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Abstract
Albert Lautman (1908–44) was a philosopher of mathematics whose views on mathematical reality and on the philosophy of mathematics parted with the dominant tendencies of mathematical epistemology of the time. Lautman considered the role of philosophy, and of the philosopher, in relation to mathematics to be quite specific. He writes that “in the development of mathematics, a reality is asserted that mathematical philosophy has as a function to recognize and describe” (Mathematics, Ideas and the Physical Real (London: Bloomsbury, 2011) 87). He goes on to characterise this reality as an “ideal reality” that “governs” the development of mathematics. The relation between mathematical problems as they arise in the historical development of mathematics and the solutions that are provided to these problems by mathematicians, in the form of new mathematical theories, definitions or axioms, are governed by what Lautman characterises as a dialectics of mathematics. The aim of this paper is to give an account of this Lautmanian dialectic and of how it can be understood to govern the development of solutions to mathematical problems.
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Notes
1 The axiomatic method is a way of developing mathematical theories by postulating certain primitive assumptions, or axioms, as the basis of the theory, while the remaining propositions of the theory are obtained as logical consequences of these axioms.
2 The Bourbaki project explicitly espoused a set-theoretic version of mathematical structuralism.
3 According to mathematical structuralism, mathematical objects are defined by their positions in mathematical structures, and the subject matter that mathematics concerns itself with is structural relationships in abstraction from the intrinsic nature of the related objects. See Hellman 556.
4 The main aim of Hilbert’s program, which was first clearly formulated in 1922, was to establish the logical acceptability of the principles and modes of inference of modern mathematics by formalising each mathematical theory into a finite, complete set of axioms, and to provide a proof that these axioms were consistent. The point of Hilbert’s approach was to make mathematical theories fully precise, so that it is possible to obtain precise results about properties of the theory. In 1931 Gödel showed that the program as it stood was not possible. Revised efforts have since emerged as continuations of the program that concentrate on relative results in relation to specific mathematical theories, rather than all mathematics. See Ferreirós chapter 2.6.3.2.
5 See Largeault 215, 264.
6 The term “metamathematics” is introduced by Hilbert in “Uber das Unendliche.”
7 See Brunschvicg.
8 A mathematical definition is impredicative if it depends on a certain set, N, being defined and introduced by appeal to a totality of sets which includes N itself. That is, the definition is self-referencing.
9 The law of the excluded middle states that every proposition is either true or false. In propositional logic, the law is written “P ∨ ¬P” (“P or not-P”).
10 See Petitot 81.
11 See ibid. 113.
12 See also Barot 6, 16 n. 1.
13 See Chevalley 61.
14 See also Mathematics 189–90, 40–42; Barot 7 n. 2.
15 See Chevalley 60.
16 Which are also referred to and operate as “dualities.” See Alunni 78.
17 Which he therefore also refers to as “logical schemas.” See Mathematics 83.
18 From Lautman’s correspondence with Fréchet dated 1 Feb. 1939.
19 See Chevalley 50.
20 See Mathematics 211.
21 A cautionary word along the lines of Cavaillès’ warning of “possible misunderstandings” of Lautman’s references to Heidegger (from correspondence dated 7 Nov. 1938, cited in Granger 299): if Lautman were to be considered Heideggerian, or to be embarking on a project of fundamental ontology, because of the few places in his work where he makes brief allusion to specific conceptual distinctions in Heidegger’s work that serve as analogies for his own undertakings in relation to mathematics, a more detailed reading of Lautman, which I hope to have provided here, should lead to revising such an understanding. It is in this vein that I briefly clarify Lautman’s relation to Heidegger.
22 This is Lautman’s gloss of Heidegger.
23 See Heidegger, The Essence of Reasons 160–61.
24 See Barot 10; Chevalley 63–64.