Meaning in Classical Mathematics: Is it at Odds with Intuitionism?

TL;DR

This paper explores the philosophical divide between classical and intuitionist mathematics, especially regarding infinitesimals, analyzing historical texts and modern implications for physical theories like the Hawking-Penrose theorem.

Contribution

It provides a comparative analysis of intuitionist and classical views on infinitesimals, and discusses their implications for physical applications and the philosophy of mathematics.

Findings

Intuitionist and classical perspectives on infinitesimals differ significantly.

Bishop's objections to Robinson's non-standard analysis are rooted in ideological and pedagogical reasons.

Constructive frameworks for variational principles in physics are still lacking.

Abstract

We examine the classical/intuitionist divide, and how it reflects on modern theories of infinitesimals. When leading intuitionist Heyting announced that "the creation of non-standard analysis is a standard model of important mathematical research", he was fully aware that he was breaking ranks with Brouwer. Was Errett Bishop faithful to either Kronecker or Brouwer? Through a comparative textual analysis of three of Bishop's texts, we analyze the ideological and/or pedagogical nature of his objections to infinitesimals a la Robinson. Bishop's famous "debasement" comment at the 1974 Boston workshop, published as part of his Crisis lecture, in reality was never uttered in front of an audience. We compare the realist and the anti-realist intuitionist narratives, and analyze the views of Dummett, Pourciau, Richman, Shapiro, and Tennant. Variational principles are important physical applications, currently lacking a constructive framework. We examine the case of the Hawking-Penrose singularity theorem, already analyzed by Hellman in the context of the Quine-Putnam indispensability thesis.