Philosophia Scientiæ Travaux d'histoire et de philosophie des sciences

18-1 | 2014 Standards of Rigor in Mathematical Practice

Gerhard Heinzmann and Jean-Jacques Szczeciniarz (dir.)

Electronic version URL: http://journals.openedition.org/philosophiascientiae/903 DOI: 10.4000/philosophiascientiae.903 ISSN: 1775-4283

Publisher Éditions Kimé

Printed version Date of publication: 15 March 2014 ISBN: 978-2-84174-665-1 ISSN: 1281-2463

Electronic reference Gerhard Heinzmann and Jean-Jacques Szczeciniarz (dir.), Philosophia Scientiæ, 18-1 | 2014, « Standards of Rigor in Mathematical Practice » [Online], Online since 15 March 2014, connection on 16 November 2020. URL : http://journals.openedition.org/philosophiascientiae/903 ; DOI : https:// doi.org/10.4000/philosophiascientiae.903

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TABLE OF CONTENTS

Preface Gerhard Heinzmann and Jean-Jacques Szczeciniarz

Mathematical Rigor, Proof Gap and the Validity of Mathematical Inference Yacin Hamami

La rigueur mathématique chez Henri Poincaré Ramzi Kebaïli

Foundation of Mathematics between Theory and Practice Giorgio Venturi

Facets and Levels of Mathematical Abstraction Hourya Benis-Sinaceur

The Transition from Formula-Centered to Concept-Centered Analysis Bolzano's Purely Analytic Proof as a Case Study Iris Loeb and Stefan Roski

D’un point de vue rigoureux et parfaitement général : pratique des mathématiques rigoureuses chez Richard Dedekind Emmylou Haffner

The Right Order of Concepts: Graßmann, Peano, Gödel and the Inheritance of Leibniz's Universal Characteristic Paola Cantù

Varia

Une nouvelle sémantique de l’itération modale Brice Halimi

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Preface

Gerhard Heinzmann and Jean-Jacques Szczeciniarz

1 From 2007 to 2011, the editors, together with Marco Panza, invited Michael Detlefsen to direct the project Ideals of Proof (Chaire d'excellence senior ANR-07-CEXC-002-01). This project focused on certain ideals, which guided mathematical reasoning throughout its history. A substantial body of publications and lectures has grown out of this project. They can be consulted under http://halshs.archives-ouvertes.fr/IP/fr/.

2 As a continuation of this project, this special issue of Philosophia Scientice focuses on standards of mathematical rigor concerning the varieties of methodological, ontological, proof-theoretical questions, tackled in the framework of the philosophy of mathematical practice. Nevertheless, this volume does not only contain contributions of participants of the project, but is the result of a separate call for papers.

3 In the last decades, many studies (an important source is [Kitcher 1984]) have aimed to overcome classical positions in mathematical ontology and epis-temology as Platonism, nominalism, formalism, but also strong anti-realism (Dummett). The failure of these efforts suggests looking at mathematical practice as a source for finding a solution to the problem these positions were willing to answer. Today, this is a crucial task for philosophy of mathematics, and for example witnessed by: C. Misak's work New Pragmatists [Misak 2007], P. Mancosu's book The Philosophy of Mathematical Practice [Mancosu 2008]; B. van Kerkhove et al. Philosophical Perspectives on Mathematical Practice [van Kerkhove & van Bendegem 2010]; volume 16 (1) of Philosophia Scientice: From Practice to Results in Logic and Mathematics [Giardino, Moktefi et al. 2012]; the constitution of the Association for the Philosophy of Mathematical Practice (2009).

4 The practical turn in philosophy of mathematics is by no means an uniform approach of anti-foundationalism but reflects different developments. There are pragmatic relativists who consider that "the only test for scientific concepts is whether they can be organized in a logically simple system that finds fruitful empirical applications" on arguing that "1° In doing science, we do not, in fact, have a leg to stand on. 2° We do not, in principle, need a leg to stand on" [Fine 2007, 59].

5 Some proponents of the practical turn pursue the heritage of Lakatos and Kuhn. Their concerns can be classified as the shift from internal to external considerations: the

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focus is then mathematical practice as a group or community phenomenon and as educational matters [Giardino, Moktefi et al. 2012, 6-7].

6 Finally, others shift the focus of ontological questions on topics previously considered only in passing, as evidence, visualization, understanding and explanation. Our topic in this volume, Standards of Rigor in Mathematical Practice, is a further issue that can be placed in this series.

7 Now, since the work of Poincaré it is a common place that formalized proofs do not provide mathematical understanding, that the notions of group and topology connect apparently unrelated concepts in other areas, that ontological questions are considered secondary to structural ones and that certain structures may be abandoned because of their unfruitfulness. Accordingly, it seems natural that Poincaré's work plays a central role in this volume, all the more in a journal edited by the Poincaré Archives.

8 Yacin Eamami discusses in his paper two possible views of the validity of mathematical inference with respect to their capacity to yield a plausible account of the intuitive notion(s) of proof gap present in mathematical practice. According to the first view, a mathematical inference is valid if and only if its conclusion can be formally derived from its premises. According to he second view an inference is valid if and only if it consists in an operation that provides a ground for its conclusion given (previously obtained) grounds for its premises (Prawitz). He concludes that the ground-based account appears of particular interest for the philosophy of mathematical practice, and he finally raises several challenges facing a full development of a ground-based account of the notions of mathematical rigor, proof gap and the validity of mathematical inference.

9 Ramzi Kebaili shows that Poincaré had implicitly in mind a personal conception of mathematical rigor that would fit with his mathematical practice. He observes Poincaré's standard of rigor in his topological work, he studies on what grounds he is opposed to some specific standards of rigor and he develops a "Poincaréan" conception of mathematical rigor.

10 Georgia Venturi proposes to look at set theory not only as a foundation of mathematics in a traditional sense but also as a foundation for mathematical practice. He distinguishes between a standard set theoretical foundation and a practical one that aims to find a set theoretical surrogate to every mathematical object. After having given some examples he argues that this distinction is relevant for the philosophy of mathematics and he proposes two different kinds of foundations: a practical one and a theoretical one.

11 Mathematical abstraction is the process of considering operations, rules, methods and concepts divested from their reference to real world phenomena, and also deprived from the content connected to particular applications. In her contribution, Hourya Bents Sinaceur investigates the mathematical practice with the aim to bring to light the fundamental thinking processes at play, and to illustrate by significant examples how much intricate and multileveled may be the combination of typical mathematical techniques.

12 Iris Loeb and Stefan Roski argue convincingly that Bolzano supports his concept-centered methodology in the development of mathematical analysis by philosophical views, which were partially shared by working mathematicians with a formula-centered approach to analysis.

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13 Emylou Haffner studies Richard Dedekind's practice of rigor in a selection of his most important works. Rigor is for him closely related to generality. The links between generality and rigor are analyzed in his theory of algebraic function, as well in his foundational essays and in his work on algebraic number theory. She analyzes the requirements for generality and the multiple conceptions of generality sustaining this demand.

14 Paola Gantu tackles the question of whether the order of concepts was still a relevant aspect of rigor in 19TH and 20TH century. Three case studies are taken into account: Graßmann, Peano, Gödel. The paper aims to question whether there is in fact such a stark contrast between the debate relating to the right order of concepts and the foundational question concerning modern axiomatics. The unity of the three case studies is to be found in their different but real relation to Leibniz's ideal of a universal characteristic.

15 We would like to thank all the contributors for their work. We are also very grateful to Sandrine Avril, assistant, for her great patience and expertise in producing the manuscript.

BIBLIOGRAPHY

FINE, Arthur [2007], Relativism, pragmatism and the practice of science, in: New Pragmatists, edited by C. Misak, Oxford: Oxford University Press, 50-67.

GIARDINO, Valeria, MOKTEFI, Amirouche, MOLS, Sandra, & VAN BENDEGEM, Jean Paul (eds.) [2012], From Practice to Results in Logic and Mathematics, Philosophia Scientice, vol. 16(1), Paris: Kime.

KITCHER, Philip [1984], The Nature of Mathematical Knowledge, New York: Oxford University Press.

MANCOSU, Paolo [2008], The Philosophy of Mathematical Practice, Oxford: Oxford University Press.

MISAK, Cheryl (ed.) [2007], New Pragmatists, Oxford: Oxford University Press.

VAN KERKHOVE, Bart & VAN BENDEGEM, Jean Paul (eds.) [2010], Perspectives on Mathematical Practices: Bringing Together Philosophy of Mathematics, Sociology of Mathematics, and Mathematics Education, Logic, Epistemology and the Unity of Science, Dordrecht: Springer.

AUTHORS

GERHARD HEINZMANN Laboratoire d'Histoire des Sciences et de Philosophie -Archives H,-Poincaré - Université de Lorraine – CNRS (UMR 7117) (France)

JEAN-JACQUES SZCZECINIARZ Université Paris Diderot-Paris 7 - UMR SPHERE (France)

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Mathematical Rigor, Proof Gap and the Validity of Mathematical Inference

Yacin Hamami

The author would like to thank Jean Paul van Bendegem and John Mumma for helpful comments and stimulating discussions relative to this paper. The author is a doctoral fellow of the Research Foundation Flanders (FWO).

1 Introduction

1 Mathematical rigor is commonly formulated by mathematicians and philosophers using the notion of proof gap1: a mathematical proof is rigorous when there is no gap in the mathematical reasoning of the proof.2 Any philosophical approach to mathematical rigor along this line requires then an account of what a proof gap is. However, the notion of proof gap makes sense only relatively to a given conception of valid mathematical reasoning. A natural way to state the connection is the following: a proof gap occurs in a mathematical proof whenever there is a failure in valid mathematical reasoning. If one considers valid mathematical reasoning to consist in chains of valid mathematical inferences, the connection becomes: a proof gap occurs in a mathematical proof whenever there is a failure in drawing a valid mathematical inference. In order to characterize the notion of proof gap in this way, one shall then (i) provide an account of what constitutes a valid mathematical inference and (ii) spell out how one can fail to draw a valid mathematical inference in the specified sense.

2 The aim of this paper is to explore two possible views of the validity of mathematical inference with respect to their capacity to yield a plausible account of the intuitive notion(s) of proof gap from mathematical practice. The first view is the one provided by the contemporary standards of mathematical rigor based on the notion of formal derivation: a mathematical inference is valid if and only if its conclusion can be formally

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derived from its premises. The second view is based on a new account of the validity of inference developed by Prawitz [Prawitz 2009, 2012a, b]: a mathematical inference is valid if and only if it consists in an operation that provides a ground for its conclusion given (previously obtained) grounds for its premises. To evaluate these two views, we will proceed as follows: (i) we will state in precise terms the two conceptions of the validity of mathematical inference they propose, (ii) we will spell out the notions of proof gap they give rise to by investigating how one can fail to draw a valid mathematical inference in the specified sense, (iii) we will evaluate the capacity of the resulting accounts to capture the intuitive notion(s) of proof gap from mathematical practice.

3 We will place our investigation within the framework of the philosophy of mathematical practice ([Mancosu 2008], [van Kerkhove & van Bendegem 2010]). This means that we will not only consider mathematical proofs and mathematical reasoning from contemporary mathematical practice, but also from different practices at various times in the history of mathematics. Consequently, by a mathematical inference, we will mean an inferential or deductive step of a mathematical proof in mathematical practice (past or present). Such proof steps are often signaled in mathematical proofs by expression such as "hence", "therefore", "it follows that", etc. Then, by a valid mathematical inference, we will mean a mathematical inference that is correct or sound by the standards of mathematical proof in mathematical practice. Finally, by the intuitive notion(s) of proof gap, we will refer to the notion(s) of proof gap that is (are) present in mathematical practice. Providing a philosophical account of these three notions appears as a central task for the philosophy of mathematical practice.

4 The paper is organized as follows. In section 2, we formulate three derivation-based accounts of the validity of mathematical inference based on three possible interpretations of the relation between the notions of rigorous mathematical proof and formal derivation. We then spell out the resulting derivation-based accounts of proof gap, and we provide arguments for their rejections as adequately capturing the intuitive notion(s) of proof gap from mathematical practice. In section 3, we present Prawitz's recent account of the validity of inference based on the notions of operation and ground. In section 4, we adapt Prawitz's account to mathematical inference, resulting in a ground-based account of the validity of mathematical inference, by specifying the notions of operation and ground in the context of mathematical proofs in mathematical practice, where the notion of mathematical practice will be provided by Kitcher's framework [Kitcher 1981, 1984]. Sections 5 and 6 are concerned with evaluating the capacity of the resulting ground-based account of proof gap to capture the intuitive notion(s) of proof gap from mathematical practice. Section 5 focuses on intra-praxis gaps—proof gaps that occur within a given mathematical practice—and shows that the ground-based account can capture the three kinds of proof gaps constituting the taxonomy proposed in [Fallis 2003] as particular types of failure that will be identified. Section 6 focuses on inter-praxis gaps—proof gaps that occur from a cross-perspective on different mathematical practices—and shows how they can as well be accommodated within the ground-based account. Section 7 concludes this paper by wrapping up the main results of our evaluation of the two views of the validity of mathematical inference through the accounts of proof gap they give rise to, and by raising several challenges facing a full development of a ground-based account of the notions of mathematical rigor, proof gap and the validity of mathematical inference.

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2 Derivation-based accounts of the validity of mathematical inference

5 According to [Detlefsen 2009], the contemporary prevailing view of what a rigorous mathematical proof is can be stated as follows: Rigorous proof [...] is reasoning all of whose inferences track purely logical relations between concepts. In the late nineteenth and early twentieth centuries, syntactical criteria for such relations were developed and these have become the basis for the currently prevailing view of formalization. [Detlefsen 2009, 17]

6 What Detlefsen refers to in this quote by 'syntactical criteria' corresponds to the modern notion of formal derivation from proof theory. The prevailing view of mathematical rigor leaves then room for different interpretations of the relation between the notions of rigorous mathematical proof and formal derivation. In this section, we will consider three possible interpretations: (i) a mathematical proof is rigorous if it is a formal derivation, (ii) a mathematical proof is rigorous if it can be turned into a formal derivation, (iii) a mathematical proof is rigorous if it can routinely be turned into a formal derivation. These three interpretations lead to three different accounts of the validity of mathematical inference, and thereby to three different notions of proof gap. We will now evaluate them in turn with respect to their capacity to capture the intuitive notion(s) of proof gap from mathematical practice.

7 The first interpretation yields the following account of the validity of mathematical inference: a mathematical inference is valid if and only if its conclusion has been formally derived from its premises. Prom this definition, it is straightforward to state what it means to fail in drawing a mathematical inference: it simply means to fail in providing a formal derivation of the conclusion of the inference from its premises. This yields the following notion of proof gap: a proof yap occurs in a mathematical proof whenever there is a mathematical inference in the proof for which no explicit formal derivation of the conclusion from the premises has been provided. This account of proof gap is over-generative, in the sense that it recognizes proof gaps in mathematical proofs that are considered as gapless in mathematical practice. The easiest way to see this is to notice that most mathematical proofs that are considered in mathematical practice are not presented under the form of formal derivations.3 In particular, these mathematical proofs contain mathematical inferences for which no formal derivation of the conclusion from the premises has been provided. If one adopts the notion of proof gap just stated, one is then forced to recognize proof gaps in most ordinary mathematical proofs. This notion is therefore over-generative and does not correspond to an intuitive notion of proof gap present in mathematical practice.

8 The second interpretation yields the following account of the validity of mathematical inference: a mathematical inference is valid if and only if its conclusion can be formally derived from its premises. This leads to the following notion of proof gap: a proof gap occurs in a mathematical proof whenever there is a mathematical inference in the proof for which no formal derivation of the conclusion from the premises can be provided. This account of proof gap is under-generative, in the sense that it does not recognize proof gaps in mathematical proofs when there are ones according to the standards of mathematical practice. To see this, consider any theorem for which a formal proof has been provided in a modern proof assistant (HOL Light, Coq, Isabelle,

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etc.), for instance the prime number theorem for which a formal proof has been provided in Isabelle [Avigad, Donnelly et al. 2007]. In this particular case, the mathematical inference consisting of the prime number theorem as conclusion, and of the axioms of higher-order logic, along with an axiom asserting the existence of an infinite set, as premises, is such that its conclusion can be formally derived from its premises. This mathematical inference is then valid according to the account just stated. Consequently, the mathematical proof consisting of this particular mathematical inference alone should be considered as gapless, which is obviously at odds with mathematical practice. We shall then conclude that this notion is under- generative and therefore does not correspond to an intuitive notion of proof gap present in mathematical practice.

9 The third interpretation yields the following account of the validity of mathematical inference: a mathematical inference is valid if and only if its conclusion can routinely be formally derived from its premises. This leads to the following notion of proof gap: a proof gap occurs in a mathematical proof whenever there is a mathematical inference in the proof for which no formal derivation of the conclusion from the premises can be routinely provided. This interpretation is probably the most faithful to the defenders of the prevailing view of mathematical rigor who argue that, even though rigorous mathematical proofs are not usually presented as formal derivations, they can routinely be turned into formal derivations.4 The main issue with this account is to provide a precise meaning of what it means for a mathematical inference to be routinely formalizable, i.e., to be turned routinely into a formal derivation of the conclusion from the premises. One way to address this issue is to look at the field of formal verification, where the main goal is to provide actual formal derivations of mathematical theorems that can be checked in a purely mechanical way by a proof assistant. What the research activity in this field seems to reveal is that it is almost never a routine affair to provide a formal derivation from an existing ordinary mathematical proof.5 In particular, there does not seem to be any direct meaningful sense that can be attributed to the idea of a routine translation of a mathematical inference into a formal derivation. Consequently, the notion of proof gap provided by the third interpretation should not be rejected as inadequate, but rather as underdetermined.

10 The three derivation-based accounts of the validity of mathematical inference proposed in this section do not yield an account of proof gap that would capture adequately the intuitive notion(s) of proof gap from mathematical practice. Yet, there is still a more general reason to doubt any derivation-based account of the validity of mathematical inference. The reason is that, from the perspective of the philosophy of mathematical practice, a philosophical account of the validity of mathematical inference and of the notion of proof gap should not only be adapted for contemporary mathematical practice, but also for other mathematical practices from the history of mathematics. In particular, such an account should be able to accommodate changes in standards of mathematical rigor over different mathematical practices, as it occurred many times in the history of mathematics.6 To this purpose, any derivation-based account would appear far too rigid, in the sense that it would provide an absolute or fixed point of reference for determining what constitutes a valid mathematical inference and a proof gap, and therefore shall fail to account for different standards.7 The second view of the validity of mathematical inference that we will discuss in the next sections promises to offer more flexibility in this respect.

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3 Prawitz on the validity of deductive inference

11 In a series of papers [Prawitz 2009, 2012a,b], Prawitz has undertaken the task to develop a new conceptualization of the notion of valid deductive inference.8 The main motivation for this project comes from the requirement that, according to Prawitz, a philosophical account of the validity of inference should be able to explain how one can acquire justifications or grounds by drawing valid inferences. More precisely, such an account should lead as a conceptual truth that by drawing a valid inference, one can acquire a ground for the conclusion given grounds for the premises that one is already in possession of. According to Prawitz, neither the notion of logical consequence from model theory nor the notion of formal derivation from proof theory have been able to address this requirement [Prawitz 2012a, 888]. Prawitz's solution is to build the grounding requirement directly into the conception of the validity of inference. More specifically, Prawitz proposes an account of the validity of inference where inferences are conceived as operations on grounds: To get a fresh approach to the concept of valid inference we should reconsider the concept of inference. As already noted, a typical way of announcing an inference is to make an assertion and state at the same time a ground for the assertion, saying for instance "B, because A" or "A, hence B". [...] Although the conclusion and the premises may be all that we make explicit, there is also some kind of operation involved thanks to which we see that the conclusion is true given that the premises are. Sometimes we vaguely refer to such an operation [...], but essentially it is left implicit. My suggestion is that in analysing the validity of inferences, we should make these operations explicit, and regard an inference as an act by which we acquire a justification or ground for the conclusion by somehow operating on the already available grounds for the premises. [Prawitz 2012b, 18] The validity of inference is then defined as follows: An individual inference is valid if and only if the given grounds for the premises are grounds for them and the result of applying the given operation to these grounds is in fact a ground for the conclusion. [Prawitz 2012b, 19]

12 There are thus two central notions to Prawitz's account of the validity of inference: operation and ground. We shall now see how they are defined, starting with the notion of ground.

13 The term ground is used by Prawitz as a synonym for justification: to have a ground for a statement A means to be justified in holding A true [Prawitz 2012a, 890]. The main issue is then to determine how the notions of ground and statement are related, i.e., to specify what constitutes a ground for a statement. Prawitz's solution is to appeal to a specific theory of meaning: The line that I shall take is [...] roughly that the meaning of a sentence is determined by what counts as a ground for the judgement expressed by the sentence. Or expressed less linguistically: it is constitutive for a proposition what can serve as a ground for judging the proposition to be true. Prom this point of view I shall specify for each compound form of proposition expressible in first order languages what constitutes a ground for an affirmation of a proposition of that form. [Prawitz 2009, 19H92]

14 For instance, in the case of conjunction, Prawitz proposes the following specification: α is a ground for the conjunction p ∧ q if and only if α = ∧G(β, γ) for some β and γ such that β is a ground for p and γ is a ground for q. Such specifications rely on grounding

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operations, such as the conjunction grounding operation ∧G, which specify how grounds for a certain type of statement shall be formed. These grounding operations are primitive in Prawitz's account. In the case of the statements expressible in first-order logic, the grounding operations closely follow the introduction rules of natural deduction. Notice, however, that this ground-based account of meaning is not restricted for Prawitz to the language of first-order logic, even though explicit descriptions of grounding operations are only provided for first-order statements. Notice also that, according to this theory of meaning, if an agent understands the meaning of a statement, she then has the capacity to recognize what constitutes a ground for it.

15 The second key notion in Prawitz's account of the validity of inference is the one of operation on grounds. Prawitz does not provide a general definition of what exactly these operations are, except that they have grounds as input and output. However, Prawitz does provide concrete examples of operations for the logical inferences corresponding to the inference rules of natural deduction for first-order logic, as well as for mathematical induction. In particular, all the grounding operations corresponding to the introduction rules of natural deduction constitute such operations. For instance, the conjunction grounding operation ∧G is an operation that takes as input a ground γ for p and a ground β for q, and outputs a ground α = ∧G(β, γ) for the conjunction p ∧ q. Regarding mathematical induction, an informal description of the associated operation has been provided by Prawitz as follows:9 Let us consider the inference form of mathematical induction, in which it is concluded that a sentence A(n) holds for an arbitrary natural number n, having established the induction base that A(0) holds and the induction step that A holds for the successor n' of any natural number n given that A holds for n. The ground for the induction step may be thought of as a chain of operations that results in a ground for A(n') when applied to a ground for A(n). The operation that is involved in this inference form may roughly be described as the operation which, for any given n, takes the given ground for A(') and then successively applies the chain of operations given as ground for the induction step n times. [Prawitz 2012b, 20]

16 We can now state what drawing a valid inference consists in for Prawitz: it means applying the given operation to the grounds of the premises and to verify that the result is indeed a ground for the conclusion. Since understanding the meaning of the statements involved in the inference is a necessary precondition for being able to draw a valid inference [Prawitz 2009, 199], the agent does have the capacity to recognize what constitutes a ground for the conclusion and can therefore evaluate whether the result of the operation is indeed a ground for the conclusion. In the example of conjunction, the inference with premises p and q and conclusion p∧q is trivially valid since, given grounds β for p and γ for q, applying the operation ∧G to p and q results in ∧G(β, γ) which is a ground for p ∧ q according to the above specification. In the example of mathematical induction, the inference with premises A(0) and A(n) → A(n+1) and conclusion ∀nA(n) is valid since the application of the above operation for mathematical induction to the grounds for the premises yields a groundfor A(n), and this for any natural number n.

17 How according to the ground-based account one can fail in drawing a valid inference? As we shall see in the next sections, there are several possible ways to fail in drawing a valid inference, which will lead in turn to different kinds of proof gaps. However, if we want to be able to evaluate the notions of proof gap resulting from Prawitz's account of

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the validity of inference, we first need to specify it in the particular case of mathematical inference.

4 A ground-based account of the validity of mathematical inference

18 Prawitz's conception of the validity of inference is not only meant as a general account of the validity of deductive inference in formal proof systems such as natural deduction, but also of the validity of what we might call informal deductive inference.10 Since mathematical inferences are par excellence a type of deductive inference which provide conclusive grounds for their conclusions, and that most mathematical inferences in mathematical proofs are informal deductive inferences, mathematical inference fits perfectly within the scope of Prawitz's account. Indeed, Prawitz confirms this application by taking as one of the key illustrative examples of his account the validity of mathematical induction, the archetypal example of a mathematical inference.

19 How can we adapt Prawitz's account to the specific context of mathematical inference? As we saw in the previous section, the main task is then to specify the central notions of this account—i.e., the ones of operation and ground—in the context of mathematical proofs in mathematical practice. To this end, it would appear particularly useful to start with a more precise conception of what a mathematical practice is. For this purpose, the most natural choice is to adopt the framework developed by Kitcher in [Kitcher 1981, 1984]. In this framework, a mathematical practice is described as a tuple 〈 L, M, Q, R, S〉 where: L is the language of the practice, M the set of meta-mathematical views, Q the set of accepted questions, R the set of accepted reasonings, and S the set of accepted statements. The two components directly relevant to the issue of the validity of mathematical inference are theset of accepted reasonings R and the set of metamathematical views M. We shall now see how Kitcher defines these two components, and how they shouldbe adapted to integrate the notions of operation and ground within the notion of mathematical practice.

20 The set of accepted reasonings R is defined by Kitcher as: “the sequences of statements mathematicians advance in support of the statements they assert” [Kitcher 1984, 180]. As we saw in the first quote of the previous section, Prawitz does not identify an inference only by its premises and conclusion, but also by some operation that should be made explicit. Consequently, we shall consider the set of accepted reasonings R as constituted of a set of mathematical inferences, where an inference is identified not only by its premises and conclusion, but also by the operation involved. For instance, inferences corresponding to mathematical induction should not only be identified by their premises A(0) and A(n) → A(n + 1) and conclusions ∀ nA(n), but also by the operation described in the previous section. Mathematical induction is here a perfect example of mathematical inferences members of the set of accepted reasonings in the mathematical practice of contemporary number theory. For other examples we can mention: the diagrammatic inferences in the mathematical practice of elementary Euclidean geometry in Euclid's Elements [Euclid ana], the computer-assisted inferences in the mathematical practice of graph theory, or the use of numerical methods—also known as experiments—in various mathematical practices such as number theory or analysis. Notice that all these examples of mathematical inferences members of the set

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of accepted reasonings do not necessarily yield valid mathematical inferences. This is an interesting and important aspect of Kitcher's framework, namely that: "the set of accepted reasonings will outrun the set of accepted proofs" [Kitcher 1984, 181]. As examples of types of accepted reasoning that do not constitute proofs, Kitcher mentions reasonings that might be used to warrant beliefs, such as inductive generalizations in number theory, or reasonings that appear unrigorous, such as various methods of reasoning with infinitesimals in the early development of the calculus. According to Kitcher, whether an accepted reasoning in a mathematical practice constitutes a proof and, in our terminology, whether a mathematical inference is valid, is determined by the set of metamathematical views.

21 The set of metamathematical views M is defined by Kitcher as containing at least the following components: ( i) standards for proof; (ii) the scope of mathematics; (iii) the order of mathematical disciplines; (iv) the relative value of particular types of inquiry. [Kitcher 1984, 189]

22 For our purposes, the only relevant component will be the standards of proof. According to Kitcher, these standards can be specified “by describing the kinds of inference which are held to be legitimate, or by indicating paradigms of the type of reasoning which is preferred” [Kitcher 1984, 190]. However, since we want to reach an account of the validity of mathematical inference, we do not want to define the correct inferences simply as elements in the set of metamathematical views. Rather, we want the set of metamathematical views to specify what constitutes a ground for a mathematical statement that can be formulated within the considered practice. In other words, we shall substitute, in the set of metamathematical views of a given mathematical practice, standards of proof for standards of justification or ground for mathematical statements.

23 Thus, our two main modifications of Kitcher's notion of mathematical practice consist in (i) identifying mathematical inferences in the set of accepted reasonings R not only by their premises and conclusions but also by the operations involved, and (ii) replacing in the set of metamathematical views M the standards of proof by a specification of what constitutes grounds for the mathematical statements of the practice. We are now in a position to state precisely the ground-based account of the validity of mathematical inference: a mathematical inference (P, C, O) is uafa'd within a given mathematical practice 〈L, M, Q, R, S〉 if and only if the operation O provides a ground for the conclusion C given grounds for the premises P, where grounds for C and P are specified by the set of metamathematical views M. The validity of mathematical inference is thus defined by components R and M together, which is in direct line with Kitcher's view: [T]he criteria [...] for correct inference are set by the background metamathematical views [...]. Those metamathematical views are intended to specify the conditions which must be met if a sequence is to fulfil the distinctive functions of proofs. [Kitcher 1984, 180]

24 To illustrate the ground-based account of the validity of mathematical inference, let us consider the different examples of accepted reasonings mentioned above. First of all, in the mathematical practice of number theory, mathematical induction leads to valid mathematical inferences since, as we have seen in the previous section, it consists in an operation that results in what is recognized as grounds for statements of the form ∀ nA(n) within this practice. For elementary Euclidean geometry, the validity of diagrammatic inferences depends on the set of metamathematical views of the

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considered practice: in the mathematical practice of Euclid's Elements, diagrammatic inferences yield grounds for their conclusions, while in contemporary mathematical practice they don't. According to the ground-based account of the validity of mathematical inference, the difference lies precisely in the different specifications of what constitutes a ground for a geometrical statement in the metamathematical views of the two practices.11 Sometimes, it is a matter of discussion whether or not a given operation yields a ground for its conclusion, as witnessed by the debate around the status of the computer-assisted proof of the four-colour theorem in the mathematical practice of graph theory. In this particular case, this reflects an indeterminacy with respect to the standards of ground within the metamathematical views of the practice, and in turn an indeterminacy with respect to what constitutes a valid mathematical inference. Finally, some mathematical inferences involve operations that do not yield (conclusive) grounds for their conclusions, and therefore should not be considered as valid according to the ground-based account. Numerical methods or inductive generalizations are examples of such mathematical inferences in the mathematical practice of number theory.

25 Following the methodology described in the introduction, we shall now spell out the notion(s) of proof gap resulting from the ground-based account of the validity of mathematical inference by investigating how one can fail to draw a valid mathematical inference in this specified sense. To this end, it will be useful to distinguish between two families of proof gaps. The reason is that, as we just saw, the validity of mathematical inference is determined by the two components R and M of a mathematical practice. Yet, mathematicians sometimes use the term proof gap to refer to situations in which some inferences within a given mathematical practice are evaluated with respect to the metamathematical views of another mathematical practice, i.e., the components R and M are taken from different practices. To distinguish between these two cases, we introduce the following terminology: if a proof gap occurs when the two components R and M are taken from the same mathematical practice, we will speak of intra-praxis gap; if a proof gap occurs when the two components R and M are taken from different mathematical practices, we will speak of inter-praxis yap. We will now investigate the different possible cases in which one can fail to draw a valid mathematical inference, focusing in turn on intra-praxis gaps and inter-praxis gaps.

5 Intra-praxis gaps

26 An intra-praxis yap occurs in a mathematical proof of a given mathematical practice whenever there is a failure in drawing a valid mathematical inference, and where validity is evaluated with respect to the practice's standards of ground. The first parameter in this definition is the notion of the validity of mathematical inference, which is given in this section by the ground-based account. The second parameter is the notion of failure. The ground-based account allows for different possible interpretations of what it could mean to fail in drawing a valid mathematical inference. We now want to evaluate whether, by fixing different interpretations of the notion of failure, one is able to capture different intuitive notions of proof gap from mathematical practice. To this end, we propose in this section the following methodology. It turns out that a fine-grained taxonomy of different notions of proof gap present in mathematical practice has been proposed by Don Fallis in [Fallis 2003].

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This taxonomy offers a perfect opportunity to evaluate the ground-based account, the question being: is it possible, by giving different interpretations of the notion of failure in the definition of intra-praxis gap, to capture the different kinds of proof gaps identified in [Fallis 2003]? To answer this question, we will now consider in turn the three kinds of proof gaps constitutive of Don Fallis' taxonomy—i.e., inferential yaps, enthymematie yaps and untraversed yaps— and see if they can be accounted for by identifying different interpretations of the notion of failure. Inferential yaps are defined by Don Fallis as follows: A mathematician has left an inferential gap whenever the particular sequence of propositions that the mathematician has in mind (as being a proof) is not a proof. [Fallis 2003, 51]

27 According to Don Fallis, the existence of inferential gaps corresponds to a form of fallibilism, in the sense that mathematicians might sometimes be mistaken in recognizing a sequence of mathematical inferences as a proof. Within the ground-based account of the validity of mathematical inference, such a failure can be represented as a mistaken evaluation of the result of the operation involved in one (or more) mathematical inference of the proof as constituting a ground for its conclusion. This possibility is indeed considered by Prawitz: To say that there are deductive inferences that give rise to conclusive and even compelling grounds or proofs is of course not to say that there are infallible roads to knowledge. One can never rule out that one is mistaken about what one thinks is a ground or a proof of a sentence. [Prawitz 2012b, 2]

28 Thus, this precise notion of failure in the evaluation of what constitutes a ground for a mathematical statement allows us to capture the notion of inferential gap as follows: an inferential gap occurs in a mathematical proof of a mathematical practice 〈L, M, Q, R, S 〉whenever there is a mathematical inference (P, C, O) in the proof such that one has mistakenly evaluated the result of the operation O, applied to grounds for the premises P, as being a ground for the conclusion C. Enthymematie gaps are defined by Don Fallis as follows: A mathematician has left an enthymematie gap whenever he does not explicitly state the particular sequence of propositions that he has in mind (as being a proof). [Fallis 2003, 54]

29 One of the main reasons for the existence of enthymematie gaps, according to Don Fallis, is to facilitate communication: by omitting in the communication of mathematical proofs the steps that anyone can easily reconstruct from common background knowledge, one can more efficiently communicate the essential aspects of new proofs. Where is the failure located in an enthymematie gap? Importantly, the failure is not on the side of the one communicating the proof: if one leaves an enthymematie gap between some premises and a conclusion, one is supposed to have successfully obtained a ground for the conclusion given grounds for the premises, and this by having carried out the operations involved in the omitted mathematical inferences. Rather, the failure lies in the communication of the proof: an enthymematie gap occurs precisely when a mathematical inference (P, C, O) in the communicated proof is such that either the operation O does not belong to the set of accepted reasonings, or O fails to provide a ground for C given grounds for P, and such that the omitted mathematical inferences can easily be reconstructed from common background knowledge in such a way that a chain of operations from the set of accepted reasonings of the practice results in a ground for C given grounds for P. Thus,

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enthymematic gaps can be captured by specifying this notion of failure in communication through a three part condition: an enthymematie gap occurs in a mathematical proof of a mathematical practice 〈L, M, Q, R, S〉whenever there is a mathematical inference (P, C, O) in the proof such that (i) either the operation O does not belong to the set of accepted reasonings R, or O fails to provide a ground for C given grounds for P, (ii) the author of the proof knows a chain of mathematical inferences in R for which she has verified that the application of the sequence of operations involved yields a ground for C given grounds for P and (iii) this chain of mathematical inferences can easily be reconstructed from the common background knowledge of the agents involved in the mathematical practice 〈L, M, Q, R, S〉. Untraversed yaps are defined by Don Fallis as follows: A mathematician has left an untraversed gap whenever he has not tried to verify directly that each proposition in the sequence of propositions that he has in mind (as being a proof) follows from previous propositions in the sequence by a basic mathematical inference. [Fallis 2003, 56-57]

30 For untraversed gaps, the failure is easily identified as a failure in performance: for some mathematical inferences in a mathematical proof, one has not performed the associated operations and verified that they resulted in grounds for their conclusions given grounds for their premises. Untraversed gaps can then be captured in the following way: an untraversed gap occurs in a mathematical proof of a mathematical practice 〈L, M, Q, R, S〉whenever there is a mathematical inference (P, C, O) in the proof such that one has not carried out the operation O on grounds for the premises P, and a fortiori has not verified that it resulted in a ground for the conclusion C.

31 Thus, the three kinds of proof gaps from Don Fallis' taxonomy can be accounted for as particular kinds of failures in drawing valid mathematical inferences, where the validity of mathematical inference is determined by the ground-based account. More specifically, inferential gaps correspond to failure in evaluation, enthymematie gaps to failure in communication and untraversed gaps to failure in performance. We shall now turn our attention to inter-praxis gaps.

6 Inter-praxis gaps

32 An inter-praxis gap occurs in a mathematical proof of a given mathematical practice whenever there is a failure in drawing a valid mathematical inference, and where validity is evaluated with respect to a different mathematical practice's standards of ground.12 Since this definition differs from the one of intra-praxis gap by simply requiring that operations and grounds be taken from different practices, the three notions of intra-praxis gap from the previous section can easily be reinterpreted as inter-praxis gap. However, none of the three resulting notions seem to correspond to an intuitive notion of proof gap from mathematical practice. Yet there is a specific kind of inter-praxis gap that is commonly mentioned in mathematical practice. Maybe one of the most illustrative examples concerns the diagrammatic inferences in Euclidean geometry, the situation being summarized by Manders as follows: In Euclidean geometry, a diagram has standing to license inference, just as do relationships recognized in the text. It is now commonly held that this is a defect of rigor. But the extraordinary career of Euclidean practice justifies a fuller consideration. It was a stable and fruitful tool of investigation across diverse cultural contexts for over two thousand years. During that time, it generally struck

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thoughtful and knowledgeable people as the most rigorous of human ways of knowing [...]. [Manders 2008, 81]

33 In this example, we are in the presence of a family of mathematical inferences— the diagrammatic inferences—that were considered as valid in the mathematical practice of Euclid's Elements [Euclid ana], but that are not considered as valid in contemporary mathematical practice. Consequently, it is now thought that diagrammatic inferences create proof gaps in the mathematical proofs of Euclid's Elements, as Manders puts it: [I]t has been commonplace for at least the last century to castigate traditional geometry for 'gaps in arguments' [...] due to 'reading off from the figure'. [Manders 2008, 87]

34 What does the failure consist in for this particular kind of proof gap? This question finds a natural answer within the ground-based account of the validity of mathematical inference, namely that the standards of what constitutes a ground for a mathematical statement have changed from Euclid to contemporary mathematical practice: even though diagrammatic inferences yield (conclusive) grounds for their conclusion according to the metamathematical views of the mathematical practice of Euclid's Elements, they do not according to the metamathematical views of contemporary mathematical practice. In other words, the results of the operations involved in diagrammatic inferences fail to meet the standards of ground from contemporary mathematical practice, while they do meet the ones of Euclid's practice. This type of proof gap seems to occur when a process of rigorization of mathematical practice is taking place,13 and can be defined in the present framework as follows: a rigorization gap occurs in a mathematical proof of a mathematical practice 〈L, M, Q, R, S〉 whenever there is a mathematical inference (P, C, O) in the proof such that the result of the operation O, applied to grounds for the premises P, does not constitute a ground for the conclusion C according to the metamathematical views M, but does constitute a ground for the conclusion C according to the metamatheniatical views M' of a mathematical practice prior to 〈L, M, Q, R, S〉 in the historical development of mathematics.

35 Two important lessons can be drawn from the existence of rigorization gaps. Firstly, rigorization gaps differ from the three kinds of proof gaps constitutive of Don Fallis' taxonomy, since they do not correspond to failure in evaluation, communication or performance. They appear thereby as a strong candidate for extending this taxonomy. Secondly, any account of the validity of mathematical inference and of the notion of proof gap which aims to be faithful to mathematical practices past and present should be able to account for rigorization gaps, as in the example of diagrammatic inferences in Euclidean geometry. It then seems that any derivation-based account is doomed to fail this requirement as it provides a fixed point of reference—i.e., the notion of formal derivation—from which to evaluate the validity of mathematical inferences. Derivation-based accounts cannot in particular explain why diagrammatic inferences are considered as valid in Euclid's practice but not in contemporary mathematical practice. As we have just seen, the ground-based account offers such a flexibility and is able to account for rigorization gaps. The reason being that the two central notions of operation and ground in Prawitz's account of the validity of inference are susceptible of different interpretations, and can in particular be relativized to specific mathematical practices.

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7 Conclusion

36 We began this paper with the observation that a philosophical account of the common gap-based formulation of mathematical rigor requires an account of proof gap, and that an account of proof gap requires in turn an account of the validity of mathematical inference. The aim of the paper was then to evaluate two possible views of the validity of mathematical inference with respect to their capacity to yield an account of proof gap that would adequately capture the intuitive notion(s) of proof gap from mathematical practice. The first view was provided by the contemporary standards of mathematical rigor which evaluate the validity of mathematical inference with respect to the notion of formal derivation. We saw that there were several possible interpretations of the relation between the notions of rigorous mathematical proof and formal derivation, leading to different derivation-based accounts of the validity of mathematical inference. We then argued that none of the resulting derivation-based accounts of proof gap adequately capture the intuitive notion(s) of proof gap from mathematical practice. The second view was based on a recent ground-based account of the validity of inference proposed by Prawitz. We first specified Prawitz's account to mathematical inference by integrating the notions of operations and grounds within Kitcher's notion of mathematical practice. In order to evaluate the resulting ground- based account of proof gap, we attempted to describe several intuitive notions of proof gap as particular types of failure in drawing valid mathematical inferences, where the validity of mathematical inference was specified by the ground-based account. Our analysis has revealed that several intuitive notions of proof gap can be accommodated in this way within this framework. More precisely, we saw how the three kinds of proof gaps comprising Don Fallis' taxonomy—inferential gaps, enthymematic gaps and untraversed gaps—can be represented respectively as failure in evaluation, communication and performance. We also saw that the ground-based account was particularly suitable for representing what we called rigorization gaps, a particular kind of proof gap that occur from a (temporal) cross-perspective on different mathematical practices. Finally, we noticed that rigorization gaps constitute a serious challenge for any derivation-based account of proof gap. We shall then conclude that the ground-based account offers a promising framework for representing different intuitive notions of proof gap present in mathematical practice, and therefore should be of particular interest for the philosophy of mathematical practice.

37 Yet, a full development of the ground-based account faces several important challenges. Some of them have already been raised in critical responses to Prawitz in [Pagin 2012] and [Murzi 2011]. Pagin identifies two problems [Pagin 2012, 881]: the first one being that the ground-based account does not fit the inferential practice of ordinary speakers; the second one that the ground-based account requires a reflection principle saying that when an agent is in possession of a ground for a statement she must be aware of it, which according to Pagin leads to problematic consequences. From the point of view of moderate inferentialism, Murzi raises two concerns [Murzi 2011, 289]: the first one regarding the unclear metaphysical nature of grounds; the second one regarding problematic consequences of conceiving the meaning of a statement as determined by what counts as a ground for it. This last point is probably the most pressing challenge facing the development of a ground-based account of the validity of mathematical inference. More specifically, such an account requires that if one

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understands the meaning of the mathematical statements involved in a given mathematical inference, one has the capacity to recognize what constitutes a ground for them. We shall then provide a full specification of (i) what constitutes a ground for a mathematical statement, and (ii) how one recognizes whether something counts as a ground for a mathematical statement. As we saw, Prawitz's response to this challenge is to adopt a theory of meaning in which the meaning of a statement is determined by what counts as a ground for holding the statement true, but this solution raises a number of problems as noticed in [Murzi 2011]. Can a ground-based theory of meaning for mathematical statements be developed along Prawitz's line while avoiding such problems? Is there an alternative way to specify what constitutes a ground for a mathematical statement? How can we account for one's capacity to recognize what counts as a ground for a mathematical statement? Addressing these questions is the next step towards a full development of a ground-based account of mathematical rigor, proof gap and the validity of mathematical inference.

BIBLIOGRAPHY

ANTONUTTI MARFORI, Marianna [2010], Informal proofs and mathematical rigour, Studio, Logica, 96(2), 261-272.

AVIGAD, Jeremy, DONNELLY, Kevin, GRAY, David, & RAFF, Paul [2007], A formally verified proof of the prime number theorem, ACM Transactions on Computational Logic (TOOL), 9(1), 1-23.

AZZOUNI, Jody [2004], The derivation-indicator view of mathematical practice, Philosophia Mathematica, 12(2), 81-106.

DETLEFSEN, Michael [2009], Proof: Its nature and significance, in: Proof and Other Dilemmas: Mathematics and Philosophy, edited by B. Gold & A. Simons, R. Washington, D.C.: The Mathematical Association of America, 3-32.

EUCLID [ana], The Thirteen Books of Euclid's Elements.

FALLIS, Don [2003], Intentional gaps in mathematical proofs, Synthese, 134(1), 45-69.

KITCHER, Philip [1981], Mathematical rigor-Who needs it?, Nous, 15(4), 469-493. − [1984], The Nature of Mathematical Knowledge, New York: Oxford

University Press.

LEITGEB, Hannes [2009], On formal and informal provability, in: New Waves in Philosophy of Mathematics, edited by 0. Linnebo & O. Bueno, New York: Palgrave, Macmillan, 263-299.

MAC LANE, Saunders [1986], Mathematics: Form and Function, New York: Springer-Verlag.

MANCOSU, Paolo [2008], The Philosophy of Mathematical Practice, Oxford: Oxford University Press.

MANDERS, Kenneth [2008], The Euclidean diagram, in: Philosophy of Mathematical Practice, edited by P. Mancosu, Oxford: Oxford University Press, 80-133.

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MURZI, Julien [2011], Inferentialism without verificationism: Reply to Prawitz, in: Logic and Knowledge, edited by C. Cellucci & E. Ippoliti, Cambridge: Cambridge Scholars Publishing, 285-290.

PAGIN, Peter [2012], Assertion, inference, and consequence, Synthese, 187(3), 869-885.

PIERPONT, James [1928], Mathematical rigor, past and present, Bulletin of the American Mathematical Society, 34, 23-53.

PRAWITZ, Dag [2009], Inference and knowledge, in: Logica Yearbook 2008, edited by M. Pelis, London: College Publications, 175-192. − [2012a], The epistemic significance of valid inference, Synthese, 187(3), 887-898. − [2012b], Validity of inference, in: Proceedings from the 2nd Launer Symposium on the Occasion of the Presentation of the Launer Prize at Bern 2006, (to appear).

RAV, Yehuda [1999], Why do we prove theorems?, Philosophia Mathematica, 7(1), 5-41.

ROBINSON, John Alan [1997], Informal rigor and mathematical understanding, in: Computational Logic and Proof Theory, Heidelberg & New York: Springer, Proceedings of the 5th annual Kurt Godel Colloquium, August 25-29, 1997, 54-64.

VAN KERKHOVE, Bart & VAN BENDEGEM, Jean Paul (eds.) [2010], Perspectives on Mathematical Practices: Bringing Together Philosophy of Mathematics, Sociology of Mathematics, and Mathematics Education, Logic, Epistemology and the Unity of Science, Dordrecht: Springer.

NOTES

1. See for instance Kitcher: "central to the idea of rigorous reasoning is that it should contain no gaps" [Kitcher 1981, 469]. 2. We do not claim that this formulation exhausts the notion of mathematical rigor, nor are we trying to evaluate it in this respect. Our standpoint in this paper is rather to start from this common formulation of mathematical rigor and to analyze what is required to determine a philosophical account of it. 3. This observation has been used to argue that "rigor and formalization are independent concerns" [Detlefsen 2009, 17], but also to argue that the prevailing view of mathematical rigor yields an implausible account of mathematical knowledge [Antonutti Marfori 2010]. This observation is also at the basis of recent discussions on the relation between formal and informal proofs (e.g., see [Azzouni 2004], [Leitgeb 2009], [Rav 1999]). 4. This view is expressed for instance by Mac Lane: "In practice, a proof is a sketch, in sufficient detail to make possible a routine translation of this sketch into a formal proof [Mac Lane 1986, 377]. 5. This point has been acknowledged by Robinson, one of the main figures in automated theorem proving, who said of the activity of formalization that: "[l]n most cases it requires considerable ingenuity, and has the feel of a fresh and separate mathematical problem in itself. In some cases [...] formalization is so elusive as to seem to be impossible" [Robinson 1997, 54]. 6. For some historical examples of changes in mathematical rigor, see [Kitcher 1981], [Pierpont 1928]. 7. We will come back to this point at the end of section 6. 8. In this section, when we will use the term inference, we will always refer to deductive inference. 9. A formal description of this operation has been proposed by Prawitz in [Prawitz 2012a, 897]. 10. As also noted by Pagin: "A main point of Prawitz's discussion [...] is that it does not solely apply to formal systems of deduction, but to informal deductive reasoning as well" [Pagin 2012, 875].

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11. We shall come back to the example of elementary Euclidean geometry in section 6. 12. There are many different ways in which one can understand 'different' mathematical practices in the definition of inter-praxis gap. For instance, one may compare mathematical practices from different domains (number theory, analysis, algebra, etc.), but also from the same domain but different perspectives such as the ones of pure and applied mathematics. In this section, we focus in particular on practices that differ along a temporal dimension—i.e., different mathematical practices from the same domain, but from different times in the historical development of mathe matics. Providing a taxonomy of the different possible kinds of inter- praxis gaps is beyond the scope of this paper. 13. See [Kitcher 1981, 1984] for a detailed analysis of rigorization processes in mathematical practice.

ABSTRACTS

Mathematical rigor is commonly formulated by mathematicians and philosophers using the notion of proof gap: a mathematical proof is rigorous when there is no gap in the mathematical reasoning of the proof. Any philosophical approach to mathematical rigor along this line requires then an account of what a proof gap is. However, the notion of proof gap makes sense only relatively to a given conception of valid mathematical reasoning, i.e., to a given conception of the validity of mathematical inference. A proof gap can in particular be conceived as a failure in drawing a valid mathematical inference. The aim of this paper is to discuss two possible views of the validity of mathematical inference with respect to their capacity to yield a plausible account of the intuitive notion(s) of proof gap present in mathematical practice. The first view is the one provided by the contemporary standards of mathematical rigor: a mathematical inference is valid if and only if its conclusion can be formally derived from its premises. We will argue that this conception does not lead to a plausible account of the intuitive notion(s) of proof gap. The second view is based on a new account of the validity of inference proposed by Prawitz: an inference is valid if and only if it consists in an operation that provides a ground for its conclusion given (previously obtained) grounds for its premises. We will first specify Prawitz's account to mathematical inference and we will then argue that the resulting ground-based account is able to capture various intuitive notions of proof gap as different types of failure in drawing valid mathematical inferences. We conclude that the ground-based account appears of particular interest for the philosophy of mathematical practice, and we finally raise several challenges facing a full development of a ground-based account of the notions of mathematical rigor, proof gap and the validity of mathematical inference.

Mathématiciens et philosophes définissent communément la rigueur mathématique de la manière suivante : une preuve mathématique est rigoureuse dès lors qu'elle ne présente aucun « trou » dans le raisonnement mathématique qui la compose. Toute approche philosophique de la rigueur mathématique formulée suivant cette conception se doit de définir la notion de « trou ». Cependant, une telle notion ne peut être pensée que relativement à une conception du raisonnement mathématique valide, i.e., de la validité de l'inférenee mathématique. Un « trou » dans une preuve mathématique peut ainsi être conçu comme un échec dans la production d'une inférence mathématique valide. L'objectif de cet article est d'évaluer deux conceptions de la validité de l'inférence mathématique par rapport à leur capacité à fournir une explication

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plausible des notions intuitives de « trou » présentes dans la pratique mathématique. La première conception est issue des standards contemporains de la rigueur mathématique : une inférence mathématique est valide si, et seulement si, sa conclusion peut être dérivée formellement à partir de ses prémisses. Nous montrerons que cette conception ne peut fournir une explication plausible des notions intuitives de « trou » dans les preuves mathématiques. La seconde conception est issue d'une nouvelle approche de la validité de l'inférence proposée par Prawitz : une inférence est valide si, et seulement si, elle consiste en une opération produisant une justification pour sa conclusion à partir de justifications pour ses prémisses. Nous adapterons tout d'abord cette conception à l'inférence mathématique et nous montrerons alors qu'elle est en mesure d'accommoder différentes notions intuitives de « trou » à travers différents types d'échecs dans la production d'inférences mathématiques valides. Nous conclurons en soulignant l'intérêt de cette conception pour la philosophie de la pratique mathématique, et nous relèverons un certain nombre de défis confrontant le développement d'une telle approche des notions de rigueur mathématique, de « trou » et de validité de l'inférence mathématique.

AUTHOR

YACIN HAMAMI Centre for Logic and Philosophy of Science Vrije Universiteit Brussel (Belgium)

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La rigueur mathématique chez Henri Poincaré

Ramzi Kebaïli

1 Introduction

1 Henri Poincaré est l’un des plus grands mathématiciens de tous les temps. Mais il est également connu pour ses erreurs célèbres, par exemple, dans le mémoire Sur le problème des trois corps et les équations de la dynamique qui lui valut l’attribution du prix Oscar II en 1889 (à ce sujet, voir [Barrow ‒Greene 1997]).

2 En plus de ces cas caractérisés d’erreurs mathématiques, son style est réputé être assez négligé. Par exemple, dans son éloge posthume de Poincaré, Gaston Darboux écrit : Mais, il ne faut pas craindre de le dire, si l’on veut donner une idée précise de la manière dont travaillait Poincaré ; bien des points demandaient des corrections ou des explications. Poincaré était un intuitif. Une fois au sommet, il ne revenait jamais sur ses pas. Il se contentait d’avoir brisé les difficultés, et laissait aux autres le soin de tracer les routes royales qui devaient conduire plus facilement au but. [Darboux 1913, 15]

3 Comme nous le verrons, Poincaré a également été sévèrement critiqué par Jean Dieudonné dans A History of Algebraic and Differential Topology pour le manque de rigueur de ses mémoires topologiques [Dieudonné 1989].

4 Or, dans l’ensemble de ses écrits philosophiques, Poincaré insiste sur le fait que les mathématiques ne doivent pas se réduire à la rigueur logique, et que « l’intuition » y joue un rôle fondamental. Au premier abord, on pourrait se contenter de penser que Poincaré était principalement intéressé par la découverte de nouvelles idées mathématiques (qu’il attribuerait à l’intuition) et qu’il se souciait peu de leur mise en forme technique, ce qui serait somme toute assez banal pour un mathématicien de son envergure. Néanmoins, comme nous allons le voir, il a dans ses écrits philosophiques refusé de se cantonner à une opposition superficielle entre intuition heuristique et rigueur démonstrative, puisqu’il a soutenu que l’intuition était nécessaire à la démonstration mathématique.

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5 En soutenant cette thèse, Poincaré s’opposait de manière virulente aux logiciens (notamment Russell et Whitehead) et reprochait à leur « logistique » d’entraver le travail du mathématicien1. En un certain sens, la rigueur — du moins telle que cette notion est comprise par les logiciens — serait donc stérilisante. Il y aurait donc dans le travail des mathématiciens une tension entre la rigueur et la fécondité. Mais en réalité, il s’agit là d’un faux dilemme : en effet, Poincaré refuse la vision logicienne de ce qui constitue l’essence même de la preuve mathématique, en dressant par exemple une comparaison avec le jeu d’échecs en 19022. Nous nous appuyons ici sur la thèse déjà développée par Michael Detlefsen dans [Detlefsen 1992], selon laquelle Poincaré aurait en tête une vision « alternative » de la rigueur. Selon Detlefsen, la vision « logicienne » de la démonstration mathématique, qui décompose celle ‒ci en une succession de déductions logiques, ne donne pas de compréhension d’ensemble de cette démonstration. Et c’est précisément ce manque de compréhension qui constituerait un manque de rigueur au sens « poincaréen ».

6 Or, comme nous allons le voir, les remarques de Poincaré sur la rigueur ne se limitent pas au problème (déjà crucial) de la compréhension globale. Notre objectif est donc de proposer une interprétation exhaustive des réflexions de Poincaré sur la rigueur, et qui découlerait de sa propre pratique des mathématiques. Fondamentalement, nous faisons donc l’hypothèse qu’il existe bien une telle conception unifiée qui serait sous‒jacente aux remarques de Poincaré, et qui permettrait de répondre (au moins partiellement) aux critiques de Dieudonné. Mais encore faut‒il préciser quels types de manquements à la rigueur sont visés par celui‒ci. Pour évaluer les normes de rigueur chez Poincaré, nous allons commencer par étudier un exemple tiré de son mémoire Analysis Situs qui marque la création d’une nouvelle discipline, la topologie, dont les concepts de base n’étaient pas a priori clairement délimités. Nous allons ensuite présenter quelles sont les critiques portées par Poincaré contre deux types de normes de rigueur, celles des logiciens et celles des analystes, et enfin proposer une interprétation de ce que serait la rigueur « poincaréenne ». Dans cette dernière partie, notre critère sera d’essayer de donner une interprétation unifiant différentes pistes suggérées par Poincaré, ainsi que sa pratique mathématique.

2 L’absence de rigueur dans l’Analysis Situs

7 Henri Poincaré lui‒même considérait ses travaux en topologie comme étant parmi les plus fondamentaux de son œuvre. Ainsi, dans l’analyse qu’il livre de ses propres travaux en 1901, et qui ne sera publiée qu’en 1921, il insiste sur « l’importance extrême de cette science3 ». De plus, en 1912, il ajoute : Et voici ce qui fait pour nous l’intérêt de cette Analysis Situs ; c’est que c’est là qu’intervient vraiment l’intuition géométrique. [Poincaré 1912, 484]

8 Le mémoire Analysis Situs de 1895 [Poincaré 1895], ainsi que ses cinq compléments, sont à l’origine de la topologie en tant que discipline propre, et en particulier de notre topologie algébrique moderne. Nous assistons donc à la naissance d’une discipline.

9 En tant que discipline en cours de formation, et dont les concepts ne sont pas encore clairement dégagés, l’Analysis Situs se prête particulièrement à une étude de la méthodologie de recherche scientifique de Poincaré. Comme nous allons le voir, on note chez Poincaré différents types de manque de rigueur dans ce travail, comme le souligne Jean Dieudonné [Dieudonné 1989].

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10 Ainsi, après avoir donné plusieurs définitions des variétés (sans préciser les liens entre elles), et plusieurs définitions des homologies, Poincaré va se retrouver à effectuer une démonstration complètement erronée du théorème de dualité. Mais nous allons expliquer pourquoi ce cas d’erreur mathématique illustre la méthodologie de Poincaré.

11 Rappelons tout d’abord l’objet du mémoire Analysis Situs. En termes modernes, il s’agit d’étudier certains sous‒espaces de ℝn en ne regardant que leurs propriétés restant inchangées par certaines déformations4, propriétés qui jouent un rôle capital dans plusieurs branches des mathématiques. Poincaré commence par définir les objets de son étude, appelés « variétés », puis il définit les déformations permises sur ces objets, appelées « homéomorphismes ». Riemann, puis Betti, avaient eu l’idée d’encoder les

propriétés d’une variété à k dimensions par la donnée de k — 1 nombres P1,…, Pk ‒ 1 (les

« nombres de Betti ») où — de manière heuristique — le nombre Pj correspondrait au nombre de trous de dimension j que possède la variété. L’espoir de départ était que cet encodage serait complet, dans le sens où deux variétés ayant les mêmes nombres de Betti seraient les mêmes du point de vue topologique (c’est ‒à ‒dire homéomorphes). Mais Poincaré va montrer que la donnée de ces nombres est insuffisante pour caractériser les variétés homéomorphes, et il propose pour y remédier d’associer également aux variétés un certain groupe5. Un des objectifs du mémoire de Poincaré est de légitimer l’importance fondamentale de ce groupe en construisant deux variétés ayant les mêmes nombres de Betti mais des groupes associés différents (et donc non ‒ homéomorphes). Mais pour parvenir à ce but, encore reste‒t‒il à définir correctement les nombres de Betti (ce qui nécessite de définir préalablement les « relations d’homologie », et d’être capable de les calculer. D’où l’importance cruciale d’obtenir

une démonstration du Théorème de Dualité (pour une variété de dimension n, Pn ‒k = Pk), qui fournit un moyen de calculer facilement les nombres de Betti.

12 Pour évaluer les normes de rigueur à l’œuvre chez Poincaré, nous allons voir différents exemples précis d’absence de rigueur : dans sa définition des variétés, dans sa définition des homéomorphismes, dans sa définition des homologies, et enfin dans la détermination de ces homologies. Enfin, nous verrons comment un de ses manques de rigueur a conduit à une véritable erreur mathématique dans la démonstration du théorème de dualité, ce qui a nécessité l’écriture d’un Complément à l’Analysis Situs [Poincaré 1899].

13 Le mémoire commence par une « première définition des variétés » (c’est le titre de la 6 première section) . Poincaré considère x1,x2, ...,xn n variables réelles regardées comme les coordonnées d’un point dans l’espace à n dimensions. Il prend ensuite un système

formé de p égalités de la forme Fk(x1,...,xn) = 0 et q inégalités de la forme Φj(x1,...,xn) > 0, 7 en imposant comme premières conditions que les fonctions Fk et Φj soient uniformes , continues et à dérivées continues. Il demande également que si on considère la matrice de taille n x p formée à la k ‒ème ligne et la i ‒ème colonne par les dérivées partielles

dFk/dx1 évaluées en un point, ses sous‒matrices de taille p x p ne soient jamais toutes de déterminant nul. Une fois tout ceci posé, Poincaré écrit : Je dirais que l’ensemble des points qui satisfont aux conditions (1), s’il y en a, ce que je suppose, forme une variété à n ‒ p dimensions. [Poincaré 1895, 196]

14 L’objet ainsi défini est très proche de notre notion moderne de variété différentielle8. Donnons un exemple pour bien comprendre ce que Poincaré a en tête. Par exemple,

dans l’espace R3 formé des points à coordonnées réelles (x1, x2, x3), le système

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définit une variété à trois dimensions, qui est une boule ouverte B, et le système définit une boule creuse K, qui a également deux dimensions (puisqu’il n’y a que des inégalités). Autres exemples, le

système définit une variété à deux dimensions, qui est la sphère S1 de centre (0, 0, 0) et de rayon 1, et le système définit une autre

sphère, que nous appellerons S2. 15 Cette définition à partir d’un système d’équations et d’inéquations présente certains avantages, le premier d’entre eux étant de pouvoir déterminer la dimension d’une variété à partir du nombre d’égalités. De plus, cette approche permet à Poincaré de définir la notion de variété « continue » comme étant une variété où l’on peut passer de n’importe quel point à n’importe quel autre en les faisant « varier d’une manière continue » et « sans que les relations (1) cessent d’être satisfaites ». Poincaré ne précise pas ce qu’il entend par une variation faite « de manière continue », mais il n’y a pas de cercle vicieux contrairement à ce que les termes paraissent suggérer : la continuité d’une variété n’est pas de même nature que la continuité d’une application, et celle ‒ci peut se définir aisément si l’on admet l’analyse réelle déjà constituée. En termes modernes, Poincaré définit ce que nous appellerions aujourd’hui un espace connexe par arcs9. Dans la suite, il ne considérera que ce type de variétés.

16 Un autre avantage de ce type de définition, c’est de pouvoir définir la « frontière » d’une variété, ainsi : pour une variété V de dimension d donnée, sa frontière est constituée par toutes les variétés de dimension d ‒ 1 obtenues en remplaçant n’importe laquelle des inégalités qui définissent V par une égalité.

17 Dans nos exemples, la frontière de B est formée par une seule variété, Si, et la frontière

de K est constituée de Si et S2. On notera au passage qu’ainsi définie, la frontière d’une variété n’est jamais incluse au sens ensembliste dans la variété, mais que Poincaré considère néanmoins que la frontière « fait partie » de la variété. Enfin, Poincaré appelle « fermée » une variété continue, bornée et dont la frontière est vide.

18 Cette définition des variétés est donc très commode pour définir des notions qui vont jouer un rôle capital par la suite. Cependant, à partir de la troisième section, il va construire de nouveaux objets qu’il va continuer de considérer comme des variétés, mais dans un sens différent. Il revendique cette démarche et le fait qu’il donne une « deuxième définition des variétés » (c’est le titre de la section). En réalité, c’est même deux nouvelles définitions qu’il donne : la première d’entre elles ne pose pas de

problème, c’est une définition paramétrique par un système de n équations x1 = θ(t1, …,

tn) et m inégalités Φj(ti,...,tn) > 0, les fonctions étant prises supposées continues et même analytiques. Il n’y a pas de difficultés pour étendre les notions données précédemment à cette nouvelle approche. Mais il ajoute ensuite une nouvelle définition, qu’il appelle le « procédé de la continuation analytique ». Son idée est que si les frontières de deux variétés (continues) possèdent des éléments en commun, alors on peut considérer le recollage de ces deux variétés le long de cette frontière commune comme étant lui ‒ même une nouvelle variété (continue), en un sens élargi. Il forme ainsi des « chaînes » de variétés, qui n’ont pas nécessairement de représentation paramétrique.

19 Poincaré aurait pu commencer par donner une seule définition générale de ce qu’est une variété, mais il ne l’a pas fait. Pourquoi ? Une hypothèse serait que du point de vue pédagogique, il préférait avancer pas à pas et élargir progressivement sa notion pour que le lecteur comprenne bien la marche du raisonnement. Néanmoins, le plus

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probable, c’est que toutes les manipulations qu’il s’autorise sur les variétés définies de la première manière nécessiteraient de pénibles justifications pour être étendues en toute généralité.

20 Or, il y a un point vraiment problématique dans l’argumentation de Poincaré : c’est qu’il définit un critère d’équivalence topologique entre deux variétés distinctes au deuxième paragraphe, puis ne revient jamais dessus et n’explique pas comment celui‒ci s’étendrait aux nouvelles définitions données dans la troisième section. Plus précisément, il définit une « équivalence du point de vue de l’Analysis Situs » qu’il appelle « homéomorphisme » (qui ne correspond pas exactement à notre notion actuelle d’homéomorphisme, mais plutôt à celle de difféomorphisme). Deux variétés sont homéomorphes au sens de Poincaré s’il existe entre elles une bijection dérivable, dont la dérivée est continue et inversible. Dans nos exemples, la boule B est

homéomorphe à toute autre boule, mais n’est pas homéomorphe à la sphère S1., ni à K. La définition qu’a donnée Poincaré est plus restrictive que la définition moderne, qui demande simplement une bijection continue d’inverse continue. Mais quoi qu’il en soit, cette définition ne joue presque aucun rôle dans la suite de son mémoire. En revanche, il invoque souvent un autre critère d’équivalence, qui ne doit pas être confondu avec l’homéomorphie, qu’il désigne par les termes de variétés « peu différentes » ou encore « infiniment proches ».

21 Or, ce critère non précisé joue un rôle capital dans la définition des nombres de Betti, comme nous allons le voir. Dans la cinquième section, Poincaré définit les relations

d’homologie en dimension m pour une variété W à d dimensions de la sorte : si V1, …, Vn sont des variétés à m dimensions formant la frontière d’une variété fixée à m+1

dimensions « faisant partie de W », on pose V1+...+Vn ~ 0. Par exemple dans la boule B, on

a vu que la sphère S1 formait sa frontière et donc on a S1 ~ 0, et dan s K on a S1 + S2 ~ 0. Ensuite à partir de ces équations posées, il stipule que « les homologies peuvent se combiner comme des équations ordinaires ». Concrètement dans la suite de son mémoire, il s’autorise les manipulations algébriques formelles suivantes :

• Si V1, V2, Vn sont n variétés « peu différentes », on peut remplacer V1 + V2 + ... + Vn par n. V1 dans la relation d’homologie.

• Si V1 + V2 + ... + Vn ~ 0, on pose V2 + ... + Vn ~ ‒V1.

• Si n.V1 ~ m.V2, où m et n entiers relatifs et n non ‒nul, on pose V1 ~ (m/n). V2. (Il est important de souligner pour la suite, que durant tout ce mémoire un flou persiste sur le caractère licite de cette opération de division).

22 Ce faisant, Poincaré commet une nouvelle entorse manifeste à la rigueur, qui est la suivante : il avait déjà défini dans la quatrième section ce que signifiait le symbole ‒V. Cette variété serait constituée des mêmes points que la variété V (elles seraient donc identiques du point de vue ensembliste), mais avec une représentation différente. En effet, pour l’obtenir à partir d’une variété définie de la première manière, on permute l’ordre de deux équations du système, et pour une variété définie paramétriquement, on permute l’ordre de deux variables (par contre, il n’explique pas pour la troisième définition). Or, il n’a pas vérifié que cette définition du symbole ‒V coincidait avec celle

donnée par le système des homologies (dans l’exemple que nous avions donné, S1 ~ ‒S2

dans K, mais ceci ne pose pas de problème car ‒S2 n’avait pas été défini auparavant).

23 Ceci posé, Poincaré définit dans la sixième section pour une variété V à n dimensions les

nombres de Betti Pk, pour k compris entre 1 et n ‒ 1. Pk est défini comme étant le nombre minimal de variétés fermées à k dimensions qui sont liées par une relation d’homologie

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dans V. Mais cette définition n’est pas adaptée pour des calculs effectifs : même dans des cas élémentaires, il serait trop fastidieux de la vérifier intégralement, et d’ailleurs

Poincaré donne des exemples sans justifier. Il affirme ainsi que pour B, P1 = 1 et P2 = 2, et

que pour K, P1 = 1 et P2 = 1, mais sans en donner de démonstration. Toutefois, ce sont à l’époque de Poincaré des exemples déjà connus, qui correspondent à l’idée intuitive que 10 Pk est le « nombre de trous à k dimensions » augmenté d’une unité . 24 Jusque‒là, le manque de rigueur de Poincaré s’est caractérisé par des définitions flottantes et par des preuves évasives voire absentes. Ceci semblait toutefois ne pas toucher à la justesse de ses idées, et présenter l’avantage de montrer comment les notions se définissent par tâtonnements et élargissements progressifs. Pourtant, Poincaré va commettre une véritable erreur mathématique qui sera relevée par Poul Heegaard, comme il l’explique en préface de son Complément à l’Analysis Situs [Poincaré 1899]. En effet, dans la neuvième section, il prétend démontrer un théorème de dualité,

à savoir que pour une variété fermée à n dimensions, Pn‒k = Pk. Or, Heegaard exhibe ce qui semble être un contre‒exemple tout à fait acceptable selon les critères de Poincaré. Celui‒ci reconnaît que sa démonstration était fausse, mais affirme que le contre‒ exemple de Heegaard n’en est un que si on interdit l’opération de division sur les homologies, ce qui donne lieu à deux notions distinctes de « nombres de Betti ». Ainsi, nous nous retrouvons avec deux notions qui semblent compatibles avec la définition originelle de Poincaré, mais qui ne coincident pas avec certaines variétés (comme celle exhibée par Heegaard). Or, ce qui est intéressant ici, c’est que Poincaré reconnaît que cette définition alternative ne correspond pas à la notion qu’il visait au départ, mais qu’il la maintient quand même comme étant légitime du fait qu’elle permet d’obtenir le théorème de dualité.

25 Notre interprétation est que confronté à un choix entre deux définitions distinctes, Poincaré a tout simplement choisi celle qui conduit à un théorème intéressant (car celui ‒ci fournit une symétrie dans les nombres de Betti), quand bien même celui ‒ci nécessite une distorsion de l’idée originelle. En ce sens, le manque de rigueur doit donc être pleinement assumé comme étant inscrit dans la pratique mathématique, puisque le flou dans les définitions employées permet justement de réajuster celles ‒ci en fonction de nouveaux objectifs qui peuvent apparaître en cours de route, comme celui de fournir un théorème important : en ce sens, ce seraient donc parfois les théorèmes qui justifieraient les définitions, et non l’inverse.

26 Ainsi, nous avons vu que l’Analysis Situs de Poincaré doit être lu plus comme une exposition de ses recherches avec une progression dans la généralité, que comme une exposition systématique. Or, cette démarche se justifie non seulement au point de vue pédagogique, mais également au niveau proprement mathématique : les concepts en question peuvent avoir plusieurs définitions possibles, et ce n’est qu’en cours de route qu’on trouvera des arguments nouveaux pour préférer l’une ou l’autre. Notre interprétation est donc qu’il y a dans la pratique de Poincaré une démarche volontaire de refus de certaines normes de rigueur, par exemple en évitant de donner des définitions précises, ou des preuves complètes, et que ceci se révèle être non pas une lubie mais une nécessité pour le développement des mathématiques. Mais pour que ce développement ne soit pas entravé, il est indispensable de distinguer soigneusement les cas de flou nécessaire et les cas d’erreurs mathématiques manifestes ; en somme il est indispensable de faire prévaloir une conception alternative de la rigueur.

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3 Le refus assumé de certaines normes de rigueur

27 La question de la rigueur est souvent abordée chez Poincaré, notamment dans [Poincaré 1898], [Poincaré 1902], [Poincaré 1904] et [Poincaré 1905]. Bien qu’il adopte une attitude très critique, il est néanmoins soucieux de critiquer les manques de rigueur des mathématiciens du passé, comme en 1904 : Longtemps les objets dont s’occupaient les mathématiciens étaient mal définis ; on croyait les connaître parce qu’on se les représentait avec les sens ou l’imagination, mais on en n’avait qu’une image grossière et non une idée précise sur laquelle le raisonnement pût avoir prise. [Poincaré 1904, 262]

28 Mais une fois ceci posé, Poincaré reste sceptique envers deux types de rigueur que nous allons caractériser : la rigueur de « l’analyste », qu’il juge mal adaptée à l’invention mathématique tout en lui reconnaissant une légitimité, et la « rigueur » du « logisticien », qu’il juge elle véritablement néfaste.

29 En 1902, Poincaré oppose deux types d’esprits mathématiques : « l’analyste » caractérisé par la rigueur de ses raisonnements et « le géomètre » caractérisé au contraire par son manque de rigueur et son appel à une intuition pensée au premier abord comme étant avant tout heuristique et ne donnant pas de certitude11 [Poincaré 1902]. Rappelons que Poincaré n’entend pas par « le géomètre » le mathématicien qui étudie la géométrie, ni par « l’analyste » celui qui étudie l’analyse, mais deux types distincts de méthodologie qui peuvent s’appliquer à toutes les branches des mathématiques, l’une étant plutôt attachée à la représentation géométrique, et l’autre à la rigueur analytique. Cette rigueur est caractérisée au premier abord comme voulant éliminer au maximum tout appel à l’intuition, mais comme nous le verrons par la suite, il y a bien selon Poincaré un type d’intuition qui joue un rôle dans la rigueur de l’analyste.

30 Pour préciser sa distinction, il compare « la conception géométrique de Riemann et la conception arithmétique de Weierstrass » en théorie des fonctions : Weierstrass ramène tout à la considération des séries et à leurs transformations analytiques ; pour mieux dire, il réduit l’Analyse à une sorte de prolongement de l’Arithmétique ; on peut parcourir tous ses Livres sans y trouver une figure. Riemann, au contraire, appelle tout de suite la Géométrie à son secours, chacune de ses conceptions est une image que nul ne peut oublier dès qu’il en a compris le sens. [Poincaré 1902, 117]

31 Mais les travaux de Riemann manqueraient de rigueur, et cette rigueur n’aurait été obtenue par Weierstrass « qu’au prix de modifications profondes et de détours compliqués » qui en dénaturent le sens géométrique et masquent l’origine des idées. Cette opposition entre la démarche de Riemann et celle de Weierstrass est assez classique à son époque, notamment dans le sillage de Félix Klein [Klein 1898]. Néanmoins, Poincaré se démarque explicitement des conceptions de Félix Klein qui, selon lui, fait reposer le succès la méthode géométrique sur une intuition de type physique, alors que ce type d’intuition « ne peut nous donner la rigueur, ni même la certitude » qui sont bien pour Poincaré des caractéristiques essentielles du raisonnement mathématique. Il est ainsi indispensable de souligner que même si Poincaré part des points de référence de son époque, il leur donne un sens nouveau. Ainsi, chez Poincaré, le clivage n’oppose pas une arithmétique ou une analyse rigoureuse mais non‒intuitive avec une géométrie moins rigoureuse mais plus intuitive et féconde. La distinction entre arithmétique et géométrie ne recoupe pas la distinction

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logique/sensible, car le raisonnement de type arithmétique pur n’est pas de la logique, et le raisonnement de type géométrique pur n’est pas empirique. Et pour cause, les mathématiques nécessitent un type de jugement qui ne soit ni empirique ni analytique, et qui repose donc sur une « intuition pure », point sur lequel Poincaré rejoint Kant. Toutefois, l’intuition chez Poincaré est très différente de chez Kant.

32 Pour clarifier le propos, il faut commencer par distinguer les intuitions de type empirique, qui fournissent des arguments heuristiques qui ne donnent pas de certitude, et que Poincaré appelle également « l’appel aux sens et à l’imagination » [Poincaré 1902]. Ceux ‒ci sont donc condamnables du point de vue de la rigueur. Mais de manière assumée, Poincaré revendique le fait qu’ils sont utiles, et donc que les normes de rigueur de l’analyste sont un peu trop strictes dans le cadre de la recherche mathématique. Nous avons ainsi vu que son Analysis Situs était bien à l’image de ce scepticisme. À ce propos, le point de vue de Jean Dieudonné dans [Dieudonné 1989] est intéressant pour nous car représentatif de celui d’un « analyste » au sens de Poincaré. Or, il porte un jugement très sévère sur la validité de ce mémoire, qu’il qualifie ainsi : « an unsuccessful attempi to give a genuine mathematica ! formulation to the intuitive ideas of Riemann and Betti ». À plusieurs reprises, Dieudonné se lamente des erreurs et des manques de rigueur qui le conduisent à ce jugement sévère. Bien sûr, il reconnaît la puissance et l’importance de ses idées, mais il n’y voit pas d’excuses pour un tel manque de précision. Ainsi, selon Dieudonné, dans l’Analysis Situs de 1895, « we should not look for precise définitions », et surtout « For many results, he simply gave no proof at ail » ou de temps en temps « an obscure and totally unconvincing argument ».

33 Une première lecture pourrait consister à dire que Poincaré reconnaît la nécessité de la rigueur, mais qu’il préfère laisser ce travail à d’autres. Et que le seul problème serait quand le manque de rigueur conduit à des erreurs mathématiques. Par exemple, Poincaré à partir de 1898 citera souvent comme véritable erreur commise par manque de rigueur l’ancienne croyance qu’une courbe continue admet nécessairement des tangentes en certains points12 [Poincaré 1898]. Selon lui, c’est ce contre ‒exemple qui motive l’arithmétisation de l’analyse, programme dont l’objectif est d’éliminer des raisonnements les appels à certaines intuitions.

34 Or, Poincaré affirme que même du point de vue de la rigueur la plus stricte, il faut l’usage d’un autre type d’intuition, qui est-elle purement mathématique, et qui serait même nécessaire pour obtenir la rigueur et la certitude. C’est donc bien une forme d’intuition totalement distincte de la précédente. L’exemple qu’il donne est « l’intuition du nombre pur » [Poincaré 1902], qui selon lui permet aux analystes « non seulement de démontrer, mais encore d’inventer ». La rigueur de l’analyste consisterait donc en réalité à réduire les mathématiques à l’intuition arithmétique pure. Ainsi, selon Poincaré, la validité du raisonnement par récurrence, qui permet de produire des connaissances arithmétiques, nous est donnée par l’intuition du nombre pur. Il n’y a donc pas à chercher à justifier ce raisonnement par des principes logiques. En affirmant cette position, principalement à partir de 1905 Poincaré s’oppose à ceux qu’il appelle les « logisticiens » et qui sont les logiciens comme Bertrand Russell ou Louis Couturat qui cherchent à réduire le nombre entier à une définition purement logique [Poincaré 1905].

35 Ainsi, Poincaré refuse la doctrine logisticienne comme étant celle qui essaie de chasser des mathématiques les intuitions pures, ce qui constitue déjà un dévoiement des objectifs de rigueur fixés par les analystes. Nous nous trouvons donc en présence de

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deux conceptions distinctes de la rigueur, chacune étant contestée par Poincaré : la conception des analystes, à laquelle il reproche de ne pas être adaptée à un travail de recherche (mais à laquelle il reconnaît le caractère rigoureux), et la conception des logisticiens qu’il juge véritablement erronée puisqu’elle élimine un type d’intuition dont les mathématiques ne peuvent pas se passer. Il suggère d’ailleurs que l’entreprise des logisticiens est vouée à l’échec même en suivant leurs propres critères. Poincaré s’amuse ainsi des cercles vicieux dans les définitions proposées des nombres entiers, bien qu’il précise que de toute façon, même si une définition logiquement satisfaisante finissait par être trouvée, cela ne changerait rien : Vous donnez du nombre une définition subtile ; puis une fois cette définition donnée, vous n’y pensez plus ; parce qu’en réalité, ce n’est pas elle qui vous a appris ce qu’est un nombre, vous le saviez depuis longtemps. [Poincaré 1905, 821]

36 Ainsi, selon notre interprétation de Poincaré, vouloir définir les nombres entiers, c’est déjà renoncer à une certaine rigueur mathématique qui impose d’accepter cette notion comme fondamentale13. La conception logisticienne de la rigueur est donc disqualifiée. Mais ne peut ‒on pas aller plus loin, et chercher chez Poincaré une conception de la rigueur qui améliorerait celle des analystes ?

4 Vers une conception alternative de la rigueur mathématique

37 En fait même en mettant de côté les considérations proprement heuristiques ou psychologiques qui doivent être disjointes de la rigueur elle ‒même, nous trouvons chez Poincaré des raisons précises conduisant à rejeter certaines normes trop strictes, au profit de l’usage assumé de certains types « d’intuitions ». Après les avoir présentées, il nous restera à proposer une interprétation liant ces raisons à ce que nous avons pu observer dans sa pratique des mathématiques.

38 Nous reprenons ici la thèse développée par Michael Detlefsen en 1992 [Detlefsen 1992]. Selon lui, Poincaré avance une conception de la « rigueur mathématique » différente de celle de la « rigueur logique ». Detlefsen distingue ainsi les inférences logiques (qui sont valides quel que soit le contexte) des inférences mathématiques (qui ne sont valides que dans un contexte mathématique précis). Il précise également : It is important to distinguish the notion of a proposition’s mathematically resting on another from that of a proposition logically resting on another. Propositions that belong to the logical basis of a proposition need not belong to its mathematical basis. [Detlefsen 1992, 367]

39 Pour expliquer cela, Detlefsen part d’un « principle of epistemic typification » qui sépare le domaine de nos connaissances en plusieurs types, dont le type mathématique et le type logique14. À cela il ajoute un « principle of epistemic conservation » qui dit que chaque domaine possède ses propres types d’inférences, et donc que pour le cas particulier des mathématiques, il faut utiliser des inférences de type mathématique. Poincaré combattrait donc la vision logicienne d’une preuve rigoureuse que Detlefsen caractérise ainsi : a rigorous proof is one in which all substantive (i.e., topic ‒specific) information has been driven out of the inference and into the axioms, thence to be explicitly registered in the premises of the proofs in which it is used. [Detlefsen 1992, 352] Au contraire, il défendrait une vision alternative de la rigueur selon laquelle :

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rigor will be achieved not by the elimination of logical or informational gaps separating premises from conclusions (hence, the elimination of substantive inference) but, rather, by the elimination of gaps in our mathematical understanding (and, hence, the elimination of inferences in which the premises do not constitute a good mathematica! reason for the conclusion). [Detlefsen 1992, 352]

40 Or, nous soutenons que cette compréhension de la marche d’un raisonnement mathématique ne peut se faire que par l’usage des intuitions. En effet, nous avons vu que selon Poincaré les mathématiques nécessitent un type de raisonnement qui donne à la fois certitude et fécondité, et sans lequel il n’est pas possible de produire de nouvelles mathématiques. Poincaré est en effet l’héritier d’une tradition kantienne selon laquelle la déduction logique, par définition, ne peut pas et ne doit pas produire d’information qui n’était pas déjà contenue dans les prémisses. Il faut donc expliquer comment peuvent se produire de nouvelles connaissances mathématiques. La logique se retrouve alors en position d’auxiliaire permettant certes d’organiser les théories mathématiques ou d’en contrôler certaines erreurs, mais perd son statut potentiel de producteur de connaissance mathématique. De plus, tout possible rôle fonda ‒tionnel de la logique est écarté : en effet, l’idée de Poincaré est que certains concepts mathématiques doivent être admis, et que la véritable rigueur nécessite de savoir les reconnaître et ne pas chercher à les démontrer (ni à tenter d’en démontrer la non‒ contradiction) sous peine de cercle vicieux. Par exemple, nous avons vu que le principe de récurrence était pour Poincaré indémontrable à partir des seuls principes logiques.

41 Or, si Poincaré insiste sur des exemples tirés de l’arithmétique (comme le principe de récurrence) pour montrer l’impossibilité d’atteindre la rigueur prônée par les logisticiens, ce ne sont pourtant pas les seuls cas d’intuitions pures qu’il nous donne. Il serait ainsi réducteur de limiter son propos à l’arithmétique : selon lui en géométrie aussi (et même plus précisément en topologie), le mathématicien fait nécessairement usage d’intuitions pures qu’il ne peut pas éliminer de ses raisonnements. Comme nous l’avons déjà rappelé en première partie, il affirme ainsi en 1912 : Et voici ce qui fait pour nous l’intérêt de cette Analysis Situs ; c’est que c’est là qu’intervient vraiment l’intuition géométrique. [Poincaré 1912, 484]

42 Et de plus, il ajoute que « nous avons tous en nous l’intuition du continu d’un nombre quelconque de dimensions ». Toutefois, il ne développera pas ces propos (il meurt la même année), mais selon notre interprétation c’est en fait l’usage de cette intuition géométrique pure qui distingue le « géomètre » de « l’analyste », et qui rend le raisonnement géométrique véritablement rigoureux. Certes, nous avons vu qu’en 1902, Poincaré pointait le manque de rigueur du géomètre, mais à cette époque il n’évoquait pas non plus d’intuition géométrique pure, comme s’il lui avait fallu quelques années pour réaliser la portée philosophique de ses travaux mathématiques en topologie. Selon nous, Poincaré a évolué sur ce point : il y a bien en réalité une rigueur proprement géométrique, distincte de la rigueur de l’analyste, et c’est celle à l’œuvre dans l’Analysis Situs. Le défaut de l’analyste serait ainsi de se limiter à la seule intuition du nombre pur, ce qui constitue bien un progrès par rapport à la méthode géométrique naive, mais qui n’est pourtant pas suffisante pour développer les mathématiques. C’est pourquoi Poincaré souligne en 1908 : les progrès de l’Arithmétique ont été plus lents que ceux de l’Algèbre et de l’Analyse, et il est aisé de comprendre pourquoi. Le sentiment de la continuité est un guide précieux qui fait défaut à l’arithméticien. [Poincaré 1908a, 934]

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43 Mais la méthode géométrique pure (ou méthode topologique) se caractérise‒t‒elle uniquement par le fait de se reposer sur l’intuition pure du Continu, de la même manière que la méthode analytique se reposait sur l’intuition pure du nombre, ou bien y a‒t‒il une différence plus profonde entre les deux méthodologies ? Nous avons vu que, pour Poincaré, la pratique des mathématiques, y compris rigoureuses, nécessite l’usage de certaines notions qui nous sont données directement par des intuitions pures qu’on ne peut pas remettre en doute. Poincaré distinguait bien ce type d’intuition de « l’appel aux sens et à l’imagination » [Poincaré 1902] auquel il reprochait son manque de rigueur, bien qu’il en soulignait l’utilité heuristique. Nous avons vu que cette distinction permettait de distinguer entre la rigueur de l’analyste (qui refuse tout appel aux intuitions sensibles) et la rigueur du logicien (qui refuse toute intuition).

44 Mais en fait, selon Poincaré, l’intuition ne sert pas uniquement à faire des généralisations heuristiques ou bien à garantir la certitude de ce que la logique ne peut pas fonder. Il y a également dans la pratique des mathématiques un usage d’un autre type d’intuition, qui correspond à la compréhension de la raison pour laquelle le mathématicien a fait tel ou tel choix. Il détaille ceci en 1908 [Poincaré 1908b], mais en avait déjà évoqué l’idée en 1902 [Poincaré 1902] : la pleine compréhension mathématique nécessite de prendre conscience du « je ne sais quoi qui fait l’unité de la démonstration ». Selon Poincaré, c’est la faculté de percevoir les ressorts cachés d’une démonstration formelle qui fait que certaines personnes sont plus douées pour les mathématiques que d’autres. Mais comment cette faculté se manifeste‒t‒elle ? Il parle de : « ce sentiment, cette intuition de l’ordre mathématique, qui nous fait deviner des harmonies et des relations cachées » nous permettant de comprendre le sens des démonstrations. Les « relations cachées » correspondent à des liens entre des théories mathématiques a priori distinctes. C’est par la mise au jour de ces liens que les mathématiques progressent. Or, ce nouveau type d’intuition correspondrait à « un sentiment esthétique de la beauté mathématique, de l’harmonie des nombres et des formes, de l’élégance géométrique ». Notre hypothèse est que c’est l’usage de ce sentiment esthétique qui caractérise la véritable rigueur géométrique, ne pouvant pas être atteinte par l’analyste se limitant à la seule intuition du nombre pur.

45 En effet, du point de vue de l’analyste, tout pourrait idéalement se réduire à des définitions et des axiomes (qui correspondraient à ce que Poincaré appelle « intuitions pures ») à partir desquels on pourrait déduire logiquement les théorèmes. Mais ceci ne suffit pas à expliquer les pratiques mathématiques que nous avons observées en première partie : il faut comprendre pourquoi on a choisi telle définition plutôt qu’une autre. Or, dans l’exemple que nous avons observé, la bonne définition n’est pas venue au début d’une théorie, mais à la fin. Ce serait donc ici que nous aurions besoin de ce type d’intuition, qui permet de comprendre où on veut aller, et de « faire des choix » comme l’explique Poincaré. Pour illustrer ses propos, nous avons les exemples topologiques étudiés en première partie : nous avons vu que pour obtenir certaines propriétés des variétés, il a fallu les définir de différentes manières. Nous avons également vu que pour obtenir le théorème de dualité, il avait fallu sacrifier une certaine notion des nombres de Betti qui était pourtant, de l’aveu de Poincaré, celle portée par Riemann, Betti et lui‒même au départ. Comment justifier qu’une notion soit sacrifiée au nom d’un théorème ? Il existe des critères justifiant ces choix implicites qui constituent le véritable cœur d’une théorie mathématique, bien plus que les démonstrations qui ne sont généralement que le déroulement des propriétés que le

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mathématicien a sélectionnées après coup, ce qui peut donner un caractère artificiel à son travail.

46 Pour désigner ces critères, nous pourrions parler de la recherche de théorèmes « beaux » ou « élégants », puisque c’est Poincaré lui‒même qui emploie cette terminologie. Mais encore faut‒il donner un sens mathématique précis à ces termes. Dans l’exemple du théorème de dualité, on peut considérer l’apparition d’une symétrie comme un trait de beauté ou d’élégance. Certes ce sentiment esthétique peut paraître subjectif, mais ce qui importe avant tout c’est de comprendre quel est le résultat auquel voulait arriver le mathématicien à l’origine de la théorie considérée. Ainsi de manière générale, il est possible qu’on touche là à une limite du langage mathématique qui nécessiterait justement de s’en remettre à l’intuition du mathématicien. Mais quoi qu’il en soit, il reste possible pour le lecteur de chercher à retrouver les raisons qui ont motivé son choix. Et ceci est non seulement possible, mais même nécessaire pour parvenir à la bonne compréhension d’une théorie. Or, nous faisons l’hypothèse que seule l’utilisation d’un langage géométrique (par exemple les symétries) permet d’exhiber cette « harmonie cachée ». Voici pourquoi la rigueur de l’analyste, quoi que valable, serait insuffisante pour prétendre rendre compte de ce qu’est la certitude mathématique.

47 Or, selon nous, c’est la topologie qui joue le rôle d’unification des mathématiques. En effet, cette conception des mathématiques semble avoir été profondément influencée par sa pratique de la topologie, à laquelle il est arrivé en explorant les bases de diverses théories mathématiques15, et qu’il a ensuite développée bien plus en se fiant à son sentiment esthétique qu’à des normes de rigueur strictes. Ainsi, la topologie fournit une nouvelle interprétation des autres disciplines mathématiques, leur donnant un sens nouveau permettant de les enrichir. Or, il faut bien préciser que dans cette perspective, la topologie ne serait pas une théorie servant à fonder les autres disciplines mathématiques. En effet, celles‒ci sont également mobilisées pour faire progresser la topologie, par exemple la théorie des groupes pour étudier les variétés : elles ne peuvent donc pas être logiquement postérieures à la topologie. La topologie serait donc le cœur des mathématiques, mais pas son fondement.

Conclusion

48 En conclusion, nous affirmons que la conception « poincaréenne » de la véritable rigueur mathématique cumule plusieurs aspects. Si Poincaré n’a pas explicitement proposé cette interprétation, du moins nous estimons que celle ‒ci permet de donner une vision unifiée et cohérente de ses différentes remarques et pratiques. Tout d’abord, il s’agit de faire la place à l’usage d’intuitions pures permettant la production de nouvelles connaissances mathématiques. Mais pour aller plus loin, pour comprendre pour quelles raisons des systèmes mathématiques a priori distincts se rejoignent et permettent la démonstration de nouveaux théorèmes, il faut employer l’intuition topologique pure. Cette intuition topologique se caractérise par le fait qu’elle pose comme acquises les propriétés fondamentales de la continuité, et également qu’elle est capable de relier entre eux des domaines mathématiques à priori séparés en dévoilant des relations considérées comme « harmoniques ». Or, c’est précisément en cherchant les conditions pour que des objets satisfassent ces relations que le mathématicien pourra adopter les bonnes définitions des notions fondamentales. Et c’est donc bien la

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compréhension profonde des motivations esthétiques présidant au choix des définitions qui permet au lecteur de saisir le véritable sens de la démonstration : non pas des définitions aux théorèmes, mais l’inverse. Et c’est pour pouvoir exprimer mathématiquement ces motivations esthétiques que nous avons besoin de développer le langage topologique. C’est la raison profonde pour laquelle la topologie se retrouve être au cœur des mathématiques.

BIBLIOGRAPHIE

BARROW‒GREENE, June [1997], Poincaré and the Three Body Problem, London: London Mathematical Society.

DARBOUX, Gaston [1913], Eloge historique d’Henri Poincaré, Paris : Gauthier ‒Villars.

DETLEFSEN, Michael [1992], Poincaré against the logicians, Synthèse, 90(3), 349 ‒378.

DIEUDONNÉ, Jean [1989], A History of Algebraic and Differential Topology 1900 ‒1960, Boston: Birkhâuser.

KLEIN, Félix [1898], Conférences sur les mathématiques, Paris : A. Hermann, trad. L. Laugel.

POINCARÉ, Henri [1895], Analysis Situs, Journal de l’École polytechnique, 1, 1 ‒123. — [1898], L’œuvre mathématique de Weierstrass, Acta Mathematica, 22, 1 ‒18. — [1899], Complément à l’Analysis Situs, Rend. Cire. Mat. Palermo, 13, 1 ‒121. — [1902], Du rôle de l’intuition et de la logique en mathématiques, dans Comptes rendus du Ile Congrès international des mathématiciens, édité par E. Duporcq, Paris : Gauthier ‒Villars, 115 ‒130. — [1903], L’espace et ses trois dimensions, Revue de métaphysique et de morale, 11, 281 ‒301. — [1904], Rapport sur les travaux de M. Hilbert, Bulletin de la société physico ‒mathématique de Kasan, 14, 10 ‒48. — [1905], Les mathématiques et la logique, Revue de métaphysique et de morale, 13, 815 ‒835. — [1906], Les mathématiques et la logique, suite, Revue de métaphysique et de morale, 14, 294 ‒317. — [1908a], L’avenir des mathématiques, Revue générale des sciences pures et appliquées, 10, 930 ‒939. — [1908b], L’invention mathématique, L’Enseignement mathématique, 10, 445 ‒459. — [1912], Pourquoi l’espace à trois dimensions, Revue de métaphysique et de morale, 20, 483 ‒504. — [1921], Analyse des travaux scientifiques de Henri Poincaré faite par lui ‒même, Acta Mathematica, 38, 3 ‒135.

NOTES

1. En 1906, il dit notamment : « Je ne vois au contraire dans la logistique que des entraves pour l'inventeur ; elle ne nous fait pas gagner en concision, loin de là, et s'il faut 27 équations pour établir que 1 est un nombre, combien en faudra-t-il pour démontrer un vrai théorème. Si nous distinguons, avec M. Whitehead, l'individu x, la classe dont le seul membre est x et qui s'appellera {x}, puis la classe dont le seul membre est la classe dont le seul membre est x et qui s'appellera {{x}}, croit-on que ces distinctions, si utiles qu'elles soient, vont beaucoup alléger notre allure ? » [Poincaré 1906, 295].

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2. « Si vous assistez à une partie d'échecs, il ne vous suffira pas, pour comprendre la partie, de savoir les règles de la marche des pièces. Cela vous permettrait seulement de reconnaître que chaque coup a été joué conformément à ces règles et cet avantage aurait vraiment bien peu de prix. C'est pourtant ce que ferait le lecteur d'un livre de mathématiques, s'il n'était que logicien. Comprendre la partie, c'est tout autre chose ; c'est savoir pourquoi le joueur avance telle pièce plutôt que telle autre qu'il aurait pu faire mouvoir sans violer les règles du jeu. C'est apercevoir la raison intime qui fait de cette série de coups successifs une sorte de tout organisé. À plus forte raison, cette faculté est-elle nécessaire au joueur lui-même, c'est-à-dire à l'inventeur » [Poincaré 1902, 125]. 3. Il précise : « Quant à moi, toutes les voies diverses où je m'étais engagé successivement me conduisaient à l’Analysis Situs. J'avais besoin des données de cette science pour poursuivre mes études sur les courbes définies par les équations différentielles et pour les étendre aux équations différentielles d'ordre supérieur et en particulier à celles du problème des trois corps. J'en avais besoin pour l'étude des fonctions non uniformes de 2 variables. J'en avais besoin pour l'étude des périodes des intégrales multiples et pour l'application de cette étude au développement de la fonction perturbatrice. Enfin, j'entrevoyais dans l’Analysis Situs un moyen d'aborder un problème important de la théorie des groupes, la recherche des groupes discrets ou des groupes finis contenus dans un groupe continu donné » [Poincaré 1921, 101]. 4. En 1903 il précise : « Les théorèmes de l’Analysis Situs ont donc ceci de particulier qu'ils resteraient vrais si les figures étaient copiées par un dessinateur malhabile qui altérerait grossièrement toutes les proportions et remplacerait les droites par des lignes plus ou moins sinueuses. En termes mathématiques, ils ne sont pas altérés par une « transformation ponctuelle » quelconque » [Poincaré 1903, 285]. 5. Cette idée est considérée comme l'acte de naissance de la topologie algébrique, discipline qui associe à chaque objet topologique un objet algébrique, généralement un groupe. Cependant, cette appellation est anachronique car pour Poincaré, la théorie des groupes ne fait pas partie de l'algèbre. Il porte en effet une conception des groupes différente de la nôtre, point que nous ne détaillerons pas ici, puisque nous allons uniquement considérer des passages traitant des nombres de Betti. 6. Ce type de définition est parfois appelé « analytique », mais nous préférons éviter ici ce terme, car il reviendra dans l'article avec des sens différents. 7. C'est-à-dire que l'image d'un point est unique, condition que nous considérons aujourd'hui comme déjà intégrée dans la définition d'une fonction. 8. Pour toutes les traductions en termes modernes, nous nous référons à Jean Dieudonné, History of Algebraic and Differential Topology [Dieudonné 1989], sauf une exception qui sera signalée. 9. Dieudonné traduit l'idée de Poincaré par espace « connexe », mais ceci nous semble incorrect. 10. Dans notre exemple, la boule creuse aurait un trou à deux dimensions, correspondant au prélèvement d'une boule intérieure, mais pas de trous à une dimension. 11. Nous pourrions donner comme exemple, non mentionné par Poincaré, le fait de compter les trous pour obtenir les nombres de Betti. 12. Nous pourrions cependant objecter à Poincaré qu'il n'y a pas ici d'erreur mathématique proprement dite, mais simplement une confusion entre deux notions distinctes de ce qu'est une courbe continue : celle définie à la manière de Weierstrass, et l'intuition originelle que nous en avions. De la même manière que la définition donnée par Poincaré des nombres de Betti ne correspondait pas à l'intention originelle. 13. Comme nous le verrons, il n'y a d'ailleurs pas que les nombres entiers qui doivent être admis pour Poincaré. 14. Il faudrait néanmoins faire attention au fait que pour Poincaré, le raisonnement logique se caractérise par ce qui n'apporte pas de nouvelle connaissance.

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15. Comme nous l'avions rappelé au début, il disait en 1901 : « Quant à moi, toutes les voies diverses où je m'étais engagé successivement me conduisaient à l’Analysis Situs. J'avais besoin des données de cette science pour poursuivre mes études sur les courbes définies par les équations différentielles et pour les étendre aux équations différentielles d'ordre supérieur et en particulier à celles du problème des trois corps. J'en avais besoin pour l'étude des fonctions non uniformes de 2 variables. J'en avais besoin pour l'étude des périodes des intégrales multiples et pour l'application de cette étude au développement de la fonction perturbatrice. Enfin j'entrevoyais dans l’Analysis Situs un moyen d'aborder un problème important de la théorie des groupes, la recherche des groupes discrets ou des groupes finis contenus dans un groupe continu donné » [Poincaré 1921, 101].

RÉSUMÉS

Henri Poincaré était réputé être un mathématicien hostile à la rigueur, aussi bien dans sa pratique mathématique que dans ses réflexions philosophiques. Or, des éléments indiquent que Poincaré se basait implicitement sur une conception personnelle de la rigueur mathématique, et qui correspondrait à sa pratique des mathématiques. Nous proposons donc de caractériser ce que serait cette conception. Tout d’abord, nous observons donc son rapport à la rigueur dans ses travaux en topologie, à partir d’exemples tirés du mémoire Analysis Situs de 1895 [Poincaré 1895]. Ensuite, nous étudions pour quelles raisons Poincaré revendique son opposition à certaines normes de rigueur. Enfin, nous développons une conception « poincaréenne » de la rigueur mathématique.

Henri Poincaré had a reputation for being a mathematician hostile to rigor, as much in his mathematical practice as in his philosophical thoughts. But some elements show that Poincaré had implicitly in mind a personal conception of mathematical rigor, that would fit with his mathematical practice. We propose to develop what this conception would be. First, we observe his standards of rigor in his topological work, with some examples taken from his 1895 paper Analysis Situs [Poincaré 1895]. Then, we study on what grounds Poincaré claims his opposition to some specific standards of rigor. Finally, we develop a “Poincarean” conception of mathematical rigor.

AUTEUR

RAMZI KEBAÏLI

Université Paris ‒ Diderot, UMR 7219 SPHERE (France)

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Foundation of Mathematics between Theory and Practice

Giorgio Venturi

1 A wonderful aspect of mathematical work is the possibility to create useful interactions between apparently different areas. This aspect, that we may call the unity of mathematics, is a distinctive aspect of modern mathematics. The tools and the ideas that come to light thanks to this global point of view are so powerful that they allow to overcome the Aristotelian caveat about the different genus, for example, between geometry and arithmetic. Moreover, the birth of modern mathematical logic and the need to keep together a very vast and disparate development of mathematics were among the reasons that allowed and pushed toward the foundational programs of the beginning of the last century. Nevertheless history frustrated these foundational efforts. Not only contradictions were discovered, but also a deep and unsolved tension between syntax and semantics: two very new branches of mathematical enquire. We can say that all foundational programs did not succeed in the sense they were conceived.

2 Nevertheless foundational enquires are still open and there are mathematical problems that have a foundational flavor. This situation calls for an explanation of what a foundation is and how it is possible to propose one nowadays. We think that among the many reasons that push for a foundation of mathematics, there is a goal that is common to every foundation, that is to shape the mathematical field. By this we mean that any kind of foundation, if it does not define, at least it distinguishes between mathematical and non—mathematical work and, in some way, characterizes mathematical practice, as being of a certain kind and obeying some specific rules. It is in this sense that we can find concerns for the unity of mathematics also in the foundational context and we believe that this is a common aspect of all different foundations of mathematics.

3 In this article we propose to look at set theory not only as a foundation of mathematics in a traditional sense, but as a foundation for mathematical practice. For this purpose, we distinguish between a standard, ontological, set theoretical foundation that aims to find a set theoretical surrogate to every mathematical object, and a practical one that

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tries to explain mathematical phenomena, giving necessary and sufficient conditions for the proof of mathematical propositions. We will present some examples of this use of set theoretical methods, in the context of mainstream mathematics, in terms of independence proofs, equiconsistency results. We will also discuss some recent results that show how it is possible to complete the structures Hand H(ℵ2). Moreover, in the central part of this article we will claim that a practical foundation of mathematics can be considered relevant not only for the practice of doing mathematics, but also from a truly philosophical perspective, showing its importance in the context of the philosophy of mathematical practice. This latter task will be done considering the explanatory role of some set theoretical axioms and discussing Kitcher's account on the matter of scientific explanation. In the end, we will propose a more general distinction between two different kinds of foundation: a practical one and a theoretical one, drawing some examples from the history of the foundations of mathematics.

1 Set theoretical foundation as unity

4 In order to explain this concept of unity we can see how it is realized in the context of the most common foundation of mathematics: set theory. This character of set theory has always been stressed by many people. We offer just one quotation for many, by Penelope Maddy: For all that, set theoretic foundations still play a strong unifying role: vague structures are made more precise, old theorems are given new proofs and unified with other theorems that previously seemed quite distinct, similar hypotheses are traced at the basis of disparate mathematical fields, existence questions are given explicit meaning, unprovable conjectures can be identified, new hypotheses can settle old open questions, and so on. That set theory plays this role is central to modern mathematics, that it is able to play this role is perhaps the most remarkable outcome of the search for foundations. [Maddy 1997, 34—35]

5 However, instead of describing the almost 'standard' set theoretical foundation following which every mathematical entity is intended to be a set, we propose to look at set theory as a means to give a foundation to mathematical practice.1 Indeed, the universality character of set theoretical language—i.e., the possibility to formalize any piece of mathematics inside set theory and to find a set theoretic surrogate for any mathematical object—has not a priori any ontological meaning. Set theory is not to be intended here as only ZFC, as it is often the case when set theory is called upon arguing for a standard foundation, but as a general method that makes use of set theoretical principles to analyze mathematical practice. As part of this method we include also reverse mathematics and all the useful set theoretical assumptions, sometimes called axioms, that extend ZFC.2 Clearly the term “theory” here is an abuse of language from a logical point of view, because, neither we think of a consistent set of sentences, nor of an intuitive theory with its intended interpretation. What we have in mind is a general method that is widely and sometimes tacitly used in mathematical practice.

1.1 Hilbertian origins

6 There are two aspects of this set theoretical point of view that we propose that model the corresponding foundation of mathematics. They explain in which sense set theory meets the requirements of unity of mathematics. These ideas can be traced back to

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Hilbert's foundational works. They follow the evolution of Hilbert's thought on foundational issues, belonging to two different periods, also chronologically distant. Hence they do not characterize his point of view on this subject.

7 The first one pertains to Hilbert's period of foundations of Geometry. In his mind the question of why a theorem is true was equivalent to the problem of elucidating the main possibility of a proof. I understand under the axiomatical exploration of a mathematical truth [or theorem] an investigation which does not aim at finding new or more general theorems being connected with this truth, but to determine the position of this theorem within the system of known truths in such a way that it can be clearly said which conditions are necessary and sufficient for giving a foundation of this truth. [Hilbert 1902—1903, 50, my italics]

8 Of course, despite Hilbert's ideas, history then showed that metamathematics can give rise to new truly mathematical results and it is a powerful method not only to determine general properties of the axiomatic setting of a formal theory. What is important to stress here is that this attitude is an attempt to give an answer to possible 'why questions' that can rise in the mathematical discourse. Indeed this is exactly what Hilbert was hoping to do in his foundation of geometry. In a letter to Frege, dated December 29th, 1899 Hilbert wrote: I wanted to make possible to understand and answer such questions as why the sum of the angles in a triangle is equal to two right angles and how this fact is connected with the parallel axiom. [Prege 1980, 38—39, italics mine]

9 The second idea that we would like to recover from Hilbert is the conviction that a good axiomatization of mathematics should be a catalog of the principles that we use in our mathematical practice. The fundamental idea of my proof theory is none other than to describe the activity of our understanding, to make a protocol of the rules according to which our thinking actually proceeds. [Hilbert 1927, 475], in [Van Heijenoort 1967]

10 In Hilbert's program, this belief was related to the expectation that few arithmetical and logical axioms were able to characterize every piece of mathematics. Since this has been shown to be impossible, we accept this suggestion to be compatible with an open —ended list. Indeed this idea of a catalog of principles could a priori involve also incompatible principles. We are not looking for a categoricity theorem that permits to define what a theory is about, but a theory that can explain our mathematical work, showing its uniformity of methods and arguments, in order to account for its unity.

11 We think that these two ideas are also able to account for the explanation of a mathematical fact, outlining the main conditions of its proof and pointing at the reasons for accepting its truth.3 Since we are dealing with a demonstrative context, what is often essential for overcoming the difficulty of an argument is a combinatorial aspect of the proof, that reveals the key ingredient for the solution of a problem. This is the reason why many set theoretical principles have a combinatorial character, but this does not prevent us from listing them in the catalog, as long as they contribute to account for the unity of mathematics—i.e., they are not ad hoc and they have many and different applications. What is relevant in showing that some principles are necessary and/or sufficient is their role in the argumentative structure of a theorem. Sometimes these principles go hand in hand with a more general understanding of a whole field.

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12 The general idea behind this conception of set theory is that it is a method that can be applied to all other branches of mathematics. Indeed this is how it was conceived, at least by Zermelo, in the thirties. Our axiom system is non—categorical after all, which, in this case, is not a disadvantage, but an advantage. For the enormous significance and unlimited applicability of set theory rests precisely on this fact. [Zermelo 2010, 427]

13 A good conceptual reason for arguing in favor of set theory as an open—ended foundation lays in the fact that—like mathematics, as we will argue in the next section —the subject matter of set theory is not sufficiently clear to immediately characterize a model and isolate its axioms. This is one of the main concern in contemporary research in set theory, but what is important to stress is the distinction between a foundational role of set theory and the research that tries to single out one true model for ZFC, out of the many we can conceive. This is not an easy task, because if we have a good intuition, for example in the case of the real line, of what are the pathological aspects we would like to avoid, like the Banach—Tarski paradox and the consequent non— measurability of some sets of reals, this is much more difficult as soon as we proceed in the hierarchy of the transfinite. Far from being a weakness, this hazy boundary is what allows set theory to account for the unity of mathematics. However, this aspect of vagueness, common to both set theory and mathematics, has been criticized and has always been subject of discussions, in the foundational context, where, exactly, to draw the dividing line between mathematics and non mathematics. There are people like Feferman and Weaver who would like to put the crossbar much lower than the level of ZFC.4 However if we are trying to explain the unity of mathematics, and therefore we are working at a foundational level, we cannot drop so easily set theory. Indeed either we discredit modern research in set theory as being mathematics, or we have to propose a sufficiently wide framework, where it is possible to place it. In a slogan: there are more things in mathematical research and mathematical practice than are dreamt of in ZFC. What we propose here is to consider the methods offered by set theory as a framework for mathematics, part of which is of course set theory.

1.2 Practical reasons

14 We now plan to show why set theory can offer a foundation for the practice of doing mathematics. Before we start we need a definition that is fundamental in what follows.

15 Definition 1.1. We say that a theory T, that extends ZFC, has consistency strength stronger than a theory S if in first order Peano arithmetic it is possible to prove Con(T) → Con(S), where Con(T) is the sentence expressing the consistency of T. Moreover, for a sentence A written in the language of set theory, we refer to Con(A) as an abbreviation for Con(ZFC + A).

16 There are three reasons that support the idea of a set theoretical foundation of mathematical practice.

17 Independence proofs. This is the main subject of modern research in set theory. Since the invention of forcing,5 in the sixties, many problems were shown to be independent from ZFC, like for example the Continuum Hypothesis (CH) and Souslin's Hypothesis. This kind of proofs is used, as Hilbert did, to prove that a set of axioms is not sufficient for a mathematical result.

18 Combinatorial principles. The discovery of the independence of a proposition does not conclude its mathematical analysis. Indeed the examination of an independent problem

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often brings together the identification of a technical impasse and the corresponding combinatorial principles that are sufficient for its solution. For a safe use of these principles, the method of forcing is used to show that they are consistent relative to some theory like, for example ZFC. But sometimes ZFC is not sufficient for this task. It is here that large cardinals come into play.

19 Large cardinals. These are hypotheses on the existence of cardinals large enough6 to prove6 Con(ZFC). They are used to determine the power of sentences stronger than ZFC, in terms of their consistency strength. Indeed many natural sentences stronger than ZFC can be proved to be equiconsistent—in the context of ZFC—with the existence of suitable large cardinals. Then large cardinals can be viewed, modulo equiconsistency, as necessary and sufficient conditions for the proof of sentences stronger than ZFC.

20 There is an important reason for using large cardinals as the backbone for the analysis of the propositions that transcend the deductive power of ZFC. Empirical fact: the order induced by the consistency strength of large cardinal hypothesis is, except in few cases, linear and well founded. The use of large cardinals in set theory is twofold: on the one hand they serve to compare different principles, using equiconsistency results and the linear order given by their consistency strength; on the other hand they supply the means to give relative consistency proofs, being the key ingredient of theorems of the form: given an independent statement A, written in the language of set theory, if the hypothesis I stating the existence of a large cardinal holds, then there is a model where A holds; in other words I implies Con(A).

21 There exists an epistemological tension between logical deduction and consistency strength and this aspect is responsible for the richness of the analysis that set theory can offer to necessary and sufficient conditions. Indeed the epistemological value of the search for necessary and sufficient conditions for the proof of a theorem consists in the discovery of its place in the logical structure of a theory. This process can work in two directions: starting from an axiomatic system and asking which of its axioms are needed for the proof of a theorem, or starting with a proposition and looking for the axioms that are needed for its (non—trivial) proof, without specifying the axiomatic context. In the first case this analysis is informative on the content of a theorem, like for example Hilbert's work on Desargue's theorem—where the aim is to clear its spatial content. But in the second case, when the goal is a context—free analysis that looks for the principles that are needed for the proof of a proposition—for example a proposition independent from ZFC—the discovery of necessary and sufficient conditions consists just in finding logically equivalent formulation of the proposition. In this latter case the progress in our knowledge may be given by a combinatorial character of an equivalent formulation, or its relevance in a different field, but it is not informative for what concern the possibility of its proof, nor for its content—i.e., we cannot give an answer to the question “Why this proposition is a theorem of set theory?” On the contrary, a result of equiconsistency is well more informative on the epistemological status of a proposition. Indeed, such a proof outlines the fact that we have to believe not only in the truth of a sentence, but also in the existence of a particular class of models of ZFC: the ones whose existence is guaranteed by the equiconsistency proof. Moreover, logical equivalence and equiconsistency cannot be assimilated, without collapsing truth and existence. While the former is a syntactical notion, the latter is semantical and expresses the fact that we need to believe in something, possibly, stronger than ZFC in

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order to believe the truth of a particular sentence. This is the reason why the use of large cardinals in the consistency proofs fills the gap between logical deduction and consistency strength, because not only we can have theorem of the form I → A, but also I → Con (A), i.e., we can show that the sentence expressing the existence of a large cardinal, logically implies not only another sentence, but also the fact that there is a model where this is true.

22 Hence, large cardinals provide a precise answer to “why” questions. In this sense, large cardinals can be seen as more fundamental principles that give more and more powerful means, not only to prove new propositions, but also to analyze our believes in their truth.7 After the discovery of the difference between, truth, provability and existence, we have to accept the original Sin of Gödel's theorem, but what large cardinals—together with the method of forcing—offer is a way to analyze and stratify the degree of incompleteness that we find in our mathematical practice.

1.3 Example of sufficient conditions

23 We would like here to present some examples that show how set theory is used to analyze mathematical problems. It is important here to stress not only the fine and deep explanation that is given by a set theoretical investigation, but also the fact that the problems discussed come from some of the characteristic fields of classical mathematics: group theory and functional analysis. This aspect is important since it acknowledges the importance of set theoretical method not only in logical or pathological context, but in classical domains of mathematical practice.

24 We shall describe the solutions given to Whitehead's problem and a recent result by Farah on operators algebras of an Hilbert space.

25 Definition 1.2. (Whitehead's problem (WP)) Is every Whitehead group (i.e., an abelian group A such that, whenever B is an abelian group and f : B → A is a surjective group homomorphism, whose kernel is isomorphic to the group of integers ℤ then there exists a group homomorphism

g : A → B with fg = idA) a free group (i.e., a group A that has a subset X, called the set of generators, such that every element of A can be written uniquely8 as afinite combination of elements in X and their inverses)? In the seventies Shelah proved the following theorems. Theorem 1.3. ([Shelah 1980]) If V = L9, then the answer to WP is yes. Theorem 1.4. ([Shelah 1977]) If Martin's Axiom, (MA)10 and the negation, of the Continuum, Hypothesis (¬CH) both, hold, then, the answer to WP is no.

26 We then have another proof of the fact that V = L and ¬CH are incompatible. Moreover, since Con(ZFC + V=L) ⇔ Con(ZFC) ⇔ Con(ZFC + MA + ¬CH) we have sufficient conditions for both answers to WP, without exceeding the consistency strength of ZFC; that is, without an overshooting that would confuse the problem. Another example is the following result in the context of functional analysis.

27 Definition 1.5. The Calkin algebra C(H) is the quotient of B(H), the ring of bounded linear operators on a separable infinite—dimensional Hilbert space H, by the ideal K(H) of compact operators.

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28 It is a natural question to ask if every automorphism is inner; i.e., if it is induced by the operation of conjugation. The answer to this question is again sensible to the background set theoretical hypothesis. Theorem 1.6. ([Phillips & Weaver 2007]) If CH holds there is an automorphism of C(H) that is not inner. Theorem 1.7. ([Farah 2010]) If the Open Coloring Axiom (OCA)11 holds all holds all automorphism of C(H) are inner.

29 It is interesting to note that, in this case, the first version of Farah's theorem used the Proper Forcing Axiom (PFA), whose consistency strength is much higher than Con(ZFC). Then, the analysis of why PFA was used in the proof led to the discovery that just OCA, that is a combinatorial consequence of PFA, was needed. Then since Con(ZFC + OCA) ⇔ Con(ZFC) ⇔ Con(ZFC + CH) we have found again sufficient conditions for the solution of a natural mathematical problem; the best possible solution with respect to consistency strength.

1.4 Examples of equivalence and equiconsistent results

30 Of course there are also examples of necessary and su‑cient conditions for su‑ciently natural mathematical problems. They indeed show equivalences between different principles that can be epistemically informative for their combinatorial content, or just useful for nding new and unexpected links between dierent areas of mathematics. Theorem 1.8. ([Fretting 1986]) The following are equivalent, over ZFC: ℵ 1. Continuum Hypothesis: 2 ° = ℵ1, 2. Axiom, of Symmetry: for any function, f that associates countable sets of real numbers to real

numbers, i.e., f : ℝ → [ℝ]ℵ° , there are x0,x1 ∈ ℝ such, that x0 ∉ f(x1) and x1 ∉ f(x0).

31 Nevertheless, the strength of the set theoretic method can be mostly appreciated in combination with large cardinals and so when necessary and sufficient conditions are such, up to equiconsistency. The best example is Solovay's model for the following very natural property for sets of reals, that started the study of descriptive set theory.

32 Definition 1.9. (LM, BP and PSP) Given X ⊆ ℝ, we say that X is Lebesgue measurable (LM) if it belongs to the σ—algebra generated by the Lebesgue measure on ℝ. We say that X had the Property of Baire (BP), if there is an open set U such that U Δ X (the symmetric difference) is a meager set (i.e., small). We say that X has the Property of the perfect set (PSP), if it is either countable or has a nonempty perfect subset: a closed set with no isolated point. Theorem 1.10. ([Solovay 1970]) The following are equivalent, over ZFC: • Con(ZF+ all sets of reals are LM and have BP and PS), • Con(There exists an inaccessible12 cardinal ).

33 The epistemological meaning of this theorem is that it explains what we need to believe in terms of consistency to accept that all subset of the reals behave very nicely with respect to some natural properties.

1.5 Necessary and sufficient conditions

34 We would like here to present a new point of view on the application of the forcing method to the general phenomenon of independence.13 They are part of a more general program that helps in making more precise the methodology suggested in Gödel's

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program and that is now called Woodin's program.14 This program aims at finding a satisfactory description of the universe of set theory step by step; that is, giving a sufficiently complete description of initial segments of the class V = {x : x = x}. We need a definition in order to make precise the “step by step” methodology of this program.

35 Definition 1.11. If x is a set we define tc(x) as the transitive closure of x: the minimal set under inclusions that contains x and that is transitive, i.e. if y ∈ tc(x) then y ⊆ tc(x).

36 Definition 1.12. (A cumulative hierarchy) We can build V stage by stage in the following way: for every λ ∈ Card the structure H(λ) consists of the sets of cardinality hereditarily less than λ. H(λ) = {x : |x| < λ and ∀y (y ∈ tc(x) ⇒ |y| < λ)}. Then Woodin's way to phrase his program is the following:

One attempts to understand in turn the structures H(ℵ0), H(ℵ1) and then H(ℵ2). A little more precisely, one seeks to find the relevant axioms for these structures. Since the 15 Continumm Hypothesis concerns the structure of H(ℵ2), any reasonably complete

collection of axioms for H(ℵ2) will resolve the Continumm Hypothesis. [Woodin 2001, 569]

37 Notice that one of the main motivation is the solution of CH. Now we need a definition in order to make precise the sense in which the program attempts to complete the initial segment of the universe of set theory, ruling out the trivial incompleteness phenomena given by Gödel's sentences and the sentences expressing the consistency of a theory.

38 Definition 1.13. ψ is called a solution of a structure M, that models enough of ZFC, iff for every sentence ϕ ∈ Th(M), ZFC + ψ ⊢ M ⊨ϕ or ZFC + ψ ⊢ M ⊨¬ϕ .

39 We now present⌜ some⌝ initial results⌜ of Woodin's⌝ program and some other by Viale, that show how the forcing axioms fit in this program. The importance of these results is to be found in the possibility of using forcing not only to give sufficient conditions, but also necessary. Indeed the slogan that motivates them is the following. Key idea: The method of forcing is a tool that allow to prove theorems over certain natural16 theories T which extend ZFC.

40 The first result of this kind is a reformulation of Cohen's forcing theorem, that shows

how any transitive model of ZFC overlaps with the Σ1 theory of H(ℵ1).

Theorem 1.14. ([Cohen 1963]) Assume T extends ZFC. Then for every Σ0-formula φ(x,p) and every parameter p such that T ⊢ p ⊂ ω the following are equivalent:

• T ⊢ H(ℵ1) ⊨ ⴺ xφ(x,p)⌝

• T ⊢⌜ There is a partial order P such that ⊨ P ⴺ xφ(x,p). 41 Thanks to large cardinals the above theorem can be extended to all formulas, with

parameters in H(ℵ1). Next theorem says that it is possible to find a solution of the theory of L(ℝ) (i.e., the class of all set that are constructible with real parameters), but L(ℝ) we have that H(ℵ1) ⊆ L(ℝ) and H(ℵ1) = H(ℵ1). So next result is really an extension of the previous one.

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Theorem 1.15. (Woodin [Larson 2004]) Assume T extends ZFC + There are class many Woodin cardinals.17 Then for every formula φ(p) and every parameter p such that T ⊢ p ⊆ ω the following are equivalent: • T ⊢ L(ℝ) ⊨ φ(p)⌝ L(ℝ) • T ⊢ ⌜There is a partial order P such that ⊨ P φ (p).

42 So, modulo the method of forcing, large cardinals are a solution for the structure H(ℵ1);

i.e., they decide the theory of H(ℵ1) with parameters in H(ℵ1). Indeed the fact of interpreting every result that we present in this section as a good solution for the corresponding theory depends heavily on the assumption that the forcing is the only way to obtain new models for set theory. This is true as far as other known model— theoretic methods are concerned, but there is no proof—and I doubt that there will ever be—of this fact. However, this is a theoretical innocuous assumption, that goes hand in hand with the general pragmatism of the methodology that is used in every mathematical research.

43 Moreover, it is possible to extend this result to a full solution of H(ℵ2), thanks to the axiom: MM+++: the strongest version of a specific type of axioms called Forcing Axioms. Theorem 1.16. ([Viale 2013]) Assume T extends ZFC + MM+++ + 18 There are class many super huge cardinals. Then for every formula φ(x) in the free variable x and every parameter p such

that T ⊢ p ∈ H(ℵ2) the following are equivalent:

• T ⊢ H(ℵ2) ⊨ φ(p)⌝ H(ℵ ) • T ⊢⌜ There is a stationary set preserving partial order P such that ⊨ P φ ² (p). (p) and P preserves MM+++.

44 Hence, thanks to the Forcing Axioms, we have a good description of the structure H(ℵ2) and, in keeping with the general idea that lies behind and foundation of mathematics— as we saw in the example of the Whitehead Problem and of the Calkin Algebra—they are capable of unifying different branches of mathematics. Indeed both MA and OCA are consequences of MM+++.

45 It is now time to come back to the main aspect of a set theoretical foundation of mathematics: unity. Indeed, we want to give more philosophical arguments to sustain the claim that set theory should be able to unify mathematics, as far as it is a foundational theory. In doing so we will also elaborate on our claim that set theory should be considered as a foundation for mathematical practice.

2 Axioms as explanations

46 So far we showed the relevance and usefulness of set theory for what concerns the practice of doing mathematics, but our claim is stronger and refers to the philosophy of mathematical practice: a recent tradition in the philosophy of mathematics, as a quick look at contemporary bibliography clearly shows.19 Our main thesis of this section is that many of the principles that are used in contemporary set theory, many of which are called axioms, manifest specific characteristics that can be assimilated to, at least, one important account of mathematical explanation—one of the more studied and developed area of the philosophy of mathematical practice. Hence, our derived claim is that some set theoretical axioms can be seen as an explanation of the mathematical phenomena.

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47 The aim of our analysis will be to link unification and explanation in a foundational context. Many authors have proposed a philosophical inquire on the notion of explanation in science in terms of unification, but few of them have proposed it as a way to understand the notion of mathematical explanation, i.e., the role of mathematics in the scientific explanation. Among them Kitcher's account, presented in terms of the unification power of scientific theories, can be compared with the unifying features that some axioms in set theory have. We will not present here his position in details, but we refer to the primary bibliography ([Kitcher 1981], [Kitcher 1989] and [Kitcher 1984]) and to Molinini's Phd thesis ([Molinini 2011]), for a good presentation of the subject, able to clear many of the obscure passages that can be found in Kitcher's work.

48 In trying to find some explanatory aspects in the concept of axioms in set theory, we connect our arguments with a long tradition in the philosophy of mathematics and in the foundational studies, that has its roots in the empiricist positions of John Stuart Mill. Paolo Mancosu, in [Mancosu 2001], has named this tradition h—inductivism: the position that sees in the success of an hypothesis of an axiom, and in its ability to give a systematization of a discipline, the main justification for its acceptance. On the same par, we can find also Russell and Gödel. In particular, the latter believed in a direct connection between the role of the axioms of set theory and their explicative power, in analogy with the process of explanation of a physical phenomenon.20

49 However, we want to be clear that we do not endorse the thesis that axioms can be seen as explanation to argue in favor of a realist conception of mathematical object. As a matter of fact we argue for a foundation of mathematics free from any ontological commitment. Instead, in what follows we will try to give arguments in favor of the thesis that, in the context of the axiomatic setting, explanation of a proof and justification of an axiom are two sides of the same coin: unification. Once this point is achieved, then, there will be no reasons for justifying the axioms in terms of an existing mathematical reality.

2.1 Applying Kitcker's account?

50 What interests us here is the role of mathematical explanation, if there is any, inside mathematics. Indeed mathematical explanation can mean both the use of mathematics in explaining physical phenomena and the use of explanatory considerations in the context of pure mathematics. Prom now on by mathematical explanation we will mean the latter: mathematical explanation of mathematical phenomena.

51 For the sake of precision, there is no precise account of mathematical explanation in any of the writings of Kitcher, but, instead, of scientific explanation of physical phenomena. Nevertheless the possibility to export this model from physic to—pure— mathematics is proposed by Kitcher himself,21 in the light of his holistic point of view on scientific knowledge. [G]iven my own views on the nature of mathematics, mathematical knowledge is similar to other parts of scientific knowledge, and there is no basis for a methodological division between mathematics and natural sciences. [Kitcher 1989, 423]

52 We will see below how weak this thesis is, but what we want to save from Kitcher's way to set the problem is the global point of view on the problem of mathematical

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explanation—i.e., to consider how a general theory can explain some of the phenomena it is able to formalize or deduce—as opposed to a local point of view that tries to look for the explanatory characters of a proof. Indeed, this latter task is much more complicated and there are hints that it is not possible to give a detailed and objective account of why a proof counts for more explanatory than others. A seminal, but isolated, case of such a work can be found in Paseau's study of the different proofs of the compactness theorem [Paseau 2010]. The main thesis of Paseau, with whom we agree, is that the explanatory virtue of a proof always depends on the context and so it is hard, if not impossible, to give an objective account of it. As we will see, one of the points we will make in this section is that also a global explanation depends on the background theory in a substantial way, contrary to a long tradition that dates back to Aristotle's times and that numbers, among his members, also Bolzano.22 Leaving aside these difficulties, we want to stress how a global account of explanation can in principle fit with a global mathematical point of view, as it is the case when dealing with a foundation of mathematics.

53 Once we are assured that the general setting of the two problems is compatible, let us see more in details Kitcher's account and his affinities with our proposed set theoretical practical foundation for mathematics. The more important conceptual similarity is, of course, the individuation of unity as the main virtue of both a practical foundation and a global account of mathematical explanation. Indeed, when discussing Hempel's model of explanation, Kitcher describes his position in these terms: This unofficial view, that regards explanation as unification, is, I think, more promising than the official view. My aim in this paper is to develop the view and present its virtues. [Kitcher 1981, 508]

54 This is done at least at two levels: making clear the aim of an explanation and clearing the nature of an explanation. For what concerns the former, Kitcher is explicit in saying that an explanation is an answer to a why question. Exactly how, echoing Hilbert's foundation of geometry, our practical foundation aims to do. I shall restrict my attention to explanation—seeking why—questions, and I shall attempt to determine the conditions under which an argument whose conclusion is S can be used to answer the question “Why is it the case that S”?23 [Kitcher 1981, 510]

55 Notice that this declaration of intention is not as narrow as Kitcher seems to argue. Indeed, when we get to mathematical explanation, the question “Why is it the case that S”? can be interpreted in different ways according to the context in which the question is asked. For example, we can take S to be Fermat's Last Theorem (FLT), but if we ask: “why is it the case that FLT” in a context where it is possible to understand and state the question, like that of number theory, we cannot even formulate a possible answer because we do not have an elementary proof of it. Contrariwise, this is a reasonable question in the context of a theory sufficiently strong to incorporate scheme theory and algebraic geometry. This example is meant to show that in the context of pure mathematics the methods of proof do have a role in the determination of an answer to a why—question. Moreover, what we suggest is that, in the particular case of mathematical explanation, an answer to a why—question can hide many different problems, like for examples considerations on the purity of methods, that bring together some controversial and less objective positions towards the nature of mathematical discourse.

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56 For what concerns the second aspect Kitcher is explicit in saying that an argument is a derivation24 and—this is the main thesis contained in [Kitcher 1981] and [Kitcher 1989] —that, given a set of sentences K, there is an explanatory store E(K), that consists in the best systematization—read formalization in the context of the axiomatic method—of K. What makes E(K) the best systematization is the possibility to associate to it a set A of arguments— called a basis—that instantiate general arguments patterns and that, better then other systematizations, unify K, in the following sense: So the criterion of unification I shall try to articulate will be based on the idea that E(K) is a set of derivations that makes the best tradeoff between minimizing the number of patterns of derivation employed and maximizing the number of conclusions generated. [Kitcher 1989, 432]

57 It is not clear, in Kitcher's work, how this minimizing—maximizing effect should act in the process of choosing an argument instead of another.25 We maintain that Kitcher is appealing here to an intuitive principle of success that E(K)

58 What is important to stress here is that, for Kitcher, what is fundamental in the analysis of explanatory unification is the notion of argument pattern. Kitcher offers a description of these arguments as detailed as vague. What is important to keep from Kitcher's idea of argument pattern is that it is a general structure of an argument—not only a logical structure—sufficiently general to be applied in many different contexts and in many different forms: it is what permits to recognize that different proofs are essentially the same. We will not give a detailed presentation of the notion of argument pattern as it can be found in [Kitcher 1981] mostly because of the lack in Kitcher's work of a clear analysis of the notion of similarity. This suggests that our conditions on unifying power should be modified, so that, instead of merely counting the number of different patterns in a basis,26 we pay attention to similarities among them. All the patterns in a basis may contain a common core pattern, that is, each of them may contain some pattern as a subpattern. [Kitcher 1981, 521] Although Kitcher tries to clear what is an argument pattern, it is better to keep this notion as intuitive, although vague, as possible.27 Then at this level of generality we could ask: which are the similarities between some axioms of set theory—especially those exceeding ZF, that we discussed in the last section— with respect to the argument patterns, as far as both are responsible for the unity of the theory? To answer this question we can recall Zermelo's idea about the “unlimited applicability of set theory”. In order to make the argument more concrete, recall the problem whether all the automorphisms of the Calkin algebra are inner. The way in which the proof works is by finding enough similarities between this algebra of operators and the structure . Then, it is possible to use the axiom OCA to perform the same argument on both sides. It is a general methodology when facing a mathematical problem and looking for its solution. Indeed, this is one of the main advantages in relaying on set theory as a foundation of mathematics: the concept of set and the methods used in set theory are so general and abstract that they can be applied—possibly—to any field of mathematical inquiry. Hence, the use of the axioms for set theory permits to show the similarity, in the arguments, of many different mathematical reasonings. When attacking a problem, the first attempt of a mathematician is to bring the difficulties to a more clean and comprehensible level, where a solution is easier to find. This operation, of cleaning a problem from the irrelevant aspect, amounts in recognizing similar patterns or making more evident the core of the problem that often has— as the

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methodology of set theory shows, in abstracting from any content—a combinatorial aspect. Here by combinatorial aspects of a proof we do not only mean the part of an argument that is performed by pure calculation without a broader overview of the structure of a proof, but also the steps of an argument that manifest a necessity character similar, for strength, to calculation, and that act like the fundamental ingredient of a theorem. In a set theoretical context, the combinatorial aspects of a proof are often found when abstracting from the particular properties of the subject matter of a theorem and when outlining the general set theoretical properties that make possible to perform an argument. Indeed the combinatorial character of some axioms, or some principles that flow from them, is capable of showing in a pure form what is needed for the proof of a sentence: they show how to overcome the main difficulty one finds in a problem, acting as the key ingredient for its solution. Indeed, this method works also in the opposite direction, from solutions to axioms and sometimes brings together the discovery of new, tacitly used, principles as it is the case of the Axiom of Choice. Even one of the more influential proponent of the more outstanding alternative foundation of mathematics: category theory, acknowledges the ability of ZFC to reduce many arguments to few ones. The rich multiplicity of mathematical objects and the proofs of theorems about them can be set out formally with absolute precision on a remarkably parsimonious base. [Mac Lane 1986, 358]

59 Moreover, whenever these principles are proved to be independent from ZFC, even if we do not have logical necessity, we have a deductively dependency of a proposition on the principles used in its proof that shows the insufficiency of other methods to solve a particular problem. Hence, as for the argument patterns, the axioms of set theory can be seen as the reasons for an argument to work. Then we can say that the axioms that extend ZFC can be considered as argument patterns.

60 The underlying notion of argument pattern is of course stretched to its limit and it could be argued that, in the context of the axioms of set theory, it is hardly recognizable. But our claim is not that argument patterns are set theoretical axioms, because we acknowledge that there are of course different methods of proof and argument patterns in different areas of mathematics that have nothing to do with a set theoretical methodology. What we argue is that, when it comes to the foundation of mathematics, some axioms of set theory explain why a given proposition is a mathematical theorem, providing its proof; not why it is a theorem of a particular theory, say geometry or analysis. Moreover, we are not claiming that any set theoretical axiom, singularly, can be seen as an instantiation of argument patterns, but that set theory as a whole can be seen as an explanation of why it is possible to prove a theorem—showing the core argument that allows a proof to work—once it has been cleared that the sense of explanation we use is related to a form of unification.

2.2 Kitcher's problem for mathematical explanation

61 We are aware of the fact that the arguing for the relevance of the role of the axioms in the context of argument patterns is a subtle and far from easy task28— even if Kitcher seems to support this view, as we will see in the only passage were he discusses the possibility to use his account for analyzing the notion of mathematical explanation— but there is a preliminary problem that needs to be cleared. Even if we give for granted that axioms act as—or instantiate—a form of argument patterns:29 what is the

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epistemological argument in favor of the coincidence between unification and explanation? In other words, even if it seems at first sight a convincing matching, what are the arguments in favor of this identification? The main argument that Kitcher advances, to hold the epistemological link between the act of unifying a theory and that of explaining why some of phenomena described by the theory hold, is that the limit goal of all science is to unveil the causal structure of the world. The growth of science is driven in part by the desire for explanation, and to explain is to fit the phenomena into a unified picture insofar as we can. What emerges in the limit of this process is nothing less than the causal structure of the world. [Kitcher 1989, 500]

62 This position of course has profound consequences, one of which is the dependency of the concept of causality from that of explanation. Kitcher is well aware of this fact. Indeed, I have been emphasizing the idea (favored by Mill, Hempel, and many other empiricists) that causal notions are derived from explanatory notions. Thus I am committed to

(2) If F is causally relevant to P, then F is explanatory relevant to P. [Kitcher 1989, 495]

63 Without entering in the discussion of the robustness of this philosophical position, we want to outline the main difficulty that this position suffers in the context of an analysis of explanation internal to mathematics: mathematics is not a causal world. There is a general agreement on this point and even a truly platonistic—minded thinker, like Gödel, has always advanced only an analogy between the physical world and the mathematical realm.

64 As we said before, Kitcher has never fully addressed the matter of the applicability of his model to the mathematical explanation. However, in [Kitcher 1989], there is a, although short, attempt to discuss the problem. For even in areas of investigation where causal concepts do not apply—such as mathematics—we can make sense of the view that there are patterns of derivation that can be applied again and again to generate a variety of conclusions. Moreover, the unification criterion seems to fit very well with the examples in which explanatory asymmetries occur in mathematics. Derivations of theorems in real analysis that start from premises about the properties of the real numbers instantiate patterns of derivation that can be used to yield theorems that are unobtainable if we employ patterns that appeal to geometrical properties. Similarly the standard set of axioms for group theory covers both the finite and the infinite groups, so that we can provide derivations of the major theorems that have a common pattern, while the alternative set of axioms for the theory of finite groups would give rise to a less unified treatment in which different patterns would be implied in the finite and in the infinite case. Lastly, what Lagrange seems to have aimed for is the incorporation of the scattered methods for solving equations within a general pattern, and this was achieved first in his pioneering memoir and later, with greater generality, in the work of Galois. The fact that the unification approach provides an account of explanation, and explanatory asymmetries, in mathematics stands to its credit. [Kitcher 1989, 437]

65 As it is clear, Kitcker argues that the bare possibility of applying the same pattern again and again is responsible for the unificatory virtue of a systematization without demanding a causal connection. This thesis is compatible with the claim that “If F is causally relevant to P, then F is explanatory relevant to P”, but it is not with the idea

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that the unication process shows, in the limit, the causal relations between phenomena, because this would imply an even stronger thesis:

(2*) F is causally relevant to P if and only if F is explanatory P.

66 But this is of course false when dealing with mathematical explanation in pure mathematics. Then if we want to argue that axioms act as explanations we should look for a framework with different motivations than Kitcker's, in order to justify the link between unification and explanation in a non causal context.

67 As a matter of fact, there is a theoretical, and not just methodological, difference between argument patterns in science and axioms in pure mathematics. What differs in the two contexts is the nature of the why—questions for which the explanatory unification looks for an answer. Indeed, if not partial, an answer to a why—question is such that it is not possible to ask any further why—question. When we get to pure mathematics and we discuss explanation in an axiomatic context we need to distinguish between the explanation of ”why is it the case that S”, for a sentence S and “why is it the case that A”, for an axiom A. Kitcher chooses to explain ”why is it the case that S” appealing to the best possible unification of all sentences of the theory to which S belongs. This strategy can work as long as we remain in the context of physical phenomena, where we do not need to ask why—questions on the physical laws. Indeed the example on Newton's theory of gravitation, proposed in Explanatory Unification, is acceptable, since nobody would ever ask why it is the case for Newton's law of Universal Gravitation. The reason is that reality serves as a bedrock in the search for the causes of a phenomenon. As a matter of fact, when Kitcher deals properly with mathematical explanation he proposes to explain group theory by means of its axioms, but what explains the axioms—as far as they are mathematical propositions—if we cannot make reference to a physical world where groups exist, i.e., if there is no causal connection between the groups and the theorems of group theory?

68 Before trying to find a solution to the problem of “why is it the case that A”, for an axiom A, let us look at the question of “why is it the case that S”, for a mathematical sentence S, without appealing to the causal structure of the world, i.e., without appealing to the bedrock of reality that can stop the rise of new why—questions. If we are not dealing with a self—evident proposition, nor we are referring to some metaphysical property of mathematical objects,30 an answer to a why question, in terms of argument patterns, can be considered satisfactory only when we are not anymore in the position to ask for the reasons that could explain why some proposition hold. Only in this case it is possible to give objective reasons for a proposition S. Then we cannot make reference to any extra—mathematical—informal—property, but we have to ground our answer on something as objective and indubitable as the logical structure of mathematics. In other words, the explanation needs to be internal to the mathematical discourse. So, if we accept that arguments are derivations—as it is the case in Kitcher's account—and the fact that axioms act as argument patterns, then, following Hilbert's suggestion, the best answer to “why is it the case that S” amounts in showing the necessary and sufficient conditions for the proof of S.

69 However, such an answer seems to be clearly unsatisfactory when we restrict it to a single sentence S, because it is often the case that logical equivalences are not explanatory at all.31 But this objection misses an important aspect of doing mathematics, because the why—questions for which we are normally seeking answers

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are not “why is it the case that S” independently from the mathematical context, but, once the background theory T is made explicit: i.e., “why is it the case that S ∈ T”. Indeed, granting that the act of explanation is a global matter, given by unification, necessary and sufficient conditions need to be given to explain the claim that S is a theorem of T.

70 Hence when we restrict to countable languages our thesis is that a set of axioms A = {Ai : i ∈ ω} can act as a unifying explanation of a theory T only if it is possible to show that, for any of its sentence S

S ∈ T ⇔ ⴺn ∈ ωⴺ Ai0, ..., Ain ∈ A such that Ai0 ⋀... ⋀ Ain ⋀ S. 71 Nevertheless, the possibility to ask a general, context—independent, why—question has value and deserves to be considered. Then the question “why is it the case that S” becomes a question about the mathematical pedigree of S. If such a context-free question can ever find an answer, this will be in a sufficiently broad framework where it is possible to ask why we can consider S as a mathematical theorem: exactly the context given by a foundational theory as it is the case for set theory. For this reason we can say that set theoretical axioms as large cardinals and MM+++ can be seen as explanations of a mathematical proposition S, for what concerns the question “why is it the case that S”, i.e., why S is a mathematical theorem.

72 Then we are finally in the position to come back to our initial problem: to give a philosophical justification of the claim that set theory can be seen as a foundation for mathematics as long as it is capable of unifying mathematical practice. The answer then is to be found in its possibility to explain mathematical phenomena, given necessary and sufficient condition, at least when it is possible to make clear reference to a theory T, that we can easily describe and recognize; as it is the case for an initial segment of the cumulative hierarchy, in terms of an H(θ), as it has been seen for Woodin's and Viale's results.

73 However, we are left with the problem “why is it the case that A”, for an axiom A, that is, the search for justifications of the axioms. And remember that our goal, at the beginning of this section was to give a philosophical analysis of the criteria of unification. The outcome of this inquire is that explanation and justification are tied together by a sort of completeness theorem that links axioms and propositions, in the attempt to unify a theory. One side of the if—and—only—if—condition, from right to left, shows how it is possible to explain that a given proposition S belongs to some theory T—this is done by showing that S is a consequence of the set of axioms A. Then, starting from the axioms, we have an implicit definition of T, as it was indeed the case for Hilbert's.

74 Axiom of Completeness. On the other hand, the implication from left to right presupposes an intuitive description of T and then asks for the axioms that can prove the whole of its theorems and, thus, unify the theory—in the sense described by Kitcher, as argument patterns. If this second implication holds then it is possible to match the intuitive theory and its axioms, and so we are able to justify the axioms in terms of their unification power. These judgments are indeed related to an informal description of a theory T and so presuppose its intuitive description. To come back to the argument we proposed at the end of our historical examples of the axiom—as— explanation position, we think that justification and explanation are two sides of the same coin: a complete unification. Indeed unification allows the proof of completeness theorem of the form we have just described, where a link is established between syntax

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and semantics. The correctness direction, from left to right, amounts to the justification of the axioms of a theory T, while the completeness direction, T from right to left, amounts to the explanation of why the sentences can be seen as proposition of T. Hence we maintain that the answer to the question “why is it the case that A”, for an axiom A, consists in its justification—whenever it is possible to give a sufficiently clear description of a theory T, for which A acts as an axiom—thanks to a completeness theorem of the form we have just outlined.

75 In the end, we want to be clear that our thesis is neither that explanation always comes, in mathematics, through axioms, nor that the explanatory unification of set theoretical principles is always granted by their acting as argument patterns. Indeed, as we argued, the epistemological importance of an equiconsistency proof gives different reasons for the explanatory role of large cardinal axioms. However, the possibility to apply Kitcher's model to some axioms of set theory is intended to show that explanation is part of the role of these axioms, but neither to make a general theory of the nature of the axioms in mathematics, nor to make a theory of mathematical explanation. On this latter aspect, we acknowledge that explanation does not always come in the context of an axiomatic setting, nor all the axioms are capable of explanation. As a matter of fact, for what concerns explanation we favor a more pluralist conception, capable of taking into account all the different nuances that can have a mathematical explanation, internal to the mathematical work.

76 Moreover, although Woodin's and Viale's results point in the direction of a complete axiomatization of set theory and we maintained that an intuitive description of a theory is needed in order to give a complete axiomatization of a theory T, we do not want to argue neither that an analysis of the concept of set, nor an intuition to this concept—a la Gödel—is needed in order to give a foundation of mathematics. Quite the contrary, as we hinted before, the vagueness of this concept is the main reason to argue for the set theoretical foundation of mathematics we proposed. As a matter of fact, we think that a foundation of set theory—with the word 'foundation' intended in the sense explained by Gödel: “a procedure aiming at establishing the truth of the relevant mathematical statements and at clarifying the meaning of the mathematical concepts involved in these theories” [Mehlberg 1960, 86]—is really a different task and we will try to make this distinction clearer in the last part of this section.

2.3 Towards a more general distinction

77 To come back to the first part of this work, we hope we have been able to show that set theory, in the extended sense considered, is a good tool in the analysis of necessary and sufficient conditions for the proof of all mathematical problems and in this sense it is to be intended as a foundation for mathematics. The instruments it provides go much beyond the possibilities that are given by the use of solely logical tools—that encouraged the vast application of the axiomatic method in the last century. As a matter of fact, thanks to equiconsistency results, it is possible to find equivalence results that are not only logical, but epistemological in character; and this analysis is a good form of explanation, in terms of the main possibility of proof—able to unveil deep combinatorial aspects. Moreover it is important to stress the difference between the set theoretical foundation we described and the standard view that sees a big ontological importance in the possibility to reduce every piece of mathematics to set theory. As a matter of fact the foundation of mathematics we argued for is ontologically and

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theoretically neutral: it does not even take a stand about the attempt to single out the true universe of set theory, in the context of the multiple alternatives offered by the method of forcing. This line of research is an interesting and fruitful subject, but it has important theoretical implications that cannot be compatible with a foundation that aims to explain practice and then follows the free and unforeseeable development of mathematics.

78 Indeed there is an important debate on the main possibility to find such a complete description of V, where platonic—minded mathematicians, like for example Woodin, are opposed to researchers that hold a multiverse point of view, like for example Hamkins [Hamkins 2012]. There are also positions in—between like Magidor's who maintains that “some set theories are more equal than others”32 or, similarly, like Sy Friedman and Sharon Shelah, who argue in favor of the possibility to find rational arguments for choosing one model instead of another.33 As a matter of fact, we think that a multiverse view on the nature of the set theoretic universe just confuses the foundational role of set theory with its nature of mathematical theory in itself for which the search for a good description of the intended model is a fundamental and natural demand. On the contrary, even if this would be found, it would not disqualify all the theorems that do not hold in that model. As a matter of fact the distinction between foundational aspects and infra—theoretical ones is meant to legitimate both analyses.

79 In conclusion, we want to stress the importance of not confusing the foundational role of set theory with its nature of mathematical theory in itself, for which the search for a good description of the intended model is a fundamental and natural demand.

3 Two different foundational ideas

80 We believe that the difference between the two set theoretical foundations of mathematics we discussed before is the appearance of a more general phenomenon: two distinct attitudes in the foundation of mathematics. Of course, we do not pretend to give an exhaustive classification, but at least to indicate that there are two areas that deal with foundational problems with distinctive perspectives: philosophy and mathematics. These two attitudes are of course well interlaced in the foundational works of the last century, but they are, in principle, autonomous. As a matter of fact, these two dispositions act in response to different needs. The choice of the terms to indicate them could be theoretical and practical. We could have called them philosophical and mathematical but this choice is somehow misleading, because on the one hand there is no sharp distinction between the two subjects at a foundational level and, on the other hand, we do not want to suggest an opposition between philosophy and mathematics, but, on the contrary, a distinction that can produce useful interactions. The antinomy that I would like to propose with this categorization is the one existing between essence and method.

81 We will use the expressions “theoretical foundation “and” practical foundation “to indicate the corresponding attitude in the foundational enterprise. We will quote same examples of these approaches, but we would like to be clear that we are not proposing a classification of philosophers and mathematicians in two separate categories. On the contrary, we just delineate a distinction for what concerns goals, approaches and, sometimes, admittedly, true predilections. Indeed, it will always be difficult to draw a

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clear line to distinguish the two kinds of foundations in the work of an author, since the reflection on mathematics is always a difficult and broad enterprise. As we outlined in the first part of this article the main concern of a foundation is unity. For this reason a theoretical foundation and a practical foundation are both stimulated by this idea and we will describe how they achieve this purpose.

3.1 Theoretical foundation

82 By a theoretical foundation we mean the attitude that sees in the foundation of mathematics the possibility of a reduction. This stance tries to answer a question on what there is in the mathematical world and how we can give a mathematical definition of our mathematical concepts. A reduction of this kind deals mostly with ontological or semantical problems; as, for example, in the case of the reduction of mathematical objects to sets (with all the problems related to the fact of assuming that everything is a set (see [Benacerraf 1965]). See for example the very beginning of [Kunen 1980], one of the most used textbook in set theory: Set theory is the foundation of mathematics. All mathematical concepts are defined in terms of the primitive notions of set and membership. In axiomatic set theory we formulate a few simple axioms about these primitive notions in an attempt to capture the basic “obviously true“ set theoretic principles. Prom such axioms, all known mathematics may be derived. [Kunen 1980, xi] Another example of this approach can be found in Russell's logicist program, for which the reduction is even more conceptual. In constructing a deductive system such as that contained in the present work ... we have to analyse existing mathematics, with a view to discovering what premisses are employed, whether these premisses are mutually consistent, and whether they are capable of reduction to more fundamental premisses. ... [T]he chief reason in favor of any theory on the principles of mathematics must always lie in the fact that the theory in question enables us to deduce ordinary mathematics. [Whitehead & Russell 1910, Preface, v]

83 In an opposite way, also, any attempt of nominalization of the mathematical discourse can be seen as a form of reduction; a reduction of the truth value of a mathematical sentence to a syntactic game that can be played uniformly within any mathematical theory.34 As a matter of fact, the answer to the question “on what there is” can be answered in many and incompatible ways, like, for example, everything or nothing. What is peculiar to this attitude is that it tries to give a comprehensive reduction of the whole of mathematical discourse, or sentences, or truths, to some objects or principles that are able to subsume or vanish any peculiar aspect of a particular mathematical field. Not only this kind of foundation tries to unify but also to disappear the differences, explaining that the various things we encounter in our mathematical experience are just diverse manifestations of the same phenomenon. What is common to the foundations that share this goal is an holistic and static view of mathematics, that sees mathematical practice as the field where to test if the reduction proposed is sufficiently comprehensive.

84 Of course there are problematic aspects of a theoretical foundation. These problems arise in trying to give a general account of mathematics and not only of its unity. First of all there is the matter of fact that mathematics is an always evolving enterprise. This makes very difficult to single out, once and for all, the very characteristic marks of mathematics and moreover to confine its existence within rigid boundaries. The

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horizon of sense and application of mathematics is always moving and follows freely the heavy burden of its history. Secondly there are problems of reference, or aboutness, as in the case of numbers and sets, as outlined in [Benacerraf 1965]. Indeed, once a reduction is proposed, there should be arguments in favor of that particular reduction instead of another, maybe, of the same kind. For example, following Benacerraf, once we admit that numbers are sets we should be able to explain which sets are the numbers. Finally, and related to this latter point, there is always a metaphysical obscurity that surrounds any reduction: how does this reduction work? what is the relationship between what is reduced and the the tools of reduction? We will not try to give an answer to these questions, because this is not a necessary task for a theoretical foundation, even if, of course, we have to admit that these questions deserve an answer, in the context of a philosophical account of mathematics. We would like here just to outline this view, clear its weaknesses and not try to defend it.

3.2 Practical foundation

85 The second attitude we would like to describe is the practical foundation: it aims to explain the unity of mathematics without proposing a reduction and it is epistemological in character. The main question that it tries to answer is: Why can we prove a theorem? Why a proposition can be seen as a theorem of a theory? The main reason for calling it practical, in contrast with theoretical, is the attention that is devoted to mathematical practice. As a matter of fact the motivation for such a foundation is the observation that doing mathematics consists essentially in trying to prove theorems. Moreover this attitude grants that one of the most important task of a serious reflection on mathematics is to explain the nature and the possibility of mathematical knowledge. In contrast with a theoretical foundation, a practical foundation of mathematics is not confined to a fixed set of axioms or to a given set of primitive principles, as in the case of the Prineipia Mathematica, but it makes use of the axiomatic method, trying to give a detailed description of the mathematical work. In this context, the unity of mathematics is suggested as a methodological uniformity. The main goal of a practical foundation is to explain in what consists the procedure that allows to recognize an argument as a proof. To qualify something as a proof has the consequence of characterizing the proposition that is proved as a piece of mathematical work. The roots of this attitude can be found in Hilbert's foundational work on geometry. Remember the letter to Frege, dated December 29th, 1899, when Hilbert said that he wanted to understand why “the sum of the angles in a triangle is equal to two right angles” [Frege 1980, 38—39].

86 In a different way, with respect to the role of the axioms in defining the basic ideas of a theory, an attitude of this kind can be found also in Prege's foundation of arithmetic. By insisting that the chains of inferences do not have any gaps we succeed in bringing to light every axiom, assumption, hypothesis or whatever else you want to call it on which a proof rests; in this way we obtain a basis for judging the epistemological nature of the theorem. [Prege 1893, Introduction]

87 We can see here what explanation means in the context of a practical foundation. The explanation that is given is internal to the theory for which the foundation is proposed. Indeed the explanation of a theorem is given in terms of its place in the logical structure of a theory, as for Hilbert, or in terms of the elucidation, step by step, of a proof, in Prege's proposal. Then the reasons that explain are to be found in the axioms

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that characterize a domain of knowledge or in the tools that we use to get from the premises of an argument to its conclusion. It is important to note that in these cases nothing depends on some metaphysical property of the subject matter nor to the recognition of a sort of similarity in the nature of the things involved in the foundational analysis. With a practical foundation mathematical practice is investigated in details and the quest for the reasons terminates only when we stop asking why—questions.

88 calculus ratiocinator. We have to pause here for a moment, because a digression is needed for the role that logic plays with respect to both types of foundations; and meanwhile to illustrate why we can find aspects of the work of the two main champions of logicism, Prege and Russell, in different horns of the dichotomy we are proposing. The reason for it is the twofold nature of logic. There is an old and venerable tradition, that can be traced back to Leibniz, that acknowledges logic, on the one hand, as a characteristica universalis and, on the other hand, as a calculus ratiocinator. The former aspect stresses the fact that logic is a universal language that can express any mathematical concept, while the latter indicates the circumstance that logic can be used to perform formal deductions. This bivalent character of logic can be found also in Prege's work but the passage quoted above shows one of the relevant feature of his Beyriffsschrift: the possibility to explain why a conclusion follows from its premises, in terms of a rigorous deduction. While Prege maintained that these two aspects of logic cannot be disentangled, he emphasized that the characteristica universalis aspect was the more important one.35 However Russell goes much further in the direction of the characteristica universalis and he says not only that logic is the language in which mathematical concepts can be expressed, but that every piece of mathematics can be defined in terms of logic. Then, while Prege has a universal view about logic but he thought that the computational aspect was really necessary for any meaningful notion of logic, on the contrary Russell sees in its work a more fundamental reductive stance.

3.2.1 Single theory vs. mathematics

89 By looking at the quotations above, it could be thought that a practical foundation is context—depending and it works only when a single theory needs a foundation, and not the whole of mathematics. Indeed Hilbert's work was on geometry, whereas Prege's was on arithmetics and, thanks to their work, it is possible not only to explain what is needed for the proof of a proposition but also why we can recognize it as a proposition of geometry or arithmetic. However, in this case: when a foundation of a particular theory is proposed, it raises the problem of the adequacy of an axiomatization to the theory that is axiomatized. We will not tackle this problem here in its generality, because this involves issues such as clarifying what a mathematical concept is, how it is possible to formalize it and how we manage to know what we formalize. Even if the attempt to give an answer to these questions is among the central tasks of the philosophy of mathematics, it is not in the scope of this work. Both Prege and Hilbert had their personal solutions to the problem of the adequacy: the former believed in the existence of a realm of concepts, while the latter discarded the problem using implicit definitions. What is important to stress here is that there is an insolvable tension between intuition and formalization, that, in the context of a theory for which we feel to have strong intuitions about its subject matter, can rise deep philosophical questions. In the case of geometry or arithmetic there is a tentative solution that comes

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directly from mathematics: a categoricity proof like Hilbert's for analytic geometry, or the one that is possible to give for natural numbers, using second order Peano axioms. Leaving aside the discussion on the significance of a categoricity proof, we just acknowledge that there are also situations where there are not even such results, as it is indeed the case for the formalization of set theory proposed by Zermelo and Praenkel, for which it is not even sufficient to use second order ZFC, as it is shown by Zermelo's theorem on the quasi—categoricity of the universes of set theory [Zermelo 1930].

90 Despite all the difficulties that emerge in the case of a practical foundation of a single theory, we would like to argue that this is not the case for a practical foundation of mathematics, as a whole—as the set theoretical one we discussed in the beginning. In this case, we do not have neither the problem of reference, as for a theoretical foundation of mathematics, nor the problem of adequacy, as for a practical foundation of a single mathematical theory. Indeed, it is not necessary to know the subject matter of mathematics before we can propose a practical foundation for it. Or, to put it in a different way, knowing why we can prove a theorem does not entail knowledge of what the theorem is about.

91 In the case of set theory, the same difficulty to develop a reliable intuition of the general concept of set was sufficient to show the independence of a practical foundation from a complete knowledge of the matter for which a foundation is sought. I would like, in conclusion, to discuss another example where, even if we feel to have strong mathematical intuition, we can still mark a conceptual distinction between a practical and a theoretical foundation. Let us consider the case of arithmetic. In general, the fact that we know which principles allow us to solve a problem in number theory does not depend on our knowledge of what natural numbers are. Indeed, there are many cases in which tools that transcend arithmetic are used to solve a problem in number theory, as in the case of Fermat's Last Theorem, while, on the contrary, just second order Peano axioms are able to fix the structure of the natural numbers. This situation could be seen—and it is often seen—as an historical accident. Indeed it is common opinion among mathematicians that for any relevant number—theoretic statement it can be found a proof in elementary number theory. This belief would involve an extensive coincidence of the set of principles that allow to give an explanation of the ”epistemological nature of a theorem“ in number theory and the set of axioms that are able, in second order logic, to characterize the structure of natural numbers. Then, this could be seen as a cause of ambiguity between the two different foundations that we are proposing. Nevertheless, even granted this quantitative coincidence, there is a qualitative difference in looking at the axioms as characterizing natural numbers and as tools that characterize the work in number theory. In the former case we are tempted to say that the truth of a proposition in number theory depends on the fact that Peano Arithmetic is the right formalization the ”natural numbers“, while in the latter that it depends on the knowledge of which principles—or axioms—we are using in its proof. This is the reason why it would be a mistake to confuse the level of explanation of why we can prove a theorem and the level of metaphysical justification of the truth of a theorem in terms of the nature of the terms involved.36

92 In the case of a practical foundation of the entire mathematics this point is even more evident, because we do not have a clear idea of what the subject matter of mathematics

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is. Quite the contrary, we have a vague and ambiguous intuition of it, whence spring our feeling that mathematics is completely free in its paths and development. To make clear the borders of mathematics is exactly the purpose of a foundation, hence shaping mathematics. This is the reason why we cannot know what mathematics is before proposing a foundations for it. Then it is clear that being able to explain the facts we encounter in our mathematical practice does not presuppose a precise knowledge of their subject matter.

93 In conclusion, we hope we have given a clear picture of these two different aims in the foundations of mathematics. However we do not want to argue in favor of a separation of a more philosophical attitude from a more mathematical one. Of course a useful interaction between these points of view is not only the best way to find a deep understanding of our mathematical experience, but also a good guide for our mathematical work. The recognition of a conceptual distinction between two different attitudes in the foundational studies does not involve a separation of them in practice as working tools in the attempt to account for the mathematical phenomena and to widen our mathematical knowledge.

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FRIEDMAN, Sy—David & Arrigoni, Tatiana [2013], The hyperuniverse program, Bulletin of Symbolic Logic, 19(1), 77—96.

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JECH, Tomás [2003], Set Theory, Berlin: Springer, 3rd edn.

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MANCOSU, Paolo & Hafner, Johannes [2008], Beyond Unification, in: The Philosophy of Mathematical Practice, edited by P. Mancosu, Oxford University Press, 151—179.

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TAPPENDEN, Jamie [2005], Proof style and understanding in mathematics I: Visualization, unification and axiom choice, in: Visualization, Explanation and Reasoning Styles in Mathematics, edited by P. Mancosu, K. Jorgensen, & S. Pedersen, Dordrecht: Springer—Verlag, 147—214.

VAN HEIJENOORT, Jean [1967], From Frege to Gödel: A Source Book in Mathematical Logic, 1879 1931, Cambridge: Harvard University Press.

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ZERMELO, Ernst [1930], On boundary numbers and domains of sets. New investigations in the foundations of set theory, in: Collected Works. Volume I, Berlin; Heidelberg: Springer—Verlag, 401— 429, 2010. — [2010], Collected Works. Volume I, Berlin; Heidelberg: Springer—Verlag.

NOTES

1. On this ground we will be inspired by Resnik's idea that mathematics is a science of patterns and that a set theoretical foundation can be seen as a macro−pattern: “There is another phenomenon which has greatly changed mathematics and which could be called a reduction. This is the set theorizing of mathematics. 1 have in mind the use of the language of set theory as the background language of working mathematics and the attendant objectification (or, in my terms, positionalization) of mathematical structures” [Resnik 1981, 540]. However, contrary to the realist structural position of [Resnik 1997] and [Resnik 1981] we will try to show how to make sense of the notion of pattern not only in an ontological, but also epistemological way. Our aim indeed will be to support a set theoretical foundation of mathematics, deprived of its standard realist proposal. 2. Among them a special status is hold by large cardinals, but we will discuss it later. 3. Later we will argue more on this point. 4. See, for example, [Weaver 2009] and [Feferman 1999]. 5. See [Kunen 1980] for a very good introduction to this subject. 6. If is large enough, so that is a model of ZFC, then we say that is a large cardinal. However this is not a definition of what is a large cardinal. 7. If we accept this point of view we also have to accept the consequence that the more fundamental principle is a contradiction. As a matter of fact, many large cardinals hypothesis can be seen as stating the existence of a non trivial elementary embedding of two universes of set theory; the more similar are these classes, the higher is the consistency strength of the corresponding large cardinal. Pushing this process at the limit, we get a statement that postulates the existence of a non trivial elementary embedding of the universal class V in itself. This statement has been shown to be inconsistent with ZFC, by Kenneth Kunen. 8. Modulo equivalence of the form ab = axx-1b. 9. The set−theoretical hypothesis V = L states that all sets are constructible. Indeed V is the standard notation that refers to the universal class of all sets that can be defined by stages

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iterating the power set operation α-many time, for every ordinal :

• V0 = ∅,

• Vα+1 = (Vα),

• Vλ = ∪α∈λ Vα for λ limit ordinal,

• V = ∪α∈Ord Vα. On the other hand L is the class of all constructible sets, that can also be presented as a cumulative hierarchy. For a set X let Def(X) be the collection of all sets definable with parameters in X. Then L is defined as follows:

• L0 = ∅

• Lα+1 = Def(Lα) • L = L , for λ limit ordinal, λ ∪α∈λ α

• L = ∪α∈Ord Lα 10. We will not give the definition of MA here. What is important to know is that it is one of the weakest Forcing Axioms. We refer, for the interested reader, to [Jech 2003]. 11. As in the case of MA, we refer to [Jech 2003] for the definition of OCA. 12. This is the weakest notion of large cardinal. We say that is inaccessible if it is regular and such that for every λ < , we have 2λ < . 13. Some of the results we quote are old, but what is new is their presentation, the context in which they are placed and consequently the meaning they assume in this new context; see [Viale 2011]. 14. See [Woodin 2001] for a presentation of this program. 15. Indeed ℝ ⊆ H(ℵ2), since (ω) ⊆ H(ℵ1) and so every subset of ℝ belongs to H(ℵ2). 16. Of course there is a an important philosophical problem behind the concept of natural, but we will came back on this later. Luckily the main concept of naturalness is sufficiently natural to be easily understood, but nevertheless it deserves a philosophical analysis. 17. This notion is again too technical to be defined here; see [Jech 2003]. What isimportant to know is just that Woodin cardinals are large cardinals. 18. See [Viale 2013] for the definition of this principle. 19. See, among others [Mancosu 2008]. 20. See Mehlberg, in this respect: “The limited effect of the failure of Hilbert's program upon the dependability of the impressive cluster of mathematical theories which he tried to place on a common 'foundation' can be clarified by reference to certain relevant views of Gödel which he informally conveyed to me, some years ago, during a discussion we had at Princeton, N. J. According to Gödel, an axiomatization of classical mathematics on a logical basis or in terms of set theory is not literally a foundation of the relevant mathematics, i.e., a procedure aiming at establishing the truth of the relevant mathematical statements and at clarifying the meaning of the mathematical concepts involved in these theories. In Gödel's view, the role of these alleged 'foundations' is rather comparable to the function discharged, in physical theory, by explanatory hypotheses. [... ] Professor Gödel suggests that so−called logical or set theoretical 'foundations' for number−theory, or any other well established mathematical theory, is explanatory, rather than really foundational, exactly as in physics” [Mehlberg 1960, 86−87]. 21. And sustained by his readers, as it is done in [Tappenden 2005, 158−159]−where Tappenden says: However, mindful of the fact that some explanations in physics andmathematics do seem to be governed by the same principles, I'll count it as an advantage of an account that it supports a uniform treatment of some mathematical andsome physical explanations. A promising candidate to support a uniform treatment of some pure mathematical cases and some non−mathematical ones is the treatment of explanation as unication as proposed in the seventies by Michael Friedman and Philip Kitcherand in [Mancosu & Hafner 2008]. 22. See [Mancosu 1999] for a detailed historical presentation of Bolzano's theory of mathematical explanation.

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23. Notice that this quotation manifests Kitcher's global point of view. Indeed, his aim is not to show which are the properties that an argument should have to be considered as explanatory, but which are the general—global—conditions under which an argument should be considered as explanatory. 24. In Kitcher words: “a sequence of statements whose status (as a premise or as following from a previous members in accordance with some specified rule) is clearly specified” [Kitcher 1989, 431]. 25. The problem, with this counting arguments, is that it can work just in the case of finitely many consequences. And not with a mathematical theory with countably many consequences. 26. Just to recall it, a basis is the set of arguments that instantiate in the more unifying way all the relevant argument patterns of a given systematization. 27. In passing let us just notice how this vagueness on the concept of argument pattern can be related to a more qualitative analysis of the notion of explanation. Indeed any account that aims to ascribe the explanatory power of an axiomatic setting that minimizes the arguments patterns, seems to be promising. But without a corresponding analysis of the concept of relevance this account would be useless. 28. Notice that, at the ontological level, also Resnik does not exclude the fundamental relevance that axioms can have with respect to patterns: “But I must elaboratea bit on this answer, since one might remark that I am saying that, in effect, the premisses to which we appeal in proving the theorem 'implicitly define' the pattern or class of patterns to which the theorem pertains. Now I have no problem per se with calling such premisses (or a more condensed set of axioms from which they might be derived) an implicit definition, so long as this is not taken to imply that the premisses are known a priori in some absolute sense. Of course the axioms constituting the clauses of an implicit definition are trivial consequences of this denition. Thus it is a matter of definition that they characterize the pattern they help specify” [Resnik 1997, 237−238]. In general, the aims and the context of the notion of pattern, in Resnik's work, is very far from Kitcher's and ours, but when we get to the more fundamental elements of doing mathematics, even in a platonic context as Resnik's, the notions of axiom and of pattern tend to collide. What is really different here is the background idea of mathematics: a science of existing structures for Resnink, while the domain of rigorous arguments for us. 29. Notice that Kitcher rejected the arguments proposed by Friedman in [Friedman 1974]−who also proposed to identify explanation and unification−saying that the major difference between his account and Friedman's consisted in what was to be assumed as the basic notion in the process of explanation: for Friedmann it was the physical laws, while for Kitcher it was the argument patterns. In Kitcher words: “Finally, I think that it is not hard to see why Friedman's theory goes wrong. Although he rightly insists on the connection between explanation and unification, Friedman is incorrect in counting phenomena according to the number of independent laws. [...] What is much more striking than the relation between these numbers is the fact that Newton's laws of motion are used again and again and that they are always supplemented by laws of the same types, to wit, laws specifying force distributions, mass distributions, initial velocity distributions, etc. Hence the unification achieved by Newtonian theory seems to consist not in the replacement of a large number of independent laws by a smaller number, but in the repeated use of a small number of types of law which relate a large class of apparently diverse phenomena to a few fundamental magnitudes and properties. Each explanation embodies a similar pattern: from the laws governing the fundamental magnitudes and properties together with laws that specify those magnitudes and properties for a class of systems, we derive the laws that apply to systems of that class” [Kitcher 1976, 212]. However Kitcher's criticism is directed to some technical points raised by M. Friedman's proposal, hence nothing prevents, in principle, to argue in favor of the possibility that axioms−or laws−can capture some essential feature of an argument pattern

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30. As for example Steiner seems to do in [Steiner 1978]. It seems that for him the question “why is it the case that S” needs to be answered referring to some essential properties of the mathematical objects to which S refers. For example, we can take S to be Fermat's Last Theorem (FLT) and ask: “why is it the case that FLT”? Steiner's answer is that it is the case that FLT if there is a property of the natural numbers such that for every n ∈ ℕ there is no positive integers a, b and c such that an + bn = cn. This type of answer recalls closely a Tarskian denition of truth: “for every n ∈ ℕ there is no positive integers a, b and c such that an + bn = cn” for every n ∈ ℕ there is no positive integers a, b and c such that an + bn = cn. However such a move, on one side, hides a strong realist position toward the existence of mathematical objects that needs to be argued and, on the other side, is tautological and hence non explanatory. 31. This is not always the case as some equivalence theorems show. Consider for example the equivalence between the Axiom of Choice and the Well−ordering Theorem. 32. This is the title of the draft of a talk that Magidor gave in Harvard in 2012. 33. On this topic see the work of Friedman [Friedman & Arrigoni 2013] and of Shelah [Shelah 2003]. 34. See, for example, the work of Hartry Field. 35. Contrary to Schroder's critiques, according to which Frege's foundational work tended mostly towards a calculus ratiocinator. See [Peckhaus 2004] for this debate. 36. This distinction echoes the disagreement on the role of the axioms between Frege and Hilbert and the point that we are trying to make is on the same line as the one of Hilbert.

ABSTRACTS

In this article I propose to look at set theory not only as a foundation of mathematics in a traditional sense, but as a foundation for mathematical practice. For this purpose I distinguish between a standard, ontological, set theoretical foundation that aims to find a set theoretical surrogate to every mathematical object, and a practical one that tries to explain mathematical phenomena, giving necessary and sufficient conditions for the proof of mathematical propositions. I will present some example of this use of set theoretical methods, in the context of mainstream mathematics, in terms of independence proofs, equiconsistency results and discussing some recent results that show how it is possible to “complete” the structures H(ℵ1) and

H(ℵ2). Then I will argue that a set theoretical foundation of mathematics can be relevant also for the philosophy of mathematical practice, as long as some axioms of set theory can be seen as explanations of mathematical phenomena. In the end I will propose a more general distinction between two different kinds of foundation: a practical one and a theoretical one, drawing some examples from the history of the foundation of mathematics.

Je me propose dans cet article de traiter de la théorie des ensembles, non seulement comme fondement des mathématiques au sens traditionnel, mais aussi comme fondement de la pratique mathématique. De ce point de vue, je marque une distinction entre un fondement ensembliste standard, d'une nature ontologique, grâce auquel tout objet mathématique peut trouver un succédané ensembliste, et un fondement pratique, qui vise à expliquer les phénomènes mathématiques, en donnant des conditions nécessaires et suffisantes pour prouver les propositions mathématiques. Je présente quelques exemples de cette utilisation des méthodes ensemblistes, dans le contexte des principales théories mathématiques, en termes de preuves

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d'indépendance et de résultats d'équiconsistance, et je discute quelques résultats récents qui montrent comment il est possible de « compléter » les structures H(ℵ1) et H(ℵ2). Ensuite, je montre que les fondements ensemblistes de mathématiques peuvent être utiles aussi pour la philosophie de la pratique mathématique, car certains axiomes de la théorie des ensembles peuvent être considérés comme des explications de phénomènes mathématiques. Dans la dernière partie de mon article, je propose une distinction plus générale entre deux différentes espèces de fondement : pratique et théorique, en tirant quelques exemples de l'histoire des fondements des mathématiques.

AUTHOR

GIORGIO VENTURI Scuola Normale Superiore di Pisa (Italy) Université Paris Diderot (France)

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Facets and Levels of Mathematical Abstraction

Hourya Benis-Sinaceur

I thank cordially the reviewers for their helpful comments to the previous version of this paper. I am indebted to them for many improvements. I thank also Jean-Pierre Marquis for discussions when he was visiting the University Paris-Diderot in 2012.

Introduction

1 Mathematical abstraction is the process of considering and manipulating operations, rules, methods and concepts divested from their reference to real world phenomena and circumstances, and also deprived from the content connected to particular applications. So the abstract concept of number does not come down to any real aggregate (of sheep, beans, pencils, etc.) nor to any conceived collection (of geometrical points, numerical elements, unspecified elements, etc.) and it includes sets of numbers with rules of calculation different from the usual ones, such as the rules for quaternions, octonions, etc.

2 Actually, in mathematics, one encounters from the very beginning not one but several abstraction processes, which constitute specific and permanent ways of developing the mathematical core. In modern times, especially from the 19th century onwards, abstraction flourishes, and various processes are more systematically piled up, concatenated, and blended for producing procedures, entities, structures, and theories at higher and higher levels of abstraction. Abstracting is an ongoing innovation processing, which expands the mathematical stuff and makes it still richer and more and more intricate and layered.

3 I am not aiming at tackling head-on the fundamental question: “What is an abstract object?” or “In which sense abstract objects 'exist'?”1 My purpose is much more modest and my method is mainly descriptive. I want to establish a picture of different and recurring procedures of mathematical abstraction. Thus, I will focus on different features of mathematical practice while I will disregard (explicit or implicit)

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ontological stands about the nature of mathematics and the status of its abstract objects. My purpose is epistemological, and it concerns the actual ways of performing abstraction in mathematical doing.2 On the way I shall inevitably display how I see the means and products of mathematical activity. I think that focussing on actual mathematical abstraction processes may afford a positive picture of what is mathematical abstraction. I mean that we may come across criteria for being abstract that are not obtained by the classical “way of negation”, an abstract object being not located in space and time and not causally active.3 I am rather taking “the way of example” despite its limits.

4 I will begin by a rapid incursion into the philosophical corpus. As a method I will focus on how the terms “abstract” and “abstraction” have been and are used; perforce I shall get information about the terms “concept” and “conceptualization”, thanks to which I can make precise my understanding of mathematical concepts. I will then try to parallel the outcome of my inquiry with specific mathematical techniques: as a result we will notice that mathematical abstraction is not reducible to logical abstraction, at least as it was understood in the Aristotelian tradition. Thirdly, I attempt to describe the fundamental thinking processes underlying the main ways to get and increase abstraction, with a special attention to recurrent mathematical actions that produce more and more abstract objects. As a specific illustration, I am giving in a fourth section significant or emblematic examples; relying on them I want to stress that in mathematical practice several abstracting processes work simultaneously and interact together, conceptualization and axiomatization being an important but only one factor in the job of systematic and uniform problem solving.

1 Philosophical background

5 Philosophers may have recourse to mathematical practice and history of mathematics for making more precise and more substantial the understanding of some fundamental thought processes, such as abstracting. A philosophical mean at hand is to focus on the changes in meaning of the terms “abstraction” and “abstract”. Such a semantic analysis provides indeed a crucial basis for contemporary linguistic, cultural, and conceptual understanding; it is largely used in “conceptual history”, which may be internal (considering the rational links between mathematical concepts and methods) or external, considering the institutional, political, and social environment which promotes or fights some typical way of thinking and acting: for instance, in mathematics abstraction has been viewed as a royal route of invention in Hilbert's and E. Noether's school and, at nearly the same time, as a degenerate trend destroying the vitality of intuition in the ideology of the “Deutsche Mathematik” championed by Ludwig Bieberbach and Oswald Teichmuller. For my part, I see no unbridgeable gap between abstraction and intuition, since insights may bring in abstraction processing and follow from it as well. As some mathematicians (E. Artin, A. Weil, and others) maintain, there is indeed a symbolic and abstract intuition. Anyway, I am not aiming at discussing here the question of axiomatic or logic versus intuition, which was the focus of intense debates during the 20th century and still is one of the main issues of the philosophy of mathematics.

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6 To the question “What is meaning?”, I will give Quine's answer: “Meaning is what essence becomes when it is divorced from the object of reference and wedded to the word.” Let me quote the whole passage: The Aristotelian notion of essence was the forerunner, no doubt, of the modern notion of intension or meaning... Things had essences for Aristotle, but only linguistic forms have meanings. Meaning is what essence becomes when it is divorced from the object of reference and wedded to the word. [Quine 1951, 22], see also [Quine 1990, 88]

7 Actually, I am taking the divorce of meaning from object of reference as a methodological device for avoiding ontological considerations and focussing on what and how we know rather than on what we believe or what we assume or must assume in order to give a philosophical account of some mathematical actions or attitudes. I am not saying that epistemological views do not commit to ontological assumptions, I am just saying that I will leave aside those possible commitments and the influence that they might have on actual knowledge processes and on our theoretical explanation of those processes.

1.1 Abstraction and concept-formation

8 Abstraction is an essential knowledge process, the process (or, to some, the alleged process) by which we form concepts. It consists in recognizing one or several common features or attributes (properties, predicates) in individuals, and on that basis stating a concept subsuming those common features or attributes. Concept is an idea, associated with a word expressing a property or a collection of properties inferred or derived from different samples. Subsumption is the logical technique to get generality from particulars.

9 This rough description complies with Aristotle's account of ἀφαίρεαις: Considering different things we subtract, remove, take away their particularities and retain only what they have in common. The concept of man applies to all humans, male or female, tall or short, blond or brown, etc.; the concept of triangle applies to any triangle, rectangle, equilateral or isosceles. According to Aristotle, concepts are immaterial ideas attached to material things; they exist within things on which they are predicated.4

10 There is a discussion about the nature of Aristotelian abstraction. Prege, and some Aristotle's experts such as David Ross and H.G. Apostle give a psychological interpretation. By contrast John Cleary claims5 that ἀφαίρεαις that he rightly translates by “subtraction”, “deprivation” (in contrast with πρὀσδεσɩς, to which corresponds “addition”), is the logical method which is used to identify and isolate the primary subject of predication for any given attributes (Posterior Analytics), and which consequently legitimates the intellectual separation of abstract objects.

11 Anyway, abstraction is the process of passing from things to ideas, properties and relations, to properties of relations and relations of properties, to properties of relations between properties, etc. Being a fundamental thinking process, abstraction has two faces: a logical face and evidently a psychological aspect that is the target of cognitive sciences.

12 John Locke (1632-1704) introduced particular ideas between individuals and general ideas. On a first step, particular ideas gather individuals into a class; on a second step, general ideas are created through the process of abstracting, drawing away, or removing the

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uncommon characteristics from several particular ideas. For example, the abstract general idea or concept that is designated by the word “red” is that characteristic which is common to the particular ideas (particular concepts) of apples, cherries, and blood. Thus, is pointed out the fact that the abstracting process forms a scale with at least two steps, and general concepts come loose from things. Locke writes indeed: General and universal belong not to the real existence of things; but are Inventions and creatures of the understanding, made by it for its own use, and concern only signs, whether words or ideas. [Locke 1689]

13 In contrast with Aristotle's ontological and logical point of view, Locke's standpoint is squarely epistemological. Note also that ideas may play the role of signs; later on, Charles Sanders Peirce (1839-1914) developed his semiotic philosophy on a very similar perspective.

1.2 Concepts

14 Developing further on Locke's approach, let us abandon Plato's and Aristotle's view that concepts are universal, unchanging ideal objects grasped by the understanding or made up in conformance with pre-existent relations in the real world. Concepts result indeed from the logical operation of subtraction but they do not have an eternal existence in some heavens of universal forms, separate from particulars as thought Plato or not separate as argued Aristotle. In my opinion, concepts are historical products of the mind's activity and their emergence depends on many theoretical, cultural, social, economical, and political data. Nevertheless concepts are or may be objective, since they help to grasp, to express in a most communicable way (at least in principle), and to master, within variable limits, phenomena of the real world.

15 To stress the objectivity of scientific concepts, the semantic tradition in philosophy [Bolzano (1781-1848), Prege (1848-1925), Husserl (1859-1938), and their followers] proposed to consider the sphere of concepts as autonomous. The aim was to separate semantic phenomena from their linguistic expressions and from their mental representations [Vorstellunyen]. But grounding the semantic sphere on itself may lead to erase the historical character of its elements and to give them an immutable ontological status. It is well known that, in order to ground the objectivity of scientific concepts, Gottlob Prege proposed to locate concepts in a “third realm”, the realm of “abstract objects”, which are neither sensible nor mental. Prege's “abstract objects” are not objects, they are meanings, more precisely timeless everlasting meanings. Given a linguistic expression F, Frege named the meaning of F its “conceptual content” [begrifflicher Inhalt]. A conceptual content is either always true or always false. Prege argues that we cannot create meanings, and that we can only grasp them; he considers also meanings as if they were a priori essences that we have to discover. “Abstract objects” are, in Prege's perspective, “meanings in themselves”, just like the old “things in themselves”. That gave birth to philosophical endless and currently ongoing discussions, with a revival of Platonic tendencies.

16 The semantic tradition was a reaction against the promotion of the Subject by Descartes, Kant, and Hegel among others, and an attempt to “save” the alleged eternal character of scientific truths. But from a more pragmatic point of view there is no need to ground semantic objectivity on objects fixed and independent from the mind, whose accessibility would then be questionable, as pointed out P. Benacerraf [Benacerraf

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1973]. The divorce of conceptual objectivity from fix and everlasting objects is not new. Georg Kreisel has pointed it out many times in his papers. Grounding on that I want to consider objectivity as resulting from a successful interaction between the rational activity of the understanding and the environment.

17 Concepts are also products and tools of thinking and reasoning; and they do not exist in the mind before the abstracting act. In Kant's terms they are a posteriori, i.e., they arise out of experience, “experience” being taken by me in an as wide as possible sense, and not in its Kantian sense, which is limited to perceptual or physical experience. I would then say that a concept is a thought-object, which results from a subtractive process that constructs the unity under which several specific thought-objects, rather than several rough physical objects, may be gathered.

18 Mathematical activity is concerned with thought-objects rather than with objects, even in the case of important impulse given by physical, biological, economic, or sociological phenomena. Mathematical entities are products of the activity of the understanding; they appear in a particular presentation [Darstellung]6, which might be modified or replaced by another one. In other words, we have access to mathematical entities only through the concepts we form for expressing some of the properties we want to take as a basis for developing our knowledge concerning those entities and many others that come to be related to them. Abstraction involves perceiving something, relating it to other things, grasping some common trait of those things, and conceiving of the common trait as to it can be related not only to those things but also to other similar things. [Locke 1689, 1, 20]

19 A mathematical concept is the association of a meaning (conceptual content) with a sign. Generally, once adopted by a mathematical community, a sign does not change, for instance, the notation dx, the notation f, or the Arabic numerals. But the meaning associated with a sign may evolve. Notably, for instance, the concept of function and the sign f have now a meaning different from the one that they had first in the 17th and 18th centuries; and they have now different meanings in set theory and in category theory. Actually, mathematical activity is concerned with the processes of continuous transformation of a given presentation into others: meaning changes, affording new concepts for the presumed same entity; new procedures are introduced at some point of time and reveal new aspects of our most familiar tools, new notations are proposed for designating the innovative concepts. Finally, a mathematical entity is the pair constituted by the idea of a supposed unique substrate designated by a name and its many actual and potential aspects or presentations, including the operations and rules of calculation set up in each case.7 In other words, a mathematical entity is the virtual referent, supposedly common to similar but distinct concepts. Dedekind-Peano concept of positive integers is not the same as Euclid's concept, even though both refer to the more or less same entity.

20 Concepts are formed gradually, through reason's indefatigable abstracting work, organizing similarities and differences, dissolving hidden links and creating links that were unnoticed. They are not obvious to whom who is not trained in this kind of work. Not everybody knows Dedekind-Peano definition or even Euclid's definition of numbers. Experience rather than pure intuition is at work. New insights are gained thanks to growing knowledge and experience.

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1.3 Abstract and concrete concepts

21 One distinguishes sometimes abstract concepts from concrete concepts. Since any concept results from an abstracting process, what is a concrete concept?

22 In Latin, “concretus” means “mixed”, “composite”, “compound”, while the Latin word “ abstractus” means “withdrawn”, “taken out of”, “extracted” (or “isolated”), “estranged”. That is all that is contained in the original etymological meaning of these words. The rest pertains to the philosophical conception that is expressed through them.

23 1.3.1. In one sense, a concrete concept is the concept of one or many concrete sensible things: so the concepts of this table, of one apple, of five pencils, as concepts of perceived things. “Concrete” pertains to the direct sensory referents understood under a concept, while “abstract” hints to non-sensory referents, which are the result of a repeated operation of extracting a general idea from more particular ideas. In this sense one usually makes a radical but rough difference between concrete and abstract, actual and unreal, perceptible and imperceptible. However in science to be given to the senses is an unsatisfying criterion for the demarcation between concrete and abstract entities: elementary particles are non-sensible entities and concrete data of physical experiment. The problem of finding a criterion satisfying in any case is a difficult one, and I will not undertake to solve it because, from an epistemological point of view, the distinction between “abstract” and “concrete” is relative and unstable: a concept F may be more abstract than a concept G, which may itself be abstract but less abstract, i.e., more concrete than F. Leibniz said in Nouveaux Essais that concreteness and abstractness are correlated; that means that concreteness and abstractness are a question of more or less rather than a question of yes or no. Cognitive scientists confirm experimentally indeed the gradation of the process beginning with a direct “categorization” on perceptual objects and continuing with categorizations at higher and higher levels on more and more abstract objects.

24 Moreover, an interesting view comes from results of psychological experiment: concreteness is mostly associated with perceptual features of some specific situation, which is generally caught in a global view, while abstractness points to a wide range of diverse situations embedding different (kinds of) entities, connected in some way, and a variety of processes attached to these (kinds of) entities. And it is suggested that there is a greater engagement of the verbal brain (left cerebral hemisphere) system for processing of abstract concepts and a greater engagement of the perceptual brain system (right cerebral hemisphere) for processing of concrete concepts. An abstract concept is understood through verbal-thinking working out, a concrete concept is visualized: I have either a direct perception or at least a mental image of a table or of five apples. That may explain how mathematical working consists partly in making easier the access to mathematical concepts and their handling through visualization on the blackboard or on a sheet of paper or in the imagination: we use symbols, we draw figures and diagrams, and we write down calculations and formulae. We may even maintain that reasoning and proving through mere analysis of symbolic formulae, as in Sturm-Liouville theory of differential equations8, or through diagrams, as in category theory9, are concrete handling with abstract constructions. We manipulate formulae and diagrams as being themselves mathematical objects, detecting properties not being otherwise discerned. It is known that prodigy people who are capable to make quickly

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calculations with great numbers “perceive” sounds or pictures emotionally associated with numbers. Daniel Tammet says that when he is calculating the decimals of π he “sees the numerals passing before his eyes like the pictures of a movie”.10 It seems indeed that gifted mathematicians “see” the world through mathematical filters. The French neuroscientist Stanislas Dehaene thinks that Presumably, one can become a mathematical genius only if one has an outstanding capacity for forming vivid mental representations of abstract mathematical concepts—mental images that soon turn into an illusion, eclipsing the human origins of mathematical objects and endowing them with the semblance of an independent existence. [Dehaene 2011, 225]

25 The irresistible leaning to a realist view of the mathematical universe of concepts and techniques has its roots in the actual process of visualizing abstract procedures.

26 1.3.2. In a second sense, “concrete” pertains to our usage and training. Familiar concepts are taken to be concrete and intuitively (visually) graspable, e.g., the positive integers, which are called “natural numbers” qua being the basic representation of the act of counting. Thus concreteness is a developed or developing character. According to Kant's Logic: The expressions abstract and concrete refer not so much to the concepts themselves —for any concept is an abstract concept—as to their usage. And this usage can again have different grades;— according as one treats a concept now more, now less abstract or concrete, that is, takes away from or adds to it now more, now fewer definitions. [Kant 1800, §16, Aninerk 1, 154]

27 In this second perspective, the distinction abstract/concrete is clearly an episternic distinction and it is relative in a sense different from that meant by Leibniz: not only abstract and concrete are correlated concepts, but an abstract concept or construction may become concrete or more concrete and it may be visualized through a symbol or image or diagram standing materially for it. That means in fact that we may form, through some kind of drawing, concrete representations of abstract concepts.

28 Hegel (1770-1831) introduced important refinements in the distinction concrete/ abstract. He assumed that any concept is always abstract, but he added that a genuine concept is not only abstract, but also concrete, in the sense that its definitions (what old logic calls features) are combined in it in a single complex expressing its individual unity. A concept is concrete because it contains all the content of its genesis within it. By contrast immediate perception is abstract in the sense that its determinations remain undeveloped.11 A concept is the concrete unity of different determinations. Thus the concreteness of a concept lies in the meaningful cohesion of its features, which may be developed at different moments of time. For instance, out of context, a verbal definition is abstract and abstract only. Immersed into the context of a scientific theoretical discourse, any abstract definition becomes concrete (in an epistemic sense). The concreteness of a concept is therefore always expressed through unfolding all its possible definitions/features in their mutual connections rather than through an isolated “definition“, and in immersing the concept into a web of interconnected concepts. It is as to say that “flesh” is given by the mutual connections between different features of the concept under consideration and by the links with other concepts. Such a consideration may well be applied to mathematics: the image of a dense network for representing the mathematical stuff has become commonplace by now.

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2 Mathematical practice

29 2.1. Mathematical concepts may generally be introduced or defined in different ways. The more presentations [Darstellungen] a concept has and the more it is embodied by different procedures performed in different areas, the more concrete it is taken to be. This may happen through two ways. a. When a concept is repeatedly used in different contextual theories, e.g., when we add numbers, vectors, vector spaces, etc., we get a meaning-generality (semantic generality), which is an extensive generality, a transversal generality of use. In the same wise, we use “products” for vector spaces, groups, topological spaces, Banach spaces, automata, etc. In each case we have to tell which properties among all the possible properties of the operation + or x, such as commutativity, associativity, etc., are preserved and which must be dropped. The fewer are the properties considered, the greater is generality. The very general concept of addition is illustrated by the structure of a monoid, which is instantiated by so many different models. The concreteness comes from the repeated use under rules specified in each case. b. The more connections a concept has, actually or potentially, with other concepts, the more intricate is its own meaning. Through its meaning-complexity (semantic complexity or richness)12 a concept gets some kind of concreteness, it looks like an individuated entity because several operations along with their properties are combined under it. Concreteness here is taken in Hegel's sense. In contrast with the traditional ratio between extension and intension of a concept, it is not the case in mathematics that increasing meaning complexity entails decreasing meaning generality; for instance, a category, let us say Grp is at the same time a more general and a more complex mathematical object than the group structure. Thus, analysis of mathematical abstraction does not give the same results as the traditional grammatical or logical analysis of concept formation.

30 2.2. Creative manipulation of mathematical concepts pertains their meaning, not just their names or nominal definitions. Names designate things, while concepts condense meaning even when they appear at first sight very abstract.

31 For instance real numbers may seem so abstract that their mathematical existence is challenged. They are, indeed, rejected by some constructivist mathematicians: e.g., instead of speaking of real roots of an algebraic equation, Kronecker considered intervals bounded by rational quantities, rational quantities being constructed by a finite number of operations from the integers. However, there is a larger notion of constructive existence, as it was made explicit by Hermann Weyl, who argued that we are entitled to claim that there exists an α only after having instantiated α [Weyl 1921, 54-55].

32 In this view, real numbers exist since we have encountered instances of them (e.g., ratio of the side of a square to its diagonal, π, the base e of the natural logarithm). The concept of real number, though abstract in the double sense that we can neither survey all its individual instances nor have a finite calculation for each instance, needs not to be eliminated; we rightly reason with the concept of real number as a set, a collection, and as a domain equipped with more than only one structure. Putting a structure on a set is stipulating relations and operations (functions) between the elements of the set and stipulating rules for working with them. In addition to algebraic structures such as groups, rings, fields, modules, vector spaces, etc., we have order structures, metric structures, topologies, differential structures, categories, among others.

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33 The structural complexity of the real number system emerged gradually (and mainly in the 19th century) through successive abstractive operations, disentangling different structures that were mixed together, dissociating especially topological notions from algebraic operations, from the order relation, and from the metric. As a bearer of several structures, the real numbers appear compound and multi-faceted, just like individuated physical objects. Actually, the set of real numbers carries the following standard structures: • an order: each number is either less or more than every other number. • an algebraic structure: multiplication and addition make it into a field. • a measure: intervals along the real line have a specific length, which can be extended to the Lebesgue measure on many of its subsets. • a metric: there is a notion of distance between points. • a geometry: it is equipped with a metric and is flat. • a topology: there is a notion of open sets. More significant is the possibility of hybrid structures; for instance: • the order and, independently, the metric structure induce the standard topology, • the order and the algebraic structure make this set into a totally ordered field, • the algebraic structure and the topology make it into a Lie group.

34 What matters with a structure, that was called “concept” by German mathematicians of the Gottingen School, is that it provides us with a new abstract concept, and, at the same time, it gives a more determined meaning to the underlying set of specified or unspecified elements. It is to say that abstraction brings a richer, not a poorer meaning, even for more general concepts. In other words, structural complexity brings simultaneously syntactic and semantic richness. As W.v.O. Quine stressed many times, the creation of abstract concepts is a semantic ascent,13 which goes hand in hand with the syntactic ascent.

35 2.3. Thus, we observe in mathematics something which is close to Hegel's description of abstract and concrete. What is of concern to us in this description is that it develops further Kant's epistcmological distinction between abstract concept and concrete concept.

36 According to Kant, very abstract concepts give little information about many things, while through concrete concepts we know much about few things.

37 To account for the fruitfulness of mathematics, Kant argues that mathematical knowledge proceeds in concrete-., i.e., presents the concept into a pure and a priori but singular intuition. Thus the division abstract/concrete integrates the division general/ particular and the division class/individual. Kant tells that the concrete usage of a concept is that which is most close to the individual.

38 By contrast, Hegel considers not only the form of knowledge, but also its content, and he detaches the concreteness from its reference to a real-world existent individual: we may have a very abstract, a very poor knowledge of an individual or of a singular situation, while a concept, as a product of knowledge, is an evolving concrete unity, which may get more and more meaning determinations and, then, become more and more concrete. We thus go from abstract to concrete and not vice versa. What matters is how much and via how many ways or viewpoints we know about something at some point of time; what matters is the knowledge-content, the increasing richness and the progressive diversification of knowledge. Knowledge-content is semantic content in its historical

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dimension. Abstractness and concreteness are not fixed forms of the subjective, empirical or transcendental, act of knowing, they are characteristics of knowledge as such, of knowledge as historical and objective product of collective activity. An important gain of that view is that it is now clear that the division abstract/concrete coincides neither with the division general/particular nor with the division class/individual.

39 2.4. Let us return to mathematical practice. If an individual thing (phenomenon, fact, entity, concept, procedure, theory) is not understood through the concrete interconnection within which it actually emerged, exists, and develops, that means that only abstract knowledge has been obtained, e.g., when one has learned what is a group by learning the group axioms without knowing the context of their emergence (history) and at least some of the different situations where they can be applied fruitfully for revealing the structure of a domain or suggesting a solution for some problem (actual practice and problem-solving). Usually, algebraic concepts produce knowledge when they are tied to facts and problems belonging to other mathematical areas, arithmetic, geometry, analysis, topology, etc., or belonging to an earlier stage of the algebraic trend itself, as it is, e.g., the case for the concept of group. If, on the other hand, an individual thing is understood in its objective links with other things forming a coherent network, that means that it has been understood, realized, known, conceived concretely. In such a perspective we can understand how we may have a concrete knowledge of a highly abstract concept, as it happens especially in modern mathematics. Thus an abstract concept becomes concrete not only through its instantiations (realizations, models), but also through the theories in which it plays a role, i.e., through the theoretical or technological applications following from it, and still through the theories to which it gives birth by being included in a more general abstract concept.

40 For instance, the algebraic concept of group is made concrete 1) through its embodiment in arithmetical and geometrical models, 2) through its use to represent symmetry in physics and to classify crystal structures in chemistry, and also 3) through the categorical construction of Grp.

3 Descriptive analysis of the fundamental thinking processes underlying the main ways of getting abstraction

41 The title of this section seems ambitious. However, I must say that since I am no expert in cognitive sciences, I am essentially relying on a more or less direct analysis of actual mathematical procedures combined with information got in cognitive scientists' readings. Cognitive scientists name “categorization” any kind of activity that involves association, comparison, analogy, and correspondence between two or more things. I will detail the actions performed in such an activity, which is in fact the task of getting abstract ideas, from the most simple to the most sophisticated.

42 3.1. Abstracting is a result of several overlapping or intertwined thought operations that I describe now. • Considering things, not necessarily physical ones, not necessarily located in space and time.

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• Comparing things not in themselves but sub specie generalitatis, i.e., comparing them as possible samples of something else, something which is not necessarily already known but only glimpsed and still relatively vague or fuzzy. Precision comes later. • Selecting one or several aspects (qualities, properties, predicates) in the things submitted to comparison and presumed to have something in common, then presumed to be classed (subsumed) under some concept. • Leaving aside or discarding all other aspects, especially specific substantial or space-time aspects. This operation has been called idealization because it comes down to extracting a form, from sundry situations; it has been especially promoted in the beginning of the 20th century by abstract algebra and abstract topology, which made familiar the study of structures not qua being associated with any specific instance. Idealization follows from seeing or guessing some invariant basic properties attached to a plurality of apparently heterogeneous situations and it leads to a unifying view of the different domains on which we perform the same type of operations: counting, addition, subtraction, compactification, etc. Idealization has also a heuristic role in suggesting a possible or unexpected connection with a situation not having been considered at first. The extracted form is not rigid; it may be affected by some controlled variation in passing from a certain type of situations to another one: the addition of two vector subspaces differs from the addition of let us say two real numbers. • Isolating some property or some set of properties of the operation(s) under consideration and viewing them on their own, i.e., transforming the selected conjunction of predicates into a thought-object (Frege's radical separation between concept and object does not fit mathematical practice). Peirce called this kind of transformation “reflective” or “hypostatic” abstraction, Husserl called it “thematization”. Cavailles popularized the term “thematization“, at least among French philosophers. Thematization is especially important in considering as a whole an infinite collection of things; it played a fundamental role in the emergence of set theory, and it has been consistently codified within different frames: Russell type theory, Zermelo-Praenkel system (ZF) and Quine's system (NF). Thematization is essential in passing from a set S of elements to its possibly many structures and from the study of a structure Σ on the set S to the study of the structure Σ in its own right, i.e., to the study of a class of homomorphisms between structures of the type Σ. The standard example is given by the passage from Dedekind's axiomatics for numbers and Hilbert's axiomatics for geometry to Emmy Ncether's style of studying classes of group's homomorphisms, classes of ring's homomorphisms, etc. Attention is paid to homomorphisms rather than to the sets that are respectively source and target of them. It is that attitude that “changed the face of algebra” (see [Artin 1962, 555] and [Weyl 1935, 433]) opening up a wide domain of research and new stuff for developing new insights and new procedures typical of the “begriffliche Mathematik“,14 which was understood as the study of algebraic or topological structures considered in and for themselves. Thematization plays also a role in transforming an abstract object (predicate, concept matching many items possessing similar structures) into a concrete object, which becomes element of some larger class, e.g., the structure of abelian groups viewed as an element of the category of groups. Thematization is still involved in analyzing a concept by breaking down its global unity into components that were formerly tightly connected. Analysis, in this chemical sense, comes out at idealization and thematization; it is disambiguation of meaning by dissociating and studying separately characters, which have been “intuitively” associated during centuries. It was, e.g., the case when Riemann showed (1854) that not every space is a metric space or when Dedekind (1872 or even sooner) showed that not every space is a continuous space.

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Thus the concept of space becomes very general, divested from any particular property, and simultaneously subject to different specifications. By space we understand now any set of elements taken as a substrate for some selected relations and functions and as specifications we get new subclasses of objects, in our examples the subclass of metrical spaces and the subclass of continuous spaces. More generally, by iterated thematization one pushes further mathematical conceptual constructions, as it is well illustrated by category theory, which is a theory of systems of structural theories, treating the notion of structure in a uniform manner: e.g., sets and usual functions form the category of sets (Set), groups with group-homomorphisms (which preserve the group-structure) form the category of groups (Grp), topological spaces and continuous functions (which preserve the topological structure) form the category of topological spaces (Top). Abstracting again, functors are structure-preserving maps between categories. Functors (arrows) are the very objects of category theory; they belong to a higher level of abstraction than morphisms, which in their turn are on a higher level of abstraction than maps. By studying categories and functors, we are not just studying a class of mathematical structures and the morphisms between them; we are studying the relationships between various classes of mathematical structures. This is a fundamental idea, which first surfaced in algebraic topology. Searching for general invariants makes up the dynamic construction of new layers of sophisticated abstraction processes. Abstracting yet again, a “natural transformation” provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved.15 Hence, natural transformations can be considered to be “rnorphisrns of functors“; they yield the usual ho-rnornorphisms of structures in the traditional set theoretical framework. And so on... The point is the endless dynamical concatenation of polysemous symbols and symbolic operations. Of course, this concatenation is not necessarily linear; it forms a kind of tree with interweaved branches at the same level and from lower to higher levels, or, as said above, a complicate and dense network. • Analogies are concurrent with idealization and thematization. Setting up, guessing, or looking for analogy16 between sundry situations is a main way to bring to light similarities, differences and possible relations between two or several thought-objects. Combined with idealization and thematization, analogy is a basic constituent of abstraction. One makes sometimes a distinction between analogy and abstraction. Grounding on the emergence of abstract group theory from 1) the theory of algebraic equations, 2) number theory, and 3) geometry, and on the conception of modern algebra as the study of algebraic structures which came after 1) abstract group theory, 2) abstract field theory and 3) abstract ring theory, Jean-Pierre Marquis argues that it is an empirical fact that analogy concerns two things, while abstraction comes only when three or more things are considered [Marquis forthcoming, 5-6]. Reasoning by analogy is indeed transferring information or meaning from a particular situation to another particular situation. A good example is given by J.-P. Marquis, namely Dedekind's and Weber's work on algebraic number theory and algebraic functions. Another example is the transfer of algebraic laws and tools to logic in the works of G. Boole, A. de Morgan, E. Schroder, etc. Abstraction comes in play when several, and not only two, domains of entities or several classes of structures are a priori in question. Indeed, at a first step a theory is abstract when it has a priori a plurality of models. The plurality criterion is indeed commonly used to distinguish between concrete or material axiomatics and abstract axiomatics, e.g., between Euclid's geometry and Hilbert's

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axiomatization of Cartesian geometry,17 which permits to construct different geometric models by selecting different sets of axioms. At a second step, domains of entities are neglected, while one considers a priori a plurality of structures along with their specific structure preserving rnorphisrns. But, even in the earlier stage of considering similarities, differences and relations between only two situations belonging to the same domain (or only two domains of different entities or only two structure types) is involved the implicit assumption that it must be some abstract framework in virtue of which the transfer from one situation to the other or from one domain to the other or from one structure type to the other is possible. Analysing how analogy works, Henri Poincaré writes that the mathematician must have a direct insight of what makes the organic unity of sundry situations.18 Analogy as a guide for mathematical invention and for great productivity with economy of thought is the chief theme of Poincaré's talk at the 1908 International Mathematical Congress. According to Poincaré, the crucial step is the passage from material to formal19 and from diversity to unification: analogy between materially different entities or procedures appears when one sees, constructs, or supposes a formal similarity between those entities or procedures.20 Formal similarity hints to a unique mould, which may serve for predicting or finding out unexpected analogies with new items and which may thus lead to a more precise view of the architecture of the whole body of mathematics, as it happened with the concept of group. Thus, searching after analogies involves an abstracting mind, if not yet a systematic use of the abstract method.21 There is actually a back-and-forth play between analogy and abstraction: setting up analogies leads to conceive of an abstract theory and, once an abstract theory is at hand, it is used to unearth more and deeper analogies.22

43 3.2. Although I have taken examples mainly from modern mathematics, it must be stressed that abstraction is there from the very first beginning. Even the most elementary notions of mathematics are abstract: the notions of number, of rectangle or triangle or circle, etc., are abstract notions, i.e., products of abstracting processes. For instance, whole positive numbers result from several abstracting processes: associating a symbol with a collection of actual things, dissociating this symbol from this particular collection and associating it with any collection of the same number of things, then establishing a one-to-one correspondence between many different collections, combining this symbol with other symbols similarly generated in order to perform operations like addition, multiplication, and so on. It is only through a long habit that we consider positive integers as given intuitive concrete objects and geometrical figures as concrete spatial visualizations supporting the proof process. Abstraction is always there and is an ongoing process, becoming more and more sophisticated. As Ch. S. Peirce, E. Husserl and J. Cavailles argued, abstraction is “constitutive of mathematical thinking and it can be repeatedly exemplified in the processes of idealizing, thematizing, extracting invariants, and setting up analogies. The more advanced the abstraction process, the more concrete the abstract objects become—classes, structures, operations as such, functions as such, morphisms, categories, etc. Thus it is not a paradox to think that, in mathematics, higher levels of abstraction produce more and more concrete thought-objects, concrete in the double sense that they are complex, individuated objects with various determinations, and that they become concretely known and manipulated through symbolic formulation, precise diagrams or even sketchy drawings. “Concrete” means simultaneously polysemous23 and daily handled.

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44 Ascent towards abstraction is not limited to the logical process of subsuming particulars or particular ideas under a more general concept. Notably Frege rejected the Aristotelian dfaipeaig as being not the only sort of logical abstraction24 and he dissented from the traditional view on concepts; he used mathematical tools, namely a functional relation and an equation for stating a putative logical definition of the concept of number. In most elementary cases indeed a mathematical concept encompasses more thought-processes than only the logical subsumption, to which corresponds the set- theoretic operation of inclusion. In practice mathematicians are dealing with many sorts of operations and calculations and many sorts of relationships between structured sets (one-to-one correspondence and equivalence relation as in the so called Hume's principle,25 linear transformations, group homomorphisms, morphisms, etc.).

45 Subsumption is a fundamental level of classification, but in mathematics fruit-fulness and new insights result from combining it with other abstraction processes, as Prege made clear in his seminal reform of logic and as I shall illustrate below by some mathematical examples. I am not saying that the ascent towards abstraction is not a logical ascent from step to step. I am just saying that, in mathematical practice, at any step, genuine mathematical stuff fills the logical move. This is why I have stressed hereinabove that the ascent is at once semantic and syntactic. Mathematical abstraction is a many-faceted and multi-leveled process and it leads to a sophisticated and branched hierarchy of mathematical concepts and operations. Moreover it is not always the case that the more abstract a concept is the more undetermined it is. For instance, with just a general concept of set as a collection of any things one does not go far. If one wants actual and effective work, one must begin by a meaning determination, i.e., by setting up the axioms ruling a consistent usage of the concept. It happens often that the more abstract is a structure the more overdetermined and stratified it is: axiomatics and category theory give many examples. The mathematical branching of concepts is simultaneously complication of concepts taken in isolation and clarification of their mutual links: bringing to light new and new structures gives more and more power to solve problems not one by one depending on their particularities but uniformly in one go grounding on the general structure fitting all of them.

3.1 Abstraction and axiomatization

46 A rapid look at the history of mathematics, especially of modern mathematics, shows that abstraction is closely tied up with symbolization and axiomatization. Mathematical thinking is thinking with and on symbols and diagrams, may they be considered as representations or as themselves mathematical objects. Anyway, creative manipulation of symbols and diagrams does not dwell only on their drawings; it pertains their meanings and meaningful connections with other symbols and diagrams. Abstract concepts (abstract structures) are usually defined by a finite set of axioms that state the relations to be satisfied by candidates for being models of those abstract concepts. But abstract concepts need not to coincide in every respect with their less abstract counterparts; the meaning changes in between,26 it becomes more sharply determined and yet more ambiguous: not every group is abelian; the multiplication of integers is symmetric, composition of permutations of three objects is not; in category theory the term “structure” has not exactly the same meaning as it has in set theory and in model theory: structures of structures do not always reduce to structures of elements (see

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[Awodey 1996]). There are really different levels of abstraction, even if there are connecting paths between levels.

47 Mathematicians differently oriented have recognized axiomatization as an effective tool for understanding and invention: Dedekind, Hilbert, Emmy Ncether, and Emil Artin, but also Poincaré, Weyl, and Brouwer,27 who did not reject the use of the axiomatic method but rather the view that it might provide a foundation or dispense with calculation and algorithmic proofs. One must distinguish between axiomatics as a fruitful mathematical method and axiomatics as a putative foundation or useless mathematical ideology, which is an epiphenomenon harmful in teaching. In practice, it would be absurd to go without the axiomatic contributions: for instance Galois' theory has been deeply and effectively understood only after Dedekind's, Weber's and Artin's axiomatic presentations. Working with axioms develops new insights and ideas: notably the study of categories is an attempt to axiomatically capture what is commonly found in various classes of related mathematical structures by relating them to the structure-preserving functions between them. A systematic study of category theory then allows us to prove general results about any of these types of mathematical structures directly from the axioms of a category. Mathematics is always aiming at more and more general results about more and more complicated structures.

48 Although axiomatization plays now an indispensable role in mathematical practice, it is not the only way to make mathematical procedures abstract. I will now give a non- exhaustive list of other mathematical abstraction processes that interplay in mathematical thinking and actually illustrate the unceasing iteration of intertwining processes of setting up invariants, idealizing entities and procedures, transforming operations into objects (thematizing), bringing to light analogies between sets, structures, categories, etc.

4 Various samples of mathematical abstraction processes

49 1. Representing an infinite numerical sequence by its law of recurrence. One gets the law by discarding concrete calculation and retaining only how one passes from any element n to its successor. One does not actually know all the elements of the sequence but one knows how to generate the sequence. Here it matters of finding out a rule of calculation, not a concept, but the rule dispenses with enumerating all the elements of the sequence like the concept of even integers dispenses with enumerating all the multiples of 2.

50 2. Discarding the specific nature of the elements forming a sequence, e.g., the sequence of positive integers, so as to characterize the order type of the sequence. For that Dedekind invented the concept of chain.28 What matters here is neither the integers themselves nor even their generative law by itself, but the ordering generated by this law (linear discrete order). The level of abstraction is higher than in the example 1, because we are not concerned with a particular calculation law valid for one particular sequence but with a law type generating an order structure suitable for integers and for sequences of unspecified elements as well. Dedekind's definition shows that integers are a particular instantiation of a general structure; it indicates one way of linking abstraction and generalization.

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51 3. Combining operations (+, x, etc.) and selecting properties of these operations (associativity, commutativity, etc.) in order to form different kinds of mathematical structures (concepts): groups, fields, rings, ideals, lattices, vector spaces, categories, etc., that connect models originating from different mathematical areas or different structures. Abstract concepts are multiply instantiated, they define not one single model nor a single structure, but classes of models and classes of structures. This kind of abstraction is really “modern”, in contrast with Euclid's axiomatic system for geometry, which concerns one single model (the real three-dimensional space) of one single structure (the structure of Euclidean space, realized for instance by the vector space Rn with the standard inner product and by the vector space of real polynomials of degree ≤ n with a convenient inner product). In the spirit of Hilbert's and Bernays' distinction one sets Euclid's material system in contrast with Hilbert's system in Die Grundlagen der Geometric (1899) or Dedekind's system for arithmetic [Dedekind 1888], which are abstract systems (informally presented); moreover one makes a difference between Dedekind/Hilbert's style and Emmy Noether's style of abstraction.

52 In the perspective of abstract set theory we are using, for instance, the following terms, which mostly appeared in the 19th century: • “abstract set”, which surfaces in Cantor's matured theory, • “abstract group”: Dedekind recognized similarities among various mathematical structures, like rotations and quaternions, and identified them as instances of the abstract notion of group [Dedekind 1855-1858, 439]. Heinrich Weber gave, in 1882, axiom systems for groups, and later on these axiom systems have been formalized and investigated in their own right. • “abstract number”, which was used, for instance, by Bolzano in the sense of number as a single entity and in contrast with concrete number, which is number associated to the things being counted [Bolzano 1851]. We owe the abstract axiomatic characterization of the sequence of positive integers to Dedekind through the definition 73 of [Dedekind 1888]: If in the consideration of a simply infinite system N set in order by a transformation ø we entirely neglect the special character of the elements; simply retaining their distinguishability and taking into account only the relations to one another in which they are placed by the order-setting transformation ø, then are these elements called natural numbers or ordinal numbers or simply numbers, and the base-element 1 is called the base-number of the number-series N. With reference to this freeing the elements from every other content (abstraction) we are justified in calling numbers a free creation of the human mind. The relations or laws which are derived entirely from the conditions α, β, γ, δ in (71) and therefore are always the same in all ordered simply infinite systems, whatever names may happen to be given to the individual elements (compare 134), form the first object of the science of numbers or arithmetic. • “abstract field”: this structure has been defined by Steinitz [Steinitz 1910]. • “abstract space”: it surfaced in Riemann's famous paper [Riemann 1854], where a topology and a metric for a space E is defined before defining the functions having their arguments and values in E. Prom 1914 onwards ([Hausdorff 1914]) it was known that a topological space was a set structured by a lattice of open subsets. But it was not until the middle thirties, with the work of Marshall Stone (1903-1989) on the topological representation of Boolean algebras and distributive lattices that this connection between topology and lattice theory began to be exploited, and it became clear that it is possible to construct topologically interesting spaces from purely algebraic data. • In the categorical perspective we are using “morphism”, which is the abstract generalization of structure-preserving mappings between two mathematical structures. In set theory,

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morphisms are functions; in linear algebra they are linear transformations; in group theory, they are group homomorphisms; in topology, they are continuous functions, in manifold theory they are smooth functions (functions having derivatives of all orders), and so on.

53 4. Classifying: This action may be direct as when one collects elements in a set—by the way it is epistemologically meaningful that for “collecting” Kronecker said “begrifflich zusammenfasseri”, expression that might have been come under Dedekind's pen, while Cantor used the presumptive ontological “zusammenseiri”—, or when one collects “material” (interpreted, embodied) structures under the head of an abstract structure of which they are models, or when one collects abstract structures in a category, or when one ranks categories under different types: abelian categories, Cartesian closed categories, complete categories, topos, etc.

54 A more stratified task consists of dividing a set in classes of equivalent elements29 and making up the identity of a class from the equivalence of its members: quotient group, quotient ring, quotient field, etc. Equivalent Cauchy sequences of rational numbers are identified for defining the concept of real number. Similarly, Prege used the process of forming equivalent, namely equinumerical classes for defining positive cardinal numbers. Russell named this kind of definition “the abstraction principle”; it is the subject of many philosophical reflections, but in mathematics even though it is systematically used, it is only one abstraction principle, only one way to perform abstraction, namely forming a quotient structure of some given structure. In particular, this way must not be confused with those listed in 2. (order structure) and 3. (algebraic structure), where, considering a material structure, we do not start by defining an equivalence relation on the underlying set of elements, but we consider the schematic structure itself, independently of the material elements, and examine which compatible relations may be matched for a characterization. Such structural definitions were not welcome in Prege's conception. Prege's abstraction principle was not a mathematical novelty; the novelty lied in introducing a typical mathematical relation, the one-to-one relation, within the scope of logic and presenting this relation as a logical tool for defining a concept.

55 More generally, the equivalence relation is involved in “classification theorems”, which answer the question: “What are the objects of a given type, up to some equivalence?” Example: the Wedderburn theorem (1908), which states that every simple ring that is finite-dimensional over a division ring (a simple algebra) is a matrix ring; it is a way to unify the real numbers, the complex numbers, the quaternions and the square matrices under the same structure. Emil Artin later (1928) generalized this result to the case of Artinian rings (rings satisfy the descending chain on ideals). Several levels of abstraction are crossed from the abstract concept of ring to Artin's theorem. Another famous example is the classification of finite simple groups: every finite simple group belongs to one of four classes (cyclic groups, alternating groups, classical Lie groups, sporadic simple groups). In category theory equivalence is very essential: one reasons on equivalent categories, i.e., categories related by a functor F, which has an inverse G, but the composition of F and G is not necessarily the identity mapping; thus equivalence of categories is less restricted than isomorphism of categories and allows to translate theorems between different kinds of structures.

56 5. Classification is a down top process. Going top down, the converse action is also a way to show the structure of an entity or a procedure by breaking it up into simple pieces: e.g., reducing, factorizing a number, a polynomial, an ideal, in order to unearth

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the basing building blocks. Generally factorization and classification blend together for producing what was named “structure theorems” in the 1930s.30 For instance, Kronecker proved that every finite abelian group is uniquely presented as a direct product of cyclic groups of prime power order; this theorem applies to Galois' theory, to number theory, and to other theories; it is generalized to finitely generated abelian groups and to finitely generated modules over a principal ideal domain;31 in the latter case the structure theorem roughly states that finitely generated modules can be uniquely decomposed in much the same way that integers have a prime factorization. That shows deep connections between arithmetic and algebra: historically that was a result of the project, shared by Kronecker, Dedekind and Weber, to arithmetize algebra, i.e., to bring to light the analogy between divisibility of the integers and divisibility of ideals in a ring.

57 6. Thinking in terms of functional relation, so as to make room for establishing other identity relations than equality of elements of some set, or equinumericity between different sets, or isomorphisms between distinct models of this or that structure. In set theory one associates frequently an element a belonging to a set S to an element α belonging to a set Σ, and one reasons on α as “representative” for a. Although one may describe this process by saying that it consists in seeing a as an α, one must underscore that what is at stake is not the mental content of an idea [Vorstelluny], which would consists in a psychological association of α with a; what is at stake is the presentation [Darstellung] of something as something different but similar in some respect, more exactly the functional association of α with a, which makes α = f(a). It may happen that it is much easier to get results by reasoning on the image α rather than directly on the source element a, and then to come back to a adjusting the obtained results. Dedekind saw a very fundamental way of mathematical thinking in “the ability of the mind to relate things to things, to let a thing correspond to a thing, or to represent a thing by a thing” [”...Dinge auf Dinge zu beziehen, einem Dinge ein Ding entsprechen zu lassen, oder ein Ding dureh ein Ding abzubilden”]. Indeed, a real number x is associated with a certain class

of equivalent Cauchy sequences (xn) of rational numbers, a rational number p/q may be identified with the equivalence class of the ordered pairs of integers (p,q) with q ≠ 0, modulo the relation (p,q) (p',q') iff pq' = qp', etc.

58 More generally, when a structure A is embedded in another structure B by an injection f, every element a of A is identified with its image f(a), in B. f(a) is another way to present a, which then has a multiple identity or, more exactly, we have for a several distinct representatives that we identify as referring to the same entity. When f is a bijection, a and f(a) are distinct but behave in the same way in the structure A and the structure B respectively, A and B being isomorphic. This process turned out to be essential in category theory. As spotted by J.-P. Marquis: “There is no unique, global, and universal relation of identity for abstract objects. [...] Abstract objects are of different sorts and this should mean, almost by definition, that there is no global, universal identity for sorts. Each sort X is equipped with an internal relation of identity but there is no identity relation that would apply to all sorts.”32

59 In mathematics, one looks permanently for new presentations of the “same” entity (or taken to be the same). The concept “real number” is thought through different presentations, actual (Cauchy's sequences, Dedekind's cuts among others) or possible, but it must not be confused with anyone of them. In good cases, different presentations for the “same” entity are provably equivalent in the sense that the meaning of

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theorems valid in one case is preserved by theorems valid in the other case. The question of the “sameness” of referent through different presentations or definitions poses a difficult epistemological problem. Mathematics faces this problem constantly and solves it pragmatically by showing, in case it is possible, an equivalence relation between the entities under consideration.

60 For instance, topological spaces can be defined in many different ways, e.g., via open sets, via closed sets, via neighbourhoods (Hausdorff), via convergent filters, and via closure operations. These definitions describe “essentially the same” objects, what Category theory expresses via the notion of concrete isomorphism.

61 7. Probably the most fundamental action is thinking in terms of invariance; it operates in any mathematical area and corresponds to the task of isolating intrinsic or stable properties of the object under study. One wants indeed to study not only the structure of some entity but also how it behaves under transformations. A few examples are below, taken from arithmetic, geometry, algebra, topology, algebraic topology, and category theory. • The cardinal number of a set is invariant under the process of counting, angles are invariant under scalings, rotations, translations and reflections; for any circle the ratio of the circumference to the diameter is invariant and equal to π. • Felix Klein characterized a geometry by a set of geometric invariants under a given group of symmetries; e.g., lengths, angles and areas are preserved with respect to the Euclidean group E(n) of isometrics (i.e., reflections, rotations, translations and combinations of these basic operations), while only the incidence structure and the cross-ratio are preserved under the most general projective transformations. • Sylvester law of inertia: certain properties of the coefficient matrix of a real quadratic form (homogeneous polynomial of degree 2 in a number n of variables) remain invariant under a change of coordinates. Expressed geometrically, the law of inertia says that all maximal subspaces on which the restriction of the quadratic form is positive definite (respectively, negative definite) have the same dimension. • In Hilbert's invariants theory the finite basis theorem states that every ideal in the ring of multivariate polynomials over a Ncetherian ring is finitely generated (invariance combined with reduction to a basis). Translated into algebraic geometry that means that every algebraic set over a field can be described as the set of common roots to a finite number of polynomial equations. • The normal subgroups of a certain group G are the subgroups of G invariant (stable) under the inner automorphisms of G. • The dimension of a topological space is invariant under homeomorphism. • Algebraic invariants are used for classifying topological spaces up to homeomorphism or, more usually, to homotopy equivalence:33 given two spaces X and Y, we say they are homotopy-equivalent or of the same homotopy type if there exist continuous maps f : X → Y and g : Y → X such that g ℴ f is homotopic to the identity map idX and f ℴ g is homotopic to idY. Going further, one defines the homotopy category as the category whose objects are topological spaces, and whose morphisms are homotopy equivalence classes of continuous maps. Two topological spaces X and Y are isomorphic in this category if and only if they are homotopy-equivalent. Then a functor on the category of topological spaces is homotopy- invariant if it can be expressed as a functor on the homotopy category.

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62 These examples show that the ideas of functional relation, invariance, equivalence, classification, factorization and some others are working together and are using tools from one area (e.g., arithmetic and algebraic tools respectively) for characterizing entities belonging to another area (e.g., algebraic number theory and topological spaces respectively).

5 Conclusion

63 Mathematical abstraction consists in various processes increasing knowledge; so I have considered it from its epistemological aspect rather than from its logical or ontological aspect. The question whether the abstraction process is logical or psychological gives rise to argument. I think the process has evidently a logical side and a psychological side, the latter being by now very much investigated by cognitive scientists and neuroscientists. Prom the point of view of mathematical practice, abstraction is an indispensable tool of work and production. I have been interested here by the multiple ways of constructing and developing mathematical abstract objects, and I have tried to show which permanent actions are involved in all those ways.

64 Abstraction is very often linked with generalization; nevertheless there are abstract and non general objects, such as Dedekind's integers, which are a particular model of a general structure, and there are concepts that are equally abstract but have a different degree of generality: e.g., the concept of group is as abstract as the concept of field and it is more general. I had personally no example of a general procedure or entity, which would not involve abstraction at some level. J.-P. Marquis gives the example of passing from the notion of continuity of a function f : R→R at a point to that of continuity over a real interval [Marquis forthcoming, 17]. That leads me to think that generalization can sometimes be made without using abstracting processes, while any process of abstraction involves generalization.

65 Mathematical abstraction has more than one way; it is not limited to Aristotelian concept formation even though conceptualization, that is to say forming concepts by various procedures, is one essential way and is very characteristic of modern mathematics. Moreover, different ways are simultaneously used in constructions of higher and higher levels.

66 Some ways are known from the beginnings: idealization (geometrical shapes), invariance (invariant ratio between lengths or integers), factorizing (integers). Other ways are more specific of modern mathematics: • making a whole from an infinite number of unspecified elements, manipulating symbols, formulae, diagrams, sets of axioms as being, rather than expressing mathematical objects, • setting up analogies between apparently different objects, sets, structures, theorems, etc., and correlatively dealing with classes of structures and theorems, • considering functional relations or correspondences between elements, structures, functors, • thematizing: ◦ viewing operations of one level as objects of the successor level, ◦ dealing with abstract structures and proving structure theorems with the help of structure-preserving maps, ◦ considering equivalent classes of elements, of structures, of mor-phisms, etc., and proving classification theorems, transferring theorems between categories, etc.

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67 The variety, wide enough, of the examples I have recalled shows that the notion of mathematical abstraction is plural and flexible. The abstraction process is open: new steps towards higher levels yielding more abstract, more sophisticated, and more encompassing concepts are to be expected.

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NOTES

1. For discussing such questions belonging to “the heroic tradition in the philosophy of mathematics” [Kreisel 1985], see among others [Burgess & Rosen 1997], [Zalta 1983], [Rosen 2012], [Parsons 2008]. 2. My approach seems to be similar to Jean-Pierre Marquis' approach in [Marquis forthcoming]. J.-P. Marquis makes fine distinctions between “symbolic”, “formal”, and “abstract”, and also between “abstraction” and “generalization”. 3. See [Burgess & Rosen 1997, 20]. The now standard expression “way of negation” was coined by Lewis in his book [Lewis 1986]. 4. A little more on Aristotle's abstraction in [Szczeciniarz 1999, 4-5]. More in [Cleary 1985, 13-45]. 5. See [Cleary 1985, 35-36]. On Aristotle's view about abstract objects as a result of subtraction: τὰ ἐξ ἀφαɩρέσεως, λεγόμενα, τὰ δι᾽ ἀφαιρἐσεως, τὰ ἐν ἀφαιρἐσει λεγόμενα, see Metaphysics, μ, 1-3. 6. It is necessary to make a distinction between the word “presentation” [Darstellung], which means the objective mathematical way of introducing or using a concept, and the word: “representation” [Vorstellung], which has here its usual meaning of a subjective mental content. Moreover, when an element a of a set E belongs to some equivalence class A ⊂ E, we say that a is a “representative” for A; that means that a stands for any element belonging to A, what again has nothing to do with a subjective (mental) representation.

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7. In my view, it is hard to isolate completely the substrate entity from the operations attached to it. From an abstract point of view operations and properties are even more important than their specific substrate. 8. Poincaré called that “qualitative analysis” [Poincaré 1928, XXI-XXII]. 9. See [Kromer 2007, especially, 81-84]: commutative diagrams play a central role. 10. [Tammet 2005]. When D. Tammet multiplies two numbers, he “see[s] two shapes. The image starts to change and evolve, and a third shape emerges. That's the answer. It's mental imagery. It's like maths without having to think”. 11. I am simplifying the very suggestive although intricate developments of Hegel's drittes Buch of [Hegel 1812-1816]. 12. I thank one of the referees who suggested to use “richness” rather than “complexity”. I am taking indeed meaning-complexity not as a kind of mathematical complexity, algorithmic or measurable in some other way, but as an expression for the polysemous character of many, if not all mathematical concepts. “Number”, for instance, has different meanings depending on whether you consider integers or rational numbers, or real numbers, or quaternions, etc. The polysemous character of mathematical concepts and symbols has been put to the fore by the rise of abstract axiomatics (comments might be found in [Benis Sinaceur 1991]). 13. The semantic ascent “is the shift from talk of miles to talk of ”mile“, it is what leads from the material (inhaltlich) mode into the formal mode, to invoke an old terminology of Carnap; [...] The strategy is one of ascending to a common part of two fundamentally disparate conceptual schemes”, [Quine 1960, 271-272]; [Quine 1990, 33]. 14. The expression “begriffliche Mathematik” was coined by Pavel Alexandroff in his obituary of Emmy Ncether [Alexandroff 1935]. 15. If F and G are functors between the categories C and D, then a natural transformation η from F to G associates to every object X in C a morphism ηxX : F(X) → G(X) between objects of D, called the component of η at X, such that for every morphism f : X → Y in C we have :

ηy ° F(f) = G(f) ° ηx This equation can conveniently be expressed by the commutative diagram:

The notion of a natural transformation states that a particular map between functors can be done consistently over an entire category. Informally, a particular map, let us say an isomorphism between individual objects (not entire categories) is referred to as a “natural isomorphism”, meaning implicitly that it is actually defined on the entire category, and defines a natural transformation of functors; formalizing this intuition was a motivating factor in the development of category theory. 16. I have analyzed different aspects of the fundamental role of analogy in the progress of mathematics in [Benis Sinaceur 2000]. 17. [Hilbert & Bernays 1934-1939, 20]. In Hilbert's and Bernays' terms the distinction is between “inhaltliche und anschauliche Axiomatik” and “formale Axiomatik”. 18. [Poincaré 1900, 127-128]. Notice that Poincaré used “unity” rather than “identity”. 19. Poincaré is using the term “formal” as the opposite of “material” and he underlines the important role of language in discovering new analogies between domains sundry at first sight, but he does not mean a logically formal language, as it is meant in Hilbert's and Bernays' Grundlagen. 20. “En mathématiques, [...] des éléments variés dont nous disposons, nous pouvons faire sortir des millions de combinaisons différentes; mais une de ces combinaisons, tant qu'elle est isolée,

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est absolument dépourvue de valeur ; [...] Il en sera tout autrement le jour où cette combinaison prendra place dans une classe de combinaisons analogues et où nous aurons remarqué cette analogie; nous ne serons plus en présence d'un fait, mais d'une loi. Et, ce jour-là, le véritable inventeur, ce ne sera pas l'ouvrier qui aura patiemment édifié quelques-unes de ces combinaisons, ce sera celui qui aura mis en évidence leur parenté. [...] Si un résultat nouveau a du prix, c'est quand en reliant des éléments connus depuis longtemps, mais jusque-là épars et paraissant étrangers les uns aux autres, il introduit subitement l'ordre là où régnait l'apparence du désordre. [...] ce n'est pas seulement l'ordre, c'est l'ordre inattendu qui vaut quelque chose” [Poincaré 1908, 168-170]. 21. See J.-P. Marquis' fine decomposition of the abstract method into four components in [Marquis 2012, 9-10]. 22. Saying analogy or similarity is not saying identity. While mathematicians are using analogies to set up isomorphisms between sets or equivalence between categories, some cognitive scientists are using the mathematical concept of isomorphism for giving a theoretical explanation of analogy (see e.g., [Gentner 1983, 155-170]). 23. On the polysemy or ambiguity of axiomatic concepts, see e.g., [Benis Sinaceur1991, 191-196]. 24. Actually, Frege thought that Aristotle's analysis was psychological. 25. The name “Hume's principle” was coined by George Boolos. This principleplays a central role in Frege's definition of numbers, and it says that the number of Fs is equal to the number of Gs if and only if there is a one-to-one correspondence (a bijection) between the Fs and the Gs. Boolos and other logicians as well have recognized that Hume's principle is not a logical truth, but from it we can logically deduce what we now call second-order arithmetic. See [Boolos 1998] or [Zalta 2013]. 26. The fact is stressed by J.-P. Marquis in the case of the passage of set and mapping to abstract set and arrow. Marquis writes : “Abstract mathematics, like the concept of mathematical structure, is open, in the sense that it denotes changes with respect to the theoretical tool used to interpret and illustrate the concept” [Marquis 2012, 2, llsqq.]. 27. See e.g., Poincaré's praise of the concept of group [Poincaré 1908]; Weyl, [Weyl 1932, 349] and [Weyl 1951, 464]; Brouwer's conception of geometrical method [Brouwer 1909]. 28. A chain is the minimal closure of a set A in a set B containing A under a function f on B (where being “minimal” is conceived of in terms of the general notion of intersection). 29. Elements of an equivalence class satisfy a relation, which is reflexive, symmetric, and transitive. 30. This expression was commonly used; one can find it for instance under Helmut Hasse's pen [Hasse 1931, 496] (see [Benis Sinaceur 1991, 187-191]). 31. Principal ideal domains (PID) behave somewhat like the integers, with respect to divisibility: any element of a PID has a unique decomposition into prime elements (so an analogue of the fundamental theorem of arithmetic holds); any two elements of a PID have a greatest common divisor, although it may not be possible to find it using the Euclidean algorithm. 32. See [Marquis 2012, 9, fn 20]: “Each sort of abstract entity, for example, monoid, group, ring, field, topological space, partial order, etc., has its criterion of identity. It is certainly a nice feature of category theory that it provides a unified analysis of these criteria of identity as being isomorphisms in the appropriate category.” 33. Two continuous functions from one topological space to another are homotopic iff one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions.

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ABSTRACTS

Mathematical abstraction is the process of considering and manipulating operations, rules, methods and concepts divested from their reference to real world phenomena and circumstances, and also deprived from the content connected to particular applications. There is no one single way of performing mathematical abstraction. The term “abstraction” does not name a unique procedure but a general process, which goes many ways that are mostly simultaneous and intertwined; in particular, the process does not amount only to logical subsumption. I will consider comparatively how philosophers consider abstraction and how mathematicians perform it, with the aim to bring to light the fundamental thinking processes at play, and to illustrate by significant examples how much intricate and multi-leveled may be the combination of typical mathematical techniques which include axiomatic method, invariance principles, equivalence relations and functional correspondences.

L'abstraction mathématique consiste en la considération et la manipulation d'opérations, règles et concepts indépendamment du contenu dont les nantissent des applications particulières et du rapport qu'ils peuvent avoir avec les phénomènes et les circonstances du monde réel. L'abstraction mathématique emprunte diverses voies. Le terme « abstraction » ne désigne pas une procédure unique, mais un processus général où s'entrecroisent divers procédés employés successivement ou simultanément. En particulier, l'abstraction mathématique ne se réduit pas à la subsomption logique. Je vais étudier comparativement en quels termes les philosophes expliquent l'abstraction et par quels moyens les mathématiciens la mettent en œuvre. Je voudrais par là mettre en lumière les principaux processus de pensée en jeu et illustrer par des exemples divers niveaux d'intrication de techniques mathématiques récurrentes, qui incluent notamment la méthode axiomatique, les principes d'invariance, les relations d'équivalence et les correspondances fonctionnelles.

AUTHOR

HOURYA BENIS-SINACEUR IHPST-CNRS-Université Paris 1-ENS Ulm (France)

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The Transition from Formula- Centered to Concept-Centered Analysis Bolzano's Purely Analytic Proof as a Case Study

Iris Loeb and Stefan Roski

The authors thank Arianna Betti and Paolo Mancosu for their comments on earlier versions of this paper, and the anonymous referees for their suggestions. They thank Lionel Mamane for the translation of the abstract into French. Work on this paper was made possible by ERC Starting Grant TRANH 203194.

1 Introduction

1 Bernard Bolzano's (1781-1848) treatise “Rein analytischer Beweis”1 ([Bolzano 1817]; henceforth [RAB]) has been the subject of many studies. Typically these studies discuss the relation between [RAB] and Cauchy's work—did Cauchy plagiarise Bolzano, or not [Grattan-Guiness 1970b], [Freudenthal 1971], [Benis Sinaceur 1973], [Grabiner 1984], [Bottazzini 1986, 97-98]?—focus on the mathematical content of [RAB] (see [Grattan- Guiness 1970a, 51-57, 71-75]; [Bottazzini 1986, 99-101]; [Rusnock 2000, 73-84]) and point out Bolzano's advanced standards of rigor of proofs (see [Bottazzini 1986, 98]) and their underlying philosophical ideas (see [Rusnock 2000, 69-73]). It is usually emphasized that Bolzano was a precursor of later developments in mathematics and that his work thus did not fit into the common mathematical practice of his time. Bottazzini writes, for example, that [t]he arguments that Bolzano brought to his demonstrations and the motives that he brought to his method of reasoning were completely unusual in the context of mathematics at the time. [Bottazzini 1986, 97]

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2 In the current paper we emphasize another side of the story, one touched upon only marginally in the studies just mentioned: we show how [RAB] reflects the transitions in the development of mathematics of Bolzano's time, especially transition from a so- called formula-centered to a concept-centered approach, i.e from an approach in which mathematics is viewed as primarily concerned with formulas and their manipulation to an approach in which concepts, conceived of as independent of any particular formal representation, take center stage.2

3 We argue that [RAB] not only presented Bolzano's remarkable ideas on proofs—ideas both absent in 18th century mathematical practice and ultimately rooted in a millenia old Aristotelian conception of deductive sciences (cf. [Betti & de Jong 2010]3)—but also that [RAB] clearly and in many places witnesses the important transitions of mathematics of the early 19th century. By discussing [RAB] from the perspective of two major transitions in analysis of the 18th and early 19th century, we contribute to a more balanced account of Bolzano's [RAB]. This paper may be considered as a detailed elaboration focused on [RAB] of Rusnock's observation:4 We find him [Bolzano] at the forefront of the movement to recast the calculus as real analysis, moving from the geometrical and algebraic understanding of the subject common in the eighteenth century to one based on logical and arithmetical concepts. [Rusnock 2000, 56]5

4 Let us briefly sketch which changes took place in analysis in the 18th and 19th century. Two major transitions can be identified: First a change from analysis based on geometrical conceptions to an algebraic approach starting in the 1740s, and second a change from that algebraic approach to an arithmetical approach at the beginning of the 19th century [Eraser 1989, 317]; [Liitzen 2003, 156]. As we will discuss below, the algebraic approach is formula-centered, whereas the arithmetical one is concept- centered.

5 A proponent of the first transition was Euler, who as early as 1740 rejected a geometrical proof of a certain theorem of differential calculus because such a proof, he argued, would be “drawn from an alien source” (see [Eraser 1989, 319]). We will see that Bolzano uses a similar argument to reject a proof in [RAB]. Instead, Euler's approach was formula-centered [Sørensen 2005].

6 According to the formula-centered approach mathematics can be said to deal primarily with analytic expressions (formulas) and algebraic manipulations (or: calculations) of these expressions: A function was usually regarded as being an analytic expression; it might be a polynomial, a rational function or an explicit algebraic function; it might involve logarithms, exponentials or trigonometric functions. It might also involve series or products or continued fractions, and it was assumed that the rules of formal algebra applied to these irrespective of any considerations of convergence. [Smithies 1986, 42] Although there was no universal consensus on the notion of function in the 18th century, the view that a function is an analytic expression can be found, e.g., in early Euler [Liitzen 2003, 156]; [Euler 1748]. A wider notion of function, also employed by Euler, allows a function to be given on different intervals by different analytic expressions [Eraser 1989, 326]. A notion of continuity that fits squarely within the formula-centered approach considers a function (in the wider sense) to be continuous if and only if it is given throughout by a single analytic expression [Smithies 1986, 43]; [Jahnke 2003, 124]. Note that this definition of continuity does not capture the property

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of being connected: is a continuous function according to it, even though the function has a vertical asymptote at x = 0.

7 The assumption that the rules of formal algebra are applicable to analysis has been baptised “the principle of the generality of algebra” (following Cauchy's terminology). Eraser characterises this algebraic approach to analysis as follows: The algebraic calculus studies functional relations, algorithms and operations on variables. The values that these variables receive, their arithmetic or geometric interpretation, are of secondary concern. [...] The calculus of Euler and Lagrange differs from later analysis in its assumptions about mathematical existence. The relation of this calculus to geometry or arithmetic is one of correspondence rather than representation. [Eraser 1989, 328]

8 Particularly illustrative in this context is the specific interpretation of equality within the formula-centered approach, sometimes called formal equality, which is a conception of equality governed by formal, algebraic rules. According to this conception, an equality does not express a relation between numbers, but rather states that certain expressions are equivalent in a certain sense— independently of whether they can be interpreted numerically. “Theorems” were hence often seen as nothing more than rules, which could hold even if numerical exceptions were known or if a numerical interpretation was lacking altogether, as in certain infinite series or in parts of infinitesimal calculus.

9 With the decline of the formula-centered approach, the emphasis shifted towards an interpretation of equality as a relation strictly between numbers. This was witnessed by broad methodological change: If one works with a notion of numerical equality rather than formal equality, and if theorems of analysis are not interpreted as rules but as truths about a certain domain of quantities, then numerical exceptions can no longer be admitted and a new quest for rigor of proofs begins.

10 This shift from formal equality to numerical equality is exemplary for the transition that took place, leading away from the central role of expressions. Instead, concepts took that role, often generalising certain traits that were earlier expressed as properties of these expressions [Lützen 2003, 165]; [Grabiner 1984, 113]. For example, while Euler's notion of continuity, mentioned above, was defined as being given by a single analytical expression, during the shift to the conceptual approach to analysis a gradual consensus arose that the notion of continuity should be independent of any particular representation of functions.

11 In what follows we will show that Bolzano's mathematical practice, and especially [RAB], can be understood in terms of these historical shifts. We will see that he rejected the geometrical approach just as his formula-centered precursors did. Moreover, we will maintain that Bolzano's concept-centered tendency appeared not so much in explicit criticisms he made of formula-centered proofs—as was the case with the geometrical approach—but was rather implied in his practice. We will further note that Bolzano does not categorically oppose the methods that are associated with the formula-centered approach. He approvingly mentions, for instance, certain proofs by Gauss that were purely algebraic [RAB, 253]. Moreover, we will see that Bolzano argued in favor of the new rigor of proofs by referring to the Aristotelian ideal of a science.

12 In order to do this we will begin by giving an overview of Bolzano's distinctively Aristotelian take on the sciences in general and on mathematics in particular as it is

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presented in his early Beyträge zu einer begründeteren Darstellung der Mathematik (henceforth [BD]).6

2 Philosophical background

13 Bolzano's [RAB] is often seen as a paradigm example of how philosophical considerations can influence mathematical practice in a fruitful way. It has been pointed out that Bolzano's take on what a proper scientific proof should accomplish led him to strive for a rigour that enabled him to achieve original results in analysis (cf. [Mancosu 1996, 93], [Mancosu 1999], [Detlefsen 2008], [Rusnock 2000]). Central to the relevant philosophical considerations of Bolzano is the idea that all true propositions stand in a unique objective explanatory order, which in later works he calls grounding (Abfolge). We will call this order also the grounding order or the order of grounds and consequences. In [BD] he gives the following description of this order: [T]his much seems to me certain: in the realm of truth, i.e., the collection of all true judgements, a certain objective connection prevails, which is independent of our accidental subjective recognition of it. As a consequence of this some of these judgements are the grounds of others and the latter are the consequences of the former. Presenting this objective connection of judgements and placing them one after another so that a consequence is represented as such and conversely, seems to me to be the real purpose to pursue in a scientific exposition. [BD, 39-40]7

14 Regardless of the basis on which we actually come to know a given truth, it occupies a place in a certain objective order. Bolzano argues that proper scientific proofs (or, in Bolzano's later terminology, demonstrations (Begründungen)) should follow this order of grounds and consequences: [B]y a scientific proof of a truth [we understand] the representation of the objective dependence of it on other truths, i.e., the derivation of it from those truths which must be considered as the ground for it—not fortuitously, but actually [an sich] and necessary—while the truth it self, in contrast, must be considered as their consequence. [BD, 64]; cf. also [Bolzano 1837, §525], henceforth [WL]

15 Bolzano conceived of scientific proofs or demonstrations as explanatory. The grounds on which a given proposition depends explain why that proposition is true [WL, §177] (cf. [Mancosu 1999]). That a proof is explanatory, though, does not mean that it is the most convincing one. And conversely, that a proof is convincing does not mean necessarily that it is explanatory. In particular, thus, that a theorem is obviously true does not deprive the investigator from searching which place it occupies in the explanatory order, i.e., from searching for the grounds that reveal why it is true.

16 Bolzano was never able to clarify exactly what this objective order looks like in a manner that fully satisfied him. In his early work—which stands in the background of [RAB]—Bolzano introduces a number of conditions that proper scientific presentations and proofs have to meet from which we can derive some fundamental properties of the order.8 First of all, proofs that explain why a certain theorem holds must not rely on that very theorem (or on other truths that rely on it) in the course of the proof. Secondly, there is at most one correct such proof for any truth in the order [BD, II § 30]. Finally, Bolzano argues that proper scientific proofs should proceed from general to specific propositions, and from simple to complex propositions [BD II, §§ 26,27] and that they should be pure, i.e., not “cross to another kind” ([BD II, §29] and [Bolzano 1804],

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second of unnumbered pages).9 These last ideas can be nicely illustrated by Bolzano's classification of the mathematical disciplines put forward in [BD], (cf. \BD I, § 20]).

A. General mathematics (deals with “things in general”)

B. Special mathematical disciplines (relative to “special things” they deal with)

I. Actiology (deals with the theory of causes and effects of "unfree things")

II. Theory of “unfree sensible [i.e., perceivable] things”...

a. ... according to their form “in abstracto”

α. Theory of time (where time is the respective form)

β. Theory of space (where space is the respective form)

b. ... according to their form “in concreto”

α. Temporal aetiology

β. Pure natural science

17 We can see a certain order with respect to the generality of the respective disciplines within the classification. The domain of general mathematics consists of things “in general”, whereas the domain of, e.g., pure natural science consists of only those things that are concrete, perceivable and situated in space. We can also note that in each of the disciplines certain notions are introduced by which the domain of the respective discipline is narrowed down (e.g., the notion of time and the notion of concrete object cf. [Rusnock 2000, 33] and the literature cited therein). As we shall see, the theorem that Bolzano sets out to prove in [RAB] belongs to general mathematics. Notably, Bolzano's general mathematics does not encompass geometry. Rather, the latter constitutes a branch of special mathematics (which Bolzano in other contexts also calls applied mathematics), namely the theory of space.

18 Bolzano's views on grounding imply that within proper scientific proofs truths that belong to disciplines that are higher in the hierarchy must not be proven by reliance on truths that belong to disciplines lower in the order (though it is allowed to prove truths lower in the order by truths higher in it). Doing so would break the prohibition on “crossing to another kind”. Note, however, that also internal to the disciplines truths are ordered with respect to their generality and complexity. Using a complex and specific truth in a proof of a more simple and more general one from the same discipline is thus also inadmissible.

19 Bolzano views mathematical truths as being part of an order that is independent of the human mind and also independent of any particular representation. Proper scientific proofs are supposed to determine the place that a given theorem occupies in that order. Since this place is determined by the specific concepts of which the truth is composed, as well as its complexity and extension, acquiring a grasp of the concepts contained in a theorem becomes an indispensable precondition for rigorous proofs [Rusnock 2000, 59]. One must find correct definitions of the concepts that occur in a

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given theorem and determine the domain of objects to which it applies rather than 'blindly' manipulate a formal representation of the theorem.

3 Purely analytic proof

20 Bolzano's [RAB] appeared in 1817, in the early days of the transition from the formula- centered to the concept-centered approach. The full title of [RAB] reads: “Purely analytic proof of the theorem that between any two values, which give results of opposite sign, there lies at least one real root of the equation”. In order to illustrate the transitions visible in Bolzano's paper, it is instructive to take a close look at how Bolzano phrases (and paraphrases) this theorem precisely, and how he sets out to prove it.

21 The theorem mentioned in the title of [RAB] refers to polynomial equations with rational coefficients (cf. [Rusnock 2000, 69]). It is stated in full explicitly in the last section of [RAB] as follows: If a function of the form xn + axn-1 + xn-2 + ... + px + q in which n denotes a positive integer, takes a positive value for x = α, but a negative value for x = β, then the equation xn + axn-1 + xn-2 + ... + px + q = 0 has at least one real root lying between α and β [RAB, 276]

22 Let's call this theorem the “Opposite Sign Theorem” (OST). A special case of OST had been used without proof by Gauss to prove the Fundamental Theorem of Algebra (henceforth FTA) in [Gauss 1815] and [Gauss 1816], which Bolzano praises as a proof that can be understood “in a purely analytical sense” [RAB, 253]. By this, Bolzano as well as Gauss mean—negatively— that the proof does not cross to another kind in making use of geometrical considerations (cf. [Gauss 1815, 33]; [Russ 2004, 144]). We shall see, though, that even though Bolzano and Gauss agree to a certain extent on what a purely analytic proof must not make use of, their opinions differ as to which methods are preferable in such a proof. While Gauss's methods in his “new” proofs of FTA can be placed squarely in a formula-centered approach, Bolzano's proof of OST clearly is a step towards a concept-centered perspective.

23 This becomes evident from the way Bolzano phrases the theorem in the foreword of [RAB] in the context of discussing other proofs for OST. Even though OST (as stated in the title of [RAB]) is a theorem about the roots of polynomial equations, Bolzano frequently refers to the theorem “which is to be proved” in a different way in the course of the foreword. He paraphrases it, for example, as follows: [(I)] (...) every continuous function of x which is positive for one value of x, and negative for another, must be zero for some intermediate value of x. [RAB, 255]

24 And in a slightly different context “the very proposition which we wish to establish” is stated thus: [(II)] (...) every continuous variable function of x, which is positive for x = α, and negative for x = β, must be zero for some value between α and β. [RAB, 258]

25 Since Bolzano does not mention any other theorem in the foreword apart from FTA, it seems that (I) and (II) are intended as paraphrases of OST. But in those paraphrases the theorem does not appear any longer as a claim about polynomial equations, but instead

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as a claim about continuous functions and their values. Note that the notion of continuity, which we find in (I) and (II), neither occurs in the title of Bolzano's treatise, nor in the fully explicit statement of OST in [RAB, § 18] quoted above.

26 In approaching a problem that stemmed from a formula-centered approach to mathematics, Bolzano thus immediately provides a reinterpretation from the point of view of a concept-centered approach. This shift in perspective necessitated a proof quite different than those given for OST beforehand, which Bolzano carefully surveyed. In particular, a rigorous definition of the notion of continuity that appears in (I) and (II) turned out to be necessary for the proof. We will discuss Bolzano's survey of other proofs of OST in the next section. But let us first offer a few remarks on the course of Bolzano's own proof. It is not difficult to see that paraphrases (I) and (II) are in their wording very close to the Intermediate Value Theorem (IVT). The latter theorem is central for Bolzano's proof of OST (cf. [RAB, § 15]) and it is this theorem for which Bolzano's paper is nowadays most famous. This theorem is stated as follows: If two functions of x, fx and øx, vary according to the law of continuity either for all values x or for all those lying between α and β, and furthermore if fα < øα and fβ > øβ, then there is always a certain value of x between α and β for which fx = øx [RAB, 273].

27 Bolzano's proof of this theorem has been discussed extensively in the literature.10 We will content ourselves with quickly pointing out how Bolzano shows that OST can be proved by means of IVT. He proceeds as follows. First, he shows that “[e]very function of the form a + bxm + cxn + ... + pxr, in which m,n,..., r denote positive integer exponents, is a quantity which varies according to the law of continuity” [RAB, 275]. In other words, he shows that the polynomials involved in the equations that OST is about are (or determine) continuous functions [RAB, § 17]. Subsequently, he makes use of IVT to show that these functions will have the value zero for some x (where α < x < β) in case their value is positive for α and negative for β, and argues that this value is then the root of the corresponding polynomial equation [RAB, § 18]. Bolzano's proof thus reduces a theorem which deals of functions “of a certain form” to a more general one concerning continuous functions, which is independent of the particular representation of the functions.

4 Rejections of other proofs

28 Bolzano saw the original contribution of his proof, as sketched in the previous section, not in the presumed fact that he had shown the theorem to be true. This would have been superflous given its general acceptance. Rather, he saw his contribution as having provided a demonstration—a proof that situates the theorem in the objective order of grounds and consequences.

29 To promote his proof, Bolzano therefore also discussed other, known proofs of OST, and explained why they were not acceptable as demonstrations. We discuss three of these rejections below, two of which concern proofs that mathematicians with a formula- centered approach would also (or might) reject. The fact that they rejected the same proofs, sometimes even for similar reasons, does not mean that Bolzano followed their mathematical practice. In one of Bolzano's rejections, for example, he argued for an arithmetical definition of continuity of functions that fits in the concept-centered approach, rather than in the formula-centered. So, although there seems to have been

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some consensus in rejecting geometrical proofs for theorems from analysis, there was no agreement concerning what they should be replaced by.

30 The last rejection that we discuss is one of a proof by Lagrange with a markedly formula-centered approach. The reason that Bolzano gave for rejecting this proof did not directly relate to issues surrounding the formula-centered approach. Rather, he supported his rejection on the basis of his ideas concerning the grounding relation, and specifically on the claim that the ground of a given truth never lies in a more complex truth.

4.1 Proof depending on a geometrical truth

31 The first proof discussed and subsequently rejected by Bolzano is one that depends on the following geometrical truth: Every continuous line of simple curvature of which the ordinates are first positive and then negative (or conversely), must necessarily intersect with the abscissae- line somewhere at a point lying between those ordinates. [RAB, 254]

32 This proposition is, according to Bolzano, correct and obvious, but using it to derive OST is not a demonstration (see Section 2). In a proper scientific proof of a truth from general mathematics such as OST, one must not appeal to truths which belong to a more specific discipline (geometry): (...) the strictly scientic proof, or the objective ground of a truth, which holds equally for all quantities, whether in space or not, cannot possibly lie in a truth which holds merely for quantities which are in space. [RAB, 254]

33 Bolzano refers to the prohibition of kind crossing when rejecting this geometrical proof of a truth from general mathematics [RAB, 254], just as, for example Euler and also Gauss had done in similar cases, as mentioned above in the Introduction and Section 3. Furthermore, Bolzano explains that kind crossing (in the current proof) leads to a circularity, if the proof would be a proper demonstration: If we adhere to this view [that the objective reason of a truth which holds equally for all quantities can lie in a truth which holds merely for quantities which are in space] we see instead that such a geometrical proof is, in this as in most cases, really circular. [RAB, 254]

34 His reasoning goes as follows. First, he argues that the geometrical truth cannot possibly be an axiom,. Hence, there will be truths that constitute its ground. Second, he claims that the most plausible candidate for being a ground for the geometrical truth in question is the “general truth” OST. Since the grounding relation is non-circular according to Bolzano, this proof of OST from the geometrical truth is rejected.

35 In other words: Bolzano links the prohibition of kind crossing to his theory of grounding, yet this is not the only way in which his rejection distinguishes itself from similar rejections by mathematicians who had a formula-centered approach. The main difference is not situated in the reasons for rejecting this geometric proof, but rather in the fact that Bolzano takes a concept-centered approach to OST, as shown by his reformulation (I) of this theorem (see previous section), which appears in this context.

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4.2 Proof using concepts of time and motion

36 The second proof that Bolzano rejects on the basis of methodological considerations reads as follows:11 'If two functions fx and øx', they say, 'vary according to the law of continuity and if for x = α, fa < øα, but for x = β, fβ > øβ, then there must be some value u, lying between α and β, for which fu = øu. For if we imagine that the variable quantity x in both these functions successively takes all values between α and β, and in both always takes the same value at the same moment, then at the beginning of this continuous change in the value of x, fx < øx, and at the end, fx > øx. But since both functions, by virtue of their continuity, must first through all intermediate states, there must be some intermediate moment at which they were both equal to one another.' [RAB, 255]

37 Clearly, temporal vocabulary is employed in the statement of the proof (“at the same moment”, “the beginning”). The proof, Bolzano continues, is then further illustrated by the example of the motion of two bodies. As notions of time and motion are “alien to general mathematics” [RAB, 255], by an argument analogous to the one given above, a proof that makes use of these notions cannot count as a demonstration. However, Bolzano does not reject this proof on that basis. He points out that expressions for time and space are used in a non-essential way, “just to avoid the constant repetition of the same word” [RAB, 255-256]. Accordingly Bolzano rephrases the proof without temporal or spatial expressions. It is exactly in his reformulation—which in the end he also rejects for methodological reasons—that we can identify his concept-centered approach.

38 For example, the notion of continuity that Bolzano distills from the proof is the following [RAB, 256]: A function fx varies according to the law of continuity for all values of x inside or outside certain limits, when f(x + n∆x) can take every value between fx and f(x + ∆x) if n is taken arbitrarily between 0 and 1.

39 Although Bolzano argues that the above is not the correct definition of continuity, but rather a theorem that is actually a special case of IVT, it is telling that he extracts the above definition (instead of one in formula-centered terms). Even more significant is his own proposal for a definition of continuity: According to a correct definition, the expression that a function fx varies according to the law of continuity for all values of x inside or outside certain limits means only that, if x is any such value the difference f(x + ω) — fx can be made smaller than any given quantity, provided ω can be taken as small as we please. [RAB, 256] 40 On first sight the above definition does not refer to analytic expressions and can thus be regarded as concept-centered. The footnote Bolzano attaches to his definition of continuity makes this impression even stronger: There are functions which are continuously variable for all values of their argument [Wurzel], e.g., α + βx. However, there are also others which vary according to the law of continuity only for values of their argument inside or outside certain limits. Thus varies continuously only for all values of x which are < +1 or > +2, but not for the values which lie between +1 and +2. [RAB, 256] Bolzano mentions several analytic expressions in this footnote, but only in order to counter formula-centered definitions like Euler's, in which, as mentioned, a function is continuous if and only if it is given by a single analytic expression. The function of the last example, is given by a single analytic expression and would thus be considered continuous tout court on Euler's definition. Bolzano, introducing a local,

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numerical notion of continuity that captures the geometric intuition of being connected, must instead specify the domain on which he considers the function to be continuous, leaving out the interval on which the function has no (real) value.

41 On the other hand, it can also be argued that the footnote shows that in [RAB] Bolzano was still partially rooted in the formula-centered approach. First, the examples in the footnote are single analytic expressions. This means it is not only unclear whether Bolzano is here considering at all functions not given by analytic expressions, but it is even unclear whether he accepts functions given on different intervals by different analytic expressions.12

42 Second, the word that Bolzano uses for “argument”, Wurzel (“root”), also bears a clear reference to functions as (analytic) expressions. It is, in fact, a value of an unknown which satisfies an equation (see [Russ 2004, 256, fn. f|),such as in the title of [RAB].

43 What we can conclude from Bolzano's definition is that he introduced a local, numerical notion of continuity that did not mention analytic expressions and that was thus in principle also broad enough to capture continuity for a wider notion of function.

4.3 Proof depending on decomposition into factors

44 Just before Bolzano wrote [RAB], Gauss had given two proofs of the fundamental theorem of algebra (FTA) in a formulation that avoids the use of complex numbers [Gauss 1815], [Gauss 1816]; see also [Cain 2005, 1]: [E]very algebraic rational integral function of one variable quantity can be decomposed into real factors of first or second degree. [RAB, 253]

45 According to Bolzano, Gauss's proofs of FTA “leave hardly anything to be desired” [RAB, 253]. Yet when subsequently Lagrange had derived OST from FTA [Lagrange 1808], Bolzano found this latter proof unacceptable.

46 The fact that Bolzano rejects Lagrange's proof is of particular interest, because of his apparent agreement with Gauss's aim to avoid crossing to another kind in his proof, i.e., to avoid any appeal to truths from geometry (see [RAB, 253]; [Gauss 1815, Sect. 1]). This means that of the rejections we discuss, this is the only one in which a proof is rejected that can be placed in the formula-centered approach.

47 Bolzano's reason for rejecting Lagrange's proof is twofold. First, he argues that a proof of OST from FTA is not acceptable as a demonstration: But the fact remains that such a derivation could not be called a demonstration, in that the second proposition [FTA] clearly expresses a much more complex truth than our present one [OST]. The second can therefore certainly be based on the first, but not, conversely, the first on the second. [RAB, 258]

48 Second, Bolzano points out that Gauss's proof of FTA, on which Lagrange's proof relies, makes tacit use of OST. So a proof of OST from FTA would actually be circular, and indeed a logical mistake.

49 Note that neither of the reasons Bolzano gives for rejecting Lagrange's proof has a direct relation to the formula-centered approach. Bolzano does not criticise Lagrange for making a wrong application of algebraic rules or holding an incorrect notion of continuity. Rather, the former is based upon Bolzano's philosophical ideas and the latter on general logical considerations.

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5 Conclusion

50 We have identified a number of elements within Bolzano's mathematical practice that show a clear tendency towards a concept-centered approach in [RAB]. This tendency is visible, in particular, in Bolzano's reformulations of OST and in his employment of a local, numerical definition of continuity. Motivated by his philosophical views on proper scientific proofs, Bolzano was trying to identify and define the concepts which occur in the main theorem of his paper. Accordingly, his proof draws on traits of those concepts rather than on traits of a particular formulation of the theorem. While Bolzano's concept-centered approach can be seen retrospectively as an advancement on the formula-centered one, his attitude towards the latter was one of non- participation rather than of overt criticism. This is witnessed by the fact that he claims Gauss's purely algebraic proofs of FTA leave “hardly anything to be desired” [RAB, 253]. This stands in stark contrast to his attitude towards the geometric approach, which he criticised more than once. It is in the rejection of the geometric approach that we even find a profound agreement between Bolzano's methodological views and the views of others more typically regarded as having favoured a formula-centered approach.

BIBLIOGRAPHY

BENIS SINACEUR, Hourya [1973], Cauchy et Bolzano, Revue d'histoire des sciences, 26, 97-112.

BETTI, Arianna [2010], Explanation in metaphysics and Bolzano's theory of ground and consequence, Logique et Analyse, 211(53), 281-316.

BETTI, Arianna & DE JONG, Willem [2010], The Classical Model of Science. A Millenia-old Model of Scientific Rationality, Synthese, 174(2), 185-203.

BOLZANO, Bernard [1804], Betrachtungen über einige Gegenstände der Elementargeometrie, Prag: Karl Barth. — [1810a], Beyträge zu einer begründeteren Darstellung der Mathematik, Prag: Caspar Widtmann. — [1810b], Contributions to a better-grounded presentation of mathematics, in: The Mathematical Works of Bernard Bolzano, edited by S. Russ, Oxford; New York: Oxford University Press, 83-138, 2004. — [1817], Rein analytischer Beweis des Lehrsatzes, daß zwischen je zwey Werthen, die ein entgegensetztes Resultat gewaehren, wenigstens eine reelle Wurzel der Gleichung liege, Prag: Gottlieb Haase. — [1833-1841], Erste Begriffe der allgemeinen Grössenlehre, in: Bernard Bolzano Gesamtausgabe, Reihe 2A, Bd. 7, edited by J. Berg, Stuttgart Bad-Cannstatt: Frommann-Holzboog, 1975. — [1837], Wissenschaftslehre, in: Bernard Bolzano Gesamtausgabe, Reihe 1, Bd. 11-14, edited by J. Berg & E. Winter, Stuttgart Bad-Cannstatt: Frommann-Holzboog, 1969ff.

BOTTAZZINI, Umberto [1986], The Higher Calculus; A History of Real and Complex Analysis from, Euler to Weierstrass, New York: Springer-Verlag, translated by W. Van Egmond.

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BUHL, Günther [1961], Ableitbarkeit und Abfolge in der Wissenschaftstheorie Bolzanos, Köln: Kölner Universitätsverlag.

CAIN, Harel [2005], C.F. Gauss's proofs of the fundamental theorem of algebra, unpublished.

CENTRONE, Stefania [2012], Strenge Beweise und das Verbot der metdbasis eis alio genos. Eine Untersuchung zu Bernard Bolzanos Beträgen zu einer begründeteren Darstellung der Mathematik, History and Philosophy of Logic, 33, 1-31.

DETLEFSEN, Michael [2008], Purity as an ideal of proof, in: The Philosophy of Mathematical Practice, edited by P. Mancosu, Oxford; New York: Oxford University Press, 179-197.

EULER, Leonhard [1748], Introductio in analysin infinitorum (2 vols), in: Opera Omnia (1), Lausanne: M. M. Bousquet, vol. 8-9.

FRASER, Craig G. [1989], The calculus as algebraic analysis: Some observations on mathematical analysis in the 18th century, Archive for History of Exact Sciences, 39, 317-335.

FREUDENTHAL, Hans [1971], Did Cauchy plagiarize Bolzano?, Archive for History of Exact Sciences, 7(5), 37-392.

GAUSS, Carl Friedrich [1815], Demonstratia nova altera theorematis om-nem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse, in: Werke, Göttingen: Königliche Gesellschaft der Wissenschaften, vol. 3, 31-56, 1876. — [1816], Theorematis de resolubilitate functionum algebraicum integrarum in factores reales demonstratio tertia. Supplementum commenta-tionis praecendentis, in: Werke, Göttingen: Königliche Gesellschaft der Wissenschaften, vol. 3, 57-64, 1876.

GRABINER, Judith [1984], Cauchy and Bolzano: Tradition and transformation in the history of mathematics, in: Transformation and Tradition in the Sciences: Essays in honor of I. Bernard Cohen, edited by E. Mendelsohn, Cambridge; New York: Cambridge University Press, 105-124.

GRATTAN-GUINESS, Ivor [1970a], Bolzano, Cauchy and the “New Analysis” of the Early Nineteenth Century, Archive for History of Exact Sciences, 6(5), 372-400. — [1970b], The Development of the Foundations of Mathematical Analysis from Euler to Riemann, Cambridge, MA; London: MIT Press.

JAHNKE, Hans Niels [2003], Algebraic analysis in the 18th century, in: A History of Analysis, edited by N. Jahnke, H. Providence, RI: American Mathematical Society, 105-136.

JOHNSON, Dale M. [1977], Prelude to dimension theory: The geometrical investigations of Bernard Bolzano, Archive for History of Exact Sciences, 17(3), 261-295.

KITCHER, Philip [1975], Bolzano's ideal of algebraic analysis, Studies In History and Philosophy of Science, 6(3), 229-269.

LAGRANGE, Joseph-Louis [1808], Traité de la Résolution des équations numériques de tous les degrés, avec des notes sur plusieurs points de la théorie des équations algébriques, Paris: Imprimerie de Huzard- Courcier.

LÛTZEN, Jesper [2003], The foundations of analysis in the 19th century, in: A History of Analysis, edited by N. Jahnke, H. Providence, RI: American Mathematical Society, 155-195.

MANCOSU, Paolo [1996], Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century, Oxford: Oxford University Press. — [1999], Bolzano and Cournot on mathematical explanation, Revue d'Histoire des Sciences, 52, 429-455.

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RUSNOCK, Paul [2000], Bolzano's Philosophy and the Emergence of Modern Mathematics, Amsterdam: Rodopi.

Russ, Steve (ed.) [2004], The Mathematical Works of Bernard Bolzano, Oxford; New York: Oxford University Press.

SEBESTIK, Jan [2011], Bolzano's logic, in: The Stanford Encyclopedia of Philosophy, Zalta, E., winter 2011 edn.

SMITHIES, Prank [1986], Cauchy's conception of rigour in analysis, Archive for History of Exact Sciences, 36, 41-61.

SØRENSEN, Henrik Kragh [2005], Exceptions and counterexamples: Understanding Abel's comment on Cauchy's Theorem, Historia Mathematica, 32, 453-480.

TATZEL, Armin [2002], Bolzano's theory of ground and consequence, Notre Dame Journal of Symbolic Logic, 43, 1-25.

NOTES

1. In full: “Rein analytischer Beweis des Lehrsatzes, daß zwischen je zwey Werthen, die ein entgegensetztes Resultat gewaehren, wenigstens eine reelle Wurzel der Gleichung liege”. 2. This terminology goes back to Sørensen, cf. [Sørensen 2005, 454ff] for a more detailed account. 3. Betti and de Jong call this the Classical Model of Science. 4. A view opposite to Rusnock's can be found in [Kitcher 1975], who puts [RAB] in the algebraic tradition and attacks the view that Bolzano inaugurated the arith-metisation of analysis (see esp. [Kitcher 1975, 244]). Kitcher's main argument is that a certain mistake in Bolzano's proof can only be explained from an algebraical point of view, and we do not deny that also certain algebraic traits can still be found in [RAB]. For further critique on [Kitcher 1975], see [Johnson 1977, 264, fn 2]. 5. Similar, more extensive remarks can be found in [Russ 2004, 141-147]. His observations, however, are not concentrated especially on [RAB]. Neither does he emphasize the transition from the formula-centered to the concept-centered approach as we will in this paper. 6. We will not go much into the details of this treatise, and refer the interested reader to the treatments in [Rusnock 2000, chap. 2] and [Centrone 2012]. 7. Page references are to the German original [Bolzano 1810a]. English translations are mostly taken from [Bolzano 1810b]. In some cases we depart somewhat from the translation. 8. In his later work Wissenschaftslehre ([WL]), Bolzano gave a much more detailed account of the grounding relation [WL, § 198-221], though we will not need to go into this for the purposes of this paper. The interested reader may consult [Tatzel 2002], [Buhl 1961], the relevant sections of [Sebestik 2011], and [Betti 2010]. Neither the idea that the realm of truths is ordered by an explanatory relation, nor the specific properties Bolzano ascribes to the order were novel inventions by him (as he acknowledges, cf. [BD, 11 §26]). The first chapter of [Mancosu 1996] offers a useful overview of related concerns by predecessors of Bolzano. 9. The question of how these conditions are precisely to be understood goes beyond the scope of this paper. Cf. [Centrone 2012] for a thorough discussion. 10. Cf. [Rusnock 2000, chap. 3.3], [Russ 2004, 148-151], and the literature cited therein. Rusnock and Russ also provide a discussion of one crucial flaw that can be found in Bolzano's proof. 11. Note that this proof is actually not directly a proof of OST, but of IVT, which is also part of general mathematics in Bolzano's sense.

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12. The acceptance of functions given on different intervals by different analytic expressions as a solution to the vibrating string equation was the subject of the famous debate between Euler and D'Alembert (see [Fraser 1989, 326]). It is clear that, in his later mathematical works, Bolzano rejected the view that functions must be expressible by an analytic expression (cf. [Bolzano 1833-1841, 231f]).

ABSTRACTS

In the 18th and 19 th centuries two transitions took place in the development of mathematical analysis: a shift from the geometric approach to the formula-centered approach, followed by a shift from the formula-centered approach to the concept-centered approach. We identify, on the basis of Bolzano's Purely Analytic Proof [Bolzano 1817], the ways in which Bolzano's approach can be said to be concept-centered. Moreover, we conclude that Bolzano's attitude towards the geometric approach on the one hand and the formula-centered approach on the other were of a different nature; the former being one of rejection, the latter of non-participation. Bolzano supports his concept-centered methodology by philosophical views, which were partially shared by mathematicians with a formula-centered approach to analysis.

Deux transitions ont eu lieu aux XVIIIe et XIXe siècles dans le développement de l'analyse mathématique : de l'approche géométrique à l'approche axée sur des formules d'une part ; de l'approche axée sur les formules à l'approche conceptuelle d'autre part. En nous appuyant sur la Preuve purement analytique de Bolzano, nous montrons qu'il adopte une approche que l'on peut qualifier de conceptuelle. Nous parvenons à la conclusion selon laquelle Bolzano n'adopte pas la même attitude selon qu'il se rapporte à l'approche géométrique d'une part, à l'approche axée sur des formules d'autre part ; dans le premier cas, il est question de rejet, dans le second cas de non- participation. Bolzano appuie sa méthodologie conceptuelle sur des opinions philosophiques partagées en partie par certains mathématiciens partisans d'une approche de l'analyse axée sur les formules.

AUTHORS

IRIS LOEB Department of Mathematics, VU University Amsterdam (The Netherlands)

STEFAN ROSKI Institut für Philosophie, Universität Duisburg-Essen (Germany)

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D’un point de vue rigoureux et parfaitement général : pratique des mathématiques rigoureuses chez Richard Dedekind

Emmylou Haffner

1 Introduction

1 Lorsque Félix Klein décrit, en 1895, le mouvement d’arithmétisation de l’analyse qui a parcouru les mathématiques de Kronecker à Peano, via Weierstrass, c’est un mouvement de rigorisation radicale qu’il dépeint. Une peinture dont il ne souhaite pas tracer les détails, et dont il retient la « rigueur weierstras-sienne » (Weierstrass’sche Strenge), le réductionnisme de Kronecker, l’insistance des tenants de l’arithmétisation sur la nécessité de la logique sous-jacente à l’arithmétique pour s’assurer de la clarté et de la rigueur des mathématiques. Bref, c’est la rigidité logique que Klein choisit de retenir de l’arithmétisation : Comme essence même de la question, ce n’est pas la forme arithmétique de la marche des idées que j’envisage, mais plutôt la rigidité logique obtenue à l’aide de cette forme. [Klein 1895, trad. Vassilieff et Laugel 1897, légèrement modifiée, 117]

2 On sait bien pourtant que si l’on interroge les motivations des grands acteurs de l’arithmétisation1, c’est un camaieu de raisons que l’on aura en main, du rejet de l’intuition à des convictions très fortes sur l’ontologie des mathématiques. Ils ont en commun une volonté de promouvoir une stricte rigueur, certes, mais au nom de quelle rigueur prêchent-ils ?

3 Dans cet article, je propose d’interroger la pratique de la rigueur dans les mathématiques de Richard Dedekind (1831-1916), dont les essais sur les fondements des mathématiques, et tout particulièrement la construction des réels au moyen des coupures en 1872 dans Stetigkeit und irrationale Zahlen [Dedekind 1872]2, illustrent parfaitement les réflexions de Klein. L’exigence de rigueur est une composante

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explicite et essentielle de sa démarche, au-delà de Stetigkeit. Aussi, je ne restreindrai pas l’étude aux travaux fondationnels. Se pose alors la question de l’enjeu plus large de l’arithmétisation. Nous verrons que Dedekind affirme fréquemment que le but de ses travaux est de donner une fondation rigoureuse et d’atteindre une plus grande généralité dans la définition de certaines notions ou dans l’exposition d’une théorie. Il met ainsi en avant une relation à éclaircir entre généralité et rigueur. En puisant dans son corpus mathématique comme dans ses écrits fondationnels, je tenterai de dénouer ces liens, d’en comprendre les raisons et ressorts. Il s’agira alors de montrer et d’expliquer de quelle manière l’idéal logique de rigueur est modulé en fonction d’une pratique commandée par une autre norme épistémique : celle de généralité.

4 Je proposerai dans un premier temps d’étudier quelques-uns de ses travaux les plus importants, dans lesquels Dedekind noue étroitement la demande de rigueur à celle de généralité. Puis, je tenterai d’éclairer sa conception de la rigueur, en discutant l’idéal de rigueur, son rôle dans la pratique du mathématicien et son articulation avec l’arithmétisation. Enfin, je me tournerai vers la question de la généralité, je montrerai que l’exigence de Dedekind de satisfaire à une demande forte de généralité recouvre une conception plurielle de la généralité et déplierai son articulation avec la pratique de mathématiques rigoureuses.

2 Généralité et rigueur

2.1 Théorie des fonctions algébriques

5 En 1882, paraît Theorie der algebraischen Funktionen einer Veränderlichen3 (« Théorie des fonctions algébriques d’une variable ») dans le Journal für reine und angewandte Mathematik, un article co-écrit par Dedekind et Heinrich Weber (1842-1913). Le but des deux mathématiciens est de mettre en place une approche nouvelle de la théorie des fonctions algébriques4 telle qu’inventée par Bernhard Riemann en 1851 [Riemann 1851, 3-45] repr. in [Riemann 18761902], et de proposer ainsi une nouvelle définition de la notion de surface de Riemann. Cet article, longtemps considéré comme une simple application de la théorie des nombres aux fonctions, prend naissance dans le souci de Dedekind et Weber de rendre les travaux de Riemann et de ses héritiers plus clairs et plus rigoureux.

6 Dedekind écrit, dans la biographie de Riemann qui accompagne ses Gesammelte Mathematische Werke parues en 1876, que celui-ci possédait une puissance de pensée et une imagination brillantes qui ne le rendaient pas toujours facile à suivre. Les intuitions fulgurantes de Riemann laissent peu de place à la rigueur. Dedekind était étudiant à Göttingen en même temps que Riemann et a suivi ses cours jusqu’en 18585. Weber, quant à lui, était un spécialiste reconnu des mathématiques riemanniennes, choisi pour seconder Dedekind dans l’édition des Œuvres de Riemann après la mort prématurée d’Alfred Clebsch en 1872. Tous deux familiers des travaux de Riemann, ils souhaitent clarifier son œuvre et lui donner la rigueur qu’ils considèrent indispensable à la bonne compréhension et au bon développement des mathématiques. Dedekind avoue même à Weber qu’il ne sera sûr de maîtriser les travaux de Riemann que lorsqu’il sera venu à bout « de toute une série d’obscurités, avec la rigueur coutumière à la théorie des nombres6 ».

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7 Dans leur article de 1882, Dedekind et Weber proposent une présentation donnant, d’après eux, un fondement plus clair et général, et plus rigoureux aux travaux de Riemann : Dans les recherches suivantes, nous nous donnons comme objectif de fonder la théorie des fonctions algébriques d’une variable, l’un des plus importants résultats de la création de Riemann, d’un point de vue qui soit à la fois simple, rigoureux et parfaitement général7. [Dedekind & Weber 1882, 238]

8 Dedekind et Weber expriment leur insatisfaction face aux traitements donnés par les mathématiciens qui avaient jusque-là travaillé sur la théorie rieman-nienne des fonctions et admettent des théorèmes sur la seule évidence de l’intuition géométrique ou font des hypothèses trop restrictives sur les singularités des fonctions. Dedekind et Weber sont parvenus à la conclusion que : il est possible d’obtenir une base plus sûre pour les notions fondamentales, ainsi que pour un traitement général et sans exception de la théorie, si l’on part d’une généralisation de la théorie des fonctions rationnelles d’une variable. [Dedekind & Weber 1882, 238]

9 Leur article, long d’une centaine de pages et extrêmement technique, est le chemin qui mène à ladite base plus sûre et à une définition qu’ils considèrent comme parfaitement précise, générale et rigoureuse de la notion de point d’une surface de Riemann. Pour atteindre cette définition, ils utilisent les méthodes introduites par Dedekind en théorie des nombres algébriques et « tirées de la création par Kummer des nombres idéaux » qui peuvent être « transférées à la théorie des fonctions » [Dedekind & Weber 1882, 238-239], puisque l’on peut définir, de manière analogue à la théorie des nombres, les notions de corps et d’idéaux de fonctions algébriques8. Ainsi, écrivent-ils que sur la base [des] idéaux premiers, on arrive à une définition parfaitement précise et générale du « point d’une surface de Riemann », c’est-à-dire un système parfaitement déterminé de valeurs numériques qui peuvent être données sans contradiction par les fonctions du corps. [Dedekind & Weber 1882, 240]

10 Le point d’une surface de Riemann, ici, est défini en termes des fonctions du corps considéré, comme une assignation de valeurs (constantes) aux fonctions. La définition de la surface de Riemann donnée par Dedekind et Weber s’appuie alors sur la possibilité d’établir une correspondance bi-univoque entre idéaux du corps et complexes de points. Elle met ainsi en jeu des notions bien définies, en ne faisant intervenir aucune intuition (géométrique ou de quelque autre sorte) et aucune notion trop vague ou mal connue, et sans considérations a priori sur les singularités des fonctions. La présentation des motivations épistémologiques faite dans l’introduction de l’article de 1882 articule sans distinction généralité et rigueur. Il apparaît alors qu’une définition n’est admissible comme définition rigoureuse que si elle satisfait des critères stricts de généralité.

11 Ces considérations ne sont toutefois pas réduites à la théorie des fonctions algébriques et à une réaction à l’encontre des héritiers de Riemann. Des exigences semblables sont avancées à de nombreuses reprises dans les textes de Dedekind.

2.2 Travaux fondationnels

12 Dedekind introduit son essai de 1888, Was sind und was sollen die Zahlen ?9, en soulignant que jusque dans les fondements de « la plus simple des sciences » [Dedekind 1888, Préface à la première édition, 133]10, un certain manque de rigueur persiste en

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mathématiques. Ainsi, on considère, à tort, certaines notions comme évidentes, simples, données par l’intuition interne ou l’intuition géométrique, et l’on omet des définitions ou même de prouver certains résultats.

13 Déjà en 1872, dans la préface à Stetigkeit, Dedekind remarquait que « l’Arithmétique manque d’un fondement réellement scientifique » [Dedekind 1872, 59]. Pour établir une base proprement scientifique et respecter ainsi les exigences strictes de Dedekind, il n’est bien sûr pas autorisé de se reposer sur des théorèmes qui ne sont pas proprement démontrés, sur des hypothèses d’existence illicites, ou encore — c’est le premier moteur de la démarche d’arithmétisation de l’analyse — d’avoir recours à l’évidence ou à l’intuition géométrique [Dedekind 1872, 60]. Dedekind s’évertue alors à trouver un « fondement purement arithmétique et parfaitement rigoureux aux principes du Calcul infinitésimal » [Dedekind 1872, 59-60], ce qu’il considère avoir accompli de manière complètement satisfaisante une fois qu’il a défini ses fameuses coupures, outils ne nécessitant de supposer que l’existence des nombres rationnels et permettant aux nombres irrationnels d’être « définis d’un seul coup et — ce qui est le plus important — dans leur complétude (continuité), propriété qui est suffisante et en même temps, indispensable pour édifier de manière absolument rigoureuse et scientifique l’arithmétique des nombres irrationnels » [Bénis Sinaceur 2008, Lettre à Lipschitz, 27 juillet 1876, 279]. Ainsi, la définition donnée aux nombres irrationnels est à la fois rigoureuse et générale : une définition unique pour tous les nombres, permettant de poser des bases solides pour le calcul avec les irrationnels, et pour les recherches sur la continuité.

14 Lorsqu’il s’agit de définir la suite des nombres naturels, les réquisits de généralité sont plus que manifestes. Ayant constaté le manque de démonstrations rigoureuses aux fondements mêmes de l’arithmétique des nombres naturels, Dedekind propose de définir la suite des entiers naturels sur la base des concepts généraux de système et de représentation (Abbildung). Dedekind, comme il l’explique à Keferstein dans sa lettre du 27 février 189011, souhaite subordonner [les propriétés fondamentales de la suite N] aux concepts généraux et aux activités de l’entendement sans lesquels nulle pensée n’est possible et grâce auxquels le fondement est donné pour des démonstrations sûres et complètes et pour la formation de définitions de concepts non contradictoires. [Bénis Sinaceur 2008, 305]

15 Ainsi « dépouillées de leur caractère spécifiquement arithmétique » [Bénis Sinaceur 2008, 305], les propriétés de la suite des nombres naturels peuvent être données, après de longues considérations techniques, par une définition ensembliste (dans le vocabulaire de Dedekind, « logique »). La définition qui en résulte est même parfaitement générale puisque Dedekind parvient à démontrer ce que l’on appelle aujourd’hui le théorème de catégoricité de (sa version de) l’arithmétique du second ordre, affirmant par là (à rebours d’une interprétation de théorie des modèles) que il est évident que toute proposition sur les nombres, Le., les éléments n du système N simplement infini ordonné par la représentation φ, à vrai dire toute proposition dans laquelle on fait totalement abstraction de la nature particulière des éléments et ne considère que les concepts issus de l’ordre φ, est valide de manière tout à fait générale pour tout autre système simplement infini. [Dedekind 1888, 201, je souligne]

16 En prenant soin de ne pas placer les propriétés arithmétiques au fondement de la définition des nombres, on assure finalement la rigueur de la définition des bases de l’arithmétique elle-même, puisque l’on évite les raisonnements circulaires.

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L’arithmétique pour Dedekind est la science des nombres, et pour qu’elle repose effectivement sur une fondation rigoureuse, il faut lui donner un objet qui soit défini sans recours aux propriétés intrinsèquement arithmétiques, en particulier celles impliquant les quatre opérations fondamentales de l’arithmétique rationnelle. En effet, pour Dedekind, les opérations arithmétiques sont définies par les facteurs sur lesquels elles agissent et nécessitent leur définition préalable.

17 On voit se dessiner plus précisément un lien tissé par Dedekind entre rigueur et généralité : une définition fondée sur des concepts généraux garantit, selon lui, des démonstrations « sûres et complètes ». Baser le concept de nombre naturel même, le plus élémentaire de l’arithmétique, sur les notions parfaitement générales de système et de représentation permet des fondations qui assurent que l’on évite les lacunes, et la mise en jeu de circularités ou de notions trop vaguement définies dans les définitions suivantes (telles celles des opérations et les extensions du concept de nombre) et les raisonnements en découlant.

2.3 Idéaux et théorie des entiers algébriques

18 En 1876, Rudolf Lipschitz invite Dedekind à publier, dans le Bulletin des sciences astronomiques et mathématiques de Darboux, une présentation de sa théorie des entiers algébriques. Dans Sur la théorie des nombres entiers algébriques12 , publié en français en quatre parties en 1876 et 1877, Dedekind propose « d’établir les lois générales de la divisibilité » [Dedekind 1876, Introduction, 278-279] qui régissent le système des entiers algébriques d’un corps de nombres algébriques13.

19 Le problème auquel s’intéresse Dedekind est l’unicité de la décomposition en facteurs premiers pour les entiers algébriques — la plus haute généralisation du concept d’entier [Dedekind 1876, Introduction]. L’entreprise de Dedekind est une généralisation des résultats de Kummer qui, dans ses recherches sur les nombres cyclotomiques, a remarqué que « l’analogie observée jusqu’ici avec l’ancienne théorie [des entiers rationnels] menace de se rompre pour toujours » [Dedekind 1876, Introduction, 281], car la décomposition n’est plus unique. Kummer a pu rétablir l’unicité pour les entiers cyclotomiques en introduisant les nombres idéaux. Dedekind souhaite répondre à la question amenée naturellement par le succès de Kummer pour les nombres cyclotomiques : ces lois de divisibilité sont-elles généralement valides, c’est-à-dire valides dans « tous les domaines numériques de l’espèce la plus générale » [Dedekind 1876, Introduction, 283] (Le., les domaines d’entiers algébriques)14 ?

20 Pour parvenir à la « théorie générale et sans exception » [Dedekind 1876, Introduction, 283], il est nécessaire d’abandonner la démarche plus formelle de Kummer, car si sa découverte est « vraiment grande et féconde » [Dedekind 1876, Introduction, 282], sa généralisation ne s’opère pas sans difficultés : [i]l est (...) à craindre d’abord que, par le mode d’expression que l’on a choisi, dans lequel on parle de nombres idéaux déterminés et de leurs produits, et aussi par l’analogie présumée avec la théorie des nombres rationnels, on ne soit entraîné à des conclusions précipitées et par là à des démonstrations insuffisantes, et en effet cet écueil n’est pas toujours complètement évité. [Dedekind 1876, Introduction, 283]

21 De plus, il est indispensable, du point de vue de Dedekind, de proposer une définition « exacte et qui soit commune à tous les nombres idéaux qu’il s’agit d’introduire dans un domaine numérique déterminé » [Dedekind 1876, Introduction, 283] — une définition

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générale, en somme. Dedekind propose alors de plutôt considérer « l’ensemble a de tous les nombres a du domaine [des entiers du corps] qui sont divisibles par un nombre idéal déterminé » [Dedekind 1876, Introduction, 285]. Un tel ensemble a est alors appelé un idéal et est déni par deux conditions nécessaires et suffisantes : I. Les sommes et les différences de deux nombres quelconques du système a sont toujours des nombres du même système a. II. Tout produit d’un nombre du système a par un nombre du système [des entiers du corps] est un nombre du système a. [Dedekind 1876, Introduction, 286]

22 Ainsi, cette définition, unique pour tous les idéaux, permet-elle de mener à bien la démonstration des « lois fondamentales qui s’appliquent également à tous les corps finis15 sans exception, et qui régissent et expliquent les phénomènes de la divisibilité » [Dedekind 1876, Section IV, 207] pour les entiers d’un tel corps. Pour démontrer la validité des lois de divisibilité dans le domaine des entiers d’un corps de nombres algébriques, Dedekind définit une relation de divisibilité pour les idéaux : pour deux idéaux a et b, a divise b signifie que a contient b ( b ⊆ a). Après avoir exhibé l’équivalence entre divisibilité des idéaux et divisibilité des entiers, il démontre alors la validité des lois pour les idéaux. En vertu de l’équivalence mentionnée, cela revient à avoir prouvé la validité générale des lois pour les entiers du corps étudié.

23 La démonstration est faite en utilisant exclusivement les relations arithmétiques définies. De cette manière, les méthodes utilisées sont essentiellement une généralisation de la notion de divisibilité, et évitent de faire appel à la nature individuelle des éléments du corps ou de s’appuyer sur des éléments trop lourds de contingence comme le choix d’une variable ou une notation particulière. Ainsi, [p]ar la théorie générale des idéaux (...) les phénomènes de la divisibilité des nombres pour tout domaine [d’entiers d’un corps de nombres algébriques] (...) ont été ramenés aux mêmes lois fixes qui régnent dans l’ancienne théorie des nombres rationnels. [Dedekind 1876, §27, 231]

24 La « certitude que ces lois générales existent réellement » [Dedekind 1876, § 27, 231] a été acquise « avec une entière rigueur » [Dedekind 1876, Introduction, 288]. La preuve, valable pour n’importe quel corps de nombres algébriques, en a été donnée sur la base seule des nombres algébriques et de la notion d’idéal, notion ensembliste se réduisant à « collecter » des nombres existants. De plus, on voit bien de quelle manière il apparaît crucial pour Dedekind que la démonstration soit non seulement valable dans n’importe quel corps de nombres algébriques, mais également développée sur la base de concepts généraux au sein de théorie étudiée.

25 Deux questions s’imposent, afin de comprendre réellement le lien tissé entre généralité et rigueur : (1) quelle conception de la rigueur sous-tend la démarche de Dedekind ? Et (2) quelle généralité recherche Dedekind et comment peut-on effectivement l’atteindre ?

3 Vers une élucidation de la conception dedekindienne de la rigueur

3.1 « Dedekind’s principle » et définition

26 L’essai de Dedekind sur les entiers naturels, dont le titre pose la question de la nature ( Was sind, que sont) et du rôle (was sollen, à quoi servent) des nombres, s’ouvre sur une

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affirmation dont le lien avec une construction des entiers naturels n’est a priori pas évident : En science ce qui est démontrable ne doit pas être admis sans démonstration. [Dedekind 1888, 133]

27 Michael Detlefsen, dans son article « Dedekind Against Intuition. Rigor, Scope and the Motives of His Logicism » [Detlefsen 2011], propose une interprétation de cette idée qu’il appelle « Dedekind’s principle », dans laquelle il suggère que ce principe peut être compris comme guidant ce qu’il est acceptable de croire, quelle justification doit être considérée comme acceptable pour une proposition scientifique. Selon lui, Dedekind embrasse la vue selon laquelle The only proper, and the best possible scientific justification of a provable proposition is a proof. More accurately, it is a proof in which ail the basic premises are unprovable. [Detlefsen 2011, 210]

28 Ainsi, explique Detlefsen, le « principe de Dedekind » nous fournit un standard de rigueur pour les mathématiques selon lequel une preuve n’est rigoureuse, et donc acceptable, que si elle repose sur des prémisses dont il est reconnu qu’elles sont improuvables. C’est ce principe qui soutient l’entreprise de Zahlen et qui, du point de vue de Detlefsen, guiderait un certain logicisme dedekindien. Ici, l’argument de Detlefsen s’articule avec l’idée, communément admise au xixe siècle, que les « lois de la pensée » sur lesquelles repose l’édifice construit par Dedekind dans Zahlen, seraient les seules vérités non-prouvables (voir [Detlefsen 2011, 208-210]). Suivant le « principe de Dedekind », l’indé-montrabilité des lois de la pensée en fait de parfaites candidates pour servir d’ultimes prémisses et répondre à la demande de rigueur. This being so, [the laws of thought] provided a similarly objective terminus for the pursuit of rigor. According to Dedekind’s Principle, a genuinely rigorous or scientific mathematics required that proof be pursued to the fullest extent logically possible — that is, to the point where every proposition susceptible of proof had been proved. [Detlefsen 2011, 211]

29 On comprend alors de quelle manière l’essai de Dedekind sur les nombres naturels répond au critère de rigueur exigeant de ne rien accepter sans preuve : la construction de la suite des entiers naturels, et l’arithmétique qui en découle, y sont, en effet, basées sur les lois de la pensée, garantissant que l’on s’appuie sur des prémisses indémontrables. C’est en vertu du principe de rigueur selon lequel il ne faut rien laisser indémontré qui soit susceptible de démonstration, que l’on définit la suite des entiers naturels avec des concepts généraux représentant les opérations de la pensée.

30 Du point de vue de Detlefsen, le standard de rigueur attaché au « principe de Dedekind », lié à l’indémontrabilité des pures lois de la pensée et à un « engagement envers un standard objectif pour déterminer lorsqu’une preuve a été menée jusqu’aux dites prémisses improuvables » [Detlefsen 2011, 211], constituent le « logicisme » de Dedekind. Ici, le « logicisme » est moins un engagement fort sur la nature des objets et vérités mathématiques, qu’une manière de normer ce qui est acceptable comme raisonnement scientifique et en particulier mathématique. De tels préceptes justifient pleinement le refus de l’utilisation de l’intuition dont Dedekind se fait l’écho et qu’il appuie, notamment, sur la possibilité de réduire des « vérités à d’autres plus simples, si longue encore et artificielle en apparence que puisse être la suite des inférences » [Dedekind 1888, 137], mais semblent omettre certains aspects essentiels de ses travaux.

31 Soulignons, en effet, une difficulté qui apparaît si l’on tente de regarder Zahlen comme essentiellement une application du « principe de Dedekind ». L’essai de Dedekind a

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pour but de définir la suite des entiers naturels, il est une réponse à la double question posée en titre de l’ouvrage. Ne devrait-on alors pas questionner le principe de Dedekind et son insertion dans le livre ? Dedekind prouve-t-il ou définit-il ? Faire de ce principe la première phrase de l’essai n’est pas anodin. On peut, semble-t-il, exclure deux possibilités. D’une part, on peut écarter une interprétation logiciste fondée seulement sur le rejet de l’intuition en arithmétique. Si Dedekind refuse ouvertement tout recours à l’intuition16, il ne présente aucun projet fondationnaliste ou réductionniste tenant d’un logicisme à proprement parler17. La logique, comme on peut le distinguer dans les principes posés par Detlefsen comme base au « logicisme », se présente chez Dedekind essentiellement comme instrument pour guider les raisonnements. D’autre part, on peut également exclure la possibilité que le principe renvoie à quelques démonstrations essentielles du livre (la preuve de la validité du raisonnement par récurrence, par exemple), car alors sa cible manquerait le cœur même du livre : la définition de la suite des entiers et des premiers éléments de l’arithmétique (nombres cardinaux, addition, multiplication, etc.). Reste alors à considérer la possibilité d’un jeu entre démontrer et définir.

32 Le principe de rigueur pourrait alors s’énoncer comme une exigence de définition, possibilité d’interprétation étayée notamment par une lettre à Lipschitz du 27 juillet 1876. Dedekind y argumente en faveur de sa définition des nombres réels, dont Lipschitz affirme qu’elle ne dit rien qui ne soit déjà chez Euclide18. Dedekind, analysant la théorie des proportions d’Euclide pour en souligner la différence avec ses propres travaux sur les irrationnels, explique que si l’on souhaite, comme c’est son cas, « édifier l’Arithmétique sur le concept du rapport de grandeur (ce qui n’était pas l’intention d’Euclide) » [Bénis Sinaceur 2008, 280], alors l’évidence de l’existence des grandeurs qui suffisait à Euclide est « absolument insuffisante » : En effet, dans cette manière de fonder l’Arithmétique, la complétude du concept de nombre dépend uniquement de la complétude du concept de grandeur ; mais comme la complétude continue des nombres réels est indispensable pour édifier scientifiquement l’Arithmétique, il est essentiel de savoir d’entrée de jeu avec exactitude le degré de complétude des grandeurs car rien n’est plus dangereux en mathématiques que d’admettre sans preuve suffisante des existences. [Bénis Sinaceur 2008, 280, je souligne la fin du paragraphe]

33 Il s’agit alors « d’hypothèses d’existence illicites » [Bénis Sinaceur 2008, 280]. C’est ici qu’intervient vraisemblablement le « principe de Dedekind » : on vérifie (ou démontre) l’hypothèse d’existence en définissant effectivement la notion dont on suppose l’existence par réduction systématique à des concepts antérieurs. Et c’est exactement ce qui est fait dans Zahlen. Dedekind le mentionne lui-même, les nombres naturels sont un concept que l’on a tendance à ne pas définir rigoureusement, car ils sont souvent considérés (à tort) comme « quelque chose de simple, évident, donné à l’intuition interne » [Dedekind 1888, 137].

34 H. Bénis Sinaceur, dans la note introductive à sa traduction de Zahlen, souligne également la quasi-équivalence entre démontrer et définir dans la phrase d’ouverture de Zahlen, ajoutant que sans une véritable définition, « les démonstrations pourraient être lacunaires, voire circulaires ou impossibles » [Bénis Sinaceur 2008, 96]. Pour prévenir les démonstrations fautives, il faut alors ne rien admettre qui soit définissable sans le définir effectivement.

35 La rigueur chez Dedekind s’articule autour de cette idée que tout ce qui peut être réduit à des vérités « plus simples » doit l’être effectivement, même si pour y parvenir une

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suite d’inférences longue et « artificielle en apparence » [Dedekind 1888, 137] est nécessaire, car il est ensuite possible de fixer la définition pour permettre un développement plus sûr et plus aisé des mathématiques. Par cette réduction, on s’assure de la cohérence du concept que l’on définit, on sécurise les inférences de sa définition et on les maîtrise toutes parfaitement : chaque étape est convenablement justifiée, et chaque étape est clairement comprise. On s’assure ainsi une définition qui permette des démonstrations complètes : rien n’est admis sans démonstration.

36 Cette démarche, à nouveau, n’est pas limitée à la définition des entiers naturels, mais il faut y apporter une nuance et en préciser l’enjeu. Il n’est pas nécessaire, et d’ailleurs il n’est sans doute pas toujours fécond, de systématiquement remonter jusqu’à ces « ultimes prémisses ». Le travail de Dedekind sur le fondement des nombres n’est pas une recherche fondationnelle pour la beauté du geste, elle vise à établir une base rigoureuse et générale utile aux développements futurs des mathématiques. Il convient, par conséquent, de contextualiser le « principe de Dedekind » présenté par Detlefsen. Si, lorsque l’on définit les nombres naturels, il est requis de baser la définition sur des prémisses improuvables et générales, c’est parce que l’on donne un fondement au premier objet de la plus simple des sciences. Dans les développements ultérieurs des mathématiques, et notamment lors de l’extension du concept de nombre, il convient de tirer avantage des fondations déjà posées et de « réduire les longues suites d’inférences » [Dedekind 1888, 138] en basant sur les concepts (antérieurs) déjà bien définis — le but avoué des travaux de définition de Dedekind étant, en effet, de poser des bases solides pour simplifier le travail du mathématicien. Ainsi, qu’il s’agisse des extensions des systèmes de nombres ou de définir/démontrer dans des travaux de mathématiques plus avancées, la réduction à des concepts antérieurs ne doit pas se comprendre comme un réductionnisme, mais comme la proposition que ce qui est bien défini dans les aires conceptuelles antérieures peut être utilisé comme outil pour des développements futurs — c’est le principe essentiel de la stratégie d’arithmétisation proposée par Dedekind.

3.2 Treppen-Verstand

37 La définition des entiers naturels proposée par Dedekind est, de son propre aveu, abstraite au point de paraître transformer les nombres, « amis fidèles et familiers », en « formes fantomatiques » [Dedekind 1888, 136]. Zahlen décompose le raisonnement à la source même de la création des entiers naturels et procède d’une « longue suite d’inférences simples » qui, si elle peut sembler laborieuse et inutile, correspond selon Dedekind à la manière de fonctionner de notre esprit : la « facture progressive de notre entendement » [Dedekind 1888, 136], « Treppen- Verstand » littéralement, « compréhension en escalier »). Dedekind reprend le terme « Treppen-Verstand » dans une lettre à Cantor du 29 août 1899 [Dugac 1976, 261] discutant les recherches de Cantor en théorie des ensembles. L’image choisie par Dedekind d’une compréhension « en escalier » reflète exactement le précepte mentionné dans la partie précédente. La rigueur vient alors assurer le raisonnement du mathématicien — rigueur indispensable, pour Dedekind, à une bonne compréhension comme le montrent ses propos sur les travaux de Riemann.

38 Mais comment, en pratique, met-on en œuvre un tel principe en dehors des fondements des mathématiques (aire finalement restreinte et qui n’est

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mathématiquement pas la plus féconde) ? L’article sur les fonctions algébriques de 1882 semble indiquer une solution possible, d’ailleurs largement exploitée par Dedekind. Il semblerait en effet que pour Dedekind mettre en évidence une construction reposant sur les concepts algébriques de corps et d’idéal et sur les opérations rationnelles de l’arithmétique pour poser des bases rigoureuses et construire un arsenal au service de théories mathématiques plus avancées, soit à même de satisfaire cette demande.

39 Née dans les cours sur la théorie de Galois que Dedekind donne entre 1856 et 185819 et pleinement exploitée à partir de la théorie des entiers algébriques publiée en 1871, la méthodologie que développe Dedekind articule l’utilisation de systèmes (infinis) d’éléments répondant à des conditions de clôture par les opérations rationnelles, et la définition d’opérations arithmétiques (en particulier la multiplication et la division) entre ces systèmes20. Par sa manière de procéder — l’utilisation exclusive de notions arithmétiques — et par ses motivations — donner un fondement logiquement rigoureux — l’approche de Dedekind est typiquement une démarche d’arithmétisation, bien qu’elle soit d’une envergure qui dépasse le cadre généralement attribué à l’arithmétisation.

40 Cette approche permet de concilier les exigences énoncées par Dedekind. On notera tout d’abord, qu’à ses yeux, l’arithmétique possède une rigueur propre, notamment reconnaissable en ce que l’intuition n’y entre pas en jeu. Cet élément essentiel est palpable dans les travaux fondationnels de Dedekind. L’arithmétique, basée sur des concepts clairement définis, procède par inférences (simples) ne recourant qu’à la logique. D’autre part, elle permet également de mener à bien la réduction à des concepts antérieurs sans pour autant remonter systématiquement aux opérations de la pensée — ce qui serait plus laborieux qu’efficace, la bonne définition des nombres et des opérations autorisant à ne pas y revenir. Enfin, avantage non négligeable, l’utilisation de l’arithmétique concilie les exigences de rigueur avec la liberté d’introduction de nouveaux concepts revendiquée par Dedekind21. En effet, l’arithmétique se laisse développer en théorie abstraite, dans la mesure où l’on peut définir des opérations arithmétiques pour des objets qui ne sont pas des nombres, et offre ainsi la possibilité d’introduire de nouveaux concepts, comme les corps ou les idéaux. Ces nouveaux concepts ouvrent de nouveaux champs d’étude en mathématiques et sont également extrêmement féconds comme outils dans d’autres aires des mathématiques — pour la théorie des formes binaires quadratiques ou celle des fonctions algébriques, par exemple.

41 La démarche d’arithmétisation est également présente, comme on le sait, dans la définition des nombres réels. Dans Stetigkeit, Dedekind affirme, déplorant le manque d’un fondement scientifique pour l’analyse infinitésimale et la continuité : Il restait seulement alors à en découvrir l’origine propre dans les éléments de l’Arithmétique et à obtenir ainsi une véritable définition de l’essence de la continuité. [Dedekind 1872, 60, je souligne]

42 Dans l’introduction à Zahlen, Dedekind mentionne également que c’est en fondant les définitions sur l’arithmétique, que l’on peut apporter une véritable clarté aux élargissements successifs du concept de nombre [Dedekind 1888, 138]. De plus, utiliser exclusivement des notions arithmétiques pour étendre l’arithmétique permet de s’assurer qu’aucun élément étranger n’intervienne dans le raisonnement, puisque l’on procède à l’extension des domaines de nombres

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en les ramenant toujours à des concepts antérieurs, et sans interférence de représentations de nature étrangère. [Dedekind 1888, 138] De manière significative, ce type d’argument réapparaît dans la théorie des fonctions algébriques : Ainsi, une partie bien délimitée et relativement étendue de la théorie des fonctions algébriques est traitée uniquement par des moyens appartenant à sa propre sphère. [Dedekind & Weber 1882, 240]

43 La nouvelle théorie est ainsi développée grâce à l’élaboration d’outils conçus au moyen d’éléments conceptuellement antérieurs et bien connus. On s’assure, ainsi, une théorie plus claire, plus précise, dont on maîtrise chacune des étapes, chacune des inférences, et dont les concepts sont définis sur la base de concepts antérieurs déjà bien définis. On s’assure, en somme, une théorie rigoureuse.

44 Pour arriver à une théorie rigoureuse, l’arme de choix de Dedekind s’avère être une combinaison de notions ensemblistes et d’arithmétique rationnelle. La théorie des idéaux, encore une fois, en est un excellent exemple : le nouveau concept d’idéal est un ensemble d’éléments (existants) vérifiant une propriété de clôture par les opérations rationnelles, et dont la théorie est développée en s’appuyant de manière essentielle sur la définition d’une relation de divisibilité entre idéaux. Cette théorie donne la possibilité de mettre en place un arsenal algébrico-arithmétique mettant en jeu des notions relativement simples, comme la divisibilité, et grâce auxquelles il est possible, par exemple, de donner à la surface de Riemann une nouvelle définition plus rigoureuse, plus générale et reposant largement sur les opérations de l’arithmétique. La définition arithmétique paraît ainsi donner, pour Dedekind, la garantie de respecter le critère essentiel de rigueur : tout ce qui peut être défini ou démontré est effectivement défini ou démontré par réduction à des concepts antérieurs grâce aux outils de l’arithmétique. L’arithmétisation comme outil de rigueur semble d’une évidence qui s’approche du truisme. Il est important, donc, de souligner que dans la démarche de Dedekind, arithmétiser ne vient pas seulement répondre à une demande de rigueur, cela vient également servir une exigence de généralité.

45 Il s’agit en effet d’un autre élément essentiel qui sous-tend la mise en œuvre de mathématiques rigoureuses chez Dedekind : la définition, de même que la preuve, doit être basée sur des concepts généraux permettant de définir (et démontrer) d’un même geste tous les éléments concernés. C’est pour cela que les outils logiques des opérations de la pensée sont adéquats pour définir les entiers naturels : on ne peut pas trouver de concepts antérieurs aux nombres entiers qui soient plus simples, et on ne peut pas en trouver de plus généraux.

4 Généralité(s) et pratique de la rigueur

46 Les travaux de Dedekind ont vu le jour dans une volonté soit de démontrer un résultat de manière parfaitement générale (la théorie des entiers algébriques), soit de (re)définir les concepts fondamentaux d’une théorie avec une complète généralité (la théorie des fonctions algébriques, les travaux fondationnels). Dans chaque cas, c’est en passant par l’arithmétique que Dedekind atteint son but22. Il faut souligner qu’arithmétique n’est pas synonyme de propriétés des entiers naturels : tout comme le concept de nombre est étendu successivement — le plus souvent pour répondre aux besoins du développement des mathématiques —, l’arithmétique prend, au fur et à

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mesure de son développement, une envergure plus grande. Avec la possibilité de définir des opérations arithmétiques entre concepts algébriques, elle atteint un stade d’abstraction qui en fait un outil d’une très grande généralité.

47 L’approche algébrico-arithmétique de Dedekind, non contente de servir le « principe de Dedekind », semble permettre également de s’assurer un traitement général — c’est-à- dire sans exception. En théorie des fonctions algébriques, Dedekind et Weber expriment leur insatisfaction face aux travaux des héritiers de Riemann en soulignant leur manque de généralité (et de rigueur) : Dans les travaux effectués sur ce sujet jusqu’ici, en règle générale, des hypothèses restrictives sur les singularités des fonctions considérées ont été faites, et les soi- disant exceptions sont alors soit mentionnées en passant comme cas limites, soit laissées complètement de côté. De même, certains théorèmes fondamentaux sur la continuité ou la développabilité sont admis sans les mentionner, leur évidence reposant sur une intuition géométrique d’une sorte ou d’une autre. [Dedekind & Weber 1882, 238]

48 Les hypothèses tacites sur la continuité ou la développabilité sont évitées chez Dedekind et Weber, car ces considérations doivent être une conséquence de la définition et n’entrent pas en jeu dans l’article. C’est par l’étude arithmétique des corps de fonctions algébriques qu’ils parviennent à une nouvelle (et satisfaisante, de leur point de vue) définition de la surface de Riemann — définition tout aussi arithmétique, puisqu’elle n’est pas seulement étroitement liée à la divisibilité dans le corps de fonctions étudié, sa construction repose également en grande partie sur des opérations arithmétiques.

4.1 Une généralité à plusieurs visages ?

49 Si le réquisit de généralité est exprimé clairement par Dedekind, la caractérisation de la généralité, elle, est beaucoup moins claire.

4.1.1 Généralité des opérations de la pensée

50 Le « principe de Dedekind » mentionné au paragraphe précédent, appliqué à la définition des entiers naturels, amène à fonder leur définition sur les lois de la pensée, ce qui en pratique revient à utiliser comme base de la définition les deux notions mathématiques formalisant les opérations fondamentales de la pensée : les ensembles et les représentations. Si cette fondation est acceptable, c’est d’une part parce que les lois de la pensée sont des prémisses indémontrables comme le souligne Detlefsen ; d’autre part parce qu’elles sont l’élément primitif auquel nous mène l’analyse conceptuelle de la suite des nombres comme le montrent les lettres de Dedekind à Keferstein, mais également parce qu’elles possèdent une parfaite généralité. En tant que logiques, les opérations de la pensée traduites en concepts ensemblistes ont le plus haut degré de généralité, car elles s’appliquent objectivement à toute pensée — et même, elles sont essentielles à la pensée. Les opérations de la pensée ont une généralité universelle, et sont alors, pour Dedekind, plus générales que le nombre. C’est ce qui les place au fondement même des mathématiques et comme seul candidat viable pour définir (rigoureusement) la suite des nombres, l’objet de « la plus simple des Sciences ». Les ensembles et les représentations sont les outils mathématiques formalisant les opérations de la pensée. Aussi sont-ils les outils les plus généraux à la disposition du

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mathématicien, et donc en pratique, les plus appropriés pour fonder la science des nombres.

51 Cette définition fondée sur les opérations de la pensée est la première source de généralité des nombres. Élaborée exclusivement à partir de concepts généraux, la définition donnée par Dedekind (dont il suffira de dire qu’elle est équivalente aux axiomes de Peano, par souci de brièveté) est la définition de tous les systèmes vérifiant les axiomes (les systèmes simplement infinis) et la suite des nombres naturels elle- même est le type abstrait des systèmes simplement infinis23.

4.1.2 Généralité de l’arithmétique

52 Science d’un objet défini de manière parfaitement générale, l’arithmétique acquiert ainsi une généralité équivalente. L’arithmétique est en effet définie de la manière suivante : [L]es relations ou lois qui sont dérivées des seules conditions [définissant les systèmes simplement infinis] et qui, pour cette raison demeurent toujours identiques dans tous les systèmes ordonnés simplement infinis, forment, quels que puissent être les noms donnés aux éléments singuliers, le premier objet de la science des nombres ou Arithmétique. [Dedekind 1888, 180]

53 De manière plus significative encore, Dedekind parvient à démontrer ce que l’on appelle aujourd’hui la catégoricité de l’arithmétique. Ce théorème est vu par Dedekind comme la preuve que les propositions de l’arithmétique sont « valide[s] de manière tout à fait générale » [Dedekind 1888, 201] pour tous les systèmes simplement infinis puisque ceux-ci sont isomorphes24. Ainsi, du point de vue de Dedekind, il est rigoureusement prouvé que l’arithmétique élémentaire est générale. Dedekind évoque même, dans la suite de la remarque qui vient d’être citée, la possibilité de transférer (übertragen) les propositions et théorèmes d’un système à l’autre.

54 Rappelons de plus, que les opérations de l’arithmétique rationnelle (addition, multiplication et leurs opérations inverses) sont définies, elles aussi, en termes d’ensembles et de représentations dans Zahlen. La définition des opérations est alors aussi générale que les développements précédents dans le livre, ce qui vient soutenir la généralité de la science des nombres (qui n’est pas réellement effective sans ses opérations). La possibilité de « transférer » les opérations arithmétiques à d’autres domaines est alors ouverte. Et en effet, dans ses travaux algébriques, Dedekind exploite pleinement la généralité de l’arithmétique et la possibilité de « transfert », en définissant, comme je l’ai déjà évoqué, des opérations arithmétiques pour des objets qui ne sont pas des nombres — et en particulier pour ses concepts algébriques.

55 La généralité de l’arithmétique est prégnante dans les travaux mathématiques les plus élaborés de Dedekind. Elle permet une application, un transfert, des opérations dans des domaines nouveaux ou que l’on n’aurait pas, a priori, considérés comme arithmétisables. Ainsi, Dedekind montre-t-il, dans sa théorie des nombres algébriques, qu’il est possible de définir et d’utiliser (efficacement) les opérations et lois de l’arithmétique rationnelle pour les idéaux de nombres. L’article avec Weber pousse la possibilité de définir des opérations arithmétiques plus loin encore, puisque les méthodes de la théorie des nombres algébriques y sont transférées à la théorie des fonctions algébriques dans laquelle les opérations rationnelles sont utilisées jusque dans la mise en place des éléments nécessaires à la définition de la surface de Riemann.

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56 La généralité de l’arithmétique est alors une généralité horizontale, transférable et valide dans différents cadres, permettant d’en faire un outil puissant répondant conjointement aux exigences de généralité et de rigueur exprimées par Dedekind : bien fondée et générale, l’arithmétique assure une parfaite rigueur.

4.1.3 Généralité comme uniformité

57 Demander que la définition donnée soit unique pour tous les éléments concernés mène à remodeler les définitions selon le cadre dans lequel on se place. Ainsi, dans le cas de la notion d’entier en théorie des nombres algébriques : La plus haute généralisation de la notion de nombre entier consiste dans ce qui suit. Un nombre θ est dit un nombre algébrique lorsqu’il satisfait à une équation n n-1 θ + a1θ + ... + an-1θ + an = 0

de degré fini n et à coefficients rationnels a1, a2,..., an ; il est dit un nombre entier algébrique, ou plus brièvement un nombre entier, lorsqu’il satisfait une équation de

la forme ci-dessus dans laquelle les coefficients a1, a2,..., an sont tous des nombres entiers rationnels. [Dedekind 1876, Introduction, 278-279]

58 Ici, le nombre entier algébrique est la plus haute généralisation, le concept de nombre entier « le plus général » car tout entier (entiers naturels, rationnels, de Gauss...) est un entier algébrique : la définition est donc valable pour tous les nombres entiers. Dans Zahlen, en revanche, ce que Dedekind propose est la définition « la plus générale » du concept de nombre naturel.

59 Pour qu’une définition soit générale, elle doit définir d’un seul geste tous les éléments qui tombent sous le concept. Il ne doit y avoir aucune exception demandant un traitement spécial, pas de stratégie impliquant la distinction de plusieurs cas, et elle doit éviter toute contingence (telle que s’appuyer sur la notation) et de prendre comme élément essentiel quelque chose d’arbitraire (comme c’est le cas lorsque l’on doit choisir une variable). Un élément crucial, et sur lequel je reviendrai, est que si l’on est capable de donner une telle définition, alors on pourra déduire des démonstrations aussi générales. Une définition répondant à ces exigences doit permettre de donner des preuves suivant les mêmes principes.

60 C’est cette généralité que Dedekind et Weber recherchent en théorie des fonctions algébriques : la possibilité de traiter des fonctions algébriques sans avoir à poser a priori des restrictions sur les singularités des fonctions ou des hypothèses sur leur comportement individuel. Mieux encore, une telle définition permet de développer toute la théorie sans exceptions. La généralité, ici, est relative au cadre de travail, ce n’est plus une universalité mais la possibilité d’un traitement uniforme de la théorie.

61 La recherche de généralité en tant que partie intégrante de la pratique de mathématiques rigoureuses chez Dedekind apparaît comme extrêmement dépendante du contexte dans lequel on travaille, dans le sens où Dedekind cherche à donner les définitions et preuves les plus générales possibles au sein de la théorie dans laquelle il travaille. Ainsi, par exemple, la théorie des idéaux développée dans la théorie des entiers algébriques est appelée « théorie générale des idéaux » parce qu’elle est la théorie des idéaux de n’importe quel corps de nombres. La dépendance au cadre de travail est un élément très important qui souligne que la généralité n’est pas une notion purement philosophique, mais à articuler avec la pratique.

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62 De plus, il n’est pas utile, ni même conseillé, de rechercher systématiquement la plus grande généralité. Au contraire, il faut être attentif à la maniabilité et la fécondité effective et immédiate de la définition que l’on propose... ainsi qu’à sa rigueur : Bien sûr, tous ces résultats peuvent être obtenus de la théorie de Riemann avec un investissement de moyens bien plus limité et comme cas particuliers d’une présentation générale beaucoup plus étendue ; mais il est alors bien connu que fonder cette théorie sur une base rigoureuse offre encore certaines difficultés, et tant qu’il n’aura pas été possible de surmonter complètement ces difficultés, il se pourrait que l’approche que nous avons adoptée, ou au moins une qui y soit apparentée, soit la seule qui puisse conduire réellement avec une rigueur et une généralité suffisantes à ce but pour la théorie des fonctions algébriques. [Dedekind & Weber 1882, 240]

4.2 La généralité, une condition pour un développement rigoureux ?

63 Le lien tissé entre généralité et rigueur apparaît donc très étroit. La généralité des définitions pourrait-elle ne pas seulement accompagner la rigueur, mais en être une condition ?

64 La généralité d’une définition, comme celle d’une théorie, dépend largement des outils utilisés pour la mettre en place. Rappelons que si l’on base une définition sur des concepts antérieurs plus généraux, on prévient le risque d’introduire des notions « étrangères » dans la théorie. On évite également les raisonnements circulaires, les démonstrations lacunaires, les hypothèses d’existence illicites. On s’assure enfin la possibilité de donner une définition générale : une définition unique pour tous les cas concernés. Il s’agit alors d’élaborer des concepts généraux, taillés sur mesure, pour assurer le bon fondement de la théorie que l’on souhaite étudier. Pour les entiers naturels, ce sont les ensembles et représentations et pour les nombres irrationnels, la notion de coupure. Pour la théorie des entiers algébriques et pour celle des fonctions algébriques, ce sont les concepts algébriques (corps, module, idéal) qui serviront cette fin.

65 Dans la théorie des nombres algébriques, évoquée en première partie de cet article, Dedekind se donne pour but de donner une « théorie générale échappant à toute exception » [Dedekind 1876, Introduction, 278] et y parvient en fondant la théorie sur la notion d’idéal. En référence à la définition des idéaux par les deux conditions nécessaires et suffisantes de clôture, Dedekind écrit : [La] constatation [que les propriétés sont des conditions nécessaires et suffisantes] m’a conduit naturellement à fonder toute la théorie des nombres du domaine [des entiers du corps] sur cette définition simple, entièrement délivrée de toute obscurité et de l’admission des nombres idéaux. [Dedekind 1876, Introduction, 287]

66 L’insistance sur la généralité des définitions et des concepts utilisés comme bases porte en creux l’idée que si l’on parvient à donner la « bonne » définition, les démonstrations sûres et complètes suivront par une suite simple d’inférences logiques élémentaires. Ainsi, souvent chez Dedekind, les théorèmes « résultent immédiatement » [Dedekind 1854] dans [Bénis Sinaceur 2008, 227], les lois sont « dérivées des seules conditions » [Dedekind 1888, 180] de la définition. Les démonstrations « découlent immédiatement des définitions » [Dedekind 1872, 81].

67 La théorie des entiers algébriques illustre particulièrement bien cette idée qu’en basant la théorie sur la « bonne » définition, générale et soigneusement choisie25 et en

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identifiant les méthodes les plus appropriées pour développer la théorie, les propositions et théorèmes et leurs démonstrations suivent facilement. Avoir identifié la bonne notion d’entier pour les nombres algébriques permet de développer la théorie des idéaux dans les corps de nombres algébriques, théorie qui donne à Dedekind la possibilité de prouver la validité générale des lois de divisibilité en suivant un développement analogue à la théorie des nombres rationnels en utilisant seulement les opérations rationnelles. C’est également le cas de la théorie des fonctions algébriques qui est développée en analogie avec les corps de nombres algébriques et repose largement sur les opérations de l’arithmétique rationnelle.

68 On voit bien ici quel avantage Dedekind a pu voir dans l’utilisation des idéaux : plutôt que s’engager dans des développements et calculs laborieux, il peut déployer une théorie extrêmement élaborée et féconde mettant en œuvre essentiellement les « principes les plus simples de l’arithmétique », dont on est assuré de la généralité et de la définition rigoureuse26.

69 Ainsi, l’exigence de généralité pour les définitions est par extension une garantie de généralité et de rigueur pour les démonstrations. Cela permet, notamment, de donner des preuves rigoureuses qui manquaient encore. Un exemple bien connu que prend souvent Dedekind pour souligner l’efficacité de sa définition des irrationnels, est qu’elle lui permet de fournir la première (d’après lui) démonstration rigoureuse d’affirmations telles que √2.√3 = √627. On peut souligner également que, dans Algebraische Funktionen, Dedekind et Weber donnent la première véritable preuve du théorème de Riemann- Roch [Dedekind & Weber 1882, §§ 28-29]28. Mais les conséquences positives vont plus loin encore que la seule possibilité de mener à bien les démonstrations au sein de la théorie étudiée, on donne également la possibilité d’assurer le développement futur des mathématiques. Une définition correcte des irrationnels est essentielle, car il s’agit de fournir une caractéristique précise de la continuité, utilisable comme base de véritables déductions. [Dedekind 1872, 72]

70 Cette remarque est également faîte également en théorie des fonctions algébriques, dans laquelle continuité et considérations topologiques n’entrent pas en jeu, mais pourront bénéficier des fondements établis dans l’article : Dans notre travail, (...) un long détour est fait pour donner un fondement algébrique à la théorie des idéaux et ainsi obtenir une définition du « point d’une surface de Riemann » totalement précise et rigoureuse, laquelle pourra servir de base pour les recherches sur la continuité et toutes les questions qui y sont liées. [Dedekind & Weber 1882, 241]

71 De cette manière, on s’assure un développement général et rigoureux de la théorie entière grâce à une définition « totalement précise et rigoureuse » du point de la surface de Riemann puisque la théorie repose de manière essentielle sur la notion de surface de Riemann. Grâce à la bonne définition du point de la surface de Riemann, une base stable est garantie au mathématicien souhaitant approfondir la théorie et mener des investigations nécessitant la mise en place de notions topologiques — de la même manière que la bonne définition des nombres irrationnels apparaît essentielle pour l’Analyse.

72 Le point essentiel des recherches de Dedekind est donc d’assurer une base générale et rigoureuse pour soutenir et porter le développement des mathématiques. Prégnant dans le cas des fonctions algébriques, cet aspect pragmatique est également présent dans les recherches fondationnelles de Dedekind qui souhaite « inciter d’autres

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mathématiciens à réduire les longues suites d’inférences à des proportions plus modestes et plus agréables » [Dedekind 1888, 138] ou donner la base de « véritables déductions » sur la continuité (cf. citation ci-dessus). La preuve de la validité générale des lois de divisibilité des entiers algébriques elle-même est surtout « d’un intérêt on ne peut plus pratique » [Dedekind 1876, §27, 231], car offrant la base de développements plus larges (arithmétique modulaire, division du cercle, formes quadratiques...) et d’après Dedekind : la certitude que ces lois générales existent réellement facilite au plus haut degré la démonstration et la découverte des phénomènes spéciaux qui se présentent dans un corps déterminé. [Dedekind 1876, §27, 231]

4.3 Conclusion

73 La rigueur, que Klein assimile à une rigidité logique des démarches d’arithmétisation, se présente dans les mathématiques de Dedekind comme un élément structurant d’une pratique des mathématiques visant à établir les théories sur des bases solides et générales, dans l’optique de développements futurs. L’arithmétisation, dans l’œuvre de Dedekind, vient servir ces objectifs en permettant de mettre en place des arsenaux mathématiques construits à partir de concepts antérieurs indémontrables ou déjà bien définis, ciselés pour atteindre ce but, et suffisamment puissants et généraux non seulement pour offrir un bon fondement à la théorie étudiée mais également pour stimuler et faciliter le développement de pans entiers des mathématiques — baser la théorie des nombres algébriques sur la notion d’idéal plutôt que sur celle de congruence supérieure, par exemple, est décisif. Le principe de rigueur s’éloigne ainsi d’un principe purement logique pour embrasser pleinement les exigences de la pratique mathématique. La rigueur rencontre alors une autre valeur épistémique tout aussi essentielle, la généralité, qui va infléchir la mise en œuvre de l’idéal de rigueur.

74 Rigueur et généralité s’articulent chez Dedekind de deux manières distinctes mais qui ne sont pas indépendantes. On les retrouve d’une part, dans une démarche de recherche de fondements pour les théories mathématiques : tout ce qui peut être prouvé ou défini doit effectivement être prouvé ou défini. C’est en basant les définitions sur des concepts antérieurs et généraux, que l’on s’assure de répondre à cette demande. D’autre part, rigueur et généralité s’allient sur ce que la définition doit pouvoir permettre d’accomplir, sur la motivation pratique d’une recherche systématique de fondements généraux et rigoureux : on souhaite que la définition soit générale afin qu’elle permette de mener chaque démonstration d’un seul geste, sans avoir à traiter de cas particulier ou s’appuyer sur des éléments contingents ou extérieurs, comme la notation.

75 Ainsi, tout ce qui est en amont de la définition doit être démontré, tout ce qui est en aval de la définition doit être (rendu) démontrable. La généralité est alors au service de la rigueur, justifiant des choix tactiques orientés pour favoriser le développement des mathématiques.

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EPPLE, Moritz [2003], The end of the science of quantity : Foundations of analysis 1860-1910, dans A History of Analysis, édité par N. Jahnke, H. Providence : American Mathematical Society, 291-323.

FERREIRÓS, José [2007], Labyrinth of Thought: A History of Set Theory and its Role in Modern Mathematics, Boston: Birkhäuser.

GEYER, Wulf-Dieter [1981], Die Theorie der algebraischen Funktionen einer Veränderlichen nach Dedekind und Weber, dans Richard Dedekind, 1831/1981 : Eine Würdigung zu seinem, 150. Geburtstag, édité par W. Scharlau, Braunschweig : Vieweg, 109-133.

HOUZEL, Christian [2002], La Géométrie algébrique. Recherches historiques, Paris : Blanchard.

JAHNKE, Hans Niels & OTTE, Michael [1981], Origins of the program of "arith-metization of mathematics", dans Social History of Nineteenth Century Mathematics, édité par H. Mehrtens, H. Bos & I. Schneider, Boston: Birkhäuser, 21-49, http://dx.doi.org/10.1007/978-1-4684-9491-4_3

KLEIN, Félix [1895], Über Arithinetisierung der Mathematik, dans Nachrichten der König. Gesel. der Wissen. Göttingen. Geschäftliche Mitteilungen, edité par R. Fricke & H. Vermeil, Berlin ; Heidelberg : Springer, 82-91, reproduit dans Felix Klein Gesammelte Mathematische Abhandlungen, 1921, vol. 1 édité par Fricke, R. et Vermeil, H., Berlin : Springer, 232-240. Trad. fr. A. V. Vassilieff, L. Laugel,

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1897, Nouvelles Annales de Mathématiques 3, 16, 114-129, URL : http://dx.doi.org/ 10.1007/978-3-642-51958-l_14.

PETRI, Birgit & SCHAPPACHER, Norbert [2007], On arithmetization, dans The Shaping of Arithmetic after G. F. Gauss’s Disquisitiones arithmeticae, edité par C. Goldstein, N. Schappacher & J. Schwermer, Berlin ; Heidelberg ; New York : Springer, 343-374.

RIEMANN, Bernhard [1851], Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse, dans Gesammelte mathematische Werke und Wissenschaftlicher Nachlass, edité par R. Dedekind, H. Weber, M. Noether & W. Wirtinger, Leipzig : Teubner, 3-46, reproduit dans [Riemann 1876-1902, 3-46]. — [1876-1902], Gesammelte mathematische Werke und Wissenschaftlicher Nachlass, Leipzig : Teubner, edited by Dedekind, R., Weber, H., Noether, M., and Wirtinger, W.

SCHAPPACHER, Norbert [2010], Rewriting points, dans Proceedings of the International Congress of Mathematicians, Hyderabad, India, 3258-3291.

SCHARLAU, Winfried [1981], Richard Dedekind, 1831/1981 : Eine Würdigung zu seinem 150. Geburtstag, Braunschweig : Vieweg.

SIEG, Wilfried & SCHLIMM, Dirk [2005], Dedekind’s analysis of number : systems and axiom, Synthese, 147(1), 121-170.

STILLWELL, John [2012], Theory of Algebraic Functions of One Variable, Providence, R.I. : American Mathematical Society, trad. angl. de [Dedekind & Weber 1882].

STROBL, Walter [1982], Über die Beziehung zwischen der Dedekindschen Zahlentheorie und der Theorie der algebraischen Funktionen von Dedekind und Weber, dans Abhandlungen der Braunschweigischen Wissenschaftlichen Gesellschaft, t. 33, 225-246.

TAPPENDEN, Jamie [1995], Geometry and generality in Frege’s philosophy of arithmetic, Synthese, 102, 319-361.

NOTES

1. Pour une présentation plus large du mouvement d’arithmétisation de l’analyse dans son sens communément accepté de rigorisation des fondements de l’analyse et rejet de l’intuition, on pourra consulter [Jahnke & Otte 1981], [Epple 2003] et [Pétri & Schappacher 2007]. Par ailleurs, Schappacher dans [Schappacher 2010] propose une interprétation intéressante de l’arithmétisation, liant la définition des réels à celle du point d’une surface de Riemann dans l’article co-écrit par Dedekind et Heinrich Weber en 1882. Le point de vue que j’adopterai concernant l’arithmétisation dans l’œuvre de Dedekind se détache des études menées jusqu’ici, puisque je proposerai de l’envisager sous un angle plus systématique et d’interroger effectivement la conception de l’arithmétique qui soutient cette démarche — question jusqu’ici passée sous silence. 2. Je désignerai par la suite cet ouvrage par Stetigkeit. 3. Je désignerai par la suite cet article par Algebraische Funktionen. Pour une présentation du contenu mathématique de cet article, on pourra se référer à [Geyer1981], [Strobl 1982], à l’introduction de [Stillwell 2012], ainsi qu’à [Houzel 2002], quioffre également une mise en perspective avec le contexte historico-mathématique en géométrie algébrique. Cependant, le contenu mathématique en lui-même n’est pas aucentre de mon argumentation, qui vise à expliciter les motivations épistémologiques qui ont incité à et guidé sa rédaction — questions qui n’intéressent les auteurs de ces études historiques que de manière tangente.

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4. Une fonction algébrique θ d’une variable indépendante z est une fonction satisfaisant une équation polynomiale F(θ, z) = 0 dont les coefficients sont des polynômes. 5. De nombreux détails historiques et épistémologiques sur les relations entre Riemann et Dedekind peuvent être trouvés dans [Ferreiŕos 2007]. 6. “I am not the profound expert on Riemann’s work that you take me to be. I certainly know those works, and I believe in them, but I do not master them, and I will not master them until having overcome in my way, with the rigor that is customary in number theory, a whole séries of obscurities”, trad. dans [Ferreiŕos 2007, 78]. 7. La traduction de Algebraische Funktionen utilisée tout au long de cet article est la mienne. Je tiens à remercier Karine Chemla pour son aide précieuse. Les pages réfèrent à la version originale reproduite dans [Dedekind 1930-1932, vol. 1, 238-350]. 8. La notion de corps en théorie des nombres est définie de la manière suivante : J’appellerai corps tout système A de nombres a (ne s’annulant pas tous), tel que les sommes, les différences, les produits et les quotients de deux quelconques de ces nombres a appartiennent au système A 1876, § 14] En théorie des fonctions algébriques, la notion de corps est présentée ainsi : De manière analogue à la théorie des nombres, on entend par corps de fonctions algébriques un système de ces fonctions de telle nature que l’application des quatre opérations fondamentales de l’arithmétique aux fonctions de ce système donne encore une fonction de ce système. [Dedekind & Weber 1882, 239] Pour la définition d’un idéal en théorie des nombres algébriques, cf. §2.3 ci-dessous. Pour la théorie des fonctions algébriques, la définition d’un idéal est similaire à celle donnée pour les nombres algébriques, avec le mot « fonction » remplaçant chaque occurrence de « nombre ». Soulignons, par ailleurs, que Dedekind parle des idéaux du corps. Bien qu’il introduise une notion formellement équivalente à celle d’anneau, les ordres (Ordnungen), celle-ci n’a pas le rôle et l’importance attribués aux anneaux aujourd’hui. De plus, les ordres ne sont pas utilisés en théorie des fonctions algébriques. Je conserverai donc cette manière de présenter, rendue possible par le cadre extrêmement restreint dans lequel travaille Dedekind. 9. Je désignerai par la suite cet ouvrage par Zahlen. 10. Toutes les traductions des travaux fondationnels et des extraits de correspondance sont celles de Hourya Bénis Sinaceur dans [Bénis Sinaceur 2008]. Toutes les références des citations de ces travaux renvoient au livre de H. Bénis Sinaceur. 11. Voir [Bénis Sinaceur 2008, 304-311]. 12. Cet article est une version remaniée du Supplément X aux Vorlesungen ùber Zahlentheorie de Dirichlet, publiées en 1871 par Dedekind et rééditées en 1879 et 1894. Dans ces trois éditions, Dedekind présente sa théorie des entiers algébriques dans les suppléments (Supplément X en 1871, Supplément XI en 1879, une version similaire à la publication en français, et 1894). Chaque version est considérablement remaniée par rapport aux précédentes. 13. θ est un nombre algébrique lorsqu’il satisfait une équation polynomiale à coefficients rationnels ; il est dit un entier algébrique lorsqu’il satisfait une équation polynomiale à coefficients entiers. 14. Kronecker, comme on le sait, aborde la même question avec une approche extrêmement différente. On pourra consulter [Edwards 1980] pour une comparaison de leurs approches. 15. Un « corps fini », pour Dedekind, est un corps finiment généré. 16. « En considérant l’Arithmétique (l’Algèbre, l’Analyse) comme une simple partie de la logique, j’exprime déjà que je tiens le concept de nombre pour totalement indépendant des représentations ou intuitions de l’espace et du temps et que j’y vois plutôt une émanation directe des pures lois de la pensée » [Dedekind 1888, 133-134]. 17. Sur le logicisme de Dedekind, on pourra consulter [Bénis Sinaceur s. d.].

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18. Lipschitz fait référence à la théorie eudoxienne des proportions dans le livre V des Éléments, voir [Bénis Sinaceur 2008, 256-258]. 19. Le cours a été publié en 1981 par Scharlau dans [Scharlau 1981]. 20. Voir la définition de la division entre idéaux donnée en § 1.3. Cette approche est adoptée pour les groupes, les corps, et les modules également et choisie car elle rend le maniement des concepts plus aisé. 21. « [Ljes progrès les plus grands et les plus féconds en mathématiques (...) sont dus avant tout à la création et à l’introduction de nouveaux concepts... » [Dedekind 1888, 139]. 22. Zahlen étant une exception notable puisque l’arithmétique n’est pas encore définie lorsqu’on définit les entiers naturels. 23. Voir Zahlen, 179-180, et la correspondance avec Keferstein. 24. Voir la citation donnée en §2.2., page 137. 25. Dans la théorie des nombres algébriques, identifier quels sont les nombres qui doivent être considérés comme les entiers est particulièrement crucial pour le succès de la généralisation des résultats de Kummer. Voir [Edwards 1980, §5]. 26. Zahlen n’est publié qu’en 1888, mais Dedekind explique dans la première préface à Zahlen qu’une première ébauche essentiellement similaire à celle publiée a été rédigée entre 1872 et 1878, dont on pourra trouver des extraits dans [Sieg & Schlimm 2005]. Cela correspond au moment où la version de 1876-1877 de la théorie des idéaux a été écrite et publiée, version qui fait un usage beaucoup plus développé des opérations arithmétiques que la version de 1871, et est transférée à la théorie des fonctions. 27. Voir la lettre à Lipschitz du 10 juin 1876 [Bénis Sinaceur 2008, 262-270] et la lettre à Weber du 19 novembre 1878 [Bénis Sinaceur 2008, 288-290]. 28. Dans [Tappenden 1995], Jamie Tappenden souligne que la nouvelle preuve du théorème de Riemann-Roch donnée par Dedekind et Weber permet une meilleure compréhension du théorème, car « différentes preuves peuvent offrir différents diagnostics sur la nature de la proposition prouvée » et une nouvelle preuve permet de délimiter une étendue appropriée pour les applications, ainsi que d’affirmer la « priorité relative des raisonnements géométriques et arithmétiques » [Tappenden 1995, 339-340].

RÉSUMÉS

Dans cet article, je considère la pratique et la conception de la rigueur chez Richard Dedekind qui se dégagent de l’étude d’une sélection de ses travaux les plus importants. Une analyse des mentions multiples de réquisits de rigueur dans les textes de Dedekind amène à constater qu’il lie très étroitement la rigueur à la généralité. La première partie de l’article donne à voir les liens serrés tissés par Dedekind entre généralité et rigueur, dans sa théorie des fonctions algébriques co-écrite avec H. Weber, ainsi que dans ses travaux fondationnels et dans ses travaux de théorie des nombres. Dans la seconde partie, j’examine les critères de rigueur qui apparaissent dans la pratique mathématique de Dedekind. Je discute l’idéal logique de rigueur dans l’essai de Dedekind sur les entiers naturels, étudié par M. Detlefsen sous l’appellation « Dedekind’s principle » ; puis je m’intéresse à la stratégie de Dedekind pour arithmétiser les mathématiques afin de mettre en évidence qu’il ne s’agit pas d’une approche guidée par un principe purement logique. Ainsi, l’idéal logique de rigueur apparaît comme intimement lié à la pratique d’une autre

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norme épistémique : la généralité, en lien avec la quête, par Dedekind, de définitions et preuves générales. Dans la dernière partie, j’analyse la demande de généralité et la pluralité de conceptions de la généralité que recouvre cette demande, et termine en mettant en avant la relation des définitions aux preuves et de quelle manière une définition générale se pose en condition de rigueur dans les mathématiques dedekindiennes.

In this paper, I study Richard Dedekind’s practice of rigor in a selection of his most important works. The analysis of the various occurrences of rigor requirements in his mathematics leads to observe that rigor, for him, is closely related to generality. The first part of the paper shows evidences of the tight links woven by Dedekind between generality and rigor in his theory of algebraic functions co-written with H. Weber, as well as in his foundational essays and his work on algebraic number theory. In the second part, I examine criteria of rigor in Dedekind’s practice of mathematics, so as to shed light on his conception of rigor. I discuss, first, the logical ideal of rigor in Dedekind’s book on natural numbers, studied by M. Detlefsen as "Dedekind’s principle", and, secondly, I look at Dedekind’s strategy of arithmetization of mathematics to exhibit how it is not solely driven by a rigid logic. Seen in this light, then, the logical ideal of rigor appears to be intimately related to the practice of another epistemic value: generality. This, I argue, is strongly linked to Dedekind’s research of general definitions and proofs. Finally, in the last part, I analyze the requirements for generality and the multiple conceptions of generality sustaining this demand, and I proceed to unfold the relationship between definitions and proofs and how a general definition appears to be a condition for rigorous mathematics according to Dedekind.

AUTEUR

EMMYLOU HAFFNER Université Paris Diderot, Sorbonne Paris Cité, SPHere, UMR 7219 (France)

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The Right Order of Concepts: Graßmann, Peano, Gödel and the Inheritance of Leibniz's Universal Characteristic

Paola Cantù

1 Introduction

1 The question of the right order of concepts has traditionally been associated with the problem of rigour in mathematics. Aristotle's distinction between ordo essendi and ordo cognoscendi and the idea that what is first for us is usually not first in itself suggested that the search for rigour in science should include an analysis of the differences between the ordo essendi and the ordo cognoscendi and some kind of activity that could lead us from what is first for us to what is first in itself: dialectics plays this role in Aristotle's system [de Jong 2010].

2 The problem of the right order of concepts was particularly evident in mathematics and gave rise to criticism and proposals of revision of Euclid's Elements, as in the case of Port Royal Logic and Pierre de la Ramee's writings. The search for the right order of concepts could not be separated from the search for the right definitions and the fundamental concepts (considered either as first in themselves or as unanalysable, or as first for us).1 It is traditionally believed that the distinction between ordo essendi and ordo cognoscendi got lost in hypothetico-deductive axiomatics, given that the primitive notions assumed in the axioms need not be concepts that are first in themselves at the ontological level nor concepts that are first for us at the epistemological level, but just concepts that can be more convenient or more fruitful. In particular, a contrast is often introduced between two ways of investigating the foundations of logic and mathematics and the role of axioms: the search for an exposition of scientific truths according to the "right" order of concepts and the presentation of scientific truths in hypothetico-deductive systems.

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3 This paper claims on the contrary that there is more in common between the two approaches than one might expect, and in particular that the search for the "right" order of concepts plays a relevant role in the works of three authors who contributed greatly to the rise of modern axiomatics, and to the investigation into the foundations of mathematics and logic from the mid-19th century to the mid-20th century: Hermann Graßmann, Giuseppe Peano and Kurt Gödel. The choice of these authors is not arbitrary: all three authors were deeply influenced by Leibniz's ideal of a universal characteristic, and shared a deep interest in the problem of primitive concepts and propositions. Given that Leibniz admitted the distinction between an ordo essendi and an ordo cognoscendi — as he distinguished for example synthesis from analysis2 — it seems natural to raise the question of whether authors who explicitly related their own work to Leibniz's characteristic might not have shared his idea that there is a right order of concepts.3 The paper isn't aimed at pointing to a historical development from Graßmann to Peano and from Peano to Gödel,4 — but rather at highlighting differences in their respective use of Leibniz's ideas either to develop new approaches or to corroborate new results by means of an appeal to Leibniz's authority. So the paper does not aim to historically investigate whether the mentioned authors actually inherited something from Leibniz or not,5 but rather inquires as to what use they made of that inheritance. The main aim of the paper is to show the inadequacy of a historical framework that tends to eliminate from modern logic relevant philosophical issues, such as the problem of the right order of concepts — judging them to be merely superfluous or outdated questions.

4 It is true that recent trends in the history and philosophy of mathematics, especially the approaches based on mathematical practice, have rehabilitated several philosophical issues that had long remained unnoticed, such as the question of the purity of method, the explanatory power of proofs, or the relevance of mathematical values, either pragmatic or aesthetic. Yet, the role of the right order of concepts has not been fully investigated in this respect either. I cannot discuss this issue here — it could be the topic of a further separate paper — but I believe that when one has shown how much importance the problem of the right order of concepts had in the works of Graßmann, Peano and Gödel, then it would be quite natural to ask the question of whether there still is a role for this issue in the contemporary debate on the foundations of mathematics.6

5 This paper will investigate what Graßmann, Peano and Gödel thought about the realisation or realisability of Leibniz's project of a universal characteristic, and in particular whether they conceived the characteristic as one or many, and how they considered primitive concepts occurring in it.7 The focus on these authors and these questions will allow us to show not only that Leibniz left behind an important legacy in that period (from the mid-19th to the mid-20th century) when the modern axiomatic was developed — a result that has already been largely investigated in the literature8 — , but also that the use of Leibniz's ideas was considerably different for each of the mentioned authors. What will be specifically investigated in this paper is the role assigned to the search for a "right" (i.e., ontologically or epistemologically prior) order of concepts: was it totally abandoned or did it survive in modified forms? The focus on this question will guide the brief exposition of certain aspects of Leibniz's project of a universal characteristic in the next section and the choice of some specific quotations that will be

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useful for a textual comparison with remarks by Graßmann, Peano and Gödel in the next sections.

2 The heritage of Leibniz's characteristic

2.1 Leibniz's characteristic and the right order of concepts

6 Leibniz's characteristic is based on the search for a small number of primitive notions that might be identified by some fundamental characters, so that the complex notions could be obtained and their features described from the combination of the fundamental characters.9 As Couturat remarked, Leibniz's idea is based on the belief that for each simple concept there might be a symbol that expresses it in the most natural way, so that all complex concepts might be expressed naturally as combinations of the former.10

7 The question of the right order of concepts is here related to the question of the distinction between simple and complex ideas. It is not mainly related to the problem of guaranteeing a sure foundation of scientific knowledge, but rather to the possibility of building a synoptic table where each idea finds its own place, and solutions to problems can be found more easily.11 The improvement in heuristic efficiency provided by the characteristic is compared to the improvement in sight guaranteed by technological instruments such as the microscope, or the telescope, but it is considered to be more powerful, because it does not limit itself to an improvement of a sensorial faculty but rather it improves the power of reason.12

8 An essential trait of Leibniz's characteristic is its general applicability to forms, i.e., not only to quantities but also to qualities. More precisely, it is the possibility of applying the characteristic to order, similitude, and relation that enables its application to quantities and not vice versa [Leibniz 1695, 61].13 The characteristic concerns geometry, and might be applied to physics. Algebra describes only quantitative aspects of things; a new geometrical analysis is needed to express position: its characters could thus represent figures, but also machines and movements.14

9 Here Leibniz seems to consider algebra and the geometrical calculus as two parallel treatments of quantities and figures respectively. Yet in other passages he considers algebra itself as an application of the combinatorial art considered as a general theory of abstract forms that concerns metaphysics, thereby basing the question of the right order of concepts on metaphysical grounds [Leibniz 1715, 24, Engl, transl. 669].

2.2 A tension in Leibniz's idea of a characteristic

10 There is a peculiar tension in what Leibniz says about the relationship between the characteristic applied to specific domains and a general characteristic.15 Is a philosophical analysis of the first metaphysical principles and fundamental ideas necessary to develop the characteristic16 or is the latter independent from true philosophy?17 In other words, is the characteristic a metaphysical instrument that should determine the absolutely primary concepts or a way to constitute specific scientific domains? For example, is binary arithmetic only useful to determine the properties of natural numbers or is it essential to describe the metaphysical fact that all things derive from God and nothing?18 This tension is clearly reflected in the

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problem of whether the concepts that are assumed as primitive in a science are merely first for us or first in some objective sense. This would imply a conception of axiomatics that does not restrict itself to the ordo cognoscendi, but should ideally converge with an axiomatics based on the ordo essendi.

11 What holds for the characteristic language holds for the characteristic calculus too: there is a tension between specific calculi, like the geometric calculus, and the idea of a general calculus — calculus ratiocinator — that should operate on real characters.19 So, on the one hand Leibniz considers that there might be as many characteristics as there are domains of investigation, while on the other hand he aims to develop a characteristic of all characteristics. On the one hand he claims that the characteristic might be developed independently from philosophy (e.g., in specific domains such as arithmetic, or geometry) and on the other hand he suggests that it is subordinated to the development of true philosophy, because only the latter can tell which concepts are really fundamental.

12 claimed that the ambiguity is to be found in Leibniz's own texts and is not only a matter of interpretation. Donald Rutherford [Rutherford 1998], although accepting Pombo's suggestion, rather presents the distinction as pertaining to two different interpretations, preferring the standard analytical reading promoted by Russell [Russell 1900]: the merit of Leibniz is certainly related to his idea of a formal symbolism, where the characters can be seen as devoid of contents and their deductive relations are made explicit.

13 These two issues are strictly combined in Leibniz's writings and the tension remains unresolved. The question of the right order of concepts applies both in the case of specific characteristics and in the case of a general characteristic, but its answer is clearly different depending on the relation with true philosophy. In the next two sections we will consider whether this tension, that is so typical of Leibniz, remained in the works of successive authors such as Graßmann, Peano and Gödel, and in particular whether they considered the order of concepts as a matter of scientific rigour, and whether philosophy played a role in it.

2.3 Three different heritages: Graimann, Peano and Gödel

14 Graßmann, Peano, and Gödel all presented themselves as inheritors of Leibniz's tradition. Yet they developed different aspects of Leibniz's philosophical project, thereby defending different conceptions of mathematical rigour and a different understanding of the role of philosophy in the search for primitive concepts and propositions. There might be opposite explanations of this fact: either they did not really take inspiration from Leibniz, but just made recourse to his authority as a precursor in order to legitimize their innovations; or they inherited only one of several ideas that were already in tension in Leibniz's thought. My claim is that the truth is somehow in-between these two interpretations. All three authors became truly interested in Leibniz's logical work, and made frequent references to Leibniz's project of a characteristic, because they were fascinated by it; yet, they developed their original mathematical and logical results quite independently from Leibniz's results. What they shared was an interest in Leibniz's philosophical project and in the possibility of accomplishing it. Yet, they had different epistemological and

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philosophical perspectives, which guided them to different readings of Leibniz, all somehow faithful to the texts.

15 The positions concerning the role of rigour in mathematics were very different, as follows: a) Hermann Graßmann was particularly keen on linguistic rigour, introducing a new technical terminology for newly introduced concepts; b) Giuseppe Peano considered rigour as a preliminary and essential condition of any mathematical discourse (without rigour, one does poetry, but not mathematics [Peano 1891, 66]): rigour is associated with the lack of contradiction and with the possibility of giving a proof; and c) Kurt Gödel's idea of rigour was related to the clarification and analysis of concepts that might make them sufficiently precise.20 The same notion of rigour, based on the idea of the clarification of concepts, applies to philosophy: for this reason, Gödel seems to be convinced of the possibility of developing philosophy into a rigourous science.21

16 The relations between mathematical and philosophical foundation were also different: a) According to Graßmann, philosophy played an essential role in the determination of the primitive concepts of mathematics: the division of mathematics itself into four branches and a general theory that precedes them is grounded on a philosophical deduction; b) Peano always tried to separate the foundational role of philosophy from the foundational role of mathematics: the former discusses the origin of concepts and the latter the choice of the smallest number of primitive concepts and propositions that allow the derivation of all truths; and c) Gödel wondered whether the search for the right primitives in mathematics and logic should not depend on the search for the right primitives in philosophy and metaphysics, thereby associating the problem caused by paradoxes in set theory with a lack of rigour (i.e., the lack of a sufficiently precise clarification of concepts) in philosophy and metaphysics.

17 In this paper I will claim that the differences in their understanding of mathematical rigour and in the conception of the relations between mathematics and philosophy can be understood as different ways of inheriting some aspects of Leibniz's idea of a characteristic, and in particular the idea of a right order of concepts. Differences between the authors might be explained by the fact that 1) some of them believed that a characteristic of all characteristics should be developed while others believed that there should be as many characteristics as there are domains of investigation; 2) some of them claimed that the characteristic should be developed independently from philosophy (e.g., in specific domains such as arithmetic, or geometry), whereas others claimed that it should be subordinated to true philosophy, which determines the fundamental concepts.

3 Graßmann

18 The tension between the strive towards a characteristic of all characteristics and the construction of different characteristics based on different domains of investigation cannot be found as such in the texts by Hermann Graßmann: his writings were aimed at distinguishing the specific geometric calculus from the more general characteristic.22 It is well known that Hermann Graßmann reacted explicitly to Leibniz's idea of a characteristic expressed in the letter to Huygens, first published in 1833, in his essay Geometrische Analyse written for the 1846 Jablonowski Prize, which asked to develop the Leibnizian idea of a geometric characteristic, and to build a calculus that might express

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Leibniz's ideas [Graßmann 1847]. I will not here enter into the discussion as to whether Graßmann's aim was the same as Leibniz's:23 for the scope of the present paper it is sufficient to remark that Graßmann declared that he had developed his work independently but tried to present it as a realization of Leibniz's project.24 Apart from the contingent application to the prize, which made it necessary to show that his own theory was somehow related to Leibniz's project, it is interesting to remark that Graßmann's defense of his new theory was based on the great number of its possible applications. As a matter of fact, extension theory included an abstract foundation of vectorial spaces and a treatment of extensive multiplicities with n dimensions that could be applied to the specific case of 3-dimensional geometry. Besides, the new calculus could be applied to the whole of physics25 and to non spatial objects too, because the calculus might become independent from spatial intuition,26 as Leibniz himself had claimed [Leibniz 1679b, 571].

19 The previous remarks suggest that Graßmann meant to develop a specific calculus and not a general characteristic, but a calculus that might have a variety of different applications. Not only did he avoid any mention of the role of philosophy in the search for the relevant elements of this calculus, but he aimed at separating the geometrical calculus from the general characteristic.27 So, Graßmann could be considered as the one who first separated (long before Russell) the project of a universal language made of real characters from the project of building a symbolical calculus.

20 Echeverría is right as he remarks upon the decrease in generality in Graßmann's geometrical calculus, but he does not consider that the idea of a general characteristic can be found elsewhere in Graßmann's works, even if Graßmann does not explicitly associate it to Leibniz nor to the idea of a characteristic of characteristics [Echeverría 1979]. What can be found in Hermann Graßmann's mathematical treatises and in the works of his brother Robert Graßmann is the concept of a unified analysis of concepts. The General Theory of Forms developed by Hermann is a preliminary investigation of the fundamental thought operations that occur in any mathematical domain (logic, arithmetic, geometry, combinatorics, extension theory) [Graßmann 1844, 33ff.], while the Theory of Forms developed by Robert is a science based on qualitative besides quantitative relations that should generally be valid for all human beings, whatever their nation or their language [Graßmann 1872, § 1, 6].

21 So, the tension between the idea of one characteristic and the development of applied or specific characteristics has not completely vanished in Graßmann's works, although, as Michael Otte rightly observed [Otte 1989], the metaphysical and ontological foundation has been abandoned. Yet, the question of the right order of concepts is still present: on the one hand it emerges in the philosophical deduction by means of which Graßmann introduces a partition of the general theory of forms into four independent but parallel branches; on the other hand the choice of the primitive concepts and the order of the proofs is not at all arbitrary in Extension Theory.

22 For example, one of the differences between Peano's calculus and Graßmann's theory concerns the choice of the notion of dimension as more primitive than the concept of base: although theorems and proofs can be compared, the philosophical idea behind Graßmann's project is lost, if one changes the order of concepts. Peano first takes a system of entities of a certain dimension as given and then introduces a way to obtain it from a subset of its elements that are linearly independent. Graßmann on the contrary first takes the operation that determines a set of (independent) generators as primitive,

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and then considers the systems it can give rise to. The right order of concepts appears thus related not only to the degree of generality or efficiency in proofs,28 but also and most importantly to the assumption of the primacy (for us and thus also per se, given Graßmann's idealism) of operations on their products.29

23 Hermann Graßmann did not introduce an axiomatic theory of extensive quantities or of arithmetic in the same sense as Peano or Dedekind, but he developed an analysis of the primitive concepts of both sciences and also of the general theory of forms. In the case of mathematics the primitive concepts are considered to be fundamental, because they are obtained by a philosophical deduction, i.e., by a dichotomic division that is similar to the Platonic procedure by diaeresis. Yet, unlike the Platonic dichotomy, Graßmann's philosophical deduction proceeds by intersection of the opposites and not only by successive divisions, and is not followed by a movement that goes back from the multiplicity of construed concepts to the unity of the starting point. The starting point of the deduction (i.e., the division of sciences into formal and real, of generating acts into continuous and discrete, and of elements into equal and different) is not verified, and the dialectic division is justified by the correspondence with acts of thought [Cantü 2003, 172]. Mathematical primitives are the couples of opposites equal/ different and continuous/discrete; the logical primitives, presented in the allgemeine Formenlehre as general forms, are: equality, difference, connection [Verknüpfung] and separation [Sonderung].

24 So, the primitive concepts occurring in the general theory of forms and in mathematics are the same and depend on philosophy: on the one hand, because they are justified by a philosophical deduction; on the other hand because they correspond to the fundamental acts of thought. The order of primitive concepts cannot be arbitrarily changed, but rather is strictly related to the correspondence between subjective and objective levels that is typical of idealism.

4 Peano

25 While Hermann Graßmann seemed mainly interested in a specific geometrical calculus (analysis situs), Giuseppe Peano explicitly described Leibniz's project of a Speeiosa Generalis as a sort of universal Language or Writing System, where the symbols guide reasoning.30 Quoting Leibniz's essay on the universal characteristic, Peano recalls that this discovery is taken to be more important than telescopes or microscopes: it is the polar star of reasoning.31 Besides, Peano shared Leibniz's aim to determine a very small number of primitives, and his concern for the identification of symbols that could naturally express ideas and their reciprocal relations. Yet, in the Formulary one finds only a specific symbolic system and calculus concerning logical and mathematical truths. Here, the symbols guide reason inasmuch as different symbols denote different ideas, whereas the same symbol is used when the difference between two words is grammatical rather than conceptual.32

26 Unlike Graßmann, Peano believes that the construction of a symbolic system should not be limited to mathematics. Yet, given that this enterprise goes beyond the possibility not only of a man but of a whole research group, needing the effort of the whole of society,33 Peano and his collaborators contented themselves with the application of the symbolic notation to the analysis of mathematics.34 So, according to Peano it is perfectly possible to develop a specific calculus without having to

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preliminarily establish a general characteristic: the advantage (both foundational and didactical) of each part is already evident before the whole is completed.35 The relative independence of specific calculi from a complete analysis of ordinary language — which needs nonetheless to be accomplished, because it is useful to distinguish ambiguities and avoid imprecise formulations — also depends on the provisional nature attributed to the Formulary: even if it is a collection of truths and not of conventions, it can be corrected and ameliorated by the comments and criticism of contributors, like a collaborative Dictionary or a Wiki that can be implemented by its own readers.36 Besides, Peano was particularly impressed by the aim of Leibniz's project: the solution of verbal controversies and the search for a unique notation. This is particularly evident in Peano's own remarks37 and in the remarks made by other members of the school. Giovanni Vailati for example considered one of the merits of Peano's enterprise and more generally of logical pragmatism that of identifying different historical theories (known under different names) as having the same content, thus avoiding sterile oppositions.38 Burali-Forti and Marcolongo remarked upon the importance of the introduction of a non-arbitrary, unique notation in order to improve the diffusion of new theories such as the vector calculus.39

27 According to Peano, a symbol is primitive with respect to a given set of symbols if it is not defined by means of those symbols. Being a primitive is a relative and not an absolute property of symbols. Primitive symbols denote ideas that are considered as primitive in a given axiomatic system: e.g. the system of natural, rational, real numbers, etc. Properties of primitive symbols are expressed in the primitive propositions (axioms), which might serve as definitions of the primitive terms. Not all ideas expressed by the primitive symbols of a system need be fundamental ideas, as already proved by Peano's remarks on the redundancy of the logical symbols introduced in the Formulary.40

28 Peano's symbols do not express exactly the same concept in all contexts: the symbol of equality is defined for example as a relation of equivalence between individuals in one section and as mutual implication between propositions in another section. If the logical symbols, although used in mathematical sections, change their meaning according to the context, they express different concepts in different sections of the Formulary: therefore they cannot be taken to express a list of fundamental concepts that ground all knowledge. This is also due to the fact that Peano always has a privileged model for his axiomatic systems and introduces local definitions for the symbols.

29 Terms might belong to specific parts of mathematics (geometry, arithmetic) or be common to all of them. Mathematical logic studies relations and operations that occur with the same properties in different branches of mathematics, and that should thus be expressed by the same symbol. Primitive terms are not fundamental: the choice of the terms used to denote the fundamental concepts might vary according to didactical needs and several alternative definitions of the primitive terms are possible. Besides, philosophy does not play any significant role in the determination of the primitive concepts.

30 Apparently, there is no interest in the question of the right order of concepts in the Formulary and in Peano's understanding of axiomatics. Yet, Peano's choice of symbolism reveals an effort to mirror the concepts by means of the symbols used to denote them. Peano, like Leibniz, insisted on a natural relation between the symbol and what it designates: this is clear in his choice of the symbol for "being a member of — a

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Greek epsilon that stands for est — , or of the inverse iota, which expresses the inverse of the operation expressed by the iota. Leibniz's idea of a characteristic containing "real" characters is not completely abandoned in Peano's perspective. It emerges with even more force in Peano's investigations into a universal language, because the latino sine flexione should be based on symbols (roots of Latin words) that should preserve the essential relation to the denoted concept, independently from grammatical variations.

5 Gödel

31 Kurt Gödel considered Prege's and Peano's mathematical logic as a realisation of Leibniz's project of a general characteristic.41 Yet, he clearly remarked that mathematical logic was but a part of Leibniz's project, even if a central part of it, given that it is a science that is prior to all others and contains the principles underlying all sciences. In particular, he saw in the unaccomplished philosophical analysis of the primitive concepts occurring in the axioms the reason for the partial success of such calculi.42 The unsatisfactory analysis of the primitive concepts is responsible, according to Gödel, for the paradoxes of set theory. Even if Russell's simple theory of types and axiomatic set theory can avoid all known paradoxes,43 Gödel seems to be unsatisfied with restriction of types and with the extensional interpretation of sets for other reasons.44

32 Such reasons are philosophical and are related to Gödel's interpretation of Leibniz's characteristica as a non-utopian project,45 based on the idea that "everything in the world has a meaning", and aiming at clarifying concepts so as to develop an intensional logical theory. The clarification of concepts is important not only from the point of view of the ordo essendi, but also from the point of view of the ordo cognoscendi, because a correct analysis of mathematical concepts might immediately lead to the solution of mathematical problems.46 How can the clarification of concepts take place? Gödel adopted the same metaphor used by Leibniz and recalled by Peano: primitive concepts have to be discerned like stars in the sky. In his conversation with Wang Gödel remarked that Leibniz had assumed seven primitive concepts in analogy with the Great Bear constellation.47 Gödel suggested that a potentiation of sight could lead to the individuation of other primitive concepts: symbolism can be used to potentiate our capacity of distinguishing concepts just as the telescope is used to discern more stars.48 The search for primitive concepts should not concern only mathematical logic, but should be extended to all concepts: Leibniz's characteristic is understood by Gödel as a general science. The following analogy from the Philosophical Manuscripts confirms this interpretation: Leibniz's scicntia yeneralis is to scientifical phenomena (sciences) as Newtonian physics is to physical phenomena. Leibniz's scicntia yeneralis is interpreted by Gödel as a characteristic of all characteristics which introduces a constellation of concepts that apply to all phenomena — i.e. to all sciences, whereas Newtonian physics introduces a constellation of concepts (point of the space, point of time, point of mass, position, force, mass) that apply only to physical phenomena.49

33 The clarification of concepts allowed by the general characteristic will grant a rigourous discussion of the foundations of mathematics50 and a mathematically rigourous analysis of metaphysical and theological concepts.51 If the analysis correctly separates concepts that are mixed up at first sight, new fundamental concepts will be discovered and their analysis will lead to the solution of scientific problems, even if the

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procedure for solving problems cannot be a completely mechanical one: according to Gödel, Leibniz was wrong on this point.52

34 Primitive concepts are, according to Gödel, the concepts we start out from, and also the concepts that cannot be derived from others.53 Axioms are the primitive propositions of a theory and express the properties of primitive concepts. Primitive concepts have to be looked for not only in mathematical or scientific disciplines, but also in philosophy and in theology. The search for primitive concepts in logic does not amount to an axiomatisation of mathematical logic, but rather concerns the basic elements of a general theory of concepts.54

35 Some primitive concepts might be fundamental in the sense that we must assume them as given in order to develop an axiomatic system. They appear as the most clear concepts that we have, as concepts that are primary according to the ordo cognoscendi.55 But there is a second sense in which primitive concepts might be fundamental. Several passages from the Philosophical Manuscripts suggest that Gödel aimed to distinguish logic from psychological concepts, objective from subjective relations between concepts: like Prege, he claimed that the two levels are relevant in order to build a general theory of concepts. Primitive concepts might be fundamental both as psychological and as logical concepts, but in the first case this just means that we cannot do without them, in the second case this means that they are the most simple concepts that enter in the composition of all other concepts. The distinction between the subjective and the objective level allows the distinction between the epistemological order of concepts (ordo cognosccndi) and the "right" or "natural" order of concepts (ordo essendi).56 Unlike Peano, Gödel attributed an important role to philosophy in the search for the primitive concepts of the general characteristic, but it is only in the interplay and the reciprocal influence between the particular sciences and philosophy that the task of finding the right order of concepts might be accomplished.57

36 Even the analysis of logical primitives involves questions that can probably be answered only by the introduction of metaphysical questions.58 Given the idea that there are some fundamental concepts that philosophy should investigate and ultimately determine, even if this task has not been accomplished yet, Gödel's conception of axiomatics shows some affinity with the Classical Model of Science mentioned at the beginning. Like Bolzano, Gödel used the distinction between ordo essendi and ordo cognoscendi to explain why a list of fundamental primitives had not yet been given. Notwithstanding the inevitable discrepancy between the two levels, the search for rigour is based on the ideal convergence between ordo essendi and ordo cognoscendi: the ultimate task is to find the primitive concepts that are also fundamental at the objective level. Yet, this task can never be fully accomplished, because the determination of the primitives and of their correct relations and properties would amount to the solution of all problems, and thus to the elimination of human incompleteness, which is on the contrary an intrinsic and essential property of our existence as finite beings.

37 So, the task of determining a general characteristic is at the same time something we should strive towards and believe in — because there is no reason to give up hope — and a task that can never be fully accomplished, given our finite nature. It is an ideal that guides axiomatics but can never be fully reached.59 It is thus not surprising that Gödel's determination of the primitives of a general theory of concepts was never definitively achieved. Yet, in his conversation with Wang and in the Philosophical

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Manuscripts he mentioned provisional lists of logical primitives,60 and evaluated several metaphysical concepts in order to understand which could be considered as most fundamental.

38 As we saw in section 2, Leibniz's project of a characteristic was based on the idea that symbols should express concepts in a natural way. Is there any inheritance of this idea in Gödel? In the procedure known as Gödelization, Gödel used the first thirteen prime numbers to represent the most relevant logical terms. The choice of designating logical symbols by numbers is not only a matter of efficiency or fruitfulness. If one analyses some remarks that occur in Gödel's Philosophical Manuscripts, it emerges that he was not insensible to the problem of an analogy between the signs and the things denoted by the signs, as in a passage where he discussed whether binary numbers could be more apt than decimal numbers to express the fundamental concepts.61 This remark about binary numbers is even more interesting if compared with other passages from the MaxPhil, where the number one is associated to God and to full existence.62 This means that Gödel's preference for the binary system is related to the capacity of its signs to express some fundamental metaphysical ideas. The right order of concepts depends thus on the choice of the right primitive metaphysical concepts.

6 Conclusion

39 Discussing the legacy of Leibniz's characteristica in the works of Graßmann, Peano and Gödel, this paper has shown that, apart from several differences, all three authors took the task suggested by Leibniz seriously, and tried to develop the idea of a general characteristic. Together with the project of the characteristic, they inherited some unresolved tensions that can be found in Leibniz's writings. Gödel and Graßmann, more than Peano, investigated the possible relations between a general and specific characteristics, and, unlike Peano, assigned a relevant role to philosophy in the search for primitive concepts and primitive propositions. A clear distinction between the ordo csscndi and the ordo cognosccndi allowed Gödel to explain how the fact that there is a unique true order of concepts might be compatible with different orders developed by axiomatic systems. Although he believed that there might be some fundamental concepts, Peano did on the contrary consider the choice of the primitives and the order of concepts as something that might be changed according to didactical needs, and never mentioned the idea of a unique "natural" order of concepts.

40 The analysis of these three case-studies shows that the choice of primitive concepts was not only a question of convenience in modern hypothetico-deductive investigations, but sometimes also the result of philosophical investigations into the foundation of scientific disciplines. The question of the "right" order of concepts became an ideal to be followed rather than a task that can be fulfilled but remained nonetheless an essential part of the axiomatic enterprise. The scientific rupture determined by the appearance of hypothetico-deductive systems in mathematics and logic should thus not be dissociated from some relevant continuities concerning the ideal of knowledge as the search for a general theory of concepts deriving from some fundamental elements.

41 The notion of mathematical rigour inherited from Leibniz concerned the philosophical analysis of concepts as well as deduction. For this reason, it was not fully dissociated from the belief in an ideal, "natural" order of concepts that should orientate the search for the most fundamental primitives.

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NOTES

1. De Jong and Betti have tried to recall those aspects of the theory of knowledge in a scheme that they called The classical model of science and which attempts to describe the conception of scientific knowledge as a cognitio ex principiis [Betti & de Jong 2010]. This paper was actually first presented at the International Conference The Classical Model of Science II. The Axiomatic Method, the Order of Concepts and the Hierarchy of Sciences from Leibniz to Tarski organized at the Vrije Universiteit Amsterdam, August 2-5, 2011. What interested me in the model was the emphasis on the distinction between the ordo essendi (conditions 1-5 of the model) and the ordo cognoscendi (conditions 6-7 of the model). 2. "Synthesis is achieved when we begin from principles and run through truths in good order, thus discovering certain progressions and setting up tables, or sometimes general formulas, in which the answers to emerging questions can later be discovered. Analysis goes back to the

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principles in order to solve the given problems only, just as if neither we nor others had discovered anything before" [Leibniz 1683-1685, 232]. 3. "As a boy I learned logic, and having already developed the habit of digging more deeply into the reasons for what I was taught, I raised the following question with my teachers. Seeing that there are categories for the simple terms by which concepts are ordered, why should there not also be categories for complex terms, by which truths may be ordered? I was then unaware that geometricians do this very thing when they demonstrate and order propositions according to their dependence upon each other" [Leibniz 1683-1685, 229]. See also [de Jong 2010, 239]. 4. I have discussed this issue in other papers. See in particular [Cantu 2003, 332-337], where I discuss some differences between Grafimann's calculus in the two editions of the Extension Theory and Peano's axiomatization in The Geometrical Calculus: the latter is limited to the case of n dimensions and is not focused on the generation of the system. See also [Cantu forthcoming], where I compare several passages from Gödel's unpublished philosophical manuscripts, the Max Phil, with relevant passages from Peano's Formulary and from Russell's Principia Mathematica on definite descriptions, definitions and functions, suggesting how intensively Gödel had worked on Peano's writings and opposing, or at least restricting, the conceptual continuity between Peano and Russell outlined in recent literature. 5. Concerning the relation of Graßmann to Leibniz, the debate — which I briefly reconstructed in [Cantù 2003, 319-320] — involved among others [Couturat 1901], [Rothe 1916], [Heath 1917], [Lotze 1923], [Barone 1968], [Freudenthal 1972] and [Muenzenmayer 1979]. Recent literature generally agrees on the idea that Grafimann's project had not been directly influenced by Leibniz's perspective, but on different grounds. Echeverría claimed that there is a huge difference in generality between Leibniz's analysis situs and Grafimann's geometrical calculus [Echeverría 1979], i.e., a different level of generality, which gets lost in Graßmann, because he introduces equality instead of congruence, whereas Otte [Otte 1989] remarked that the difference concerns the abandonment of the ontological foundation of classical epistemol-ogy. De Risi [De Risi 2007, 111-112] recalls Grafimann's opportunism, because he clearly adapted his previous work for the 1846 Jablonowski Prize, but mentions also some aspects where Grafimann's perspective is truly Leibnizian. I agree with the idea that Graßmann had not been directly influenced by Leibniz's writings, but I also claim that the effort to present his own work in relation to Leibniz's project had some effects on his philosophical approach (see further section 3). 6. My intuition would be that the epistemological question can be dealt with from the perspective of the inquiries into mathematical values and mathematical styles, rather than on the basis of investigations into the kind of mathematical rigour granted by axiomatics. Yet, the question would be whether some metaphysical traits of the question might fail to be adequately analysed from this perspective, and might require an interdisciplinary approach that takes into account the relations between scientific, philosophical and theological domains. 7. The emphasis on these issues was suggested to me by the interpretation presented by Francesco Barone in the introduction to an Italian edition of Leibniz's "logical" writings which, although largely unknown, presents several reasons of interest [Barone 1968]. 8. See for example [Heinekamp 1986], [Krömer & Chin-Drian 2012], and especially [De Risi 2007] for the history and the success of the analysis situs. 9. "As a matter of fact, when thinking about these matters a long time ago, it was already clear to me that all human thoughts may be resolved into very few primitive notions; and that, if characters are assigned to them, it will then be possible to form characters for the derived notions, from which it will always be possible to extract all their conditions, as well as the primitive notions they contain, and — let me say explicitly — their definitions or values, and therefore, the properties, which may be deduced from the definitions as well" [Leibniz 1684, vol. 7, 223, Engl, transl. 182].

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10. "Leibniz's characteristic is the search for the right and natural symbols to express an idea as decomposed in its fundamental parts" [Couturat 1901, 76]. 11. See [Leibniz 1683-1685, 232], quoted above in footnote 2. 12. "Once the characteristic numbers of most notions are formed, humankind will have a new type of instrument which will enlarge the mind's power to a far greater degree than the eyes' power was increased by optical lenses, an instrument as superior to microscopes and telescopes as reason is superior to sight. No magnetic needle ever offered greater comfort to seamen than this Little Dipper (cynosura) shall offer those traversing a sea of experiences" [Leibniz 1679c, 268], Engl, transl. in [Leibniz 2008, 124]. 13. See also the following passage: "This art is distinct from common algebra, which deals with formulas applied to quantity only or to equality and inequality. This algebra is thus subordinate to the art of combinations and constantly uses its rules. But these rules of combination are far more general and find application not only in algebra but in the art of deciphering, in various games, in geometry itself when it is treated linearly in the manner of the ancients, and finally, in all matters involving relations of similarity" [Leibniz 1683-1685, 233]. 14. "But in spite of the progress which I have made in these matters, I am still not satisfied with algebra, because it does not give the shortest methods or the most beautiful constructions in geometry. This is why I believe that, so far as geometry is concerned, we still need another analysis which is distinctly geometrical or linear and which will express situation [situs] directly as algebra expresses magnitude directly. And I believe that I have found the way and that we can represent figures and even machines and movements by characters, as algebra represents numbers or magnitudes" [Leibniz 1679b, 568-569, Engl, transl. 248-249]. 15. This point was clearly made by Francesco Barone [Barone 1968, lxix-lxxi]. A similar distinction has been recently introduced by O. Pombo [Pombo 1988], who 16. This is what Leibniz claimed in a letter to Burnet dated 24 August 1697 [Leibniz 1875-1890, vol. 3, 216]. 17. See Leibniz's unpublished remark [Leibniz 1966, 27-28]. 18. This example is based on a passage from De organo sive arte magna cogitandi [Leibniz 1698, 239]. 19. This is another point made by Francesco Barone [Barone 1968, lxxii-lxxiii]. 20. "In the second case it must be possible, after making the concepts in question precise, to give a rigourous proof for the existence of that necessity" [Gödel 1972, 274]. 21. "I am under the impression that after sufficient clarification of the concepts in question it will be possible to conduct these discussions with mathematical rigour and that the result then will be that (under certain assumptions which can hardly be denied [in particular the assumption that there exists at all something like mathematical knowledge]) the Platonistic view is the only one tenable" [Gödel 1951, 322-323]. 22. This is in particular the point where my interpretation differs from that of Echeverría [Echeverría 1979]. Graßmann was fascinated by Leibniz's strive for generality, but was interested in the construction of a specific calculus: the geometric one. 23. See footnote 5. 24. "In order to let the scientific meaning of [Leibniz's] peculiar characteristic come into light also otherwise, and in order to make the scientific gain in this domain more intuitive from another point of view, I will take the following line in the derivation and development of the new analysis. I will assume the Leibniz's characteristic, and show how the analysis that I am inclined to see as a realisation, even if only a partial one, of Leibniz's idea of a geometrical analysis emerges from this nucleus — by implementation and further development, by an appropriate elimination of what is extraneous and by fertilization with the ideas of geometrical affinity. That this is not the path along which I have arrived at this analysis does not even need to be mentioned here" [Graßmann 1847, 327-28].

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25. "So, I think I have shown in the application to mechanics introduced above how mechanics can be effectively treated in a pure geometrical way by means of this analysis [... ] I could have easily given other examples from optics, acoustics, electrodynamics and other branches of physics" [Graßmann 1847, 397-398]. 26. "Finally there is at the end of Leibniz's presentation still a peculiar place where he clearly enunciated the applicability of this analysis also to objects that are not of spatial nature [...]. And one can easily see, once one has accepted this idea of a pure conceptually grasped passage, that also the laws developed in this section are capable of being conceived independently from spatial intuition. In this way Leibniz's thought is realized" [Graßmann 1847, 398]. 27. "Leibniz himself definitely distinguished his thoughts about a pure geometric analysis, whose development and achievement floated before his eyes as a far objective, even if he fully recognised its importance, from his search for a new characteristic, which he connected to the former in order to make the possibility of the realisation of those thoughts more believable and to leave a monument to posterity, in case he should be hindered from its achievement. The two need to be sharply separated, if one wants to rightly appreciate the merit of Leibniz in the development of the geometrical analysis" [Graßmann 1847, 326]. Cf. also the passage quoted in footnote 24 on page 166, where Graßmann remarks that Leibniz's geometrical calculus needs to be separated from what is extraneous to it. 28. For a more detailed comparison of these aspects in Peano and Graßmann, see [Cantù 2003, 332ff.]. 29. For an analysis of the role played by the operation of multiplication in Graßmann's mathematical theory and its epistemological and philosophical import concerning the difference between numbers and magnitudes, the relation between geometry and extension theory, and the development of a constructivist approach to mathematics, see [Cantù 2010, 98-100]. 30. "Gottfried Wilhelm Leibniz during all his life (1646-1716) was concerned with a kind of 'Speeiosa generalis' where all truths of reason are reduced to a sort of calculus. At the same time it could be a sort of universal language or writing system, but infinitely different from those that have been planned until now, because the characters and even the words would thereby direct reason" (Opera philosophica a. 1840, 701) [Leibniz 1679a], [Peano 1896, 1]. 31. "He considers this discovery as more important than the invention of telescopes and microscopes; it is the North Star of reasoning" [Peano 1896, 1]. 32. "The study of different properties of ideas represented by the symbols ∈ and ⊃ prevents us from representing them by the same symbol, even if they correspond in language to more or less the same word 'to be'. The identity of the expressions 'it is contained' and 'one deduces' shows us that there is only a grammatical difference between them and leads us to denote them by the same symbol ⊃. And so on. Changing the forms of the signs ∈, ⊃, … does not change those truths" [Peano 1896, 1]. 33. "This project is undoubtedly beautiful. Unfortunately its execution goes beyond the energy, not only of a man, but of several men. Only a numerous and well organized society could accomplish it" [Peano 1896, 4]. 34. "We have already applied those results both to enunciate certain propositions precisely, and to analyse some complete theories, especially relative to the still controversial principles of mathematics" [Peano 1896, 3]. 35. "Because it is not necessary that all this work be achieved in order to be fruitful. Each published part is already useful to students of those particular subjects" [Peano 1896, 4]. 36. "Does one want to study a topic whatsoever? One just needs to open the Formulary at the right page, because it is possible to order the topics according to the signs that compose them, just as one orders words in a dictionary according to the letters that constitute them. In a few pages one will find all known truths on that topic, together with their proofs and historical information. Should the reader know any proposition that he might have discovered or found in some book, or

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should he notice any inaccuracy in those propositions, he might convey those additions and those corrections to the Editorial Board of the Formulary: they will be announced in some periodical publication and will be taken into account for the next edition" [Peano 1896, 2]. 37. See for example the numerous quotations given in [Luciano 2012], who claims that Peano's interest for Leibniz was mainly guided by the search for a precursor of his own work. 38. "A third point of contact between pragmatists and mathematical logicians consists in the interest shown on both sides for historical research into the development of scientific theories. [... ] To this tendency to recognise the identity of theories, beyond or under differences of expression, symbolism, language, representative conventions and the rest, is to be attributed also the constant interest of the mathematical logicians in linguistic questions — from Grafimann, at once the author of the Ausdehnungslehre and of the Worterbuch zum Rig-Veda, to Nagy, student of the transmission of Greek thought through the Syriac and Arabic commentaries; from Couturat, joint author with Leau of a history of the projects of 'Universal Language', to Peano, inventor and propagandist of one of the most practical among them: the 'latino non flexo'" [Vailati 1906, 691-692]. 39. "The reason why this admirable means of research and presentation [the vector calculus] spread slowly and is still accepted suspiciously, is the fact that different authors use different names and signs to indicate the same vectorial entities" [Burali-Forti & Marcolongo 1907-1908, I, 324]. 40. Peano's logical primitives terms are: ∈, ⊃, =, ∪, ∩, — ⋀. Some terms in this list are redundant but useful for reasons of clarity and simplification of the derivations. 41. "On the other hand, [mathematics] is a science prior to all others, which contains the ideas and principles underlying all sciences. It was in this second sense that mathematical logic was first conceived by Leibniz in his characteristica universalis, of which it would have formed a central part. But it was almost two centuries after his death before his idea of a logical calculus really sufficient for the kind of reasoning occurring in the exact sciences was put into effect (in some form at least, if not the one Leibniz had in mind) by Frege and Peano" [Gödel 1944, 119]. 42. "Many symptoms show only too clearly, however, that the primitive concepts need further elucidation. It seems reasonable to suspect that it is this incomplete understanding of the foundations which is responsible for the fact that mathematical logic has up to now remained so far behind the high expectations of Peano and others who (in accordance with Leibniz's claims) had hoped that it would facilitate theoretical mathematics to the same extent as the decimal system of numbers has facilitated numerical computations. For how can one expect to solve mathematical problems systematically by mere analysis of the concepts occurring if our analysis so far does not even suffice to set up the axioms?" [Gödel 1944, 140]. 43. "Major among the attempts in this direction (some of which have been quoted in this essay) are the simple theory of types (which is the system of the first edition of Principia in an appropriate interpretation) and axiomatic set theory, both of which have been successful at least to this extent, that they permit the derivation of modern mathematics and at the same time avoid all known paradoxes" [Gödel 1944, 140]. 44. For a survey of Gödel's readings of Leibniz, see [Crocco 2012], who presents — in opposition to [Parsons 1990] — a detailed interpretation of the 1944 paper as focused on several Leibnizian issues that Russell had failed to solve adequately, in the belief that a good solution might only come from a return to logic as the science of all sciences. For a critical remark on the effectiveness of the analogy with mon-adology used by Gödel in order to justify the reflection principle in set theory, see [van Atten 2009]. 45. "But there is no need to give up hope. Leibniz did not, in his writings about the characteristica universalis, speak of a Utopian project" [Gödel 1944, 140].

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46. "The epistemological problem is to set the primitive concepts of our thinking right. For example, even if the concept of set becomes clear, even after satisfactory axioms of infinity are found, there would remain more technical (i.e., mathematical) questions of deciding the continuum hypothesis from the axioms. This is because epistemology and science (in particular, mathematics) are far apart at present. It need not necessarily remain so. True science in the Leibnizian sense would overcome this apartness. In other words, there may be another way of analyzing concepts (e.g., like Hegel's) so that true analysis will lead to the solution of the problem" [Wang 1996, 237]. 47. "The fundamental principles are concerned with what the primitive concepts are and also their relationship. The axiomatic method goes step by step. We continue to discover new axioms; the process never finishes. Leibniz used formal analogy: in analogy with the seven stars in the Great Bear constellation, there are seven concepts. One should extend the analogy to cover the fact that by using the telescope we [now] see more stars in the constellation' [Wang 1996, 297]. Actually Leibniz used the term cynosura (see above p. 162), which might mean either the constellation containing the Polar Star, i.e., the Ursa Minor or Little Bear constellation, or the Polar star itself, as interpreted by Peano (see footnote 31 on page 168). Gödel's confusion might have arisen from the fact that both constellations contain seven stars. 48. "The undefined concepts are those that are so bright (clear) that it is enough to say: look approximately in this or that direction (of the sky). In the other concepts the word is constructed by means of definitions. The feeling that only mathematical concepts and propositions are precise derives from the fact that those concepts are the most simple (bright), and therefore they are the first to be seen precisely" [Die Undefinierten Begriffe sind die, welche so hell (deutlich) sind, dass es genügt zu sagen: Schaue ungefähr in diese oder jene Richtung des Himmels. Bei den anderen wird das Wort erst durch Def konstruiert. Das Gefühl, dass nur die mat Begriffe und Sätze präzise sind, kommt daher, dass diese Begriffe die einfachsten (hellsten) sind und daher am ersten präzise gesehen werden] [Gödel forthcoming, IX, 89-90]. Passages from Gödel's Max Phil are quoted also in the German original, given that they have been recently transcribed from handwritten notes, and — being still unpublished — are not easily accessible to the reader. 49. "Leibniz's scicntia generalis is clearly something similar with respect to the whole domain of phenomena, i.e., all sciences — including mathematics — as Newtonian physics is with respect to physical phenomena. The 'Cynosura notionum' consists there of point of the space, point of time, point of mass, position, force, mass. Projecting all physical phenomena onto this system, i.e., trying to use them to interpret phenomena, the possibilities that subsist a priori are limited and predictions become possible" [Die scicntia generalis des Leibniz ist offenbar etwas Ähnliches hinsichtlich des ganzen Gebiets der Erscheinung d.h. aller Wissenschaften, inkl Math wie die Newtonsche Physik hinsichtlich der physikalischen Erscheinungen. Die 'Cynosura notionum' besteht dort aus Raumpunkt, Zeitpunkt, Massepunkt, Lage auf, Kraft, Masse. Dadurch, dass man alle physikalischen Erscheinungen auf dieses System 'projiziert', d.h. es durch sie zu 'interpretieren' sucht, werden die a priori bestehenden Möglichkeiten eingeschränkt, und es sind daher Voraussagen möglich] [Gödel forthcoming, X, 67-68]. See also other passages from the Philosophical Manuscripts: [Gödel forthcoming, IX, 85; IX, 90 and X, 2-3]. 50. "I am under the impression that after sufficient clarification of the concepts in question it will be possible to conduct these discussions with mathematical rigour and that the result then will be that (under certain assumptions which can hardly be de- med, in particular the assumption that there exists at all something like mathematical knowledge) the Platonistic view is the only one tenable" [Gödel 1951, 323]. 51. "The famed philosopher and mathematician Leibniz attempted to do this as long as 250 years ago, and this is also what I tried to do in my last letter. The thing that I call the theological worldview is the concept that the world and everything in it has meaning and sense [Sinn und

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Vernunft], and in particular a good and unambiguous [zweifellosen] meaning. From this it follows directly that our presence on Earth, because it has of itself at most a very uncertain meaning, can only be the means to the end [Mittel zum Zweck] for another existence. The idea that everything in the world has a meaning is, by the way, exactly analogous to the principle that everything has a cause, which is the basis of the whole of science" [Wang 1996, 108]. 52. "In 1678 Leibniz made a claim of the universal characteristic. In essence it does not exist: any systematic procedure for solving problems of all kinds must be nonmechanical" [Wang 1996, 202]. 53. "Given any set of conceptions, in the sense of concepts with associated beliefs about them, we can try to determine what the reliable basic beliefs about each concept are; whether some of the concepts can be defined in terms of others; and whether some beliefs can be derived from others. Often we find that some concepts can be defined by other concepts, so that we can arrive at a subset of primitive concepts and construe all the beliefs in the set as concerned with them. Those beliefs in the initial set of beliefs which cannot be derived from other beliefs in the set are then taken as the axioms" [Wang 1996, 334-335]. 54. "Gödel often speaks of an axiomatic theory or system in quite a loose way, so that he considers it necessary to find axioms for arithmetic, for geometry, for physics, but also for philosophy, and for theology. He also aims at finding the primitive concepts of logic as a general theory of concepts", see [Wang 1996, 334]. 55. “The notion of existence is one of the primitive concepts with which we must begin as given. It is the clearest concept we have. Even 'all', as studied in predicate logic, is less clear, since we don't have an overview of the whole world. We are here thinking of the weakest and the broadest sense of existence. For example, things which act are different from things which don't. They all have existence proper to them” [Wang 1996, 150]. 56. "Even if we might not have access to it, there seems to be a right or natural order of primitive concepts and propositions that we should look for. The search for the right primitives and axioms is a philosophical task, based on the decomposition of concepts in simpler parts. The faculty that allows us to perceive concepts might be helped by instruments such as symbolism, just as our faculty of sight is improved by instruments such as the telescope (cf. Leibniz's passage)" [Wang 1996, 234]. 57. "My work with respect to philosophy should consist in an analysis of higher concepts (logical and psychological), i.e., what should be done is to write a list of those concepts and to consider the possible axioms, theorems and definitions for them (of course together with the application to the empirically given reality). But in order to do that, one should first obtain through (half understood) philosophical lectures, a 'feeling' of what one might assume. On the other hand, the understanding of an axiomatic would also increase the understanding of philosophical authors (so there is a reciprocal action from 'top' and 'bottom', whereby the correct behavior is important)." [Meine Arbeit in Bezug auf Phil soll in einer Analyse der obersten Begriffe bestehen (der logischen und psychol); d. h. was letzten Endes zu tun ist, ist eine Liste dieser Begriffe aufschreiben und die möglichen Ax, Th und Def für sie überlegen (selbstverständlich samt Anwendung auf die empirisch gegebene Wirklichkeit). Um das aber tun zu können, muss man zuerst durch (halb verstandene) phil Lektüre ein 'Gefühl' dafür erwerben, was man annehmen kann. Andererseits wieder wird das Verstehen einer Axiomat das Verständnis der phil Schriftsteller erhöhen (also Wechselwirkung von 'oben' und von 'unten', wobei das richtige Verhältnis wichtig ).] [Gödel forthcoming, IX, 78-79]. 58. "Logical questions that are not mathematical and not psychological are those concerning logical primitive concepts, for example: belongs to, concept, proposition, class, ⊃, relation. So, e.g.: if there is a concept for each propositional function, if there are classes that contain themselves, if all concepts are everywhere defined. These questions trespass into the domain of

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metaphysics and can probably be decided only by the introduction of mere metaphysical concepts." [Logische Fragen, die einerseits nicht mathematisch und nicht psychologisch sind, sind solche, welche die logischen Grundbegriffe betreffen, z. B. e, Begriff, Satz, Klasse, ⊃, Relation. Also zum Beispiel: Gibt es zu jeder Aussagefunktion einen Begriff, gibt es Klassen, die sich selbst enthalten, ist jeder Begriff überall definiert. Diese Fragen greifen in das Gebiet der Metaphysik über und können wahrscheinlich nur mit Einführung rein metaphysischer Begriffe entschieden werden.] [Gödel forthcoming, IX, 62]. 59. Analogously, Bolzano had claimed that there are some fundamental concepts from which all other concepts and propositions on them can be derived, although a list of these fundamental concepts cannot be given once and for all. See for example Bolzano's remarks on the concepts "having" and "quality": one is simple and one is composed from the other, but which one is simple cannot be determined with certainty [Bolzano 1837, §80, 184]. 60. "Gödel mentioned the following list of logical primitives of a general theory of concepts: negation, conjunction, existence, universality, object, the concept of concept, which all belong to predicate logic, and the relation of application (which is specific to a theory of concepts)" [Wang 1996, 277]. 61. "The designation of numbers in the dual system is more similar to a real 'ideog-raphy' (i.e., there are more properties that can be deployed from the symbols and there is less arbitrariness in the designation) than the decimal system. In the latter for example all numbers from 1 to 10 are designated in a fully arbitrary way, whereas in the dual system this is the case only for 0 and |, but one can prescind from this too when one considers the mere sequential structure. The less arbitrary designation is certainly ||||, and this apparently gives the most faithful 'image' of numbers." [Die Bezeichnung der Zahlen im Dualsystem kommt einer wirklichen 'Begriffsschrift' näher (d.h., es sind mehr Eigenschaften unmittelbar aus den Symbolen abzulösen, und es herrscht weniger Willkürlichkeit in der Bezeichnung) als die Dezimale. In dieser z.B. alle Zahlen von 1 bis 10 völlig willkürlich bezeichnet, in der dualen nur 0 und |, aber auch von dieser abzusehen, wenn man die bloße Reihenstruktur betrachtet. Am wenigsten willkürlich ist freilich die Bezeichnung ||||, und diese gibt scheinbar das treueste 'Bild' der Zahlen] [Gödel forthcoming, X, 80]. See also the following passage from the Philosophical Manuscripts: XI, 112-113. Note that this example is the same one as mentioned by Leibniz (see above p. 163). 62. "Onlv God exists, God is One" [Gott allein ist, Gott ist Eines] [Gödel forthcoming, IX, 51]

ABSTRACTS

This paper tackles the question of whether the order of concepts was still a relevant aspect of scientific rigour in the 19th and 20th centuries, especially in the case of authors who were deeply influenced by the Leibnizian project of a universal characteristic. Three case studies will be taken into account: Hermann Graßmann, Giuseppe Peano and Kurt Gödel. The main claim will be that the choice of primitive concepts was not only a question of convenience in modern hypothetico- deductive investigations, but sometimes also the result of philosophical investigations onto the foundation of scientific disciplines. The question of the "right" order of concepts is an ideal to be followed rather than a task that can be fulfilled, but remains nonetheless an essential part of the axiomatic enterprise. This paper aims to question whether there is in fact such a stark contrast, as there is often claimed to be in the literature, between the debates relating to the right order of concepts and the foundational questions concerning modern axiomatics. The scientific rupture

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determined by the appearance of hypothetico-deductive systems in mathematics and logic should thus not be dissociated from some relevant continuities concerning the ideal of knowledge as the search for a general theory of concepts deriving from some fundamental elements.

L'article aborde la question suivante : est-ce que le bon ordre des concepts peut être considéré un élément essentiel de rigueur scientifique dans la logique et les mathématiques de XIXe et XXe siècle, en particulier quand il s'agit d'auteurs qui ont été influencés profondément par le projet leibnizien de la caractéristique? L'article prend en considération trois exemples : Hermann Graßmann, Giuseppe Peano et Kurt Gödel. Selon notre thèse, le choix des concepts primitifs dans les théories hypothético-déductives n'était pas seulement une question d'opportunité, mais parfois aussi le résultat d'une investigation philosophique sur les fondements des disciplines scientifiques. La question du « bon » ordre des concepts n'est plus considérée comme une tâche réalisable, maïs elle est devenue un idéal à suivre ; néanmoins elle reste une partie essentielle du travail axiomatique. L'article vise donc à critiquer l'opposition trop nette qu'on trouve dans la littérature entre l'investigation de l'âge classique sur le bon ordre des concepts et la création de l'axiomatique moderne. La rupture scientifique déterminée par la création des systèmes axiomatico-deductifs en mathématiques et logique doit donc être associée à certains éléments de continuité qui regardent l'idéal de la connaissance en tant que recherche d'une théorie générale des concepts à obtenir par composition de certains éléments fondamentaux.

AUTHOR

PAOLA CANTÙ Aix-Marseille Université CNRS - CEPERC UMR 7304 (France)

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Varia

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Une nouvelle sémantique de l’itération modale

Brice Halimi

1 La sémantique moderne de la logique modale a repris de Leibniz la notion de monde possible, tout en cherchant à s’affranchir du cadre métaphysique dans lequel cette notion est ancrée chez Leibniz. Le terme de « monde possible » est, pour beaucoup d’auteurs, un simple moyen commode et imagé de désigner les unités d’interprétation des formules de la logique modale. On trouve néanmoins en logique modale un résidu essentiel de la métaphysique leibnizienne, dont même les auteurs les moins réalistes sont les héritiers : le fait que les mondes possibles forment un tout défini une fois pour toutes, dont la variation est hors de propos. Certes, à différentes interprétations d’un système de logique modale correspondront, dans la sémantique kripkéenne, différents ensembles de mondes possibles. Néanmoins, pour chaque interprétation donnée, l’ensemble de tous les mondes possibles est fixé et d’emblée clos. Or, si ce présupposé est naturel dans le contexte du système de Leibniz, il n’a pas de justification claire dans le contexte de la logique modale contemporaine, où les modalités itérées sont la règle.

2 En effet, Leibniz n’envisage à aucun moment de qualifier tel ou tel trait de notre monde de nécessairement contingent ou d’accidentellement nécessaire : une vérité est nécessaire ou contingente, mais dans l’un comme l’autre cas elle l’est absolument. Dans cette perspective, le fait que l’ensemble des mondes possibles forme une totalité close et absolue est dans l’ordre des choses. En revanche, les modalités itérées sont l’objet même des systèmes formels de logique modale, dans la mesure où la nécessité comme la possibilité y sont représentées par des opérateurs indéfiniment itérables. Or, dès lors que toute nécessité peut elle-même être posée comme contingente (à tort ou à raison, peu importe — seul importe que la contingence de la nécessité d’une vérité ait un sens), il devient naturel de considérer que l’ensemble des mondes possibles relativement auquel une vérité est dite nécessaire soit lui-même un ensemble contingent de mondes possibles, c’est-à-dire un ensemble de mondes possibles parmi d’autres ensembles possibles de mondes possibles. Et par là toute totalisation des mondes possibles est remise en cause, ou devrait du moins pouvoir l’être. Ce qui suit est l’exploration d’une telle voie.

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1 Les deux sens de la nécessite

3 Par itération modale, il faut entendre toute superposition de qualifications modales. Dire, par exemple, d’une proposition qu’elle est « nécessairement nécessairement vraie », ou bien encore d’une proposition qu’elle est « nécessairement possiblement nécessairement vraie », c’est à chaque fois produire un cas d’itération modale. Le principe de l’itération modale remonte historiquement au moins à Clarence I. Lewis, c’est-à-dire à la tradition algébrique de la logique modale. En revanche, l’ouvrage de Rudolf Carnap intitulé Meaning and Neeessity, qui a marqué un véritable renouveau de la logique modale, n’envisage à aucun moment la perspective d’une itération modale. Ceci s’explique par la conception que Carnap se fait de la nécessité comme validité logique. L’interprétation sémantique que Carnap propose de la nécessité renvoie en effet à l’espace logique de tous les mondes possibles définis comme ensembles maximalement consistants de propositions (pour un langage fixé) ; dans le cadre d’une telle analyse, une vérité peut bien être dite logiquement valide (vraie dans tous les mondes possibles), mais il est dénué de sens de se demander si cette validité logique est elle- même logiquement valide. Pour le dire dans les termes de Wittgenstein, l’énoncé qu’une proposition est possible ou nécessaire ne constitue pas lui-même l’énoncé d’une proposition ; il n’est donc, à proprement parler, ni vrai ni faux, et, a fortiori, ne saurait être dit possiblement ou nécessairement vrai : cf. [Wittgenstein 1922, 5.525, 6.124]. Cette idée, avec celle d’espace logique, a été reprise par Bas van Praassen dans [van Praassen 1969]. On peut dès lors distinguer deux grandes conceptions opposées de la nécessité : selon la première, toute qualification modale est fondamentalement itérable (et exprimée à cet effet par un opérateur du langage) ; selon la seconde, le redoublement de la nécessité (en particulier lorsque cette dernière exprime la notion métalinguistique de validité logique) n’a aucun sens1. S’il est tout à fait cohérent de combiner, comme Leibniz, une conception absolue des mondes possibles et un rejet des modalités itérées — la première n’étant que la traduction sémantique du second —, il l’est moins, en revanche, d’autoriser des modalités itérées de degré arbitrairement grand tout en maintenant un ensemble absolu de mondes possibles.

4 La sémantique modale introduite par Saul Kripke en 1963 dans [Kripke 1963], fondée sur la considération de relations d’accessibilité entre mondes possibles, présente l’avantage d’une forme de conciliation de ces deux voies. Selon cette sémantique, un

cadre pour la logique modale propositionnelle est une structure Ғ =〈W, R, wo〉 formée par un ensemble W dont les éléments sont appelés par convention « mondes

possibles », par un élément distingué wo de W qui représente le « monde actuel », et par une relation binaire définie sur W, le sens intuitif de la condition wRw’ étant que le monde w’ représente une variante admissible de w relativement au type de possibilité (physique, métaphysique, logique, etc.) en jeu. Pour un cadre donné, une valuation est une correspondance V qui assigne, à chaque variable propositionnelle p du langage, une interprétation V(p) consistant en un sous-ensemble de W formé par tous les mondes possibles dans lesquels p est réalisée. Un modèle de Kripke est un quadruplet ℳ = 〈W, R,

wo, V〉 obtenu à partir d’un cadre par la donnée supplémentaire d'une valuation V sur les variables propositionnelles.

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5 Tout système déductif pour la logique modale propositionnelle (dont on suppose ici la syntaxe connue) est caractérisé par un certain groupe d’axiomes. Un système est dit normal s’il inclut • toutes les tautologies de la logique propositionnelle non modale : (pV ¬ p), ((p → q) → (¬q → ¬p)), etc. • l’axiome K : □(p → q) → (□p → □q), et s’il a pour règles d’inférence • le modus ponens • la règle de nécessitation : si Φ est un théorème, alors □Φ également (de sorte que toute tautologie est automatiquement comptée comme nécessaire) • la règle de substitution uniforme : si une formule χ(p) contenant p est un théorème, alors il en va de même de χ[Φ/p], pour toute formule Φ du langage.

6 Le système qui résulte de ces axiomes et règles est appelé K. L’addition de l’axiome T, □p → p (ou, de façon équivalente, p → ◇p), définit le système KT. L’addition supplémentaire de l’axiome 4, □p → □□p, définit le système S4 = KT4 ; celle de l’axiome 5, ◇p → □◇p, définit le système S5 = KT5.

7 La sémantique modale kripkéenne consiste dans les clauses inductives suivantes pour la satisfaction : • ℳ, w ⊨ p ssi w ∈ V (p) ; • ℳ, w ⊨ ¬Φ ssi ℳ, w ⊭ Φ ; • ℳ, w ⊨ (Φ ∧ ψ) ssi ℳ, w ⊨ Φ et ℳ, w ⊨ ψ ; • ℳ, w ⊨ (Φ ∨ ψ)ssi ℳ, w ⊨ Φ ou ℳ, w ⊨ ψ ; • ℳ, w ⊨ □Φ ssi, pour tout w’ tel que wRw’ ℳ, w’ ⊨ Φ • ℳ, w ⊨ ◇Φ ssi il existe w’ tel que wRw’ ℳ, w’ ⊨ Φ 8 Une formule Φ est dite valide dans ℳ ssi ℳ, w ⊨ Φ pour tout w ∈ V. Elle est dite valide dans un cadre Ғ ssi elle est valide dans tout modèle issu de Ғ. Enfin, elle est dite valide (tout court) si elle est valide dans tout cadre d’interprétation du langage.

9 Il est relativement aisé de voir qu’un cadre valide le système S4 (resp. S5) ssi sa relation d’accessibilité est réflexive et transitive (resp. une relation d’équivalence). La sémantique de Kripke est particulièrement propice à l’obtention de tels résultats, dits de complétude. Son avantage est également d’autoriser l’itération modale tout en la cantonnant à certaines limites, en la linéarisant. Ainsi, dans le cas de la nécessité redoublée : ℳ, w ⊨ □□p ssi ∀υ tel que wRυ, ∀u tel que υRu, ℳ, u ⊨ p 10 Comme on le voit, l’exploration du graphe défini par la relation d’accessibilité R permet d'interpréter l'itération modale : plus haut est le degré modal d'une formule, plus longs sont les chemins à considérer sur le graphe. La sémantique kripkéenne traduit ainsi toute itération modale par une ramification.

11 Elle constitue en cela une façon optimale d’interpréter l’itération modale au moyen d’un seul et même ensemble de mondes possibles W explicitement donné dès le départ. Mais s’agit-il de l’unique façon d’interpréter l’itération modale ? La suite de cet article voudrait suggérer une autre voie pour restituer toute la force de l’itération modale. Il s’agira moins, toutefois, de critiquer la sémantique de Kripke qu’au contraire de la prolonger et, en un certain sens qu’on précisera, de la généraliser.

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12 Revenons au préalable sur la signification des modalités itérées. Admettons par exemple comme douée de sens l’assertion qu’une vérité nécessaire l’est pour des raisons contingentes. Un enchâssement de plans de validité est par là même suggéré : selon le plan le plus immédiat, la proposition considérée est tenue pour nécessairement vraie, mais, selon un plan plus profond qui relativise le premier, la nécessité attribuée à cette proposition apparaît elle-même comme un trait contingent. De même, permettre l’invocation de situations qui sont seulement possiblement possibles, c’est autoriser une marge de variation bien plus ample que celle représentée par le simple possible, puisque c’est accepter de faire varier le système de variation même sur lequel est fondée la notion de possible, et resituer ce système de possibilité comme étant seulement un parmi d’autres possibles, à l’intérieur d’un système de possibilité d’une échelle supérieure. Corrélativement, affirmer qu’une proposition est nécessairement nécessaire, c’est affirmer incomparablement plus que la simple nécessité de cette proposition ; c’est en effet, intuitivement, affirmer qu’elle est nécessaire quelle que puisse être la donnée de tous les possibles.

13 La signification de l’itération modale demande donc d’introduire des contextes variables d’évaluation de ce qui est possible ou impossible. Comment le faire dans les termes de la sémantique des mondes possibles ? Si la vérité d’une proposition donnée p

signifie sa satisfaction dans le monde actuel wo, sa nécessité (c’est-à-dire la vérité de la proposition □p énonçant la nécessité de p) signifie sa satisfaction, non seulement dans le monde actuel, mais dans tout monde possible w ∈ W (pour simplifier, on suppose ici

que tout monde possible est accessible depuis wo, ce qui revient à adopter l’acception la plus stricte possible de la nécessité). On passe ainsi de la considération d’un monde

distingué wo à celle d’un ensemble de mondes W dont wo n’est qu’un membre parmi d’autres possibles. De la même manière, affirmer qu’une proposition est nécessairement nécessaire devrait engager la considération d’un ensemble d'ensembles de mondes dont W ne serait qu'un membre parmi d'autres possibles, et qui jouerait relativement à W le

rôle joué par W relativement à wo. 14 Cet ensemble d’ensembles de mondes possibles devrait à son tour, dans le cas d’une nouvelle itération modale, jouer le rôle de monde actuel relativement à un ensemble d’ensembles d’ensembles de mondes possibles, etc. On appellera démultiplication modale chacune de ces étapes : le passage d’un monde à un ensemble de mondes de niveau supérieur, puis à un ensemble d’ensembles de mondes de niveau encore supérieur, et ainsi de suite.

15 Une telle progression n’est pas entièrement étrangère à la sémantique modale héritée de Kripke. En effet, l’itération syntaxique de l’opérateur de nécessité induit, dans la

sémantique kripkéenne, un changement dans le lieu d’évaluation des formules : wo pour

les formules non modales, Ew0 : = {w ∈ W : W0Rw} pour les formules de degré 1,ᑗw∈Ew Ew

pour les formules de degré 2 (où Ew : = {w’ ∈ W : wRw’}), et ainsi de suite. On aboutit donc bien à un emboîtement de systèmes de possibilité, au sens où chacun des mondes

possibles w appartenant à un certain système de possibilité Ew* (si w*Rw) définit lui-

même de son côté un nouveau système Ew = {w’ ∈ W : wRw’}. En cela, la sémantique kripkéenne constitue une prise en compte véritable de l’itération modale.

16 On peut cependant rechercher une sémantique qui fasse davantage droit à la portée démultiplicative des modalités itérées. En effet, la sémantique kripkéenne confine d'avance tout ensemble de mondes à un sous-ensemble de W, ensemble clos,

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explicitement donné dès le début, de tous les mondes possibles en général. Certes, la collection de tous les mondes actuellement possibles est bien elle-même

contrefactualisable — en ce sens que l’ensemble Ew0 des mondes possibles relativement

au monde actuel w0 peut être remplacé par un autre ensemble Ew —, mais ce dans les limites d’un stock W fixé. Ensuite, Kripke ne fait pas d’un monde possible un monde intrinsèquement relatif à un système de possibilité : le même monde w peut en général être accessible à partir de mondes différents et se trouver au terme de branches de longueurs différentes. Par exemple, un même monde w peut être 1 - accessible à partir

d’un certain monde w1 (si w1Rw), mais seulement 3 -accessible à partir d’un autre

monde w2 (si w2Rw', w'Rw'' et w''Rw pour un certain w' et un certain w'' sans que w2-Rw,

w2-Rw" ni w'Rw). Autrement dit, il est bien vrai que, dans la sémantique de Kripke, les possibilités varient d’un monde à un autre — certains mondes n’étant possibles (accessibles) que relativement à certains mondes ; néanmoins, on ne peut pas distinguer de réels niveaux de possibilité : un monde possible, bien qu’accessible relativement à un autre monde, n’est pas en lui-même relatif à ce monde. Enfin, deux

systèmes Ew et Ew' n’ont en général rien à voir entre eux : s’il existe des relations entre les mondes, il n’en existe pas en général entre les ensembles de mondes, à l’échelle de ce qui serait un système de possibilité d’ordre supérieur. Sans doute, aucun des traits qu’on vient de souligner ne compromet la sémantique kripkéenne. Mais, précisément, l’objet de ce qui suit ne sera pas de critiquer cette dernière, il sera au contraire de radicaliser l’écart qu’elle marque avec la conception leibnizienne des mondes possibles, et également de généraliser d’une certaine manière le cadre qu’elle propose, d’un point de vue qui sera précisé à la fin.

17 Il ne s’agira pas non plus de contester tel ou tel axiome modal. Il faut en effet distinguer deux versants tout à fait indépendants l’un de l’autre : le choix d’un certain type de sémantique, et le choix d’un certain système axiomatique. Etant donné le premier, le second revient à sélectionner, parmi toutes les interprétations possibles, celles qui vérifient certains axiomes, mais précisément il ne change rien au principe général de la sémantique dans laquelle on décide de se placer, c’est-à-dire au sens donné aux opérateurs modaux en tant que tels, et par suite à l’itération modale. Le but du présent travail n’est donc pas de rejeter le système modal S5 correspondant à la conception leibnizienne de la nécessité. Le système S5 affirme que l’itération modale nous reconduit à l’éventail de possibles pris pour point de départ, que les possibilités actuelles épuisent d’emblée toutes les possibilités possibles. Il s’agit là d’une thèse qui ne va pas de soi. Mais cela signifie simplement que le système S5 correspond sémantiquement à une classe très particulière de structures au regard des structures d’interprétation en général.

18 Une dernière question préjudicielle reste à aborder. Elle concerne le concept même de monde possible d’ordre supérieur. Comment concevoir qu’un ensemble de mondes possibles puisse être considéré à son tour comme un monde actuel vis-à-vis d’un ensemble d’ensembles de mondes possibles ? Comment un ensemble de mondes possibles pourrait-il lui-même constituer un monde possible de mondes possibles : comment plusieurs mondes possibles disjoints pourraient-ils former un monde, c’est-à- dire une totalité structurée et unifiée ?

19 En réponse à cette difficulté, il convient de penser un monde possible d’ordre supérieur, non comme un ensemble de mondes possibles de niveau inférieur, mais à l’inverse comme un ensemble de mondes possibles de niveau supérieur, en faisant de

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chaque monde de niveau supérieur une certaine spécification du monde de niveau inférieur auquel il est attaché. Dans cette perspective, tout monde d’ordre supérieur émerge, non par synthèse, mais par analyse : tout monde de niveau inférieur devient un monde d’ordre supérieur du fait d’être regardé comme l’ensemble de toutes ses spécifications possibles au niveau supérieur — exactement comme l’analyse mathématique identifie une valeur a à l’ensemble de tous les développements limités possibles d’une fonction ayant cette valeur ; ou bien exactement comme la géométrie rapporte un point à l’ensemble des petites portions de courbes passant par ce point (autrement dit, à l’ensemble des façons de passer par ce point).

20 L’ordre d’un monde est donc à penser sur le modèle d’un ordre de dérivation. Selon ce modèle, tout monde possible recouvre un ensemble de mondes pour peu qu’il soit examiné de façon plus fine et apparaisse ainsi au carrefour de virtualités situées à un autre niveau, mais il ne cesse pas par ailleurs de constituer un monde, ce qui répond à la question de savoir comment une simple pluralité de mondes peut elle-même former un monde. Ce modèle présente en outre l’avantage technique qu’il est plus aisé d’analyser une structure interne en un monde que de construire une superstructure de mondes en l’absence de tout moyen canonique pour le faire.

21 Un monde de mondes possibles, ou monde possible d’ordre supérieur, n’est donc rien d’autre qu’un monde possible, c’est-à-dire un point d’un certain espace de possibilité, pris comme index d’un nouvel espace de possibilité, dont les points sont appelés mondes possibles de niveau supérieur. Parler d’un « monde de mondes possibles » n’est qu’une façon de désigner cette juxtaposition d’un monde possible et d’un système de mondes possibles relatifs à ce monde. Le problème n’est plus de faire en sorte qu’une pluralité de mondes possibles fasse monde, mais il reste à préciser à quelles conditions un espace suffisamment cohérent de mondes peut être introduit et attaché en propre à chaque monde possible pris comme point de départ. C’est ici que la géométrie intervient.

2 Géométrisation

2.1 Démultiplication modale

22 Dans tout ce qui suit, l’opérateur modal primitif sera, non la nécessité, mais la possibilité. La sémantique modale recherchée est une sémantique dans laquelle chaque introduction de l’opérateur ‘◇’ déplace l’évaluation des formules à un niveau supérieur au sein d’un étagement de systèmes de mondes possibles : partant des mondes possibles usuels de premier niveau, on doit ainsi aboutir à des mondes possibles de second niveau relatifs à tel ou tel monde de premier niveau, puis à des mondes possibles de troisième niveau dont chacun est à son tour relatif à un monde de deuxième niveau lui-même relatif à un monde de premier niveau, et ainsi de suite — le passage d'un monde w à l'ensemble des mondes relatifs à w constituant une opération de démultiplication indéfiniment itérable.

23 Dans cette représentation des choses, chaque monde donné est l’index du système des mondes possibles (de niveau supérieur) qui lui sont relatifs. Il est donc important d’avoir à l’esprit la distinction entre l’ordre et le niveau d’un monde possible. Un monde possible d’ordre 1 est un monde possible de niveau n auquel est relatif un ensemble de mondes possibles de niveau n + 1. Un monde possible d’ordre 2 est un monde de niveau

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n auquel sont relatifs des mondes possibles de niveau n + 1 eux-mêmes chacun d’ordre 1. Et ainsi de suite. Comme on se ramène ultimement aux mondes de niveau 0, les mondes possibles d’ordre i (requis pour l’interprétation d’une formule modale de degré i), dorénavant, seront les mondes possibles de niveau 0 à chacun desquels est relative une succession de mondes possibles emboîtés de niveaux respectifs 1, 2,..., i et d’ordres respectifs i — 1, i — 2,..., 0. L’ordre correspond à la profondeur attachée aux mondes possibles de base, en fonction du degré modal de la formule à interpréter.

24 Une structure mathématique apte à servir de modèle doit donc schématiquement associer à chaque point d’un certain espace (point représentant un monde possible pris pour base) un nouvel espace qui représente le système des mondes relatifs à ce monde de base et dont chaque point pourra à son tour se voir associer un espace propre, et

ainsi de suite. L’hypothèse directrice de tout ce qui suit est que l’espace tangent TxM à une variété différentiable M en l’un de ses points x répond à cette description, menant ainsi à une interprétation géométrique de la logique modale. (Une variété n’est rien d’autre que la généralisation mathématique de l’idée de surface, comme par exemple une sphère ; l’espace tangent, la généralisation de l’image du plan tangent à une sphère en l’un de ses points.)

25 x L’union disjointe ∐x∈MT M de tous les espaces tangents à une variété M donne elle- même lieu à une nouvelle variété, appelée le « fibré tangent » TM de la variété initiale

M, et accompagnée d’une projection naturelle p : TM → M définie par p(νx) = x, pour tout

vecteur vx tangent à M en x. Comme le fibré tangent TM est lui-même une variété, on peut en considérer le propre fibré tangent, TTM = :T²M, puis à nouveau TTTM = : T3M, etc., autrement dit TiM pour tout i ≥ 0 (avec T0M = M et T1M = TM). Pour une explication précise des notions géométriques mentionnées (variété différentiable, espace tangent, fibré tangent, etc.), voir [Lafontaine 1996, II, III].

26 Dans l'hypothèse qui est ici adoptée, une variété M représente donc l'ensemble des

mondes possibles de niveau 0 ; TxM, l’ensemble des mondes possibles de niveau 1 relatifs x à x ∈ M, et TM = ∐x∈MT M, l’ensemble des mondes possibles de niveau 1 en général. Pris en soi, un monde possible x ∈ M de niveau 0 est un monde possible au sens usuel ; pris

en tant que point de tangence entre TxM et M, ce même monde x devient l’index d’un système de possibilité, un monde de mondes possibles. Pour un vecteur vx tangent à M en x, T(x, vx)TM, représente l’ensemble des mondes possibles de niveau 2 relatifs à (x,vx) ∈ TM, et TTM, l’ensemble des mondes possibles de niveau 2 en général ; et ainsi de suite. Le niveau d'un monde possible x peut donc être défini comme l’indice i pour lequel x ∈ TiM, et l’ordre de ce même monde comme la différence j — i, où j représente le niveau maximum des mondes relatifs à x qui sont considérés. Dans ce cadre, l’évaluation d’une formule non modale prend place en n'importe quel monde de niveau 0, c'est-à-dire en n'importe quel point de M ; celle de ◇Φ fait appel à un ensemble

d’ensembles de mondes possibles, à savoir l’ensemble TM de tous les TxM (x ∈ M) ; de même, l’évaluation de ◇◇Φ met en jeu TTM ; et ainsi de suite. Le passage de M à TM, puis à TTM correspond ainsi au passage de Φ à ◇Φ, puis à ◇◇Φ. L’idée géométrique de démultiplication correspond au fait que la dimension de Ti+1M est à chaque fois le double de celle de TiM.

27 La nouvelle sémantique de l’itération modale qui est recherchée relève ainsi d’un projet de géométrie modale consistant finalement à comprendre toute itération modale selon un fibré géométrique. Dans une telle perspective, il est assez naturel d'interpréter

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chaque variable propositionnelle p par un ensemble de courbes2 γ : ℝ → M sur M, puis de déclarer p vraie en x ssi l’une de ces courbes γ passe par x. De même, on va le voir, la relation d’accessibilité devient un ensemble de chemins continus et peut ainsi prendre tout son sens géométrique. Toutefois, l’idée d’une telle géométrie modale se heurte à un certain nombre de difficultés de construction, qu’il s’agit à présent d’analyser.

2.2 Tâches

28 Trois tâches sont en effet à prendre en considération. Il faut tout d’abord indiquer inductivement les moyens d’interpréter ¬p, (p ∧ q) et (p ∨ q) par des ensembles de courbes respectifs, en partant des deux ensembles de courbes interprétant p et q. Il faut ensuite pouvoir interpréter ◇p au moyen d'un ensemble de courbes se situant, non plus au niveau de M, mais au niveau de TM : on peut parler à ce propos de relèvement modal. Il faut, en outre, pouvoir définir un relèvement d’un autre type pour n’importe quelle famille de courbes sur M Considérons en effet une formule aussi simple que (◇p ∧ q) : si ◇p est interprétée par une famille de courbes sur M, l'ensemble des courbes sur M interprétant q doit pouvoir être transporté en un ensemble de courbes sur TM, de façon que l'interprétation de q puisse être combinée à celle de ◇p — mais sans qu’un tel transport ne se confonde avec un relèvement modal, puisque, en l'occurrence, ◇q ne figure pas dans la formule considérée. On qualifiera donc ce transport de relèvement non modal.

29 Les tâches à remplir sont dès lors les suivantes. La première (Tl) consiste à définir le relèvement modal de l’interprétation de n’importe quelle formule, faisant passer d’une famille de courbes sur TiM (interprétant une formule Φ de degré modal i) à une famille de courbes sur Ti+1M (interprétant la formule ◇Φ). La deuxième tâche (T2) est celle de définir le relèvement non modal de l’interprétation de n’importe quelle formule Φ. Du fait des écarts en degré modal entre sous-formules d’une même formule, toute famille de courbes sur TiM doit posséder un relevé non modal à Ti+1M. La troisième tâche (T3) est corrélative des deux premières : le relèvement (modal ou non modal) de n’importe quelle famille de courbes (interprétant une certaine formule) appelle la possibilité de relever n’importe quel monde possible. Supposons en effet que la variable p soit interprétée par une famille de courbes sur une variété M. L'interprétation de ◇p sera alors une famille de courbes sur TM, et de même l'interprétation de ◇◇p, une famille de courbes sur TTM. La formule ◇◇p sera vraie en x ∈ M si l'une des courbes sur TTM interprétant cette formule passe, non par x (c'est impossible, puisque x ne se situe pas sur TTM), mais par un certain élément de TTM relié de manière précise à x. Cela i suppose, de façon générale, qu'à chaque point x i de T M corresponde un ensemble de points de Ti+1M, de sorte que tout point x ∈ M donne finalement naissance à une suite π 0(x) = {x}, π1(x) ⊆ TM, π2(x) ⊆ TTM,... de contreparties de x. La dernière tâche (T4) a pour objet de vérifier que les axiomes élémentaires de toute logique modale normale sont vérifiés. On aboutit ainsi à la définition suivante : un cadre modal géométrique consiste en une

suite (pi+1 : Mi+1 → Mi)i ≥ 0 de fibrés sur la base d’une variété , permettant de définir : 1° ) le relevé

modal λ(Γ) à Mi+1 de toute famille de courbes Γ sur Mi ; 2° ) le relevé non modal L(Γ) à Mi+1 1 de toute famille de courbes r sur Mi ; 3° ) une suite ({x}, π (x) ⊆ M1, π²(x) ⊆ M2,...) au- i+1 i dessus de chaque point x de M vérifiant pi+1(π (x)) ⊆ π (x) pour tout i ≥ 0. Une valuation

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sur un tel cadre est l'assignation, à chaque variable propositionnelle p, d'un ensemble de courbes sur M. Un cadre muni d'une valuation s'appelle un modèle modal géométrique.

2.3 Modèles modaux métriques

30 Le traitement des tâches (T1)-(T3) requiert l'ajout à M d'une structure supplémentaire suffisamment riche pour permettre la définition relativement canonique des différents relèvements à opérer. Pour des raisons qui apparaîtront plus clairement au fur et à mesure, une voie assez naturelle consiste à se tourner vers la géométrie dite « riemannienne », qui représente le noyau de la géométrie différentielle moderne, et qui est une source très riche de constructions géométriques.

31 Le plan des développements qui vont suivre et mener jusqu’à la conclusion est ainsi le suivant. L’introduction d’un certain nombre de notions de géométrie riemannienne est tout d’abord nécessaire à la présentation du nouveau cadre sémantique qui vient d’être esquissé. Ce cadre, qui concrétise le projet de géométrie modale, sera ensuite défini en détail, compte tenu, notamment, des trois premières tâches qui ont été dégagées. Enfin, un certain nombre de premiers résultats seront présentés.

32 Une variété riemannienne est une variété munie d’une « métrique » g, qui rend possible

de mesurer la longueur gx(v,v) de tout vecteur tangent v ∈ TxM, autrement dit la vitesse de tout mouvement ayant lieu dans M, et par suite la distance entre deux points le long d'une courbe donnée sur M. La donnée d’une métrique g sur M induit la « connexion de Levi-Civita » ∇ associée à g3. Intuitivement, l’opérateur ∇ associe à deux champs de 4 vecteurs X et Y, définis au voisinage d’un point de M, la déviation infinitésimale (∇xY)

(x0) de Y en x0 par rapport à la direction définie par Xx0. Un champ de vecteurs X le long

d’une courbe γ est dit parallèle si ∇xX = 0 en chaque point de γ. Une courbe γ dont le champ de vecteurs tangents γ' est parallèle s’appelle une géodésique : c’est une courbe dont le vecteur vitesse ne dévie jamais de lui-même. Dans le cas de l’espace euclidien, une géodésique n’est rien d’autre qu’une ligne droite. En général, c’est une courbe qui minimise localement la distance entre deux points par lesquels elle passe (ainsi des méridiens sur une sphère).

33 γ La connexion ∇ associée à g permet de définir le transport parallèle J t,t'(v) d’un vecteur v

∈ Tγ(t)M le long d’une courbe γ, du point γ(t) de paramètre t au point γ(t') de paramètre γ t'. Intuitivement, J t,t'(v) est, comme son nom l'indique, le résultat du déplacement de v le long de γ (de γ(t) à γ(t')) lorsqu’on évite tout glissement comme tout pivotement, ce qui fournit le moyen de rapporter l’un à l’autre des vecteurs appartenant pourtant à des espaces tangents différents.

34 Par ailleurs, toute métrique g supposée donnée sur une variété M induit une métrique 5 naturelle gT sur TM, appelée la métrique de Sasaki de TM . La construction de gT est itérable : en partant d’une variété riemannienne (M,g), on définit une tour canonique de

projections, à savoir ⟨pn+1 : (Mn+1,gn+1) → (Mn,gn)⟩ n ≥ 0, avec M0 = M,g0 = g et Mn+1 = TMn,gn+1

= (gn)T 35 A présent, tout champ de vecteurs X défini le long d’une courbe γ sur M constitue lui- même une courbe sur TM définie au voisinage de γ(0) = x : il suffit pour le voir d’écrire ‘

X(t)’ au lieu de Xγ(t), ce qui fait bien de X une courbe X :] → ϵ, ϵ [ → TM sur TM. Si le

champ de vecteurs X est parallèle le long de γ, X'(0) ∈ TX(0)TM, vecteur tangent à la

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courbe X, est dit horizontal. C’est notamment le cas si γ est une géodésique et que X = γ'. Une courbe δ : ℝ → TM sur TM est dite horizontale si δ'(t) l’est pour tout t ∈ ℝ. L’ensemble des vecteurs horizontaux de TTM forment un sous-fibré HTM de TTM tel que 6 HvxTM est isomorphe à TxM selon Tvxp1, et ce pour tout vx ∈ TxM. Par suite , pour toute géodésique γ de (M,g) et tout vecteur v ∈ Tγ(0)M, il existe localement une unique courbe v v horizontale γ̃ sur TM passant par (γ(0), v) et telle que p1 ° γ̃ = γ. Cette courbe s’appelle « le relevé horizontal de γ passant par v en t = 0 » et est elle même une géodésique de (TM, gT). En particulier, pour toute géodésique γ de (M,g) (γ,γ') est le relevé horizontal de γ passant par γ'(0), ainsi qu’une géodésique de (TM,gT). Pour définir une structure de démultiplication conforme à la définition abstraite qu’on en a donnée, il convient d’introduire des mondes possibles de niveaux croissants, à chaque fois relatifs à un monde possible de niveau immédiatement inférieur. Pour cela, on suppose donnée une sélection = {γi : i ∈ I} de courbes sur M, appelées courbes d’accessibilité, qui représentent autant de chemins d’accessibilité entre mondes possibles. Ceci étant, on note γ l’ensemble {γ(t) : t ∈ ℝ} des points d’une courbe quelconque γ et, pour x ∈ γ̅ on note tx le paramètre correspondant à x (autrement dit : 7 1 γ(tx) = x) . Pour tout x ∈ γ̅i, A (x) : ∪=xg∈γ̅i γ̅ i représente l’ensemble des mondes accessibles depuis x. Ces mondes, de niveau 0, ne sont pas encore les contreparties de x au niveau 1. Les contreparties de x peuvent cependant se déduire des mondes accessibles depuis x, au moyen d'une application permettant de coder les points situés au voisinage d'un point donné x d'une variété riemannienne M, par des vecteurs tangents de TxM. Pour tout vecteur v ∈ TxM, on peut en effet noter cv, la courbe géodésique de (M,g) telle que 8 cv(0) = x et c'v(0) = v . L’application v ∈ TxM → cv(1) ∈ M constitue un difféomorphisme entre un voisinage Vx du vecteur nul de TxM et un voisinage Ux de x dans M. Le plus grand réel rx > 0 tel que la boule Bx : = B( , rx) de centre et de rayon rx soit incluse dans Vx s’appelle le rayon de convexité de M en x. Pour deux vecteurs quelconques v et w distincts dans Bx, on a donc toujours : cv(1) ≠ cw(1). On peut alors introduire 1 1 l’ensemble P (x) : = {v ∈ Bx : cv(1) ∈ A (x)} des représentants de x dans TxM, puis, au-delà 1 1 9 de TxM, à l'échelle de TM en totalité, π (x) : = {c'v(t) : v ∈ P (x),t ∈ ℝ} . Le passage de P1(x) à π1(x) étend naturellement des mondes possibles (de niveau 1) isolés en des courbes de mondes possibles (de niveau 1). Les éléments de π1(x) seront appelés les contreparties de x au niveau 1. (Les mondes possibles de niveau 1 relatifs à x seront, quant

à eux, tous les éléments de TxM, et par extension de TM, contenus dans le relèvement modal d'une courbe passant par x.) Il est à présent nécessaire d’indiquer l’itération de la construction de π = π1 le long de la tour de variétés riemanniennes de base M. Pour n = 0, on pose simplement : π0(x) = {x}. 2 Comment définir π (x) ? Toute courbe γi sur M définit un transport parallèle

et par conséquent, pour tout v ∈ Tγi(tv)M une courbe

sur TM. On peut donc construire successivement A2(α), P2(α) et π2(α) pour α ∈ TM comme on a défini A1(x), P1(x) et π1(x) pour x ∈ M, en T remplaçant les géodésiques c de ( M,g) par celles c de ( TM, gT), et les courbes de l’accessibillté γi depuis x ∈ M (pour un certain x donné) par toutes les courbes de 2-

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accessibilité depuis un élément quelconque w ∈ Tx0M de π(x). Plus précisément, si

l’on note I0 l’ensemble des indices j de I tels que x0 ∈ γ̅j (et tj le nombre tel que x0 = γj(tj),

pour chaque j ∈ I0), l’ensemble des mondes possibles accessibles depuis w est

. L’ensemble des représentants de w est alors P(w) := {u ∈ Bw : (1) ∈ A(w)} (en notant la géodésique de (TM,gT) telle que (0) = w et '(0) = u), et l’ensemble complet des contreparties de w est π(w) :={( )'(t) : u ∈ P(w), t ∈ ℝ}. L’ensemble des contreparties de x au niveau 2 est l’ensemble des contreparties des 2 1 contreparties de x au niveau 1, autrement dit π (x) : = ∪w∈π (x) π(w). La, définition de π n(x) pour π ≥ 3 est analogue.

36 Il reste à définir les relevés respectivement modal et non modal de n'importe quelle courbe γ de M à TM, à TTM, et ainsi de suite. Pour cela on introduit, pour chaque n, une n sélection Cn de courbes sur T M, appelées courbes admissibles, qui sont les courbes dont l’interprétation d’une proposition de degré modal n est astreinte à se composer. On

demande seulement que (i) si δ ∈ Cn+1, alors pn+1(δ) ∈ Cn et que (ii) pour toute γ ∈ Cn, il

existe δ ∈ Cn+1 relevant γ, c’est-à-dire telle que pn+1(δ) = γ. Les relevés γ̃γ'(t) sont les extensions naturelles à Tn+1M d’une courbe γ sur TnM, et sont donc un choix naturel pour le relèvement modal de γ10. Par ailleurs, comme le vecteur

nul est le représentant naturel de x dans TxM ( correspond visuellement au point

de contact de l’espace tangent TxM avec M), il est également naturel d’introduire, pour n toute courbe γ sur T M, la section nulle de Pn+1 le long de γ, autrement dit la courbe γ̂¨ n+1 sur T M située au-dessus de γ qui est donnée par : γ̂(t) = γ(t) pour tout t ∈ ℝ. On introduit alors, pour toute courbe γ̂ sur TnM, les deux ensembles suivants de courbes sur Tn+1M :

ν λ(γ) = {γ̃ ∈ Cn+1 : ∃t; k ∈ ℝ tels que v = kγ'(t)}

(γ) = {γ̂} ∩ Cn+1 Comme l’écart en degré modal de deux sous-formules d’une même formule peut être strictement supérieur à 1, on dénit l'itération de l'opérateur de relèvement non modal :

On peut à présent définir un cadre modal géométrique F̲ de type métrique comme un

quadruplet (M, g, , (Cn)n≥0). Une valuation V dans ce cadre consiste en l’assignation, à toute variable propositionnelle p, d’un ensemble V(p) de portions continues de 11 membres de C0 . À toute courbe correspond l’ensemble γ̅ de ses points, et pour tout

ensemble Γ de courbes on note : Γ̅ = ∪γ∈Γ γ̅ et par ailleurs λ(Γ) = ∪ γ∈Γ λ(γ) et L(Γ) = ∪γ∈Γ L(γ). On suppose que chaque V(p) est un ensemble clos par passage aux sous- courbes : toute courbe de V(p), restreinte au voisinage de l'un de ses points, donne une courbe appartenant encore à V(p). La conséquence de cette dernière hypothèse est que

et que . La valuation V(Φ) de toute formule Φ est alors définie par induction : • V(¬Φ) = Cn \ V (Φ) (pour deg(Φ) = n) ;ˆ • V(Φ ᐱ ψ) = V(Φ) ∩ (V ( )) (pour deg(Φ) = n et deg(ψ) = m < n) ;ˆ • V(Φ ᐱ ψ) = V (Φ) ∪ (V(ψ)) (sous la même hypothèse que ci-dessus) ;

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• V(◇Φ) = λ(V(Φ)). 37 Un modèle modal métrique M̲ est un cadre modal métrique muni d’une valuation V.

38 Etant donné un tel modèle M̲, on pose finalement, pour toute formule Φ de degré modal n et tout x ∈ M :

39 Une formule Φ est donc vraie en x si l’un des mondes possibles relatifs à x contenus dans V(Φ) coïncide avec une contrepartie de x au niveau fixé par le degré modal de Φ. On peut prendre un exemple simple pour illustrer les principales définitions : le plan euclidien P2 muni d’un repère, avec pour courbes d’accessibilité toutes les droites horizontales (c’est-à-dire toutes les droites parallèles à l’axe des abscisses). Soit O = (0,0) le point de P2 pris pour origine. L’unique courbe d’accessibilité passant par O est l’axe des abscisses : A1(O) = {(a, 0) : a ∈ ℝ}. On a par conséquent :

, car un vecteur appliqué en O ne peut avoir pour pointe un point de l’axe des abscisses qu’à la condition d’être horizontal. Par suite,

2 : a ∈ ℝ, α ∈ ℝ}. Soit à présent γp une courbe de P

faisant partie de la valuation de la formule p. Par définition, λ(γp) est l’ensemble des

courbes de la forme t → (γp(t), kγ'p(t0)) (pour un certain k ∈ ℝ et un certain t0 ∈ ℝ). Pour que P2, O ⊨ ◇p, il faut et il suffit qu’il existe une courbe γp dans V(p) vérifiant

(pour au moins un quintuplet ( t,k,to,a,α) de nombres réels). Comme on peut toujours prendre k = α = 0, on obtient finalement : 2 P , O ⊨ ◇p ssi il existe une courbe γp dans V(p) coupant l’axe des abscisses en au moins un point (qui peut être autre que O).

40 On retrouve ainsi une clause sémantique très proche des conditions de satisfaction de ◇p dans un modèle de Kripke : la sémantique modale géométrique fondée sur les modèles métriques permet de ressaisir la sémantique des modèles de Kripke comme une forme de cas particulier. La sémantique de Kripke correspond en effet au cas où l’espace topologique sous-jacent au graphe de la relation d’accessibilité est un espace discret. La sémantique fondée sur les modèles métriques en est ainsi à la fois une continuisation et une généralisation.

41 Un modèle modal métrique M̲ donne en général à l’évaluation des formules un caractère essentiellement local. En particulier, la forme du codage de A1(x) par P1(x) dépend à chaque fois de la structure géodésique de (M,g) au voisinage de x ∈ M, et il en

va à nouveau de même aux niveaux supérieurs (Mn, gn) pour n ≥ 1. L’esprit de la sémantique kripkéenne (de la clause donnée pour la satisfaction de ◇p) est donc conservé, mais mis en œuvre de façon à chaque fois propre au point considéré, ce qui est conforme à l’idée fondamentale que chaque monde, en cas d’itération modale, devient l’index d’un système de possibilité qui lui est propre. L’axiome T, p → ◇p, est bien valide. Supposons en effet (en se restreignant au cas où p 1 ̅ est une formule de degré modal nul) que π (x) ∩ γ̂ ≠ Φ, autrement dit que γ(tx) ∈ π 1 (x) pour une certaine courbe γ ∈ V(p) et un certain paramètre tx ∈ ℝ. On a alors : γ(

(tx) tx) = γ̃ (0) ∈ , et donc . Par conséquent

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implique , et donc M̲, x ⊨ p → ◇p. De plus, la définit ion de □Φ entraîne qu’une courbe γ (sur Tn+1M si Φ est de degré n) appartient à V(□Φ) ssi γ est disjointe de toute courbe (sur Tn+1M) relevant une courbe (sur TnM) disjointe de . On vérifie ainsi que l’axiome K, □(p → q) → (□p → □q), est également valide. 42 L’axiome 4 n’est généralement pas valide. En effet, la vérification de M̲, x ⊨ ◇p suppose que λ(V(p)) passe par au moins un point de π(x) ; en revanche, la vérification de M̲, x ⊨ ◇◇p suppose simplement que λ(λ(V(p))) passe par au moins un point de π(w) pour un élément w de π(x) qui peut tout à fait être autre que x. En revanche, on peut supposer que 4 devient valide si l'on se restreint à une classe très spéciale de cadres modaux métriques. Supposons en effet que (M,g) soit une variété riemannienne simple (c'est-à-dire telle

que deux points quelconques de M sont reliés par une unique géodésique), que et C0

soient uniquement composés de géodésiques, que, pour tout n ≥ 0, Cn+1 soit l’ensemble

des relevés horizontaux des courbes de Cn, et enfin que deux courbes quelconques de soient ou bien identiques à reparamétrisation près, ou bien disjointes. Un cadre modal métrique F̲ vérifiant ces conditions sera dit minimal. Proposition 1. L’axiome 4 est valide dans tout cadre métrique minimal.

Démonstration. Soit F̲ = 〈M,g, ,(Cn)n>0〉 un cadre métrique minimal, qu’on suppose kc'(t) muni d’une valuation V. Pour toute géodésique c de M, λ(c) ⊆ C1. En outre, c̃ = kc', qui

est une géodésique de TM. Par induction, on montre ainsi que, pour toute courbe c de Co n et pour tout n ≥ 1, λ (c) ⊆ Cn et que, pour tout n ≥ 0, Cn se compose uniquement de géodésiques. Dans ce qui suit, pour plus de simplicité, on restreint l’axiome 4 à une variable propositionnelle p, mais la démonstration est analogue pour une formule Φ quelconque, c’est-à-dire pour ◇◇Φ → ◇Φ.

Pour x ∈ M, supposons qu’il existe une courbe cx ∈ passant par x. Cette courbe est par hypothèse une géodésique unique à reparamétrisation près. Tous les mondes

accessibles depuis x sont les points de cx et, pour tout point y de cx distinct de x, cx est la seule géodésique reliant x et y, car (M,g) est supposée simple. On a donc : P1(x) =

(k0c'x(0) ∈ Bx : k0 ∈ ℝ} et πi(x) = (k0c'x(t0) : k0c'x(0) ∈ Bx, k0, t0 ∈ ℝ }. Par suite :

Mais, pour k0 fixé, et k0c'x sont, à reparamétrisation près, deux courbes géodésiques

de TM ayant deux points d’intersection, puisque k0c'x(t0) et que (1) =

k0c'x(t1). Ces deux points sont bien distincts, car sinon on aurait (1) = w (1), alors

que ∈ Bw et u ∈ Bw. On en déduit que k0c'x (à reparamétrisation près), et ainsi

1 1 que π²(x) = ∪k0∈ℝ . Si aucune géodésique de ne passe par x, alors P (x) = π (x) =

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et π²(x) = , donc l’égalité précédente reste valable : il suffit de prendre k0

= 0. Si une (unique) géodésique de passe par x mais en se réduisant à {x}, alors c'x =

1 et c''x = , d’une part, d’autre part π (x) = et π²(x) = , donc l’égalité est encore valable.

De plus, comme on l’a dit, pour λp ∈ V(p), λ(γp) est composée (à reparamétrisation

près) de toutes les courbes kγ'p, et toutes ces courbes sont des géodésiques contenues

dans C1. De même, λ(λ(γp)) ⊆ C2 est composée (à reparamétrisation près) de toutes les 2 courbes kγ̋p. Supposons à présent que 〈F̲,V〉, x ⊨ ◇◇p, c’est-à-dire que π (x) ∩

λ(λ(V(p)) ≠ Ø. Cela signifie qu’il existe γp ∈ V(p) et k,k0,t,t' ∈ ℝ tels que k0c''x(t) = kγ̋p(t'). 12 Il s’ensuit que k0c'x(t) = kγ'p(t'), donc que

. En outre,

avec , donc

. Par conséquent . Ainsi,

implique que , et donc 〈F◌̲,V〉 ⊨ ◇◇p → ◇p.

43 Soit S4* le système S4 privé de la règle de nécessitation, et dont la règle de substitution uniforme est restreinte au cas où le substituens possède un degré modal au plus égal à celui du substituendum. On peut montrer que le système S4* est complet relativement à la classe de tous les cadres métriques minimaux. La neutralisation de l’itération modale — neutralisation incarnée par l’axiome 4 — correspond ainsi, dans le cadre de la sémantique géométrique qui vient d’être proposée, à des modèles métriques très particuliers, ce qui est conforme au diagnostic que l’itération modale n’a pas un sens aisément neutralisable.

44 On pourrait objecter que la complexité de la sémantique fondée sur les cadres modaux géométriques rend cette dernière difficilement praticable d’un point de vue logique. Toutefois, ce serait adopter des critères techniques, et non philosophiques. Pour cette même raison, la facilité avec laquelle divers résultats de complétude peuvent être obtenus dans le cadre de la sémantique kripkéenne ne saurait être un argument. Seul importe ici le sens à donner à l’itération modale, et la possibilité d’exhiber une sémantique modale démultiplicative fidèle à ce sens. A cet égard, la sémantique géométrique qui vient d’être proposée est bien conforme à l’idée d’un déploiement progressif d’une hiérarchie ouverte de mondes possibles : elle introduit des mondes possibles d’ordres croissants au fur et à mesure que le degré modal de la formule considérée augmente.

Conclusion

45 La compréhension de l’itération modale défendue dans cet article repose sur l’idée suivante : à supposer qu’on raisonne en termes de mondes possibles pour analyser les notions modales aléthiques, la donnée de l’ensemble actuel des mondes possibles doit

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elle-même pouvoir être contrefactualisée, c’est-à-dire replacée dans un ensemble d’ensembles de mondes possibles — monde de deuxième ordre lui-même contrefactualisable, et ainsi de suite. C’est cette structure indéfiniment ouverte du possible que vise à représenter la géométrie modale. Dire qu’il aurait été possible qu’un certain état de choses fût possible, c’est dire que cet état de choses aurait été possible si le système de possibilité défini par l’ensemble actuel des mondes possibles avait été différent.

46 Il ne va pas de soi de considérer l’itération modale comme douée de sens, et différents auteurs ont précisément refusé de le faire. Mais, dès lors que l’itération modale est acceptée, elle devrait être interprétée comme un saut radical, en rupture avec l’héritage leibnizien d’une totalité absolue de mondes possibles donnée une fois pour toutes. Passer d’une possibilité à une possibilité de possibilité (ou bien d’une nécessité à une nécessité nécessaire), ne devrait pas revenir simplement à augmenter d’un cran la longueur des explorations d’un même système de possibilité fixe, mais devrait au contraire impliquer une démultiplication, c’est-à-dire le passage d’un système de possibilité à un autre système de possibilité, au sein d’un système de possibilité de deuxième ordre. Telle a été la thèse conceptuelle défendue dans la première partie de cet article.

47 La seconde partie a consisté à proposer un cadre sémantique conforme à cette thèse. Il n’est pas seulement conceptuellement justifié, il est également techniquement possible de traduire l’itération modale par une démultiplication fondée sur une collection ouverte de mondes possibles articulés selon des niveaux indéfiniment croissants. La géométrie différentielle et la géométrie riemannienne fournissent des outils naturels au service d’une telle intuition, le passage d’une variété à son fibré tangent fournissant une image fidèle et intuitive de la démultiplication modale. D’autres raisons justifient de se tourner vers la géométrie : en particulier, la possibilité d’exploiter pleinement le sens géométrique de la notion d’accessibilité, et la possibilité de comparer un opérateur modal à un opérateur différentiel.

48 La sémantique géométrique proposée dans cet article est certes techniquement plus complexe que la sémantique kripkéenne, ce qui rend moins simple son emploi, et plus ardue l’obtention de théorèmes de complétude. Toutefois, un premier résultat a pu être obtenu, concernant l’axiome 4. En outre, le premier but de la géométrie modale est de faire droit à la signification démultiplicative des modalités. Il est de montrer précisément comment cela est possible, et par là, en retour, de rendre plus précise l’idée de démultiplication elle-même. Or l’interprétation de la logique modale fondée sur les modèles métriques suffit à ce propos, tout en ouvrant à une approche plus géométrique de la logique modale.

BIBLIOGRAPHIE

GALLOT, Sylvestre, HULIN, Dominique & LAFONTAINE, Jacques [2004], Riemannian Geometry, Berlin : Springer.

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KRIPKE, Saul [1963], Semantical considérations on modal logic, Acta Phüosophica Fennica, 16, 83-94.

LAFONTAINE, Jacques [1996], Introduction aux variétés différentielles, Grenoble : Presses Universitaires de Grenoble.

LEHMANN, Daniel & SACRÉ, Carlos [1982], Géométrie différentielle des courbes et des surfaces, Paris : PUF.

VAN FRAASSEN, Bas [1969], Meaning relations and modalities, Nous, 3(2), 155-167.

WITTGENSTEIN, Ludwig [1922], Tractatus Logico-Philosophicus, London: Routledge.

NOTES

1. À cet égard, une certaine interprétation de l'axiome 4, ▫p → ▫▫p (toute proposition n'est nécessaire qu'à la condition de l'être nécessairement), peut être considérée comme un croisement paradoxal des deux conceptions qu'on vient de distinguer. En effet, une justification souvent avancée de l'axiome 4 est la restitution de l'idée de validité logique. Mais l'interprétation de ‘▫’ comme exprimant la validité logique vaut pour une modalité simple (▫ø), non pour une modalité redoublée (▫▫ø). Par conséquent, de deux choses l'une : ou bien ‘▫’ est interprété par la validité logique, et dans ce cas l'axiome 4 est naturel mais informulable, car l'itération syntaxique de l'opérateur ‘▫’ n'a plus de sens clair ; ou bien l'itération modale est admise, mais dans ce cas l'interprétation de ‘▫’ par la validité logique, et avec elle la justification naturelle de l'axiome 4, devient problématique. 2. Une courbe sur une variété M n'est rien d'autre qu'une fonction γ : ℝ → M supposée infiniment dérivable. La naturalité et l'importance des courbes sur une variété M sont dues au fait que les seuls sous-ensembles naturels de M sont les ensembles sous-jacents aux sous-variétés de M, et que, parmi ces sous-variétés, les courbes sont celles de plus petite dimension, et en cela les plus simples. Dans le contexte de la géométrie riemannienne, il est d'autant plus naturel de considérer les courbes d'une variété, que la considération des courbes, et notamment des géodésiques, est ce qui permet de définir la distance entre deux points. 3. Cf. [Lehmann & Sacré 1982, III] et [Gallot, Hulin et al. 2004, 2B] pour une définition de ∇. 4. Un champ de vecteurs sur une variété M est une application X qui à tout point x de M associe un certain vecteur Xx de TxM. 5. Cf. [Gallot, Hulin et al. 2004, 80]. 6. Cf. [Gallot, Hulin et al. 2004, 100-102]. 7. On supposera qu'aucune courbe d'accessibilité ne se coupe elle-même, et par suite que toute courbe d'accessibilité ne passe qu'une seule fois par chacun de ses points, ce qui permet de définir tx de façon univoque. 8. L'existence et l'unicité de cette géodésique sont démontrables, voir [Gallot, Hulin et al. 2004, 81].

9. L'application (v,t) → c'v(t) s'appelle le « flot géodésique » de la variété (M,g). 10. . L'existence et l'unicité du relevé horizontal local de γ ne sont assurées que si γ est une n géodésique de (T M,gn). C'est une raison pour limiter les sélections Cn à des ensembles de géodésiques. 11. Par « portion », il faut comprendre qu'une courbe de V(p) peut ne décrire qu'une partie connexe de l'ensemble des points d'une courbe admissible, et avoir un vecteur vitesse variable (nul lorsqu'on atteint les extrémités de la courbe). On appellera « courbes admissibles », en un sens élargi, de telles portions de courbes admissibles au sens strict.

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12. En effet, pour toute courbe γ, le vecteur tangent γ'(t) est par définition attaché au point γ(t) : si donc deux vecteurs tangents y1 (t) et y2(u) sont identiques, ils appartiennent a fortiori au même espace tangent, ce qui implique que y 1 (t) = y 2(u).

RÉSUMÉS

Dire d’une proposition que, nécessairement, elle est nécessairement vraie, c’est affirmer incomparablement plus que ce que l’on affirme en disant simplement qu’elle est nécessairement vraie. C’est en effet, intuitivement, affirmer qu’elle est nécessaire quelle que puisse être la donnée de tous les mondes possibles à l’aune de laquelle sa nécessité est établie. C’est faire de cette donnée elle-même un possible parmi d’autres, et faire ainsi référence à des mondes possibles d’ordre supérieur. Cet article vise à formaliser la notion de monde possible d’ordre supérieur au moyen d’outils empruntés à la géométrie riemannienne. Le cadre sémantique proposé repose sur une collection ouverte de mondes possibles de niveaux croissants, par opposition à l’héritage leibnizien d’une totalité close de mondes possibles.

Saying that a proposition is necessarily necessarily true claims incomparably more than simply saying that this proposition is necessarily true. Indeed, it amounts, intuitively, to saying that the proposition is necessarily true whatever the range of all the possible worlds may be. This range, upon which the assessment of simple necessity relies, then becomes a possible datum among others, which triggers the reference to higher-order possible worlds. This article aims at formalizing such a notion of high-order possible world, by using tools coming from Riemannian geometry. The semantical framework that is finally put forward involves an open-ended collection of possible worlds lying at higher and higher levels, in sharp contrast to the Leibnizian heritage of a fixed closed totality of possible worlds.

AUTEUR

BRICE HALIMI

Université Paris Ouest (IREPH) & SPHERE (France)

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