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| Criticism of nonstandard analysis | 2/4 | https://en.wikipedia.org/wiki/Criticism_of_nonstandard_analysis | reference | science, encyclopedia | 2026-05-05T04:17:38.400334+00:00 | kb-cron |
=== Bishop's review === Bishop reviewed the book Elementary Calculus: An Infinitesimal Approach by Howard Jerome 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 nonstandard 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 1960, 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 [Keisler's] obfuscation and devitalization of those wonderful ideas [of standard calculus]. 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!
and
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.
=== Responses === In his response in The Notices, Keisler (1977, p. 269) asked:
why did Paul Halmos, the Bulletin book review editor, choose a constructivist as the reviewer? Comparing the use of the law of excluded middle (rejected by constructivists) to wine, Keisler likened Halmos' choice with "choosing a teetotaller to sample wine". Bishop's book review was subsequently criticized in the same journal by Martin Davis, who wrote on p. 1008 of Davis (1977):
Keisler's book is an attempt to bring back the intuitively suggestive Leibnizian methods that dominated the teaching of calculus until comparatively recently, and which have never been discarded in parts of applied mathematics. A reader of Errett Bishop's review of Keisler's book would hardly imagine that this is what Keisler was trying to do, since the review discusses neither Keisler's objectives nor the extent to which his book realizes them. Davis added (p. 1008) that Bishop stated his objections
without informing his readers of the constructivist context in which this objection is presumably to be understood. Physicist Vadim Komkov (1977, p. 270) wrote:
Bishop is one of the foremost researchers favoring the constructive approach to mathematical analysis. It is hard for a constructivist to be sympathetic to theories replacing the real numbers by hyperreals. Whether or not nonstandard analysis can be done constructively, Komkov perceived a foundational concern on Bishop's part. Philosopher of Mathematics Geoffrey Hellman (1993, p. 222) wrote:
Some of Bishop's remarks (1967) suggest that his position belongs in [the radical constructivist] category ... Historian of Mathematics Joseph Dauben analyzed Bishop's criticism in (1988, p. 192). After evoking the "success" of nonstandard analysis
at the most elementary level at which it could be introduced—namely, at which calculus is taught for the first time, Dauben stated:
there is also a deeper level of meaning at which nonstandard analysis operates. Dauben mentioned "impressive" applications in
physics, especially quantum theory and thermodynamics, and in economics, where study of exchange economies has been particularly amenable to nonstandard interpretation. At this "deeper" level of meaning, Dauben concluded,
Bishop's views can be questioned and shown to be as unfounded as his objections to nonstandard analysis pedagogically. A number of authors have commented on the tone of Bishop's book review. Artigue (1992) described it as virulent; Dauben (1996), as vitriolic; Davis and Hauser (1978), as hostile; Tall (2001), as extreme. Ian Stewart (1986) compared Halmos' asking Bishop to review Keisler's book, to inviting Margaret Thatcher to review Das Kapital. Katz & Katz (2010) point out that
Bishop is criticizing apples for not being oranges: the critic (Bishop) and the criticized (Robinson's non-standard analysis) do not share a common foundational framework. They further note that
Bishop's preoccupation with the extirpation of the law of excluded middle led him to criticize classical mathematics as a whole in as vitriolic a manner as his criticism of non-standard analysis. G. Stolzenberg responded to Keisler's Notices criticisms of Bishop's review in a letter, also published in The Notices. Stolzenberg argues that the criticism of Bishop's review of Keisler's calculus book is based on the false assumption that they were made in a constructivist mindset whereas Stolzenberg believes that Bishop read it as it was meant to be read: in a classical mindset.
== Connes' criticism == In "Brisure de symétrie spontanée et géométrie du point de vue spectral", Journal of Geometry and Physics 23 (1997), 206–234, Alain Connes wrote: