Criticism of non-standard analysis
Encyclopedia
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. Abraham Robinson
's non-standard analysis does not need any axioms beyond Zermelo–Fraenkel set theory
(ZFC) (as shown explicitly by Wilhelmus Luxemburg
's ultrapower construction of the hyperreal
s), while its variant by Edward Nelson
, known as IST
, is similarly a conservative extension
of ZFC. It provides an assurance that the newness 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 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 (2002) by Diane Ravitch:
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
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).
, non-constructive mathematics, which includes Robinson's approach to nonstandard analysis, was deficient in numerical meaning . Bishop was particularly concerned about the use of non-standard analysis in teaching as he discussed in his essay "Crisis in mathematics" . Specifically, after discussing Hilbert's formalist program he wrote:
Katz & Katz (2010) note that a number of criticisms were voiced by the participating mathematicians and historians following Bishop's "Crisis" talk, at the AAAS 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.
The fact that Bishop viewed the introduction of non-standard 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, as follows:
In Foundations of Constructive Analysis (1967, page ix), Bishop wrote:
by 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
while discussing the use of non-standard analysis in education. Tall wrote:
Bishop's review supplied several quotations from Keisler's book, such as:
and
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:
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 . Martin Davis
wrote (p. 1008):
Davis added (p. 1008) that Bishop stated his objections
Physicist Vadim Komkov
(1977, p. 270) wrote:
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:
Historian of Mathematics Joseph Dauben
analyzed Bishop's criticism in (1988, p. 192). After evoking the "success" of nonstandard analysis
Dauben stated:
Dauben mentioned "impressive" applications in
At this "deeper" level of meaning, Dauben concluded,
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
Katz & Katz (2010) point out that
They further note that
Connes wrote:
In his '95 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 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.
In the view of M. Katz and K. Katz Connes' comments are critical of non-standard analysis, and they challenge these specific claims. 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.
writes in "Invariant subspaces", American Mathematical Monthly
85 ('78) 182–183 as follows:
Halmos writes in (Halmos '85) as follows (p. 204):
While commenting on the "role of non-standard analysis in mathematics", Halmos writes (p. 204):
Halmos concludes his discussion of non-standard analysis as follows (p. 204):
Katz & Katz (2010) note that
F. Medvedev (1998) further points out that
Non-standard analysis
Non-standard analysis is a branch of mathematics that formulates analysis using a rigorous notion of an infinitesimal number.Non-standard analysis was introduced in the early 1960s by the mathematician Abraham Robinson. He wrote:...
and its offshoot, non-standard calculus
Non-standard calculus
In mathematics, non-standard calculus is the modern application of infinitesimals, in the sense of non-standard analysis, to differential and integral calculus...
, have been criticized by several authors. The evaluation of non-standard analysis in the literature has varied greatly. Joseph Dauben
Joseph Dauben
Joseph W. Dauben is a Herbert H. Lehman Distinguished Professor of History at the Graduate Center of the City University of New York. He obtained his Ph.D...
described it as a scientific revolution
Revolutions in Mathematics
Revolutions in Mathematics is an influential collection of essays in the history and philosophy of mathematics.-Contents:*Michael J. Crowe, Ten "laws" concerning patterns of change in the history of mathematics ;*Herbert Mehrtens, T. S...
, while Paul Halmos
Paul Halmos
Paul Richard Halmos was a Hungarian-born American mathematician who made fundamental advances in the areas of probability theory, statistics, operator theory, ergodic theory, and functional analysis . He was also recognized as a great mathematical expositor.-Career:Halmos obtained his B.A...
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. Abraham Robinson
Abraham Robinson
Abraham Robinson was a mathematician who is most widely known for development of non-standard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were incorporated into mathematics....
's non-standard analysis does not need any axioms beyond Zermelo–Fraenkel set theory
Zermelo–Fraenkel set theory
In mathematics, Zermelo–Fraenkel set theory with the axiom of choice, named after mathematicians Ernst Zermelo and Abraham Fraenkel and commonly abbreviated ZFC, is one of several axiomatic systems that were proposed in the early twentieth century to formulate a theory of sets without the paradoxes...
(ZFC) (as shown explicitly by Wilhelmus Luxemburg
Wilhelmus Luxemburg
Wilhelmus Luxemburg is Professor of Mathematics, Emeritus at the California Institute of Technology. He received his B.A. from the University of Leiden in 1950; his M.A., in 1953; his Ph.D., from the Delft Institute of Technology, in 1955. He was Assistant Professor at Caltech during 1958-60;...
's ultrapower construction of the hyperreal
Hyperreal number
The system of hyperreal numbers represents a rigorous method of treating the infinite and infinitesimal quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form1 + 1 + \cdots + 1. \, Such a number is...
s), while its variant by Edward Nelson
Edward Nelson
Edward Nelson is a professor in the Mathematics Department at Princeton University. He is known for his work on mathematical physics and mathematical logic...
, known as IST
Internal set theory
Internal set theory is a mathematical theory of sets developed by Edward Nelson that provides an axiomatic basis for a portion of the non-standard analysis introduced by Abraham Robinson. Instead of adding new elements to the real numbers, the axioms introduce a new term, "standard", which can be...
, is similarly a conservative extension
Conservative extension
In mathematical logic, a logical theory T_2 is a conservative extension of a theory T_1 if the language of T_2 extends the language of T_1; every theorem of T_1 is a theorem of T_2; and any theorem of T_2 which is in the language of T_1 is already a theorem of T_1.More generally, if Γ is a set of...
of ZFC. It provides an assurance that the newness 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 a theory of infinitesimals (see Smooth infinitesimal analysis
Smooth infinitesimal analysis
Smooth infinitesimal analysis is a mathematically rigorous 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 being expressed in terms of discrete...
). Philip J. Davis
Philip J. Davis
Philip J. Davis is an American applied mathematician.Davis was born in Lawrence, Massachusetts. He is known for his work in numerical analysis and approximation theory, as well as his investigations in the history and philosophy of mathematics...
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.
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
Influence of non-standard analysis
The influence of Abraham Robinson's theory of non-standard analysis has been felt in a number of fields.-Probability theory:"Radically elementary probability theory" of Edward Nelson combines the discrete and the continuous theory through the infinitesimal approach...
. 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 BishopErrett Bishop
Errett Albert Bishop was an American mathematician known for his work on analysis. He is the father of constructive analysis, because of his 1967 Foundations of Constructive Analysis, where he proved most of the important theorems in real analysis by constructive methods.-Life:Errett Bishop's...
, non-constructive mathematics, which includes Robinson's approach to nonstandard analysis, was deficient in numerical meaning . Bishop was particularly concerned about the use of non-standard 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 AAAS 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.
The fact that Bishop viewed the introduction of non-standard 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, 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 review
Bishop reviewed the book Elementary Calculus: An Infinitesimal ApproachElementary Calculus: An Infinitesimal Approach
Elementary Calculus: An Infinitesimal approach is a textbook by H. Jerome Keisler. The subtitle alludes to the infinitesimal numbers of Abraham Robinson's non-standard analysis...
by Keisler
Howard Jerome Keisler
H. Jerome Keisler 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...
which presented elementary calculus using the methods of nonstandard analysis. The review appeared in the Bulletin of the American Mathematical Society
Bulletin of the American Mathematical Society
The Bulletin of the American Mathematical Society is a quarterly mathematical journal published by the American Mathematical Society...
in 1977. This article is referred to by David O. Tall
David O. Tall
David Orme Tall is a mathematics education theorist at the University of Warwick. One of his most influential works is the joint paper with Vinner Concept image and concept definition.... The "concept image" is a notion in cognitive theory. It consists of all the cognitive structure in the...
while discussing the use of non-standard analysis in education. Tall wrote:
- 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 . 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 HalmosPaul HalmosPaul Richard Halmos was a Hungarian-born American mathematician who made fundamental advances in the areas of probability theory, statistics, operator theory, ergodic theory, and functional analysis . He was also recognized as a great mathematical expositor.-Career:Halmos obtained his B.A...
, the Bulletin book review editor, choose a constructivistConstructivism (mathematics)In the philosophy of mathematics, constructivism asserts that it is necessary to find a mathematical object to prove that it exists. When one assumes that an object does not exist and derives a contradiction from that assumption, one still has not found the object and therefore not proved its...
as the reviewer?
Comparing the use of the law of excluded middle
Law of excluded middle
In logic, the law of excluded middle is the third of the so-called three classic laws of thought. It states that for any proposition, either that proposition is true, or its negation is....
(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 . Martin Davis
Martin Davis
Martin David Davis, is an American mathematician, known for his work on Hilbert's tenth problem . He received his Ph.D. from Princeton University in 1950, where his adviser was Alonzo Church . He is Professor Emeritus at New York University. He is the co-inventor of the Davis-Putnam and the DPLL...
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 constructivistConstructivism (mathematics)In the philosophy of mathematics, constructivism asserts that it is necessary to find a mathematical object to prove that it exists. When one assumes that an object does not exist and derives a contradiction from that assumption, one still has not found the object and therefore not proved its...
context in which this objection is presumably to be understood.
Physicist Vadim Komkov
Vadim Komkov
Vadim Komkov obtained a Ph.D. from the University of Utah in 1965. He was a professor of Mathematics at Texas Tech University from 1969-1980. He also taught at the University of Utah, the University of Wisconsin, Florida State University, West Virginia University, and Winthrop University in...
(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 hyperrealHyperreal numberThe system of hyperreal numbers represents a rigorous method of treating the infinite and infinitesimal quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form1 + 1 + \cdots + 1. \, Such a number is...
s.
Whether or not non-standard analysis can be done constructively, Komkov perceived a foundational concern on Bishop's part.
Philosopher of Mathematics Geoffrey Hellman
Geoffrey Hellman
Geoffrey Hellman is an American professor and philosopher. He is Chairman of the Department of Philosophy at the University of Minnesota in Minneapolis, Minnesota.-Education:He obtained his B.A. and Ph.D...
(1993, p. 222) wrote:
- Some of Bishop's remarks (1967) suggest that his position belongs in [the constructivist] category [...]
Historian of Mathematics Joseph Dauben
Joseph Dauben
Joseph W. Dauben is a Herbert H. Lehman Distinguished Professor of History at the Graduate Center of the City University of New York. He obtained his Ph.D...
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 theoryQuantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
and thermodynamicsThermodynamicsThermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...
, and in economicsEconomicsEconomics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
, 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
Ian Stewart (mathematician)
Ian Nicholas Stewart FRS is a professor of mathematics at the University of Warwick, England, and a widely known popular-science and science-fiction writer. He is the first recipient of the , awarded jointly by the LMS and the IMA for his work on promoting mathematics.-Biography:Stewart was born...
(1986) compared Halmos' asking Bishop to review Keisler's book, to
- inviting Margaret ThatcherMargaret ThatcherMargaret Hilda Thatcher, Baroness Thatcher, was Prime Minister of the United Kingdom from 1979 to 1990...
to review Das KapitalDas KapitalDas Kapital, Kritik der politischen Ökonomie , by Karl Marx, is a critical analysis of capitalism as political economy, meant to reveal the economic laws of the capitalist mode of production, and how it was the precursor of the socialist mode of production.- Themes :In Capital: Critique of...
.
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.
Connes' criticism
In "Brisure de symétrie spontanée et géométrie du point de vue spectral", Journal of Geometry and Physics 23 ('97), 206–234, Alain ConnesAlain Connes
Alain Connes is a French mathematician, currently Professor at the Collège de France, IHÉS, The Ohio State University and Vanderbilt University.-Work:...
Connes 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 [0, 1], so that it is impossible (Stern, 1985) to exhibit a single [nonstandard real number]. The formalism that we propose will give a substantial and computable answer to this question."
In his '95 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 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.
In the view of M. Katz and K. Katz Connes' comments are critical of non-standard analysis, and they challenge these specific claims. With regard to (1), Connes' own infinitesimals similarly rely on non-constructive foundational material, such as the existence of a Dixmier trace
Dixmier trace
In mathematics, the Dixmier trace, introduced by , is a non-normal trace on a space of linear operators on a Hilbert space larger than the space of trace class operators.Some of its applications to noncommutative geometry are described in .-Definition:...
. With regard to (2), Connes presents the independence of the choice of infinitesimal as a feature of his own theory.
Halmos' remarks
Paul HalmosPaul Halmos
Paul Richard Halmos was a Hungarian-born American mathematician who made fundamental advances in the areas of probability theory, statistics, operator theory, ergodic theory, and functional analysis . He was also recognized as a great mathematical expositor.-Career:Halmos obtained his B.A...
writes in "Invariant subspaces", American Mathematical Monthly
American Mathematical Monthly
The American Mathematical Monthly is a mathematical journal founded by Benjamin Finkel in 1894. It is currently published 10 times each year by the Mathematical Association of America....
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 BishopErrett BishopErrett Albert Bishop was an American mathematician known for his work on analysis. He is the father of constructive analysis, because of his 1967 Foundations of Constructive Analysis, where he proved most of the important theorems in real analysis by constructive methods.-Life:Errett Bishop's...
), 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.
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 [...] [H]is 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
The foremost Leibniz historian H. Bos is quoted by Katz & Katz (2010) as acknowledging that Robinson's hyperreals provide- [a] 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
- [n]onstandard 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 [have] serve[d] as a basis for the construction of so [magnificent] an edifice of one of the most important mathematical disciplines?