Criticism of non-standard analysis
Non-standard analysis and its offshoot, non-standard calculus, have been criticized by several authors, notably Errett Bishop, Paul Halmos, and Alain Connes. These criticisms are analyzed below.
The evaluation of non-standard analysis in the literature has varied greatly. Paul Halmos described it as a technical special development in mathematical logic. Fields medalist 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 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 hyperreals), 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:
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