6.1 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Criticism of nonstandard analysis | 1/4 | https://en.wikipedia.org/wiki/Criticism_of_nonstandard_analysis | reference | science, encyclopedia | 2026-05-05T04:17:38.400334+00:00 | kb-cron |
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
allows one to rigorously manipulate things such as "the set of all small numbers", or to rigorously say things like "η1 is smaller than anything that involves η0", while greatly reducing epsilon management issues by automatically concealing many of the quantifiers in one's argument. 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 although usually strongly dependent on choice. Abraham Robinson's nonstandard analysis does not need any axioms beyond Zermelo–Fraenkel set theory (ZFC) (as shown explicitly by Wilhelmus Luxemburg's ultrapower construction of the hyperreals), 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 (see Smooth infinitesimal analysis). Philip J. Davis wrote, in a book review of Left Back: A Century of Failed School Reforms 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. 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 (1994), page 172; Chihara (2007); and Dauben (1988).
== 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 (Feferman 2000). Bishop was particularly concerned about the use of nonstandard 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. Katz & Katz (2010) 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 (1985, p. 1) in his text Schizophrenia in contemporary mathematics (first distributed in 1973), as follows:
Brouwer's criticisms of classical mathematics were concerned with what I shall refer to as "the debasement of meaning". 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 (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 remarks are supported by the discussion following his lecture:
George Mackey (Harvard): "I don't want to think about these questions. I have faith that what I am doing will have some kind of meaning...." Garrett Birkhoff (Harvard): "...I think this is what Bishop is urging. We should keep track of our assumptions and keep an open mind." Shreeram Abhyankar: (Purdue): "My paper is in complete sympathy with Bishop's position." J. P. Kahane (U. de Paris): "...I have to respect Bishop's work but I find it boring...." Bishop (UCSD): "Most mathematicians feel that mathematics has meaning but it bores them to try to find out what it is...." Kahane: "I feel that Bishop's appreciation has more significance than my lack of appreciation."