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Non-standard analysis and its offshoot, non-standard calculus, have been criticized by several authors. The evaluation of non-standard analysis in the literature has varied greatly. Joseph Dauben described it as a scientific revolution, while Paul Halmos described it as a technical special development in mathematical logic.

The nature of the criticisms is not directly related to the logical status of the results proved using non-standard analysis. In terms of conventional mathematical foundations in classical logic, such results are quite acceptable. Robinson's non-standard analysis does not need any axioms beyond ZFC, while its variant by Edward Nelson, known as IST, is similarly a conservative extension of ZFC[1]. 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[2].

G. Schubring (2005, p. 153) provides a more positive assessment. After discussing an alternative approach to calculus developed in Germany, he writes that the alternative approach

has been [...] unable to win as much celebrity and as many adherents as [non-standard analysis].

Non-standard calculus in the classroom has been analysed in the Chicago study by K. 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 (1994), page 172; Chihara (2007); and Dauben (1988).

Contents

Bishop's criticism

In the view of Errett Bishop, non-constructive mathematics, which includes Robinson's approach to nonstandard analysis, was deficient in numerical meaning (Feferman 2000). 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 wrote:

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[3]. The term was clarified by Bishop (1985, p. 1) in his text Schizophrenia in contemporary mathematics, as follows:

Brouwer's criticisms of classical mathematics were concerned with what I shall refer to as "the debasement of meaning".

In Foundations of Constructive Analysis (1967, page ix), Bishop wrote:

Our program is simple: To give numerical meaning to as much as possible of classical abstract analysis. Our motivation is the well-known scandal, exposed by Brouwer (and others) in great detail, that classical mathematics is deficient in numerical meaning.

Bishop's comments have been reflected in secondary literature, as when logician Solomon Feferman wrote:

Bishop criticized both non-constructive classical mathematics and intuitionism. He called non-constructive mathematics "a scandal", particularly because of its "deficiency in numerical meaning".

According to Feferman,

What [Bishop] simply meant was that if you say something exists you ought to be able to produce it, and if you say there is a function which does something on the natural numbers then you ought to be able to produce a machine which calculates it out at each number. [1]
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Bishop's review

Bishop reviewed the book Elementary Calculus: an infinitesimal approach by H. Jerome Keisler which presented elementary calculus using the methods of nonstandard analysis. 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 non-standard analysis in education. Tall wrote:

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 supplied 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.

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!

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 equated Halmos' choice with "choosing a teetotaller to sample wine".

Bishop's book review was subsequently criticized in the same journal by Davis (1977). Martin Davis wrote (p. 1008):

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 [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.

Connes' comments

Before his major research on von Neumann algebras, Alain Connes had worked on nonstandard analysis in the group of Gustave Choquet[4][5][6]. 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."[7] In "Brisure de symétrie spontanée et géométrie 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 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

Notes

  1. ^ This is shown in Edward Nelson's AMS 1977 paper in an appendix written by William Powell.
  2. ^ http://www.siam.org/news/news.php?id=527
  3. ^ in Donald Gillies, Revolutions in Mathematics (1992), p. 76.
  4. ^ Connes, Alain (1970), "Détermination de modèles minimaux en analyse non standard et application", C. R. Acad. Sci. Paris, Sér. A-B 271: A969-A971  
  5. ^ Connes, Alain (1970), Ultrapuissances et applications dans le cadre de l'analyse non standard, Séminaire Choquet : 1969/70  
  6. ^ Connes' web comments about nonstandard analysis
  7. ^ Goldstein, Catherine; Skandalis, Geroges (2007), "An interview with Alain Connes" (PDF), European Mathematical Society Newsletter, http://www.ems-ph.org/journals/newsletter/pdf/2007-03-63.pdf  

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, http://projecteuclid.org/euclid.bams/1183538669  
  • Bishop, E. (1985) Schizophrenia in contemporary mathematics. Errett Bishop: reflections on him and his research (San Diego, Calif., 1983), 1--32, Contemp. Math. 39, Amer. Math. Soc., Providence, RI.
  • Chihara, C. (2007) The Burgess-Rosen critique of nominalistic reconstructions. Philos. Math. (3) 15, no. 1, 54--78.
  • Connes, A.: Noncommutative geometry and reality. J. Math. Phys. 36 (1995), no.~11, 6194—6231.
  • Dauben, J. (1988) Abraham Robinson and Nonstandard Analysis: History, Philosophy, and Foundations of Mathematics. In William Aspray and Philip Kitcher, eds. History and philosophy of modern mathematics (Minneapolis, MN, 1985), 177--200, Minnesota Stud. Philos. Sci., XI, Univ. Minnesota Press, Minneapolis, MN, 1988. Online here.
  • Dauben, J. (1996) Arguments, logic and proof: mathematics, logic and the infinite. History of mathematics and education: ideas and experiences (Essen, 1992), 113--148, Stud. Wiss. Soz. Bildungsgesch. Math., 11, Vandenhoeck & Ruprecht, G"ottingen.
  • Davis, Martin (1977), "Review: J. Donald Monk, Mathematical logic", Bull. Amer. Math. Soc. 83: 1007–1011, http://projecteuclid.org/euclid.bams/1183539465  
  • Davis, M.; Hausner, M. (1978) Book review. The Joy of Infinitesimals. J. Keisler's Elementary Calculus. Mathematical Intelligencer 1, 168-170.
  • Feferman, Solomon (2000), "Relationships between constructive, predicative and classical systems of analysis", Synthese Library (Kluwer Academic Publishers Group) (292)  ; online PDF.
  • Halmos, Paul R.: I want to be a mathematician. An automathography. Springer-Verlag, New York, 1985. xvi+421 pp. ISBN 0-387-96078-3
  • Hellman, Geoffrey (1993) Constructive Mathematics and Quantum Mechanics: Unbounded Operators and the Spectral Theorem, Journal of Philosophical Logic 12, 221-248.
  • Keisler, H. Jerome (1977) Letter to the editor. Notices Amer. Math. Soc. 24, p. 269.
  • Komkov, Vadim (1977) Letter to the editor, Notices Amer. Math. Soc. 24, no. 5, 269--271.
  • 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  
  • Stewart, Ian (1986) Frog and Mouse revisited. Mathematical Intelligencer, p. 78-82.
  • Sullivan, Kathleen (1976), "The Teaching of Elementary Calculus Using the Nonstandard Analysis Approach", The American Mathematical Monthly 83: 370–375, http://www.jstor.org/stable/2318657  
  • Tall, David (1980) (PDF), Intuitive infinitesimals in the calculus (poster), Fourth International Congress on Mathematics Education, Berkeley, http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1980c-intuitive-infls.pdf  
  • Tall, David (2001), "Natural and Formal Infinities", Educational Studies in Mathematics (Springer Netherlands) 48 (2-3)  

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