Terence Tao summed up the advantage of the hyperreal framework by noting that it allows one to rigorously manipulate things such as "the set of all small numbers", or to rigorously say things like "η1 is smaller than anything that involves η0", while greatly reducing epsilon management issues by automatically concealing many of the quantifiers in one's argument.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 although usually strongly dependent on choice.
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.
Philip J. Davis wrote, in a book review of Left Back: A Century of Failed School Reforms[3] by Diane Ravitch:[4] 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.
[citation needed] In the view of Errett Bishop, classical mathematics, which includes Robinson's approach to nonstandard analysis, was nonconstructive and therefore deficient in numerical meaning (Feferman 2000).
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!
(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.)
In "Brisure de symétrie spontanée et géométrie du point de vue spectral", Journal of Geometry and Physics 23 (1997), 206–234, Alain Connes wrote: In his 1995 article "Noncommutative geometry and reality" Connes develops a calculus of infinitesimals based on operators in Hilbert space.
[8] With regard to (1), Connes' own infinitesimals similarly rely on non-constructive foundational material, such as the existence of a Dixmier trace.
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'".