Abstract
This paper argues in favor of a nonreductionist and nonlocal approach to the philosophy of mathematics. Understanding of mathematics can be achieved neither by studying each of its parts separately, nor by trying to reduce them to a unique common ground which would flatten their own specificities. Different parts are inextricably interwined, as emerges in particular from the practice of working mathematicians. The paper has two topics. The first one concerns the conundrum of the unity of mathematics. We present six concepts of unity. The second topic focuses on the question of reflexivity in mathematics. The thesis we want to defend is that an essential motor of the unity of the mathematical body is this notion of reflexivity we are promoting. We propose four kinds of reflexivity. Our last argument deals with the unity of both of the above topics, unity and reflexivity. We try to show that the concept of topos is a very powerful expression of reflexivity, and therefore of unity.
Envoi. This essay is a friendly and grateful tribute to Roshdi Rashed. Needless to say, this article will deal with the philosophy of mathematics and particularly what we call recent mathematics. As a historian of mathematics, Roshdi Rashed (like the great Neugebauer) is a tireless reader of contemporary mathematics. He knows how to draw, for example from category theory examples and ways of thinking that serve as benchmarks for exploring the conceptual history of mathematics.
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Notes
- 1.
Alian Connes, 2008 A View of Mathematics: Concepts and Foundations vol. 1 www.colss.net/ or Eolss. http://www.eolss.net/sample-chapters/c02/E6-01-01-00.pdf.
- 2.
Plato, Rep. VI, 508–509, Platon, Œuvres complètes Texte établi par Auguste Diès, Paris, Les Belles Lettres, Budé T. 7-1, Platonis Opera John Burnet, Oxford Classical Texts, Clarendon Press 1900–1907, G. Leroux, Garnier Flammarion, Paris 2002, nelle édition de la République 2003.
- 3.
Marco Panza, Newton et les origines de l’analyse: 1664–1666, Blanchard, Paris, 2005.
- 4.
Alain Connes, A view of mathematics. ibid.
- 5.
Alexandre Grothendieck, Reaping and Sowing 1985 Récoltes et Semailles Part 1. The life of a mathematician. Reflections and Bearing Witness. Alexander Grothendieck 1980, English Translation by Roy Lisker, Begun December 13, 2002.
- 6.
Robin Hartshorne, Algebraic Geometry, Springer, New York, 1977.
- 7.
Actually he concept of SpecA is based on two types of earlier constructions: the correspondence between the points of an affine algebraic variety \(X \subset kn\), on an algebraically closed field k and the maximal ideals of \(k(X) = k[t_{1} \cdots t_{n}]/I\) I being the ideal of null polynomials on X: Zariski’s topology on X 2) correspondence between the points of a compact topological space K and the maximal ideals of C(K) algebra of continuous functions on K with complex values and reconstruction of the topology from this algebra (Gel’fand).
- 8.
The great innovation of Grothendieck (he proceeds in this way very often) consists in considering a geometrical object associated with any commutative ring A; to have a spectrum that depends functorially on A it is necessary to replace the maximal ideals by the primes ideals.
- 9.
EGA I, Le langage des schémas. Publ. Math. IHES 4, 1960.
- 10.
It could be noticed : Ouv(X)
- 11.
For the concept of functor see Sect. 5.3.3(ii).
- 12.
More deeply, (\(SpecA, \mathcal {O}\)) solves a universal problem: the ring A defines a contravariant functor \(F_{A}\) of the category of locally ringed spaces in the category of sets such that \(F_{A} (X, \mathcal {O}_{X})\) is the set of ringed space morphisms of (\(X \mathcal {O}_{X}\) in (•, A) (•, is the punctual space); an element of \(F_{A} (X, \mathcal {O}_{X})\) is a rings morphism \(A \rightarrow \Gamma (X, \mathcal {O}_{X})\) . For any locally ringed space \((X, \mathcal {O}_{X})\) any \(\phi \): \(A \rightarrow \Gamma (X, \mathcal {O}_{X})\) defines a unique morphism \(\Phi \) : \(\Gamma (X, \mathcal {O}_{X}) \rightarrow (SpecA, \mathcal {O})\) of locally ringed spaces such as \(\Phi \)* \((O) \rightarrow (\mathcal {O}_{X})\) defines an homomorphism \(A \simeq \Gamma (SpecA, \mathcal {O})\) \(\rightarrow \Gamma (X, \mathcal {O}_{X})\) that identifies itself to \(\phi \).
- 13.
Récoltes et Semailles Part I, The life of a mathematician. Reflection and Bearing Witness, Alexander Grothendieck, English translation by Roy Lisker, Begun December 13, 2002.
- 14.
[Lautman 2006, pp. 52–53] Les mathématiques, les idées et le réel physique Vrin, Paris. Introduction and biography by Jacques Lautman; introductory essay by Fernando Zalamea. Preface to the 1977 edition by Jean Dieudonné. Translated in Brandon Larvor Dialectics in Mathematics. Foundations of the Formal Science, 2010.
- 15.
Lautman (2006) ibid.
- 16.
ibid., (Lautman 2006, p. 130).
- 17.
Lautman means completeness in the sense of completion. The system is said to be completed if any proposition of the theory is either demonstrable or refutable by the demonstration of its negation. The property of completion is said to be structural because its attribution to a system or a proposal requires an internal study of all the consequences of the considered system.
- 18.
Recall that I am analyzing the philosophical architectonic unity of mathematics. This was illustrated by Hilbert’s position. He stressed the dominant role of metamathematical notions compared to those of the mathematical notions they serve to formalize. On this view, a mathematical theory receives its value from the metamathematical properties that embody its structure in some generic sense. We recognize in this approach one (very influential) structural conception of mathematics.
- 19.
[Lautman 2006, p. 30].
- 20.
David Hilbert, Gesammelte Abhandlungen , Verlag Julius Springer Berlin, 1932.
- 21.
Paul Bernays, Hilberts Untersuchungen über die Grundlagen der Arithmetik, Springer, 1934.
- 22.
Lautman, Essai sur les notions de structure et d’existence, Hermann, Paris 1937: the structural point of view to which we must also refer is that of Hilbert’s metamathematics etc.
- 23.
[Lautman 2006, pp. 48–49].
- 24.
Lautman, ibid.
- 25.
Brendon Larvor, Albert Lautman: Dialectics in Mathematics, Foundations of formal Science, 2010.
- 26.
Brendan Larvor, Albert Lautman, ibid.
- 27.
Lautman (2006).
- 28.
Hermann Weyl, Gruppentheorie and Quantenmechanik, Hirzel, Leipzig, 1928.
- 29.
Larvor (2010).
- 30.
Brendan Larvor, ibid.
- 31.
[Lautman 2006, p. 84].
- 32.
Larvor (2010), Lautman, 2005, p. 196.
- 33.
Lautman (2006), pp. 86–87.
- 34.
[Lautman 2006, p. 87].
- 35.
Gilles-Gaston Granger, Formes oṕerations, objets Paris, Vrin, 1994, pp. 290–292.
- 36.
Hermann Weyl, The Concept of a Riemann Surface, Addison and Wesley, 1964, First Edition die Idee der Riemanschen Fläche,Teubner, Berlin, 1912.
- 37.
Immanuel Kant, Kritik der reinenVernunft, Hartnoch Transl. N. Kemp Smith (1929) as Critique of Pure Reason, Mcmillan.
- 38.
Colin McLarty, Elementary Categories, Elementary Toposes, Clarendon Press Oxford, 1992, p. 5 “Compare the self, the universe and God in Kant 1781”.
- 39.
William Lawvere, The category of categories as a foundation of mathematics by, Proceedings of the Conference on Categorical Algebra, La Jolla Calif. 1965, pp. 1–20, Springer Verlag, New York, 1966.
- 40.
Mathieu Belanger, La vision unificatrice de Grothendieck: au-delà de l’unité (méthodologique?) de Lautman Philosophiques vol 37 Numéro 1–2010.
- 41.
Grothendieck, Récoltes et Semailles, 1985, Reaping and Sowing my translation.
- 42.
ibid., my translation.
- 43.
A. Grothendieck, Récoltes et Semailles, traduction Roy Lisker p. 66.
- 44.
Ibid.
- 45.
Mathieu Belanger [Belanger p. 15 online].
- 46.
Reaping and Sowing, 1985, my translation.
- 47.
These morphisms are étales morphisms in étales topology. Grothendieck has built other topologies in his theory of descent
- 48.
See below Sect. 6.1.7.
- 49.
Saunders Mac lane, Jeke Moerbijk, Sheaves and Geometry, Springer, New York, 1992, pp. 110, 111.
- 50.
Mac Lane, Moerbijk, ibid.
- 51.
Grothendieck, 1985, Reaping and Sowing, my translation.
- 52.
Grothendieck, 1985 ibid.
- 53.
Olivia Caramello developed a deep and powerful work on the “topos-theoretic background, and on the concept of a bridge” see “the bridge-building technique” in Olivia Caramello, ‘Topos-theoretic background” IHES, September, 2014.
- 54.
This does not implies a revision of mathematics but the following position. There are structural principles of demonstration that most mathematicians use when demonstrating. These principles, if used alone, define a constructive or intuitionistic mathematics. It has structural rules, models, an essential notion of context, soundness, etc. It can be shown that the excluded middle can not be deduced from it etc. the job is to show if this logic is sufficient or to specify what of the classical logic should be available to do some demonstrations.
- 55.
Saunders Mac Lane and Ieke Moerdijk, Sheaves in Geometry and Logic A first introduction to Topos Theory, p. 296 sq.
- 56.
John L. Bell, Toposes and Local Set Theories, Clarendon Press, Oxford, 1988, p. 236 sq.
- 57.
John L Bell, ibid., p. 23.
- 58.
John Bell, ibid.
- 59.
Robert Goldblatt, Topoi, the categorial analysis of logic, North-Holland publishing company, Amsterdam New York Oxford, 1977.
- 60.
Goldblatt, 1977, p. 382.
- 61.
These brief remarks are borrowed from the course by Alain Prouté in Paris VII Introduction to categorical logic and Robert Glodblatt (vs). Let \(\mathbf {C}\) be a small category with a Grothendieck J topology seen as a sub-object \(\Omega \in \) Ob \((\mathcal {\hat{O}})\). \(\Omega \) is the “object of truth values”, the characteristic arrow of \(J, j: \Omega \rightarrow \Omega \) has the following properties
-
•
\(j \circ T= T\)
-
•
\(j \circ j= j\)
-
•
\(\wedge \circ (j \times j)= j \circ \wedge \)
This Grothendieck topology corresponds to Lawvere Tierney topology on a presheaf category. Moreover a topos has an internal language that allows to manipulate the logical or mathematical objects already present in it. This language is given to us by the work by Bénabou Mitchel. And in this language we get a translation of Lawvere-Tiernay topology. The translation by the adverb locally corresponds intuitively to the way in which the Lawvere Tierney topologies were constructed as generalizations of Grothendieck topologies. Take the presheaf \(\zeta \) that associates to the open set U all functions \(f: U \rightarrow \mathbf R \) and the subpresheaf \(\eta \) that associates to the open set U to the functions \(f : U \rightarrow \mathbf R \) which are the functions that are bounded on U. Let \(\phi : \zeta \rightarrow \Omega \) the characteristic arrow of inclusion \(\eta \subset \zeta \). Above the open \(U \phi \) sends a function f defined on U on the set of open sets included in U, on which f is bounded, while \(j \circ \phi \) sends f on all open which are covered by open on where f is bounded, in other words, on all open sets on which f is locally bounded. The effect of the modality j on the internal predicate \(\phi \) is to attach the adverblocally to it.
-
•
- 62.
William Goldblatt, Topoi The categorical analysis of Logic North–Holland, Amsterdam-New York-Oxford, 1979, p. 286 sq.
- 63.
We refer not only to second order logics but also to other logics.
- 64.
Robert Goldblatt, ibid. p. 287.
- 65.
Michael P. Fourman, Connections between category theory and logic D. Phil. Thesis Oxford University, 1974.
- 66.
André Boileau, Types versus Topos, Thèse de Philosophie Doctor Université de Montréal, 1975.
- 67.
William Lawvere, Introduction and ed. for Toposes, Algebraic Geometry and Logic, Lectures Notes in Mathematics, Vol. 274, Springer Verlag, 1972.
- 68.
[Lautman 2006, p. 228].
- 69.
Brendan Larvor, p. 201.
- 70.
Albert Lautman, Les mathématiques les idées et le réel physique, p. 79 my translation.
- 71.
A. Lautman, ibid. p. 263, my translation.
- 72.
A. Lautman, ibid. p. 263, my translation.
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Szczeciniarz, JJ. (2018). For a Continued Revival of the Philosophy of Mathematics. In: Tahiri, H. (eds) The Philosophers and Mathematics. Logic, Epistemology, and the Unity of Science, vol 43. Springer, Cham. https://doi.org/10.1007/978-3-319-93733-5_12
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