Errett Bishop, review of Keisler's Elementary Calculus (Bull. AMS 83(2), 1977, pp. 205–208)

The one text in this corpus that considered teaching calculus on infinitesimals and said no in print. Paul Halmos, as the Bulletin's review editor, gave Keisler's book to Errett Bishop — a constructivist — and got four pages that are still the standard citation for the case against.

What was read. The review sits behind Project Euclid's paywall. This page rests on the passages quoted verbatim in the secondary literature (Wikipedia's Criticism of nonstandard analysis, the Davis and Keisler replies of the same year); the full argument between those sentences has not been read here, and the tier grades the source, not this page's reach into it.

The argument, in Bishop's sentences

• The complaint is not technical: "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]."
• It is against the axioms doing the work: "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!" (V* and VI* are Keisler's extension and transfer axioms, the hyperreals supplied by fiat.)
• And it is a claim about what students lose: "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 is central to the important practical problems of approximation and estimation.) They do not believe me. … Now we have a calculus text that can be used to confirm their experience of mathematics as an esoteric and meaningless exercise in technique."

The defence the spoke's growth edge asked for is in that parenthesis: the standard definition is defended not as rigorous but as meaningful — it names an approximation problem a student can recognise — while the transfer principle hands over a result whose reason is elsewhere.

The reception, which is part of the record

• Keisler (Notices, 1977) answered the choice of reviewer rather than the review: "why did Paul Halmos … choose a constructivist as the reviewer?" — like "choosing a teetotaller to sample wine."
• Martin Davis (Bull. AMS, 1977) objected that a reader of the review "would hardly imagine Keisler's actual objectives", since it discusses neither them nor how far the book meets them, and that Bishop stated his objections without disclosing the constructivist position they rest on.
• Sullivan (1976) had already run the classroom comparison: students taught from infinitesimals did better than controls at interpreting the calculus formalism. Bishop does not engage it.

So the defence is a philosopher's, and its opponents said so at once. It is still the only one.

Related

keisler-elementary-calculus · h-jerome-keisler · infinitesimal-calculus · real-analysis · klazar-elementary-real-analysis · synthesis