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Fermat, Leibniz, Euler, and the Gang: the True History of the Concepts of Limit and Shadow

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Fermat, Leibniz, Euler, and the Gang: the True History of the Concepts of Limit and Shadow Fermat, Leibniz, Euler, and the Gang: The True History of the Concepts of Limit and Shadow Tiziana Bascelli, Emanuele Bottazzi, Frederik Herzberg, Vladimir Kanovei, Karin U. Katz, Mikhail G. Katz, Tahl Nowik, David Sherry, and Steven Shnider he theories as developed by European Fermat determined the optimal value by impos- mathematicians prior to 1870 differed ing a condition using his adequality of quantities. from the modern ones in that none of But he did not really think of quantities as func- them used the modern theory of limits. tions, nor did he realize that his method produced Fermat develops what is sometimes only a necessary condition for his optimization Tcalled a “precalculus” theory, where the optimal condition. For a more detailed general introduc- value is determined by some special condition such tion, see chapters 1 and 2 of the volume edited by as equality of roots of some equation. The same Grattan-Guinness (Bos et al. 1980 [19]). can be said for his contemporaries like Descartes, The doctrine of limits is sometimes claimed to Huygens, and Roberval. have replaced that of infinitesimals when analysis Leibniz’s calculus advanced beyond them in was rigorized in the nineteenth century. While working on the derivative function of the variable x. it is true that Cantor, Dedekind and Weierstrass He had the indefinite integral whereas his prede- attempted (not altogether successfully; see Ehrlich cessors only had concepts more or less equivalent 2006 [32], Mormann & Katz 2013 [79]) to eliminate to it. Euler, following Leibniz, also worked with infinitesimals from analysis, the history of the limit such functions, but distinguished the variable (or concept is more complex. Newton had explicitly variables) with constant differentials dx, a status written that his ultimate ratios were not actually that corresponds to the modern assignment that x ratios but, rather, limits of prime ratios (see Russell is the independent variable, the other variables of 1903 [89, item 316, pp. 338-339]; Pourciau 2001 [84]). In fact, the sources of a rigorous notion of the problem being dependent upon it (or them) limit are considerably older than the nineteenthth functionally. century. Tiziana Bascelli is an independent researcher in history In the context of Leibnizian mathematics, the and philosophy of science. Her email address is tiziana. limit of f (x) as x tends to x0 can be viewed as [email protected]. the “assignable part” (as Leibniz may have put Emanuele Bottazzi is a Ph.D. student at the Università di it) of f (x0 + dx) where dx is an “inassignable” Trento, Italy. His email address is Emanuele.Bottazzi@ infinitesimal increment (whenever the answer is unitn.it. independent of the infinitesimal chosen). A modern Frederik Herzberg is an assistant professor of mathemati- formalization of this idea exploits the standard cal economics at Bielefeld University, Germany, as well as part principle (see Keisler 2012 [67, p. 36]). an external member of the Munich Center for Mathematical Philosophy, Germany. His email address is fherzberg@uni- Mikhail G. Katz is professor of mathematics at Bar Ilan Uni- bielefeld.de. versity, Israel. His email is [email protected]. Vladimir Kanovei is professor of mathematics at IPPI, Tahl Nowik is professor of mathematics at Bar Ilan Univer- Moscow, and MIIT, Moscow, Russia. His email address is sity, Israel. His email is [email protected]. [email protected]. David Sherry is professor of philosophy at Northern Arizona Karin U. Katz teaches mathematics at Bar Ilan University, University. His email address is [email protected]. Israel. Her email address is [email protected]. Steven Shnider is professor of mathematics at Bar Ilan DOI: http://dx.doi.org/10.1090/noti1149 University, Israel. His email is [email protected]. 848 Notices of the AMS Volume 61, Number 8 In the context of ordered fields E, the standard −1 0 1 2 3 4 part principle is the idea that, if E is a proper extension of the real numbers R, then every finite (or limited) element x 2 E is infinitely close to a suitable x0 2 R. Such a real number is called the standard part (sometimes called the shadow) of x, or in formulas, st(x) = x . Denoting by E the 0 f r collection of finite elements of E, we obtain a map r + α r + β r + γ st : Ef ! R: Here x is called finite if it is smaller (in absolute value) than some real number (the term finite is immediately comprehensible to a wide mathemati- cal public, whereas limited corresponds to correct st technical usage); an infinitesimal is smaller (in absolute value) than every positive real; and x is −1 0 1 r 2 3 4 infinitely close to x0 in the sense that x − x0 is infinitesimal. Figure 1. The standard part function, st, “rounds Briefly, the standard part function “rounds off” off” a finite hyperreal to the nearest real number. a finite element of E to the nearest real number The function st is here represented by a vertical (see Figure 1). projection. An “infinitesimal microscope” is used The proof of the principle is easy. A finite to view an infinitesimal neighborhood of a standard real number r , where α, β, and γ element x 2 E defines a Dedekind cut on the represent typical infinitesimals. Courtesy of subfield R ⊂ E (alternatively, on Q ⊂ R), and Wikipedia. the cut in turn defines the real x0 via the usual correspondence between cuts and real numbers. One sometimes writes down the relation Methodological Remarks x ≈ x0 To comment on the historical subtleties of judging to express infinite closeness. or interpreting past mathematics by present-day We argue that the sources of such a relation, standards,1 note that neither Fermat, Leibniz, Euler, and of the standard part principle, go back to nor Cauchy had access to the semantic founda- Fermat, Leibniz, Euler, and Cauchy. Leibniz would tional frameworks as developed in mathematics + discard the inassignable part of 2x dx to arrive at the end of the nineteenthth and first half of at the expected answer, 2x, relying on his law of the twentieth centuries. What we argue is that homogeneity (see the section entitled “Leibniz’s their syntactic inferential moves ultimately found Transcendental Law of Homogeneity”). Such an modern proxies in Robinson’s framework, thus inferential move is mirrored by a suitable proxy in placing a firm (relative to ZFC)2 semantic foun- the hyperreal approach, namely the standard part dation underneath the classical procedures of function. these masters. Benacerraf (1965 [10]) formulated Fermat, Leibniz, Euler, and Cauchy all used a related dichotomy in terms of mathematical one or another form of approximate equality, or the idea of discarding “negligible” terms. Their practice vs. mathematical ontology. inferential moves find suitable proxies in the For example, the Leibnizian laws of continuity context of modern theories of infinitesimals, and (see Knobloch 2002 [69, p. 67]) and homogene- specifically the concept of shadow. ity can be recast in terms of modern concepts The last two sections present an application of such as the transfer principle and the standard the standard part to decreasing rearrangements part principle over the hyperreals, without ever of real functions and to a problem on divergent appealing to the semantic content of the technical integrals due to S. Konyagin. development of the hyperreals as a punctiform con- This article continues efforts in revisiting the tinuum; similarly, Leibniz’s proof of the product history and foundations of infinitesimal calculus rule for differentiation is essentially identical, at and modern nonstandard analysis. Previous efforts the syntactic level, to a modern infinitesimal proof in this direction include Bair et al. (2013 [6]), (see, again, the section “Leibniz’s Transcendental Bascelli (2014 [7]), Błaszczyk et al. (2013 [15]), Law of Homogeneity”). Borovik et al. (2012 [16], [17]), Kanovei et al. (2013 [55]), Katz, Katz & Kudryk (2014 [61]), Mormann 1Some reflections on this can be found in Lewis (1975 [76]). et al. (2013 [79]), Sherry et al. (2014 [92]), Tall et 2The Zermelo–Fraenkel Set Theory with the Axiom of al. (2014 [97]). Choice. September 2014 Notices of the AMS 849 A-track and B-track formulated a condition, in terms of the mean value 5 The crucial distinction between syntactic and se- theorem, for what would qualify as a successful mantic aspects of the work involving mathematical theory of infinitesimals, and concluded: continua appears to have been overlooked by I will not say that progress in this direction R. Arthur who finds fault with the hyperreal proxy is impossible, but it is true that none of of the Leibnizian continuum, by arguing that the the investigators have achieved anything latter was non-punctiform (see Arthur 2013 [5]). positive (Klein 1908 [68, p. 219]). Yet this makes little difference at the syntactic Klein was referring to the current work on level, as explained above. Arthur’s brand of the syn- infinitesimal-enriched systems by Levi-Civita, Bet- categorematic approach following Ishiguro (1990 tazzi, Stolz, and others. In Klein’s mind, the [52]) involves a reductive reading of Leibnizian infinitesimal track was very much a current re- infinitesimals as logical (as opposed to pure) fic- search topic; see Ehrlich (2006 [32]) for a detailed tions involving a hidden quantifier à la Weierstrass, coverage of the work on infinitesimals around 1900. ranging over “ordinary” values. This approach was critically analyzed in (Katz & Sherry 2013 [65]), Formal Epistemology: Easwaran on Hyperreals (Sherry & Katz 2013 [92]), and (Tho 2012 [101]). Some recent articles are more encouraging in that Robinson’s framework poses a challenge to they attempt a more technically sophisticated ap- traditional historiography of mathematical analysis.
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