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 ABC’s 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 (A) the Weierstrassian approach (in the context du problème: of an Archimedean continuum) and footnote [A] Ou bien l’on élimine du (B) the approach with indivisibles and/or in- langage mathématique le terme d’infiniment finitesimals (in the context of what we will petit, et l’on établit, en termes finis, le sens refer to as a Bernoullian continuum).1 à donner à la notion intuitive de ‘valeur limite’. … Jacques Bair is professor of mathematics at the University [B] Ou bien l’on accepte de maintenir, of Liege, Belgium. His email address is [email protected]. tout au long du Calcul, la présence d’objets Piotr Błaszczyk is professor of mathematics at the Peda- portant ouvertement la marque de l’infini, 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 Na- Semen S. Kutateladze is professor of mathematics at the mur, Belgium. Her email address is [email protected] Sobolev Institute of Mathematics, Novosibirsk State Univer- be. sity, Russia. His email address is [email protected]. Vladimir Kanovei is professor of mathematics at IPPI, Thomas McGaffey is professor of mathematics at Rice Uni- Moscow, and MIIT, Moscow. His email address is kanovei@ versity. His email address is thomasmcgaffey@sbcglobal. rambler.ru. net. Karin U. Katz is professor of mathematics at Bar Ilan Univer- David M. Schaps is professor of classical studies at the Bar sity, Israel. Her email address is karin.usadi.katz@gmail. Ilan University, Israel. His email address is dschaps@mail. com. biu.ac.il. 1Systems of quantities encompassing infinitesimal ones were David Sherry is professor of philosophy, 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/noti1001 il. 2 Notices of the AMS Volume 60, Number 7 from physics to economics. One example is the vi- B-continuum sionary work of Enriques exploiting infinitesimals, recently analyzed in an article by David Mumford, who wrote: A-continuum In my own education, I had assumed [that Figure 1. Parallel tracks: a thick continuum and a Enriques and the Italians] were irrevocably thin continuum. stuck.…As I see it now, Enriques must be credited with a nearly complete geometric proof using, as did Grothendieck, higher mais en leur conférant un statut propre qui order infinitesimal deformations.…Let’s be les insère dans un système dont font aussi careful: he certainly had the correct ideas partie les grandeurs finies.… about infinitesimal geometry, though he C’est dans cette seconde voie que les had no idea at all how to make precise vues philosophiques de Leibniz l’ont orienté. definitions. (Mumford 2011 [89]) 2 (Granger 1981 [43, pp. 27–28]) Another example is important work by Cauchy Thus we have two parallel tracks for con- (see the subsection “Cauchy, Augustin-Louis” be- ceptualizing infinitesimal calculus, as shown in low) on singular integrals and Fourier series using Figure 1. infinitesimals and infinitesimally defined “Dirac” At variance with Granger’s appraisal, some of delta functions (these precede Dirac by a century), the literature on the history of mathematics tends which was forgotten for a number of decades be- to assume that the A-approach is the ineluctably cause of shifting foundational biases. The presence “true” one, while the infinitesimal B-approach was, of Dirac delta functions in Cauchy’s oeuvre was at best, a kind of evolutionary dead end or, at worst, noted in (Freudenthal 1971 [40]) and analyzed by altogether inconsistent. To say that infinitesimals Laugwitz (1989 [74]), (1992a [75]); see also (Katz provoked passions would be an understatement. and Tall 2012 [69]) and (Tall and Katz 2013 [107]). Parkhurst and Kingsland, writing in The Monist, Recent papers on Leibniz (Katz and Sherry [67], proposed applying a saline solution (if we may be [68]; Sherry and Katz [100]) argue that, contrary allowed a pun) to the problem of the infinitesimal: to widespread perceptions, Leibniz’s system for [S]ince these two words [infinity and infini- infinitesimal calculus was not inconsistent (see the tesimal] have sown nearly as much faulty subsection “Mathematical Rigor” for a discussion logic in the fields of mathematics and meta- of the term). The significance and coherence of physics as all other fields put together, Berkeley’s critique of infinitesimal calculus have they should be rooted out of both the fields been routinely exaggerated. Berkeley’s sarcastic which they have contaminated. And not only tirades against infinitesimals fit well with the should they be rooted out, lest more errors ontological limitations imposed by the A-approach be propagated by them: a due amount of salt favored by many historians, even though Berkeley’s should be ploughed under the infected ter- opposition, on empiricist grounds, to an infinitely ritory, that the damage be mitigated as well divisible continuum is profoundly at odds with the as arrested. (Parkhurst and Kingsland 1925 A-approach. [91, pp. 633–634]) [emphasis added—the A recent study of Fermat (Katz, Schaps, and authors] Shnider 2013 [66]) shows how the nature of his Writes P. Vickers: contribution to the calculus was distorted in So entrenched is the understanding that recent Fermat scholarship, similarly due to an the early calculus was inconsistent that “evolutionary dead-end” bias (see the subsection many authors don’t provide a reference to “Fermat, Pierre”). support the claim, and don’t present the The Marburg school of Hermann Cohen, Cassirer, set of inconsistent propositions they have Natorp, and others explored the philosophical foun- in mind. (Vickers 2013 [108, section 6.1]) dations of the infinitesimal method underpinning the mathematized natural sciences. Their versatile, Such an assumption of inconsistency can influ- 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 3 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.