---
title: "Criticism of nonstandard analysis"
chunk: 3/4
source: "https://en.wikipedia.org/wiki/Criticism_of_nonstandard_analysis"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T04:17:38.400334+00:00"
instance: "kb-cron"
---

"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 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).  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 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 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 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 [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 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 [...] [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 ==
Leibniz historian Henk Bos (1974) acknowledged 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?

== See also ==
Constructive nonstandard analysis
Influence of nonstandard analysis

== Notes ==