Is Mathematical History Written by the Victors?
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Is Mathematical History Written by the Victors? Jacques Bair, Piotr Błaszczyk, Robert Ely, Valérie Henry, Vladimir Kanovei, Karin U. Katz, Mikhail G. Katz, Semen S. Kutateladze, Thomas McGaffey, David M. Schaps, David Sherry, and Steven Shnider The ABCs of the History of Infinitesimal Klein’s sentiment was echoed by the philosopher Mathematics G. Granger, in the context of a discussion of Leibniz, The ABCs of the history of infinitesimal mathemat- in the following terms: ics are in need of clarification. To what extent does Aux yeux des détracteurs de la nouvelle the famous dictum “history is always written by Analyse, l’insurmontable difficulté vient de the victors” apply to the history of mathematics as ce que de telles pratiques font violence well? A convenient starting point is a remark made aux règles ordinaires de l’Algèbre, tout en by Felix Klein in his book Elementary Mathematics conduisant à des résultats, exprimables en from an Advanced Standpoint (Klein [72, p. 214]). termes finis, dont on ne saurait contester Klein wrote that there are not one but two separate l’exactitude. Nous savons aujourd’hui que tracks for the development of analysis: deux voies devaient s’offrir pour la solution du problème: (A) the Weierstrassian approach (in the context [A] Ou bien l’on élimine du langage of an Archimedean continuum) and mathématique le terme d’infiniment petit, (B) the approach with indivisibles and/or in- et l’on établit, en termes finis, le sens à finitesimals (in the context of what we will donner à la notion intuitive de ‘valeur limite’. refer to as a Bernoullian continuum).1 … [B] Ou bien l’on accepte de maintenir, Jacques Bair is professor of mathematics at the University tout au long du Calcul, la présence d’objets of Liege, Belgium. His email address is [email protected]. portant ouvertement la marque de l’infini, Piotr Błaszczyk is professor of mathematics at the Peda- mais en leur conférant un statut propre qui gogical University of Cracow, Poland. His email address is [email protected]. Mikhail G. Katz is professor of mathematics at Bar Ilan Uni- Robert Ely is professor of mathematics at the University of versity, Israel. His email address is [email protected]. Idaho. His email address is [email protected]. il. Valérie Henry is professor of mathematics, University of Semen S. Kutateladze is professor of mathematics at the Namur and of Liege, Belgium. Her email address is valerie. Sobolev Institute of Mathematics, Novosibirsk State Univer- [email protected]. sity, Russia. His email address is [email protected]. Vladimir Kanovei is professor of mathematics at IPPI, Thomas McGaffey teaches mathematics at Rice University. Moscow, and MIIT, Moscow. His email address is kanovei@ His email address is [email protected]. rambler.ru. David M. Schaps is professor of classical studies at Bar Karin U. Katz teaches mathematics at Bar Ilan University, Ilan University, Israel. His email address is dschaps@mail. Israel. Her email address is karin.usadi.katz@gmail. biu.ac.il. 1Systems of quantities encompassing infinitesimal ones were David Sherry is professor of philosophy at Northern Arizona used by Leibniz, Bernoulli, Euler, and others. Our choice of University. His email address is [email protected]. the term is explained in the subsection “Bernoulli, Johann”. Steven Shnider is professor of mathematics at Bar Ilan Uni- It encompasses modern non-Archimedean systems. versity, Israel. His email address is [email protected]. DOI: http://dx.doi.org/10.1090/noti1026 il. 886 Notices of the AMS Volume 60, Number 7 les insère dans un système dont font aussi from physics to economics (see Herzberg 2013 partie les grandeurs finies.… [110]). One example is the visionary work of En- C’est dans cette seconde voie que les riques exploiting infinitesimals, recently analyzed vues philosophiques de Leibniz l’ont orienté. in an article by David Mumford, who wrote: 2 (Granger 1981 [43, pp. 27–28]) In my own education, I had assumed that Thus we have two parallel tracks for con- Enriques [and the Italians] were irrevocably ceptualizing infinitesimal calculus, as shown in stuck.…As I see it now, Enriques must be Figure 1. credited with a nearly complete geometric proof using, as did Grothendieck, higher B-continuum order infinitesimal deformations.…Let’s be careful: he certainly had the correct ideas about infinitesimal geometry, though he had no idea at all how to make precise A-continuum definitions. (Mumford 2011 [89]) Figure 1. Parallel tracks: a thick continuum and a Another example is important work by Cauchy thin continuum. (see the subsection “Cauchy, Augustin-Louis” be- low) on singular integrals and Fourier series using At variance with Granger’s appraisal, some of infinitesimals and infinitesimally defined “Dirac” the literature on the history of mathematics tends delta functions (these precede Dirac by a century), to assume that the A-approach is the ineluctably which was forgotten for a number of decades be- “true” one, while the infinitesimal B-approach was, cause of shifting foundational biases. The presence at best, a kind of evolutionary dead end or, at worst, of Dirac delta functions in Cauchy’s oeuvre was altogether inconsistent. To say that infinitesimals noted in (Freudenthal 1971 [40]) and analyzed by provoked passions would be an understatement. Laugwitz (1989 [74]), (1992a [75]); see also (Katz Parkhurst and Kingsland, writing in The Monist, and Tall 2012 [69]) and (Tall and Katz 2013 [107]). proposed applying a saline solution (if we may be Recent papers on Leibniz (Katz and Sherry [67], allowed a pun) to the problem of the infinitesimal: [68]; Sherry and Katz [100]) argue that, contrary [S]ince these two words [infinity and infini- to widespread perceptions, Leibniz’s system for tesimal] have sown nearly as much faulty infinitesimal calculus was not inconsistent (see the logic in the fields of mathematics and meta- subsection “Mathematical Rigor” for a discussion physics as all other fields put together, of the term). The significance and coherence of they should be rooted out of both the fields Berkeley’s critique of infinitesimal calculus have which they have contaminated. And not only been routinely exaggerated. Berkeley’s sarcastic should they be rooted out, lest more errors tirades against infinitesimals fit well with the be propagated by them: a due amount of salt ontological limitations imposed by the A-approach should be ploughed under the infected ter- favored by many historians, even though Berkeley’s ritory, that the damage be mitigated as well opposition, on empiricist grounds, to an infinitely as arrested. (Parkhurst and Kingsland 1925 divisible continuum is profoundly at odds with the [91, pp. 633–634]) [emphasis added—the A-approach. authors] A recent study of Fermat (Katz, Schaps, and Writes P. Vickers: Shnider 2013 [66]) shows how the nature of his So entrenched is the understanding that contribution to the calculus was distorted in the early calculus was inconsistent that recent Fermat scholarship, similarly due to an many authors don’t provide a reference to “evolutionary dead-end” bias (see the subsection support the claim, and don’t present the “Fermat, Pierre de”). set of inconsistent propositions they have The Marburg school of Hermann Cohen, Cassirer, in mind. (Vickers 2013 [108, section 6.1, p. Natorp, and others explored the philosophical foun- 146]) dations of the infinitesimal method underpinning Such an assumption of inconsistency can influ- the mathematized natural sciences. Their versatile, ence one’s appreciation of historical mathematics, and insufficiently known, contribution is analyzed make a scholar myopic to certain significant devel- in (Mormann and Katz 2013 [88]). opments due to their automatic placement in an A number of recent articles have pioneered “evolutionary dead-end” track, and inhibit potential a reevaluation of the history and philosophy fruitful applications in numerous fields ranging of mathematics, analyzing the shortcomings of received views, and shedding new light on the 2Similar views were expressed by M. Parmentier in (Leibniz deleterious effect of the latter on the philosophy, 1989 [79, p. 36, note 92]). the practice, and the applications of mathematics. August 2013 Notices of the AMS 887 Some of the conclusions of such a reevaluation are Next, it appears in the papers of Archimedes as the presented below. following lemma (see Archimedes [2, I, Lamb. 5]): Of unequal lines, unequal surfaces, and un- Adequality to Chimeras equal solids [ a; b; c ], the greater exceeds the Some topics from the history of infinitesimals illus- lesser [ a < b ] by such a magnitude [b − a] trating our approach appear below in alphabetical as, when added to itself [ n(b − a) ], can order. be made to exceed any assigned magnitude [c] among those which are comparable with Adequality one another. (Heath 1897 [47, p. 4]) Adequality is a technique used by Fermat to solve This can be formalized as follows: problems of tangents, problems of maxima and minima, and other variational problems. The term (2) (8a; b; c)(9n 2 N) [a < b ! n(b − a) > c]. adequality derives from the παρισ oτης´ of Dio- Note that Euclid’s definition V.4 and the lemma of phantus (see the subsection “Diophantus”). The Archimedes are not logically equivalent (see the technique involves an element of approximation subsection “Euclid’s Definition V.4”, footnote 11). and “smallness”, represented by a small varia- The Archimedean axiom plays no role in the + − tion E, as in the familiar difference f (A E) f (A). plane geometry as developed in Books I–IV of The El- Fermat used adequality in particular to find the ements.4 Interpreting geometry in ordered fields, or tangents of transcendental curves such as the in geometry over fields in short, one knows that F2 cycloid that were considered to be “mechanical” is a model of Euclid’s plane, where (F; +; ·; 0; 1; <) is curves off-limits to geometry by Descartes.