Non-standard analysis and its offshoot, non-standard calculus, have been criticized by several authors. The nature of such criticisms is not directly related to the logical status of the results proved using non-standard analysis. In terms of conventional mathematical foundations, such results are quite acceptable. In the technical language of mathematical logic, IST is a conservative extension of ZFC.

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  • Non-standard analysis and its offshoot, non-standard calculus, have been criticized by several authors. The nature of such criticisms is not directly related to the logical status of the results proved using non-standard analysis. In terms of conventional mathematical foundations, such results are quite acceptable. In the technical language of mathematical logic, IST is a conservative extension of ZFC. It provides an assurance that the novelty 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 the theory. Philip J. Davis 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. Non-standard calculus in the classroom has been analysed in the Chicago study by 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 in "NSA and its weak impact on education", page 172.
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  • December 2008
  • January 2009
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  • Davis
  • 1977 (xsd:integer)
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  • Non-standard analysis and its offshoot, non-standard calculus, have been criticized by several authors. The nature of such criticisms is not directly related to the logical status of the results proved using non-standard analysis. In terms of conventional mathematical foundations, such results are quite acceptable. In the technical language of mathematical logic, IST is a conservative extension of ZFC.
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  • Criticism of non-standard analysis
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