A footnote to The crisis in contemporary mathematics

TL;DR

This paper analyzes Errett Bishop's 1975 critique of non-standard analysis and his efforts to reconcile differing views on logic in mathematics, highlighting his concerns about the foundational implications.

Contribution

It provides historical and contextual insights into Bishop's critique of non-standard analysis and his stance on the Law of Excluded Middle in mathematical foundations.

Findings

Bishop's critique of Robinson's Non-standard Analysis and its impact.

Historical context of Bishop's comments on the 'debasement' of meaning.

Relationship between Bishop's views and the Law of Excluded Middle.

Abstract

We examine the preparation and context of the paper "The Crisis in Contemporary Mathematics" by Errett Bishop, published 1975 in Historia Mathematica. Bishop tried to moderate the differences between Hilbert and Brouwer with respect to the interpretation of logical connectives and quantifiers. He also commented on Robinson's Non-standard Analysis, fearing that it might lead to what he referred to as 'a debasement of meaning.' The 'debasement' comment can already be found in a draft version of Bishop's lecture, but not in the audio file of the actual lecture of 1974. We elucidate the context of the 'debasement' comment and its relation to Bishop's position vis-a-vis the Law of Excluded Middle. Keywords: Constructive mathematics; Robinson's framework; infinitesimal analysis.