# 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;
• “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.

## Inadequacies for constructive mathematics

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.

More generally, Bishop worked with metric spaces but rejected more general topological spaces as unnecessary for mathematics. From the perspective of modern constructive mathematics, Bishop may be praised for reviving the discipline, but much of his work must be rejected as inadequate in favour of locale theory. In particular, the definition of continuous function on the locale of real numbers agrees with neither of Bishop's books; the same goes for locally compact spaces and measure theory.

Summarising, Bishop studied pointwise analysis, but constructive pointwise analysis appears to be unworkable after all (without the fan theorem). Although Bishop went farther with it than previously thought possible, one can go farther yet with a localic approach.

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Revised on March 30, 2015 22:03:30 by David Roberts (150.101.89.149)