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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).
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]
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.
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.
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
- ^
This is shown in Edward Nelson's AMS 1977 paper in an
appendix written by William Powell.
- ^
http://www.siam.org/news/news.php?id=527
- ^
in Donald
Gillies, Revolutions in
Mathematics (1992), p. 76.
- ^
Connes, Alain (1970),
"Détermination de modèles minimaux en analyse non standard et
application", C. R. Acad. Sci. Paris, Sér. A-B
271: A969-A971
- ^
Connes, Alain (1970),
Ultrapuissances et applications dans le cadre de l'analyse non
standard, Séminaire Choquet : 1969/70
- ^
Connes' web comments about
nonstandard analysis
- ^
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
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(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.
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reconstructions. Philos. Math. (3) 15, no. 1, 54--78.
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36 (1995), no.~11, 6194—6231.
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Aspray and Philip Kitcher, eds. History and philosophy of modern
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Philos. Sci., XI, Univ. Minnesota Press, Minneapolis, MN, 1988.
Online here.
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(1977), "Review: J. Donald Monk,
Mathematical logic", Bull. Amer. Math. Soc.
83: 1007–1011, http://projecteuclid.org/euclid.bams/1183539465
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Intelligencer 1, 168-170.
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and classical systems of analysis", Synthese Library
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; online PDF.
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automathography. Springer-Verlag, New York, 1985. xvi+421 pp. ISBN
0-387-96078-3
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Mechanics: Unbounded Operators and the Spectral Theorem, Journal of
Philosophical Logic 12, 221-248.
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Math. Soc. 24, p. 269.
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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
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Intelligencer, p. 78-82.
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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
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(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
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"Natural and Formal Infinities", Educational Studies in
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(2-3)
External
links