Nonstandard analysis
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using (ε, δ)-definitio ...
and its offshoot,
nonstandard calculus
In mathematics, nonstandard calculus is the modern application of infinitesimals, in the sense of nonstandard analysis, to infinitesimal calculus. It provides a rigorous justification for some arguments in calculus that were previously considered m ...
, have been criticized by several authors, notably
Errett Bishop
Errett Albert Bishop (July 14, 1928 – April 14, 1983) was an Americans, American mathematician known for his work on analysis. He expanded constructive analysis in his 1967 ''Foundations of Constructive Analysis'', where he Mathematical proof, p ...
,
Paul Halmos
Paul Richard Halmos ( hu, Halmos Pál; March 3, 1916 – October 2, 2006) was a Hungarian-born American mathematician and statistician who made fundamental advances in the areas of mathematical logic, probability theory, statistics, operator ...
, and
Alain Connes
Alain Connes (; born 1 April 1947) is a French mathematician, and a theoretical physicist, known for his contributions to the study of operator algebras and noncommutative geometry. He is a professor at the , , Ohio State University and Vand ...
. These criticisms are analyzed below.
Introduction
The evaluation of nonstandard analysis in the literature has varied greatly.
Paul Halmos
Paul Richard Halmos ( hu, Halmos Pál; March 3, 1916 – October 2, 2006) was a Hungarian-born American mathematician and statistician who made fundamental advances in the areas of mathematical logic, probability theory, statistics, operator ...
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
Abraham Robinson (born Robinsohn; October 6, 1918 – April 11, 1974) was a mathematician who is most widely known for development of nonstandard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were reincorp ...
's nonstandard analysis does not need any axioms beyond
Zermelo–Fraenkel set theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such ...
(ZFC) (as shown explicitly by
Wilhelmus Luxemburg
Wilhelmus Anthonius Josephus Luxemburg ( Delft, 11 April 1929 – 2 October 2018) was a Dutch American mathematician who was a professor of mathematics at the California Institute of Technology.
He received his B.A. from the University of Leide ...
's ultrapower construction of the
hyperreal
Hyperreal may refer to:
* Hyperreal numbers, an extension of the real numbers in mathematics that are used in non-standard analysis
* Hyperreal.org, a rave culture website based in San Francisco, US
* Hyperreality, a term used in semiotics and po ...
s), while its variant by
Edward Nelson
Edward Nelson (May 4, 1932 – September 10, 2014) was an American mathematician. He was professor in the Mathematics Department at Princeton University. He was known for his work on mathematical physics and mathematical logic. In mathematical ...
, known as
internal set theory
Internal set theory (IST) is a mathematical theory of sets developed by Edward Nelson that provides an axiomatic basis for a portion of the nonstandard analysis introduced by Abraham Robinson. Instead of adding new elements to the real numbers, N ...
, is similarly a
conservative extension In mathematical logic, a conservative extension is a supertheory of a theory which is often convenient for proving theorems, but proves no new theorems about the language of the original theory. Similarly, a non-conservative extension is a superthe ...
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
Smooth infinitesimal analysis is a modern reformulation of the calculus in terms of infinitesimals. Based on the ideas of F. W. Lawvere and employing the methods of category theory, it views all functions as being continuous and incapable of bein ...
).
Philip J. Davis
Philip J. Davis (January 2, 1923 – March 14, 2018) was an American academic applied mathematician.
Davis was born in Lawrence, Massachusetts. He was known for his work in numerical analysis and approximation theory, as well as his investigati ...
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
Errett Albert Bishop (July 14, 1928 – April 14, 1983) was an Americans, American mathematician known for his work on analysis. He expanded constructive analysis in his 1967 ''Foundations of Constructive Analysis'', where he Mathematical proof, p ...
, 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:
: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
The American Academy of Arts and Sciences (abbreviation: AAA&S) is one of the oldest learned societies in the United States. It was founded in 1780 during the American Revolution by John Adams, John Hancock, James Bowdoin, Andrew Oliver, a ...
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."
Bishop's review
Bishop reviewed the book ''
Elementary Calculus: An Infinitesimal Approach'' by
Howard Jerome Keisler
Howard Jerome Keisler (born 3 December 1936) is an American mathematician, currently professor emeritus at University of Wisconsin–Madison. His research has included model theory and non-standard analysis.
His Ph.D. advisor was Alfred Tarski a ...
, which presented elementary calculus using the methods of nonstandard analysis. Bishop was chosen by his
advisor
An adviser or advisor is normally a person with more and deeper knowledge in a specific area and usually also includes persons with cross-functional and multidisciplinary expertise. An adviser's role is that of a mentor or guide and differs categor ...
Paul Halmos
Paul Richard Halmos ( hu, Halmos Pál; March 3, 1916 – October 2, 2006) was a Hungarian-born American mathematician and statistician who made fundamental advances in the areas of mathematical logic, probability theory, statistics, operator ...
to review the book. The review appeared in the ''
Bulletin of the American Mathematical Society
The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society.
Scope
It publishes surveys on contemporary research topics, written at a level accessible to non-experts. I ...
'' in 1977. This article is referred to by
David O. Tall
David Orme Tall (born 15 May 1941) is Emeritus Professor in Mathematical Thinking at the University of Warwick. One of his early influential works is the joint paper with Vinner " Concept image and concept definition in mathematics with particula ...
while discussing the use of nonstandard analysis in education. Tall wrote:
:the use of the
axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
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 1960, 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 . The review ended as follows:
The technical complications introduced by Keisler's approach are of minor
importance. The real damage lies in eisler'sobfuscation and devitalization of those
wonderful ideas f 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!
and
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
Although the function (sin ''x'')/''x'' is not defined at zero, as ''x'' becomes closer and closer to zero, (sin ''x'')/''x'' becomes arbitrarily close to 1. In other words, the limit of (sin ''x'')/''x'', as ''x'' approaches z ...
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
Paul Richard Halmos ( hu, Halmos Pál; March 3, 1916 – October 2, 2006) was a Hungarian-born American mathematician and statistician who made fundamental advances in the areas of mathematical logic, probability theory, statistics, operator ...
, the ''Bulletin'' book review editor, choose a
constructivist as the reviewer?
Comparing the use of the
law of excluded middle
In logic, the law of excluded middle (or the principle of excluded middle) states that for every proposition, either this proposition or its negation is true. It is one of the so-called three laws of thought, along with the law of noncontradi ...
(rejected by constructivists) to wine, Keisler likened Halmos' choice with "choosing a
teetotaller
Teetotalism is the practice or promotion of total personal abstinence from the psychoactive drug alcohol, specifically in alcoholic drinks. A person who practices (and possibly advocates) teetotalism is called a teetotaler or teetotaller, or is ...
to sample wine".
Bishop's book review was subsequently criticized in the same journal by
Martin Davis, who wrote on p. 1008 of :
: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
Vadim Komkov (August 18, 1919 – May 14, 2008) was born in Moscow, Russia, and raised in Poland after his parents died in the Bolshevik Revolution. He was a member of the Polish RAF during WWII. He was stationed at Hucknall Aerodrome, and was sent ...
(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
hyperreal
Hyperreal may refer to:
* Hyperreal numbers, an extension of the real numbers in mathematics that are used in non-standard analysis
* Hyperreal.org, a rave culture website based in San Francisco, US
* Hyperreality, a term used in semiotics and po ...
s.
Whether or not nonstandard analysis can be done constructively, Komkov perceived a foundational concern on Bishop's part.
Philosopher of Mathematics
Geoffrey Hellman
Geoffrey Hellman (born August 16, 1943) is an American professor and philosopher. He is Professor of Philosophy at the University of Minnesota in Minneapolis, Minnesota. He obtained his B.A. (1965) and Ph.D. (1972) degrees in philosophy from Harv ...
(1993, p. 222) wrote:
:Some of Bishop's remarks (1967) suggest that his position belongs in
he radical constructivistcategory ...
Historian of Mathematics
Joseph Dauben
Joseph Warren Dauben (born 29 December 1944, Santa Monica) is a Herbert H. Lehman Distinguished Professor of History at the Graduate Center of the City University of New York. He obtained his PhD from Harvard University.
His fields of expertise ...
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
Quantum theory may refer to:
Science
*Quantum mechanics, a major field of physics
*Old quantum theory, predating modern quantum mechanics
* Quantum field theory, an area of quantum mechanics that includes:
** Quantum electrodynamics
** Quantum ...
and
thermodynamics
Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of th ...
, and in
economics
Economics () is the social science that studies the production, distribution, and consumption of goods and services.
Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics analyzes ...
, 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
Margaret Hilda Thatcher, Baroness Thatcher (; 13 October 19258 April 2013) was Prime Minister of the United Kingdom from 1979 to 1990 and Leader of the Conservative Party from 1975 to 1990. She was the first female British prime ...
to review ''
Das Kapital
''Das Kapital'', also known as ''Capital: A Critique of Political Economy'' or sometimes simply ''Capital'' (german: Das Kapital. Kritik der politischen Ökonomie, link=no, ; 1867–1883), is a foundational theoretical text in materialist phi ...
''.
Katz & Katz (2010) point out that
:Bishop is criticizing apples for not being oranges: the critic (Bishop) and the criticized (Robinson's non-standard analysis) do not share a common foundational framework.
They further note that
:Bishop's preoccupation with the extirpation of the law of excluded middle led him to criticize classical mathematics as a whole in as vitriolic a manner as his criticism of non-standard analysis.
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 (1997), 206–234,
Alain Connes
Alain Connes (; born 1 April 1947) is a French mathematician, and a theoretical physicist, known for his contributions to the study of operator algebras and noncommutative geometry. He is a professor at the , , Ohio State University and Vand ...
wrote:
:"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
, 1
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
so that it is impossible (Stern, 1985) to exhibit a single
onstandard real number The formalism that we propose will give a substantial and computable answer to this question."
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:
(1) a nonstandard hyperreal "cannot be exhibited" (the reason given being its relation to nonmeasurable 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.
Katz & Katz analyze Connes' criticisms of nonstandard analysis, and challenge the specific claims (1) and (2).
[See ] With regard to (1), Connes' own infinitesimals similarly rely on non-constructive foundational material, such as the existence of a
Dixmier trace. With regard to (2), Connes presents the independence of the choice of infinitesimal as a ''feature'' of his own theory.
Kanovei et al. (2012) 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
Robert Martin Solovay (born December 15, 1938) is an American mathematician specializing in set theory.
Biography
Solovay earned his Ph.D. from the University of Chicago in 1964 under the direction of Saunders Mac Lane, with a dissertation on ' ...
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
Vladimir G. Kanovei (born 1951) is a Russian mathematician working at the Institute for Information Transmission Problems in Moscow, Russia. His interests include mathematical logic and foundations, as well as mathematical history.
Selected p ...
and
Saharon Shelah (2004). Kanovei et al. (2012) 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 (2013) 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 (approvingly) about his own infinitesimal approach."
Halmos' remarks
Paul Halmos
Paul Richard Halmos ( hu, Halmos Pál; March 3, 1916 – October 2, 2006) was a Hungarian-born American mathematician and statistician who made fundamental advances in the areas of mathematical logic, probability theory, statistics, operator ...
writes in "Invariant subspaces", ''
American Mathematical Monthly'' 85 (1978) 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 1985) as follows (p. 204):
:The Bernstein–Robinson proof
invariant_subspace_conjecture_of_Halmos.html" ;"title="Invariant subspace problem">invariant subspace conjecture of Halmos">Invariant subspace problem">invariant subspace conjecture of Halmos uses non-standard models of higher order predicate languages, and when
obinsonsent 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
Errett Albert Bishop (July 14, 1928 – April 14, 1983) was an Americans, American mathematician known for his work on analysis. He expanded constructive analysis in his 1967 ''Foundations of Constructive Analysis'', where he Mathematical proof, p ...
), it's an equally emotional issue...
Halmos concludes his discussion of nonstandard 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.
Katz & Katz (2010) note that
:Halmos's anxiousness to evaluate Robinson's theory may have involved a conflict of interests
..Halmos invested considerable emotional energy (and ''sweat'', as he memorably puts it in his autobiography) into his translation of the Bernstein–Robinson result
.. s blunt unflattering comments appear to retroactively justify his translationist attempt to deflect the impact of one of the first spectacular applications of Robinson's theory.
Comments by Bos and Medvedev
Leibniz historian
Henk Bos (1974) acknowledged that Robinson's hyperreals provide
:
preliminary explanation of why the calculus could develop on the insecure foundation of the acceptance of infinitely small and infinitely large quantities.
F. Medvedev (1998) further points out that
:
nstandard analysis makes it possible to answer a delicate question bound up with earlier approaches to the history of classical analysis. If infinitely small and infinitely large magnitudes are regarded as inconsistent notions, how could they
ave
''Alta Velocidad Española'' (''AVE'') is a service of high-speed rail in Spain operated by Renfe, the Spanish national railway company, at speeds of up to . As of December 2021, the Spanish high-speed rail network, on part of which the AVE s ...
serve
as a basis for the construction of so
agnificentan edifice of one of the most important mathematical disciplines?
See also
*
Constructive nonstandard analysis In mathematics, constructive nonstandard analysis is a version of Abraham Robinson's nonstandard analysis, developed by Moerdijk (1995), Palmgren (1998), Ruokolainen (2004). Ruokolainen wrote:
:The possibility of constructivization of nonstanda ...
*
Influence of nonstandard analysis
Notes
References
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online PDF
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External links
*
ttp://www.math.nsc.ru/LBRT/g2/english/ssk/teaching_e.html S. Kutateladze "Teaching Calculus"
{{Infinitesimals
Nonstandard analysis
Scientific controversies