Meaning in Classical Mathematics: is it at odds with Intuitionism?

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 à 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.