Errett Bishop

Errett Bishop (1928-1983) was a mathematical constructivist who, unlike the intuitionist Arend Heyting, held a dim view of classical mathematics in general and Robinson's infinitesimals in particular. Discouraged by the apparent non-constructivity of his early work in functional analysis under Halmos, he believed to have found the culprit in the law of excluded middle (LEM). He spent the remaining 18 years of his life in an effort to expunge the reliance on LEM from analysis, and sought to define meaning itself in mathematics in terms of such LEM-extirpation. Accordingly, he described classical mathematics as both a debasement of meaning and sawdust, and did not hesitate to speak of both crisis and schizophrenia in contemporary mathematics, predicting an imminent demise of classical mathematics. His criticism (subcontracted by Halmos) of calculus pedagogy based on Robinson's infinitesimals was a natural outgrowth of his general opposition to the logical underpinnings of classical mathematics.

A preliminary version of Bishop's 1975 lecture contained violent rhetoric against calculus pedagogy based on Abraham Robinson's work. Such rhetoric indeed appears in the published version of his lecture. However, an examination of an actual audio recording of his lecture reveals that in real time Bishop chickened out and said nothing at all about Robinson; see the article in Historia Mathematica linked above.

Bishop's article "A general language"
Bishop's article "How to compile mathematics into Algol"

See also
MO discussion of Bishop's new articles
Heyting
Keisler
Bishop Connes Halmos Sergeyev
More on infinitesimals
SPOT as a conservative extension of Zermelo-Fraenkel (MO)
ABC's of infinitesimals
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