For a continued revival of the philosophy of mathematics

Jean-Jacques Szczeciniarz

pp. 263-295

in: Hassan Tahiri (ed), The philosophers and mathematics, Berlin, Springer, 2018

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.