Criticism of nonstandard analysis

Non-standard analysis and its offshoot, non-standard calculus, have been criticized by several authors.

The nature of such criticisms is not directly related to the logical status of the results proved using non-standard analysis. In terms of conventional mathematical foundations, such results are quite acceptable. In the technical language of mathematical logic, IST is a conservative extension of ZFC. It provides an assurance that the novelty of non-standard analysis is entirely as a strategy of proof, not in range of results. Further, model theoretic non-standard 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 non-standard analysis as developed is not the only candidate to fulfill the aims of the theory. Philip J. Davis wrote, in a book review of Left Back: A Century of Failed School Reforms (2002) by Diane Ravitch:

There was the nonstandard analysis movement for teaching elementary calculus. Its stock rose a bit before the movement collapsed from inner complexity and scant necessity.

Non-standard calculus in the classroom has been analysed in the Chicago study by Sullivan, as reflected in secondary literature at Influence of non-standard analysis. Sullivan showed that students following the NSA 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 in "NSA and its weak impact on education", page 172.

Bishop's criticism

In the view of Errett Bishop non-constructive mathematics, which includes Robinson's approach to nonstandard analysis, was a scandal that was deficient in numerical meaning (Feferman 2000). Errett Bishop was particularly concerned about the use of non-standard analysis in teaching as he discussed in his essay "Crisis in mathematics" (Bishop 1975). Specifically, after discussing Hilbert's formalist program he writes:

A more recent attempt at mathematics by formal finesse is non-standard analysis. I gather that it has met with some degree of success, whether at the expense of giving significantly less meaningful proofs I do not know. My interest in non-standard analysis is that attempts are being made to introduce it into calculus courses. It is difficult to believe that debasement of meaning could be carried so far.

The fact that Bishop viewed the introduction of non-standard analysis in the classroom, as a "debasement of meaning", was noted by J. Dauben.

Bishop reviewed the book Elementary Calculus: an infinitesimal approach by H. Jerome Keisler which presented elementary calculus using the methods of nonstandard analysis. Bishop's reviewed 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. He writes:

"Criticism of 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

In his review Bishop supplies 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."

Then he criticizes Keisler for not providing evidence to support these statements.

Bishop proceeds to criticize Keisler's text for not adopting an axiomatic approach when it is not clear to the students there is any system that satisfies the axioms (Tall 1980).

Toward the very end of the review, Bishop writes:

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!

At the end of his review, Bishop writes:

"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.)"

He states that this point of view is met with skepticism because of the students previous experience. He goes on to say that Keisler's book will confirm their previous experiences in mathematics.

Responses

As a response, Keisler published a 10-page practical guide describing the success of Elementary Calculus: an infinitesimal approach in the classroom.

Bishop's book review was subsequently criticized in the same journal by Davis (1977).

Connes' comments

Before his major research on von Neumann algebras, Alain Connes had worked on nonstandard analysis in the group of Gustave Choquet. He was sent by Choquet to a physics summer school at Les Houches in 1970, where he realised he had "found a catch in the theory." In "Brisure de symetrie spontanee et geometrie du point de vue spectral", Journal of Geometry and Physics 23 ('97), 206-234, Connes writes as follows on page 211:

"The answer given by non-standard analysis, namely a nonstandard real, is equally disappointing: every non-standard real canonically determines a (Lebesgue) non-measurable subset of the interval [0, 1], so that it is impossible (Stern, 1985) to exhibit a single one [such number]. The formalism that we propose will give a substantial and computable answer to this question."

The general formalism Connes proposed involves the Dixmier traces, whose importance in Noncommutative geometry was noted by Albeverio et al ('96). Meanwhile, Dixmier's construction of his traces involves the choice of an ultrafilter on the integers, the existence of which is dependent on the Axiom of choice, but there are other constructions. In his '95 article "Noncommutative geometry and reality", Connes gives a detailed account of the role of the Dixmier trace in his theory. On page 6207, Connes states as the goal of section II, to develop a calculus of infinitesimals based on operators in Hilbert space. He proceeds to "explain why the formalism of nonstandard analysis is inadequate". Connes points out the following three aspects of Robinson's hyperreals:

(1) a nonstandard hyperreal "cannot be exhibited" (the reason given being its relation to non-measurable sets);

(2) "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 [...]".

(3) the hyperreals are commutative.

Halmos' remarks

Paul Halmos writes in "Invariant subspaces", American Mathematical Monthly 85 ('78) 182–183 as follows:

"the extension to polynomially compact operators was obtained by Bernstein and Robinson (1966). They presented their result in the metamathematical language called non-standard analysis, but, as it was realized very soon, that was a matter of personal preference, not necessity."

Halmos wrote he had been criticized for making the selection of Bishop as reviewer for Keisler's textbook, but said it was intentional and he did it in other cases, that philosophically opposed reviewers would be likely to produce thought-provoking reviews, essentially, that there is no light without heat.

Halmos writes in (Halmos '85) as follows (p. 204):

The Bernstein-Robinson proof [of the invariant subspace conjecture of Halmos'] uses non-standard models of higher order predicate languages, and when [Robinson] sent me his reprint I really had to sweat to pinpoint and translate its mathematical insight.

While commenting on the "role of non-standard analysis in mathematics", Halmos writes (p. 204):

For some other[... mathematicians], who are against it (for instance Errett Bishop), it's an equally emotional issue...

Halmos concludes his discussion of non-standard analysis as follows (p. 204):

it's a special tool, too special, and other tools can do everything it does. It's all a matter of taste.

See also

• Controversy over Cantor's theory
• Brouwer-Hilbert controversy
• Influence of non-standard analysis

References

• Albeverio, S.; Guido, D.; Ponosov, A.; Scarlatti, S.: Singular traces and compact operators. J. Funct. Anal. 137 (1996), no. 2, 281--302.
• Artigue, Michèle (1994), Analysis, Advanced Mathematical Thinking (ed. David O. Tall), Springer-Verlag, p. 172, ISBN 0792328124
• Bishop, Errett (1975), "The crisis in contemporary mathematics", Historia Math., 2 (4): 507–517
• Bishop, Errett (1977), "Review: H. Jerome Keisler, Elementary calculus", Bull. Amer. Math. Soc., 83: 205–208
• Connes, A.: Noncommutative geometry and reality. J. Math. Phys. 36 (1995), no.~11, 6194--6231.
• Davis, Martin (1977), "Review: J. Donald Monk, Mathematical logic", Bull. Amer. Math. Soc., 83: 1007–1011
• Feferman, Solomon (2000), "Relationships between constructive, predicative and classical systems of analysis", Synthese Library (292), Kluwer Academic Publishers Group

• Halmos, Paul R.: I want to be a mathematician. An automathography. Springer-Verlag, New York, 1985. xvi+421 pp. ISBN: 0-387-96078-3

• Schubring, Gert (2005), Conflicts Between Generalization, Rigor, and Intuition: Number Concepts Underlying the Development of Analysis in 17th–19th Century France and Germany, Springer, p. 153, ISBN 0387228365
• Sullivan, Kathleen (1976), "The Teaching of Elementary Calculus Using the Nonstandard Analysis Approach", The American Mathematical Monthly, 83: 370–375
• Tall, David (1980), Intuitive infinitesimals in the calculus (poster) (PDF), Fourth International Congress on Mathematics Education, Berkeley
• Tall, David (2001), "Natural and Formal Infinities", Educational Studies in Mathematics, 48 (2–3), Springer Netherlands

External links

• Online version of "Elementary Calculus: An Approach Using Infinitesimals"