Criticism of nonstandard analysis


and its offshoot, nonstandard calculus, have been criticized by several authors, notably Errett Bishop, Paul Halmos, and Alain Connes. These criticisms are analyzed below.

Introduction

The evaluation of nonstandard analysis in the literature has varied greatly. Paul Halmos described it as a technical special development in mathematical logic. Terence Tao summed up the advantage of the hyperreal framework by noting that it
The nature of the criticisms is not directly related to the logical status of the results proved using nonstandard analysis. In terms of conventional mathematical foundations in classical logic, such results are quite acceptable. Abraham Robinson's nonstandard analysis does not need any axioms beyond Zermelo–Fraenkel set theory , while its variant by Edward Nelson, known as internal set theory, is similarly a conservative extension of ZFC. It provides an assurance that the newness of nonstandard analysis is entirely as a strategy of proof, not in range of results. Further, model theoretic nonstandard analysis, for example based on superstructures, which is now a commonly used approach, does not need any new set-theoretic axioms beyond those of ZFC.
Controversy has existed on issues of mathematical pedagogy. Also nonstandard analysis as developed is not the only candidate to fulfill the aims of a theory of infinitesimals. Philip J. Davis wrote, in a book review of Left Back: A Century of Failed School Reforms by Diane Ravitch:
Nonstandard calculus in the classroom has been analysed in the study by K. Sullivan of schools in the Chicago area, as reflected in secondary literature at Influence of nonstandard analysis. Sullivan showed that students following the nonstandard analysis course were better able to interpret the sense of the mathematical formalism of calculus than a control group following a standard syllabus. This was also noted by Artigue, page 172; Chihara ; and Dauben.

Bishop's criticism

In the view of Errett Bishop, classical mathematics, which includes Robinson's approach to nonstandard analysis, was nonconstructive and therefore deficient in numerical meaning. Bishop was particularly concerned about the use of nonstandard analysis in teaching as he discussed in his essay "Crisis in mathematics". Specifically, after discussing Hilbert's formalist program he wrote:
Katz & Katz note that a number of criticisms were voiced by the participating mathematicians and historians following Bishop's "Crisis" talk, at the American Academy of Arts and Sciences workshop in 1974. However, not a word was said by the participants about Bishop's debasement of Robinson's theory. Katz & Katz point out that it recently came to light that Bishop in fact said not a word about Robinson's theory at the workshop, and only added his debasement remark at the galley proof stage of publication. This helps explain the absence of critical reactions at the workshop. Katz & Katz conclude that this raises issues of integrity on the part of Bishop whose published text does not report the fact that the "debasement" comment was added at galley stage and therefore was not heard by the workshop participants, creating a spurious impression that they did not disagree with the comments.
The fact that Bishop viewed the introduction of nonstandard analysis in the classroom as a "debasement of meaning" was noted by J. Dauben. The term was clarified by Bishop in his text Schizophrenia in contemporary mathematics, as follows:
Thus, Bishop first applied the term "debasement of meaning" to classical mathematics as a whole, and later applied it to Robinson's infinitesimals in the classroom. In his Foundations of Constructive Analysis, Bishop wrote:
Bishop's remarks are supported by the discussion following his lecture:
Bishop reviewed the book 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 while discussing the use of nonstandard analysis in education. Tall wrote:
Bishop's review supplied several quotations from Keisler's book, such as:
and
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. 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!

and

Although it seems to be futile, I always tell my calculus students that mathematics is not esoteric: It is common sense. 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 asked:
Comparing the use of the law of excluded middle 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 added that Bishop stated his objections
Physicist Vadim Komkov wrote:
Whether or not nonstandard analysis can be done constructively, Komkov perceived a foundational concern on Bishop's part.
Philosopher of Mathematics Geoffrey Hellman wrote:
Historian of Mathematics Joseph Dauben analyzed Bishop's criticism in. After evoking the "success" of nonstandard analysis
Dauben stated:
Dauben mentioned "impressive" applications in
At this "deeper" level of meaning, Dauben concluded,
A number of authors have commented on the tone of Bishop's book review. Artigue described it as virulent; Dauben, as vitriolic; Davis and Hauser, as hostile; Tall, as extreme.
Ian Stewart compared Halmos' asking Bishop to review Keisler's book, to inviting Margaret Thatcher to review Das Kapital.
Katz & Katz point out that
They further note that
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, 206–234, Alain Connes wrote:
In his 1995 article "Noncommutative geometry and reality" Connes develops a calculus of infinitesimals based on operators in Hilbert space. He proceeds to "explain why the formalism of nonstandard analysis is inadequate" for his purposes. Connes points out the following three aspects of Robinson's hyperreals:
a nonstandard hyperreal "cannot be exhibited" ;
"the practical use of such a notion is limited to computations in which the final result is independent of the exact value of the above infinitesimal. This is the way nonstandard analysis and ultraproducts are used ".
the hyperreals are commutative.
Katz & Katz analyze Connes' criticisms of nonstandard analysis, and challenge the specific claims and. With regard to, Connes' own infinitesimals similarly rely on non-constructive foundational material, such as the existence of a Dixmier trace. With regard to, Connes presents the independence of the choice of infinitesimal as a feature of his own theory.
Kanovei et al. analyze Connes' contention that nonstandard numbers are "chimerical". They note that the content of his criticism is that ultrafilters are "chimerical", and point out that Connes exploited ultrafilters in an essential manner in his earlier work in functional analysis. They analyze Connes' claim that the hyperreal theory is merely "virtual". Connes' references to the work of Robert Solovay suggest that Connes means to criticize the hyperreals for allegedly not being definable. If so, Connes' claim concerning the hyperreals is demonstrably incorrect, given the existence of a definable model of the hyperreals constructed by Vladimir Kanovei and Saharon Shelah. Kanovei et al. also provide a chronological table of increasingly vitriolic epithets employed by Connes to denigrate nonstandard analysis over the period between 1995 and 2007, starting with "inadequate" and "disappointing" and culminating with "the end of the road for being 'explicit'".
Katz & Leichtnam note that "two-thirds of Connes' critique of Robinson's infinitesimal approach can be said to be incoherent, in the specific sense of not being coherent with what Connes writes about his own infinitesimal approach."

Halmos' remarks

writes in "Invariant subspaces", American Mathematical Monthly 85 182-183 as follows:
Halmos writes in as follows :
While commenting on the "role of non-standard analysis in mathematics", Halmos writes :
Halmos concludes his discussion of nonstandard analysis as follows :
Katz & Katz note that

Comments by Bos and Medvedev

Leibniz historian Henk Bos acknowledged that Robinson's hyperreals provide
F. Medvedev further points out that