# Errett Bishop

Errett Bishop (1928–1983) was responsible for reviving interest in constructive mathematics as a branch of mathematics. Before this, he had a career in areas of classical (i.e., non-constructive) analysis, including approximation theory?, several complex variables, function algebras, and other fields.

## Successes in constructive mathematics

While Brouwer had insisted that mathematics was primary to logic, the intuitionists who followed him went more into intuitionistic foundations and philosophy than into mathematics as such. The Russian school of constructivism focussed on recursion theory? and was far removed from modern Bourbaki-style mathematics. There was a pervasive feeling that constructive mathematics, particularly constructive analysis, was simply unworkable; Bishop attacked this idea, not by arguing against it, but by simply demonstrating how constructive analysis could be done.

Bishop's important works in constructive mathematics include:

• Foundations of Constructive Analysis (1967), the work which made him famous in the foundations of mathematics circles;
• “Schizophrenia in contemporary mathematics” (1973), a philosophical attack on nonconstructive mathematics;
• Constructive Measure Theory (1972, with Henry Cheng?), featuring a better measure theory than in FCA;
• Constructive Analysis (1985, with Douglas Bridges), a revised version of FCA, incorporating the measure theory of CMT.

Bishop's constructive mathematics was weakly predicative and can be formulated within any locally cartesian closed pretopos with a natural numbers object and dependent choice. (The measure theory from FCA, before the revised version from CMT, is an exception; it can be formulated in any topos with an NNO.)

Bishop's work was primarily in analysis, because constructive mathematics is more difficult at an elementary level there. Bishop's style was extended to constructive algebra by his disciple Fred Richman.

There is some confusion about the notion of continuous function (from $\mathbb{R}$ to $\mathbb{R}$) in Bishop's work, diagnosed by Frank Waaldijk in On the foundations of constructive mathematics - especially in relation to the theory of continuous functions (2001). In particular, FCA and CA use different definitions. These can be reconciled using the fan theorem (which Bishop rejected). There is a similar problem with the notion of locally compact space.