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Connes' criticism[edit]
In "Brisure de symetrie spontanee et geometrie du point de vue spectral", Journal of Geometry and Physics 23 (1997), 206?234, Alain 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 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 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.

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