# Criticism of Non-standard Analysis - Bishop's Criticism - Bishop's Review

Bishop's Review

Bishop reviewed the book Elementary Calculus: An Infinitesimal Approach by Keisler which presented elementary calculus using the methods of nonstandard analysis. Bishop was chosen by his advisor Paul Halmos to review the book. The review appeared in the Bulletin of the American Mathematical Society in 1977. This article is referred to by David O. Tall (Tall 2001) while discussing the use of non-standard analysis in education. Tall wrote:

the use of the axiom of choice in the non-standard approach however, draws extreme criticism from those such as Bishop (1977) who insisted on explicit construction of concepts in the intuitionist tradition.

Bishop's review supplied several quotations from Keisler's book, such as:

In '60, Robinson solved a three hundred year old problem by giving a precise treatment of infinitesimals. Robinson's achievement will probably rank as one of the major mathematical advances of the twentieth century.

and

In discussing the real line we remarked that we have no way of knowing what a line in physical space is really like. It might be like the hyperreal line, the real line, or neither. However, in applications of the calculus, it is helpful to imagine a line in physical space as a hyperreal line.

The review criticized Keisler's text for not providing evidence to support these statements, and for adopting an axiomatic approach when it was not clear to the students there was any system that satisfied the axioms (Tall 1980). The review ended as follows:

The technical complications introduced by Keisler's approach are of minor importance. The real damage lies in obfuscation and devitalization of those wonderful ideas . No invocation of Newton and Leibniz is going to justify developing calculus using axioms V* and VI*-on the grounds that the usual definition of a limit is too complicated!

Although it seems to be futile, I always tell my calculus students that mathematics is not esoteric: It is common sense. (Even the notorious (ε, δ)-definition of limit is common sense, and moreover it is central to the important practical problems of approximation and estimation.) They do not believe me. In fact the idea makes them uncomfortable because it contradicts their previous experience. Now we have a calculus text that can be used to confirm their experience of mathematics as an esoteric and meaningless exercise in technique.

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