A FOOTNOTE TO THE CRISIS IN CONTEMPORARY MATHEMATICS

BORIS KATZ, MIKHAIL G. KATZ, AND SAM SANDERS

摘要：我们研究了Errett Bishop 1975年在《数学历史》上发表的论文《当代数学的危机》的准备和背景。毕晓普（Bishop）试图调和希尔伯特（Hilbert）和布劳威尔（Brouwer）在逻辑连接词和量词的解释上的差异。他还对罗宾逊的非标准分析发表了评论，担心这会导致他所说的“意义的贬低”。在毕晓普（Bishop）演讲的草稿中已经可以找到“贬低”的评论，但在1974年实际演讲的音频文件中却找不到。我们阐明了“贬低”评论的背景及其与毕晓普（Bishop）对排中律的立场的关系。

关键词：构造性数学；罗宾逊（Robinson）的框架；无穷小分析。

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.

https://mathscinet.ams.org/mathscinet-getitem?mr=3802555​mathscinet.ams.orghttps://arxiv.org/abs/1804.02645​arxiv.org