Is Mathematical History Written by the Victors? - American

Is Mathematical HistoryWritten 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 InfinitesimalMathematicsThe ABCs of the history of infinitesimal mathemat-ics are in need of clarification. To what extent doesthe famous dictum “history is always written bythe victors” apply to the history of mathematics aswell? A convenient starting point is a remark madeby Felix Klein in his book Elementary Mathematicsfrom an Advanced Standpoint (Klein [72, p. 214]).Klein wrote that there are not one but two separatetracks for the development of analysis:

(A) the Weierstrassian approach (in the contextof an Archimedean continuum) and

(B) the approach with indivisibles and/or in-finitesimals (in the context of what we willrefer to as a Bernoullian continuum).1

Jacques Bair is professor of mathematics at the Universityof Liege, Belgium. His email address is [email protected].

Piotr Błaszczyk is professor of mathematics at the Peda-gogical University of Cracow, Poland. His email address [email protected].

Robert Ely is professor of mathematics at the University ofIdaho. His email address is [email protected].

Valérie Henry is professor of mathematics, University ofNamur and of Liege, Belgium. Her email address is [email protected].

Vladimir Kanovei is professor of mathematics at IPPI,Moscow, and MIIT, Moscow. His email address is [email protected].

Karin U. Katz teaches mathematics at Bar Ilan University,Israel. Her email address is [email protected] of quantities encompassing infinitesimal ones wereused by Leibniz, Bernoulli, Euler, and others. Our choice ofthe term is explained in the subsection “Bernoulli, Johann”.It encompasses modern non-Archimedean systems.

DOI: http://dx.doi.org/10.1090/noti1001

Klein’s sentiment was echoed by the philosopherG. Granger, in the context of a discussion of Leibniz,in the following terms:

Aux yeux des détracteurs de la nouvelleAnalyse, l’insurmontable difficulté vient dece que de telles pratiques font violenceaux règles ordinaires de l’Algèbre, tout enconduisant à des résultats, exprimables entermes finis, dont on ne saurait contesterl’exactitude. Nous savons aujourd’hui quedeux voies devaient s’offrir pour la solutiondu problème:

[A] Ou bien l’on élimine du langagemathématique le terme d’infiniment petit,et l’on établit, en termes finis, le sens àdonner à la notion intuitive de ‘valeur limite’.…

[B] Ou bien l’on accepte de maintenir,tout au long du Calcul, la présence d’objetsportant ouvertement la marque de l’infini,mais en leur conférant un statut propre qui

Mikhail G. Katz is professor of mathematics at Bar Ilan Uni-versity, Israel. His email address is [email protected].

Semen S. Kutateladze is professor of mathematics at theSobolev Institute of Mathematics, Novosibirsk State Univer-sity, Russia. His email address is [email protected].

Thomas McGaffey teaches mathematics at Rice University.His email address is [email protected].

David M. Schaps is professor of classical studies at BarIlan University, Israel. His email address is [email protected].

David Sherry is professor of philosophy at Northern ArizonaUniversity. His email address is [email protected].

Steven Shnider is professor of mathematics at Bar Ilan Uni-versity, Israel. His email address is [email protected].

886 Notices of the AMS Volume 60, Number 7

les insère dans un système dont font aussipartie les grandeurs finies.…

C’est dans cette seconde voie que lesvues philosophiques de Leibniz l’ont orienté.(Granger 1981 [43, pp. 27–28])2

Thus we have two parallel tracks for con-ceptualizing infinitesimal calculus, as shown inFigure 1.

B-continuum

A-continuum

Figure 1. Parallel tracks: a thick continuum and athin continuum.

At variance with Granger’s appraisal, some ofthe literature on the history of mathematics tendsto assume that the A-approach is the ineluctably“true” one, while the infinitesimal B-approach was,at best, a kind of evolutionary dead end or, at worst,altogether inconsistent. To say that infinitesimalsprovoked passions would be an understatement.Parkhurst and Kingsland, writing in The Monist,proposed applying a saline solution (if we may beallowed a pun) to the problem of the infinitesimal:

[S]ince these two words [infinity and infini-tesimal] have sown nearly as much faultylogic in the fields of mathematics and meta-physics as all other fields put together,they should be rooted out of both the fieldswhich they have contaminated. And not onlyshould they be rooted out, lest more errorsbe propagated by them: a due amount of saltshould be ploughed under the infected ter-ritory, that the damage be mitigated as wellas arrested. (Parkhurst and Kingsland 1925[91, pp. 633–634]) [emphasis added—theauthors]

Writes P. Vickers:

So entrenched is the understanding thatthe early calculus was inconsistent thatmany authors don’t provide a reference tosupport the claim, and don’t present theset of inconsistent propositions they havein mind. (Vickers 2013 [108, section 6.1, p.146])

Such an assumption of inconsistency can influ-ence one’s appreciation of historical mathematics,make a scholar myopic to certain significant devel-opments due to their automatic placement in an“evolutionary dead-end” track, and inhibit potentialfruitful applications in numerous fields ranging

2Similar views were expressed by M. Parmentier in (Leibniz1989 [79, p. 36, note 92]).

from physics to economics (see Herzberg 2013[110]). One example is the visionary work of En-riques exploiting infinitesimals, recently analyzedin an article by David Mumford, who wrote:

In my own education, I had assumed thatEnriques [and the Italians] were irrevocablystuck.…As I see it now, Enriques must becredited with a nearly complete geometricproof using, as did Grothendieck, higherorder infinitesimal deformations.…Let’s becareful: he certainly had the correct ideasabout infinitesimal geometry, though hehad no idea at all how to make precisedefinitions. (Mumford 2011 [89])

Another example is important work by Cauchy(see the subsection “Cauchy, Augustin-Louis” be-low) on singular integrals and Fourier series usinginfinitesimals and infinitesimally defined “Dirac”delta functions (these precede Dirac by a century),which was forgotten for a number of decades be-cause of shifting foundational biases. The presenceof Dirac delta functions in Cauchy’s oeuvre wasnoted in (Freudenthal 1971 [40]) and analyzed byLaugwitz (1989 [74]), (1992a [75]); see also (Katzand Tall 2012 [69]) and (Tall and Katz 2013 [107]).

Recent papers on Leibniz (Katz and Sherry [67],[68]; Sherry and Katz [100]) argue that, contraryto widespread perceptions, Leibniz’s system forinfinitesimal calculus was not inconsistent (see thesubsection “Mathematical Rigor” for a discussionof the term). The significance and coherence ofBerkeley’s critique of infinitesimal calculus havebeen routinely exaggerated. Berkeley’s sarcastictirades against infinitesimals fit well with theontological limitations imposed by the A-approachfavored by many historians, even though Berkeley’sopposition, on empiricist grounds, to an infinitelydivisible continuum is profoundly at odds with theA-approach.

A recent study of Fermat (Katz, Schaps, andShnider 2013 [66]) shows how the nature of hiscontribution to the calculus was distorted inrecent Fermat scholarship, similarly due to an“evolutionary dead-end” bias (see the subsection“Fermat, Pierre de”).

The Marburg school of Hermann Cohen, Cassirer,Natorp, and others explored the philosophical foun-dations of the infinitesimal method underpinningthe mathematized natural sciences. Their versatile,and insufficiently known, contribution is analyzedin (Mormann and Katz 2013 [88]).

A number of recent articles have pioneereda reevaluation of the history and philosophyof mathematics, analyzing the shortcomings ofreceived views, and shedding new light on thedeleterious effect of the latter on the philosophy,the practice, and the applications of mathematics.

August 2013 Notices of the AMS 887

Some of the conclusions of such a reevaluation arepresented below.

Adequality to ChimerasSome topics from the history of infinitesimals illus-trating our approach appear below in alphabeticalorder.

Adequality

Adequality is a technique used by Fermat to solveproblems of tangents, problems of maxima andminima, and other variational problems. The termadequality derives from the παρισoτης of Dio-phantus (see the subsection “Diophantus”). Thetechnique involves an element of approximationand “smallness”, represented by a small varia-tion E, as in the familiar difference f (A+E)− f (A).Fermat used adequality in particular to find thetangents of transcendental curves such as thecycloid that were considered to be “mechanical”curves off-limits to geometry by Descartes. Fermatalso used it to solve the variational problem ofthe refraction of light so as to obtain Snell’s law.Adequality incorporated a procedure of discardinghigher-order terms in E (without setting themequal to zero). Such a heuristic procedure wasultimately formalized mathematically in terms ofthe standard part principle (see the subsection“Standard Part Principle”) in Robinson’s theory ofinfinitesimals starting with (Robinson 1961 [94]).Fermat’s adequality is comparable to Leibniz’s tran-scendental law of homogeneity (see the subsection“Lex homogeneorum transcendentalis”).

Archimedean Axiom

What is known today as the Archimedean axiomfirst appears in Euclid’s Elements, Book V, asDefinition 4 (Euclid [34, Definition V.4]). It isexploited in (Euclid [34, Proposition V.8]). Weinclude bracketed symbolic notation so as toclarify the definition:

Magnitudes [ a, b ] are said to have a ratiowith respect to one another which, beingmultiplied [ na ] are capable of exceedingone another [ na > b ].

It can be formalized as follows:3

(1)(∀a, b)(∃n ∈ N) [na > b], where na=a+ · · · + a︸︷︷︸

n−times

.

3See, e.g., the version of the Archimedean axiom in (Hilbert1899 [51, p. 19]). Note that we have avoided using “0” informula (1), as in “∀a > 0”, since 0 was not part of the con-ceptual framework of the Greeks. The term “multiplied” inthe English translation of Euclid’s definition V.4 correspondsto the Greek term πoλλαπλασιαζoµενα. A commonformalization of the noun “multiple”, πoλλαπλασιoν,is na = a+ · · · + a.

Next, it appears in the papers of Archimedes as thefollowing lemma (see Archimedes [2, I, Lamb. 5]):

Of unequal lines, unequal surfaces, and un-equal solids [a, b, c ], the greater exceeds thelesser [ a < b ] by such a magnitude [b− a]as, when added to itself [ n(b − a) ], canbe made to exceed any assigned magnitude[c] among those which are comparable withone another. (Heath 1897 [47, p. 4])

This can be formalized as follows:

(2) (∀a, b, c)(∃n ∈ N) [a < b → n(b − a) > c].Note that Euclid’s definition V.4 and the lemma ofArchimedes are not logically equivalent (see thesubsection “Euclid’s Definition V.4”, footnote 11).

The Archimedean axiom plays no role in theplane geometry as developed in Books I–IV of The El-ements.4 Interpreting geometry in ordered fields, orin geometry over fields in short, one knows that F2

is a model of Euclid’s plane, where (F,+, ·,0,1, <) isa Euclidean field, i.e., an ordered field closed underthe square root operation. Consequently, R∗ ×R∗(where R∗ is a hyperreal field) is a model ofEuclid’s plane as well (see the subsection belowon modern implementations). Euclid’s definitionV.4 is discussed in more detail in the subsection“Euclid’s Definition V.4”.

Otto Stolz rediscovered the Archimedean axiomfor mathematicians, making it one of his axiomsfor magnitudes and giving it the following form:if a > b, then there is a multiple of b suchthat nb > a (Stolz 1885 [106, p. 69]).5 At thesame time, in his development of the integers,Stolz implicitly used the Archimedean axiom.Stolz’s visionary realization of the importance ofthe Archimedean axiom and his work on non-Archimedean systems stand in sharp contrastto Cantor’s remarks on infinitesimals (see thesubsection below on mathematical rigor ).

In modern mathematics, the theory of or-dered fields employs the following form of theArchimedean axiom (see, e.g., Hilbert 1899 [51,p. 27]):

(∀x > 0) (∀ε > 0) (∃n ∈ N) [nε > x]or equivalently

(3) (∀ε > 0) (∃n ∈ N) [nε > 1].

A number system satisfying (3) will be referred toas an Archimedean continuum. In the contrary case,there is an element ε > 0 called an infinitesimal

4With the exception of Proposition III.16, where so-calledhorn angles appear that could be considered as non-Archimedean magnitudes relative to rectilinear angles.5See (Ehrlich [32]) for additional historical details concern-ing Stolz’s account of the Archimedean axiom.


such that no finite sum ε + ε + · · · + ε will everreach 1; in other words,

(4) (∃ε > 0) (∀n ∈ N)[ε ≤ 1

n

].

A number system satisfying (4) is referred to asa Bernoullian continuum (i.e., a non-Archimedeancontinuum); see the subsection “Bernoulli, Johann”.

Berkeley, George

George Berkeley (1685–1753) was a cleric whoseempiricist (i.e., based on sensations, or sensa-tionalist ) metaphysics tolerated no conceptualinnovations, such as infinitesimals, without anempirical counterpart or referent. Berkeley wassimilarly opposed, on metaphysical grounds, toinfinite divisibility of the continuum (which hereferred to as extension), an idea widely takenfor granted today. In addition to his outdatedmetaphysical criticism of the infinitesimal calculusof Newton and Leibniz, Berkeley also formulated alogical criticism.6 Berkeley claimed to have detecteda logical fallacy at the basis of the method. In termsof Fermat’s E occurring in his adequality (see thesubsection “Adequality”), Berkeley’s objection canbe formulated as follows:

The increment E is assumed to be nonzeroat the beginning of the calculation, but zeroat its conclusion, an apparent logical fallacy.

However, E is not assumed to be zero at the endof the calculation, but rather is discarded at theend of the calculation (see the subsection “Berke-ley’s Logical Criticism” for more details). Such atechnique was the content of Fermat’s adequality(see the subsection “Adequality”) and Leibniz’stranscendental law of homogeneity (see the sub-section “Lex homogeneorum transcendentalis”),where the relation of equality has to be suitablyinterpreted (see the subsection “Relation [\ ”). Thetechnique is equivalent to taking the limit (of a typ-ical expression such as f (A+E)−f (A)

E for example) inWeierstrass’s approach and to taking the standardpart (see the subsection “Standard Part Principle”)in Robinson’s approach.

Meanwhile, Berkeley’s own attempt to explainthe calculation of the derivative of x2 in TheAnalyst contains a logical circularity: namely,Berkeley’s argument relies on the determination ofthe tangents of a parabola by Apollonius (which isequivalent to the calculation of the derivative). Thiscircularity in Berkeley’s argument was analyzed in(Andersen 2011 [1]).

6Berkeley’s criticism was dissected into its logical andmetaphysical components in (Sherry 1987 [98]).

Berkeley’s Logical Criticism

Berkeley’s logical criticism of the calculus amountsto the contention that the evanescent increment isfirst assumed to be nonzero to set up an algebraicexpression and then is treated as zero in discardingthe terms that contained that increment when theincrement is said to vanish. In modern terms,Berkeley was claiming that the calculus was basedon an inconsistency of type

(dx 6= 0)∧ (dx = 0).

The criticism, however, involves a misunderstand-ing of Leibniz’s method. The rebuttal of Berkeley’scriticism is that the evanescent increment need notbe “treated as zero” but, rather, is merely discardedthrough an application of the transcendental lawof homogeneity by Leibniz, as illustrated in the sub-section “Product Rule” in the case of the productrule.

While consistent (in the sense of the subsection“Mathematical Rigor”, level (2)), Leibniz’s systemunquestionably relied on heuristic principles, suchas the laws of continuity and homogeneity, andthus fell short of a standard of rigor if measured bytoday’s criteria (see the subsection “MathematicalRigor”). On the other hand, the consistency andresilience of Leibniz’s system is confirmed throughthe development of modern implementations ofLeibniz’s heuristic principles (see the subsection“Modern Implementations”).

Bernoulli, Johann

Johann Bernoulli (1667–1748) was a disciple ofLeibniz’s who, having learned an infinitesimalmethodology for the calculus from the master,never wavered from it. This is in contrast to Leibnizhimself, who throughout his career used both

(A) an Archimedean methodology (proof byexhaustion) and

(B) an infinitesimal methodology

in a symbiotic fashion. Thus Leibniz relied onthe A-methodology to underwrite and justifythe B-methodology, and he exploited the B-methodology to shorten the path to discovery(Ars Inveniendi). Historians often name Bernoullias the first mathematician to have adhered sys-tematically to the infinitesimal approach as thebasis for the calculus. We refer to an infinitesimal-enriched number system as a B-continuum, asopposed to an Archimedean A-continuum, i.e., acontinuum satisfying the Archimedean axiom (seethe subsection “Archimedean Axiom”).


Bishop, Errett

Errett Bishop (1928–1983) was a mathematicalconstructivist who, unlike his fellow intuition-ist7 Arend Heyting (see the subsection “Heyting,Arend”), held a dim view of classical mathematics ingeneral and Robinson’s infinitesimals in particular.Discouraged by the apparent nonconstructivity ofhis early work in complex analysis, Bishop believedhe had found the culprit in the law of the excludedmiddle (LEM), the key logical ingredient in everyproof by contradiction. He spent the remainingeighteen years of his life in an effort to expunge thereliance on LEM (which he dubbed “the principleof omniscience” in [11]) from analysis and soughtto define meaning itself in mathematics in termsof such LEM-extirpation.

Accordingly, he described classical mathematicsas both a debasement of meaning (Bishop 1973[13, p. 1]) and sawdust (Bishop 1973 [13, p. 14]),and he did not hesitate to speak of both crisis(Bishop 1975 [11]) and schizophrenia (Bishop 1973[13]) in contemporary mathematics, predicting animminent demise of classical mathematics in thefollowing terms:

Very possibly classical mathematics willcease to exist as an independent discipline.(Bishop 1968 [10, p. 54])

His attack in (Bishop 1977 [12]) on calculuspedagogy based on Robinson’s infinitesimals was anatural outgrowth of his general opposition to thelogical underpinnings of classical mathematics, asanalyzed in (Katz and Katz 2011 [63]). Robinsonformulated a brief but penetrating appraisal ofBishop’s ventures into the history and philosophyof mathematics, when he noted that

the sections of [Bishop’s] book that attemptto describe the philosophical and historicalbackground of [the] remarkable endeavor[of Intuitionism] are more vigorous thanaccurate and tend to belittle or ignore theefforts of others who have worked in thesame general direction. (Robinson 1968 [95,p. 921])

See the subsection “Chimeras” for a relatedcriticism by Alain Connes.

7Bishop was not an intuitionist in the narrow sense of theterm, in that he never worked with Brouwer’s continuum or“free choice sequences”. We are using the term “intuitionism”in a broader sense (i.e., mathematics based on intuitionisticlogic) that incorporates constructivism, as used for exampleby Abraham Robinson in the comment quoted at the end ofthis subsection.

Cantor, Georg

Georg Cantor (1845–1918) is familiar to the modernreader as the underappreciated creator of the“Cantorian paradise”, out of which David Hilbertwould not be expelled, as well as the tragic hero,allegedly persecuted by Kronecker, who endedhis days in a lunatic asylum. Cantor historianJ. Dauben notes, however, an underappreciatedaspect of Cantor’s scientific activity, namely, hisprincipled persecution of infinitesimalists:

Cantor devoted some of his most vituper-ative correspondence, as well as a portionof the Beiträge, to attacking what he de-scribed at one point as the ‘infinitesimalCholera bacillus of mathematics’, which hadspread from Germany through the work ofThomae, du Bois-Reymond, and Stolz, to in-fect Italian mathematics.…Any acceptanceof infinitesimals necessarily meant thathis own theory of number was incomplete.Thus to accept the work of Thomae, duBois-Reymond, Stolz, and Veronese was todeny the perfection of Cantor’s own cre-ation. Understandably, Cantor launched athorough campaign to discredit Veronese’swork in every way possible. (Dauben 1980[27, pp. 216–217])

A discussion of Cantor’s flawed investigationof the Archimedean axiom (see the subsection“Archimedean Axiom”) may be found in thesubsection “Mathematical Rigor”.8

Cauchy, Augustin-Louis

Augustin-Louis Cauchy (1789–1857) is often viewedin the history of mathematics literature as a pre-cursor of Weierstrass. Note, however, that contraryto a common misconception, Cauchy never gave anε, δ definition of either limit or continuity (see thesubsection “Variable Quantity” for Cauchy’s defini-tion of limit). Rather, his approach to continuitywas via what is known today as microcontinuity(see the subsection “Continuity”). Several recentarticles, (Błaszczyk et al. [14]; Borovik and Katz[16]; Bråting [20]; Katz and Katz [62], [64]; Katz andTall [69]), have argued that a proto-Weierstrassianview of Cauchy is one-sided and obscures Cauchy’simportant contributions, including not only hisinfinitesimal definition of continuity but also suchinnovations as his infinitesimally defined (“Dirac”)delta function, with applications in Fourier anal-ysis and evaluation of singular integrals, and hisstudy of orders of growth of infinitesimals thatanticipated the work of Paul du Bois-Reymond,

8Cantor’s dubious claim that the infinitesimal leads to con-tradictions was endorsed by no less an authority thanB. Russell; see footnote 15 in the subsection “MathematicalRigor”.


Borel, Hardy, and ultimately (Skolem [102], [103],[104]) Robinson.

To elaborate on Cauchy’s “Dirac” delta function,note the following formula from (Cauchy 1827 [23,p. 188]) in terms of an infinitesimal α:

(5)12

∫ a+εa−ε

F(µ)α dµ

α2 + (µ − a)2 =π2F(a).

Replacing Cauchy’s expression αα2+(µ−a)2 by δa(µ),

one obtains Dirac’s formula up to trivial modifica-tions (see Dirac [30, p. 59]):∫∞

−∞f (x)δ(x) = f (0).

Cauchy’s 1853 paper on a notion closely relatedto uniform convergence was recently examinedin (Katz and Katz 2011 [62]) and (Błaszczyk etal. 2012 [14]). Cauchy handles the said notionusing infinitesimals, including one generated bythe null sequence ( 1

n ).

Chimeras

Alain Connes (1947–) formulated criticisms ofRobinson’s infinitesimals between the years 1995and 2007 on at least seven separate occasions(see Kanovei et al. 2012 [57, Section 3.1, Table 1]).These range from pejorative epithets such as“inadequate,” “disappointing,” “chimera,” and “irre-mediable defect,” to “the end of the rope for being‘explicit’.”

Connes sought to exploit the Solovay model S(Solovay 1970 [105]) as ammunition againstnonstandard analysis, but the model tends toboomerang, undercutting Connes’s own earlierwork in functional analysis. Connes describedthe hyperreals as both a “virtual theory” and a“chimera”, yet acknowledged that his argumentrelies on the transfer principle (see the subsection“Modern Implementations”). In S, all definable setsof reals are Lebesgue measurable, suggesting thatConnes views a theory as being “virtual” if it is notdefinable in a suitable model of ZFC. If so, Connes’sclaim that a theory of the hyperreals is “virtual” isrefuted by the existence of a definable model ofthe hyperreal field (Kanovei and Shelah [59]). Freeultrafilters aren’t definable, yet Connes exploitedsuch ultrafilters both in his own earlier work onthe classification of factors in the 1970s and 80sand in his magnum opus Noncommutative Geom-etry (Connes 1994 [26, Ch. V, Sect. 6.δ, Def. 11]),raising the question whether the latter may notbe vulnerable to Connes’s criticism of virtuality.The article [57] analyzed the philosophical under-pinnings of Connes’s argument based on Gödel’sincompleteness theorem and detected an apparentcircularity in Connes’s logic. The article [57] alsodocumented the reliance on nonconstructive foun-dational material, and specifically on the Dixmier

trace −∫

(featured on the front cover of Connes’smagnum opus) and the Hahn–Banach theorem, inConnes’s own framework; see also [65].

See the subsection “Bishop, Errett” for a relatedcriticism by Errett Bishop.

Continuity to Indivisibles

Continuity

Of the two main definitions of continuity of afunction, Definition A (see below) is operative ineither a B-continuum or an A-continuum (satisfy-ing the Archimedean axiom; see the subsection“Archimedean Axiom”), while Definition B worksonly in a B-continuum (i.e., an infinitesimal-enrichedor Bernoullian continuum; see the subsection“Bernoulli, Johann”).

• Definition A (ε, δ approach): A real func-tion f is continuous at a real point x if andonly if

(∀ε > 0) (∃δ > 0) (∀x′)[|x− x′| < δ→ |f (x)− f (x′)| < ε

].

• Definition B (microcontinuity): A real func-tion f is continuous at a real point x if andonly if

(6) (∀x′)[x′ [\x → f (x′) [\ f (x)

].

In formula (6) the natural extension of f is stilldenoted f , and the symbol “ [\ ” stands for therelation of being infinitely close; thus, x′ [\x if andonly if x′ − x is infinitesimal (see the subsection“Relation [\ ”).

Diophantus

Diophantus of Alexandria (who lived about 1,800years ago) contributed indirectly to the develop-ment of infinitesimal calculus through the tech-nique called παρισoτης, developed in his workArithmetica, Book Five, problems 12, 14, and 17.The termπαρισoτης can be literally translated as“approximate equality”. This was rendered as adae-qualitas in Bachet’s Latin translation [4] and adé-galité in French (see the subsection “Adequality”).The term was used by Fermat to describe thecomparison of values of an algebraic expression,or what would today be called a function f , atnearby points A and A + E and to seek extremaby a technique closely related to the vanishing off (A+E)−f (A)

E after discarding the remaining E-terms;see (Katz, Schaps, and Shnider 2013 [66]).

Euclid’s Definition V.4

Euclid’s Definition V.4 has already been discussedin the subsection “Archimedean Axiom”. In additionto Book V, it appears in Books X and XII and isused in the method of exhaustion (see Euclid [34,


Propositions X.1, XII.2]). The method of exhaustionwas exploited intensively by both Archimedesand Leibniz (see the subsection “Leibniz’s DeQuadratura” on Leibniz’s work De Quadratura). Itwas revived in the nineteenth century in the theoryof the Riemann integral.

Euclid’s Book V sets the basis for the theoryof similar figures developed in Book VI. Greatmathematicians of the seventeenth century, suchas Descartes, Leibniz, and Newton, exploitedEuclid’s theory of similar figures of Book VI whilepaying no attention to its axiomatic background.9

Over time Euclid’s Book V became a subject ofinterest for historians and editors alone.

To formalize Definition V.4, one needed aformula for Euclid’s notion of “multiple” and anidea of total order. Some progress in this directionwas made by Robert Simson in 1762.10 In 1876Hermann Hankel provided a modern reconstructionof Book V. Combining his own historical studieswith an idea of order compatible with additiondeveloped by Hermann Grassmann (1861 [44]), hegave a formula that to this day is accepted as aformalization of Euclid’s definition of proportionin V.5 (Hankel 1876 [46, pp. 389–398]). Euclid’sproportion is a relation among four “magnitudes”,such as

A : B :: C : D.It was interpreted by Hankel as the relation

(∀m,n)[(nA>1mB → nC >2 mD)∧ (nA =mB → nC =mD)∧ (nA <1 mB → nC <2 mD)

],

where n,m are natural numbers. The indiceson the inequalities emphasize the fact that the“magnitudes” A,B have to be of “the same kind”,e.g., line segments, whereas C,D could be ofanother kind, e.g., triangles.

In 1880 J. L. Heiberg, in his edition of Archimedes’Opera omnia in a comment on a lemma ofArchimedes, cites Euclid’s Definition V.4, not-ing that these two are the same axioms (Heiberg1880 [49, p. 11]).11 This is the reason why Eu-clid’s Definition V.4 is commonly known as theArchimedean axiom. Today we formalize Euclid’sDefinition V.4 as in (1), while the Archimedeanlemma is rendered by formula (2).

9Leibniz and Newton apparently applied Euclid’s conclu-sions in a context where the said conclusions did not,technically speaking, apply: namely, to infinitesimal fig-ures such as the characteristic triangle, i.e., triangle withsides dx, dy , and ds.10See Simson’s axioms that supplement the definitions ofBook V as elaborated in (Simson 1762 [101, p. 114–115]).11In point of fact, Euclid’s axiom V.4 and Archimedes’lemma are not equivalent from the logical viewpoint. Thus,the additive semigroup of positive appreciable limitedhyperreals satisfies V.4 but not Archimedes’ lemma.

Euler, Leonhard

Euler’s Introductio in Analysin Infinitorum (1748[35]) contains remarkable calculations carried outin an extended number system in which the basicalgebraic operations are applied to infinitely smalland infinitely large quantities. Thus, in Chapter 7,“Exponentials and Logarithms Expressed throughSeries”, we find a derivation of the power seriesfor az starting from the formula aω = 1 + kω,forω infinitely small, and then raising the equationto the infinitely great power12 j = z

ω for a finite(appreciable) z to give

az = ajω = (1+ kω)j

and finally expanding the right-hand side as apower series by means of the binomial formula.In the chapters following, Euler finds infiniteproduct expansions factoring the power seriesexpansion for transcendental functions (see thesubsection “Euler’s Infinite Product Formula forSine”). By Chapter 10 he has the tools to sum theseries for ζ(2) where ζ(s) =

∑n n−s . He explicitly

calculates ζ(2k) for k = 1, . . . ,13 as well as manyother related infinite series.

In Chapter 3 of his Institutiones Calculi Differen-tialis (1755 [37]), Euler deals with the methodologyof the calculus, such as the nature of infinitesimaland infinitely large quantities. We will cite theEnglish translation [38] of the Latin original [37].Here Euler writes that

even if someone denies that infinite num-bers really exist in this world, still inmathematical speculations there arise ques-tions to which answers cannot be givenunless we admit an infinite number. (ibid.,§82) [emphasis added—the authors]

Euler’s approach, countenancing the possibilityof denying that “infinite numbers really exist,” isconsonant with a Leibnizian view of infinitesimaland infinite quantities as “useful fictions” (see Katzand Sherry [67]; Sherry and Katz [100]). Euler thennotes that “an infinitely small quantity is nothingbut a vanishing quantity, and so it is really equalto 0.” (ibid., §83)

Similarly, Leibniz combined a view of infinitesi-mals as “useful fictions” and inassignable quantities,with a generalized notion of “equality” that wasan equality up to an incomparably negligible term.Leibniz sought to codify this idea in terms of histranscendental law of homogeneity (TLH); see thesubsection “Lex homogeneorum transcendentalis”.Thus, Euler’s formulas such as

(7) a+ dx = a

12Euler used the symbol i for the infinite power. Blanton re-placed this by j in the English translation so as to avoid anotational clash with the standard symbol for

√−1.


(where a “is any finite quantity” ibid., §§86, 87)are consonant with a Leibnizian tradition (cf. for-mula (17) in the subsection “Lex homogeneorumtranscendentalis”). To explain formulas like (7),Euler elaborated two distinct ways (arithmetic andgeometric) of comparing quantities in the followingterms:

Since we are going to show that an infinitelysmall quantity is really zero, we mustmeet the objection of why we do notalways use the same symbol 0 for infinitelysmall quantities, rather than some specialones…[S]ince we have two ways to comparethem, either arithmetic or geometric, let uslook at the quotients of quantities to becompared in order to see the difference.

If we accept the notation used in theanalysis of the infinite, then dx indicatesa quantity that is infinitely small, so thatboth dx = 0 and adx = 0, where a is anyfinite quantity. Despite this, the geomet-ric ratio adx : dx is finite, namely a : 1.For this reason, these two infinitely smallquantities, dx and adx, both being equalto 0, cannot be confused when we considertheir ratio. In a similar way, we will dealwith infinitely small quantities dx and dy .(ibid., §86, pp. 51–52) [emphasis added—theauthors]

Euler proceeds to clarify the difference between thearithmetic and geometric comparisons as follows:

Let a be a finite quantity and let dx beinfinitely small. The arithmetic ratio ofequals is clear: Since ndx = 0, we have

a± ndx− a = 0.

On the other hand, the geometric ratio isclearly of equals, since

(8)a± ndxa

= 1.

From this we obtain the well-known rule thatthe infinitely small vanishes in comparisonwith the finite and hence can be neglected.(Euler 1755 [38, §87]) [emphasis in theoriginal—the authors]

Like Leibniz, Euler considers more than one way ofcomparing quantities. Euler’s formula (8) indicatesthat his geometric comparison is identical withthe Leibnizian TLH; namely, Euler’s geometriccomparison of a pair of quantities amounts to theirratio being infinitely close to 1. The same is truefor TLH. Thus one has a + dx = a in this sensefor an appreciable a 6= 0, but not dx = 0 (which istrue only arithmetically in Euler’s sense). Euler’s“geometric” comparison was dubbed “the principleof cancellation” in (Ferraro 2004 [39, p. 47]).

Euler proceeds to present the usual rules ofinfinitesimal calculus, which go back to Leibniz,L’Hôpital, and the Bernoullis, such as

(9) adxm + b dxn = adxm

provided m < n “since dxn vanishes comparedwith dxm” (ibid., §89), relying on his “geometric”equality. Euler introduces a distinction betweeninfinitesimals of different order and directlycomputes13 a ratio of the form

dx± dx2

dx= 1± dx = 1

of two particular infinitesimals, assigning thevalue 1 to it (ibid., §88). Euler concludes:

Although all of them [infinitely small quan-tities] are equal to 0, still they must becarefully distinguished one from the otherif we are to pay attention to their mutualrelationships, which has been explainedthrough a geometric ratio. (ibid., §89).

The Eulerian hierarchy of orders of infinitesimalsharks back to Leibniz’s work (see the subsection“Nieuwentijt, Bernard” for a historical dissentingview).

Euler’s Infinite Product Formula for Sine

The fruitfulness of Euler’s infinitesimal approachcan be illustrated by some of the remarkableapplications he obtained. Thus, Euler derived aninfinite product decomposition for the sine andsinh functions of the following form:

sinhx = x(

1+ x2

π2

) (1+ x2

4π2

)(10)

·(

1+ x2

9π2

) (1+ x2

16π2

), . . .

sinx = x(

1− x2

π2

) (1− x2

4π2

)(11)

·(

1− x2

9π2

) (1− x2

16π2

). . .

Decomposition (11) generalizes an infinite productformula for π

2 due to Wallis [109]. Euler also

summed the inverse square series: 1 + 14 +

19 +

116 + · · · =

π2

6 (see [86]) and obtained additional

identities. A common feature of these formulasis that Euler’s computations involve not onlyinfinitesimals but also infinitely large naturalnumbers, which Euler sometimes treats as if they

13Note that Euler does not “prove that the expression isequal to 1”; such indirect proofs are a trademark of the ε, δapproach. Rather, Euler directly computes (what would to-day be formalized as the standard part of) the expression,illustrating one of the advantages of the B-methodology overthe A-methodology.


were ordinary natural numbers.14 Similarly, Eulertreats infinite series as polynomials of a specificinfinite degree.

The derivation of (10) and (11) in (Euler 1748[35, §156]) can be broken up into seven steps asfollows.

Step 1. Euler observes that

(12) 2 sinhx = ex − e−x =(

1+ xj

)j−(

1− xj

)j,

where j (or “i” in Euler [35]) is an infinitely largenatural number. To motivate the next step, notethat the expression xj − 1 = (x − 1)(1 + x + x2 +· · · + xj−1) can be factored further as

∏j−1k=0(x −

ζk), where ζ = e2πi/j ; conjugate factors can thenbe combined to yield a decomposition into realquadratic terms.

Step 2. Euler uses the fact that aj−bj is the productof the factors

(13) a2 + b2 − 2ab cos2kπj, where k ≥ 1 ,

together with the factor a− b and, if j is an evennumber, the factor a+ b as well.

Step 3. Setting a = 1 + xj and b = 1 − x

j in (12),

Euler transforms expression (13) into the form

(14) 2+ 2x2

j2− 2

(1− x

2

j2

)cos

2kπj.

Step 4. Euler then replaces (14) by the expression

(15)4k2π2

j2

(1+ x2

k2π2− x

2

j2

),

justifying this step by means of the formula

(16) cos2kπj= 1− 2k2π2

j2.

Step 5. Next, Euler argues that the difference ex −e−x is divisible by the expression

1+ x2

k2π2− x

2

j2

from (15), where “we omit the term x2

j2 since even

when multiplied by j , it remains infinitely small.”(English translation from [36].)

Step 6. As there is still a factor of a − b = 2x/j ,Euler obtains the final equality (10), arguing thatthen “the resulting first term will be x” (in order toconform to the Maclaurin series for sinhx).

Step 7. Finally, formula (11) is obtained from (10)by means of the substitution x, ix. �

We will discuss modern formalizations of Euler’sargument in the next subsection.

14Euler’s procedure is therefore consonant with the Leib-nizian law of continuity (see the subsection “Lex continui-tatis”), though apparently Euler does not refer explicitly tothe latter.

Euler’s Sine Factorization Formalized

Euler’s argument in favor of (10) and (11) wasformalized in terms of a “nonstandard” proof in(Luxemburg 1973 [82]). However, the formalizationin [82] deviates from Euler’s argument, beginningwith Steps 3 and 4, and thus circumvents the moreproblematic Steps 5 and 6.

A proof in the framework of modern non-standard analysis formalizing Euler’s argumentstep-by-step throughout appeared in (Kanovei 1988[56]); see also (McKinzie and Tuckey 1997 [86])and (Kanovei and Reeken 2004 [58, §2.4a]). Thisformalization interprets problematic details ofEuler’s argument on the basis of general principlesof modern nonstandard analysis, as well as generalanalytic facts that were known in Euler’s time. Suchprinciples and facts behind some early proofs ininfinitesimal calculus are sometimes referred toas “hidden lemmas” in this context; see (Laugwitz[73], [74]), (McKinzie and Tuckey 1997 [86]). Forinstance, the “hidden lemma” behind Step 4 aboveis the fact that, for a fixed x, the terms of theMaclaurin expansion of cosx tend to 0 faster thana convergent geometric series, allowing one toinfer that the effect of the transformation of Step4 on the product of the factors (14) is infinitesimal.Some “hidden lemmas” of a different kind, relatedto basic principles of nonstandard analysis, arediscussed in [86, pp. 43ff.].

What clearly stands out from Euler’s argumentis his explicit use of infinitesimal expressionssuch as (14) and (15), as well as the approximateformula (16), which holds “up to” an infinitesimalof higher order. Thus, Euler used infinitesimalspar excellence, rather than merely ratios thereof,in a routine fashion in some of his best work.

Euler’s use of infinite integers and their associ-ated infinite products (such as the decompositionof the sine function) were interpreted in Robin-son’s framework in terms of hyperfinite sets. ThusEuler’s product of j-infinitely many factors in (11)is interpreted as a hyperfinite product in [58,formula (9), p. 74]. A hyperfinite formalization ofEuler’s argument involving infinite integers andtheir associated products illustrates the successfulremodeling of the arguments (and not merely theresults) of classical infinitesimal mathematics, asdiscussed in the subsection “Mathematical Rigor”.

Fermat, Pierre de

Pierre de Fermat (1601–1665) developed a pio-neering technique known as adequality (see thesubsection “Adequality”) for finding tangents tocurves and for solving problems of maxima andminima. Katz, Schaps, and Shnider (2013 [66])analyze some of the main approaches in the lit-erature to the method of adequality, as well as


its source in the παρισoτης of Diophantus (seethe subsection “Diophantus”). At least some ofthe manifestations of adequality, such as Fermat’streatment of transcendental curves and Snell’s law,amount to variational techniques exploiting a small(alternatively, infinitesimal) variation E. Fermat’streatment of geometric and physical applicationssuggests that an aspect of approximation is inher-ent in adequality, as well as an aspect of smallnesson the part of E.

Fermat’s use of the term adequality relied onBachet’s rendering of Diophantus. Diophantuscoined the term parisotes for mathematical pur-poses. Bachet performed a semantic calque inpassing from par-isoo to ad-aequo. A historicallysignificant parallel is found in the similar roleof, respectively, adequality and the transcenden-tal law of homogeneity (see the subsection “Lexhomogeneorum transcendentalis”) in the work of,respectively, Fermat and Leibniz on the problemsof maxima and minima.

Breger (1994 [21]) denies that the idea of“smallness” was relied upon by Fermat. However, adetailed analysis (see [66]) of Fermat’s treatment ofthe cycloid reveals that Fermat did rely on issuesof “smallness” in his treatment of the cycloidand reveals that Breger’s interpretation thereofcontains both mathematical errors and errors oftextual analysis. Similarly, Fermat’s proof of Snell’slaw, a variational principle, unmistakably relies onideas of “smallness”.

Cifoletti (1990 [25]) finds similarities betweenFermat’s adequality and some procedures usedin smooth infinitesimal analysis of Lawvere andothers. Meanwhile, J. Bell (2009 [9]) seeks the his-torical sources of Lawvere’s infinitesimals mainlyin Nieuwentijt (see the subsection “Nieuwentijt,Bernard”).

Heyting, Arend

Arend Heyting (1898–1980) was a mathematicalintuitionist whose lasting contribution was theformalization of the intuitionistic logic underpin-ning the Intuitionism of his teacher Brouwer. WhileHeyting never worked on any theory of infinites-imals, he had several opportunities to presentan expert opinion on Robinson’s theory. Thus,in 1961, Robinson made public his new idea ofnonstandard models for analysis and “communi-cated this almost immediately to…Heyting” (seeDauben [28, p. 259]). Robinson’s first paper on thesubject was subsequently published in Proceedingsof the Netherlands Royal Academy of Sciences [94].Heyting praised nonstandard analysis as “a stan-dard model of important mathematical research”(Heyting 1973 [50, p. 136]). Addressing Robinson,he declared:

[Y]ou connected this extremely abstract partof model theory with a theory apparentlyso far apart as the elementary calculus. Indoing so you threw new light on the historyof the calculus by giving a clear sense toLeibniz’s notion of infinitesimals. (ibid)

Intuitionist Heyting’s admiration for the appli-cation of Robinson’s infinitesimals to calculuspedagogy is in stark contrast with the views of hisfellow constructivist E. Bishop (subsection “Bishop,Errett”).

Indivisibles versus Infinitesimals

Commentators use the term infinitesimal to referto a variety of conceptions of the infinitely small,but the variety is not always acknowledged. It isimportant to distinguish the infinitesimal methodsof Archimedes and Cavalieri from those employedby Leibniz and his successors. To emphasize thisdistinction, we will say that tradition prior toLeibniz employed indivisibles. For example, in hisheuristic proof that the area of a parabolic segmentis 4/3 the area of the inscribed triangle with thesame base and vertex, Archimedes imagines bothfigures to consist of perpendiculars of variousheights erected on the base. The perpendicularsare indivisibles in the sense that they are limits ofdivision and so one dimension less than the area.In the same sense, the indivisibles of which a lineconsists are points, and the indivisibles of which asolid consists are planes.

Leibniz’s infinitesimals are not indivisibles, forthey have the same dimension as the figures thatcompose them. Thus, he treats curves as comprisinginfinitesimal line intervals rather than indivisiblepoints. The strategy of treating infinitesimals asdimensionally homogeneous with the objects theycompose seems to have originated with Robervalor Torricelli, Cavalieri’s student, and to have beenexplicitly arithmetized in (Wallis 1656 [109]).

Zeno’s paradox of extension admits resolutionin the framework of Leibnizian infinitesimals(see the subsection “Zeno’s Paradox of Extension”).Furthermore, only with the dimensionality retainedis it possible to make sense of the fundamentaltheorem of calculus, where one must think aboutthe rate of change of the area under a curve, anotherreason why indivisibles had to be abandonedin favor of infinitesimals so as to enable thedevelopment of the calculus (see Ely 2012 [33]).

Leibniz to Nieuwentijt

Leibniz, Gottfried

Gottfried Wilhelm Leibniz (1646–1716), the co-inventor of infinitesimal calculus, is a key player inthe parallel infinitesimal track referred to by Felix


Klein [72, p. 214] (see the section “The ABCs of theHistory of Infinitesimal Mathematics”).

Leibniz’s law of continuity (see the subsection“Lex continuitatis”) and his transcendental law ofhomogeneity (which he had already discussed inhis response to Nieuwentijt in 1695, as noted byM. Parmentier [79, p. 38], and later in greater detailin a 1710 article [78] cited in the seminal studyof Leibnizian methodology by H. Bos [17]) form abasis for implementing the calculus in the contextof a B-continuum.

Many historians of the calculus deny significantcontinuity between infinitesimal calculus of theseventeenth century and twentieth-century devel-opments such as Robinson’s theory (see furtherdiscussion in Katz and Sherry [67]). Robinson’s hy-perreals require the resources of modern logic; thusmany commentators are comfortable denying a his-torical continuity. A notable exception is Robinsonhimself, whose identification with the Leibniziantradition inspired Lakatos, Laugwitz, and othersto consider the history of the infinitesimal in amore favorable light. Many historians have over-estimated the force of Berkeley’s criticisms (see thesubsection “Berkeley, George”) by underestimatingthe mathematical and philosophical resourcesavailable to Leibniz.

Leibniz’s infinitesimals are fictions—not logicalfictions, as (Ishiguro 1990 [54]) proposed, butrather pure fictions, like imaginaries, which are noteliminable by some syncategorematic paraphrase;see (Sherry and Katz 2012 [100]) and the subsection“Leibniz’s De Quadratura” below.

In fact, Leibniz’s defense of infinitesimals is morefirmly grounded than Berkeley’s criticism thereof.Moreover, Leibniz’s system for differential calculuswas free of logical fallacies (see the subsection“Berkeley’s Logical Criticism”). This strengthens theconception of modern infinitesimals as a formaliza-tion of Leibniz’s strategy of relating inassignableto assignable quantities by means of his transcen-dental law of homogeneity (see the subsection “Lexhomogeneorum transcendentalis”).

Leibniz’s De Quadratura

In 1675 Leibniz wrote a treatise on his infinitesi-mal methods, On the Arithmetical Quadrature ofthe Circle, the Ellipse, and the Hyperbola, or DeQuadratura, as it is widely known. However, thetreatise appeared in print only in 1993 in a textedited by Knobloch (Leibniz [80]).

De Quadratura was interpreted by R. Arthur[3] and others as supporting the thesis that Leib-niz’s infinitesimals are mere fictions, eliminableby long-winded paraphrase. This so-called syn-categorematic interpretation of Leibniz’s calculushas gained a number of adherents. We believethis interpretation to be a mistake. In the first

place, Leibniz wrote the treatise at a time wheninfinitesimals were despised by the French Acad-emy, a society whose approval and acceptancehe eagerly sought. More importantly, as (Jesseph2013 [55]) has shown, De Quadratura depends oninfinitesimal resources in order to construct anapproximation to a given curvilinear area less thanany previously specified error. This problem isreminiscent of the difficulty that led to infinitesimalmethods in the first place. Archimedes’ method ofexhaustion required one to determine a value forthe quadrature in advance of showing, by reductioargument, that any departure from that valueentails a contradiction. Archimedes possesseda heuristic, indivisible method for finding suchvalues, and the results were justified by exhaustion,but only after the fact. By the same token, theuse of infinitesimals is “just” a shortcut only if itis entirely eliminable from quadratures, tangentconstructions, etc. Jesseph’s insight is that this isnot the case.

Finally, the syncategorematic interpretation mis-represents a crucial aspect of Leibniz’s mathemat-ical philosophy. His conception of mathematicalfiction includes imaginary numbers, and he oftensought approbation for his infinitesimals by com-paring them to imaginaries, which were largelyuncontroversial. There is no suggestion by Leibnizthat imaginaries are eliminable by long-windedparaphrase. Rather, he praises imaginaries fortheir capacity to achieve universal harmony by thegreatest possible systematization, and this char-acteristic is more central to Leibniz’s conceptionof infinitesimals than the idea that they are mereshorthand. Just as imaginary roots both unified andextended the method for solving cubics, likewiseinfinitesimals unified and extended the methodfor quadrature so that, e.g., quadratures of generalparabolas and hyperbolas could be found by thesame method used for quadratures of less difficultcurves.

Lex continuitatis

A heuristic principle called The law of continuity(LC) was formulated by Leibniz and is a keyto appreciating Leibniz’s vision of infinitesimalcalculus. The LC asserts that whatever succeeds inthe finite succeeds also in the infinite. This formof the principle appeared in a letter to Varignon(Leibniz 1702 [77]). A more detailed form of LC interms of the concept of terminus appeared in histext Cum Prodiisset :

In any supposed continuous transition,ending in any terminus, it is permissibleto institute a general reasoning, in whichthe final terminus may also be included.(Leibniz 1701 [76, p. 40])


{assignablequantities

}LC�

{inassignablequantities

}TLH�

{assignablequantities

}

Figure 2. Leibniz’s law of continuity (LC) takesone from assignable to inassignable quantities,while his transcendental law of homogeneity(TLH; the subsection “Lex homogeneorumtranscendentalis”) returns one to assignablequantities.

To elaborate, the LC postulates that whateverproperties are satisfied by ordinary or assignablequantities should also be satisfied by inassignablequantities (see the subsection “Variable Quantity”)such as infinitesimals (see Figure 2). Thus thetrigonometric formula sin2 x+ cos2 x = 1 shouldbe satisfied for an inassignable (e.g., infinitesi-mal) input x as well. In the twentieth centurythis heuristic principle was formalized as thetransfer principle (see the subsection “ModernImplementations”) of Łos–Robinson.

The significance of LC can be illustrated by thefact that a failure to take note of the law of conti-nuity often led scholars astray. Thus, Nieuwentijt(see the subsection “Nieuwentijt, Bernard”) was ledinto something of a dead end with his nilpotentinfinitesimals (ruled out by LC) of the form 1

∞ .J. Bell’s view of Nieuwentijt’s approach as a pre-cursor of nilsquare infinitesimals of Lawvere (seeBell 2009 [9]) is plausible, though it could be notedthat Lawvere’s nilsquare infinitesimals cannot beof the form 1

∞ .

Lex homogeneorum transcendentalis

Leibniz’s transcendental law of homogeneity, orlex homogeneorum transcendentalis in the origi-nal Latin (Leibniz 1710 [78]), governs equationsinvolving differentials. Leibniz historian H. Bosinterprets it as follows:

A quantity which is infinitely small withrespect to another quantity can be neglectedif compared with that quantity. Thus allterms in an equation except those of thehighest order of infinity, or the lowest orderof infinite smallness, can be discarded. Forinstance,

a+ dx = a(17)

dx+ ddy = dx

etc. The resulting equations satisfy this…re-quirement of homogeneity. (Bos 1974 [17,p. 33])

For an interpretation of the equality sign in theformulas above, see the subsection “Relation [\ ”.

The TLH associates to an inassignable quantity(such as a+ dx) an assignable one (such as a); seeFigure 2 for a relation between LC and TLH.

Mathematical Rigor

There is a certain lack of clarity in the historicalliterature with regard to issues of fruitfulness,consistency, and rigorousness of mathematicalwriting. As a rough guide and to be able toformulate useful distinctions when it comes toevaluating mathematical writing from centuriespast, we would like to consider three levels ofjudging mathematical writing:

(1) potentially fruitful but (logically) inconsis-tent,

(2) (potentially) consistent but informal,(3) formally consistent and fully rigorous ac-

cording to currently prevailing standards.

As an example of level (1) we would cite the workof Nieuwentijt (see the subsection “Nieuwentijt,Bernard” for a discussion of the inconsistency).Our prime example of level (2) is provided by theLeibnizian laws of continuity and homogeneity (seesubsections “Lex continuitatis” and “Lex homoge-neorum transcendentalis”), which found rigorousimplementation at level (3) only centuries later(see the subsection “Modern Implementations”).

A foundational rock of the received history ofmathematical analysis is the belief that mathemat-ical rigor emerged starting in the 1870s throughthe efforts of Cantor, Dedekind, Weierstrass, andothers, thereby replacing formerly unrigorouswork of infinitesimalists from Leibniz onward. Thephilosophical underpinnings of such a belief wereanalyzed in (Katz and Katz 2012a [64]), where itwas pointed out that in mathematics, as in othersciences, former errors are eliminated through aprocess of improved conceptual understanding,evolving over time, of the key issues involved inthat science.

Thus no scientific development can be claimedto have attained perfect clarity or rigor merely onthe grounds of having eliminated earlier errors.Moreover, no such claim for a single scientificdevelopment is made either by the practitionersor by the historians of the natural sciences. It wasfurther pointed out in [64] that the term mathemat-ical rigor itself is ambiguous, its meaning varyingaccording to context. Four possible meanings forthe term were proposed in [64]:

(1) it is a shibboleth that identifies the speakeras belonging to a clan of professionalmathematicians;

(2) it represents the idea that, as a scien-tific field develops, its practitioners attaingreater and more conceptual understand-ing of key issues and are less prone toerror;


(3) it represents the idea that a search forgreater correctness in analysis inevitablyled Weierstrass specifically to epsilontics(i.e., the A-approach) in the 1870s;

(4) it refers to the establishment of what areperceived to be the ultimate foundationsfor mathematics by Cantor, eventuallyexplicitly expressed in axiomatic form byZermelo and Fraenkel.

Item (1) may be pursued by a fashionableacademic in the social sciences but does not get tothe bottom of the issue. Meanwhile, item (2) wouldbe agreed upon by historians of the other sciences.

In this context it is interesting to comparethe investigation of the Archimedean property asperformed by the would-be rigorist Cantor, on theone hand, and the infinitesimalist Stolz on theother. Cantor sought to derive the Archimedeanproperty as a consequence of those of a linearcontinuum. Cantor’s work in this area was not onlyunrigorous but actually erroneous, whereas Stolz’swork was fully rigorous and even visionary. Namely,Cantor’s arguments “proving” the inconsistency ofinfinitesimals were based on an implicit assumptionof what is known today as the Kerry-Cantor axiom(see Proietti 2008 [92]). Meanwhile, Stolz wasthe first modern mathematician to realize theimportance of the Archimedean axiom (see thesubsection “Archimedean Axiom”) as a separateaxiom in its own right (see Ehrlich 2006 [32])and, moreover, to develop some non-Archimedeansystems (Stolz 1885 [106]).

In his Grundlagen der Geometrie (Hilbert 1899[51]), Hilbert did not develop a new geometry,but rather remodeled Euclid’s geometry. Morespecifically, Hilbert brought rigor into Euclid’sgeometry in the sense of formalizing both Euclid’spropositions and Euclid’s style of procedures andstyle of reasoning.

Note that Hilbert’s system works for geometriesbuilt over a non-Archimedean field, as Hilbertwas fully aware. Hilbert (1900 [52, p. 207]) citesDehn’s counterexamples to Legendre’s theoremin the absence of the Archimedean axiom. Dehnplanes built over a non-Archimedean field wereused to prove certain cases of the independenceof Hilbert’s axioms (see Cerroni 2007 [24]).15

Robinson’s theory similarly formalizedseventeenth- and eighteenth-century analysis

15It is a melancholy comment to note that fully three yearslater the philosopher-mathematician Bertrand Russell wasstill claiming, on Cantor’s authority, that the infinitesimal“leads to contradictions” (Russell 2003 [97, p. 345]). Thisset the stage for several decades of anti-infinitesimal vitriol,including the saline solution of Parkhurst and Kingsland(see the section “The ABCs of the History of InfinitesimalMathematics”).

by remodeling both its propositions and its pro-cedures and reasoning. Using Weierstrassian ε, δtechniques, one can recover only the propositionsbut not the proof procedures. Thus, Euler’s resultgiving an infinite product formula for sine (see thesubsection “Euler’s Infinite Product Formula forSine”) admits numerous proofs in a Weierstrassiancontext, but Robinson’s framework provides asuitable context in which Euler’s proof, relying oninfinite integers, can also be recovered. This is thecrux of the historical debate concerning ε, δ versusinfinitesimals. In short, Robinson did for Leibnizwhat Hilbert did for Euclid. Meanwhile, epsilontistsfailed to do for Leibniz what Robinson did forLeibniz, namely, formalizing the procedures andreasoning of the historical infinitesimal calculus.This theme is pursued further in terms of theinternal/external distinction in the subsection“Variable Quantity”.

Modern Implementations

In the 1940s Hewitt [48] developed a modern imple-mentation of an infinitesimal-enriched continuumextending R by means of a technique referredto today as the ultrapower construction. We willdenote such an infinitesimal-enriched continuumby the new symbol IR (“thick-R”).16 In the nextdecade, Łos (Łos 1955 [81]) proved his celebratedtheorem on ultraproducts, implying in particularthat elementary (more generally, first-order) state-ments over R are true if and only if they are trueover IR, yielding a modern implementation of theLeibnizian law of continuity (see the subsection“Lex continuitatis”). Such a result is equivalent towhat is known in the literature as the transferprinciple; see Keisler [70]. Every finite element of IRis infinitely close to a unique real number; see thesubsection “Standard Part Principle”. Such a princi-ple is a mathematical implementation of Fermat’sadequality (see the subsection “Adequality”), ofLeibniz’s transcendental law of homogeneity (seethe subsection “Lex homogeneorum transcenden-talis”), and of Euler’s principle of cancellation (seethe discussion between formulas (7) and (9) in thesubsection “Euler, Leonhard”).

Nieuwentijt, Bernard

In Nieuwentijt’s Analysis Infinitorum (1695), theDutch philosopher (1654–1718)17 proposed a sys-tem containing an infinite number, as well asinfinitesimal quantities formed by dividing finitenumbers by this infinite one. Nieuwentijt postu-lated that the product of two infinitesimals shouldbe exactly equal to zero. In particular, an infini-tesimal quantity is nilpotent. In an exchange of

16A more traditional symbol is ∗R or R∗.17Alternative spellings are Nieuwentijdt or Nieuwentyt.


publications with Nieuwentijt on infinitesimals (seeMancosu 1996 [84, p. 161]), Leibniz and Hermannclaimed that this system is consistent only ifall infinitesimals are equal, rendering differentialcalculus useless. Leibniz instead advocated a sys-tem in which the product of two infinitesimalsis incomparably smaller than either infinitesimal.Nieuwentijt’s objections compelled Leibniz in 1696to elaborate on the hierarchy of infinite and infini-tesimal numbers entailed in a robust infinitesimalsystem.

Nieuwentijt’s nilpotent infinitesimals of theform 1

∞ are ruled out by Leibniz’s law of continuity(see the subsection “Lex continuitatis”). J. Bell’sview of Nieuwentijt’s approach as a precursor ofnilsquare infinitesimals of Lawvere (see Bell 2009[9]) is plausible, though it could be noted thatLawvere’s nilsquare infinitesimals cannot be of theform 1

∞ .

Product Rule to Zeno

Product Rule

In the area of Leibniz scholarship, the receivedview is that Leibniz’s infinitesimal system was logi-cally faulty and contained internal contradictionsallegedly exposed by the cleric George Berkeley(see the subsection “Berkeley, George”). Such a viewis fully compatible with the A-track-dominatedoutlook, bestowing supremacy upon the recon-struction of analysis accomplished through theefforts of Cantor, Dedekind, Weierstrass, and theirrigorous followers (see the subsection “Mathemati-cal Rigor”). Does such a view represent an accurateappraisal of Leibniz’s system?

The articles (Katz and Sherry 2012 [67], [68];Sherry and Katz [100]), building on the earlier work(Sherry 1987 [98]), argued that Leibniz’s systemwas in fact consistent (in the sense of level (2) of thesubsection “Mathematical Rigor”)18 and featuredresilient heuristic principles such as the law ofcontinuity (see the subsection “Lex continuitatis”)and the transcendental law of homogeneity (TLH)(see the subsection “Lex homogeneorum transcen-dentalis”), which were implemented in the fullnessof time as precise mathematical principles guidingthe behavior of modern infinitesimals.

How did Leibniz exploit the TLH in developingthe calculus? We will now illustrate an application ofthe TLH in the particular example of the derivation

18Concerning the status of Leibniz’s system for differentialcalculus, it may be more accurate to assert that it was notinconsistent, in the sense that the contradictions allegedby Berkeley and others turn out not to have been there inthe first place once one takes into account Leibniz’s gen-eralized notion of equality and his transcendental law ofhomogeneity.

st

��

B-continuum

A-continuum

Figure 3. Thick-to-thin: applying the law ofhomogeneity or taking standard part (thethickness of the top line is merely conventional).

of the product rule. The issue is the justificationof the last step in the following calculation:

(18)

d(uv) = (u+ du)(v + dv)− uv= udv + vdu+ dudv= udv + vdu.

The last step in the calculation (18), namely,

udv + vdu+ dudv = udv + vdu,is an application of the TLH.19

In his 1701 text Cum Prodiisset [76, pp. 46–47],Leibniz presents an alternative justification of theproduct rule (see Bos [17, p. 58]). Here he dividesby dx and argues with differential quotients ratherthan differentials. Adjusting Leibniz’s notation tofit with (18), we obtain an equivalent calculation:20

d(uv)dx

= (u+ du)(v + dv)− uvdx

= udv + vdu+ dudvdx

= udv + vdudx

+ dudvdx

= udv + vdudx

.

Under suitable conditions the term(dudvdx

)is

infinitesimal, and therefore the last step

(19)udv + vdu

dx+ dudv

dx= u dv

dx+ v du

dxis legitimized as a special case of the TLH. The TLHinterprets the equality sign in (19) and (17) as therelation of being infinitely close, i.e., an equalityup to infinitesimal error.

Relation [\Leibniz did not use our equality symbol but ratherthe symbol “ [\ ” (see McClenon 1923 [85, p. 371]).Using such a symbol to denote the relation of beinginfinitely close, one could write the calculation

19Leibniz had two laws of homogeneity: one for dimen-sion and the other for the order of infinitesimalness. Bosnotes that they “disappeared from later developments” [17,p. 35], referring to Euler and Lagrange. Note, however, thesimilarity to Euler’s principle of cancellation (see Bair etal. [5]).20The special case treated by Leibniz is u(x) = x. Thislimitation does not affect the conceptual structure of theargument.


Figure 4. Zooming in on infinitesimal εεε (herest(±ε) = 0(±ε) = 0(±ε) = 0). The standard part function

associates to every finite hyperreal the uniquereal number infinitely close to it. The bottom

line represents the “thin” real continuum. Theline at top represents the “thick” hyperreal

continuum. The “infinitesimal microscope” isused to view an infinitesimal neighborhood of 000.

The derivative f ′(x)f ′(x)f ′(x) of f (x)f (x)f (x) is then defined bythe relation f ′(x) [\

f (x+ε)−f (x)εf ′(x) [\

f (x+ε)−f (x)εf ′(x) [\

f (x+ε)−f (x)ε .

of the derivative of y = f (x) where f (x) = x2 asfollows:

f ′(x) [\dydx

= (x+ dx)2 − x2

dx

= (x+ dx+ x)(x+ dx− x)dx

= 2x+ dx[\2x.

Such a relation is formalized by the standardpart function; see the subsection “Standard PartPrinciple” and Figure 3.

Standard Part Principle

In any totally ordered field extension E of R, everyfinite element x ∈ E is infinitely close to a suitableunique element x0 ∈ R. Indeed, via the total order,the element x defines a Dedekind cut on R, and thecut specifies a real number x0 ∈ R ⊂ E. The numberx0 is infinitely close to x ∈ E. The subring Ef ⊂ Econsisting of the finite elements of E thereforeadmits a map

st : Ef → R, x, x0,

called the standard part function.The standard part function is illustrated in

Figure 3. A more detailed graphic representationmay be found in Figure 4.21

The key remark, due to Robinson, is that the limitin the A-approach and the standard part functionin the B-approach are essentially equivalent tools.More specifically, the limit of a sequence (un)can be expressed, in the context of a hyperreal

21For a recent study of optical diagrams in nonstandardanalysis, see (Dossena and Magnani [31], [83]) and (Bair andHenry [8]).

enlargement of the number system, as the standardpart of the value uH of the natural extension of thesequence at an infinite hypernatural index n = H.Thus,

(20) limn→∞

un = st(uH).

Here the standard part function “st” associates toeach finite hyperreal the unique finite real infinitelyclose to it (i.e., the difference between them isinfinitesimal). This formalizes the natural intuitionthat for “very large” values of the index, the termsin the sequence are “very close” to the limit valueof the sequence. Conversely, the standard part of ahyperreal u = [un] represented in the ultrapowerconstruction by a Cauchy sequence (un) is simplythe limit of that sequence:

(21) st(u) = limn→∞

un.

Formulas (20) and (21) express limit and standardpart in terms of each other. In this sense, theprocedures of taking the limit and taking thestandard part are logically equivalent.

Variable Quantity

The mathematical term µεγεϑoς in ancient Greekhas been translated into Latin as quantitas. Inmodern languages it has two competing counter-parts: in English, quantity, magnitude;22 in French,quantité, grandeur ; in German, Quantität, Grösse.The term grandeur with the meaning real numberis still in use in (Bourbaki 1947 [19]). Variablequantity was a primitive notion in analysis aspresented by Leibniz, l’Hôpital, and later Carnotand Cauchy. Other key notions of analysis weredefined in terms of variable quantities. Thus, inCauchy’s terminology a variable quantity becomesan infinitesimal if it eventually drops below anygiven (i.e., constant) quantity (see Borovik and Katz[16] for a fuller discussion). Cauchy notes that thelimit of such a quantity is zero. The notion of limititself is defined as follows:

Lorsque les valeurs successivement at-tribuées à une même variable s’approchentindéfiniment d’une valeur fixe, de manièreà finir par en différer aussi peu que l’onvoudra, cette dernière est appelée la lim-ite de toutes les autres. (Cauchy, Coursd’Analyse [22])

Thus, Cauchy defined both infinitesimals andlimits in terms of the primitive notion of avariable quantity. In Cauchy, any variable quantity qthat does not tend to infinity is expected to

22The term “magnitude” is etymologically related toµεγεϑoς. Thus, µεγεϑoς in Greek and magnitudo in Latinboth mean “bigness”, “big” being mega (µεγα) in Greekand magnum in Latin.


decompose as the sum of a given quantity c andan infinitesimal α:

(22) q = c +α.In his 1821 text [22], Cauchy worked with ahierarchy of infinitesimals defined by polynomialsin a base infinitesimal α. Each such infinitesimaldecomposes as

(23) αn(c + ε)for a suitable integern and infinitesimal ε. Cauchy’sexpression (23) can be viewed as a generalizationof (22).

In Leibniz’s terminology, c is an assigna-ble quantity, while α and ε are inassignable.Leibniz’s transcendental law of homogeneity (seethe subsection “Lex homogeneorum transcen-dentalis”) authorized the replacement of theinassignable q = c +α by the assignable c, since αis negligible compared to c:

(24) q [\ c

(see the subsection “Relation [\ ”). Leibniz empha-sized that he worked with a generalized notion ofequality where expressions were declared “equal”if they differed by a negligible term. Leibniz’s pro-cedure was formalized in Robinson’s B-approachby the standard part function (see the subsection“Standard Part Principle”), which assigns to eachfinite hyperreal number the unique real number towhich it is infinitely close. As such, the standardpart allows one to work “internally” (not in thetechnical NSA sense but) in the sense of exploitingconcepts already available in the toolkit of thehistorical infinitesimal calculus, such as Fermat’sadequality (see the subsection “Adequality”), Leib-niz’s transcendental law of homogeneity (see thesubsection “Lex homogeneorum transcendentalis”),and Euler’s principle of cancellation (see Bair etal. [5]). Meanwhile, in the A-approach as formalizedby Weierstrass, one is forced to work with “exter-nal” concepts such as the multiple-quantifier ε, δdefinitions (see the subsection “Continuity”) whichhave no counterpart in the historical infinitesimalcalculus of Leibniz and Cauchy.

Thus, the notions of standard part and epsilonticlimit, while logically equivalent (see the subsec-tion “Standard Part Principle”), have the followingdifference between them: the standard part princi-ple corresponds to an “internal” development ofthe historical infinitesimal calculus, whereas theepsilontic limit is “external” to it.

Zeno’s Paradox of Extension

Zeno of Elea (who lived about 2,500 years ago)raised a puzzle (the paradox of extension, whichis distinct from his better-known paradoxes ofmotion) in connection with treating any continuous

magnitude as though it consists of infinitely manyindivisibles; see (Sherry 1988 [99]), (Kirk et al. 1983[71]). If the indivisibles have no magnitude, thenan extension (such as space or time) composedof them has no magnitude; but if the indivisibleshave some (finite) magnitude, then an extensioncomposed of them will be infinite. There is a furtherpuzzle: If a magnitude is composed of indivisibles,then we ought to be able to add or concatenatethem in order to produce or increase a magnitude.But indivisibles are not next to one another: aslimits or boundaries, any pair of indivisibles isseparated by what they limit. Thus, the concept ofaddition or concatenation seems not to apply toindivisibles.

The paradox need not apply to infinitesimalsin Leibniz’s sense however (see the subsection“Indivisibles versus Infinitesimals”) for, havingneither zero nor finite magnitude, infinitely manyof them may be just what is needed to producea finite magnitude. And in any case, the additionor concatenation of infinitesimals (of the samedimension) is no more difficult to conceive of thanadding or concatenating finite magnitudes. Thisis especially important, because it allows one toapply arithmetic operations to infinitesimals (seethe subsection “Lex continuitatis” on the law ofcontinuity). See also (Reeder 2012 [93]).

AcknowledgmentsThe work of Vladimir Kanovei was partially sup-ported by RFBR grant 13-01-00006. M. Katz waspartially funded by the Israel Science Foundation,grant No. 1517/12. We are grateful to ReubenHersh and Martin Davis for helpful discussionsand to the anonymous referees for a numberof helpful suggestions. The influence of HiltonKramer (1928–2012) is obvious.

References[1] K. Andersen, One of Berkeley’s arguments on

compensating errors in the calculus, HistoriaMathematica 38 (2011), no. 2, 219–231.

[2] Archimedes, De Sphaera et Cylindro, in ArchimedisOpera Omnia cum Commentariis Eutocii, vol. I, ed.J. L. Heiberg, B. G. Teubner, Leipzig, 1880.

[3] R. Arthur, Leibniz’s Syncategorematic Infinitesi-mals, Smooth Infinitesimal Analysis, and Newton’sProposition 6 (2007). See http://www.humanities.mcmaster.ca/∼rarthur/papers/LsiSiaNp6.rev.pdf

[4] Bachet, Diophanti Alexandrini, ArithmeticorumLiber V (Bachet’s Latin translation).

[5] J. Bair, P. Błaszczyk, R. Ely, V. Henry, V. Kanovei,K. Katz, M. Katz, S. Kutateladze, T. McGaffey,D. Schaps, D. Sherry, and S. Shnider, InterpretingEuler’s infinitesimal mathematics (in preparation).

[6] J. Bair and V. Henry, From Newton’s fluxionsto virtual microscopes, Teaching Mathematics andComputer Science V (2007), 377–384.


http://www.humanities.mcmaster.ca/~rarthur/papers/LsiSiaNp6.rev.pdf



[7] , From mixed angles to infinitesimals, TheCollege Mathematics Journal 39 (2008), no. 3,230–233.

[8] , Analyse infinitésimale - Le calculus redécou-vert, Editions Academia Bruylant Louvain-la-Neuve(Belgium) 2008, 189 pages. D/2008/4910/33.

[9] J. L. Bell, Continuity and infinitesimals, StanfordEncyclopedia of Philosophy, revised 20 July 2009.

[10] E. Bishop, Mathematics as a Numerical Language,1970 Intuitionism and Proof Theory (Proc. Conf.,Buffalo, NY, 1968), pp. 53–71. North-Holland,Amsterdam.

[11] , The crisis in contemporary mathematics. Pro-ceedings of the American Academy Workshop onthe Evolution of Modern Mathematics (Boston, Mass.,1974), Historia Mathematica 2 (1975), no. 4, 507–517.

[12] E. Bishop, Review: H. Jerome Keisler, ElementaryCalculus, Bull. Amer. Math. Soc. 83 (1977), 205–208.

[13] , Schizophrenia in contemporary mathematics.In Errett Bishop: Reflections on Him and His Research(San Diego, Calif., 1983), 1–32, Contemp. Math., vol.39, Amer. Math. Soc., Providence, RI, 1985 [originallydistributed in 1973].

[14] P. Błaszczyk, M. Katz, and D. Sherry,Ten misconceptions from the history ofanalysis and their debunking, Foundationsof Science 18 (2013), no. 1, 43–74. Seehttp://dx.doi.org/10.1007/s10699-012-9285-8 and http://arxiv.org/abs/1202.4153

[15] P. Błaszczyk and K. Mrówka, Euklides, Elementy,Ksiegi V–VI, Tłumaczenie i komentarz [Euclid, Ele-ments, Books V–VI, Translation and commentary],Copernicus Center Press, Kraków, 2013.

[16] A. Borovik and M. Katz, Who gave youthe Cauchy–Weierstrass tale? The dual his-tory of rigorous calculus, Foundations ofScience 17 (2012), no. 3, 245–276. Seehttp://dx.doi.org/10.1007/s10699-011-9235-x and http://arxiv.org/abs/1108.2885

[17] H. J. M. Bos, Differentials, higher-order differentialsand the derivative in the Leibnizian calculus, Archivefor History of Exact Sciences 14 (1974), 1–90.

[18] H. Bos, R. Bunn, J. Dauben, I. Grattan-Guinness,T. Hawkins, and K. Pedersen, From the Calculusto Set Theory, 1630–1910. An Introductory History ,edited by I. Grattan-Guinness, Gerald Duckworth andCo. Ltd., London, 1980.

[19] N. Bourbaki, Théorie de la mesure et de l’intégration,Introduction, Université Henri Poincaré, Nancy, 1947.

[20] K. Bråting, A new look at E. G. Björling and theCauchy sum theorem. Arch. Hist. Exact Sci. 61 (2007),no. 5, 519–535.

[21] H. Breger, The mysteries of adaequare: a vindicationof Fermat, Archive for History of Exact Sciences 46(1994), no. 3, 193–219.

[22] A. L. Cauchy, Cours d’Analyse de L’École Roy-ale Polytechnique. Première Partie. Analysealgébrique. Paris: Imprimérie Royale, 1821. On-line at http://books.google.com/books?id=_mYVAAAAQAAJ&dq=cauchy&lr=&source=gbs_navlinks_s

[23] , Théorie de la propagation des ondes à la sur-face d’un fluide pesant d’une profondeur indéfinie,(published 1827, with additional Notes), Oeuvres,Series 1, Vol. 1, 1815, pp. 4–318.

[24] C. Cerroni, The contributions of Hilbert and Dehn tonon-Archimedean geometries and their impact on theItalian school, Revue d’Histoire des Mathématiques13 (2007), no. 2, 259–299.

[25] G. Cifoletti, La méthode de Fermat: son statutet sa diffusion, Algèbre et comparaison de figuresdans l’histoire de la méthode de Fermat, Cahiersd’Histoire et de Philosophie des Sciences, NouvelleSérie 33, Société Française d’Histoire des Sciences etdes Techniques, Paris, 1990.

[26] A. Connes, Noncommutative Geometry, AcademicPress, Inc., San Diego, CA, 1994.

[27] J. Dauben, The development of the Cantorian settheory. In (Bos et al. 1980 [18]), pp. 181–219.

[28] , Abraham Robinson. 1918–1974, BiographicalMemoirs of the National Academy of Sciences82 (2003), 243-284. Available at the addresseshttp://www.nap.edu/html/biomems/arobinson.pdf and http://www.nap.edu/catalog/10683.html

[29] A. De Morgan, On the early history ofinfinitesimals in England, Philosophical Mag-azine, Ser. 4 4 (1852), no. 26, 321–330.See http://www.tandfonline.com/doi/abs/10.1080/14786445208647134

[30] P. Dirac, The Principles of Quantum Mechanics, 4thedition, Oxford, The Clarendon Press, 1958.

[31] R. Dossena and L. Magnani, Mathematics throughdiagrams: Microscopes in non-standard and smoothanalysis, Studies in Computational Intelligence (SCI)64 (2007), 193–213.

[32] P. Ehrlich, The rise of non-Archimedean mathe-matics and the roots of a misconception. I. Theemergence of non-Archimedean systems of magni-tudes, Archive for History of Exact Sciences 60 (2006),no. 1, 1–121.

[33] R. Ely, Loss of dimension in the history of cal-culus and in student reasoning, The MathematicsEnthusiast 9 (2012), no. 3, 303–326.

[34] Euclid, Euclid’s Elements of Geometry, editedand provided with a modern English transla-tion by Richard Fitzpatrick, 2007. See http://farside.ph.utexas.edu/euclid.html

[35] L. Euler, Introductio in Analysin Infinitorum, Tomusprimus, SPb and Lausana, 1748.

[36] , Introduction to Analysis of the Infinite. Book I,translated from the Latin and with an introductionby John D. Blanton, Springer-Verlag, New York, 1988[translation of (Euler 1748 [35])].

[37] , Institutiones Calculi Differentialis, SPb, 1755.[38] , Foundations of Differential Calculus, English

translation of Chapters 1–9 of (Euler 1755 [37]) byD. Blanton, Springer, New York, 2000.

[39] G. Ferraro, Differentials and differential coefficientsin the Eulerian foundations of the calculus, HistoriaMathematica 31 (2004), no. 1, 34–61.

[40] H. Freudenthal, Cauchy, Augustin-Louis, in Dictio-nary of Scientific Biography, ed. by C. C. Gillispie,vol. 3, Charles Scribner’s Sons, New York, 1971, pp.131–148.

[41] C. I. Gerhardt (ed.), Historia et Origo calculi dif-ferentialis a G. G. Leibnitio conscripta, Hannover,1846.

[42] (ed.), Leibnizens mathematische Schriften,Berlin and Halle: Eidmann, 1850–1863.


http://www.nap.edu/html/biomems/arobinson.pdf

http://www.nap.edu/html/biomems/arobinson.pdf

http://www.tandfonline.com/doi/abs/10.1080/14786445208647134

http://www.tandfonline.com/doi/abs/10.1080/14786445208647134

http://dx.doi.org/10.1007/s10699-012-9285-8

http://dx.doi.org/10.1007/s10699-012-9285-8

http://arxiv.org/abs/1202.4153

http://dx.doi.org/10.1007/s10699-011-9235-x

http://dx.doi.org/10.1007/s10699-011-9235-x


http://books.google.com/books?id=_mYVAAAAQAAJ&dq=cauchy&lr=&source=gbs_navlinks_s



http://farside.ph.utexas.edu/euclid.html

http://farside.ph.utexas.edu/euclid.html

http://www.nap.edu/catalog/10683.html

http://www.nap.edu/catalog/10683.html

[43] G. Granger, Philosophie et mathématique leibnizi-ennes, Revue de Métaphysique et de Morale 86eAnnée, No. 1 (Janvier-Mars 1981), 1–37.

[44] Hermann Grassmann, Lehrbuch der Arithmetik,Enslin, Berlin, 1861.

[45] Rupert Hall and Marie Boas Hall, eds., UnpublishedScientific Papers of Isaac Newton, Cambridge Uni-versity Press, 1962, pp. 15–19, 31–37; quoted in I.Bernard Cohen, Richard S. Westfall, Newton, NortonCritical Edition, 1995, pp. 377–386.

[46] Hermann Hankel, Zur Geschichte der Mathematikin Alterthum und Mittelalter, Teubner, Leipzig, 1876.

[47] T. Heath (ed.), The Works of Archimedes, CambridgeUniversity Press, Cambridge, 1897.

[48] E. Hewitt, Rings of real-valued continuous functions.I, Trans. Amer. Math. Soc. 64 (1948), 45–99.

[49] J. Heiberg (ed.), Archimedis Opera Omnia cumCommentariis Eutocii, Vol. I. Teubner, Leipzig, 1880.

[50] A. Heijting, Address to Professor A. Robin-son, at the occasion of the Brouwer memoriallecture given by Professor A. Robinson onthe 26th April 1973, Nieuw Arch. Wisk. (3)21 (1973), 134–137. MathSciNet Review athttp://www.ams.org/mathscinet-getitem?mr=434756

[51] D. Hilbert, Grundlagen der Geometrie, Festschriftzur Enthüllung des Gauss-Weber Denkmals, Göttin-gen, Leipzig, 1899.

[52] , Les principes fondamentaux de la géométrie,Annales scientifiques de l’E.N.S. 3e série 17 (1900),103–209.

[53] O. Hölder, Die Axiome der Quantität und dieLehre vom Mass, Berichte über die Verhandlungender Königlich Sächsischen Gesellschaft der Wis-senschaften zu Leipzig, Mathematisch-PhysischeClasse, 53, Leipzig, 1901, pp. 1–63.

[54] H. Ishiguro, Leibniz’s Philosophy of Logic and Lan-guage, second edition, Cambridge University Press,Cambridge, 1990.

[55] D. Jesseph, Leibniz on the elimination of infinitesi-mals: Strategies for finding truth in fiction, 27 pages.In Leibniz on the Interrelations between Mathematicsand Philosophy, edited by Norma B. Goethe, Philip Bee-ley and David Rabouin, Archimedes Series, SpringerVerlag, 2013.

[56] V. Kanovei, The correctness of Euler’s method forthe factorization of the sine function into an infiniteproduct, Russian Mathematical Surveys 43 (1988),65–94.

[57] V. Kanovei, M. Katz, and T. Mormann, Tools, ob-jects, and chimeras: Connes on the role of hyperrealsin mathematics, Foundations of Science (online first).See http://dx.doi.org/10.1007/s10699-012-9316-5 and http://arxiv.org/abs/1211.0244

[58] V. Kanovei and M. Reeken, Nonstandard Analysis,Axiomatically, Springer Monographs in Mathematics,Springer, Berlin, 2004.

[59] V. Kanovei and S. Shelah, A definable nonstandardmodel of the reals, Journal of Symbolic Logic 69(2004), no. 1, 159–164.

[60] K. Katz and M. Katz, Zooming in on infinitesi-mal 1 − .9.. in a post-triumvirate era, EducationalStudies in Mathematics 74 (2010), no. 3, 259–273.See http://rxiv.org/abs/arXiv:1003.1501

[61] , When is .999 . . . less than 1? The MontanaMathematics Enthusiast 7 (2010), no. 1, 3–30.

[62] , Cauchy’s continuum. Perspectiveson Science 19 (2011), no. 4, 426–452. Seehttp://www.mitpressjournals.org/toc/posc/19/4 and http://arxiv.org/abs/1108.4201

[63] , Meaning in classical mathematics: Is it atodds with intuitionism? Intellectica 56 (2011), no. 2,223-302. See http://arxiv.org/abs/1110.5456

[64] , A Burgessian critique of nominalistictendencies in contemporary mathematics andits historiography, Foundations of Science 17(2012), no. 1, 51–89. See http://dx.doi.org/10.1007/s10699-011-9223-1 and http://arxiv.org/abs/1104.0375

[65] M. Katz and E. Leichtnam, Commuting andnoncommuting infinitesimals, American Mathemat-ical Monthly 120 (2013), no. 7, 631–641. Seehttp://arxiv.org/abs/1304.0583

[66] M. Katz, D. Schaps, and S. Shnider, Almost equal:The method of adequality from Diophantus to Fermatand beyond, Perspectives on Science 21 (2013), no. 3,283–324. See http://arxiv.org/abs/1210.7750

[67] M. Katz and D. Sherry, Leibniz’s infinitesimals:Their fictionality, their modern implementations,and their foes from Berkeley to Russell and beyond,Erkenntnis 78 (2013), no. 3, 571–625; see http://dx.doi.org/10.1007/s10670-012-9370-y andhttp://arxiv.org/abs/1205.0174

[68] , Leibniz’s laws of continuity and homogene-ity, Notices of the American Mathematical Society59 (2012), no. 11, 1550–1558. See http://www.ams.org/notices/201211/ and http://arxiv.org/abs/1211.7188

[69] M. Katz and D. Tall, A Cauchy-Dirac delta function.Foundations of Science 18 (2013), no. 1, 107–123.See http://dx.doi.org/10.1007/s10699-012-9289-4 and http://arxiv.org/abs/1206.0119

[70] H. J. Keisler, The ultraproduct construction. Ultrafil-ters across mathematics, Contemp. Math., vol. 530,pp. 163–179, Amer. Math. Soc., Providence, RI, 2010.

[71] G. S. Kirk, J. E. Raven, and M. Schofield, PresocraticPhilosophers, Cambridge Univ. Press, 1983, §316.

[72] F. Klein, Elementary Mathematics from an AdvancedStandpoint. Vol. I, Arithmetic, Algebra, Analysis.Translation by E. R. Hedrick and C. A. Noble [Macmil-lan, New York, 1932] from the third German edition[Springer, Berlin, 1924]. Originally published as El-ementarmathematik vom höheren Standpunkte aus(Leipzig, 1908).

[73] D. Laugwitz, Hidden lemmas in the early history ofinfinite series, Aequationes Mathematicae 34 (1987),264–276.

[74] , Definite values of infinite sums: Aspects ofthe foundations of infinitesimal analysis around1820, Archive for History of Exact Sciences 39 (1989),no. 3, 195–245.

[75] , Early delta functions and the use of infinites-imals in research, Revue d’histoire des sciences 45(1992), no. 1, 115–128.

[76] G. Leibniz Cum Prodiisset…mss “Cum prodiissetatque increbuisset Analysis mea infinitesimalis. . . ”in Gerhardt [41, pp. 39–50].

[77] , Letter to Varignon, 2 Feb 1702, in Gerhardt[42, vol. IV, pp. 91–95].

[78] , Symbolismus memorabilis calculi algebraiciet infinitesimalis in comparatione potentiarum


http://www.mitpressjournals.org/toc/posc/19/4

http://www.mitpressjournals.org/toc/posc/19/4



http://dx.doi.org/10.1007/s10699-011-9223-1

http://dx.doi.org/10.1007/s10699-011-9223-1



http://www.ams.org/mathscinet-getitem?mr=434756

http://www.ams.org/mathscinet-getitem?mr=434756


http://dx.doi.org/10.1007/s10670-012-9370-y

http://dx.doi.org/10.1007/s10670-012-9370-y


http://www.ams.org/notices/201211/

http://www.ams.org/notices/201211/



http://dx.doi.org/10.1007/s10699-012-9289-4

http://dx.doi.org/10.1007/s10699-012-9289-4


http://dx.doi.org/10.1007/s10699-012-9316-5

http://dx.doi.org/10.1007/s10699-012-9316-5


http://arxiv.org/abs/arXiv:1003.1501


et differentiarum, et de lege homogeneorumtranscendentali, in Gerhardt [42, vol. V, pp. 377-382].

[79] , La naissance du calcul différentiel, 26 articlesdes Acta Eruditorum. Translated from the Latin andwith an introduction and notes by Marc Parmentier.With a preface by Michel Serres. Mathesis, LibrairiePhilosophique J. Vrin, Paris, 1989.

[80] , De quadratura arithmetica circuli ellipseoset hyperbolae cujus corollarium est trigonome-tria sine tabulis. Edited, annotated, and with aforeword in German by Eberhard Knobloch, Ab-handlungen der Akademie der Wissenschaften inGöttingen, Mathematisch-Physikalische Klasse, Folge3 [Papers of the Academy of Sciences in Göt-tingen, Mathematical-Physical Class. Series 3], 43,Vandenhoeck and Ruprecht, Göttingen, 1993.

[81] J. Łos, Quelques remarques, théorèmes et problèmessur les classes définissables d’algèbres, in Mathemat-ical Interpretation of Formal Systems, pp. 98–113,North-Holland Publishing Co., Amsterdam, 1955.

[82] W. Luxemburg, What is nonstandard analysis? Pa-pers in the foundations of mathematics, AmericanMathematical Monthly 80 (1973), no. 6, part II, 38–67.

[83] L. Magnani and R. Dossena, Perceiving the infiniteand the infinitesimal world: Unveiling and opticaldiagrams in mathematics, Foundations of Science 10(2005), no. 1, 7–23.

[84] P. Mancosu, Philosophy of Mathematics and Math-ematical Practice in the Seventeenth Century, TheClarendon Press, Oxford University Press, New York,1996.

[85] R. McClenon, A contribution of Leibniz to the his-tory of complex numbers, American MathematicalMonthly 30 (1923), no. 7, 369–374.

[86] M. McKinzie and C. Tuckey, Hidden lemmas in Eu-ler’s summation of the reciprocals of the squares,Archive for History of Exact Sciences 51 (1997), 29–57.

[87] H. Meschkowski, Aus den Briefbuchern Georg Can-tors, Archive for History of Exact Sciences 2 (1965),503–519.

[88] T. Mormann and M. Katz, Infinitesimals as an issueof neo-Kantian philosophy of science, HOPOS: TheJournal of the International Society for the History ofPhilosophy of Science, to appear.

[89] D. Mumford, Intuition and rigor and Enriques’squest, Notices Amer. Math. Soc. 58 (2011), no. 2,250–260.

[90] I. Newton, Methodus Fluxionum (1671); English ver-sion, The Method of Fluxions and Infinite Series(1736).

[91] W. Parkhurst and W. Kingsland, Infinity and theinfinitesimal, The Monist 35 (1925), 633–666.

[92] C. Proietti, Natural numbers and infinitesimals: adiscussion between Benno Kerry and Georg Cantor,History and Philosophy of Logic 29 (2008), no. 4, 343–359.

[93] P. Reeder, Infinitesimals for Metaphysics: Conse-quences for the Ontologies of Space and Time, Ph.D.thesis, Ohio State University, 2012.

[94] A. Robinson, Non-standard analysis, Nederl. Akad.Wetensch. Proc. Ser. A 64 = Indag. Math. 23 (1961),432–440 [reprinted in Selected Works; see (Robinson1979 [96, pp. 3–11]).

[95] , Reviews: Foundations of Constructive Analy-sis, American Mathematical Monthly 75 (1968), no. 8,920–921.

[96] , Selected Papers of Abraham Robinson, Vol. II.Nonstandard Analysis and Philosophy, edited andwith introductions by W. A. J. Luxemburg and S.Körner, Yale University Press, New Haven, CT, 1979.

[97] B. Russell, The Principles of Mathematics, Vol. I,Cambridge Univ. Press, Cambridge, 1903.

[98] D. Sherry, The wake of Berkeley’s analyst: Rigormathematicae? Stud. Hist. Philos. Sci. 18 (1987), no. 4,455–480.

[99] , Zeno’s metrical paradox revisited, Philosophyof Science 55 (1988), no. 1, 58–73.

[100] D. Sherry and M. Katz, Infinitesimals, imaginaries,ideals, and fictions, Studia Leibnitiana, to appear. Seehttp://arxiv.org/abs/1304.2137

[101] R. Simson, The Elements of Euclid: Viz. the First SixBooks, together with the Eleventh and Twelfth. TheErrors by which Theon and others have long agovitiated these Books are corrected and some of Eu-clid’s Demonstrations are restored. Robert & AndrewFoulis, Glasgow, 1762.

[102] T. Skolem, Über die Unmöglichkeit einer vollständi-gen Charakterisierung der Zahlenreihe mittels einesendlichen Axiomensystems, Norsk Mat. ForeningsSkr., II. Ser. No. 1/12 (1933), 73–82.

[103] , Über die Nicht-charakterisierbarkeit derZahlenreihe mittels endlich oder abzählbar un-endlich vieler Aussagen mit ausschliesslich Zahlen-variablen, Fundamenta Mathematicae 23 (1934),150–161.

[104] , Peano’s axioms and models of arithmetic,in Mathematical Interpretation of Formal Systems,pp. 1–14, North-Holland Publishing Co., Amsterdam,1955.

[105] R. Solovay, A model of set-theory in which ev-ery set of reals is Lebesgue measurable, Annals ofMathematics (2) 92 (1970), 1–56.

[106] O. Stolz, Vorlesungen über Allgemeine Arithmetik,Teubner, Leipzig, 1885.

[107] D. Tall and M. Katz, A cognitive analysis of Cauchy’sconceptions of function, continuity, limit, and infini-tesimal, with implications for teaching the calculus,Educational Studies in Mathematics, to appear.

[108] P. Vickers, Understanding Inconsistent Science,Oxford University Press, Oxford, 2013.

[109] J. Wallis, The Arithmetic of Infinitesimals (1656).Translated from the Latin and with an introduc-tion by Jacqueline A. Stedall, Sources and Studiesin the History of Mathematics and Physical Sciences,Springer-Verlag, New York, 2004.

[110] Frederik S. Herzberg, Stochastic calculus with in-finitesimals, Lecture Notes in Mathematics, 2067,Springer, Heidelberg, 2013.



Is Mathematical History Written by the Victors? - American

Documents