Criticism of Non-standard Analysis - Bishop's Criticism - Responses

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 equated 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 non-standard 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 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 contended in a letter published in The Notices that constructivists are capable of the rational minded inquiry necessary to objectively review a textbook that is not constructive. Meanwhile, a recent study notes the vitriolic tone of Stolzenberg's own letter. Thus, his short letter contains five occurrences of the root "dogma", culminating in a final "spouting of dogma" (attributed to Keisler), whereas the root is absent from Keisler's own letter.

Read more about this topic:  Criticism Of Non-standard Analysis, Bishop's Criticism

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