This paper is a response by several historians of mathematics to a series of papers published from 2012 onwards by Mikhail Katz and various co-authors, the latest of which was recently published in the Mathematical Intelligencer, “Two-Track Depictions of Leibniz’s Fictions” (Katz, Kuhlemann, Sherry, Ugaglia, and van Atten, 2021). At issue is a question of fundamental methodology. These authors take for granted that non-standard analysis provides the correct framework for historical interpretation of the calculus, and castigate rival interpretations as having had a deleterious effect on the philosophy, practice, and applications of mathematics. Rather than make this case by reasoned historical argument, however, these papers proceed largely by piecemeal confutation of isolated quotations of their opponents taken out of their argumentative context, juxtaposed with extracts from the primary sources that are again decontextualized and presented as testifying to the unique viability of the nonstandard interpretation. Instead of attempting a full rebuttal, in this paper we limit ourselves to showing how a line of argument that has been a central theme of Katz and coauthors since 2012, concerning the so-called “A” and “B” tracks, has undergone a radical change of meaning between 2012 and 2021, finally attaining a full-blown inconsistency in the 2021 paper.
Archibald, T., Arthur, R.T.W., Ferraro, G. et al. A Question of Fundamental Methodology: Reply to Mikhail Katz and His Coauthors. Math Intelligencer (2022). https://doi.org/10.1007/s00283-022-10217-7