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) (∀a, b, c)(∃n ∈ 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. Fermat a Euclidean field, i.e., an ordered field closed under also used it to solve the variational problem of the square root operation. Consequently, R∗ × R∗ the refraction of light so as to obtain Snell’s law. (where R∗ is a hyperreal field) is a model of Adequality incorporated a procedure of discarding Euclid’s plane as well (see the subsection below higher-order terms in E (without setting them on modern implementations). Euclid’s definition equal to zero). Such a heuristic procedure was V.4 is discussed in more detail in the subsection ultimately formalized mathematically in terms of “Euclid’s Definition V.4”. the standard part principle (see the subsection “Standard Part Principle”) in Robinson’s theory of Otto Stolz rediscovered the Archimedean axiom infinitesimals starting with (Robinson 1961 [94]). for mathematicians, making it one of his axioms Fermat’s adequality is comparable to Leibniz’s tran- for magnitudes and giving it the following form: if a > b, then there is a multiple of b such scendental law of homogeneity (see the subsection 5 “Lex homogeneorum transcendentalis”). that nb > a (Stolz 1885 [106, p. 69]). At the same time, in his development of the integers, Archimedean Axiom Stolz implicitly used the Archimedean axiom. Stolz’s visionary realization of the importance of What is known today as the Archimedean axiom the Archimedean axiom and his work on non- first appears in Euclid’s Elements, Book V, as Archimedean systems stand in sharp contrast Definition 4 (Euclid [34, Definition V.4]). It is to Cantor’s remarks on infinitesimals (see the exploited in (Euclid [34, Proposition V.8]). We include bracketed symbolic notation so as to subsection below on mathematical rigor). clarify the definition: In modern mathematics, the theory of or- dered fields employs the following form of the Magnitudes [ a, b ] are said to have a ratio Archimedean axiom (see, e.g., Hilbert 1899 [51, with respect to one another which, being p. 27]): multiplied [ na ] are capable of exceeding one another [ na > b ]. (∀x > 0)(∀ > 0)(∃n ∈ N) [n > x] 3 It can be formalized as follows: or equivalently (1) (∀a, b)(∃n ∈ N) [na > b], where na=a + · · · + a . (3) (∀ > 0)(∃n ∈ N) [n > 1]. | {z } n−times A number system satisfying (3) will be referred to as an Archimedean continuum. In the contrary case, 3See, e.g., the version of the Archimedean axiom in (Hilbert there is an element  > 0 called an infinitesimal 1899 [51, p. 19]). Note that we have avoided using “0” in formula (1), as in “∀a > 0”, since 0 was not part of the con- 4 ceptual framework of the Greeks. The term “multiplied” in With the exception of Proposition III.16, where so-called the English translation of Euclid’s definition V.4 corresponds horn angles appear that could be considered as non- to the Greek term πoλλαπλασ ιαζoµενα´ . A common Archimedean magnitudes relative to rectilinear angles. formalisation of the noun “multiple”, πoλλαπλασ´ ιoν, 5See (Ehrlich [32]) for additional historical details concern- is na = a + · · · + a. ing Stolz’s account of the Archimedean axiom.

4 Notices of the AMS Volume 60, Number 7 such that no finite sum  +  + · · · +  will ever Berkeley’s Logical Criticism reach 1; in other words, Berkeley’s logical criticism of the calculus amounts h i (4) (∃ > 0)(∀n ∈ N)  ≤ 1 . to the contention that the evanescent increment is n first assumed to be nonzero to set up an algebraic A number system satisfying (4) is referred to as expression and then is treated as zero in discarding a Bernoullian continuum (i.e., a non-Archimedean the terms that contained that increment when the continuum); see the subsection “Bernoulli, Johann”. increment is said to vanish. In modern terms, Berkeley was claiming that the calculus was based Berkeley, George on an inconsistency of type George Berkeley (1685–1753) was a cleric whose empiricist (i.e., based on sensations, or sensa- (dx 6= 0) ∧ (dx = 0). tionalist) metaphysics tolerated no conceptual innovations, such as infinitesimals, without an The criticism, however, involves a misunderstand- empirical counterpart or referent. Berkeley was ing of Leibniz’s method. The rebuttal of Berkeley’s similarly opposed, on metaphysical grounds, to criticism is that the evanescent increment need not infinite divisibility of the continuum (which he be “treated as zero” but, rather, is merely discarded referred to as extension), an idea widely taken through an application of the transcendental law for granted today. In addition to his outdated of homogeneity by Leibniz, as illustrated in the sub- metaphysical criticism of the infinitesimal calculus section “Product Rule” in the case of the product of Newton and Leibniz, Berkeley also formulated a rule. logical criticism.6 Berkeley claimed to have detected While consistent (in the sense of the subsection a logical fallacy at the basis of the method. In terms “Mathematical Rigor”, level (2)), Leibniz’s system of Fermat’s E occurring in his adequality (see the unquestionably relied on heuristic principles, such subsection “Adequality”), Berkeley’s objection can as the laws of continuity and homogeneity, and be formulated as follows: thus fell short of a standard of rigor if measured by The increment E is assumed to be nonzero today’s criteria (see the subsection “Mathematical at the beginning of the calculation, but zero Rigor”). On the other hand, the consistency and at its conclusion, an apparent logical fallacy. resilience of Leibniz’s system is confirmed through the development of modern implementations of However, E is not assumed to be zero at the end of the calculation, but rather is discarded at the Leibniz’s heuristic principles (see the subsection end of the calculation (see the subsection “Berke- “Modern Implementations”). ley’s Logical Criticism” for more details). Such a technique was the content of Fermat’s adequality Bernoulli, Johann (see the subsection “Adequality”) and Leibniz’s Johann Bernoulli (1667–1748) was a disciple of transcendental law of homogeneity (see the sub- Leibniz’s who having learned an infinitesimal section “Lex homogeneorum transcendentalis”), methodology for the calculus from the master, where the relation of equality has to be suitably never wavered from it. This is in contrast to Leibniz interpreted (see the subsection “Relation [\ ”). The himself, who throughout his career used both technique is equivalent to taking the limit (of a typ- (A) an Archimedean methodology (proof by ical expression such as f (A+E)−f (A) for example) in E exhaustion) and Weierstrass’s approach and to taking the standard (B) an infinitesimal methodology part (see the subsection “Standard Part Principle”) in Robinson’s approach. in a symbiotic fashion. Thus, Leibniz relied on Meanwhile, Berkeley’s own attempt to explain the A-methodology to underwrite and justify the calculation of the derivative of x2 in The the B-methodology, and he exploited the B- Analyst contains a logical circularity: namely, methodology to shorten the path to discovery Berkeley’s argument relies on the determination of (Ars Inveniendi). Historians often name Bernoulli the tangents of a parabola by Apollonius (which is as the first mathematician to have adhered sys- equivalent to the calculation of the derivative). This tematically to the infinitesimal approach as the circularity in Berkeley’s argument was analyzed in basis for the calculus. We refer to an infinitesimal- (Andersen 2011 [1]). enriched number system as a B-continuum, as opposed to an Archimedean A-continuum, i.e., a 6Berkeley’s criticism was dissected into its logical and continuum satisfying the Archimedean axiom (see metaphysical components in (Sherry 1987 [98]). the subsection “Archimedean Axiom”).

August 2013 Notices of the AMS 5 Bishop, Errett Cantor, Georg Errett Bishop (1928–1983) was a mathematical Georg Cantor (1845–1918) is familiar to the modern constructivist who, unlike his fellow intuition- reader as the underappreciated creator of the ist7 Arend Heyting (see the subsection “Heyting, “Cantorian paradise”, out of which David Hilbert Arend”), held a dim view of classical mathematics in would not be expelled, as well as the tragic hero, general and Robinson’s infinitesimals in particular. allegedly persecuted by Kronecker, who ended Discouraged by the apparent nonconstructivity his days in a lunatic asylum. Cantor historian J. Dauben notes, however, an underappreciated of his early work in functional analysis under aspect of Cantor’s scientific activity, namely, his P. Halmos, Bishop believed he had found the principled persecution of infinitesimalists: culprit in the law of excluded middle (LEM), the key Cantor devoted some of his most vituper- logical ingredient in every proof by contradiction. ative correspondence, as well as a portion He spent the remaining eighteen years of his life of the Beiträge, to attacking what he de- in an effort to expunge the reliance on LEM (which scribed at one point as the ‘infinitesimal he dubbed “the principle of omniscience” in [11]) Cholera bacillus of mathematics’, which had from analysis and sought to define meaning itself spread from Germany through the work of in mathematics in terms of such LEM-extirpation. Thomae, du Bois Reymond and Stolz, to in- Accordingly, he described classical mathematics fect Italian mathematics.…Any acceptance as both a debasement of meaning (Bishop 1973 of infinitesimals necessarily meant that [13, p. 1]) and sawdust (Bishop 1973 [13, p. 14]), his own theory of number was incomplete. and he did not hesitate to speak of both crisis Thus to accept the work of Thomae, du (Bishop 1975 [11]) and schizophrenia (Bishop 1973 Bois-Reymond, Stolz and Veronese was to [13]) in contemporary mathematics, predicting an deny the perfection of Cantor’s own cre- imminent demise of classical mathematics in the ation. Understandably, Cantor launched a following terms: thorough campaign to discredit Veronese’s work in every way possible. (Dauben 1980 Very possibly classical mathematics will [27, pp. 216–217]) cease to exist as an independent discipline. A discussion of Cantor’s flawed investigation (Bishop 1968 [10, p. 54]) of the Archimedean axiom (see the subsection His attack in (Bishop 1977 [12]) on calculus “Archimedean Axiom”) may be found in the subsection “Mathematical Rigor”.8 pedagogy based on Robinson’s infinitesimals was a natural outgrowth of his general opposition to the Cauchy, Augustin-Louis logical underpinnings of classical mathematics, as analyzed in (Katz and Katz 2011 [63]). Robinson Augustin-Louis Cauchy (1789–1857) is often viewed in the history of mathematics literature as a pre- formulated a brief but penetrating appraisal of cursor of Weierstrass. Note, however, that contrary Bishop’s ventures into the history and philosophy to a common misconception, Cauchy never gave an of mathematics, when he noted that , δ definition of either limit or continuity (see the the sections of [Bishop’s] book that attempt subsection “Variable Quantity” for Cauchy’s defini- to describe the philosophical and historical tion of limit). Rather, his approach to continuity background of [the] remarkable endeavor was via what is known today as microcontinuity [of Intuitionism] are more vigorous than (see the subsection “Continuity”). Several recent articles, (Błaszczyk et al. [14]; Borovik and Katz accurate and tend to belittle or ignore the [16]; Bråting [20]; Katz and Katz [62], [64]; Katz and efforts of others who have worked in the Tall [69]), have argued that a proto-Weierstrassian same general direction. (Robinson 1968 [95, view of Cauchy is one-sided and obscures Cauchy’s p. 921]) important contributions, including not only his See the subsection “Chimeras” for a related infinitesimal definition of continuity but also such criticism by Alain Connes. innovations as his infinitesimally defined (“Dirac”) delta function, with applications in Fourier anal- ysis and evaluation of singular integrals, and his 7Bishop was not an intuitionist in the narrow sense of the study of orders of growth of infinitesimals that term, in that he never worked with Brouwer’s continuum or anticipated the work of Paul du Bois-Reymond, “free choice sequences”. We are using the term “intuitionism” in a broader sense (i.e., mathematics based on intuitionistic 8Cantor’s dubious claim that the infinitesimal leads to con- logic) that incorporates constructivism, as used for example tradictions was endorsed by no less an authority than by Abraham Robinson in the comment quoted at the end of B. Russell; see footnote 15 in the subsection “Mathematical this subsection. Rigor”.

6 Notices of the AMS Volume 60, Number 7 R Borel, Hardy, and ultimately (Skolem [102], [103], trace − (featured on the front cover of Connes’s [104]), and Robinson. magnum opus) and the Hahn–Banach theorem, in To elaborate on Cauchy’s “Dirac” delta function, Connes’s own framework; see also [65]. note the following formula from (Cauchy 1827 [23, See the subsection “Bishop, Errett” for a related p. 188]) in terms of an infinitesimal α: criticism by Errett Bishop. 1 Z a+ α dµ π (5) F(µ) = F(a). 2 2 Continuity to Indivisibles 2 a− α + (µ − a) 2 α Continuity Replacing Cauchy’s expression α2+(µ−a)2 by δa(µ), one obtains Dirac’s formula up to trivial modifica- Of the two main definitions of continuity of a tions (see Dirac [30, p. 59]): function, Definition A (see below) is operative in Z ∞ either a B-continuum or an A-continuum (satisfy- f (x)δ(x) = f (0). ing the Archimedean axiom; see the subsection −∞ “Archimedean Axiom”), while Definition B works Cauchy’s 1853 paper on a notion closely related only in a B-continuum (i.e., an infinitesimal-enriched to uniform convergence was recently examined or Bernoullian continuum; see the subsection in (Katz and Katz 2011 [62]) and (Błaszczyk et “Bernoulli, Johann”). al. 2012 [14]). Cauchy handles the said notion • Definition A (, δ approach): A real func- using infinitesimals, including one generated by tion f is continuous at a real point x if and the null sequence ( 1 ). n only if

Chimeras (∀ > 0)(∃δ > 0)(∀x0) Alain Connes (1947–) formulated criticisms of ×  |x − x0| < δ → |f (x) − f (x0)| <  . Robinson’s infinitesimals between the years 1995 • Definition B (microcontinuity): A real func- and 2007 on at least seven separate occasions tion f is continuous at a real point x if and (see Kanovei et al. 2012 [57, Section 3.1, Table 1]). only if These range from pejorative epithets such as 0  0 0  “inadequate,” “disappointing,” “chimera,” and “irre- (6) (∀x ) x [\ x → f (x ) [\ f (x) . mediable defect,” to “the end of the rope for being In formula (6) the natural extension of f is still ‘explicit’.” denoted f , and the symbol “ [\ ” stands for the Connes sought to exploit the Solovay model S 0 relation of being infinitely close; thus, x [\ x if and (Solovay 1970 [105]) as ammunition against only if x0 − x is infinitesimal (see the subsection nonstandard analysis, but the model tends to “Relation ”). boomerang, undercutting Connes’s own earlier [\ work in functional analysis. Connes described Diophantus the hyperreals as both a “virtual theory” and a “chimera,” yet acknowledged that his argument Diophantus of Alexandria (who lived about 1,800 relies on the transfer principle (see the subsection years ago) contributed indirectly to the develop- “Modern Implementations”). In S, all definable sets ment of infinitesimal calculus through the tech- of reals are Lebesgue measurable, suggesting that nique called παρισ oτης´ , developed in his work Connes views a theory as being “virtual” if it is not Arithmetica, Book Five, problems 12, 14, and 17. definable in a suitable model of ZFC. If so, Connes’s The term παρισ oτης´ can be literally translated as claim that a theory of the hyperreals is “virtual” is “approximate equality”. This was rendered as adae- refuted by the existence of a definable model of qualitas in Bachet’s Latin translation [4] and adé- the hyperreal field (Kanovei and Shelah [59]). Free galité in French (see the subsection “Adequality”). ultrafilters aren’t definable, yet Connes exploited The term was used by Fermat to describe the such ultrafilters both in his own earlier work on comparison of values of an algebraic expression, the classification of factors in the 1970s and 80s or what would today be called a function f , at and in his magnum opus Noncommutative Geom- nearby points A and A + E and to seek extrema etry (Connes 1994 [26, Ch. V, Sect. 6.δ, Def. 11]), by a technique equivalent to the vanishing of f (A+E)−f (A) raising the question whether the latter may not E after discarding the remaining E-terms; be vulnerable to Connes’s criticism of virtuality. see (Katz, Schaps and Shnider 2013 [66]). The article [57] analyzed the philosophical under- pinnings of Connes’s argument based on Gödel’s Euclid’s Definition V.4 incompleteness theorem and detected an apparent Euclid’s Definition V.4 has already been discussed circularity in Connes’s logic. The article [57] also in the subsection “Archimedean Axiom”. In addition documented the reliance on nonconstructive foun- to Book V, it appears in Books X and XII and is dational material, and specifically on the Dixmier used in the method of exhaustion (see Euclid [34,

August 2013 Notices of the AMS 7 Propositions X.1, XII.2]). The method of exhaustion Euler, Leonhard was exploited intensively by both Archimedes Euler’s Introductio in Analysin Infinitorum (1748 and Leibniz (see the subsection “Leibniz’s De [35]) contains remarkable calculations carried out Quadratura” on Leibniz’s work De Quadratura). It in an extended number system in which the basic was revived in the nineteenth century in the theory algebraic operations are applied to infinitely small of the Riemann integral. and infinitely large quantities. Thus, in Chapter 7, Euclid’s Book V sets the basis for the theory “Exponentials and Logarithms Expressed through of similar figures developed in Book VI. Great Series”, we find a derivation of the power series mathematicians of the seventeenth century, such for az starting from the formula aω = 1 + kω, as Descartes, Leibniz, and Newton, exploited for ω infinitely small, and then raising the equation Euclid’s theory of similar figures of Book VI while to the infinitely great power12 j = z for a finite paying no attention to its axiomatic background.9 ω (appreciable) z to give Over time Euclid’s Book V became a subject of interest for historians and editors alone. az = ajω = (1 + kω)j To formalize Definition V.4, one needed a and finally expanding the right-hand side as a formula for Euclid’s notion of “multiple” and an power series by means of the binomial formula. idea of total order. Some progress in this direction 10 In the chapters following, Euler finds infinite was made by Robert Simson in 1762. In 1876 product expansions factoring the power series Hermann Hankel provided a modern reconstruction expansion for transcendental functions (see the of Book V. Combining his own historical studies subsection “Euler’s Infinite Product Formula for with an idea of order compatible with addition Sine”). By Chapter 10 he has the tools to sum the developed by Hermann Grassmann (1861 [44]), he series for ζ(2) where ζ(s) = P n−s . He explicitly gave a formula that to this day is accepted as a n calculates ζ(2k) for k = 1,..., 13 as well as many formalization of Euclid’s definition of proportion other related infinite series. in V.5 (Hankel 1876 [46, pp. 389–398]). Euclid’s In Chapter 3 of his Institutiones Calculi Differen- proportion is a relation among four “magnitudes”, tialis (1755 [37]), Euler deals with the methodology such as of the calculus, such as the nature of infinitesimal A : B :: C : D. and infinitely large quantities. We will cite the It was interpreted by Hankel as the relation English translation [38] of the Latin original [37].  (∀m, n) (nA >1 mB → nC >2 mD) Here Euler writes that ∧ (nA = mB → nC = mD) even if someone denies that infinite num- ∧ (nA < mB → nC < mD), bers really exist in this world, still in 1 2 mathematical speculations there arise ques- where n, m are natural numbers. The indices tions to which answers cannot be given on the inequalities emphasize the fact that the unless we admit an infinite number. (ibid., “magnitudes” A, B have to be of “the same kind”, §82) [emphasis added—the authors] e.g., line segments, whereas C,D could be of Euler’s approach, countenancing the possibility another kind, e.g., triangles. of denying that “infinite numbers really exist,” is In 1880 J. L. Heiberg, in his edition of Archimedes’ consonant with a Leibnizian view of infinitesimal Opera omnia in a comment on a lemma of and infinite quantities as “useful fictions” (see Katz Archimedes, cites Euclid’s Definition V.4, not- and Sherry [67]; Sherry and Katz [100]). Euler then ing that these two are the same axioms (Heiberg notes that “an infinitely small quantity is nothing 1880 [49, p. 11]).11 This is the reason why Eu- but a vanishing quantity, and so it is really equal clid’s Definition V.4 is commonly known as the Archimedean axiom. Today we formalize Euclid’s to 0.” (ibid., §83) Definition V.4 as in (1), while the Archimedean Similarly, Leibniz combined a view of infinitesi- lemma is rendered by formula (2). mals as “useful fictions” and inassignable quantities, with a generalized notion of “equality” that was 9Leibniz and Newton apparently applied Euclid’s conclu- an equality up to an incomparably negligible term. sions in a context where the said conclusions did not, Leibniz sought to codify this idea in terms of his technically speaking, apply: namely, to infinitesimal fig- transcendental law of homogeneity (TLH); see the ures such as the characteristic triangle, i.e., triangle with subsection “Lex homogeneorum transcendentalis”. sides dx, dy, and ds. Thus, Euler’s formulas such as 10See Simson’s axioms that supplement the definitions of Book V as elaborated in (Simson 1762 [101, p. 114–115]). (7) a + dx = a 11In point of fact, Euclid’s axiom V.4 and Archimedes’ lemma are not equivalent from the logical viewpoint. Thus, 12Euler used the symbol i for the infinite power. Blanton re- the additive semigroup of positive appreciable limited placed this by j in the English translation so as√ to avoid a hyperreals satisfies V.4 but not Archimedes’ lemma. notational clash with the standard symbol for −1.

8 Notices of the AMS Volume 60, Number 7 (where a “is any finite quantity” ibid., §§86, 87) Euler proceeds to present the usual rules of are consonant with a Leibnizian tradition (cf. for- infinitesimal calculus, which go back to Leibniz, mula (17) in the subsection “Lex homogeneorum L’Hôpital, and the Bernoullis, such as transcendentalis”). To explain formulas such as (7), (9) a dxm + b dxn = a dxm Euler elaborated two distinct ways (arithmetic and geometric) of comparing quantities in the following provided m < n “since dxn vanishes compared terms: with dxm” (ibid., §89), relying on his “geometric” equality. Euler introduces a distinction between Since we are going to show that an infinitely infinitesimals of different order and directly small quantity is really zero, we must 13 meet the objection of why we do not computes a ratio of the form always use the same symbol 0 for infinitely dx ± dx2 = 1 ± dx = 1 small quantities, rather than some special dx ones…[S]ince we have two ways to compare of two particular infinitesimals, assigning the them, either arithmetic or geometric, let us value 1 to it (ibid., §88). Euler concludes: look at the quotients of quantities to be Although all of them [infinitely small quan- compared in order to see the difference. tities] are equal to 0, still they must be If we accept the notation used in the carefully distinguished one from the other analysis of the infinite, then dx indicates if we are to pay attention to their mutual a quantity that is infinitely small, so that relationships, which has been explained both dx = 0 and a dx = 0, where a is any through a geometric ratio. (ibid., §89). finite quantity. Despite this, the geomet- The Eulerian hierarchy of orders of infinitesimals ric ratio a dx : dx is finite, namely a : 1. harks back to Leibniz’s work (see the subsection For this reason, these two infinitely small “Nieuwentijt, Bernard” for a historical dissenting quantities, dx and a dx, both being equal view). to 0, cannot be confused when we consider their ratio. In a similar way, we will deal Euler’s Infinite Product Formula for Sine with infinitely small quantities dx and dy. (ibid., §86, pp. 51–52) [emphasis added—the The fruitfulness of Euler’s infinitesimal approach authors] can be illustrated by some of the remarkable applications he obtained. Thus, Euler derived an Euler proceeds to clarify the difference between the infinite product decomposition for the sine and arithmetic and geometric comparisons as follows: sinh functions of the following form: Let a be a finite quantity and let dx be ! ! x2 x2 infinitely small. The arithmetic ratio of (10) sinh x = x 1 + 1 + equals is clear: Since ndx = 0, we have π 2 4π 2 ! ! ± − = x2 x2 a ndx a 0. · 1 + 1 + ,... 9π 2 16π 2 On the other hand, the geometric ratio is ! ! clearly of equals, since x2 x2 (11) sin x = x 1 − 1 − 2 2 a ± ndx π 4π (8) = 1. ! ! a x2 x2 · 1 − 1 − ... From this we obtain the well-known rule that 9π 2 16π 2 the infinitely small vanishes in comparison Decomposition (11) generalizes an infinite product with the finite and hence can be neglected. π formula for 2 due to Wallis [109]. Euler also (Euler 1755 [38, §87]) [emphasis in the 1 1 summed the inverse square series: 1 + 4 + 9 + original—the authors] 2 1 + · · · = π (see [86]) and obtained additional Like Leibniz, Euler considers more than one way of 16 6 comparing quantities. Euler’s formula (8) indicates identities. A common feature of these formulas that his geometric comparison is identical with is that Euler’s computations involve not only the Leibnizian TLH; namely, Euler’s geometric infinitesimals but also infinitely large natural comparision of a pair of quantities amounts to numbers, which Euler sometimes treats as if they their ratio being infinitely close to 1. The same is 13 true for TLH. Thus one has a + dx = a in this sense Note that Euler does not “prove that the expression is equal to 1’;’ such indirect proofs are a trademark of the , δ for an appreciable a 6= 0, but not dx = 0 (which is approach. Rather, Euler directly computes (what would to- true only arithmetically in Euler’s sense). Euler’s day be formalized as the standard part of) the expression, “geometric” comparison was dubbed “the principle illustrating one of the advantages of the B-methodology over of cancellation” in (Ferraro 2004 [39, p. 47]). the A-methodology.

August 2013 Notices of the AMS 9 were ordinary natural numbers.14 Similarly, Euler Euler’s Sine Factorization Formalized treats infinite series as polynomials of a specific Euler’s argument in favor of (10) and (11) was infinite degree. formalized in terms of a “nonstandard” proof in The derivation of (10) and (11) in (Euler 1748 (Luxemburg 1973 [82]). However, the formalization [35, §156]) can be broken up into seven steps as in [82] deviates from Euler’s argument, beginning follows. with Steps 3 and 4, and thus circumvents the more Step 1. Euler observes that problematic Steps 5 and 6. !j !j A proof in the framework of modern non- x x (12) 2 sinh x = ex − e−x = 1 + − 1 − , standard analysis formalizing Euler’s argument j j step-by-step throughout appeared in (Kanovei 1988 where j (or “i” in Euler [35]) is an infinitely large [56]); see also (McKinzie and Tuckey 1997 [86]) natural number. To motivate the next step, note and (Kanovei and Reeken 2004 [58, §2.4a]). This that the expression xj − 1 = (x − 1)(1 + x + x2 + formalization interprets problematic details of j−1 Qj−1 · · · + x ) can be factored further as k=0(x − Euler’s argument on the basis of general principles ζk), where ζ = e2πi/j ; conjugate factors can then of modern nonstandard analysis, as well as general be combined to yield a decomposition into real analytic facts that were known in Euler’s time. Such quadratic terms. principles and facts behind some early proofs in Step 2. Euler uses the fact that aj −bj is the product infinitesimal calculus are sometimes referred to of the factors as “hidden lemmas” in this context; see (Laugwitz [73], [74]), (McKinzie and Tuckey 1997 [86]). For 2 2 2kπ (13) a + b − 2ab cos , where k ≥ 1 , instance, the “hidden lemma” behind Step 4 above j is the fact that for a fixed x, the terms of the together with the factor a − b and, if j is an even Maclaurin expansion of cos x tend to 0 faster than number, the factor a + b as well. a convergent geometric series, allowing one to x x Step 3. Setting a = 1 + j and b = 1 − j in (12), infer that the effect of the transformation of Step Euler transforms expression (13) into the form 4 on the product of the factors (14) is infinitesimal. x2  x2  2kπ Some “hidden lemmas” of a different kind, related (14) 2 + 2 − 2 1 − cos . to basic principles of nonstandard analysis, are j2 j2 j discussed in [86, pp. 43ff.]. Step 4. Euler then replaces (14) by the expression What clearly stands out from Euler’s argument 4k2π 2  x2 x2  is his explicit use of infinitesimal expressions (15) 1 + − , j2 k2π 2 j2 such as (14) and (15), as well as the approximate justifying this step by means of the formula formula (16), which holds “up to” an infinitesimal of higher order. Thus, Euler used infinitesimals 2kπ 2k2π 2 (16) cos = 1 − . par excellence, rather than merely ratios thereof, 2 j j in a routine fashion in some of his best work. Step 5. Next, Euler argues that the difference ex − Euler’s use of infinite integers and their associ- − e x is divisible by the expression ated infinite products (such as the decomposition x2 x2 of the sine function) were interpreted in Robin- 1 + − k2π 2 j2 son’s framework in terms of hyperfinite sets. Thus, 2 Euler’s product of j-infinitely many factors in (11) from (15), where “we omit the term x since even j2 is interpreted as a hyperfinite product in [58, when multiplied by j, it remains infinitely small.” formula (9), p. 74]. A hyperfinite formalization of (English translation from [36].) Euler’s argument involving infinite integers and Step 6. As there is still a factor of a − b = 2x/j, their associated products illustrates the successful Euler obtains the final equality (10), arguing that remodeling of the arguments (and not merely the then “the resulting first term will be x” (in order to results) of classical infinitesimal mathematics, as conform to the Maclaurin series for sinh x). discussed in the subsection “Mathematical Rigor”. Step 7. Finally, formula (11) is obtained from (10) Fermat, Pierre by means of the substitution x , ix.  Pierre de Fermat (1601–1665) developed a pio- We will discuss modern formalizations of Euler’s neering technique known as adequality (see the argument in the next subsection. subsection “Adequality”) for finding tangents to curves and for solving problems of maxima and 14Euler’s procedure is therefore consonant with the Leib- nizian law of continuity (see the subsection “Lex continui- minima. (Katz, Schaps, and Shnider 2013 [66]) tatis”), though apparently Euler does not refer explicitly to analyzes some of the main approaches in the the latter. literature to the method of adequality, as well as

10 Notices of the AMS Volume 60, Number 7 its source in the παρισ oτης´ of Diophantus (see [Y]ou connected this extremely abstract part the subsection “Diophantus”). At least some of of model theory with a theory apparently the manifestations of adequality, such as Fermat’s so far apart as the elementary calculus. In treatment of transcendental curves and Snell’s law, doing so you threw new light on the history amount to variational techniques exploiting a small of the calculus by giving a clear sense to (alternatively, infinitesimal) variation E. Fermat’s Leibniz’s notion of infinitesimals. (ibid) treatment of geometric and physical applications Intuitionist Heyting’s admiration for the appli- suggests that an aspect of approximation is inher- cation of Robinson’s infinitesimals to calculus ent in adequality, as well as an aspect of smallness pedagogy is in stark contrast with the views of his on the part of E. fellow constructivist E. Bishop (subsection “Bishop, Fermat’s use of the term adequality relied on Errett”). Bachet’s rendering of Diophantus. Diophantus coined the term parisotes for mathematical pur- Indivisibles versus Infinitesimals poses. Bachet performed a semantic calque in Commentators use the term infinitesimal to refer passing from par-iso¯o to ad-aequo. A historically to a variety of conceptions of the infinitely small, significant parallel is found in the similar role but the variety is not always acknowledged. It is of, respectively, adequality and the transcenden- important to distinguish the infinitesimal methods tal law of homogeneity (see the subsection “Lex of Archimedes and Cavalieri from those employed homogeneorum transcendentalis”) in the work of, by Leibniz and his successors. To emphasize this respectively, Fermat and Leibniz on the problems distinction, we will say that tradition prior to of maxima and minima. Leibniz employed indivisibles. For example, in his Breger (1994 [21]) denies that the idea of heuristic proof that the area of a parabolic segment “smallness” was relied upon by Fermat. However, a is 4/3 the area of the inscribed triangle with the detailed analysis (see [66]) of Fermat’s treatment of same base and vertex, Archimedes imagines both the cycloid reveals that Fermat did rely on issues figures to consist of perpendiculars of various of “smallness” in his treatment of the cycloid heights erected on the base. The perpendiculars and reveals that Breger’s interpretation thereof are indivisibles in the sense that they are limits of contains both mathematical errors and errors of division and so one dimension less than the area. textual analysis. Similarly, Fermat’s proof of Snell’s In the same sense, the indivisibles of which a line law, a variational principle, unmistakably relies on consists are points, and the indivisibles of which a ideas of “smallness”. solid consists are planes. Cifoletti (1990 [25]) finds similarities between Leibniz’s infinitesimals are not indivisibles, for Fermat’s adequality and some procedures used they have the same dimension as the figures that in smooth infinitesimal analysis of Lawvere and compose them. Thus, he treats curves as comprising others. Meanwhile, (J. Bell 2009 [9]) seeks the his- infinitesimal line intervals rather than indivisible torical sources of Lawvere’s infinitesimals mainly points. The strategy of treating infinitesimals as in Nieuwentijt (see the subsection “Nieuwentijt, dimensionally homogeneous with the objects they Bernard”). compose seems to have originated with Roberval Heyting, Arend or Torricelli, Cavalieri’s student, and to have been explicitly arithmetized in (Wallis 1656 [109]). Arend Heyting (1898–1980) was a mathematical Zeno’s paradox of extension admits resolution intuitionist whose lasting contribution was the in the framework of Leibnizian infinitesimals formalization of the intuitionistic logic underpin- (see the subsection “Zeno’s Paradox of Extension”). ning the Intuitionism of his teacher Brouwer. While Furthermore, only with the dimensionality retained Heyting never worked on any theory of infinites- is it possible to make sense of the fundamental imals, he had several opportunities to present theorem of calculus, where one must think about an expert opinion on Robinson’s theory. Thus, the rate of change of the area under a curve, another in 1961, Robinson made public his new idea of reason why indivisibles had to be abandoned nonstandard models for analysis and “communi- in favor of infinitesimals so as to enable the cated this almost immediately to…Heyting” (see development of the calculus (see Ely 2012 [33]). Dauben [28, p. 259]). Robinson’s first paper on the subject was subsequently published in Proceedings Leibniz to Nieuwentijt of the Netherlands Royal Academy of Sciences [94]. Heyting praised nonstandard analysis as “a stan- Leibniz, Gottfried dard model of important mathematical research” Gottfried Wilhelm Leibniz (1646–1716), the co- (Heyting 1973 [50, p. 136]). Addressing Robinson, inventor of infinitesimal calculus, is a key player he declared: in the parallel infinitesimal track referred to by

August 2013 Notices of the AMS 11 Felix Klein [72, p. 214] (see the section “The ABC’s place, Leibniz wrote the treatise at a time when of the History of Infinitesimal Mathematics”). infinitesimals were despised by the French Acad- Leibniz’s law of continuity (see the subsection emy, a society whose approval and acceptance “Lex continuitatis”) and his transcendental law of he eagerly sought. More importantly, as (Jesseph homogeneity (which he had already discussed in 2013 [55]) has shown, De Quadratura depends on his response to Nieuwentijt in 1695, as noted by infinitesimal resources in order to construct an M. Parmentier [79, p. 38], and later in greater detail approximation to a given curvilinear area less than in a 1710 article [78] cited in the seminal study any previously specified error. This problem is of Leibnizian methodology by H. Bos [17]) form a reminiscent of the difficulty that led to infinitesimal basis for implementing the calculus in the context methods in the first place. Archimedes’ method of of a B-continuum. exhaustion required one to determine a value for Many historians of the calculus deny significant the quadrature in advance of showing, by reductio continuity between infinitesimal calculus of the argument, that any departure from that value seventeenth century and twentieth-century devel- entails a contradiction. Archimedes possessed opments such as Robinson’s theory (see further a heuristic, indivisible method for finding such discussion in Katz and Sherry [67]). Robinson’s hy- values, and the results were justified by exhaustion, perreals require the resources of modern logic; thus but only after the fact. By the same token, the many commentators are comfortable denying a his- use of infinitesimals is “just” a shortcut only if it torical continuity. A notable exception is Robinson is entirely eliminable from quadratures, tangent himself, whose identification with the Leibnizian constructions, etc. Jesseph’s insight is that this is tradition inspired Lakatos, Laugwitz, and others not the case. to consider the history of the infinitesimal in a Finally, the syncategorematic interpretation mis- more favorable light. Many historians have over- represents a crucial aspect of Leibniz’s mathemat- estimated the force of Berkeley’s criticisms (see the ical philosophy. His conception of mathematical subsection “Berkeley, George”) by underestimating fiction includes imaginary numbers, and he often the mathematical and philosophical resources sought approbation for his infinitesimals by com- available to Leibniz. paring them to imaginaries, which were largely Leibniz’s infinitesimals are fictions—not logical uncontroversial. There is no suggestion by Leibniz fictions, as (Ishiguro 1990 [54]) proposed, but that imaginaries are eliminable by long-winded rather pure fictions, like imaginaries, which are not paraphrase. Rather, he praises imaginaries for eliminable by some syncategorematic paraphrase; their capacity to achieve universal harmony by the see (Sherry and Katz 2012 [100]) and the subsection greatest possible systematization, and this char- “Leibniz’s De Quadratura” below. acteristic is more central to Leibniz’s conception In fact, Leibniz’s defense of infinitesimals is more of infinitesimals than the idea that they are mere firmly grounded than Berkeley’s criticism thereof. shorthand. Just as imaginary roots both unified and Moreover, Leibniz’s system for differential calculus extended the method for solving cubics, likewise was free of logical fallacies (see the subsection infinitesimals unified and extended the method “Berkeley’s Logical Criticism”). This strengthens the for quadrature so that, e.g., quadratures of general conception of modern infinitesimals as a formaliza- parabolas and hyperbolas could be found by the tion of Leibniz’s strategy of relating inassignable same method used for quadratures of less difficult to assignable quantities by means of his transcen- curves. dental law of homogeneity (see the subsection “Lex homogeneorum transcendentalis”). Lex continuitatis Leibniz’s De Quadratura A heuristic principle called The law of continuity In 1675 Leibniz wrote a treatise on his infinitesi- (LC) was formulated by Leibniz and is a key mal methods, On the Arithmetical Quadrature of to appreciating Leibniz’s vision of infinitesimal the Circle, the Ellipse, and the Hyperbola, or De calculus. The LC asserts that whatever succeeds in Quadratura, as it is widely known. However, the the finite succeeds also in the infinite. This form treatise appeared in print only in 1993 in a text of the principle appeared in a letter to Varignon edited by Knobloch (Leibniz [80]). (Leibniz 1702 [77]). A more detailed form of LC in De Quadratura was interpreted by R. Arthur terms of the concept of terminus appeared in his [3] and others as supporting the thesis that Leib- text Cum Prodiisset: niz’s infinitesimals are mere fictions, eliminable In any supposed continuous transition, by long-winded paraphrase. This so-called syn- ending in any terminus, it is permissible categorematic interpretation of Leibniz’s calculus to institute a general reasoning, in which has gained a number of adherents. We believe the final terminus may also be included. this interpretation to be a mistake. In the first (Leibniz 1701 [76, p. 40])

12 Notices of the AMS Volume 60, Number 7 ( ) ( ) ( ) assignable LC inassignable TLH assignable The TLH associates to an inassignable quantity Ž Ž quantities quantities quantities (such as a + dx) an assignable one (such as a); see Figure 2 for a relation between LC and TLH. Figure 2. Leibniz’s law of continuity (LC) takes one from assignable to inassignable quantities, Mathematical Rigor while his transcendental law of homogeneity There is a certain lack of clarity in the historical (TLH; the subsection “Lex homogeneorum literature with regard to issues of fruitfulness, transcendentalis”) returns one to assignable consistency, and rigorousness of mathematical quantities. writing. As a rough guide and to be able to formulate useful distinctions when it comes to evaluating mathematical writing from centuries To elaborate, the LC postulates that whatever past, we would like to consider three levels of properties are satisfied by ordinary or assignable judging mathematical writing: quantities should also be satisfied by inassignable (1) potentially fruitful but (logically) inconsis- quantities (see the subsection “Variable Quantity’) tent, such as infinitesimals (see Figure 2). Thus, the (2) (potentially) consistent but informal, trigonometric formula sin2 x+cos2 x = 1 should be (3) formally consistent and fully rigorous ac- satisfied for an inassignable (e.g., infinitesimal) in- cording to currently prevailing standards. put x as well. In the twentieth century this heuristic As an example of level (1) we would cite the work principle was formalized as the transfer principle of Nieuwentijt (see the subsection “Nieuwentijt, (see the subsection “Modern Implementations”) of Bernard” for a discussion of the inconsistency). Ło´s–Robinson. Our prime example of level (2) is provided by the The significance of LC can be illustated by the fact Leibnizian laws of continuity and homogeneity (see that a failure to take note of the law of continuity subsections “Lex continuitatis” and “Lex homoge- often led scholars astray. Thus, Nieuwentijt (see neorum transcendentalis”), which found rigorous the subsection “Nieuwentijt, Bernard”) was led implementation at level (3) only centuries later into something of a dead end with his nilpotent (see the subsection “Modern Implementations”). 1 A foundational rock of the received history of infinitesimals (ruled out by LC) of the form ∞ . J. Bell’s view of Nieuwentijt’s approach as a mathematical analysis is the belief that mathemat- precursor of nilsquare infinitesimals of Lawvere ical rigor emerged starting in the 1870s through (see Bell 2009 [9]) is plausible, though it could the efforts of Cantor, Dedekind, Weierstrass, and be noted that Lawvere’s nilsquare infinitesimals others, thereby replacing formerly unrigorous 1 work of infinitesimalists from Leibniz onward. The cannot be of the form . ∞ philosophical underpinnings of such a belief were analyzed in (Katz and Katz 2012a [64]), where it Lex homogeneorum transcendentalis was pointed out that in mathematics, as in other Leibniz’s transcendental law of homogeneity, or sciences, former errors are eliminated through a lex homogeneorum transcendentalis in the origi- process of improved conceptual understanding, nal Latin (Leibniz 1710 [78]), governs equations evolving over time, of the key issues involved in involving differentials. Leibniz historian H. Bos that science. interprets it as follows: Thus, no scientific development can be claimed A quantity which is infinitely small with to have attained perfect clarity or rigor merely on respect to another quantity can be neglected the grounds of having eliminated earlier errors. if compared with that quantity. Thus all Moreover, no such claim for a single scientific terms in an equation except those of the development is made either by the practitioners highest order of infinity, or the lowest order or by the historians of the natural sciences. It was of infinite smallness, can be discarded. For further pointed out in [64] that the term mathemat- instance, ical rigor itself is ambiguous, its meaning varying according to context. Four possible meanings for (17) a + dx = a the term were proposed in [64]: dx + ddy = dx (1) it is a shibboleth that identifies the speaker as belonging to a clan of professional etc. The resulting equations satisfy this…re- mathematicians; quirement of homogeneity. (Bos 1974 [17, (2) it represents the idea that as a scientific field p. 33]) develops, its practitioners attain greater For an interpretation of the equality sign in the and more conceptual understanding of key formulas above, see the subsection “Relation [\ ”. issues and are less prone to error;

August 2013 Notices of the AMS 13 (3) it represents the idea that a search for by remodeling both its propositions and its pro- greater correctness in analysis inevitably cedures and reasoning. Using Weierstrassian , δ led Weierstrass specifically to epsilontics techniques, one can recover only the propositions (i.e., the A-approach) in the 1870s; but not the proof procedures. Thus, Euler’s result (4) it refers to the establishment of what are giving an infinite product formula for sine (see the perceived to be the ultimate foundations subsection “Euler’s Infinite Product Formula for for mathematics by Cantor, eventually Sine”) admits numerous proofs in a Weierstrassian explicitly expressed in axiomatic form by context, but Robinson’s framework provides a Zermelo and Fraenkel. suitable context in which Euler’s proof, relying on Item (1) may be pursued by a fashionable infinite integers, can also be recovered. This is the academic in the social sciences but does not get to crux of the historical debate concerning , δ versus infinitesimals. In short, Robinson did for Leibniz the bottom of the issue. Meanwhile, item (2) would what Hilbert did for Euclid. Meanwhile, epsilontists be agreed upon by historians of the other sciences. failed to do for Leibniz what Robinson did for In this context it is interesting to compare Leibniz, namely, formalizing the procedures and the investigation of the Archimedean property as reasoning of the historical infinitesimal calculus. performed by the would-be rigorist Cantor, on the This theme is pursued further in terms of the one hand, and the infinitesimalist Stolz on the internal/external distinction in the subsection other. Cantor sought to derive the Archimedean “Variable Quantity”. property as a consequence of those of a linear continuum. Cantor’s work in this area was not only Modern Implementations unrigorous but actually erroneous, whereas Stolz’s work was fully rigorous and even visionary. Namely, In the 1940s Hewitt [48] developed a modern imple- Cantor’s arguments “proving” the inconsistency of mentation of an infinitesimal-enriched continuum infinitesimals were based on an implicit assumption extending R by means of a technique referred to today as the ultrapower construction. We will of what is known today as the Kerry-Cantor axiom denote such an infinitesimal-enriched continuum (see Proietti 2008 [92]). Meanwhile, Stolz was by the new symbol IR (“thick-R”).16 In the next the first modern mathematician to realize the decade, Ło´s(Ło´s1955 [81]) proved his celebrated importance of the Archimedean axiom (see the theorem on ultraproducts, implying in particular subsection “Archimedean Axiom”) as a separate that elementary (more generally, first-order) state- axiom in its own right (see Ehrlich 2006 [32]) ments over R are true if and only if they are true and, moreover, to develop some non-Archimedean over IR, yielding a modern implementation of the systems (Stolz 1885 [106]). Leibnizian law of continuity (see the subsection In his Grundlagen der Geometrie (Hilbert 1899 “Lex continuitatis”). Such a result is equivalent to [51]), Hilbert did not develop a new geometry, what is known in the literature as the transfer but rather remodeled Euclid’s geometry. More principle; see Keisler [70]. Every finite element of IR specifically, Hilbert brought rigor into Euclid’s is infinitely close to a unique real number; see the geometry in the sense of formalizing both Euclid’s subsection “Standard Part Principle”. Such a princi- propositions and Euclid’s style of procedures and ple is a mathematical implementation of Fermat’s style of reasoning. adequality (see the subsection “Adequality”), of Note that Hilbert’s system works for geometries Leibniz’s transcendental law of homogeneity (see built over a non-Archimedean field, as Hilbert the subsection “Lex homogeneorum transcenden- was fully aware. Hilbert (1900 [52, p. 207]) cites talis”), and of Euler’s principle of cancellation (see Dehn’s counterexamples to Legendre’s theorem the discussion between formulas (7) and (9) in the in the absence of the Archimedean axiom. Dehn subsection “Euler, Leonhard”). planes built over a non-Archimedean field were used to prove certain cases of the independence Nieuwentijt, Bernard of Hilbert’s axioms (see Cerroni 2007 [24]).15 In Nieuwentijt’s Analysis Infinitorum (1695), the Robinson’s theory similarly formalized 17 Dutch philosopher (1654–1718). proposed a seventeenth- and eighteenth-century analysis system containing an infinite number, as well

15 as infinitesimal quantities formed by dividing It is a melancholy comment to note that fully three years finite numbers by this infinite one. Nieuwentijt later the philosopher-mathematician Bertrand Russell was postulated that the product of two infinitesimals still claiming, on Cantor’s authority, that the infinitesimal “leads to contradictions” (Russell 2003 [97, p. 345]). This should be exactly equal to zero. In particular, an set the stage for several decades of anti-infinitesimal vitriol, infinitesimal quantity is nilpotent. In an exchange of including the saline solution of Parkhurst and Kingsland 16 ∗ ∗ (see the section “The ABC’s of the History of Infinitesimal A more traditional symbol is R or R . Mathematics”). 17Alternative spellings are Nieuwentijdt or Nieuwentyt.

14 Notices of the AMS Volume 60, Number 7 publications with Nieuwentijt on infinitesimals (see B-continuum Mancosu 1996 [84, p. 161]), Leibniz and Hermann st claimed that this system is consistent only if all infinitesimals are equal, rendering differential  A-continuum calculus useless. Leibniz instead advocated a Figure 3. Thick-to-thin: applying the law of system in which the product of two infinitesimals homogeneity or taking standard part (the is incomparably smaller than either infinitesimal. thickness of the top line is merely conventional). Nieuwentijt’s objections compelled Leibniz in 1696 to elaborate on the hierarchy of infinite and infinitesimal numbers entailed in a robust of the product rule. The issue is the justification infinitesimal system. of the last step in the following calculation: Nieuwentijt’s nilpotent infinitesimals of the 1 d(uv) = (u + du)(v + dv) − uv form ∞ are ruled out by Leibniz’s law of continuity (see the subsection “Lex continuitatis”). J. Bell’s (18) = udv + vdu + du dv view of Nieuwentijt’s approach as a precursor of = udv + vdu. nilsquare infinitesimals of Lawvere (see Bell 2009 The last step in the calculation (18), namely, [9]) is plausible, though it could be noted that Lawvere’s nilsquare infinitesimals cannot be of the udv + vdu + du dv = udv + vdu, 1 19 form ∞ . is an application of the TLH. In his 1701 text Cum Prodiisset [76, pp. 46–47], Product Rule to Zeno Leibniz presents an alternative justification of the Product Rule product rule (see Bos [17, p. 58]). Here he divides by dx and argues with differential quotients rather In the area of Leibniz scholarship, the received than differentials. Adjusting Leibniz’s notation to view is that Leibniz’s infinitesimal system was logi- fit with (18), we obtain an equivalent calculation:20 cally faulty and contained internal contradictions d(uv) (u + du)(v + dv) − uv allegedly exposed by the cleric George Berkeley = (see the subsection “Berkeley, George”). Such a view dx dx udv + vdu + du dv is fully compatible with the A-track-dominated = outlook, bestowing supremacy upon the recon- dx udv + vdu du dv struction of analysis accomplished through the = + efforts of Cantor, Dedekind, Weierstrass, and their dx dx udv + vdu rigorous followers (see the subsection “Mathemati- = . cal Rigor”). Does such a view represent an accurate dx   appraisal of Leibniz’s system? du dv Under suitable conditions the term dx is The articles (Katz and Sherry 2012 [67], [68]; infinitesimal, and therefore the last step Sherry and Katz [100]), building on the earlier work udv + vdu du dv dv du (Sherry 1987 [98]), argued that Leibniz’s system (19) + = u + v dx dx dx dx was in fact consistent (in the sense of level (2) of the is legitimized as a special case of the TLH. The TLH subsection “Mathematical Rigor”)18 and featured interprets the equality sign in (19) and (17) as the resilient heuristic principles such as the law of relation of being infinitely close, i.e., an equality continuity (see the subsection “Lex continuitatis”) up to infinitesimal error. and the transcendental law of homogeneity (TLH) (see the subsection “Lex homogeneorum transcen- Relation [\ dentalis”), which were implemented in the fullness Leibniz did not use our equality symbol but rather of time as precise mathematical principles guiding the symbol “ ” (see McClenon 1923 [85, p. 371]). the behavior of modern infinitesimals. [\ Using such a symbol to denote the relation of being How did Leibniz exploit the TLH in developing infinitely close, one could write the calculation the calculus? We will now illustrate an application of the TLH in the particular example of the derivation 19Leibniz had two laws of homogeneity: one for dimen- sion and the other for the order of infinitesimalness. Bos 18Concerning the status of Leibniz’s system for differential notes that they “disappeared from later developments” [17, calculus, it may be more accurate to assert that it was not p. 35], referring to Euler and Lagrange. Note, however, the inconsistent, in the sense that the contradictions alleged similarity to Euler’s principle of cancellation (see Bair et by Berkeley and others turn out not to have been there in al. [5]). the first place once one takes into account Leibniz’s gen- 20The special case treated by Leibniz is u(x) = x. This eralized notion of equality and his transcendental law of limitation does not affect the conceptual structure of the homogeneity. argument.

August 2013 Notices of the AMS 15 enlargement of the number system, as the standard part of the value uH of the natural extension of the sequence at an infinite hypernatural index n = H. Thus,

(20) lim un = st(uH ). n→∞ Here the standard part function “st” associates to Figure 4. Zooming in on infinitesimal ε (here each finite hyperreal the unique finite real infinitely st(±ε) = 0). The standard part function close to it (i.e., the difference between them is associates to every finite hyperreal the unique infinitesimal). This formalizes the natural intuition real number infinitely close to it. The bottom that for “very large” values of the index, the terms line represents the “thin” real continuum. The in the sequence are “very close” to the limit value line at top represents the “thick” hyperreal of the sequence. Conversely, the standard part of a continuum. The “infinitesimal microscope” is hyperreal u = [un] represented in the ultrapower used to view an infinitesimal neighborhood of 0. construction by a Cauchy sequence (un) is simply The derivative f 0(x) of f (x) is then defined by the limit of that sequence: 0 f (x+ε)−f (x) the relation f (x) [\ ε . (21) st(u) = lim un. n→∞ Formulas (20) and (21) express limit and standard of the derivative of y = f (x) where f (x) = x2 as part in terms of each other. In this sense, the follows: procedures of taking the limit and taking the dy standard part are logically equivalent. f 0(x) [\ dx (x + dx)2 − x2 Variable Quantity = dx The mathematical term µεγεϑoς´ in ancient Greek (x + dx + x)(x + dx − x) has been translated into Latin as quantitas. In = dx modern languages it has two competing counter- = 2x + dx parts: in English, quantity, magnitude;22 in French, [\ 2x. quantité, grandeur; in German, Quantität, Grösse. The term grandeur with the meaning real number Such a relation is formalized by the standard is still in use in (Bourbaki 1947 [19]). Variable part function; see the subsection “Standard Part quantity was a primitive notion in analysis as Principle” and Figure 3. presented by Leibniz, l’Hôpital, and later Carnot and Cauchy. Other key notions of analysis were Standard Part Principle defined in terms of variable quantities. Thus, in In any totally ordered field extension E of R, every Cauchy’s terminology a variable quantity becomes finite element x ∈ E is infinitely close to a suitable an infinitesimal if it eventually drops below any unique element x0 ∈ R. Indeed, via the total order, given (i.e., constant) quantity (see Borovik and Katz the element x defines a Dedekind cut on R, and the [16] for a fuller discussion). Cauchy notes that the cut specifies a real number x0 ∈ R ⊂ F. The number limit of such a quantity is zero. The notion of limit x0 is infinitely close to x ∈ E. The subring Ef ⊂ E itself is defined as follows: consisting of the finite elements of E therefore Lorsque les valeurs successivement at- admits a map tribuées à une même variable s’approchent st : Ef → R, x , x0, indéfiniment d’une valeur fixe, de manière called the standard part function. à finir par en différer aussi peu que l’on The standard part function is illustrated in voudra, cette dernière est appelée la lim- Figure 3. A more detailed graphic representation ite de toutes les autres. (Cauchy, Cours may be found in Figure 4.21 d’Analyse [22]) The key remark, due to Robinson, is that the limit Thus, Cauchy defined both infinitesimals and in the A-approach and the standard part function limits in terms of the primitive notion of a in the B-approach are essentially equivalent tools. variable quantity. In Cauchy, any variable quantity q More specifically, the limit of a sequence (un) that does not tend to infinity is expected to can be expressed, in the context of a hyperreal 22The term “magnitude” is etymologically related to 21For a recent study of optical diagrams in nonstandard µεγεϑoς´ . Thus, µεγεϑoς´ in Greek and magnitudo in Latin analysis, see (Dossena and Magnani [31], [83]) and (Bair and both mean “bigness”, “big” being mega (µεγα´ ) in Greek Henry [8]). and magnum in Latin.

16 Notices of the AMS Volume 60, Number 7 decompose as the sum of a given quantity c and magnitude as though it consists of infinitely many an infinitesimal α: indivisibles; see (Sherry 1988 [99]), (Kirk et al. 1983 [71]). If the indivisibles have no magnitude, then (22) q = c + α. an extension (such as space or time) composed In his 1821 text [22], Cauchy worked with a of them has no magnitude; but if the indivisibles hierarchy of infinitesimals defined by polynomials have some (finite) magnitude, then an extension in a base infinitesimal α. Each such infinitesimal composed of them will be infinite. There is a further decomposes as puzzle: If a magnitude is composed of indivisibles, (23) αn(c + ε) then we ought to be able to add or concatenate them in order to produce or increase a magnitude. for a suitable integer n and infinitesimal ε. Cauchy’s But indivisibles are not next to one another: as expression (23) can be viewed as a generalization limits or boundaries, any pair of indivisibles is of (22). separated by what they limit. Thus, the concept of In Leibniz’s terminology, c is an assigna- addition or concatenation seems not to apply to ble quantity, while α and ε are inassignable. indivisibles. Leibniz’s transcendental law of homogeneity (see The paradox need not apply to infinitesimals the subsection “Lex homogeneorum transcen- in Leibniz’s sense however (see the subsection dentalis”) authorized the replacement of the “Indivisibles versus Infinitesimals”) for having inassignable q = c + α by the assignable c, since α neither zero nor finite magnitude, infinitely many is negligible compared to c: of them may be just what is needed to produce a finite magnitude. And in any case, the addition (24) q [\ c or concatenation of infinitesimals (of the same (see the subsection “Relation [\ ”). Leibniz empha- dimension) is no more difficult to conceive of than sized that he worked with a generalized notion of adding or concatenating finite magnitudes. This equality where expressions were declared “equal” is especially important, because it allows one to if they differed by a negligible term. Leibniz’s pro- apply arithmetic operations to infinitesimals (see cedure was formalized in Robinson’s B-approach the subsection “Lex continuitatis” on the law of by the standard part function (see the subsection continuity). See also (Reeder 2012 [93]). “Standard Part Principle”), which assigns to each finite hyperreal number the unique real number to Acknowledgments which it is infinitely close. As such, the standard M. Katz was partially funded by the Israel Science part allows one to work “internally” (not in the Foundation, grant No. 1517/12. We are grateful technical NSA sense but) in the sense of exploiting to Reuben Hersh and Martin Davis for helpful concepts already available in the toolkit of the discussions and to the anonymous referees for a historical infinitesimal calculus, such as Fermat’s number of helpful suggestions. The influence of adequality (see the subsection “Adequality”), Leib- Hilton Kramer (1928–2012) is obvious. niz’s transcendental law of homogeneity (see the subsection “Lex homogeneorum transcendentalis”), References and Euler’s principle of cancellation (see Bair et [1] K. Andersen, One of Berkeley’s arguments on al. [5]). Meanwhile, in the A-approach as formalized compensating errors in the calculus, Historia by Weierstrass, one is forced to work with “exter- Mathematica 38 (2011), no. 2, 219–231. nal” concepts such as the multiple-quantifier , δ [2] Archimedes, De Sphaera et Cylindro, in Archimedis Opera Omnia cum Commentariis Eutocii, vol. 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20 Notices of the AMS Volume 60, Number 7