (p) = 1 iff νw (p) = 1; ΩM,w (¬α) = 1 iff ℧M,w (α) = 0; ℧M,w (¬α) = 1 iff ΩM,w (α) = 0; ΩM,w (α → β) = 1 iff ΩM,w (α) = 0 or ΩM,w (β) = 1; ℧M,w (α → β) = 1 iff ΩM,w (α) = 0 or ℧M,w (β) = 1; ΩM,w (α ∧ β) = 1 iff ΩM,w (α) = 1 and ΩM,w (β) = 1; ℧M,w (α ∧ β) = 1 iff ℧M,w (α) = 1 and ℧M,w (β) = 1; ΩM,w (α ∨ β) = 1 iff ΩM,w (α) = 1 or ΩM,w (β) = 1; 3 Due to K1, one might think that ? or ! could be introduced as definitions and that either K2 or K3 could be obtained as theorems. Unfortunately this is not so, the reason being the non-standard behavior of →, which shall become clear later. 4 For a textbook-style presentation of modal logic that uses global and local premises in the definition of the relations of deductibility and logical consequence see (Fitting 1993).

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℧M,w (α ∨ β) = 1 ΩM,w (α?) = 1 iff ℧M,w (α?) = 1 iff ΩM,w (α!) = 1 iff ℧M,w (α!) = 1 iff

iff ℧M,w (α) = 1 or ℧M,w (β) = 1; for some w′ ∈ W such that wRw′ , ΩM,w′ (α) = 1; for all w′ ∈ W such that wRw′ , ℧M,w′ (α) = 1; for all w′ ∈ W such that wRw′ , ΩM,w′ (α) = 1; for some w′ ∈ W such that wRw′ , ℧M,w′ (α) = 1.

Formula φ is satisfied in model M and world w (in symbols: M, w ⊩ φ) iff ΩM,w (α) = 1; if φ is satisfied in all worlds w of M we say that M satisfies φ (in symbols: M ⊩ φ). I say that φ is a logical consequence of A and B, A being the global premises and B the local ones (in symbols: A ⊹ B ⊧ φ) iff, given a specific set of frames F (which in K? is the set of all frames), for every model M based on F, if M satisfies all members of A, then for every world w of M such that M, w ⊩ β, for every β ∈ B, M, w ⊩ φ.5 Ω and ℧ are evaluation functions which, depending on the modal operator at hand, maximize or minimize the truth-value of formulas: while Ω minimizes and ℧ maximizes !-marked formulas, Ω maximizes and ℧ minimizes ?-marked ones. As we have shown above, it is Ω which is used in the definition of the notion of satisfaction. The need of these two functions lies on the interpretation of the negation symbol ¬: the result of Ω applied to ¬α is defined in the function of ℧, and vice-versa. This in fact is the semantic key of paranormal modal logic’s non-classical behavior. It is easy to see that while there is a model M and world w such that M, w ⊩ φ? and M, w ⊩ ¬(φ?), there is a model M and world w such that neither M, w ⊩ φ! nor M, w ⊩ ¬(φ!). In other words, while ? is a paraconsistent modal operator, ! is a paracomplete one. As I have said, exactly in the same way as it happens with normal modal logic, there is a semantic and axiomatic relation between the several paranormal modal systems. If, for instance, we restrict ourselves to the class of serial frames, we obtain system D? , which is syntactically obtained by adding α! → α? to the axiomatics of K? ; considering the class of all reflexive frames, we have the logic T? , which is syntactically obtained by adding α! → α to the axioms of K? ; taking into account the class of all reflexive and symmetric frames, we obtain the system B? , which is the same as T? plus axiom α → α?!; and so on and so forth. K? is sound and complete, as are all other paranormal modal systems (Silvestre 2013).

5 For more on the formalization of the notions of deduction and logical consequence of paranormal modal logic see Silvestre (2012, 2013).

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A Logic of Plausibility and Certainty

What I call the logic of plausibility and certainty is a specific paranormal modal logic which, in addition to ? and !, has two extra modal operators: the classically behaved operators ◻ and ⬨, meant to represent the notions of certainty and epistemological possibility. While ◻α means “α is certain”, ⬨α means “α is epistemologically possible”. Similarly to ! and ?, ◻ and ⬨ are primitive symbols of the language. In addition to the axiomatic and semantic stipulations, the structure of the inferential relations of the logic of plausibility and certainly also plays a role in the semantic interpretation of formulas. More specifically, the place in which formulas in general and non-modal formulas in particular appear in the relation of deductibility or logical consequence is important. Suppose that A ⊹ B ⊧ φ. While an arbitrary formula α belonging to the set of global premises A is seen as meaning “α is true” or “α is a true hypothesis”, a formula β belonging to the set B of local premises means simply “β is a hypothesis.” This is why one can apply the N rules (α/α! and α/◻α; see below) only to the global premises: since α is a true hypothesis, we surely can claim it to be skeptically plausible (α!) as well as to be certain about its truth (◻α). Looking at the other way round, the fact that we can semantically conclude ◻α and α! from α (which from a semantic point of view is because all models taken into account are exactly those in which α is true in all of its worlds) reflects the idea that α is being taken as a true hypothesis and not just as a certain or accepted one. In its turn, B helps to select, out of the multitude of worlds belonging to some of these models, the individual worlds that will be used to evaluate the conclusion φ. It therefore functions like a local, hypothetical premise whose truth is guaranteed not in all, but only in a few possible worlds of the models in question. In addition to the axioms and inference rules of K? shown above, the logic of plausibility and certainty has the following axioms and inference rules: Paranormal Modal Axioms D? : α! → α? B? : α → α?! Normal Modal Axioms K: ◻(α → β) → (◻α → ◻β) D: ◻α → ⬨α B: α → ◻⬨α 4: ◻α → ◻◻α NP: ⬨α ↔ ¬◻¬α

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NN: ~◻~α ↔ ¬◻¬α Multimodal Axioms PC: ◻α → α! Rules of Inference N: α/◻α As it can be seen, the logic of plausibility and certainty is an extension of both paranormal modal logic DB? (the logic obtained by adding axiom B? to logic D? ) and normal modal logic DB4 (the logic obtained by adding axioms B and 4 to logic D). NP is there to guarantee ◻ and ⬨ as the dual of each other (recall that both are primitive symbols). In its turn, NN is needed in order to set the normal and classical behavior of ◻ (and, consequently, of ⬨). D and D? guarantee, respectively, that what is certain is also epistemically possible and what is skeptically plausible is also credulously plausible. B and B? say, respectively, that if α is true then it is certain that α is epistemologically possible and that it is skeptically plausible that α is credulously plausible. The reasonableness of these principles is self-evident in the case where α is a true hypothesis and belongs to the global premises. Concerning the local, unqualified hypothesis case, B and B? state a sort of minimal rationality principle about the hypotheses we are allowed to consider: even though they may be neither plausible nor epistemologically possible, they must be so from a second-order point of view. 4 is a sort of principle of positive introspection: if we know that α, then we know that we know that α. From B and 4 we deduce 5, ¬◻α → ◻¬◻α, which is a principle of negative introspection: if we are not certain about α, then we are certain that we are not certain about α. PC or the plausibility-certainty axiom states that if α is certain then it is also an accepted hypothesis. From it, along with MP and K1, we obtain α? → ⬨α, that is to say, that if α is (credulously) plausible then it is epistemically possible. The reason why we have excluded axioms T (◻α → α) and T? (α! → α) is that they represent a kind of principle of epistemic arrogance undesirable in the case of both certainty and skeptical plausibility. Taking α as meaning “α is true,” while T means that if we are certain that α is true then it is true, T? means that accepting α as true entails that it is true. On similar grounds, T? and T cannot be accepted if we take α as representing an unqualified, local hypothesis. While from T? along with K1 we conclude α → α?, which means that every conceivable hypothesis is automatically a plausible one, from T we derive α → ¬◻¬α, which means that every conceivable hypothesis is an irrevocable one. 4? (α! → α!!) was not included on account of the desirableness of allowing gradations of credulous plausibility (T? along with K1 entails α?? → α?), from which it is possible to develop, as we shall see below, a quantitative theory of plausibility.

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About the relation between our modal operators, we have that the following axioms are valid in the logic of plausibility and certainty: ◻α → α!, that is, from certainty we obtain acceptance, α! → α?, that is, from acceptance we obtain (credulous) plausibility, and α? → ⬨α, that is, from plausibility we get epistemic possibility. A frame in the logic of plausibility and certainty is a triple ⟨W, R? , R⬨ ⟩ where W is a non-empty set of worlds, R? is a binary relation on W called the plausibility-accessibility relation and R⬨ is a binary relation on W called the certainty-accessibility relation. R? and R⬨ satisfy the following conditions: (i) for every w, w′ ∈ W if wR⬨ w′ then wR? w′ , (ii) for every w ∈ W there is at least one w′ ∈ W and at least one w′′ ∈ W such that wR⬨ w′ and wR? w′′ , (iii) for every w, w′ ∈ W if wR⬨ w′ then w′ R⬨ w and if wR? w′ then w′ R? w, and (iv) for every w, w′ , w′′ ∈ W, if wR⬨ w′ and w′ R⬨ w′′ then wR⬨ w′′ . A model is a quadruple ⟨W, R? , R⬨ , ν⟩ where F = ⟨W, R? , R⬨ ⟩ is a frame and ν is function mapping elements of P and W to truth-values 0 and 1. For the evaluation functions Ω and ℧ we have the following modifications on what has been shown in the previous section: ΩM,w (α?) = 1 iff for some w′ ∈ W such that wR? w′ , ΩM,w′ (α) = 1; ℧M,w (α?) = 1 iff for all w′ ∈ W such that wR? w′ , ℧M,w′ (α) = 1; ΩM,w (α!) = 1 iff for all w′ ∈ W such that wR? w′ , ΩM,w′ (α) = 1; ℧M,w (α!) = 1 iff for some w′ ∈ W such that wR? w′ , ℧M,w′ (α) = 1; ΩM,w (⬨α) = 1 iff for some w′ ∈ W such that wR⬨ w′ , ΩM,w′ (α) = 1; ℧M,w (⬨α) = 1 iff for some w′ ∈ W such that wR⬨ w′ , ℧M,w′ (α) = 1; ΩM,w (◻α) = 1 iff for all w′ ∈ W such that wR⬨ w′ , ΩM,w′ (α) = 1; ℧M,w (◻α) = 1 iff for all w′ ∈ W such that wR⬨ w′ , ℧M,w′ (α) = 1. The definitions of satisfiability and logical consequence are the same as in the last section. About the peculiarities of the semantics of the logic of plausibility and certainty, I first note that given a frame ⟨W, R⬨ , R? ⟩ and a world w ∈ W, the sets R⬨ (w) = {w′ |wR⬨ w′ } and R? (w) = {w′ |wR? w′ } represent, respectively, what we may call the epistemically possible worlds of w and the plausible worlds of w. Second, every plausible world is also an epistemic possible world (in symbols: R? (w) ⊆ R⬨ (w)); this is restriction (i) of the frame structure, which from a proof-theoretical point of view corresponds to axiom PC. Third, all frames considered are serial frames; this is restriction (ii), which in the axiomatics corresponds to axioms D and D? . Fourth, while R⬨ is a symmetric and transitive relation, R? is only symmetric; this, which is stated in restrictions (iii) and (iv), corresponds, respectively, to axioms B and 4 and axiom B? .

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The logic of plausibility and certainty is sound and complete (Silvestre 2010).

4

A Logic of Inductive Implication

The purpose of a logic of induction has traditionally to do with confirmation: given a piece of evidence e and a hypothesis h, it should say whether (and possibly to what extent) e confirms or gives evidential support to h (Carnap 1950) (Hempel 1945). About the status of h, despite the diversity of approaches, most people agree on one basic point: given that e confirms h and that e is true, whatever we conclude about h, it should reflect the uncertainty inherent to inductive inferences. Almost invariably some probability notion has been chosen to do the job: even though from “e confirms h” and “e is true” we cannot conclude that h is true, we can conclude that it is probable. This notion of probability should not be confounded with Carnap’s logical probability (Carnap 1950): while the latter is supposed to be a purely logical notion connecting two sentences, the former must be seen as an epistemic label we attach to inductive conclusions in order to make explicit their defeasible character. Carnap calls this non-logical notion of probability pragmatical probability (Carnap 1946); as we have seen, I prefer the qualitative and hopefully less problematic term “plausibility”, or “inductive plausibility”. This characterization of induction in terms of pragmatical probability or inductive plausibility is significant for three reasons. First, considering that the truth of e warrants us to inductively conclude not the truth but the plausibility of h, we can say that what e confirms or evidentially supports is not the truth of h, but its plausibility. Therefore, rather than saying that e confirms or inductively supports h, we should say that e confirms or inductively supports the plausibility of h. Given that “h is plausible” will possibly be inferred, the whole thing might be read as “e inductively implies the plausibility of h.” I shall call such sort of statements inductive implications. Second, as I have mentioned in the Introduction, the contradictions that are sure to arise from the use of inductive inferences force us to consider two different but complementary approaches to induction. A consequence of this is that sentences like “e confirms or evidentially supports h” shall mention the approach according to which the confirmation is being made. This is easily done by qualifying the plausibility notion appearing in the consequent of inductive implications: while “e inductive implies the credulous plausibility of h” characterizes a credulous approach, “e inductive implies the skeptical plausibility of h” characterizes a skeptical approach.

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Third, attaching an epistemic label to the conclusions of inductive inferences leaves the door open to taking the whole notion of induction as an epistemic one. In the same way that what is confirmed or evidentially supported is not the truth of h but its plausibility, we may say that what confirms the plausibility of h is not the truth of e, but something like the certainty of h.6 As far as my formalization of these points is concerned, I shall use a version of Reiter’s default logic (Reiter 1980) to represent the notion of inductive implication. We may quite naturally read default α : φ/β as “α inductively implies β unless ¬φ”. I shall represent this by α ≻ β ⋨ ¬φ. Second, the monotonic basis of this default logic shall be exactly the logic of plausibility and certainty introduced in the previous section. Third, in order to capture the epistemological nature of inductive implications just mentioned, I shall force the components of my inductive implications to be marked with the correspondent modal operator. For instance, an inductive inference made according to a credulous approach might be represented as ◻α ≻ β? ⋨ φ, which shall be read as “the certainty of α inductively implies the (credulous) plausibility of β, unless φ′′ . Let ℑ be the language of the logic of plausibility and certainty. The inductive language ℑ≻ built over ℑ is defined as follows: (i) If α ∈ ℑ then α ∈ ℑ≻ ; (ii) If α, β, φ ∈ ℑ then α ≻ β ⋨ φ ∈ ℑ≻ ; (iii) Nothing else belongs to ℑ≻ . I call α ≻ β ⋨ φ an inductive implication, being α its antecedent, β its consequent and φ its exception. α ≻ β is an abbreviation for α ≻ β ⋨ ⊥, β ⋨ φ an abbreviation for ⊤ ≻ β ⋨ φ and β° an abbreviation for ⊤ ≻ β ⋨ ⊥, where ⊥ is an abbreviation for p ∧ ¬p and ⊤ is an abbreviation for p ∨ ¬p (p is an arbitrary propositional symbol). Any formula that is not an inductive implication is called an ordinary formula. With the help of ℑ≻ I can define the corresponding notion of extension: Let A ⊆ ℑ≻ be a set of the inductive language and S ∈ ℑ a set of formulas of the multimodal language. Γ(S) ⊆ ℑ is the smallest set satisfying the following conditions: (i) A ⊆ Γ(S); (ii) If Γ(S) ⊹ ∅ ⊢ α then α ∈ Γ(S); (iii) If α ≻ β ⋨ φ ∈ A, α ∈ Γ(S), φ ∉ S and ~β ∉ S then β ∈ Γ(S). A set of formulas E is an extension of A iff Γ(E) = E, that is, iff E is a fixed point of the operator Γ. I first note that this language ℑ≻ is a mixed language containing ordinary formulas as well as inductive implications. Therefore, the set used as the parameter in the definition of P-extension plays the role of both components of a default theory: it contains both a set of ordinary formulas as well as a set of inductive implications. Second, in mentioning the deduction relation of the

6 For a full description of the theory of induction sketched here see (Silvestre 2007) and (Silvestre 2010).

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logic of plausibility and certainty ⊢ I make use exclusively of global premises, the reason for this being that I want the notion of extension to incorporate the autoepistemic principle according to which we know everything that our inductive mechanism infers (see below).7 Finally, I make the test of consistency of the consequent (in terms of ~) inside the very definition of extension, turning then α ≻ β ⋨ φ into an equivalent of default α : β ∧ ¬φ/β; this has the advantage of preventing so-called abnormal defaults (Morris 1988). This inductive language does not yet incorporate the epistemological considerations I have made about inductive inferences. As I have advanced, one way to incorporate the theory of induction sketched here is to require the antecedent of inductive implications to be marked with the ◻ symbol and the consequent with the ? symbol. We thus have what I call the epistemic inductive language ℑE≻ : (i) If α ∈ ℑ then α ∈ ℑE≻ ; (ii) If α, β, φ ∈ ℑ then ◻α ≻ β? ⋨ φ ∈ ℑE≻ ; (iii) Nothing else belongs to ℑE≻ . Trivially ℑE≻ ⊂ ℑ≻ . In order to use this ℑE≻ language, we have to slightly change our definition of extension and introduce what I shall call a Δ-extension: Let Δ ∈ ℑ≻ − ℑ be a set of inductive implications, A ⊆ ℑE≻ a set of formulas of the epistemic inductive language and S ∈ ℑ a set of ordinary formulas. Γ(S) ⊆ ℑ is the smallest set satisfying the following conditions: (i) A ⊆ Γ(S); (ii) If Γ(S) ⊹ ∅ ⊢ α then α ∈ Γ(S); (iii) If α ≻ β ⋨ φ ∈ A ∪ Δ, α ∈ Γ(S), φ ∉ S and ~β ∉ S then β ∈ Γ(S). A set of formulas E is a Δ-extension of A iff Γ(E) = E, that is, iff E is a fixed point of the operator Γ. The idea here is that while A behaves like a default theory where its defaults satisfy the above mentioned epistemic restrictions, Δ is a set of inductive implications meant to function like axioms able to nonmonotonically extend the inferential power of our logic of plausibility and certainty. Regarding which inductive implications compose Δ we have as follows. First of all, there is the serious limitation of the logic of plausibility and certainty that we cannot conjunct plausible formulas: from α? and β? we cannot conclude (α ∧ β)?. The reason for this is obvious: it might be that α and β contradict each other in a strong sense (~(α ∧ β) or α ∧ β → ⊥), in which case (α ∧ β)? trivializes the theory (~((α ∧ β)?) or (α ∧ β)? → ⊥). However, for cases where there is no contradiction between α and β it is desirable to be able to conclude (α ∧ β)? from α? and β?. In order to deal with this we introduce the schema of inductive implications below C?∧ : α? ∧ β? ≻ (α ∧ β)?

7 To see a formulation in terms of both global and local premises see Silvestre (2010).

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and set all instances of C?∧ as belonging to Δ. See that if we have α? ∧ β? as belonging to Γ(S) and α and β contradict each other (that is to say, ~((α∧β)?) ∈ S) then (α ∧ β)? shall not be included in Γ(S). Second, axiom 4, theorem 5 and rule N embody a sort of autoepistemic principle: while 4 and 5 say that we are aware of the facts we know, as well as of the facts we do not know, respectively, N says that we are aware of all those propositions we take as true. But how about those statements whose truth we have no hint about? Suppose that Th(A) is all we can conclude from knowledge situation A. By N, for each α ∈ Th(A) we will have that we know that α (◻α.) But how about those statements which do not belong to Th(A)? It seems reasonable that for all β such that β ∉ Th(A) we conclude ¬◻β. This is what we could call a negative autoepistemic principle. It is trivially a nonmonotonic principle: if from A we infer ¬◻β, from A ∪ {β} the same inference cannot be done. It therefore might be formalized only with the help of an inductive implication: NA: ((¬◻α)?)0 NA, all instances of which belong to Δ, is the axiom which transforms our system into a truly autoepistemic logic. Note that ((¬◻α)?)0 is an abbreviation for ⊤ ≻ (¬◻α)? ⋨ ⊥. Therefore, independently of the knowledge situation at hand, if it does not contain ~((¬◻α)?) we will be able to infer nonmonotonically that ¬◻α is plausible. The purpose of this is, of course, to make explicit that our agent does not know about the truthfulness of those formulas whose certainty cannot be inferred from his knowledge base: in the cases where ◻α does not belong to the logical theory, that is to say, α is not known, (¬◻α)? will be the case. One may think that because what we conclude through NA is (¬◻α)? and not ¬◻α, NA does not in fact perform the task we are claiming it performs. Not quite so. Since α? → ⬨α (which is obtained from PP, K1 and ⬨α ↔ ~◻~α), from (¬◻α)? we get ⬨¬◻α. From this, along with NP, we get ¬◻¬¬◻α, which is equivalent to ¬◻◻α. Since ¬◻◻α → ¬◻α, we have then that ¬◻α. Third, some have defended what might be called the error-prone feature of inductive reasoning (Perlis 1987): since inductive conclusions may be mistaken even when its premises are true (something the very past use of such sort of inference has shown), any fair account of inductive reasoning should have an axiom saying that independently of the circumstances we are working on, it is plausible that one of the beliefs we now take as rational is false. This can be formalized by the following axiom:

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I? : α1 ? ∧ ⋯ ∧ αn ? ≻ (¬(α1 ∧ ⋯ ∧ αn ))? ⋨ β?, are different basic formulas

wherein α1 , …, αn and β

A basic formula is an atomic formula (a propositional symbol) or the negation of an atomic formula. All instances of I? belong to Δ. I? says that if n basic formulas are plausible, then it is also plausible that some of them are false (or, as we wrote, that the negation of their conjunction is plausible.) The exception part of I? is meant to guarantee that no plausible atomic formula will be out of the conjunction α1 ? ∧ ⋯ ∧ αn ?: if this is the case, then the induction implication at hand cannot be used. I have not spoken yet about skeptically plausible formulas. First, if we are allowed to use inductive implications only in connection with credulously plausible formulas (that is to say, inductive implications belonging to the epistemic inductive language ℑE≻ ), how are we to nonmonotonically introduce skeptically plausible formulas? Second, how are we to deal, in terms of inductive implications, with the relation we know there is between ?-marked formulas and !-marked ones? One way to answer these questions is to use a simple confirmation by enumeration approach according to which α will be taken as accepted (α!) only after it has gotten enough credulous confirmation. It is as if, by observing one black raven we turn the hypothesis “all ravens are black” into a very weakly plausible one; by observing another black raven we increase its degree of plausibility a little bit; and so on and so forth, until that, after having observed a certain number of black ravens, say n, we raise the hypothesis in question to the status of an accepted or skeptically plausible statement. In order to formalize this, we need of course to quantify how much a hypothesis was weakly confirmed or, in the context of taking weak confirmation and credulous plausibility as the same, how credulously plausible a hypothesis is. The most straightforward way to do this is to count the number of plausible worlds in which a hypothesis is true. If α is true in at least one plausible world we write α?1 ; if it is true in at least two plausible worlds we write α?2 … until it is true in at least n plausible worlds, in which case we write α?n or α!. This can be done by defining the following abbreviations: (i) α?1 =def α?; (ii) α?2 =def (α ∧ q)? ∧ (α ∧ ¬q)?, where q is an arbitrary atomic formula of ℑ; (iii) α?n =def (α∧p1 ∧q)?∧⋯∧(α∧pm ∧q)?∧(α∧p1 ∧¬q)?∧⋯∧(α∧pm ∧¬q)?, where n = 2k+1 , m = 2k , k > 0, α?m ≡ (α ∧ p1 )? ∧ ⋯ ∧ (α ∧ pm )? and q is an arbitrary atomic formula of ℑ which does not occur in p1 ;

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(iv) α?n =def (α ∧ p1 )? ∧ ⋯ ∧ (α ∧ pn )?, where 2k+1 > n > 2k and α?n+1 ≡ (α ∧ p1 )? ∧ ⋯ ∧ (α ∧ pn )? ∧ (α ∧ pn+1 )?. α?n may be understood as meaning “the degree of plausibility of α is n.” As we have mentioned above, such meaning is achieved by counting in how many plausible worlds α is true, which is performed with the help of the classical feature of worlds. Given an atomic formula q, we know that q and ¬q cannot be true at the same time in world w. Therefore, if (α ∧ q)? and (α ∧ ¬q)? are true, then the plausible worlds which make these two formulae true cannot be the same. Consequently, α is true in at least two worlds. Similarly, given an atomic formula p distinct from q, (α ∧ q ∧ p)? ∧ (α ∧ ¬q ∧ p)? ∧ (α ∧ q ∧ ¬p)? means that α is true in at least three worlds, (α ∧ q ∧ p)? ∧ (α ∧ ¬q ∧ p)? ∧ (α ∧ q ∧ ¬p)? ∧ (α ∧ ¬q)? that α is true in at least four worlds, and so on and so forth. With the help of this abbreviation we can nonmonotonically obtain skeptically plausible formulas thought credulously plausible ones according to the confirmation by enumeration philosophy mentioned above: !n : α?n ≻ α! ⋨ (¬α)? All instances of !n , for some specific n, belong to Δ. Note that, according to !n , even if α?n is true (that is, α is true in at least n plausible worlds) two situations might prevent α! from being inferred: if α! implies a contradiction or if (¬α)? is the case. This second situation is significant, for it illustrates how the exception part of inductive implications can be used to set priority between inductive implications. For instance, imagine that we somehow have gotten α?n and that the inductive implication β ≻ (¬α)? belongs to A. Suppose further that we have gotten β. In this case, because of the exception part of !n , α! shall not be inferred: β ≻ (¬α)? has priority over α?n ≻ α! ⋨ (¬α)?.

5

Conclusion

In this paper, I have elaborated on how one might extend paranormal modal logic in such a way as to use the notions of plausibility along with an inductive reasoning mechanism, which seriously takes into consideration the epistemic nature of inductive reasoning. More specifically, a non-classical multimodal logic of plausibility and certainty has been introduced, in which the plausibility operators ? and ! behave paraconsistently and paracompletely, respectively, and on the other hand the operators of certainty and epistemic possibility behave classically. Along with a version of Reiter’s default logic, I was able

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to use this logic of plausibility and certainty to formalize a very simple theory of induction. It should be noted that this formalization is just one among the several possibilities in which we can use the logic of plausibility and certainty along with a nonmonotonic reasoning mechanism to formalize a theory of induction. For an illustration of some of these possibilities along with the formalization of less naïve theories of induction, see (Silvestre 2010).

References Béziau, J. (2002) ‘S5 is a paraconsistent logic and so is first-order classical logic’, Logical Studies, 9, 301–309. Carnap, R (1946) ‘Remarks on Induction and Truth’, Philosophy and Phenomenological Research, 6, 590–602. Carnap, R. (1950), Logical Foundations of Probability, Chicago: University of Chicago Press. Carnielli, W. and Coniglio, M. (2016), ‘Combining Logics’, IN Zalta, E. (ed.), The Stanford Encyclopedia of Philosophy (Winter 2016 Edition), URL = https://plato.stanford.edu/ archives/win2016/entries/logic-combining/. da Costa, N. C., Carnielli, W. (1986) ‘On paraconsistent deontic logic’, Philosophia, 16, 293–305. Epstein, R. (1995) The Semantic Foundations of Logic, Oxford: Oxford University Press, 2nd edition. Volume 1: Propositional logics, with the assistance and collaboration of W. Carnielli, I. D’Ottaviano, S. Krajewski, and R. Maddux. Fitting, M. (1993) ‘Basic modal logic’, IN Gabbay, D., Hogger, D., Robinson, J. (eds.), Handbook of Logic in Artificial Intelligence and Logic Programming, Vol. 1, Logical Foundations, Oxford: Oxford University Press, pp. 368–448. Fuhrmann, A (1990) ‘Models for relevant modal logics’, Studia Logica, 49, 501–514. Gabbay, D. and Hunter, A. (1991), ‘Making inconsistency respectable: A logical framework for inconsistency in reasoning’, IN Jorrand, P., Kelemen, J., (eds.), Foundations of Artificial Intelligence Research (LNCS 535), Springer-Verlag, New York (1991), 19–32. Goble, L. (2006) ‘Paraconsistent modal logic’, Logique et Analyse, 193, 3–29. Hempel (1960). ‘Inductive inconsistencies’, Synthese, 12, 439–469. Hempel, C. (1945) ‘Studies in the Logic of Confirmation’, Mind, 54, pp. 1–26, 97–121. Loparíc, A. da Costa, N. C. (1984), ‘Paraconsistency, paracompletenes and valuations’, Logique et Analyse, 106, 119–131. Morris, P. (1988) ‘The Anomalous Problem in Default Reasoning’, Artificial Intelligence, 35, 383–399.

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Pequeno, T., Buchsbaum, A. (1991) ‘The logic of epistemic inconsistency’, in Allen, J., Fikes, R., Sandewall, E. (eds.) Principles of Knowledge Representation and Reasoning: Proceedings of Second International Conference (1991) San Mateo: Morgan Kaufmann, pp. 453–460. Perlis, D. (1987) ‘On the Consistency of Commonsense Reasoning’, Computational Intelligence, 2, 180–190. Reiter, R. (1980) ‘A Logic for Default Reasoning’, Artificial Intelligence, 13, 81–132. Silvestre, R. (2006) ‘Modality, paraconsistency and paracompleteness’, IN Governatori, G., Hodkinson, I., Venema, Y. (eds.), Advances in Modal Logic 6 (2006) London: College Publications, pp. 459–477. Silvestre, R. (2007) ‘Ambiguidades Indutivas, Paraconsistência, Paracompletude e as Duas Abordagens da Indução’, Manuscrito, 30, 101–134. Silvestre, R. (2010) Induction and Plausibility: A Conceptual Analysis from the Standpoint of Nonmonotonicity, Paraconsistency and Modal Logic, Berlim: LAP Lambert Academic Publishing. Silvestre, R. (2012). ‘Paranormal Modal Logic – Part I: The System K? and the Foundations of the Logic of Skeptical and Credulous Plausibility’, Logic and Logical Philosophy, 21, 65–95. Silvestre, R. (2013) ‘Paranormal Modal Logic – Part II: K?, K and Classical Logic and other Paranormal Modal Systems’, Logic and Logical Philosophy, 22, 89–130.

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Chapter 7

On the Reception of Cantor’s Theory of Infinity (Mathematicians vs. Theologians) Roman Murawski

Abstract The aim of this paper is to consider the reception of Cantor’s theory of sets, in particular his theory of infinity/infinite sets. After recalling the basic facts concerning the development of the concept of infinity and the concept of sets, we present Cantor’s theory of infinity. His theory did not find acceptance among mathematicians – the only exception here was Richard Dedekind. In this situation Cantor turned to Catholic philosophers and theologians who were interested in his theory hoping that it would provide a useful tool for their considerations of the concept of God. Cantor himself was interested in philosophy and theology and was convinced of the meaning of his set theory outside of mathematics. We describe Cantor’s contacts with Catholic theologians as well as the philosophical modifications he made in his theory of infinity as a result of the influence of theologians, in particular Cardinal Johann Franzelin. The paper ends with a description of Cantor’s religious views and an attempt to explain why he – being a Protestant – collaborated only with Catholic theologians.

Keywords infinity – actual and potential infinity – absolute infinity – set theory – theology – Cantor – God

1

Introduction1

The concept of infinity belongs to one of the most important concepts of mathematics. It appeared in various forms and contexts from the very begin1 This text is based on my paper “Filozoficzne i teologiczne tło Cantora teorii mnogości”, in: P. Polak, J. Mączka, W.P. Grygiel (Eds.), Oblicza filozofii w nauce, Copernicus Center Press, Kraków 2017, 51–84.

© Roman Murawski 2021 | DOI:10.1163/9789004445956_009

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ning. And it was a difficult and troublesome concept – not only for mathematicians but also for philosophers and theologians. Recall the paradoxes of Zeno of Elea or the troubles with infinity experienced by Aristotle or Euclid. One attempted to avoid this by distinguishing potential and actual infinity and claiming that the former (being in fact not-proper infinity but only an unbounded finite magnitude) suffices and the latter (which was the real source of trouble) is in fact superfluous in mathematics. The real turning-point in considerations on infinity was Georg Cantor’s (1845–1918) theory of sets developed in 1874–1897. It showed that infinity can be treated as a genuine mathematical concept and can be investigated using mathematical tools. In this way the anxiety of antinomies was overcome in a certain sense. The very concept of sets had already appeared before Cantor. For example some elements of the algebra of sets can be found in works by Gottfried Wilhelm Leibniz (1646–1716). His ideas were developed by Johann Heinrich Lambert (1728–1777) and Leonard Euler (1707–1783). Euler proposed, for example, a geometrical interpretation of the relations between sets. William Hamilton (1788–1856) and Augustus De Morgan (1806–1878) attempted to develop a formal theory of relations, and George Boole (1815–1864) developed the calculus of sets in the form of a fully formalized theory, which today is called the theory of Boolean algebra (note that it can be interpreted in various ways, not only as an algebra of sets). The calculus of relations together with the algebra of sets was developed by Charles Sanders Peirce (1839–1914) and Ernst Schröder (1841–1902). A general concept of sets (understood in the sense of ontology) is found in an explicit form only in the first half of the 19th century in the first volume of Wissenschaftslehre (1837) by Bernard Bolzano (1781–1848). Some advanced considerations on infinite sets can be found in the works of Richard Dedekind (1831–1916) and Paul Du Bois Reymond (1831–1889) concerning mathematical analysis. All these studies were in fact only fragmentary and it was Cantor who developed a theory that could be called the theory of sets. Using the concept of the equipollence of sets,2 Cantor introduced cardinal numbers and using the concept of the similarity of orders3 – ordinal numbers. Infinite cardinal and ordinal numbers were, of course, the most interesting. Having proved the theorem stating that the power set of any set (i.e., the set of all its subsets) is always of a greater cardinality than the set itself, he in2 Two sets A and B are said to be equipollent if and only if there exists a one-one mapping of A onto B. 3 Two ordered sets are said to be similar if and only if there exists a one-one mapping of A onto B that preserves orders.

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troduced an infinite hierarchy of infinite cardinal numbers, hence an infinite hierarchy of greater and greater infinities. It is worth noting here that his considerations had not only a mathematical character but also a philosophical and even theological one.4

2

Cantor’s Concept of Infinity

Before discussing Cantor’s philosophical and theological considerations of infinity and the reception of his ideas, let us recall the basic distinctions he accepted. First, it should be noted that he followed Aristotle and distinguished potential and actual infinity. In a letter of 28 February 1886 to A. Eulenberg from Berlin (included in the paper “Mitteilungen zur Lehre vom Transfiniten”, 1887–1888; cf. also Cantor, Gesammelte, 401) he wrote: I. We can say about P-I (P-I = potential infinity, ἄπιερον) when a certain indefinite variable finite quantity occurs which increases above all finite bounds […] or becomes smaller than any finite bound […]. II. A-I (A-I = actual infinity, άφορισμένον) should be understood as a quantity, which on the one hand is invariable, constant and definite in all its parts, which is a real constant and which simultaneously exceeds any finite quantity of the same type.5 It should be stressed that in Cantor’s view, potential infinity is in fact no infinity. In some of his papers he called it improper infinity. In the letter to Eulenberg mentioned above, he wrote (cf. Cantor, Gesammelte, 404): […] P-I is an easier, nicer and more superficial, less independent concept with which a certain nice illusion of having something proper and

4 Among the recent contributions on Cantor’s ideas concerning set theory and theology are: Newstead, “Cantor on Infinity”, Achtner, “Infinity in Science” and Drozdek, “Number and Infinity”. 5 “I. Das P.-U. (d.h. das potentiale Unendliche (ἄπιερον)) wird vorzugsweise dort ausgesagt, wo eine unbestimmte, veränderliche endliche Größe vorkommt, die entweder überalle endlichen Grenzen hinaus wächst […] oder unter jede endliche Grenze der Kleinheit abnimmt. II. Unter einem A.-U. (d.h. das aktuale Unendliche (άφορισμένον)) ist dagegen ein Quantum zu verstehen, das einerseits nicht veränderlich, sondern vielmehr in allen seinen Teilen fest und besatimmt, eine richtige Konstante ist, zugleich aber andrerseits jede endliche Größe deselben Art an Größe übertrifft.”

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really infinite is connected; in fact P-I possesses only a borrowed existence.6 Cantor distinguished three forms of actual infinity: absolute infinity, infinity appearing in the world of things and infinity as a mathematical quantity. In “Mitteilungen” he wrote (cf. Cantor, Gesammelte, 378): I have distinguished three forms of actual infinity: first that which is realized in the highest perfection, in a completely independent being existing outside of the world, in God (in Deo) and which I call absolute infinity or shortly the Absolute; second that which appears in a dependent and created world, and third that which can be comprehended in thought in abstracto as a mathematical quantity, number or ordinal type.7 In both latter cases where infinity appears as a bounded quantity that can be extended and that is connected with the finite, Cantor calls it transfinity and strictly sets it against the Absolute. Absolute infinity cannot be extended whereas both other types of infinity are extendable. They point to the Absolute, which “surpasses in a certain sense, the human ability of comprehension and cannot be described mathematically”8 (cf. Cantor, Gesammelte, 405). Transfinity expresses God’s magnificence. However Cantor adds (cf. Cantor, Gesammelte, 406): This fact must still wait a long time to be commonly acknowledged and accepted especially by theologians – it will appear to be a valuable auxiliary mean in developing the domain they represent (religion).9

6 “[…] P.-U. der leichtere, angenehmere, oberflächlichere, unselbstständigere Begriff [ist] und die schmeichlerische Illusion zumeist mit ihm verknüpft ist, als hätte man daran was Rechtes, was richtig Unendliches; während doch in Wahrheit das P.-U. nur eine geborgte Realität hat.” 7 “Es wurde das A.-U. [= das aktuale Unendliche] nach drei Beziehungen unterschieden: erstens sofern es in der höchsten Vollkommenheit, in völlig unabhängigen, außerweltlichen Sein, in Deo realsiert ist, wo ich es Absolutunendliches oder kurzweg Absolutes nenne; zweitens sofern es in der abhängigen, kreatürlichen Welt vertreten ist; drittens sofern es als mathematische Größe, Zahl oder Ordnungstypus vom Denken in abstracto aufgefaßt werden kann.” 8 “übersteigt gewissermaßen die menschliche Fassungskraft und entzieht sich namentlich mathematischer Determination” 9 “Dies wird aber auf allgemeine Anerkennung noch lange zu warten haben, zumal bei den Theologen, so wertvoll auch diese Erkenntnis als Hilfsmittel zur Förderung der von ihnen vertretenen Sache (der Religion) sich erweisen würde.”

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The study of absolute infinity is – according to Cantor – the task of the speculative theology whereas the transfinity should be the object of study of metaphysics and mathematics. In “Mitteilungen” he wrote (cf. Cantor, Gesammelte, 378): As the study of absolute infinity and the determination of what the human mind is able to state about it is the task of speculative theology, then on the other hand, problems concerning transfinity belong to the domain of metaphysics and mathematics.10 Hence the study of infinity belongs to various domains. Consequently, Cantor looked for a justification of his theory of sets also beyond mathematics. Herbert Meschkowski, a biographer of Cantor writes (cf. Meschkowski, Biographical, 404): For Cantor the theory of sets was not only a mathematical discipline. He also integrated it into metaphysics, which he respected as a science. He also sought to tie it in with theology, which used metaphysics as its ‘scientific tool’. Cantor was convinced that actual infinity really existed ‘both concretely and abstractly’. On the other hand, he was convinced that “without a bit of methaphysics it is impossible to establish any strict science”11 – as he wrote in a note (made in pencil) in 1913 and never published. It should be added that this note recalls the following sentence by Kant from his Prolegomena: “There will therefore be metaphysics in the world at every time, and what is more, in every human being, and especially the reflective ones”12 (cf. Kant, Prolegomena, 367; English translation – 118). Cantor was interested in philosophy and had good knowledge of it. His library included the works of classical writers from Plato and Aristotle through Augustine, Boëthius, Thomas of Aquinas, Descartes, Spinoza, Locke, Leibniz to Kant. He not only possessed them but read them and knew them quite well. 10

11 12

“Liegt es besonders der spekulativen Teologie ob, dem Absolutunendlichen nachzuforschen und dasjenige zu bestimmen, was menschlicherseits von ihm gesagt werden kann, so fallen andrerseits die auf das Transfinite hingerichteten Frage hauptsächlich in die Gebiete der Metaphysik und der Mathematik.” “Ohne ein Quentchen Metaphysik läßt sich, meiner Überzeugung nach, keine exacte Wissenschaft begründen.” “Es wird also in der Welt jederzeit und, was noch mehr, bei jedem vornehmlich nachdenkenden Menschen Metaphysik sein.”

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In fact, many quotations are found in his writings from the works of famous philosophers as well as discussions with their views in connection with his own concepts concerning, for example, the way mathematical objects exist or infinity. He also conducted classes in philosophy for students – they were devoted, among others, to Leibniz’s philosophy (this was an occasion for him to compare his own views on infinity with those of Leibniz). Philosophers were also among his friends – let us mention here E. Husserl and H. Schwarz, who received their habilitation at the university in Halle where Cantor was a professor between 1872 and 1913. He also highly valued theology and treated it as a rational discipline – he was convinced that its basic concept, the concept of God, can be clarified by the concept of infinity and by distinguishing its various types. He also claimed that theology holds the principle of contradiction, which provides a rational justification of the theses proclaimed in it. Cantor was especially attracted to scholastics. And this did not happen by accident. Scholastics led him to problems and considerations similar to those that fascinated him. He saw the similarity and a certain relationship between his theoretical abstract set considerations on the one hand, and the considerations of scholastic logic and theology on the other. In fact, for him philosophy was not something foreign and external with respect to mathematics. He saw certain strong and internal connections between them. His remarks from the introduction to one of his papers indicate the importance of his conviction that readers of his papers should have knowledge of both mathematics and philosophy. He said there that he in fact wrote this paper for two circles of readers: “for philosophers who follow the development of mathematics to modern time and for mathematicians who know the most important old and new philosophical ideas” (cf. Cantor, Grundlagen einer allgemeinen Mannigfaltigkeitslehre, 1883, 545; see also Cantor, Gesammelte, 165). In searching for the philosophical justification of his theory of infinite sets, Cantor referred to metaphysics and theology. He was convinced that just there one should look for the foundations. In a letter of 1 February 1896 to Th. Esser, a Dominican friar active in Rome, he wrote (cf. Meschkowski, Probleme, 111 and 123; see also Meschkowski, “Aus den Briefbüchern”): The general theory of sets […] belongs entirely to metaphysics. […] The founding of the principles of mathematics and the natural sciences belongs to metaphysics; hence it should treat them as its own children as well as servants and assistants that should not be lost, on the contrary – it should constantly take care of them; as a queen bee residing in a beehive sends thousands of diligent bees to gardens to suck out nectar from flowers and to convert it then under its supervision into excellent honey,

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so metaphysics – in a similar way – should take care that mathematics and the natural sciences bring from the far kingdom of the corporal and spiritual nature the building blocks to complete its own palace.13 Moreover, Cantor was convinced of the great meaning and significance of his set theory for metaphysics and theology. In particular he was of the opinion that it will help to overcome the various mistakes appearing in them, for example the mistake of pantheism. In a letter to the Dominican friar Th. Esser from February 1896 he wrote (cf. Dauben, Georg Cantor, 147): Thanks to my works Christian philosophy has now for the first time in history the true theory of the infinite. In a letter of 19 December 1895 to Esser he wrote (cf. Tapp, Kardinalität, 303): It would be my greatest joy if my works would appear to be helpful to Christian philosophy, philosophiae perennis, which is so close to my heart. This would then become possible if they were carefully and in detail investigated and verified in the old school, so admirably renewed now and re-established by His Holiness Leo XIII.14

3

Reception of Cantor’s Set Theory by Mathematicians

Note that in Cantor’s days prejudices prevailed among mathematicians (but also among philosophers and theologians) against (actual) infinity. Philoso-

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“Die allgemeine Mengenlehre […] gehört durchaus zur Metaphysik. […] Die Begründung der Prinzipien der Math[ematik] und der Naturwiss[enschaften] fällt der Metaphysik zu; sie hat daher jene als ihre Kinder sowohl wie auch als Diener und Gehülfen anzusehen, die sie nicht aus dem Auge verlieren darf sondern stets zu bewachen und controllieren hat und wie die in einem Apiarium residierende Bienenkönigin tausende von fleißigen Bienen in den Garten hinausschickt, damit się überall aus den Blumen Saft aussaugen und ihn dann gemeinsam und unter ihrer Aufsicht in köstlichen Honig verarbeiten, die ihr aus dem weiten Reiche der körperlichen und geistigen Natur die Bausteine zur Vollendung ihres eigenes Palastes herbeibringen müßten.” “Am Meisten würde es mich aber freuen, wenn meine Arbeiten der meinem Herzen am Nächsten stehenden christlichen Philosophie, der “philosophia perennis”, zugute kämen, was nur dann denkbar und möglich wäre, wenn sie von der alten, nun durch S. Heiligkeit Leo XIII so herlich erneuerten, widererstandenen Schule genau und eingehend untersucht und geprüft würden.”

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phers noted that antinomies were connected with it. Theologians rejected infinity because they saw in it a challenge to the idea of the unique and absolutely infinite nature of God. Mathematicians were against applications of actual infinity because they feared paradoxes. A good illustration of such a position is the letter of 12 July 1831 written by Carl Friedrich Gauss to Heinrich Christian Schumacher, in which he wrote (cf. Gauss, Carl, 268–271): First of all I protest against applying infinite quantities as something being complete and closed, which is never allowed in mathematics. Infinity is only a façon de parler in which one actually speaks of limits […].15 One should take into account this intellectual atmosphere when considering Cantor’s set theory work and his attempts to find a justification for this theory. Note that his work was not of interest to mathematicians. The only exception here was Richard Dedekind who recognized the meaning of set theory, moreover, he exchanged letters with Cantor and discussed several problems connected with it (cf. Cantor, Gesammelte, 443–451; see also Noether and Cavaillès, Briefwechsel). It is worth noting here that each had a different approach to sets and understanding of them. Let us quote an anecdote (cf. Dedekind, Gesammelte, vol. 2, 449; see also Becker, Grundlagen, 316): Dedekind spoke out with respect to the concept of a set: he imagines a set as a closed sack containing quite specific things which one does not see and of which one knows nothing except that they are given and determined. Some time later, Cantor provided his image of a set: He raised his immense figure, described a circle with his arm and said looking far away: “I imagine a set as an abyss”.16 Cantor wrote about this difference in understanding sets in a letter of 27 January 1900 to Hilbert (cf. Cantor, Briefe, 427): 15

16

“Ich protestiere zuvörderst gegen den Gebrauch einer unendlichen Grösse als einer Vollendeten, welche in der Mathematik niemals erlaubt ist. Das Unendliche ist nur eine Façon de parler, in dem man eigentlich von Grenzen spricht […].” “Dedekind äußerte sich hinsichtlich des Begriffs der Menge: er stelle sich eine Menge vor wie einen geschlossenen Sack, der ganz bestimmte Dinge enthalte, die man aber nicht sehe, und von denen man nichts wissse, außer daß sie bestimmt und vorhanden seien. Einige Zeit später gab Cantor seine Vorstellung einer Menge zu erkennen: Er richtete seine kolossale Figur auf, beschrieb mit erhobenem Arm eine großartige Geste und sagte mit einem ins Unbestimmte gerichteten Blick: ‘Eine Menge stelle ich mir vor wie einen Abgrund’”.

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Another difference between me and Dedekind is, as you certainly know, his conviction that any determined multiplicity is consistent, i.e., he does not admit the difference between inconsistent and consistent multiplicities.17 The problem of consistent and inconsistent multiplicities appeared to be important because it was close to the problem of antinomies. Cantor proposed a solution based on several philosophical assumptions. In 1895, Cantor discovered the antinomy of the set of all ordinal numbers and wrote about it in an 1896 letter to Hilbert. Independently of Cantor the antinomy was discovered by Cesare Burali-Forte and published by him in 1897. In a letter of 31 August 1899 to Dedekind, Cantor informed him about the discovery of another antinomy, i.e. the antinomy of the set of all sets. Cantor’s proposal to solve both these antinomies (described in a letter of 28 July 1899 to Dedekind18) was based just on the distinction of consistent and inconsistent multiplicities: Cantor stated that the set of all ordinal numbers as well as the set of all sets are just inconsistent multiplicities, i.e., multiplicities that cannot be thought of as one whole, as a unity. He also called them absolute infinite multiplicities. On the other hand consistent multiplicities are multiplicities that can be grasped as one whole, as a unit – he called them sets. It is worth noting here that the proposed distinction of consistent and inconsistent multiplicities or – as is said today – sets and classes is an explicit reference to the concept that was pivotal in Leibniz’s metaphysics – namely the concept of co-possibility, co-existence or co-consistency. Leibniz claimed that only God is able to know when collections of properties are co-possible. Cantor understood the richness and complexity of infinity. On the other hand, in the Western tradition, infinity was always linked to God – God was in a certain sense the complement of infinity, the Absolute. Hence Cantor gives us a separate type of infinity “which is realized in the highest perfection, in a completely independent being existing outside of the world, in God (in Deo) and which I call absolute infinity or in short, the Absolute”.19 According to Western theology, the Absolute can never be adequately described. Hence, the existence of inconsistent multiplicities could appear to Cantor as something natural and normal – in fact the finite and bounded human mind is not able to 17

18 19

“Mein anderer Gegensatz zu Dedekind besteht, wie Sie ja wissen, darin, daß er jede bestimmte Vielheit für consistent [widerspruchsfrei] hält, also den Unterschied von consistenten und inconsistenten Vielheiten nicht zugiebt.” See Cantor, Gesammelte, 443–444. Cantor, “Mitteilungen”; see also Cantor, Gesammelte, 378.

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completely and fully grasp the infinity of God and of God’s creations in their existence. It is worth adding here that Dedekind differed in his attitude towards antinomies. When Cantor communicated to him in a letter dated 20 July 1877 that he proved the equipollence of some geometrical sets of different dimensions, Dedekind was extremely surprised and did not dare to express his amazement and consternation in his mother language, but did so in French: Je le vois, maix je ne le croix pas! (I see it but I cannot believe it!) (cf. Fraenkel, Das Leben, 458).20 Leopold Kronecker serves as an example of the attitude of mathematicians towards Cantor’s set theory. He claimed that mathematics does not need either infinite sets or the other concepts considered in Cantor’s theory. In his project of the “arithmetization” of algebra and analysis, he proposed to base them on the fundamental concept of a number, more exactly, on the intuition of a natural number. He summarized this in the best way with his sentence: “Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk” (The integers were made by God, everything else is the work of man). He claimed: “I study mathematics only as an abstraction of arithmetical reality.” He probably said that Cantor is corrupting young people. He did not avoid various intrigues to discredit Cantor and his concepts (cf. for example Dauben, Georg Cantor, 134–137). Another example of misunderstanding Cantor’s set theory ideas is the rejection of his paper by Magnus Gustaw Mittag-Leffler, editor-in-chief of the journal Acta Mathematica. At the end of 1884 and beginning of 1885, Cantor sent Mittag-Leffler a manuscript of his new work “Principien einer Theorie der Ordnungstypen. Erste Mitteilung”. The latter refused to accept it. He communicated his decision to Cantor in a letter dated 9 March 1885 in which he advised him to wait with publication until he obtains positive results in the theory of order types, particularly results leading to the solution of the continuum hypothesis. He compared Cantor’s investigations with Gauss’ studies on non-Euclidean geometries and stressed that Gauss was slow to proclaim his ideas as he was convinced that mathematicians were not ready yet to accept them. Mittag-Leffler believed that the situation was similar in the case of Cantor’s set-theoretical ideas. On the other hand, it should be stressed that Mittag-Leffler understood the significance of Cantor’s works. In a letter to him dated 11 March 1883, he wrote (cf. Meschkowski and Nilson, Georg Cantor, 118): 20

This fact was confirmed by Wilhelm Stahl, grandson of Cantor (cf. Thiele, “Georg Cantor”, 542).

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Your work will more easily find acceptance in the mathematical world if it appears now without philosophical and historical remarks. In particular, French and Italian mathematicians do not understand anything from philosophy and they will just be interested in the mathematical parts of your work.21 In the letter of 9 March 1885 mentioned above, Mittag-Leffler wrote (cf. Meschkowski and Nilson, Georg Cantor, 241): I am afraid that most scholars will be shocked by your new terminology and your very general philosophical way of speaking. I personally think that you write very well and I very much like the general character of your research. […] However, if your theories are negatively received then a long time will be needed before they attract the attention of the mathematical world again. It can even happen that justice will not be granted to them during our lifetime. So the theories will be discovered by someone in another 100 years or more and then they will find out afterwards that you already had all this and then justice will be done to you in the end, but in this way you will not have had a significant influence on the development of our discipline.22 A consequence of Mittag-Leffler’s rejection was the fact that Cantor distanced himself from mathematical journals and turn his attention to philosophical considerations. From his correspondence with Gustaf Eneström (he helped Mittag-Leffler in editing Acta Mathematica) and Paul Tannery, it follows that already in November 1885, Cantor prepared a “more philosophical than math21

22

“Ihre Arbeit wird viel leichter in der Mathematischen Welt Anerkennung finden, wenn sie jetzt auch ohne die philosophischen und historischen Auslegungen erscheint. Besonders verstehen französischen und italienischen Mathematiker gar nichts von Philosophie und diese sind doch jedoch, welche sonst für das Mathematische in Ihrer Arbeit das grösste Verständnis haben werden.” “Wie es jetzt ist, fürchte ich dass die meisten sich sehr erschrecken warden wegen Ihre neue Terminologie und Ihre sehr allgemeine philosophische Ausdrucksweise. Was mich persönlich anbetrifft, finde ich dass Sie hier immer sehr gut schreiben und ich liebe sehr Ihre allgemeine Art die Untersuchungen anzustellen. […] Aber wenn Ihre Theorien auf diese Weise in Misscredit kommen, wird es sehr lange dauern bis sie wieder die Aufmerksamkeit der mathematischen Welt an sich ziehen. Ja es kann wohl sein dass man Ihnen und Ihre Theorien nie in unserer Lebenszeit Gerechtigkeit zu Theil kommen lässt. So werden die Theorien wieder einmal 100 Jahren oder mehr von Jemanden entdeckt und dann findet man wohl nachträglich aus, dass Sie doch schon das alles hatten und dann thut man Ihnen zuletzt Gerechtigkeit, aber auf diese Weise werden Sie keinen bedeutenden Einfluss auf die Entwicklung unserer Wissenschaft ausgeübt haben.”

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ematical” paper on order types. This paper came to the attention of Felix Klein, editor-in-chief of Mathematische Annalen – he was ready to publish it in this journal. However Cantor refused – he preferred to publish it in a French philosophical journal! If one wants to understand Cantor’s reaction, it should be said that he treated any critique of his works very seriously, deeply and personally. In an answer to Mittag-Leffler’s rejection Cantor demanded that all manuscripts in his possession should be returned. It should be added that Mittag-Leffler was not against set theory as such (as, for example, Kronecker), moreover, he was one of the first mathematicians who applied Cantor’s ideas in his own works. He also suggested that earlier works of Cantor and a part of Grundlagen einer allgemeinen Mannigfaltigkeitslehre. Ein mathematisch-philosophischer Versuch in der Lehre des Unendlichen should be translated into French. However, MittagLeffler’s rejection came at a critical moment in Cantor’s life. He was still working – without success – on the problem of continuum and was not able do determine the truth of the continuum hypothesis. In a letter to Mittag-Leffler from May 1883 he wrote that in every proof he constructed he always discovered errors or mistakes which forced him to start from the beginning. All these circumstances were the reason that Cantor turned his attention and interests more towards the domains outside of mathematics, i.e., towards metaphysics and theology. And he found scholars who were willing to collaborate with him among Catholic philosophers and theologians.

4

Cantor’s Contacts with Catholic Theologians

The paper that Cantor sent to theologians and that was the starting point of their discussions was “Grundlagen einer allgemeinen Mannigfaltigkeitslehre” published in 1883 as a fifth part of the series of papers “Über unendliche, lineare Punktmannigfaltigkeiten” and as an independent small monograph (cf. Cantor, 1883a i 1883b; see also Cantor, 1932, 139–246, in particular pages 165–209 where “Grundlagen” can be found). Cantor’s contacts with theologians were possible thanks to a new climate in the Roman Catholic theology after the promulgation of the encyclical Aeterni Patris (in English: Of the Eternal Father) issued by Pope Leo XIII on 4 August 1879. It was subtitled “On the Restoration of Christian Philosophy in Catholic Schools in the Spirit (ad mentem) of the Angelic Doctor, St. Thomas Aquinas”. It was the first encyclical entirely devoted to philosophy, its aim was to advance a revival of scholastic philosophy, in particular Thomism as a most perfect philosophical system from the point of view of Catholicism. In fact, this

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encyclical led to an increase of the meaning of this doctrine. Attempting to reconcile new discoveries and achievements in science with the Holy Bible and the doctrine of the Roman Catholic Church, the Pope stressed the role of reason.23 Appreciating the natural sciences, he did not suggest subordinating them to philosophy, in particular Thomism, but he claimed that the results of scientific research can be reconciled with Christian philosophy, and that moreover, the sciences can profit from philosophical considerations.24 He also recommended that the results of the natural sciences should be taken into account in teaching students of theology. It should be added that Pope Leo XIII appreciated the natural sciences – it is worth mentioning here that he always ensured that the Vatican astronomical observatory was provided with modern equipment as well as a professional level of specialists working there. All these facts gave a certain impetus to intellectuals connected with the Roman Catholic Church to undertake scientific research. The encyclical espe-

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“[…] if men be of sound mind and take their stand on true and solid principles, there will result a vast amount of benefits for the public and private good.” – translation according to: http://www.vatican.va/content/leo-xiii/en/encyclicals/documents/hf_l-xiii _enc_04081879_aeterni-patris.html (access: 29.05.2020) [si sana mens hominum fuerit, et solidis verisque principiis firmiter insistat, tum vero in publicum privatumque commodum plurima beneficia progignit], URL: http://www.vatican.va/content/leo-xiii/la/ encyclicals/documents/hf_l-xiii_enc_04081879_aeterni-patris.html (access: 29.05.2020). “Nor will the physical sciences themselves, which are now in such great repute, and by the renown of so many inventions draw such universal admiration to themselves, suffer detriment, but find very great assistance in the restoration of the ancient philosophy. For, the investigation of facts and the contemplation of nature is not alone sufficient for their profitable exercise and advance; but, when facts have been established, it is necessary to rise and apply ourselves to the study of the nature of corporeal things, to inquire into the laws which govern them and the principles whence their order and varied unity and mutual attraction in diversity arise. To such investigations it is wonderful what force and light and aid the Scholastic philosophy, if judiciously taught, would bring.” (29) http://www.vatican.va/content/leo-xiii/en/encyclicals/ documents/hf_l-xiii_enc_04081879_aeterni-patris.html (access: 29.05.2020) [Quapropter etiam physicae disciplinae quae nunc tanto sunt in pretio, et tot praeclare inventis, singularem ubique cient admirationem sui, ex restituta veterum philosophia non modo nihil detrimenti, sed plurimum praesidii sunt habiturae. Illarum enim fructuosae exercitationi et incremento non sola satis est consideration factorum, contemplatioque naturae; sed, cum facta constiterint, altius assumendum est, et danda solerter opera naturis rerum corporearum agnoscendis, investigandisque legibus, quibus parent, et principiis, unde ordo illarum et unitas in varietate, et mutua affinitas in diversitate proficiscuntur. Quibus investigationibus mirum quantam philosophia scholastica vim et lucem, et opem, est allatum, si sapienti ratione tradatur], URL: http://www.vatican.va/content/leo-xiii/en/ encyclicals/documents/hf_l-xiii_enc_04081879_aeterni-patris.html (access: 29.05.2020).

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cially aroused interest among the Germans, in particular, it encouraged many Catholic theologians to attempt to reconcile Cantor’s ideas on actual infinity with Catholic doctrine. One should first mention here Constantin Gutberlet (1837–1928), professor of philosophy, apologetics and dogmatic theology in the theological seminary in Fulda, the founder (in 1888) of the journal Philosophisches Jahrbuch der Görres-Gesellschaft, which was the leading journal connected with the neo-Thomism being developed after the promulgation of Leo XIII’s encyclical. In 1886, Gutberlet published a paper “Das Problem des Unendlichen” in which he referred to Cantor’s set theory in order to defend his own theological and philosophical views connected with the nature of the infinite. He claimed there that the research on the infinite entered a new phase after Cantor’s set theory had been developed. The main issue here was the problem of reconciling mathematical infinity with the absolute infinity of God’s existence. Gutberlet claimed that it is God who secures the ideal existence of Cantor’s transfinite numbers as well as of other infinite objects of mathematics (such as irrational numbers, the number π, etc.). God is also able to solve the continuum hypothesis as well as to determine the cardinality of the set of all real numbers. Since God’s mind is infinite and invariable, so Cantor’s transfinite numbers possess real existence. In fact, either one accepts the existence and reality of actual infinity or one should resign from the thesis on infinity and the eternality of God’s absolute mind. Such theses of Gutberlet encouraged Cantor’s interests in the philosophical and theological aspects of his concepts. Other adherents of Thomism also became interested in Cantor’s works and started to write commentaries on them. One should mention here the Jesuits Tilman Pesch (1836–1899) and Joseph Hontheim (1858–1929), the Dominican friar Thomas Esser (1850–1926) who was active in Rome, the Italian Franciscan theologian Ignatius Jeiler (1823–1904) and last but not least Cardinal Johann B. Franzelin (1816–1886), a leading Jesuit philosopher and one of the Pope’s principal theologians of the First Vatican Council (Concilium Vaticanum Primum). All of them were engaged in the task of renewing scholastic philosophy in the spirit of the encyclical Aeterni Patris. Pesch and Hontheim were connected with the Maria-Laach Abbey in Rhineland (Germany) where many works devoted to neo-Thomism were published under the joint title Philosophia Lacensis. Pesch attempted to formulate the foundations of Thomistic cosmology, Hontheim published the work on mathematics and logic Der logische Algorithmus (1895). Esser belonged to a group of Dominican friars engaged in the study of the theological consequences of Cantor’s set-theoretical works. In a letter dated 22 February 1896 to Jeiler, Cantor wrote (cf. Tapp, Kardinalität, 431):

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All that concerns this issue (and I am saying this in confidence) will be now verified by Dominican friars in Rome who correspond with me on this subject. All will be directed by Father Thomas Esser OP.25 What initiated Cantor’s contacts with Catholic theologians? Cantor’s correspondence provides the following picture. The first Jesuit with whom Cantor corresponded was probably the Belgian scientist and theologian Ignace Carbonnelle – their contacts began in November 1885. It is not clear what made this possible. Maybe Cantor read Carbonnelle’s papers or Tilman Pesch told him about Carbonnelle – it is probable that Cantor knew Pesch already at that time. Cantor became acquainted with Pesch in a Catholic parish in Halle (Germany) – unfortunately it is impossible to determine when they met for the first time. Pasch was recommended to Cantor by a parish priest, Franz Wilhelm Woker, with whom Cantor was a friend, and told of his attempts to find philosophical justifications for his scientific ideas and his need to discuss them with someone proficient in philosophical and theological matters. Then Pesch introduced Cantor to Joseph Hontheim who was at that time his assistant. The latter knew already Cantor’s works.

5

Correspondence with Cardinal Franzelin and Its Results

In 1885, Cantor wrote to Cardinal Johann B. Franzelin upon the advice of Constantin Gutberlet. Contacts with the latter began when Cantor read his book about the infinity Das Unendliche, metaphysisch und mathematisch betrachtet (1878). What called the attention of Catholic theologians to set theory was the problem of whether transfinite numbers actually exist in concreto. Gutberlet and Franzelin rejected the thesis on the concrete objective existence of the transfinite. Cantor on the other hand believed that infinity does exist in natura naturata. Franzelin claimed that this view “cannot be defended and in a certain sense […] can lead to the pantheism”26 (cf. Cantor, Gesammelte, 385). On the other hand pantheism has been declared to be an erroneous doctrine and was formally condemned by Pope Pius IX in the Syllabus of Errors (Syllabus Er25

26

“Es wird nämlich (im Vertrauen theile ich Ihnen dies mit) jetzt Alles auf diese Frage Bezügliche in Rom von den Dominikanern geprüft, die mit mir darüber in einer wissenschaftlichen Correspondenz stehen, die durch P. Thomas Esser O.Pr. geführt wird.” “[…] sich nicht verteidigen ließe und in einem gewissen Sinne […] den Irrtum des Pantheosmus enthalten würde […].”

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rorum) – an annex to the Quanta cura encyclical issued on 8 December 1864. Among the errors enumerated in Syllabus, one finds at first the following view: PANTHEISM, NATURALISM AND ABSOLUTE RATIONALISM 1. There exists no Supreme, all-wise, all-provident Divine Being, distinct from the universe, and God is identical with the nature of things, and is, therefore, subject to changes. In effect, God is produced in man and in the world, and all things are God and have the very substance of God, and God is one and the same thing with the world, and, therefore, spirit with matter, necessity with liberty, good with evil, justice with injustice.27 Pantheism was the view proclaimed for example by Baruch Spinoza – a Dutch philosopher of Portuguese descent raised in the Sephardi Jewish community of Amsterdam. He claimed that the entities of the “natural world” (natura naturata) are modifications of substance (natura naturans). It should be added that Spinoza was one of the philosophers whose works were carefully studied by Cantor. The problem of infinity was the point where the error of pantheism could be eventually discovered – in fact, attempts to connect the infinity of God with a certain concrete infinity existing in time could suggest pantheism.28 Every actual infinity in concreto, in natura naturata can be identified with the infinity of God, in natura naturans. And Cantor claimed that the infinite numbers exist in concreto. Thus the reaction and objection of Cardinal Franzelin who proclaimed himself rather in favour of “possible” infinity only. In answering Franzelin’s objections Cantor made a subtle distinction between the two types of the infinity. In a letter dated 22 January 1886 to Franzelin, Cantor explained that to the distinction between infinity in natura naturata and in natura naturans he added the distinction between Infinitum aeternum increatum sive Absolutum

27

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“Nullum supremum, sapientissimum, providentissimumque Numen divinum exsistit, ab hac rerum universitate distinctum, et Deus idem est ac rerum natura et idcirco immutationibus obnoxius, Deusque reapse fit in homine et mundo, atque omnia Deus sunt et ipsissimam Dei habent substantiam; ac una eademque res est Deus cum mundo et proinde spiritus cum materia, necessitas cum libertate, verum cum falso, bonum cum malo et iustum cum iniusto.” (compare: Denzinger-Schönmetzer, Enchiridion symbolorum, definitionum et declarationum de rebus fidei et morum, 2901; http://catho.org/9.php?d=byh#da1; access: 29.05.2020). English translation from: http://www.rosingsdigitalpublications .com/pius_ix_pope_quanta_cura_and_the_syllabus_of_errors.pdf (access: 29.05.2020). On Cantor’s reception of Spinoza’s concept of infinity see Bussotti and Tapp, “The influence”.

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reserved only for God and His attributes and Infinitum creatum sive Transfinitum whose explication is, for example, the actually infinite number of objects in the Universe. He wrote (cf. Tapp, Kardinalität, 322): In connection with this I distinguish between “Infinitum aeternum sive Absolutum” which refers only to God and His attributes and “Infinitum creatum sive Transfinitum” about which one can speak of in any place in which there is actual infinity in created nature, as for example in connection with – I am strongly convinced – the actually infinite number of created objects, both in the Universe as well as on our Earth or – most probably – in any arbitrary small part of space (here I am fully in accordance with Leibniz).29 These explanations set Cardinal Franzelin’s mind at rest – as follows from his response to Cantor’s letter mentioned above of 26 January 1886. In this letter he stressed that the absolute infinity is the proper one whereas actual infinity (transfinity) should be considered to be an improper infinity in a certain sense. He wrote (cf. Tapp, Kardinalität, 326): […] with satisfaction I state that you distinguish absolute infinity and that which you call actual infinity in the created world. As you claim that the latter is “still extendable” (of course indefinitum, i.e. it cannot become non-extendable) and you contrast it with absolute infinity which is “essentially non-extendable” what obviously must refer both to the possibility as well as to the impossibility of diminution or subtraction, hence both concepts of absolute infinity and actual infinity in the created world or transfinity are essentially different. Consequently when they both are compared, only one of them can be treated as proper infinity and the other as an improper one and ambiguously called infinity. Thus understood, as far as I can see up to now, there is no danger to religious truths in your concept of the transfinite. In one thing, however, you are certainly going astray against the undoubted truth; but this straying does not fol-

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“Dementsprechend unterscheide ich ein “Infinitum aeternum sive Absolutum”, das sich auf Gott und seine Attribute bezieht, und ein “Infinitum creatum sive Transfinitum”, das überall da ausgesagt wird, wo in der Natura creata ein Actualunendliches constatiert werden muss, wie beispielsweise in Beziehung auf die, meiner feste Überzeugung nach actual unendliche Zahl der geschaffenen Einzelwesen, sowohl im Weltall, wie auch schon auf unsrer Erde und, aller Wahrscheinlichkeit nach, selbst in jedem noch so kleinen ausgedehnten Theil des Raumes, worin ich mit Leibniz ganz übereinstimme.”

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low from your concept of the transfinite, but from a defective concept of the Absolute.30 This explains the source of Cantor’s – slightly odd – distinction between the two types of actual infinity. Cantor’s views concerning pantheism are also documented in another opinion expressed in a letter dated 22 January 1886 (mentioned above) to Cardinal Franzelin where he wrote (cf. Tapp, Kardinalität, 324–325): No system is further from my main beliefs than pantheism, if I disregard materialism, with which I have absolutely no fellowship. However, I believe that pantheism, and perhaps only through my understanding of things, can be completely overcome in time.31 Cantor was convinced that not only does the transfinite not diminish anything to the nature of God, but on the contrary, it adds glory to it. The real existence of the transfinite reflects the infinite nature of God. He also proposed two arguments in favour of the existence of the infinite (more exactly: of transfinite numbers) in concreto. The first – a priori argument – says that the possibility and necessity of the creation of the transfinite can be deduced directly from the perfection of God’s nature. The second argument – a posteriori – claims that since it is impossible to completely and fully explain the natural phenomena without assuming the existence of the transfinite in natura naturata, then the transfinite does exist. Cantor wrote in a letter of 22 January 1886 to Cardinal Franzelin (cf. Tapp, Kardinalität, 324): 30

31

“[…] ersehe ich mit Genugthuung, wie Sie das Absolut-Unendliche und das, was Sie das Akt. Unendliche im Geschaffenen nennen, sehr wohl unterscheiden. Da Sie das letztere für ein “noch Vermehrbares” (natürlich in indefinitum, d.h. ohne je ein nicht mehr Vermehrbares werden zu können) ausdrücklich erklären und dem Absoluten als “wesentlich Unvermehrbaren” entgegenstellen, was selbstverständlich ebenso von der Möglichleit und Unmöglichkeit der Verkleinerung und Abnahme gelten muß; so sind die beiden Begriffe des Absolut Unendlichen und des Aktual- Unendlichen im geschaffenen oder Transfinitum wesentlich verschieden, so daß man im Vergeiche beider nur das Eine als eigentlich Unendliches, das andere als uneigentlich und aequivoce Unendliches bezeichnen muß. So aufgefaßt liegt, soweit ich bis jetzt sehe, in Ihrem Begriffe des Transfinitum keine Gefahr für religiöse Wahrheiten. Jedoch in Einem gehen Sie ganz gewiß irre gegen die unzweifelhafte Wahrheit; dieser Irrthum folgt aber nicht aus Ihrem Begriffe des Transfinitum, sondern aus der mangelhaften Auffassung des Absoluten.” “[K]ein System ist weiter von meinen Haupüberzeugungen entfernt als der Pantheismus, wenn ich vom Materialismus absehe, mit dem ich durchaus keine Gemienschaft habe. Vom Pantheismus glaube ich jedoch, dass er, und vielleicht nur durch meine Auffassung der Dinge, mit der Zeit ganz überwunden werden kann.”

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The proof starts from the concept of God and first deduces the possibility of creating the ordered transfinite from the highest perfection of God, and next from His infinite goodness and magnificence the necessity of the actual creation of the transfinite. Another argument shows a posteriori that the assumption of the “Transfinitum in natura naturata” makes it possible to explain phenomena in a better, more perfect way – in particular the organisms and psychic phenomena – than the contrary hypothesis.32 Cardinal Franzelin accepted the first part of the proof but he criticized its second part. In his opinion, Cantor’s argument is correct concerning the possibility of creating ordered transfinity – under the assumption that the concept of actual transfinity does not contain a contradiction. However his argument is incorrect when he concludes the necessity of such creation. In fact it is an error when one deduces the necessity of creating something from the assumption of God’s perfection, from “the infinite goodness and magnificence” of God and from the possibility of only such an act of creation. Such necessity reduces in a certain sense the freedom of God and consequently diminishes His perfection. In a letter of 26 January 1886 he wrote (cf. Tapp, Kardinalität, 326): If one assumes that your [concept of] transfinity does not contain a contradiction then your conclusion on the possibility of creating the transfinite deduced from the concept of God’s omnipotence is fully correct. […] Since God is in Himself an actual infinite goodness and absolute magnificence which can be neither enlarged nor diminished so the necessity of creating something whatever it would be is a contradiction and the freedom of creating is a necessary perfection of God in the same degree as all of His perfections, or better [expressed] as God’s Infinite perfection (according to our necessary distinctions), both freedom as well as omnipotence, wisdom, justice, etc.33 32

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“Ein Beweis geht vom Gottesbegriff aus und schliesst zunächst aus der höchsten Vollkommenheit Gottes Wesens auf die Möglichkeit der Schöpfung eines Transfinitum ordinatum, sodann aus seiner Allgüte und Herrlichkeit auf die Notwendigkeit einer thatsächlich erfolgten Schöpfung des Transfinitum. Ein andrer Beweis zeigt a posteriori, dass die Annahme eines “Transfinitum in natura naturata” eine bessere, weil vollkommenere Erklärung der Phänomene, im Besondern der Organismen und psychischen Erscheinungen ermöglicht, als die entgegengesetzte Hypothese.” “In der Voraussetzung, daß Ihre Transfinitum actuale in sich keinen Widerspruch enthält, ist Ihre Schluß auf die Möglichkeit der Schöpfung eines Transfinitum aus dem Begriffe

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Franzelin adds that “deducing the necessity of creating something from the infinite goodness and magnificence of God, one must consequently state that everything created has been in fact created immemorially and in God’s eyes there is nothing possible that could still be created by God’s omnipotence”34 (cf. Tapp, Kardinalität, 327). Cantor was convinced that all sets exist as ideas in God’s mind. In a letter of 13 October 1895 to Jeiler he wrote in the spirit of St. Augustin (cf. Tapp, Kardinalität, 427): Transfinity can undergo various specifications, formations or individualizations. In particular, there exist transfinite cardinal numbers and transfinite ordinal types which have – similarly as finite numbers and forms – particular regularities that are discovered by human beings. All these particular types of transfinity exist eternally as ideas in God’s mind.35 However in the work “Mittteilungen zur Lehre von Transfiniten”, he wrote (cf. Cantor, “Mitteilungen”; Gesammelte, 405): Transfinity in the full variety of its shapes and forms necessarily points to the Absolute, the “real infinity”, which in no way can be enlarged or diminished and which therefore should be quantitatively treated as an absolute maximum.36

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von Gottes Allmacht ganz richtig. […] Gerade weil Gott an sich das absolute unendliche Gut und die absolute Herrlichkeit ist, welchem Gute und welcher Herrlichkeit nichts zu wachsen und nichts abgehen kann, ist die Nothwendigkeit einer Schöpfung welche immer diese sein mag, ein Widerspruch, und die Freiheit der Schöpfung eine ebenso nothwendige Vollkommenheit Gottes wie alle seine anderen Vollkommenheiten, oder besser, Gottes unendliche Vollkommenheit ist (nach unsern nothwendigen Unterscheidungen) ebenso Freiheit als, Allmacht, Weisheit, Gerechtigkeit etc.” “[W]er die Nothwendigkeit einer Schöpfung aus der Unendlichkeit der Güte und Herrlichkeit Gottes erschließt, der muß behaupten, daß Alles Erschaffenbare wirklich von Ewigkeit erschaffen ist; und daß es vor Gottes Auge kein Mögliches giebt, das Seine Allmacht ins Dasein rufen könnte.” “Das Transfinitum ist der mannigfaltigsten Formationen, Specificationen und Individuationen fähig. Im Besonderen giebt es transfinite Cardinalzahlen und transfinite Ordnungstypen, die eine ebenso bestimmte, vom Menschen erforschbare mathematische Gesetzmäßigkeit haben, wie die endlichen Zahlen und Formen. Alle diese besonderen Modi des Transfiniten existiren von Ewigkeit her als Ideen in intellectu divino.” “Das Transfinite mit seiner Fülle von Gestaltungen und Gestalten weist mit Notwendigkeit auf ein Absolutes hin, auf das “wahrhaft Unendliche”, an Essen Größe keinerlei Hinzufügung oder Abnahme sttathaben kann und welches daher quantitative als absolutes Maximum anzusehen ist.”

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Simultaneously he stressed that one must distinguish between the actual infinity in a mathematical sense – hence transfinity – and absolute infinity. In a letter of 3 February 1884 to Kurd Laßwitz he wrote (cf. Cantor, Briefe, 174): It is clear by itself that under [transfinity] one cannot understand here the Absolute, i.e., the absolutely greatest (sive Deus) which can be determined only by itself and not by us.37 In a letter of 20 June 1908 to Grace Chisholm-Young, he wrote (cf. Cantor, Briefe, 454): I never assumed the “Genus supremum” of actual infinity. Just the contrary, I am continuously claiming that the “Genus supremum” of actual infinity does not exist. What transcends all finite and transfinite is not “Genus”, it is the unique, fully individualized unity in which all is, which embodies all, the “Absolute” which cannot be comprehended by the human mind and hence is not in any way an object of mathematics, non-measurable, “ens simplicissimus”, “Actus perissimus”, by many called “God”.38

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Remarks on Cantor’s Religious Views

The engagement of Cantor in dialogue with Catholic theologians and the importance attached by him to their opinions and appreciation may be astonishing if one takes into account that he was in fact a Protestant and belonged to the Lutheran church. However the situation is more complicated.

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“Es versteht sich von selbst, daß hierunter das Absolut d.h. das absolute Größte (sive Deus) nicht zu verstehen ist, welches nur durch sich selbst, nicht aber von uns determinirt werden kann.” “Ich bin niemals von einem “Genus supremum” des actualen Unendlichen ausgegangen. Ganz im Gegentheil habe ich streng bewiesen, daß es ein “Genus supremum” des actualen Unendlichen gar nicht gibt. Was über allem Finiten und Transfiniten liegt, ist kein “Genus”; es ist die einzige, vŏllig individuelle Einheit, in der Alles ist, die Alles umfasst, das “Absolute”, für den menschlichen Verstand Unfassbare, also der Mathematik gar nicht unterworfene, Unmessbare, das “ens simplicissimum”, der “Actus purissimus”, der von Vielen “Gott” genannt wird.”

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Cantor’s mother, Maria Cantor (maiden name: Böhm) (1819–1896), was from a Catholic family. On his father’s side, Cantor had Jewish ancestors.39 His parents married in St. Petersburg in a Protestant church. Georg Cantor himself was baptized as a child in the Lutheran church. His funeral was officiated by a Protestant minister. Cantor was – at least formally – a Protestant throughout his life. In his 28 July 1887 letter to Carl Friedrich Heman – a Protestant clergyman and professor of philosophy – he wrote: “I am not a Catholic”40 (cf. Tapp, Kardinalität, 160). However, Cantor increasingly distanced himself from the Protestant church and its representatives. On the other hand, he maintained contacts with Protestant professors of theology at the university in Halle where he himself was a professor. His relations towards Protestantism and its clergymen became increasingly critical. This stood in contrast with his ever more positive relations towards Catholicism. Maybe this was the result of the fact that his mother was Roman Catholic, even more, that she undoubtedly had a great influence on his religious education in childhood. His aunt on his father’s side, Anastasia Grimm – a Jew who married a Greek-Catholic and belonged to the Greek-Catholic church, also had a big influence on him. In the letter to Heman mentioned above Cantor described his relations to Catholicism in the following way (cf. Tapp, Kardinalität, 164): I am not a Catholic, […] but my relations – as a positive Christian – to Catholicism whose head I revere and respect, is internally, and if necessary – also externally, friendly. I do not share the essential hostility towards everything belonging to Catholicism expressed by most Protestants. […] In particular I am of the opinion that one should keep a criticalfriendly attitude towards Catholic doctrine and philosophy. In this sense, you will acknowledge me […] not as an adherent but even as an antagonist (however animated by the best intentions) of the doctrine of infinity, almost fully accepted in certain circles, namely by the order of Jesuits.41 39

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This follows from a letter (recently discovered) from Cantor to the Jesuit Alexander Baumgartner in which he wrote (cf. Tapp, Kardinalität, 157): “[…] my beloved father of blessed memory was a native Copenhagen (belonging to the local Portuguese Jewish orthodox community […](“[…] mein geliebter Pater, hochseligen Andenkens, ein geborener Kopenhagener (der dortigen portug[iesisch] jüd[isch] orthodoxen Gemeinde angehörig) war […]”). However, it should be added that Cantor’s father changed his religious affiliation and together with his mother, joint the Protestant church. “Ich bin nicht Katholik.” “Ich bin nicht Katolik, […] stehe [aber] als positiver Christ inerlich und, wenn nöthig, auch nach Außen hin, freundschftlich zum Katholicismus, dessen jetziges Oberhaupt ich

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Jesuits formed the largest group of persons with whom Cantor exchanged letters concerning the theological problems relating in a certain sense to the issue of infinity – among 30 of Cantor’s correspondents 11 belonged to the Society of Jesus. This correspondence with Jesuits concerned not only problems concerning infinity, but also the history of literature or the history of the Jesuit order. Cantor viewed his correspondence with Cardinal Franzelin as especially significant. Generally, Cantor was very interested and anxious to receive positive opinions and recognition for his set-theoretical ideas from Catholic philosophers. The extent to which he valued their opinions is evidenced by the fact that their letters containing remarks about his concepts of infinity were included in his own works on set theory – an example is “Mitteilungen zur Lehre vom Transfiniten” (1887–1888). However, it should be noted that Cantor kept a certain explicit distance from institutionalized forms of religiousness. In a letter of 7 March 1896 to Constance Mary Pott, an English investigator of Bacon (note that Cantor himself also studied Bacon and his works), he wrote (cf. Tapp, Kardinalität, 181): I do not belong to any […] church in which an earth-born man stands above me or commands me. I am responsible only to God.42 In another letter of 13 March 1896 sent to the same person he wrote (cf. Tapp, Kardinalität, 181): In religious questions and issues, my point of view is not a confessional one, because I do not belong to any existing and organized church.43 It should be added that Cantor also studied theology privately. However, he did not feel himself to be in any way bound by the theses it officially proclaimed. In the course of time, one can observe that Cantor took a certain turn towards religion. This is seen, for example, in his letter of 22 January 1894 to

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verehre und hochachte und ich theile daher nicht mit der Mehrzahl der Protestanten die grundsätzliche Feindschaft gegen alles zum Catholicismus gehörige. […] Im Besonderen glaube ich, daß der katholischen Wissenschaft und Philosophie gegenüber ein kritisch-wohlwollendes Verhältniß gestattet sein dürfe. In diesem Sinne werden Sie mich […] nicht als Anhänger, sondern sogar als Gegner (aber als wohlmeinender Gegner) der in jenen Kreisen, namentlich von dem Orden S.J., fast allgemein acceptirten Lehre vom Unendlichen erkennen.” “Ich gehöre keiner […] Kirche an, in welcher ein Erdgeborener über mir stünde oder mir etwas zu befehlen hätte. Ich bin nur vor Gott verantwortlich.” “In religiösen Fragen und Beziehungen ist mein Standpunkt kein confessioneller, da ich keiner der bestehenden organisierten Kirchen angehöre.”

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his friend, the known French mathematician Charles Hermite. He writes (in French!) (cf. Tapp, Kardinalität, 165–166): Mathematics is no longer the sole and still a less essential love of my soul.44 And he adds (in German!) (cf. Tapp, Kardinalität, 165–166): Metaphysics and theology have, I will confess it openly, seized my soul to such an extent that I have relatively little time left for my first passion. If fifteen years ago, even eight years ago, I had been given – in accordance with my wishes – a broader field of activity, for example at the University of Berlin or Göttingen, I would have probably performed my duties no worse than Fuchs, Schwarz, Frobenius, Felix Klein, Heinrich Weber, etc. But now I thank God, the best and omniscient, for denying me those wishes, because in this way he forced me through deeper penetration of theology to serve Him and His holy Roman Catholic Church, to serve better than I would be able to do working exclusively in mathematics.45 It should be added that the remark about the universities in Berlin and Göttingen was an allusion to the fact that Cantor attempted for a long time to obtain a position of professor at those universities. He did not succeed and his whole life was spent as a professor at the provincial university in Halle. Cantor was convinced that his set theory is absolutely secure and correct – he claimed that it had been revealed to him, he saw himself as a messenger of

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“Les mathematiques ne sont plus le seul et encore moins elles l’essentiel amour de mon âme.” “Metaphysik und Theologie haben, ich will es offen bekennen, meine Seele in solchem Grade ergriffen, daß ich verhältnißmäßig wenig Zeit für meine erste Flamme übrig habe. Wäre es nach meinen Wüschen vor fünfzehn, ja sogar vor acht Jahren gegangen, so hätte man mir einen größeren mathematischen Wirkungskreis, etwa an der Universitât Berlin oder in Göttingen gegeben und ich würde vielleicht meine Sache dort nicht schlechter gemacht haben als die Fuchs, Schwarz, Frobenius, Felix Klein, Heinrich Weber etc. etc. Allein nun danke ich Gott, dem Allweisen und Allgütigen, daß er mir die Erfüllung dieser Wünsche für immer versagt hat, den so hat er mich gezwungen, durch ein tieferes Eindrigen in die Theologie Ihm und seiner heiligen rŏmisch-katholischen Kirche beßer zu dienen, als ich es, nach meinen wahrscheinlich schwachen mathematischen Talenten durch die ausschließliche Beschäftigung mit der Mathematik hätte thun können.”

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God. In a letter dated 20 May 1888 to the Franciscan friar I. Jeiler, he wrote (cf. Tapp, Kardinalität, 413): I have no doubts concerning the truth of transfinity, which I recognized thanks to God’s help and which I have studied in its variety, polymorphism and unity for more than twenty years; each year and almost every day lead me further in this discipline.46 Cantor tried to interpret his results in a religious way. G. Kowalewski,47 one of Cantor’s biographers, recalls that he perceived the increasing sequence of cardinal numbers as “something holy, it formed in a certain sense, the steps leading to the throne of the Infinite” (cf. Kowalewski, Bestand, 201). On another page, Kowalewski mentions Cantor’s statement on the antinomy of the set of all sets (cf. Kowalewski, Bestand, 120): In the speech of Solomon on the occasion of the consecration of the temple (1 Kings 8, 27; 2 Chronicles 6, 18) we read: “But will God indeed dwell on the earth? Behold the heaven and heaven of heavens cannot contain thee; how much less this house that I have built?” “Heaven of heavens” – does it not recall “the set of all sets”? What Solomon had said can be translated into the language of mathematics as follows: God, the highest infinity can be grasped (erfasst werden) neither by a set nor by the set of all sets. And Kowalewski adds that Cantor “liked such religious thoughts very much”. However, such remarks from Cantor should be considered with care. One should take into account that Cantor had a nervous breakdown in 1884 and later suffered from manic depression (bipolar disorder), which required psychiatric treatment. Statements of a religious character made by him appeared in the course of time with increasing frequency after 1884.

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“I[ch] hege keinerlei Zweifel a[n] d[er] Wahrh[eit] d[es] Transf[initen], das i[ch] m[it] Gottes Hilfe erkannt habe u[nd] nach seiner Mannigfaltigkeit, Vielgestaltigkeit und Einheit seit mehr als zwanzig Jahren studiere; jedes Jahr und fast jeder Tag bringt mich in dieser Wissenschaft weiter.” In the period when he worked in Leipzig (Germany), Gerhard Kowalewski (1876–1950) met with Cantor regularly during seminars that took place every two weeks. These seminars were organized jointly by mathematicians from Leipzig and Halle.

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References Achtner, Wolfgang, “Infinity in science and religion. The creative role of thinking about infinity”, Neue Zeitschrift für Systematische Theologie und Religionsphilosophie 47 (4) (2005), 392–411. Becker, Oskar. Grundlagen der Mathematik in geschichtliche Entwicklung. FreiburgMünchen: Verlag Karl Alber, 1954. Bussetti, Paolo and Christian Tapp, “The Influence of Spinoza’s Concept of Infinity on Cantor’s Set Theory”, Studies in History and Philosophy of Science Part A 40 (1) (2009): 25–35. Cantor, Georg. “Grundlagen einer allgemeinen Mannigfaltigkeitslehre,” Mathematische Annalen, 21 (1883): 545–586. Also as: Grundlagen einer allgemeinen Mannigfaltigkeitslehre. Ein mathematisch-philosophischer Versuch in der Lehre des Unendlichen. Leipzig: Teubner, 1883. Reprinted in Cantor, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts. Hrsg. E. Zermelo, Berlin: Verlag Julius Springer, 1932. (Reprinted: Berlin–Heidelberg–New York: Springer Verlag, 1980), 165–209. Cantor, Georg. Principien einer Theorie der Ordnungstypen. Erste Mitteilung. 1884. First published by I. Grattan-Guinness as: “An unpublished paper by Georg Cantor: Principien …”, Acta Mathematica, 124 (1970): 65–107 (Cantor’s paper: 83–101). Cantor, Georg. “Mitteilungen zur Lehre vom Transfiniten”. Zeitschrift für Philosophie und philoophische Kritik 91 (1887): 81–125 and 92 (1888): 240–265. Reprinted in Cantor, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts. Hrsg. E. Zermelo, Berlin: Verlag Julius Springer, 1932. (Reprinted: Berlin–Heidelberg–New York: Springer Verlag, 1980), 378–439. Cantor, Georg. Gesammelte Abhandlungen mathematischen und philosophischen Inhalts. Hrsg. E. Zermelo, Berlin: Verlag Julius Springer, 1932. (Reprinted: Berlin– Heidelberg–New York: Springer Verlag, 1980). Cantor, Georg. Briefe. Hrsg. H. Meschkowski, W. Nilson, Berlin: Springer Verlag, 1991. Dauben, Joseph W. Georg Cantor, His Mathematics and Philosophy of the Infinite. Cambridge, Mass.–London: Harvard Univ. Press, 1979. Dedekind, Richard. Gesammelte mathematische Werke. Braunschweig: Friedrich Vieweg und Sohn, 1932. Drozdek, Adam. “Number and Infinity: Thomas and Cantor”, International Philosophical Quarterly 39 (1) (1999), 35–46. Fraenkel, Abraham. Das Leben Georg Cantors, in Cantor, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts. Hrsg. E. Zermelo, Berlin: Verlag Julius Springer, 1932. (Reprinted: Berlin–Heidelberg–New York: Springer Verlag, 1980), 452–483.

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Gauss, Carl Friedrich. Carl Friedrich Gauss – H.C. Schumacher Briefwechsel. Hildesheim: Georg Olms, 1975. Gutberlet, Constantin. Das Unendliche, metaphysisch und mathematisch betrachtet. Mainz: G. Faber, 1878. Gutberlet, Constantin. “Das Problem des Unendlichen.” Zeitschrift für Philosophie und philosophische Kritik 88 (1886): 179–223. Hontheim, Joseph. Der logische Algoritmus in seinem Wesen, in seiner Anwendung und in seiner philosophischen Bedeutung. Berlin: F. Dames, 1895. Kant, Immanuel. Prolegomena zu einer jeden künftigen Metaphysik, die als Wissenscahft wird auftreten können. Riga: Hartknoch, 1783. English translation: Prolegomena to Any Future Metaphysics That Will Be Able to Come Forward as Science with Selections from the Critique of Pure Reason. Translated and edited by Gary Hatfield, Revised Edition, Cambridge: Cambridge University Press, 1997. Kowalewski, Gerhard. Bestand und Wandel: Meine Lebenserinnerungen, zugleich ein Beitrag zur neueren Geschichte der Mathematik. München: Oldenbourg, 1950. Mason, J. “Expressing Generality and Roots of Algebra.” In Approaches to Algebra. Perspectives for Research and Teaching, edited by N. Bednarz, C. Kieran and L. Lee, 65–86. Dordrecht–Boston–London: Kluwer Academic Publishers, 1996. Meschkowski, Herbert. “Aus dem Briefbüchern Georg Cantors.” Archive for History of Exact Sciences 21 (1962–1966): 503–519. Meschkowski, Herbert. Probleme des Unendlichen. Werk und Leben Georg Cantors. Braunschweig: Friedrich Vieweg & Sohn, 1967. Meschkowski, Herbert. Biographical Dictionary of Mathematicians, vol. 1, Edited by C. Gillespie, New York: Scribner’s Sons, 1991. Meschkowski, Herbert and Nilson, Winfried (eds). Georg Cantor: Briefe. Berlin: Springer Verlag, 1991. Newstead, Anne. “Cantor on Infinity in Nature, Number, and the Divine Mind”, American Catholic Philosophical Quarterly 83 (4) (2009), 533–553. Noether, Emmy and Cavaillès, J., (eds.). Briefwechsel Cantor–Dedekind. Paris: Hermann, 1937. Tapp, Christian. 2005. Kardinalität und Kardinäle. Wissenschaftliche Aufarbeitung der Korrespondenz Georg Cantor und katholischen Theologen seiner Zeit. Wiesbaden: Franz Steiner Verlag. Thiele, Rüdiger. “Georg Cantor (1845–1918).” In Mathematics and the Divine. A Historical Study, edited by Teun Koetsier and Luc Bergmans, 523–547. Amsterdam: Elsevier, 2005.

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Chapter 8

How Is the World Mathematical? Michael Heller

Abstract How can a formal language, in particular the language of mathematics, be effective in the physical world? This long disputed question, when confronted with logical semantics, places us at the heart of the interplay between syntax and semantics. In principle, all mathematical structures can be translated into the language of category theory, and in this form they can be analysed as far as their syntactic and semantic aspects are concerned. Let T be a formalized mathematical theory. It is known from categorical semantics that such a theory T uniquely determines a certain category C, of which one can say that it provides the semantics for T (T is about C). Moreover, it turns out that every sufficiently rich category C ′ determines a theory T ′ (T ′ is a syntax of C ′ ). In categorical semantics, this interaction between syntax and semantics is modelled by the pair of adjoint functors, called Lan and Syn, acting between a category of theories and a category of categories of some class (strictly defined). The adjoint action of these functors shows how syntax creates its semantics and how semantics shapes its syntax. It is proposed, in analogy with the above, to interpret “the unreasonable effectiveness of mathematics in the natural world” in terms of an adjoint action between mathematical structures and their domain of reference. Of course, by doing so, we go beyond the purely formal analysis and ascribe to it some ontological significance.

Keywords effectiveness of mathematics – syntax and semantics – categorical semantics – functors Lan and Syn

1

Introduction

It is common knowledge that mathematics is a formal science. It uses, or rather it is, a language which can be fully formalized and which refers to nothing that lies outside of it. In spite of this, the most outstanding and peculiar feature of mathematics is that if it is put in contact with empirical data (which

© Michael Heller 2021 | DOI:10.1163/9789004445956_010

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are expressed in numbers), it interacts with them and is able not only to place them inside its own structures, but also, more often than not, is able to suggest new empirical tests and predict their numerical outputs. Here are some of the more outstanding examples. In 1915, Einstein set down his gravitational field equations and deduced from them three rather small effects in which his new gravity theory differed from that of Newton. When one of these effects (the bending of starry light by the gravitational field of the Sun, visible during the Sun eclipse) had been detected, Einstein became a world-famous star overnight. Today, a century later, we are accustomed to the fact that Einstein’s equations contain a tremendous amount of information about the structure of the world, something that was never even suspected by Einstein himself and his contemporaries. The first example is the expansion of the universe. When his equations suggested that the universe is unstable, Einstein preferred to modify the equations rather than to accept the large-scale evolution of the world. Several years needed to pass before Einstein acknowledged his error and for modern cosmology to be born as a result. More or less the same story repeated itself on a number of occasions, with even more solutions to Einstein’s equations being found. Some of them represented stationary and rotating black holes, while others gravitational waves. Einstein stubbornly claimed that they were unphysical; nowadays, all these phenomena (and many others) are standard textbook material. The recent detection of gravitational waves from the collision of two black holes was unanimously proclaimed the greatest triumph of general relativity. And by no means is the story finished; the richness of Einstein’s equations is still full of promise. There is a saying, attributed to Hertz,1 that “the equations are wiser than those who invented them”. This seems to be true not only as far as Einstein’s equations are concerned, but also with respect to the equations of Schrödinger, Dirac and many others. In this context, not to quote Wigner’s (Wigner 1960) famous saying about “the unreasonable effectiveness of mathematics in the natural sciences”2 would be at least a gaffe. A question thus poses itself: How can purely formal science achieve all of this? The question is often reformulated in the following manner: Why is the world mathematical? Quite obviously, the question is addressed to three do1 The original quotation from Hertz reads as follows: “It is impossible to study this wonderful theory without feeling as if the mathematical equations had an independent life and intelligence of their own, as if they were wiser than ourselves, indeed wiser than their discoverer, as if they gave forth more than he put into them” (Hertz 1896, 313). 2 This is, in fact, the title of Wigner’s article (Wigner 1960).

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mains: to mathematics, to the world and, last but not least, to the human mind, which “intermediates” between the two. The “intermediates” is in quotation marks, because it is a strange kind of intermediation. The human mind seems to create mathematics, or at least its formal language, but it exercises no power in creating the world. Nevertheless, some mathematical structures, if cleverly chosen and correctly interpreted, are able to disclose the inner layers of the world’s structure, even if they are deeply hidden from our senses. Of course, when we hunt for suitable equations that would reproduce a new aspect of the world, we engage in the process all our knowledge we have collected so far which might be relevant in a given field, and then we try to put it into mathematical structures. If we succeed (as Einstein did, tout proportion gardée), mathematics rewards us with new information concerning the world we had not put into our original mathematical structures. It seems as if mathematics (or at least some of its structures) are a blueprint for the world, and if we pick up the right structure, we can discern information about the world from it which is completely new for us. In this paper, I focus on the following question: How can a formal language (such as that of mathematicians) be effective and, in particular, how can it be effective in the physical world? When formulated in this way, the question demands the precision which should be sought in the logic of language. This branch of logic is usually divided into three chapters: syntax, semantics and pragmatics. Syntax investigates relations between expressions of a given language. Semantics investigates relations between a given language and what the language refers to. Pragmatics investigates relations between a given language and its users. The topic of this paper places us at the heart of the interplay between syntax and semantics. Some aspects of the problem related to pragmatics will be touched upon, albeit only marginally. Rigorous language control is only possible for purely formal languages. Such a language is reduced to a formal game of symbols and is devoid of any reference “to something external”. However, as Tarski taught us (Tarski 1933), it can be equipped with meaning (i.e. it can be given semantics) by carefully manipulating various levels of a given language (language and meta-language). Since everything is done in a purely formal manner, we retain full control over the process. However, outside formal languages we are sentenced to a “learned intuition” and analogies with strict theory. Mathematics, as it is usually presented in mathematical textbooks and academic lectures, is seldom displayed in a purely formalized manner, but mathematical theories can in principle be given the shape of formal systems. Moreover, practically all mathematical structures can be translated into the language of category theory, and in this form they can be analysed precisely

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as far as their syntactic and semantic aspects are concerned. I will follow this path when considering the question of what mathematics is talking about. The idea is as follows. Let us consider a mathematical theory T and let us suppose that it is presented in a fully formalized language L, i.e. L has a well defined syntax. If this is done in the right way, then there is a set of precise rules (all of them given in a constructive way) that translate the elements (expressions) of language L into elements of a certain category. As is well known, a category consists of objects, morphisms between objects, identity objects, satisfying certain axioms (see Appendix A). Elements of language L correspond to all these elements in such a way that dependencies between the expressions of L reflect axioms of a certain category C. In other words, every formalized theory T determines a certain category C. We can say that T is about C or that C provides the semantics for T. Everything is purely formal; nowhere do we go beyond strict formalism. But this is only half of the story. If we suitably reverse the translation rules from theory T to category C, then it turns out that every sufficiently rich category C ′ determines a theory T ′ .3 Interestingly, the processes of “translating” and “going back” are not, in general, the reverse of one another; i.e., if we first go from T to C, and then from C to some T ′ (sticking to the previous but reversed translation rules) then, in general, T ′ does not coincide with T. However, there is a strict relationship between them (T ′ is Morita equivalent to T, see section 3 below). This shows that the interaction between syntax and semantics is not trivial; both of them influence each other. This sort of interaction is mathematically modelled with the help of adjoint functors. All this is well known in categorical semantics (Lambek, Scott 1986; Mac Lane, Moerdijk 1992) and will be presented in more detail in sections 2 and 3. If we ascribe some kind of independent existence to mathematical structures, and take into account the (unreasonable) effectiveness of mathematics in the natural world, we might speculate (in analogy with what was said above on the categorical syntax and semantics) that mathematical theories effectively create their own domain of reference, i.e., their own semantics, and that the interaction between a given theory and its domain of reference is of the “adjoint type”. “Effectively”, in this context, means that the interaction between syntax and semantics is not only purely formal, but also has some ontological significance. This speculative hypothesis is developed in section 4.4

3 Sufficiently rich, since if a category is too poor (e.g., consisting of one object and one identity morphism), then the corresponding theory is trivial. 4 An outline of this hypothesis was first presented in (Heller 2018).

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“Some kind of independent existence” ascribed to mathematical structures (or theories) suggests a Platonic-like standpoint in the philosophy of mathematics, but possibly it could also be reconciled with other views on the status of mathematical entities. There is no necessity to claim that independently existing mathematics fully determines the structure of the physical world (the physical world is a shadow of mathematical structures), but rather that the dependencies of mathematical structures on the physical world, and vice versa, are “adjointly reciprocal”. It also goes without saying that it is not our formal theories that have “some kind of independent existence”, but rather some “abstract reality” or “abstract forms” of which our formal theories are representations. We thus have, on the one hand, a formal mathematical theory and, on the other hand, a domain of the physical world providing a semantics for this theory. An ingenious correspondence exists between the mathematical theory and its domain (its semantics). Both of them truly co-respond to each other, and this cor-respondence is dynamic. The dynamical interactions go both ways: the domain of the world in-forms the theory about its own internal structure, i.e., it forms this theory from within; and the theory answers by prescribing how the domain should behave. As required by adjointness, this interaction is quasi-reciprocal. We thus should also say that it is a theory which internally informs the structure of its domain in the physical world, and the behaviour of the latter prescribes some aspects of the internal architecture of the theory. Moreover, this interaction is effective: it is not only prescribed by formal rules, but also produces effects in the real world. A similar dynamical nexus between syntactic-like and semantic-like realms can be identified in two other phenomena, both of which are crucial for cosmic evolution, namely in the origin of life and in the origin of consciousness. In both these phenomena, the interaction between what we can regard as syntax and what we can relate to semantics can readily be recognized. In the phenomenon of life, the transition from information encoded in DNA to its implementation in biochemical machinery is decisive. When dealing with this transition, one has to take into account the fact that DNA itself is a “biochemical machinery”. The interaction is clearly intricately looped. The problem of consciousness is even more complex. The question that has to be asked is: How to make a transition from the electric signals propagating in the brain along the nerves from neuron to neuron across synaptic clefts to something that is so concrete and elusive as human consciousness? In section 5, it is argued that the idea of “adjoint interactions” (see Appendix C) could prove fruitful when dealing with these problems.

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As we have mentioned, one might discern some note of mathematical Platonism in the approach adopted in this essay. However, it should be noted that it is not assumed as an a priori condition of the hypothesis proposed here, but rather that the effectiveness of mathematics in the real world is taken as a fact (which is corroborated by the entire history of science), and our speculative hypothesis should only be understood as an attempt to justify it.

2

Categorical Semantics

In this section, in a somewhat simplified manner, we present those elements of categorical semantics which were only outlined in the introduction and are indispensable for the development of our hypothesis. First, we must determine the language. Our model language is the language of mathematics based on standard first order logic. To make the language more suitable for mathematical purposes, we shall assume that it is a many typed language. This means that to each term, variable, etc. (see below) there is assigned a type. For instance, in linear algebra, some quantities are of the scalar type, whereas some others are of the vector type [for details, see (Mac Lane, Moerdijk, 1992)]. In such a language, many mathematical theories can be presented. The following elements can be distinguished in it: – constants, denoted by: 0, 1, 2, …, a, b, c, … and variables: x, y, z, …, – they can be combined to obtain terms, such as: x + y, x2 , …; to each term there corresponds a function symbol; for instance to the term x + y, where the variables x and y are of the types X , Y , respectively, there corresponds the function symbol f : X × Y → Z, where f (x, y) = x + y is of type Z, – terms can be combined, with the help of primitive relations such as =,