L.J.E. Brouwer : Topologie et constructivisme

Abstract

SUMMARY. — Contrary to the received view, a close connection exists between Brouwer's topological works and his philosophy of mathematics. Brouwer arrived at his main results by abandoning the abstract methods of general (or point-set) topology and opting instead for a combinatorial approach that seeks to make topological concepts arithmetic. In this article I investigate the genesis of Brouwer's notion of the « degree » of a mapping and the discovery of the equivalence of the existence of fixed points in a topological mapping and the existence of singularities in the field of vectors tangent to the sphere.

A generalization of the Takeuti–Gandy interpretation

“…The situation is however not completely satisfactory since this justification takes place in ZFC, and uses crucially non-effective arguments (a combination of classical logic and the axiom of choice), while one of the ambitions of type theory is to be a language for constructive mathematics (Martin-Löf 1973). It is a weird fact that this unexpected connection between homotopy theory/algebraic topology and type theory (Awodey and Warren 2009;Kapulkin et al 2012;Voevodsky 2010) involves non-effective reasoning since algebraic topology has its historical root in combinatorial topology, which can be thought of as a constructive counterpart of general topology (Dubucs 1988). It is thus natural to try to justify the univalent axiom in a constructive way.…”