# PhilMath Intersem 5.2014

The fifth annual meetings of the PhilMath Intersem (PhilMath Intersem 5) will take place this coming June (June 2014).

There will be eight meetings of the seminar this year.

Meeting dates are June 3rd, June 5th, June 10th, June 12th, June 17th, June 19th, June 24th and June 26th.

Some of the sessions are single sessions (one presentation and discussion), and some double (two presentations and discussions).

All meetings will take place on the Rive Gauche campus of the University of Paris 7-Diderot, in the Condorcet Building.

All but one of the meetings, that of Thursday, June 19th, will take place in the salle Klimt (Condorcet 366A). The meeting of June 19th will take place in the salle Malevitch (Condorcet 483A).

See schedule below for fuller information, including titles, summaries and suggested readings.

## Theme

The topic of this year's seminar is definition in mathematics. Definition has had a prominent place in mathematics throughout its history. What this prominence might best be taken to consist in is less clear. So too is the question of what rational warrant there may be for it. A prime goal of this seminar is to get clearer on these and related matters and to identify what if any problems may stand in the way of such clarification.

Particular concerns that may figure in our discussions include:

• The variety of ends that definition has been taken to serve in the history of mathematics

• Specific means by which these ends have been pursued in mathematical practice

• Changes that have been proposed concerning the ends of mathematical definition

• Other types of changes (i.e. other than changes of end or purpose) that are in evidence in the history of mathematics

• Significant practical and/or theoretical limitations on the attainability of the supposed ends of mathematical definition

• Significant clarificative role(s) for definition in mathematics, what such clarification might consist in and how it might be achieved

• Compelling phenomena of "depth" or "penetration" that serve as ideals for mathematical definition, how these might relate to the identification and proper treatment of primitive terms or concepts, and how they may relate to ideals of "depth" or "penetration" concerning proof

## Schedule

#### June 3. Salle Klimt (366A)

##### 16h00--18h00

**Aldo Antonelli** (Philosophy, U of California-Davis), "Definitions by Abstraction and Reference to Abstract Objects"

Summary: Definitions by abstraction, of the sort championed by the neo-logicists, are usually construed as introducing particular kinds of abstract objects, such as numbers, directions, etc. The advantage of this way of proceeding is that the abstract objects so introduced are available as referents of the corresponding sort of singular terms. As a result they can be captured by the range of ordinary first-order quantifiers, and can stand in various relations to ordinary objects, thereby underpinning the applicability of mathematics and other formal disciplines to empirical science. The aim of this talk is to argue that these aims can be achieved under a "thin" notion of abstraction that does not require the introduction of sui-generis first-order objects, while preserving the mathematical role that they play, e.g., with respect to empirical science. We rehearse the proposal in the case of arithmetic, pointing out the conceptual difference between postulating a separate realm of first-order objects delivered by abstraction and the independent issue of commitment to higher-order entities (sets, functions, relations, quantifiers, etc.).

Reading:

• Antonelli, A. "The Nature and Purpose of Numbers"

• Antonelli, A. "Notions of Invariance for Abstraction Principles"

#### June 5. Salle Klimt (366A). Double Session

##### i. 14h00--16h00

**Michael Rathjen** (Pure Mathematics, U of Leeds), "Indefiniteness of mathematical problems?"

Summary: Feferman proposed a logical framework T for what’s definite and what’s not. Roughly speaking, what’s definite is the domain of classical logic, what’s not is that of intuitionistic logic. He also conjectured that certain mathematical problems such as CH would be indefinite in T. Recently his conjecture has been verified and in the talk I will sketch a proof of it.

Reading:

• Feferman, S. "Is the Continuum Hypothesis a definite mathematical problem?"

##### ii. 16h15--18h15

**Sean Walsh** (Logic & Philosophy of Science, U of California-Irvine) & **Tim Button** (Philosophy, U of Cambridge), "Categoricity and Knowledge of Structure"

Summary: Categoricity arguments loom large in contemporary philosophy of mathematics and contemporary foundations of mathematics. Roughly, a categoricity argument for a specific mathematical axiomatization says that any two models of the axioms are isomorphic (or perhaps can otherwise be fit together in some precise fashion). These kinds of arguments are thought to serve a variety of different ends: dissipating mathematical skepticism (cf. Kreisel, "Informal Rigor & Completeness Proofs" (1967)), distinguishing foundational from non-foundational axiomatizations (cf. Grzegorzyk, "On the Concept of Categoricity", *Studia Logica** *13 (1962)), isolating certain strains of distinctively geometrical thought (cf. Zilber, *Zariski Geometries: Geometry from a Logician's Point of View* (2010)). To serve these ends, it's natural to conceive of the mathematical axiomatization in question as encapsulating some crucial part of our knowledge of the mathematical domain. Our contention is that to be successful, these arguments inevitably end up suggesting that there is also some knowledge of mathematical structures that goes above and beyond knowledge of mathematical theories. Whether or not this is fatal to these projects thus depends on our estimation of the pressures to relegate mathematical knowledge to knowledge of mathematical axioms.

Reading:

• McGee, V. "How We Learn Mathematical Language", *Philosophical Review* (1997): 35-68

• Grzegorczyk. A. "On the Concept of Categoricity", *Studia Logica* (1962): 39-66

• Zilber, B. "Geometric Stability Theory and the Trichotomy Conjecture (Appendix B.2)" In *Zariski Geometries: Geometry from a Logician’s Point of View*, London Mathematical Society Lecture Note Series, vol. 360 (CUP 2010), 236-243

#### June 10. 14h00. Salle Klimt (366A). Double Session

##### i. 14h00--16h00

**Norbert Schappacher** (IRMA, Université de Strasbourg), "Definitions in flux---Remarks on the evolution of terminology in modern Algebraic Geometry"

Summary: A few examples from the history of Algebraic Geometry between the 1870s and the 1950s will serve to illustrate typical strategies of definitory evolutions in modern mathematics. Specific notions discussed will include: "intersection" (and its avatars), "normal variety", and "arithmetic." In the last part of the talk we will try to assess these strategies of evolution of mathematical notions from an abstract point of view.

Reading:

Schappacher, N. "Rewriting Points"

##### ii. 16:15--18h15

**Davide Crippa** (HPS, U of Paris 7-Diderot), "The Role of Impossibility Arguments in Defining the Nature of Problems (an early modern endeavor)"

Summary: I will argue that a certain type of impossibility result emerged in early modern geometry with respect to the methodological demand to solve a problem by the most adequate means given its nature. I will argue that such a requirement, inherited from ancient geometry, is pervasive in XVIIth century geometrical thinking. A related problem was therefore how to search for the best, simplest or most adequate method for a given problem, and correlatively to exclude inapproriate means. A promising direction of research might be to envisage Descartes' problem-solving strategy, and in particular its inherent method of analysis, as an instrument for understanding the nature or structure of a problem. I will substantiate my claim using case studies.

Reading:

• Sefrin-Weis, H.,* Pappus of Alexandria. Book IV of The Collection*, Springer, 2010. Pp. 144-155, 271-291, 301-309

• Descartes, R., *The Geometry of René Descartes*, ed. D. E. Smith and M. L. Latham, Open Court, 1952 Pp. 152-192

• Bos, H., *Redefining Geometrical Exactness*, Springer-Verlag, 2001. Chapters 27 and 28

#### June 12. Salle Klimt (366A)

##### 16h00

**Paolo Mancosu** (Philosophy, U of California-Berkeley), "Frege's *Grundlagen* §64 and the mathematical practice of definitions by abstraction in the nineteenth century"

Summary: In *Grundlagen* §64, Frege discusses some peculiar definitions that are nowadays called definitions by abstraction and conveys the impression that they are still quite rare in mathematical practice. The goal of that section is to explore, through an extended comparison with geometrical examples, whether one can obtain the concept of number by arriving at it through the definition of an equality between numbers obtained by means of the relation "the objects falling under the concepts C and D are in one-to-one correspondence." The definition in question is now dubbed Hume's principle: "Num(C)=Num(D) iff the objects falling under the concepts C and D are in one-to-one correspondence". In §64, Frege discusses examples from the geometrical literature such as: "the direction of a = the direction of b iff a and b are parallel". Much of the attention in the analytic literature has been devoted to the important issue of establishing in what sense the left hand side and the right hand side of Hume's principle, or of a definition by abstraction in general, have the same meaning/content. But there are other interesting claims made by Frege in §64 that have hitherto found no satisfactory treatment. In the first part of the paper I argue that, contrary to the impression conveyed by Frege in *Grundlagen*, definitions by abstraction were rather common in nineteenth-century mathematics and find their roots in classical mathematics. To investigate the extent of such definitions I discuss some paradigmatic cases from number theory, foundations of the number systems (complex numbers and irrationals), set theory, geometry, and vector theory. Using the broad canvas of the first part of the paper, I then focus on Frege’s section 64 of the *Grundlagen*. It has always been a matter of wonder to me that with so much ink spilled on that section, the sources of Frege’s discussion of abstraction principles have remained elusive. I hope to have filled this gap by providing textual evidence coming from a specific tradition in geometry and from the tradition of textbooks in geometry for secondary schools. In addition, I explain why Frege never appeals to analogous definitional techniques from number theory and I put Frege’s considerations in the context of a wide debate in Germany on ‘directions’ as a central notion in the theory of parallels.

Reading:

• Frege, G. *Foundations of Arithmetic*, §§63-69

• Draft chapter from forthcoming book, circulated privately. Request at: PM.Paper

#### June 17. Salle Klimt (366A). Double Session

##### i. 14h00--16h00

**Sébastien** **Gandon** (Philosophy, Université Blaise Pascal), "Russell against frivolous definition. The case of the definition of distance in *The Principles of Mathematics* (1903)"

Summary: I will contrast what Russell explicitly said about the logical adequacy of a good definition in *The Principles of Mathematics* and the way he used definitions in this work. In this respect, the case of metrical geometry is especially interesting, because, although Russell developed many ‘logically irreproachable’ definitions of the notion of a (linear and angular) distance, he ultimately rejected them all, on the pretext that they were ‘extremely frivolous’ (§408). Then, although he had the technical means to derive metrical geometry from logic, Russell ultimately preferred to consider metrical geometry as an empirical science. This shows that Russell asked more from a definition than mere logical adequacy. But how should we characterize this additional demand? I will end my talk by some remarks on this question.

Reading:

• Russell, B. *Principles of Mathematics*, chs. 47, 48

• Gandon, S. *Russell's Unknown Logicism*, ch. 2

##### ii. 16h15--18h15

**Timothy Bays** (Philosophy, U of Notre Dame), "The Implicit Definition of the Natural Numbers"

Summary: There’s a natural idea that our understanding of the structure of the natural numbers somehow stems from our understanding of the axioms of arithmetic. In this talk, I'll try to clarify this natural idea and defend it against some recent criticisms. I'll then explain the real problem with this idea.

Reading:

• Quinon & Zdanowski: “The Intended Model of Arithmetic. An Argument from Tennenbaum’s Theorem”

• Dean, W. "Models and Computability”

• Button, T. & Smith, P. “The Philosophical Significance of Tennenbaum’s Theorem”

#### June 19. Salle Malevitch (483A). Double Session

##### i. 14h00--16h00

**Sébastien Maronne** (Insitut de Mathématiques de Toulouse, Université de Toulouse Paul Sabatier), "Origins and practice of nominal definition in the medieval tradition and in early modern geometry"

Summary:

I will first sketch Aristotle's theory of definition and the origins of nominal definition in the works of medieval authors. Then I will study the practice of nominal definition in early modern geometry by Descartes, Desargues and Pascal, by considering in particular the treatment of infinity, for instance in *De l'Esprit Géométrique*.

Reading:

• Pascal, B. *De l'Esprit Géométrique* (éd. Mesnard) in *Oeuvres Complètes*, vol. III, 390-428 (in particular, 393-401)

• Gardies, J-L *Pascal entre Eudoxe et Cantor*, ch. IV, 85-108

##### II. 16h15--18h15

**Dominique Descotes** (Institut d'Histoire de la Pensée Classique, Université Blaise Pascal), "Good and base use of nominal definition"

Summary:

I will compare nominal definition in Pascal and in *La Logique ou art de penser* and then show, by mainly studying the *Pensées*, that a bad use of nominal definition can be fruitful.

Reading:

• Same as readings for immediately preceding presentation

#### June 24. Salle Klimt (366A).

##### 16h00

**Michael Potter** (Philosophy, U of Cambridge), "Frege on implicit definition"

Summary: Frege famously objected to Hilbert's view that the primitive terms in a theory are implicitly defined by the axioms of the theory. I shall argue that at the base of Frege's objection was an unclarity in his conception of logic -- an unclarity which many more recent authors have not avoided.

Reading:

• Frege, Draft letter to Jourdain 1914 in his *Philosophical and Mathematical Correspondence* (Blackwell, 1980)

• M. Potter & P. Sullivan, "What is wrong with abstraction?", *Philosophia Mathematica* 13 (2005): 187-93

#### June 26. Salle Klimt (366A)

##### 16h00

**Katherine Dunlop** (Philosophy, U of Texas-Austin), "Definitions and the Role of Intuition in Kant’s Philosophy of Mathematics"

Summary: Kant maintains that the definition of any mathematical concept includes a construction of a corresponding object. This seems to conflict with the most distinctive feature of his philosophy of mathematics: his view that mathematical cognition derives some justification from “intuition”, which Kant understands (on a close analogy with ordinary sense-perception) as “immediate” representation of particular objects. For it would seem that definitions are formulated and understood using purely conceptual resources, making intuition unnecessary. The “construction-in-definition” thesis appears in the Critique of Pure Reason and comes to the fore in writings of the early 1790s, in which Kant defends his Critical view against Leibnizian objections from followers of Christian Wolff (and the idiosyncratic rationalist Salomon Maimon). This dispute is of special interest because it concerns the applicability of Kant’s view to “modern” branches of mathematics, whose subject-matter is specified algebraically rather than in terms of constructions. Here, Kant contends that constructions are included even in the definitions of, e.g., conic sections given by equations (rather than constructive procedures). Interpreters face the dual challenge of explaining (1) what Kant could mean and (2) how the constructive aspect of definition (however it is understood) leaves a role for intuition. I explain the constructive character of definitions by arguing that definitions involve schemata, which Kant defines as rules or procedures by which the imagination generates intuition corresponding to concepts. I argue that the role of the sensible faculty (i.e. the faculty of intuition) is to constrain the formation of mathematical concepts, and with them definitions, by restricting schemata.

Reading:

• Kant, *Critique of Pure Reason*, beginning of the "Transcendental Doctrine of Method". A707/B735--A739/B767

• Kant's reply to rationalist critics (Draft). Circulated privately. Request at: KD. Paper