arxiv:1108.2885v3 [math.ho] 16 oct 2012 · · 2012-10-17arxiv:1108.2885v3 [math.ho] 16 oct 2012...

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WHO GAVE YOU THE CAUCHY–WEIERSTRASS

TALE? THE DUAL HISTORY OF RIGOROUS

CALCULUS

ALEXANDRE BOROVIK AND MIKHAIL G. KATZ0

Abstract. Cauchy’s contribution to the foundations of analysis isoften viewed through the lens of developments that occurred somedecades later, namely the formalisation of analysis on the basis ofthe epsilon-delta doctrine in the context of an Archimedean con-tinuum. What does one see if one refrains from viewing Cauchyas if he had read Weierstrass already? One sees, with Felix Klein,a parallel thread for the development of analysis, in the context ofan infinitesimal-enriched continuum. One sees, with Emile Borel,the seeds of the theory of rates of growth of functions as devel-oped by Paul du Bois-Reymond. One sees, with E. G. Bjorling,an infinitesimal definition of the criterion of uniform convergence.Cauchy’s foundational stance is hereby reconsidered.

Contents

1. Introduction 22. A reappraisal of Cauchy’s foundational stance 42.1. Cauchy’s theory of orders of infinitesimals 42.2. How does a null sequence become a Cauchy infinitesimal? 82.3. An analysis of Cours d’Analyse and its infinitesimals 112.4. Microcontinuity and Cauchy’s sum theorem 142.5. Five common misconceptions in the Cauchy literature 163. Comments by modern authors 183.1. Gilain’s limit 183.2. Felscher’s Bestiarium infinitesimale 193.3. D’Alembert or de La Chapelle? 243.4. Brillouet-Belluot: epsilon-delta, period 253.5. Bos on preliminary explanation 25

0Supported by the Israel Science Foundation grant 1294/06.2000 Mathematics Subject Classification. 01A85; Secondary 26E35, 03A05,

97A20, 97C30 .Key words and phrases. Archimedean axiom, Bernoulli, Cauchy, continuity, con-

tinuum, du Bois-Reymond, epsilontics, Felix Klein, hyperreals, infinitesimal, Stolz,sum theorem, transfer principle, ultraproduct, Weierstrass.

1

http://arxiv.org/abs/1108.2885v3

2 ALEXANDRE BOROVIK AND MIKHAIL G. KATZ0

3.6. Medvedev’s delicate question 253.7. Grabiner’s “deep insight” 263.8. Devlin: Will the real Cauchy–Weierstrass please stand up? 273.9. Grattan-Guinness on beautiful mathematics 304. Timeline of modern infinitesimals from Cauchy onward 304.1. From Cauchy to du Bois-Reymond 314.2. From du Bois-Reymond to Robinson 314.3. Nelson 335. Conclusion 33Acknowledgments 34Appendix A. Rival continua 34References 38

1. Introduction

When the second-named author first came across a recent book byAnderson et al entitled Who Gave You the Epsilon? And Other Tales ofMathematical History [2], he momentarily entertained a faint glimmerof hope. The book draws its title from an older essay, entitled WhoGave You the Epsilon? Cauchy and the origins of rigorous calculus [61].The faint hope was that the book would approach the thesis implied bythe title of the older essay, in a critical spirit, namely, as an ahistorical1

tale in need of re-examination. Anderson et al not having undertakenthe latter task, such an attempt is made here.

Cauchy based his definitions of both limits and infinitesimals on theconcept of a variable quantity.2 The variable quantities used in hisCours d’Analyse in 1821 are generally understood by scholars as being(discrete) sequences of values. Cauchy wrote that a variable quantitytending to zero becomes infinitely small. How do his null sequencesbecome infinitesimals? In 1829, Cauchy developed a detailed theoryof infinitesimals of arbitrary order (not necessarily integer), based onrates of growth of functions. How does his theory connect with the work

1See Grattan-Guinness’s comment on ahistory in the main text around foot-note 65.

2C. Gilain’s claims to the effect that “Cauchy definissait le concept d’infinimentpetit a l’aide du concept de limite, qui avait le premier role” [54, footnote 67] areboth in error; see Subsection 3.1 below. Already in 1973, Hourya Benis Sinaceurwrote: “on dit trop rapidement que c’est Cauchy qui a introduit la ‘methodedes limites’ entendant par la, plus ou moins vaguement, l’emploi systematique del’epsilonisation” [123, p. 108], and pointed out that Cauchy’s definition of limit re-sembles, not that of Weierstrass, but rather that of Lacroix dating from 1810 [123,p. 109].

WHO GAVE YOU THE CAUCHY–WEIERSTRASS TALE? 3

of later authors? In 1902, E. Borel elaborated on du Bois-Reymond’stheory of rates of growth, and outlined a general “theory of increase”of functions, as a way of implementing an infinitesimal-enriched con-tinuum. Borel traced the lineage of such ideas to Cauchy’s text. Weexamine several views of Cauchy’s foundational contribution in analy-sis.

Cauchy’s foundational stance has been the subject of an ongoingcontroversy. The continuing relevance of Cauchy’s foundational stancestems from the fact that Cauchy developed some surprisingly modernmathematics using infinitesimals. The following four items deserve tobe mentioned:

(1) Cauchy’s proof of the binomial formula (series) for arbitrary ex-ponents . The proof exploits infinitesimals. D. Laugwitz [90, p. 266]argues that this is the first correct proof of the formula.3

(2) Cauchy’s use of the Dirac delta function. Over a century beforeDirac, Cauchy exploited “delta functions” to solve problems in Fourieranalysis and in the evaluation of singular integrals. Such functions weredefined in terms of an infinitesimal parameter, see Freudenthal [51],Laugwitz [91, p. 219], [92].

(3) Cauchy’s definition of continuity . Cauchy defined continuity of afunction y = f(x) as follows: an infinitesimal change α of the indepen-dent variable x always produces an infinitesimal change f(x + α) − f(x)of the dependent variable y [24, p. 34]. Nearly half a century later,Weierstrass reconstructed Cauchy’s infinitesimal definition in the fol-lowing terms: for every ǫ > 0 there exists a δ > 0 such that for everyreal α, if |α| < δ then |f(x + α) − f(x)| < ǫ. Many historians havesought to interpret Cauchy’s definition as a proto-Weierstrassian defi-nition of continuity in terms of limits.4

(4) Cauchy’s “sum theorem”. This result asserts the convergence toa continuous function, of a series of continuous functions under a suit-able condition of convergence. The sum theorem has been the subjectof a historical controversy, ever since A. Robinson [112, p. 271-273] pro-posed a novel reading of the sum theorem that would make it correct.

3except for Bolzano’s proof in 1816, see [93, p. 657].4Thus, Smithies [125, p. 53, footnote 20] cites the page in Cauchy’s book where

Cauchy gave the infinitesimal definition, but goes on to claim that the conceptof limit was Cauchy’s “essential basis” for his concept of continuity [125, p. 58].Smithies looked in Cauchy, saw the infinitesimal definition, and went on to writein his paper that he saw limits (see Subsection 3.1 for an analysis of a similarmisconception in Gilain). Such automated translation has been common at leastsince Boyer [19, p. 277].


The controversy hinges on the question whether the convergence con-dition was meant by Cauchy to hold at the points of an Archimedeancontinuum, or at the points of a Bernoullian continuum, namely, aninfinitesimal-enriched continuum (see Subsection 2.4). Lakatos pre-sented a paper5 in 1966 where he argues that the 1821 result is cor-rect as stated.6 Laugwitz concurs, see e.g. his 1989 text [91]. Post-Weierstrassian historians tend both (a) to reject Cauchy’s infinitesi-mals, claiming they are merely shorthand for limits, and (b) to claimthat his 1821 sum theorem was false. The 1821 formulation of the sumtheorem may in the end be too ambiguous to know what Cauchy’sintention was at the time, if Cauchy himself knew.7

The four examples given above hopefully illustrate the interest inunderstanding the nature of Cauchy’s foundational contribution.

In Section 2, we provide a re-appraisal of Cauchy’s foundationalstance, including Cauchy’s theory of orders of infinitesimals based onorders of growth of functions (Subsection 2.1); how Cauchy null se-quences become infinitesimals (Subsection 2.2); a detailed textual studyof the related portions of the Cours d’Analyse (Subsection 2.3); ananalysis of Cauchy’s key term toujours , strengthening the hypothesisof his sum theorem (Subsection 2.4); an examination of some commonmisconceptions in the Cauchy literature (Subsection 2.5).

In Section 3, we analyze the views of some modern authors. In Sec-tion 4, we provide a timeline of the development of theories of infinitesi-mals based on orders of growth of functions, from Cauchy through Pauldu Bois-Reymond to modern times. Some conclusions and outlook forthe future may be found in Section 5. Appendix A outlines the rel-evant mathematical material on the rival continua, Archimedean andBernoullian.

2. A reappraisal of Cauchy’s foundational stance

2.1. Cauchy’s theory of orders of infinitesimals. Cauchy’s 1821Cours d’Analyse [24] presented only a theory of infinitesimals of poly-nomial rate of growth as compared to a given “base” infinitesimal α.

The shortcoming of such a theory is its limited flexibility. SinceCauchy only considers infinitesimals behaving as polynomials of a fixedinfinitesimal, called the “base” infinitesimal in 1823 [25], his framework

5The paper was published posthumously in 1978 and edited by J. Cleave [89].6Lakatos thereby reversed his position as presented in his Proofs and Refuta-

tions [88].7See more details in Subsection 2.4.


imposes obvious limitations on what can be done with such infinites-imals. Thus, one typically can’t extract the square root of such a“polynomial” infinitesimal.

What is remarkable is that Cauchy did develop a theory to overcomethis shortcoming. Cauchy’s theory of infinitesimals of arbitrary order(not necessarily integer) is noted by A. Borel [16, p. 35-36] (see below).

In 1823, and in more detail in 1829, Cauchy develops a more flexibletheory, where an infinitesimal is represented by an arbitrary function(rather than merely a polynomial) of a base infinitesimal, denoted “i”,see Chapter 6 in [27]. The title of the chapter is significant. Indeed,the title refers to functions as representing the infinitesimals; moreprecisely, “fonctions qui representent des quantites infiniment petites”.Here is what Cauchy has to say in 1829:

Designons par a un nombre constant, rationnel ou irra-tionnel; par i une quantite infiniment petite, et par r unnombre variable. Dans le systeme de quantites infini-ment petites dont i sera la base, une fonction de i rep-resentee par f(i) sera un infiniment petit de l’ordre a,si la limite du rapport f(i)/ir est nulle pour toutes lesvaleurs de r plus petites que a, et infinie pour toutes lesvaleurs de r plus grandes que a [27, p. 281].

Laugwitz [90, p. 271] explains this to mean that the order a of theinfinitesimal f(i) is the uniquely determined real number (possibly +∞,

as with the function e−1/t2) such that f(i)/ir is infinitesimal for r < aand infinitely large for r > a.

Laugwitz [90, p. 272] notes that Cauchy provides an example of func-tions defined on positive reals that represent infinitesimals of orders ∞and 0, namely

e−1/i and1

log i(see Cauchy [27, p. 326-327]).

The development of non-Archimedean systems based on orders ofgrowth was pursued in earnest at the end of the 19th century by suchauthors as Stolz, du Bois-Reymond, Levi-Civita, and Borel.8 Suchsystems have an antecedent in Cauchy’s theory of infinitesimals asdeveloped in his texts dating from 1823 and 1829. Thus, in 1902,E. Borel [16, p. 35-36] elaborated on du Bois-Reymond’s theory ofrates of growth, and outlined a general “theory of increase” of func-tions, as a way of implementing an infinitesimal-enriched continuum.Borel traced the lineage of such ideas to an 1829 text of Cauchy’s on

8See P. Ehrlich’s [39] for a more detailed discussion, as well as Section 4 below.


the rates of growth of functions, see Fisher [47, p. 144] for details. In1966, A. Robinson pointed out that

Following Cauchy’s idea that an infinitely small or in-finitely large quantity is associated with the behaviorof a function f(x), as x tends to a finite value or toinfinity, du Bois-Reymond produced an elaborate the-ory of orders of magnitude for the asymptotic behaviorof functions . . . Stolz tried to develop also a theory ofarithmetical operations for such entities [112, p. 277-278].

Robinson traces the chain of influences further, in the following terms:

It seems likely that Skolem’s idea to represent infinitelylarge natural numbers by number-theoretic functionswhich tend to infinity (Skolem [1934]),9 also is related tothe earlier ideas of Cauchy and du Bois-Reymond [112,p. 278].

Cauchy’s approach is by no means a variant of Robinson’s approach.10

Similarly, the notion of Cauchy as a pre-Weierstrassian who allegedly“tried to dislodge infinitesimals from analysis” is just as dubious. Takento its logical conclusion, the dogma of Cauchy as a pre-Weierstrassiancan assume comical proportions. Thus, the fashionable Stephen Hawk-ing comments that Cauchy

was particularly concerned to banish infinitesimals [68,p. 639],

yet on the very same page 639, Hawking quotes Cauchy’s infinitesimaldefinition of continuity in the following terms:

the function f(x) remains continuous with respect to xbetween the given bounds, if, between these bounds, aninfinitely small increment in the variable always pro-duces an infinitely small increment in the function itself[68, p. 639].

Did Cauchy “banish” infinitesimals? Using infinitely small incrementsis an odd way of doing so. Similarly, historian J. Gray lists continuityamong concepts Cauchy allegedly defined

9The reference is to Skolem’s 1934 work [124]. The evolution of modern infinites-imals is traced further in the main text in Section 4 following footnote 71.

10More specifically, Cauchy was not in the possession of the mathematical toolsrequired to either formulate or justify the ultrapower construction, requiring asit does a set-theoretic framework (dating from the end of the 19th century) to-gether with the existence of ultrafilters (not proved until 1930 by Tarski [131]), seeAppendix A.


using careful, if not altogether unambiguous, limiting

arguments [66, p. 62] [emphasis added–authors],

whereas in point of fact, limits appear in Cauchy’s definition only inthe sense of the endpoints of the domain of definition. Analogous mis-conceptions in Gilain are analyzed in Subsection 3.1.11 Similar post-Weierstrassian sentiments were expressed by editor Michel Blay’s ref-eree for the periodical Revue d’histoire des sciences where the presentarticle was submitted for publication in 2010, and rejected based onthe following assessment by referee 1:12

Our author interprets A. Cauchy’s approach as a forma-tion of the idea of an infinitely small - a variant of theapproach which was developed in the XXth century inthe framework of the nonstandard analysis (a hyperrealversion of E. Hewitt, J. Los, A. Robinson).

The referee’s summary is an inaccurate description of the text refer-eed. Cauchy’s approach is not a variant of Robinson’s, and was neverclaimed to be in this text. Based on such a strawman version of thearticle’s conception, the referee came to the following conclusion:

From my point of view the author’s arguments to sup-port this conception are quite unconvincing.

Indeed, we find such such a strawman conception unconvincing, but theconception was the referee’s, not the authors’. The referee concludedas follows:

The fact that the actual infinitesimals lived somewherein the consciousness of A. Cauchy (as in many anothermathematicians of XIXth - XXth centuries as, for exam-ple, N.N. Luzin) does not abolish his (and theirs) con-stant aspiration to dislodge them in the subconscious-ness and to found the calculus on the theory of limit.13

Did Cauchy have a “constant aspiration to dislodge infinitesimals in thesubconsciousness and to found the calculus on the theory of limit”?Certainly not. Felix Klein knew better: fifty years before Robinson,Klein clearly realized the potential of the infinitesimal approach tothe foundations. Having outlined the developments in real analysisassociated with Weierstrass and his followers, Klein pointed out that

11See [77, 75] for further discussion of the post-Weierstrassian bias in Cauchyscholarship.

12The pdf of the version submitted to Blay, as well as the two referee reports,may be found at http://u.cs.biu.ac.il/∼katzmik/straw.html

13Opposing infinitesimals and limits is in itself a conceptual error; see footnote 33.

http://u.cs.biu.ac.il/~katzmik/straw.html


The scientific mathematics of today is built upon theseries of developments which we have been outlining.But an essentially different conception of infinitesimalcalculus has been running parallel with this [conception]through the centuries [86, p. 214].

Such a different conception, according to Klein, “harks back to oldmetaphysical speculations concerning the structure of the continuumaccording to which this was made up of [. . . ] infinitely small parts”[86, p. 214].14

Cauchy did not aspire to dislodge infinitesimals from analysis; on thecontrary, he used them with increasing frequency in his work, includinghis 1853 article [28] where he relied on infinitesimals to express theproperty of uniform convergence, as we analyze in Subsection 2.4.

2.2. How does a null sequence become a Cauchy infinitesimal?

The nature of Cauchy’s infinitesimals has been the subject of an on-going debate for a number of decades. Post-Weierstrassian historianstend to dismiss Cauchy’s infiniment petits as merely a linguistic de-vice masking Cauchy’s use of the limit concept. From this perspective,Cauchy’s infinitesimals are alleged to represent an early anticipation ofthe more rigorous methods developed in the second half of the 19th cen-tury, namely epsilontic analysis. Hourya Benis Sinaceur [123] presenteda critical analysis of the proto-Weierstrassian approach to Cauchy in1973, pointing out in particular that Cauchy’s notion of limit is for themost part a kinetic one rather than an epsilontic one.15 Other historianshave taken Cauchy’s infinitesimals at face value, see e.g., Lakatos [89],Laugwitz [91]. The studies in the past decade include Sad, Teixeira,and Baldino [116], Brating [22], and Katz & Katz [75].

14Klein also formulated a criterion of what it would take for a theory of infinites-imals to be successful. Namely, one must be able to prove a mean value theorem forarbitrary intervals, including infinitesimal ones. In 1928, A. Fraenkel [49, pp. 116-117] formulated a similar requirement in terms of the mean value theorem. Sucha Klein-Fraenkel criterion is satisfied by the Hewitt- Los-Robinson theory by thetransfer principle, see Appendix A.

15Pourciau [109] argues that Newton possessed a clear kinetic conception of limitsimilar to Cauchy’s, and cites Newton’s lucid statement to the effect that “Thoseultimate ratios . . . are not actually ratios of ultimate quantities, but limits . . . whichthey can approach so closely that their difference is less than any given quantity. . . ”See Newton [106, p. 39] and [107, p. 442]. The same point, and the same passagefrom Newton, appeared a century earlier in Russell [114, item 316, p. 338-339]. Seealso footnote 33.


To what extent did Cauchy intend the process that he described asa null sequence “becoming” an infinitesimal, to involve some kind of acollapsing?

Cauchy did not have access to the modern set theoretic mentality(currently dominant in the area of mathematics), where equivalencerelation and quotient space constructions are taken for granted. Onecan still ponder the following question: to what extent may Cauchyhave anticipated such collapsing phenomena?

Part of the difficulty in answering such a question is Cauchy’s hands-on approach to foundations. Cauchy was less interested in founda-tional issues than, say, Bolzano.16 To Cauchy, getting your sedan outof the garage was the only justification for shoveling away the snowthat blocks the garage door. Once the door open, Cauchy is in topgear within seconds, solving problems and producing results. To him,infinitesimals were the asphalt under the snow, not the snow itself.Bolzano wanted to shovel away all the snow from the road. Half acentury later, the triumvirate17 shovel ripped out the asphalt togetherwith the snow, intent on consigning the infinitesimal to the dustbin ofhistory.

For these reasons, it is not easy to gauge Cauchy’s foundationalstance precisely. For instance, is there evidence that he felt that twonull sequences that coincide except for a finite number of terms, would“generate” the same infinitesimal?18 Support for this idea comes fromseveral sources, including a very unlikely one, namely Felscher’s es-say [46] in The American Mathematical Monthly from 2000, where heattacks both Laugwitz’s interpretation and the idea that infinitesimalsplay a foundational role in Cauchy.19

In his zeal to do away with Cauchy’s infinitesimals, Felscher seeksto describe them in terms of the modern terminology of germs . Heretwo sequences are in the same germ if they agree at infinity, i.e., forall sufficiently large values of the index n of the sequence 〈un : n ∈ N〉,

16Thus, Hourya Benis Sinaceur writes: “Bolzano est avant tout preoccupe derigueur theorique, d’ou le soucit constant de demonstrations formelles [. . . ] Cauchy,au contraire, donne rarement des demonstrations en forme [. . . ] Son expose adavantage des qualites de synthese que de rigueur formelle” [123, p. 102].

17Boyer [19, p. 298] refers to Cantor, Dedekind, and Weierstrass as “the greattriumvirate”.

18Note that the term “generate” was used by K. Brating [22] to describe thepassage from sequence to infinitesimal in Cauchy.

19Felscher’s text is examined in more detail in Subsection 3.2.


that is, are equal almost everywhere.20 Felscher’s maneuver success-fully eliminates the term “infinitesimal” from the picture, but has theeffect of undermining Felscher’s own thesis, by lending support to thepresence of such “collapsing” in Cauchy.

Indeed, the idea of reading germs of sequences into Cauchy is pre-cisely Laugwitz’s thesis in [90, p. 272]. Germs of sequences are also thebasis of Laugwitz’s Ω-calculus [117], a ring constructed using a Frechetfilter.21

In an editorial footnote to Lakatos’s essay, J. Cleave outlines a non-Archimedean system developed by Chwistek [29], involving a quotientby a Frechet filter (similarly to the Schmieden–Laugwitz construc-tion). Cleave then concludes that the relation of such infinitesimalsto Cauchy’s is “obvious”.22 To us it appears that the said relationrequires additional argument.

Cauchy’s published work contains evidence that he intuitively senseda “collapsing” involved in the passage from a null sequence to an in-finitesimal. Thus, the second chapter of the Cours d’Analyse [24] of1821 contains a series of theorems (eight of them) whose main purposeappears to be to emphasize the importance of the asymptotic behaviorof the sequence (i.e., as the index tends to infinity). This thread ispursued in more detail in Subsection 2.3. Furthermore, he refers tohis infinitesimals as “quantities”, a term he uses in the context of anordered number system, as opposed to the complex numbers which arealways “expressions” but never “quantities”. In fact, the recent Eng-lish translation [21] of the Cours d’Analyse erroneously translates one

20Equality “almost everywhere” is a 20th century concept; Felscher seems tosuggest that Cauchy thought of the relation between his variable quantities and hisinfinitesimals in a way that would be later described as equality almost everywhere.

21In the Schmieden–Laugwitz construction, a Frechet filter is used where anultrafilter would be used in a hyperreal construction; see Appendix A.

22Cleave writes in his footnote 32*: “A construction of non-standard analysis isgiven in Chwistek [29] (1948) which is derived from a paper published in 1926. Itis basically the reduced power RN/F where F is the Frechet filter on the naturalnumbers (the collection of cofinite sets of natural numbers) (see Frayne, Morel,and Scott [50]) [. . . ] This particular construction is not an elementary extensionof R but there are sufficiently powerful transfer properties to enable some non-standard analysis to be performed. It may be observed that the elements of RN/Fare equivalence classes of sequences of reals, two sequences s1, s2 . . . and t1, t2, . . .being counted equal if for some n, sm = tm for all m ≥ n. The relation of theseclasses to Cauchy’s variables is obvious” [89, p. 160].


of Cauchy’s complex “expressions” as a “quantity”, and the reviewerfor Zentralblatt dutifully notes this error.23

A second aspect of collapsing is the mental transformation of theprocess of “tending to zero” into a concept/noun (as a null sequence istransformed into an infinitesimal), thought of as a reification, or encap-sulation, of a highly compressed process.24 In the successive theoremsin his Chapter 2, Cauchy seeks to collapse the initial articulation of hisinfinitesimal α as a temporally-deployed process , by deliberately leav-ing out the implied index (i.e. label of the terms of the sequence), andby explicitly specifying and emphasizing a rival index: the exponentin a power αn of the infinitesimal, thought of as an infinitesimal of ahigher and higher order.

We will examine Cauchy’s approach further in Section 2.3 by meansof a detailed analysis of his text.

2.3. An analysis of Cours d’Analyse and its infinitesimals. Wewill refer to the pages in the Cours d’Analyse [24] itself, rather thanthe collected works.

Cauchy’s Chapter 2, section 1 starts on page 26. Here Cauchy writesthat a variable quantity becomes infinitely small if, etc. Here “infinitelysmall” is an adjective, and is not used as a noun-adjective pair.

On page 27, Cauchy employs the noun-adjective combination, byreferring to “infinitely small quantities” (quantites infiniment petites ,still in the feminine). He denotes such a quantity α. Note that theindex in the implied sequence is suppressed (namely, α appears withouta lower index).25

On page 28, he introduces a competing numerical index, namely theexponent, by forming the infinitesimals

α, α2, α3, . . .

By the time we reach the bottom of the page (fourth line from thebottom), he is already employing “infinitely small” as a noun in itsown right: infiniment petits (in the masculine plural).

On page 29, Theorem 1 asserts that a highest-order infinitesimal willbe smaller than all the others (infinitesimals are consistently referred

23The reviewer, Reinhard Siegmund-Schultze, was far more sensitive to Cauchy’sinfinitesimals on this occasion than in an earlier instance, occasioned by a reviewof Felscher’s text; see footnote 56.

24In the education literature, such a compressing phenomenon is studied un-der the name of procept (process+concept) [65], encapsulation [38], and reifica-tion [120].

25Note that Cauchy uses lower indices to indicate terms in a sequence in hisproof of the intermediate value theorem [24, Note III, p. 462].


independent variableincrement (∆x)

dependent variableincrement (∆y)

Cauchy’s firstdefinition

infinitesimal variable tending tozero

Cauchy’s seconddefinition

infinitesimal infinitesimal

Table 1. Cauchy’s first two definitions of continuity in

1821 are of the form “if ∆x is . . ., then ∆y is . . .”. Note

the prevalence of the term “infinitesimal”.

to in the masculine). The theorem has not yet chosen a letter label forthe competing index (i.e. order of infinitesimal).

Still on page 29, Theorem 2 for the first time introduces a label forthe order of the infinitesimal, namely the letter n, as in

αn.

On page 30, Theorem 3 introduces several different letter indices:

n, n′, n′′ . . .

for the orders of his infinitesimals. The theorem concerns the order ofthe sum, again compelling the student to focus on the competing or-ders (at the expense of the suppressed index of the “variable quantity”itself).

Still on page 30, Theorem 4 introduces the terminology of polynomi-als in α, and describes their orders n, n′, n′′ . . . for the first time as a“sequence”. We now have two “sequences”: (the variable quantity) αitself, whose index is implicit (was never labeled), and the sequenceof orders, which are both emphasized and elaborately labeled using“primes” ′ and double primes ′′.

In all these theorems, it is the asymptotic behavior of null sequencesthat is constantly emphasized, which suggests that Cauchy might havefound it perfectly natural to identify/collapse sequences that agree al-most everywhere. Terminology finit par etre, finit par devenir (sugges-tive of such collapse) is employed repeatedly.

On pages 31 and 32, three additional theorems and one corollary arestated, for a total of eight results on the asymptotic behavior of suchnull sequences.


By the time Cauchy reaches Section 2 of Chapter 2 on page 34(concerning continuity of functions), he has already encapsulated theprocess implied in the notion of “variable quantity”, into the con-cept/masculine noun infiniment petit . When he evokes an infinitelysmall x-increment α, only a stubborn Weierstrassian will refuse to in-terpret his α as a concept/noun. The second definition (out of the threedefinitions of continuity given here) is the one Cauchy italicizes, imply-ing it is the main one. Here both the x-increment and the y-incrementare described as infinitely small increments (see Table 1).

In this context, the verb “become” is being used in two differentsenses:

(a) the terms in the sequence become smaller than any number;(b) the encapsulating sense of a process being compressed into (and

thus becoming) a concept/noun,

as analyzed by Sad, Teixeira, and Baldino [116], who employed the ter-minology of a transformation of essence. It is interesting to note thatBolzano fought against the sense (a), by suppressing the parametrisa-tion altogether, and viewing the null sequence as a set (perhaps he wasinfluenced by Zeno paradoxes), but not against the sense (b).

Note Cauchy’s emphasis on the noun aspect of his infinitesimals:

Lorsque les valeurs numeriques26 successives d’une memevariable decroissent indefiniment, de maniere a s’abaisserau-dessous de tout nombre donne, cette variable devientce qu’on nomme un infiniment petit ou une quantite in-finiment petite. Une variable de cette espece a zero pourlimite [24, p. 4].

The use of the noun, un infiniment petit , makes it difficult to interpretthe “becoming” in the sense (a) above; rather, the definition requiressense (b) to be grammatically coherent. Here the variable [quantity]becomes a masculine noun: un infiniment petit . Cauchy is very precisehere: it is the limit of the variable that’s zero. The variable itselfbecomes an infiniment petit . Cauchy wrote neither that a variable isan infinitesimal, nor that the limit of the infinitesimal is zero, but ratherthat the limit of the variable is zero, cf. [116, p. 301-302].

Once the use of infiniment petit as a noun is established, Cauchyfreely uses the term interchangeably as a noun or as an adjective.

Cauchy’s Cours d’Analyse presented only a theory of infinitesimalsof polynomial rate of growth as compared to a given α. His theory ofinfinitesimals of more general order, and its influence on later authors

26The meaning of the expression valeur numerique is subject to debate; see nextsection and footnote 42.


such as E. Borel, was already discussed in Section 1;27 see Section 4 fora broader historical perspective. In Subsection 2.4, we will discuss aspecific application Cauchy makes of his infinitesimals, in the contextof the sum theorem.

2.4. Microcontinuity and Cauchy’s sum theorem. How is Cauchyable to define concepts such as uniform continuity and uniform conver-gence in terms of a single variable, unlike standard definitions thereofwhich call for a pair of variables?

Let x be in the domain of a function f , and consider the followingcondition, which we will call microcontinuity at x:

“if x′ is in the domain of f and x′ is infinitely close to x,then f(x′) is infinitely close to f(x)”.

Then ordinary continuity of f is equivalent to f being microcontinuouson the Archimedean continuum (A-continuum for short), i.e., at everypoint x of its domain in the A-continuum. Meanwhile, uniform conti-nuity of f is equivalent to f being microcontinuous on the Bernouliancontinuum (B-continuum for short), i.e., at every point x of its domainin the B-continuum (the relation of the two continua is discussed inmore detail in Appendix A).

Consider, for example, the function sin(1/x) defined for positive x.The function fails to be uniformly continuous because microcontinuityfails at a positive infinitesimal x (due to ever more rapid oscillation).The function x2 fails to be uniformly continuous because of the failureof microcontinuity at an infinite member of the B-continuum.

A similar distinction exists between pointwise convergence and uni-form convergence.28 Which condition did Cauchy have in mind in 1821?Abel interpreted it as convergence on the A-continuum, and presented“exceptions” (what we would call today counterexamples) in 1826. Ad-ditional such exceptions were published by Seidel and Stokes in the1840s.

Cauchy’s contemporary E. G. Bjorling anticipated the B-continuumdefinition in a remarkable passage dating from 1852:

when one comes to show that in a series, of which theterms are functions of a quantity x, converges [. . . ] foreach given value of x up to a certain limit X , it is notnecessary to believe that the series continues necessarilyto converge [. . . ] for values of x indefinitely close to that

27See also the main text following footnote 8.28See e.g., Goldblatt [59, Theorem 7.12.2, p. 87].


limit (Bjorling [12, p. 455], as cited by Grattan-Guinness[63, p. 232]).

Here the “limit” X is clearly a member of the usual Archimedeancontinuum; but what is the nature of the x mentioned at the endof Bjorling’s phrase? Bjorling’s reference to convergence at x impliesthat x is understood to be an individual/atomic element of the domain;yet speaking of x as being “indefinitely close” to X is meaningless inthe context of an A-continuum. One faces a stark choice of either toe-ing the triumvirate line on the “true” continuum and therefore findingBjorling’s arguments “absurd”, as Pringsheim did [110, p. 345];29 or,removing the blinders so as to envision a mid-19th century anticipationof an infinitesimal-enriched continuum.

Cauchy clarified/modified his position on the sum theorem in 1853.30

In his text [28], he specified a stronger condition of convergence on theB-continuum, including at x = 1

n(explicitly mentioned by Cauchy).

The stronger condition bars Abel’s counterexample.To give a more detailed explanation of Cauchy’s 1853 text, note that

Cauchy’s approach is based on two assumptions which can be statedin modern terminology as follows:

(1) when you have a closed expression for a function, then its valuesat “variable quantities” (such as x = 1

n) are calculated by using

the same closed expression as at real values;(2) to evaluate a function at a variable quantity generated by a

sequence, one evaluates term-by-term.

Now in 1853 Cauchy analyzed Abel’s counterexample (without men-tioning Abel’s name) by first writing down the closed form of the sumof the remainder of the series. This is given by a certain integral. Heproceeded to evaluate it at an infinitesimal x = 1

nusing assumption (1).

Concretely, he substituted the values 1n

into the closed expression (in-tegral) using assumption (2). The sequence he got was not a nullsequence. He concluded that the remainder term at this particular in-finitesimal is not an infinitesimal. Hence, he concluded, the “always”part of his hypothesis is not satisfied by Abel’s example.

Thus, Cauchy’s clarification/modification from 1853 amounts to re-quiring convergence on an incipient form of a B-continuum. Some his-torians acknowledge his clarification/modification, and interpret it asthe addition of the condition of uniform convergence, while adhering toan A-continuum framework. The recent text [75] argued that such an

29As cited by Grattan-Guinness [63, p. 233].30Grattan-Guinness [63] and Brating [22] argue that Cauchy was influenced by

Bjorling.


interpretation is problematic. Namely, Cauchy states the condition interms of a single variable, whereas the traditional definition of uniformcontinuity or convergence in the context of an A-continuum necessarilyrequires a pair of variables. Cauchy specifically evokes x = 1

nwhere

Abel’s “exception/counterexample” fails to converge. The matter isdiscussed in detail by K. Brating [22].

2.5. Five common misconceptions in the Cauchy literature.

Both Laugwitz and Hourya Benis Sinaceur [123] have exposed a numberof misconceptions in the literature concerning Cauchy’s foundationalwork. Five of the most common ones are reproduced below in italics.

1. Bolzano, Cauchy, and Weierstrass were all gardeners who con-tributed to the ripening of the fruit of the notion of limit.

Here the implicit assumption is that the Weierstrassian epsilonticnotion of “limit” in the context of an Archimedean continuum is thecenterpiece of any possible edifice of analysis. Such an assumptionis questionable on two counts. First, as Felix Klein pointed out in1908, there are two parallel threads in the development of analysis,one based on an Archimedean continuum, and the other exploitingan infinitesimal-enriched continuum.31 One risks pre-judging the out-come of any analysis of Cauchy by postulating that he is working inthe Archimedean thread. The second implicit assumption is that theWeierstrassian notion of limit is central in Cauchy. Thus, Boyer [19,p. 277] postulates that Cauchy is working with a notion of limit simi-lar to the Weierstrassian one. This requires further argument, and atleast at first glance is incorrect: Cauchy emphasizes infinitesimals as afoundational notion, but he never emphasizes limits as a foundationalnotion. Thus, in Cauchy’s definition of continuity the word “limit”does occur, but only in the sense of the “endpoint” of the interval ofdefinition of the function, rather than the behavior of its values.32

2. Cauchy, along with other mathematicians, abandoned infinitesimalsin favor of other more rigorous notions.

During the period 1814-1820, Cauchy appears to have been ambiva-lent about infinitesimals. Starting in about 1821, he uses them withincreasing frequency both in his textbooks and his research publica-tions, and insists on the centrality of infinitesimals as a foundationalnotion; see Section 4 for a related chronology.

31See material in the main text around footnote 14.32The misconception may be found, for instance, in J. Gray, see main text around

footnote 11.


3. Cauchy was forced to teach infinitesimals at the Ecole.

The intended implication appears to be that Cauchy only used in-finitesimals because of the pressure from the Ecole administration.During the period 1814-1820 there were some tensions with the admin-istration over the delayed appearance of infinitesimals in the syllabus,see [54]. At any rate, Cauchy continued using infinitesimals thoughouthis career and long after completing his teaching stint at the Ecole in1830. Thus he reproduces his 1821 definition of continuity (in terms ofinfinitesimals) as late as 1853, in his text on the sum theorem [28], seeSubsection 2.4.

4. Cauchy based his infinitesimals on the notion of limit.

This is an ambiguous claim, and essentially a play on words on theterm “limit”. The modern audience understands “limit” as a Weier-strassian epsilontic notion. If this is what is claimed, then the claim isfalse. As far as the kinetic notion of limit that Cauchy does mention indiscussing a variable quantity approaching a limit, it is conspicuouslyabsent in Cauchy’s discussion of infinitesimals. Thus, rather than in-finitesimals being based on the notion of limit, it is the notion of avariable quantity that’s primitive, and both infinitesimals and limitsare defined in terms of it (see Subsection 3.1 for a more detailed dis-cussion). Note again that the term “limit” does appear in Cauchy’sdefinition of continuity, but in an entirely different sense, namely end-point of the interval where the function is defined.33

5. Cauchy introduced rigor into calculus that anticipates the rigor ofWeierstrass.

While Cauchy certainly emphasizes rigor, postulating a continuitybetween Cauchy’s rigor and Weierstrassian rigor is a methodological

33To elaborate, we will note that the purported opposition between infinitesimalsand limits (a frequently found claim in the literature) is a conceptual error in itsown right. The opposition is not between limits and infinitesimals. Limits arepresent in both approaches. In the infinitesimal approach, limits can be defined interms of the standard part function, and the latter in terms of limits. The trueopposition is between infinitesimals, on the one hand, and epsilon-delta with itsquantifier complications, on the other. The preoccupation with the word “limit”can be exposed as a purely linguistic one by the following thought experiment, basedon Pourciau’s lucid analysis, see footnote 15. Imagine that Newton had developed

an “ult” notation for his ultimate ratios, e.g., ulth→0f(x+h)−f(x)

hfor the derivative,

and ultn→∞

∑n

i=1 f(xi)∆x for the integral. Post-Weierstrassian historians wouldthen perhaps be more moderate in their enthusiasm for the vague musings aboutlimits (incorrectly) attributed to d’Alembert, see Subsection 3.3.


error.34 To Cauchy, rigor meant abandoning the principle of the “gen-erality of algebra” as practiced by Euler, Lagrange, and others, and itsreplacement by geometry–and by infinitesimals.

3. Comments by modern authors

In this section, we analyze some comments by modern authors re-lated to Cauchy’s foundational contribution.

3.1. Gilain’s limit. C. Gilain affirms the following:

On sait que Cauchy definissait le concept d’infinimentpetit a l’aide du concept de limite, qui avait le premierrole35 (voir Analyse algebrique, p. 19 . . . )” [54, foot-note 67].

Here Gilain is referring to Cauchy’s Collected Works, Serie 2, Tome 3,p. 19, corresponding to [24, p. 4]. Both of Gilain’s claims are erroneous,as we now show. Cauchy starts by discussing variable quantities as aprimary notion, in the following terms:

On nomme quantite variable celle que l’on considerecomme devant recevoir successivement plusieurs valeursdifferentes les unes des autres.

Next, Cauchy exploits his primary notion to evoke his kinetic conceptof limit36 in the following terms:

Lorsque les valeurs successivement attribuees a une memevariable s’approchent indefiniment d’une valeur fixe, demaniere a finir par en differer par aussi peu qu l’onvoudra, cette derniere est appelee la limite de toutesles autres.

Finally, Cauchy proceeds to define infinitesimals in the following terms:

Lorsque les valeurs numeriques successives d’une memevariable decroissent indefiniment, de maniere a s’abaisserau-dessous de tout nombre donne, cette variable devientce qu’on nomme un infiniment petit ou une quantite in-finiment petite. Une variable de cette espece a zero pourlimite.37

34See Klein’s comments discussed in the main text around footnote 14, concern-ing the two parallel strands for the development of analysis.

35“We know that Cauchy defined the concept of infinitely small by means of theconcept of limit, which played the primary role.”

36Such a concept is similar to Newton’s, see footnote 15.37This definition was already analyzed in Subsection 2.3, see main text around

footnote 26.


Thus, Cauchy defined both infinitesimals and limits in terms of variablequantities. Neither is the limit concept primary, nor are infinitesimalsdefined in terms of limits, contrary to Gilain’s claims.38

3.2. Felscher’s Bestiarium infinitesimale. We have argued for aninterpretation of Cauchy’s foundational stance that endeavors to takeCauchy’s infinitesimals at their face value. Such an interpretation hasnot been without its detractors. A decade ago, W. Felscher [46] setout to investigate Cauchy’s continuity, in an 18-page text, marred byan odd focus on d’Alembert.39 To be sure, it is both legitimate andnecessary to examine Cauchy’s predecessors, including d’Alembert, ifone wishes to understand Cauchy himself. Indeed, a debate of longstanding (over a century long, in fact) had opposed two rival method-ologies in the study of the foundations of the new science of Newtonand Leibniz:

(A) a methodology eschewing infinitesimals; and(B) a methodology favoring them.40

It is legitimate to ask which of the two methodologies is the one thatunderpins Cauchy’s oeuvre. However, Felscher’s conceptual frameworkis flawed in a fundamental way. The outcome of his investigation ispredetermined from the outset by the following two factors:

(1) Felscher’s exclusive focus on d’Alembert,41 one of the radicaladherents of the A-methodology, and

(2) Felscher’s postulating a methodological continuity between thework of d’Alembert and Cauchy (see also Subsection 3.3).

Displaying a masterly command of scholarly Latin, Felscher offers thereader a glimpse of the bestiarium infinitesimale in section 6 of hisessay, starting on page 856. The punchline comes in the middle ofpage 857, where Felscher points out that Cauchy refers specifically tonumerical values of his variables, the latter being described by Cauchyas becoming infinitesimals.

The adjective numerical is linked etymologically to the noun number .Cauchy’s numbers (unlike his quantities) are certainly appreciable (i.e.,neither infinitesimal nor infinite, nor even negative), as can be seenby reading the first page Preliminaires of his 1821 Cours D’Analyse.

38For a similar misconception in Smithies, see footnote 4.39Felix Klein [86, p. 103] discusses the error in d’Alembert’s proof of the funda-

mental theorem of algebra, first noticed by Gauss.40See Felix Klein’s comments discussed in the main text around footnote 14.41Felscher mentions Euler and the Bernoullis in his section entitled D’Alembert’s

program, but says not a word about them in his article.


Felscher concludes that the variables assume only appreciable values,but not non-Archimedean ones.42

Felscher’s etymological insight offers a refutation of the Luzin hy-pothesis,43 to the effect that Cauchy variables may pass through non-Archimedean values on their way to zero. Has Felscher shown thatCauchy’s bestiarium infinitesimale is in fact uninhabited?

Hardly so. While chasing out the infinitesimal mouse of Luzin’s hy-pothesis44 for the Cauchy variable, Felscher missed the elephant of thepossibility of the variable becoming an infinitesimal through a processof reification (see Subsection 2.2).

On page 846, Felscher quotes an agitated passage from d’Alembert’s1754 article. D’Alembert attacks the obscurity , and even the falsehoodof a definition of infinitesimals attributed to unnamed geometers, andsums up his thesis by accusing such geometers of charlatanerie, a termably translated as quackery by Felscher, who sums up as follows:

Reading these words today we may get the impressionthat they were written at the time of Weierstrass orCantor,45 or even by a contemporary mathematician.46

42Note that Schubring [118, p. 446], in footnote 14, explains Cauchy’s term nu-merical value as what we would call today the absolute value. Fisher [47, p. 262]interprets Cauchy’s definition accordingly, so as to allow room for infinitesimalvalues of Cauchy’s variables. See also footnote 44 on Cleave.

43Luzin himself, in fact, similarly rejected non-Archimedean time, as discussedby Medvedev [102].

44Luzin was probably not the first and surely not the last to formulate a hy-pothesis to the effect that Cauchy’s variable quantities pass through infinitesimalvalues on their way to zero. Lakatos [89, p. 153] speculates that Cauchy’s vari-ables “ran through Weierstrassian real numbers and infinitesimals”, while J. Cleave(who edited Lakatos’s essay for publication in the Mathematical Intelligencer) infootnote 18* in [89, p. 159] disagrees, limiting Cauchy variables to sequences ofWeierstrassian reals (Cleave quotes the relevant passage on numerical values butdoes not analyze it here). Cleave alludes to the etymological point in [30, p. 268],where he disagrees with Fisher on this point.

45Cantor was indeed a worthy heir to d’Alembert’s anti-infinitesimal vitriol. Can-tor dubbed infinitesimals the cholera bacillus of mathematics, see J. Dauben [33,p. 353], [34, p. 124], H. Meschkowski [104, p. 505]. This was perhaps the mostvitriolic opposition to the B-continuum (see Appendix A) prior to Errett Bishop’sdebasement of meaning, a term he applied to classical mathematics in general in1973 [11], and to infinitesimal calculus a la Robinson in particular, in 1975 [10], see[77] and [76] for details.

46It is worth pondering which contemporary mathematician (known for anti-infinitesimal vitriol) Felscher may have had in mind here, given his interest inintuitionistic logics [44, 45], see also [76].


It is sobering to realize that, forty years after A. Robinson, a logiciannamed Walter Felscher still conceived of the history of analysis in termsof a triumphant march out of the dark age of the infinitesimal, andtoward the yawning heights of Weierstrassian epsilontics.

D’Alembert’s verbal excesses merely put in relief the fact that nosuch rhetoric is to be found anywhere in either Cauchy or Bolzano.Felscher presents a convincing case that d’Alembert was opposed toinfinitesimals. Felscher’s title Bolzano, Cauchy, epsilon, delta47 couldtherefore have pertinently been replaced by D’Alembert, Weierstrass,epsilon, delta, as the case for Bolzano’s opposition to infinitesimals cansimilarly be challenged.48 Indeed, as Lakatos points out,

[Bolzano] was possibly the only one to see the problemsrelated to the difference between the two continuums:the rich Leibnizian continuuum and, as he called it, its‘measurable’ subset–the set of Weierstrassian real num-bers. Bolzano makes it very clear that the field of ‘mea-surable numbers’49 constitutes only an Archimedean sub-set of a continuum enriched by non-measurable - infin-itely small or infinitely large - quantities [89, p. 154].

D. Kurepa [87, p. 664] provides some details on Bolzano’s use of in-finitesimals. As far as epsilon-delta techniques are concerned, a case forBolzano’s anticipation thereof is far more convincing than for Cauchy.50

47Apparently, a kind of a mantra: Bolzano, Cauchy, epsilon, delta; Bolzano,Cauchy, epsilon, delta; . . . which, repeated sufficiently many times, would leadone to accept Felscher’s reduction of Cauchy’s continuum to an A-continuum (seeAppendix A).

48In discussing Bolzano’s attitude toward infinitesimals, we have to distinguishbetween the early Bolzano and the late Bolzano. The early Bolzano defines the“Infinitely small” as “variable quantities” in the following terms: A quantity isinfinitely small if it becomes less than any given quantity (here Bolzano does notspeak of “values” [but naturally he thought of them]). The late Bolzano definiesinfinitely small (and infinitely large) numbers ; one of them is 1/(1 + 1 + 1 + . . .)(infinitely many terms). We are grateful to D. Spalt for this historical clarification.

49Measurable number is Bolzano’s term for appreciable number (no relation toLebesgue-measurability). Here Bolzano foreshadows Bjorling’s dichotomy (see [22]),which can be analyzed in terms of A- and B-continua (see Appendix A).

50Hourya Benis Sinaceur wrote: “les oeuvres de Bolzano et de Cauchyrepresentent deux courants distincts, heterogenes a l’origine, et qui ne se rencon-trent pas avant les annees 1870, avec les travaux de Weierstrass, Cantor, etc. Seulel’illusion retrospective et une connaissance indirecte des textes permettent de lesamalgamer dans un meme cours ininterrompu suppose traverser tout le XIXe siecle”[123, p. 111], and cites a letter of H. A. Schwarz to Cantor [119] to the effect that “lamethode de demonstration [. . . ] de Weierstrass est un developpement des principesde Bolzano” [123, p. 112].


Felscher’s intriguing parenthetical remark indicates that he was moresensitive to Cauchy’s language than numerous Cauchy historians:

it is left open whether a quantite variable, with an as-signment converging to zero, actually is or only becomesa quantite infiniment petite [46, p. 850].

On page 851, Felscher presents an analysis of Cauchy’s use of an in-finitesimal quantity, denoted i, in differentiating an exponential func-tion. Here Felscher’s additional parenthetical remark, to the effect that“notational confusion arises from denoting both the variable i and itsvalues by the same letter”, is an unjustified criticism of Cauchy. Thecriticism underscores Felscher’s insensitivity toward the dynamic as-pect of Cauchy’s infinitesimal i, when individual values are irrelevantin the context of the dynamism of the encapsulation taking place when-ever Cauchy evokes an infinitesimal. Otherwise Felscher’s analysis isunexceptionable, save for a non-sequitur of a conclusion:

“No ‘infinitesimal’ non-Archimedean numbers are everused by Cauchy for his quantites infiniment petites .”

In reality, Cauchy’s discussion of the derivative of the exponential func-tion admits a number of possible interpretations.

On page 852, Felscher analyzes Cauchy’s infinitesimals in modernterms:

Using today’s terminology, one would describe Cauchy’sforms to be filled by assignments as functions, but in or-der to distinguish them from the actual functions sub-sequently considered by Cauchy, one might call themfunctional germs . [Emphasis in the original–authors]

Felscher mentions Cauchy’s use of functional germs again on page 855.In his zeal to rename Cauchy’s infinitesimals by employing a modernnotion, so as bashfully to avoid the distasteful infi term, Felscher comesclose to endorsing Laugwitz’s “Cauchy numbers”, similarly defined interms of germs.

Thus, Laugwitz wrote:

Every real function f(u) defined on an interval 0 < u <p represents a Cauchy number. Two such functions f(u)and g(u) represent the same Cauchy number if and onlyif there is an interval 0 < u < q in which f(u) = g(u)[93, p. 659].

Note that Laugwitz essentially defines his “Cauchy numbers” by ex-ploiting the concept of the germ of a function; Laugwitz explicitlymentions function germs in [90, p. 272]. Felscher [46, p. 858] leaves


very little doubt as to how he felt with regard to Laugwitz infinitesi-mals:

In this connection one must also mention certain articlesand books by D. Laugwitz, in which [. . . ] he developshis own ‘mathematics of the infinitesimal’ and uses it tointerpret skillfully various aspects of the mathematicsof the period from Euler to Cauchy. And so we haveglanced at the bestiarium infinitesimale.51

On page 853, Felscher omits a crucial first phrase used by Cauchy informulating his first definition of continuity. Namely, he omits Cauchy’sphrase stating that α is an infinitesimal , corresponding to the upper-left entry in Table 1.52

On pages 854-855, Felscher makes the following statement concerningCauchy’s notion of continuity :

Both Bolzano and Cauchy gave definitions of continu-ity which express today’s [. . . ] continuity. Both madetheir definitions precise and used them in today’s sense;both employed them by comparing numbers and theirdistances with the help of inequalities in order to proveimportant theorems in analysis. However, Cauchy de-fined and used the notion of limit, whereas Bolzano didnot.

Felscher’s assertion concerning continuity and inequalities is mislead-ing. Generally speaking, Cauchy’s limit concept is a kinetic one (seeHourya Benis Sinaceur [123]), and is a derived notion, depending forits definition on the primary notion of a variable quantity . On occa-sion, Cauchy worked with real inequalities in proofs (see, e.g., Grabiner[61]), but he never gave such a definition of continuity .

On page 855, line 9, Felscher alleges that Cauchy’s first definitionof continuity is similar to Bolzano’s, with the implication that the ter-minology of “infinitesimal” is not employed by Cauchy. Now the formin which Cauchy’s definition was quoted by Felscher two pages earlier(see our comment above concerning Felscher’s page 853) did not em-ploy infinitesimals. But the form in which it appears in Cauchy didemploy infinitesimals.53

51The reviewer for MathSciNet concurred, see Section 3.4.52This crucial detail leads Felscher to a further error of a conceptual nature,

discussed below.53See Table 1 (for a summary of Cauchy’s definitions) and footnote 52. Schubring

[118, p. 465] writes that J. Lutzen’s is the best analysis of continuity in Cauchy.Meanwhile, Lutzen [97, p. 166] states: “Cauchy [. . . ] gives two definitions, first


The most remarkable aspect of Felscher’s, unfortunately seriouslyflawed, essay is how close he comes to sensing the cognitive view ofcompression/encapsulation outlined in Subsection 2.2:

We speak of a function or a variable approaching somevalue indefiniment (indefinitely); we imagine a limitingprocess. [. . . ] Thus far, ǫ and δ (and in case of sequencesalso n and N) appear as handles affixed to the stages ofthose infinite processes. It seems that if appropriatelyhandled in our mental exercises, they enable us to usefinitely many arguments to prove statements that, in theend, speak about all the stages of the infinite process [46,p. 858].

3.3. D’Alembert or de La Chapelle? Felscher’s text analyzed inSection 3.2 describes d’Alembert as “one of the mathematicians rep-resenting the heroic age of calculus” [46, p. 845]. Felscher buttresseshis claim by a lengthy quotation concerning the definition of the limitconcept, from the article Limite from the Encyclopedie ou DictionnaireRaisonne des Sciences, des Arts et des Metiers (volume 9 from 1765):

On dit qu’une grandeur est la limite d’une autre grandeur,quand la seconde peut approcher de la premiere pluspres que d’une grandeur donnee, si petite qu’on la puissesupposer, sans pourtant que la grandeur qui approche,puisse jamais surpasser la grandeur dont elle approche;ensorte que la difference d’une pareille quantite a sa lim-ite est absolument inassignable (Encyclopedie, volume 9,page 542).

One recognizes here a kinetic definition of limit already exploited byI. Newton.54 Whatever the merits of attributing visionary status tothis quote, what Felscher overlooked is the fact that the article Limitewas written by two authors. In reality, the above passage definingthe concept of “limit” (as well as the two propositions on limits) didnot originate with d’Alembert, but rather with the encyclopedist Jean-Baptiste de La Chapelle. De la Chapelle was recruited by d’Alembertto write 270 articles for the Encyclopedie.

The section of the article contaning these items is signed (E) (atbottom of first column of page 542), known to be de La Chapelle’s“signature” in the Encyclopedie. Felscher had already committed a

one without infinitesimals, and then one using infinitesimals.” The second claim iscorrect, but not the first.

54See footnote 15 on Pourciau’s analysis.


similar error of attributing de la Chapelle’s work to d’Alembert, in his1979 work [43].55 Note that Robinson [112, p. 267] similarly misat-tributes this passage to d’Alembert.

3.4. Brillouet-Belluot: epsilon-delta, period. An instructive casestudy in a bestiarium-consignment attitude toward infinitesimals is thereview of Felscher’s text for Math Reviews, by one Nicole Brillouet-Belluot [23]. The review contains not an inkling of the fact that thetext in question is a broadside attack on scholars attempting to analyzeCauchy’s infinitesimals seriously. The reviewer mentions the “epsilon-delta technique”, and notes that Cauchy made his “definitions [of con-tinuity] precise and used them in today’s sense”, but fails to mentionthat the definitions in question are infinitesimal ones, a fact not deniedby Felscher (at least in the case of one of the definitions). Her reviewmentions d’Alembert, who does not appear in Felscher’s summary, in-dicating that she had read the body of Felscher’s text itself (ratherthan merely Felscher’s summary).

Brillouet-Belluot notes that Felscher reports on how “limit was ex-plained and used by d’Alembert and Cauchy”. She reports neither onFelscher’s extensive, and vitriolic, quotes from d’Alembert (includingthe dramatic phrase “the metaphysics and the infinitely small quanti-ties, whether larger or smaller than one another, are totally useless inthe differential calculus”), nor d’Alembert’s colorful epithets like char-latanerie/quackery. The reviewer identified with Felscher’s conclusionsto such an extent that she chose to spare the Math Reviews readerthe burden of infinitesimal quackery, judging that Felscher carried thatburden once and for all, for the rest of us.56

3.5. Bos on preliminary explanation. Leibniz historian H. Bos ac-knowledged that Robinson’s hyperreals provide a

preliminary explanation of why the calculus could de-velop on the insecure foundation of the acceptance ofinfinitely small and infinitely large quantities [18, p. 13].

3.6. Medvedev’s delicate question. Reviewer Cooke [31] notes thatMedvedev [101] “devotes considerable space to refuting the point ofview of modern analysis as a thing waiting to be born, which earliermathematicians were striving unsuccessfully to find and for lack of

55We are grateful to D. Spalt for this historical clarification.56Similar remarks apply to the review of Felscher’s text by Reinhard Siegmund-

Schultze [122] for Zentralblatt Math. Siegmund-Schultze was far less myopic ina later instance occasioned by a review of an English translation of the Coursd’analyse, see footnote 23.


which they were merely groping in the dark. In particular he arguesthat the approaches of Cauchy and Weierstrass were so different fromeach other that it would be more accurate to speak of two systems ofanalysis.”

F. Medvedev further points out that nonstandard analysis

makes it possible to answer a delicate question bound upwith earlier approaches to the history of classical anal-ysis. If infinitely small and infinitely large magnitudesare [to be] regarded as inconsistent notions, how couldthey [have] serve[d] as a basis for the construction ofso [magnificent] an edifice of one of the most importantmathematical disciplines? [101, 103]

A powerful question, indeed. How do historians answer Medvedev’squestion?

3.7. Grabiner’s “deep insight”. Not all scholars are satisfied withthe amazing-intuition-and-deep-insight answer offered by J. Grabinerwho writes:

[M]athematicians like Euler and Laplace had a deep in-sight into the basic properties of the concepts of thecalculus, and were able to choose fruitful methods andevade pitfalls [61, p. 188]

How can deep insight manage to “evade pitfalls” if the foundations areregarded as inconsistent? Grabiner further claims that,

[s]ince an adequate response to Berkeley’s objectionswould have involved recognizing that an equation in-volving limits is a shorthand expression for a sequence ofinequalities—a subtle and difficult idea—no eighteenthcentury analyst gave a fully adequate answer to Berkeley[61, p. 189].

This is an astonishing claim, which amounts to reading back into his-tory, developments that came much later. Such a claim amounts topostulating the inevitability of a triumphant march, from Berkeley on-ward, toward the radiant future of Weierstrassian epsilontics.57 Theclaim of such inevitability in our opinion is an assumption that requiresfurther argument.

57Hourya Benis Sinaceur similarly mocks “l’hypothese d’un developpementmonolithique, non differencie, continue, univoque de l’analyse au XIXe siecle. Lefilon, perdu avec Bolzano, est heureusement retrouve par Cauchy, qui le redecouvre,l’elargit, l’exploite plus amplement, le transmet a Weierstrass, etc.” [123, p. 103],and names Boyer [20, p. 271] as one of the culprits in perpetuating such a myth.


Berkeley was, after all, attacking the coherence of infinitesimals .58

He was not attacking the coherence of some kind of incipient form ofWeierstrassian epsilontics and its inequalities. Isn’t there a simpleranswer to Berkeley’s query, in terms of a passage from a point of B-continuum (see Appendix A), to the infinitely close point of the A-continuum, namely passing from a variable quantity to its limitingconstant quantity?

A related attitude on the part of Felscher is discussed in Section 2.3.Like Felscher, Grabiner [61, p. 190] suppresses Cauchy’s reference

to an infinitesimal increment of the independent variable when citingCauchy’s first definition (see Table 1 in Section 2.3 above), therebymanaging to avoid discussing Cauchy’s infinitesimals altogether. Weencounter the oft-repeated claim about “the same confusion betweenuniform and point-wise convergence”59 [61, p. 191]. Her discussion ofCauchy’s rigor does not mention that what rigor meant to Cauchy wasthe replacement of the principle of the generality of algebra, by geome-try, including infinitesimals. Grabiner correctly points out [61, p. 193]that “Mathematicians are used to taking the rigorous foundations forcalculus for granted.” She concludes: “What I have tried to do as a his-torian is to reveal what went into making up that great achievement.”What we have tried to do is to introduce a necessary correction toa post-Weierstrassian reading of Cauchy, influenced by an automatedinfinitesimal-to-limits translation originating no later than Boyer [19,p. 277], see [75] for additional details.

3.8. Devlin: Will the real Cauchy–Weierstrass please stand

up? The following exchange between the second-named author andKeith Devlin, co-founder and Executive Director of Stanford Univer-sity’s H-STAR institute, took place in may 2011. The exchange wasoccasioned by Devlin’s article “Will the real continuous function pleasestand up?”60 The article refers to an alleged “Cauchy–Weierstrass def-inition of continuity”.

MK: I read with interest your online article on “continuous functionplease stand up”. I am not sure which Cauchy–Weierstrass definitionyou are referring to. What exactly was Cauchy’s definition of continu-ity? I think I know what Weierstrass’s was.

58Berkeley’s criticism is dissected into its metaphysical and logical componentsby D. Sherry [121].

59See Subsection 2.4 for a detailed discussion of the controversy over the “sumtheorem”.

60The article is online at http://www.maa.org/devlin/devlin 11 06.html

http://www.maa.org/devlin/devlin_11_06.html


KD: See61 http://www.maa.org/pubs/Calc articles/ma002.pdf

MK: But Grabiner does not say that Cauchy gave an epsilontic defini-tion of continuity. I wonder if there is a reason for that. Do you knowwhat Cauchy’s definition of continuity was?

KD: Actually, she does say that: “Delta-epsilon proofs are first foundin the works of Augustin-Louis Cauchy (1789-1867). This is not alwaysrecognized, since Cauchy gave a purely verbal definition of limit, whichat first glance does not resemble modern definitions.” In the UK whenI was a student, the standard term to refer to the usual ǫ-δ definitionwas “the Cauchy–Weierstrass definition”, and by my read of Grabinerthat description seems appropriate.

MK: But my question concerned the definition of continuity . In theUK, did they refer to a Cauchy–Weierstrass definition of continuity?

KD: It’s the same. f(x) is continuous at a if and only if f(a) is definedand limx→a f(x) = f(a).

MK: Keith, this is your definition of continuity. It is not Cauchy’sdefinition of continuity, for several reasons: (1) Cauchy did not at anytime work with “continuity at a point”, it is always continuity “betweentwo limits”, i.e. on an interval. (2) Cauchy’s definition is invariablyan infinitesimal one: “an infinitesimal x-increment always producesan infinitesimal y-increment”. (3) Elsewhere, Cauchy is working forthe most part with a kinetic definition of limit, akin to that found inNewton.62 Thus, the primary notion is that of a variable quantity.Both infinitesimals and limits are defined in terms of variable quantity.Could the notion of a “Cauchy–Weierstrass definition of continuity”be an ahistorical blunder, regardless of whether it was adopted by themathematicians at a UK college?

KD: That was actually the way continuity was taught in the UK. Tomy mind, once you have the basic epsilon-delta idea, which is a cleverstatic capture of the dynamic concept, everything just drops out. Itcomes down to that one idea. And according to that Grabiner article,that was due to Cauchy and then Weierstrass. Of course, once we hadthat definition, our entire conception changed, and it may be that it isonly from a modern perspective that we can clearly see that there isreally just one fundamental notion there. That happens all the time,of course. Our present day conception of Newton’s differential calculus

61The link provided is a link to an electronic version of Grabiner’s article [61].62See footnote 15.

http://www.maa.org/pubs/Calc_articles/ma002.pdf


is not the one Newton had.63 Sounds like you are trying to get at howthe people at the time conceived things.

MK: What you seem to be saying is that, even though Cauchy himselfsaid nothing of the sort regarding continuity, from a certain “modernperspective” is it proper to interpret Cauchy in a certain way, nowthat in retrospect we know that the epsilontic idea is the “fundamen-tal notion”. This is precisely the ideology that I seek to refute. FelixKlein pointed out in 1908 (in the original edition of his “Elementarymathematics from an advanced viewpoint”) that there are really twoparallel strands to the development of analysis, the epsilontic one andthe infinitesimal one. Emile Borel in 1902 developed further du Bois-Reymond’s theory of infinitesimal-enriched continua in terms of growthrates of functions, and specifically refers to Cauchy’s work in this di-rection from 1829 as inspiration. Note that Klein and Borel said thiswell before Robinson. Viewed as the “father” of the infinitesimal ap-proach ultimately vindicated by Robinson, Cauchy needn’t suffer aforced explanation as a proto-Weierstrassian. Rather, his approach toinfinitesimals was further developed by Stolz, du Bois-Reymond, Borel,Levi-Civita, and others around the turn of the century. When Skolemdeveloped first non-standard models of arithmetic in 1934, he was in-spired partly by du Bois-Reymond’s work, according to Robinson. In1948, Hewitt first developed hyperreal fields using ultraproducts. In1955, Los proved his theorem for ultraproducts, which implies thetransfer principle, which is the mathematical implementation of Leib-niz’s heuristic “law of continuity”: whatever succeeds for the finite,should also succeed for the infinite. Isn’t this lineage of infinitesimalcalculus more plausible than an alleged “Cauchy–Weierstrass definitionof continuity”?

KD: Ah, I assume you are a historian of mathematics. For sure myperspective is very much that of mathematicians, who if they haveany interest in history (and I do) it is precisely from the perspectiveyou describe, tracing back the origins of the now accepted concepts interms that we now use. For instance, we interpret Newton’s work oncalculus as applied to continuous functions of a real variable, when thatis not at all what Newton was doing. (Actually, I don’t think we canreally understand what he was doing, since we are rooted in a modernperspective.) Good luck with your investigations.64

63Pourciau refutes this view; see footnote 15.64Devlin’s article acknowledges that it is based on Nunez et al [108], who similarly

employ the term “Cauchy–Weierstrass definition of continuity”.


To put the above exchange in perspective, it may be useful to recallGrattan-Guinness’s articulation of a historical reconstruction projectin the name of H. Freudenthal [52], in the following terms:

it is mere feedback-style ahistory to read Cauchy (andcontemporaries such as Bernard Bolzano) as if they hadread Weierstrass already. On the contrary, their ownpre-Weierstrassian muddles65 need historical reconstruc-tion [64, p. 176].

3.9. Grattan-Guinness on beautiful mathematics. On the sub-ject of Robinson’s theory, Grattan-Guinness comments as follows:

I made no mention of non-standard analysis in my book,for it was obvious to me that this very beautiful pieceof mathematics had nothing to tell us historically [62,p. 247].

When Grattan-Guinness announced that Robinson’s construction of anon-Archimedean extension of the reals “bears no resemblance to pastarguments in favor of infinitesimals” [62, p. 247], he was only tellingpart of the story. True, the construction favored by Robinson exploitedpowerful compactness theorems66 originating with Malcev [100] andeschewed the sequential approach. On the other hand, the ultrapowerconstruction of the hyperreals, pioneered by E. Hewitt [69] in 1948and popularized by Luxemburg [98] in 1962, is firmly rooted in thesequential approach, and hence connects well with the kinetic vision ofCauchy, shared by L. Carnot.67

4. Timeline of modern infinitesimals from Cauchy onward

The historical sequence of events, as far as continuity is concerned,was as follows:68

(1) first came Cauchy’s infinitesimal definition, namely “infinitesi-mal x-increment always produces an infinitesimal y-increment”;

(2) then came the Dirichlet/Weierstrass-style nominalistic recon-struction of the original definition in terms of real inequalities,dispensing with infinitesimals;

65Grattan-Guinness’s term “muddle” refers to an irreducible ambiguity of his-torical mathematics such as Cauchy’s sum theorem of 1821.

66Robinson uses instead the term “finiteness principle of lower predicate calculus”[112, p. 48] for what is known today as the compactness theorem.

67Some details on the ultrapower construction appear in Appendix A.68We leave out Bolzano’s contribution which, while prior to Cauchy’s, did not

exert any influence until the 1860s.


(3) Cauchy’s work influenced investigations in infinitesimal-enrichedcontinua by du Bois-Reymond, E. Borel, and others at the turnof the century, see Table 2 (a more detailed account appearsbelow).

The historical priority of the infinitesimal definition is clear; whatis open to debate is the role of the modern definition in interpret-ing the historical definition. The common element here is the nullsequence, a basis both for Cauchy infinitesimals, and, via intermedi-ate developments in growth rates of functions as explained below, forultrapower-based infinitesimals.69

4.1. From Cauchy to du Bois-Reymond. Cauchy’s theory of arbi-trary orders of magnitude for his infinitesimals was a harbinger, not ofWeierstrassian epsilontics, but of later theories of infinitesimal-enrichedcontinua as developed by Stolz [126], du Bois-Reymond [37], and others.Emile Borel’s appreciation of the essential continuity between Cauchy’stheory and that of his late 19th century heirs was already discussed inSubsection 2.3.70 Robinson [112, p. 277-278] traces the evolution ofthe infinitesimal ideal from Cauchy to du Bois-Reymond and Stolz, toSkolem.71

4.2. From du Bois-Reymond to Robinson. Du Bois-Reymond’sinvestigations were in turn pursued further by such mathematicians asEmile Borel. In 1902, Borel [16, p. 35-36] cites Cauchy’s definition ofsuch “order of infinitesimal”72 as inspiration for du Bois-Reymond’stheory. Hardy [67] provided a firm foundation for du Bois-Reymond’sorders of infinity in 1910. Artin and Schreier [6] developed the theoryof real closed fields in 1926. Skolem [124] constructed the first non-standard models of arithmetic in 1934. Hewitt’s hyperreals of 1948and Los’s theorem of 1955 have already been discussed. In the 1960s,Robinson proposed an infinitesimal theory [112] as an alternative toWeierstrassian epsilontic analysis. The evolution of infinitesimal fromCauchy to Robinson is summarized in Table 2.

69Modulo suitable foundational material, one can ensure that every hyperrealinfinitesimal is represented by a null sequence; an appropriate ultrafilter (called aP-point) will exist if one assumes the continuum hypothesis, or even the weakerMartin’s axiom (see Cutland et al [32] for details).

70See the main text following footnote 8.71 See the main text around footnote 9.72This definition appears in Oeuvres de Cauchy, serie 2, tome 4, p. 181, corre-

sponding to Cauchy’s Lecons sur le calcul differentiel from 1829, appearing in asection entitled “Preliminaires”; see G. Fisher [48, p. 144].


years author contribution

1821 Cauchy Infinitesimal definition of continuity

1827 Cauchy Infinitesimal delta function

1829 Cauchy Defined “order of infinitesimal” interms of rates of growth of functions

1852 Bjorling Dealt with convergence at points “in-definitely close” to the limit

1853 Cauchy Clarified hypothesis of “sum theorem”by requiring convergence at infinitesi-mal points

1870-1900

Stolz, du Bois-Reymond, andothers

Infinitesimal-enriched number systemsdefined in terms of rates of growth offunctions

1902 Emile Borel Elaboration of du Bois-Reymond’ssystem

1910 G. H. Hardy Provided a firm foundation for duBois-Reymond’s orders of infinity

1926 Artin–Schreier Theory of real closed fields

1930 Tarski Existence of ultrafilters

1934 Skolem Nonstandard model of arithmetic

1948 Edwin Hewitt Ultrapower construction of hyperreals

1955 Los Proved Los’s theorem forshadowingthe transfer principle

1961,1966

Abraham Robinson Non-Standard Analysis

1977 Edward Nelson Internal Set Theory

Table 2. Timeline of modern infinitesimals from Cauchy

to Nelson.

Edward Nelson [105] in 1977 proposed an axiomatic theory parallel toRobinson’s theory, see Subsection 4.3. P. Ehrlich recently constructedan isomorphism of maximal hyperreal and surreal fields, resulting in a“unification of all numbers great and small” [40].


4.3. Nelson. Edward Nelson [105] in 1977 proposed an axiomatic the-ory parallel to Robinson’s theory. Nelson’s axiomatisation, called Inter-nal Set Theory (IST), takes the form of an enrichment of the Zermelo-Fraenkel set theory (ZFC) [49]. IST amounts to a more stratified ax-iomatisation for set theory, more congenial to infinitesimals.

In Nelson’s system, the usual construction of R, when interpretedwith respect to the foundational background provided by IST, producesa number system already possessing entities behaving as infinitesimals.

In more detail, Nelson’s approach is a re-thinking of the foundationalmaterial with a view to allowing a more stratified (hierarchical) numberline. Thus, the canonical set theory, namely ZFC, is modified by theintroduction of a unary predicate “standard”. Then what is known asthe usual construction of the “real” line produces a line that bears astriking resemblance to the Hewitt- Los-Robinson hyperreals.

To illustrate the power of Nelson’s approach, we quote Alain Robertwho starts his book on IST [111] with a homage to Leonard Euler:

Here is how [Euler] deduces the expansion of the cosine func-tion [. . . ] He starts from the de Moivre formula

cosnz =1

2[(cos z + i sin z)n + (cos z − i sin z)n]

= cosn z −n(n− 1)

1 · 2cosn−2 z sin2 z

+n(n− 1)(n − 2)(n − 3)

1 · 2 · 3 · 4cosn−4 z sin4 z + · · ·

and writes [. . . ]sit arcus z infinite parvus; erit cos ·z = 1, sin ·z =z; sit autem n numerus infinite magnus, ut sitarcus nz finitae magnitudinis, puta nz = v

cos ·v = 1 −v2

2!+

v4

4!—etc.

In the context of internal set theory, this argument becomes fully rig-orous.

5. Conclusion

As we have seen, Cauchy experimented with a range of founda-tional approaches, including infinitesimal methodologies. He antici-pated a number of mathematical developments that occurred decadeslater. One such development was the study of the asymptotic behav-ior and rates of growth of functions, culminating in the constructionof infinitesimal-enriched continua by du Bois-Reymond, Emile Borel,G. H. Hardy, and others. Another development anticipated by Cauchy


was the Dirac delta function, of which Cauchy gave an infinitesimaldefinition. Similarly, Bjorling and Cauchy both articulated single-variable definitions of stronger forms of continuity and convergence(today called uniform), in the context of infinitesimal-enriched con-tinua. One avenue for further exploration is Cauchy’s infinitesimalapproach to the degrees of contact among curves.73 Such visionaryanticipations in Cauchy are studiously obfuscated by scholars sportingprotective blinders, conditioned by a conceptual framework stemmingfrom a nominalistic reconstruction of analysis set in motion by the“great triumvirate” of Cantor, Dedekind, and Weierstrass. Accord-ing to the rules of such feedback-style ahistory, to borrow Grattan-Guinness’s term,74 Cauchy’s infinitesimals are subjected to an auto-mated translation to limits, the latter being promoted to first fiddle,Cauchy’s explicit statements to the contrary notwithstanding (see Sub-section 3.1). Yet, many historians have refused to toe the triumvirateline. Cauchy’s anticipations are highlighted in insightful studies byFreudenthal [51], Robinson [112], Hourya Benis Sinaceur [123], Lakatos[89], Cleave [30], Cutland et al [32], Medvedev [101], Laugwitz [91], Sadet al [116], Brating [22], and others.

It bears pointing out that the the calculus of Newton and Leibniz wasnot the same as ours, since they did not have a continuum that lives upto modern standards. Yet historians routinely attribute the inventionof the calculus to them, implicitly taking it for granted that the calculusof Newton and Leibniz can be interpreted in terms of modern calculus,and vice versa. Isn’t it time we applied such bi-interpretability toCauchy’s infinitesimals, as well?

Acknowledgments

We are grateful to Hourya Benis Sinaceur, David Sherry, and DetlefSpalt for a careful reading of an earlier version of the manuscript andfor their helpful comments.

Appendix A. Rival continua

A Leibnizian definition of the derivative as the infinitesimal ratio∆y

∆x,

whose logical weakness was criticized by Berkeley, was modified byA. Robinson by exploiting a map called the standard part , denoted “st”,

73Cauchy does not hesitate to characterize the center of curvature as the meetingpoint of a pair of infinitely close normals [26, p. 91].

74See main text at footnote 65.


st

B-continuum

A-continuum

Figure 1. Thick-to-thin: taking standard part (thethickness of the top line is merely conventional)

from the finite part of a B-continuum (for “Bernoullian”), to the A-continuum (for “Archimedean”), as illustrated in Figure 1.75

We will denote such a B-continuum by a new symbol IIR. We willalso denote its finite part, by

IIR<∞ = x ∈ IIR : |x| < ∞ ;

namely, the difference IIR \ IIR<∞ consists of the inverses of nonzeroinfinitesimals. The map “st” sends each finite point x ∈ IIR, to the realpoint st(x) ∈ R infinitely close to x:

IIR<∞

st

R

Robinson’s answer to Berkeley’s logical criticism (see D. Sherry [121])is to define the derivative of y = f(x) as

f ′(x) = st

(

∆y

∆x

)

,

rather than the infinitesimal ratio ∆y/∆x itself, as in Leibniz. “How-ever, this is a small price to pay for the removal of an inconsistency”[112, p 266].

We illustrate the construction by means of an infinite-resolution mi-croscope in Figure 2.

Note that both the term “hyper-real field”, and an ultrapower con-struction thereof, are due to E. Hewitt in 1948, see [69, p. 74]. Thetransfer principle allowing one to extend every first-order real state-ment to the hyperreals, is due to J. Los in 1955, see [96]. Thus, the

75In the context of the hyperreal extension of the real numbers, the map “st”sends each finite point x to the real point st(x) ∈ R infinitely close to x. In otherwords, the map “st” collapses the cluster of points infinitely close to a real number x,back to x. A comparative study of continua from a predicative angle may be foundin Feferman [42].


Figure 2. Zooming in on infinitesimal ǫ

Hewitt- Los framework allows one to work in a B-continuum satisfyingthe transfer principle.

Hewitt’s construction of hyper-real fields has roots in functional anal-ysis, including works by Gelfand and Kolmogorov [53]. In 1990, Hewittreminisced about his

“efforts to understand the ring of all real-valued continu-ous [not necessarily bounded] functions on a completelyregular T0-space. I was guided in part by a casual re-mark made by Gel’fand and Kolmogorov76 [. . . ] Alongthe way I found a novel class of real-closed fields thatsuperficially resemble the real number field and havesince become the building blocks for nonstandard anal-ysis. I had no luck in talking to Artin about these hy-perreal fields, though he had done interesting work onreal-closed fields in the 1920s. (My published ‘proof’that hyperreal fields are real-closed is false: John Is-bell earned my gratitude by giving a correct proof someyears later.) [. . . ] My ultra-filters also struck no respon-sive chords. Only Irving Kaplansky seemed to think myideas had merit. My first paper on the subject was pub-lished only in 1948” [70]

(Hewitt goes on to detail the influence of his 1948 text). Here Hewittis referring to Isbell’s 1954 paper [71], proving that Hewitt’s hyper-realfields are real closed. Note that a year later, Los [96] proved the generaltransfer principle for such fields, implying in particular the property ofbeing real closed.

76Here Hewitt cites Gelfand and Kolmogoroff [53].


(

QN/

Fu

)

<∞

//

st

IIR<∞

st

Q //)

66♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠

R≃

// R

Figure 3. An intermediate field QN/

Fu is built directly

out of Q

To elaborate on the ultrapower construction of the hyperreals, let QN

denote the space of sequences of rational numbers. Let(

QN)

Cdenote

the subspace consisting of Cauchy sequences. The reals are by defini-tion the quotient field

R :=(

QN)

C

/

Fnull,

where the ideal Fnull contains all the null sequences. Meanwhile, aninfinitesimal-enriched field extension of Q may be obtained by formingthe quotient

QN/

Fu,

see Figure 3. Here a sequence 〈un : n ∈ N〉 is in Fu if and only if theset

n ∈ N : un = 0

is a member of a fixed ultrafilter.77 To give an example, the se-

quence⟨

(−1)n

n

⟩

represents a nonzero infinitesimal, whose sign depends

on whether or not the set 2N is a member of the ultrafilter. To obtaina full hyperreal field, we replace Q by R in the construction, and forma similar quotient

IIR := RN/

Fu.

A more detailed discussion of the ultrapower construction can befound in M. Davis [35]. See also B laszczyk [14] for some philosophi-cal implications. More advanced properties of the hyperreals such assaturation were proved later, see Keisler [84] for a historical outline.A helpful “semicolon” notation for presenting an extended decimal ex-pansion of a hyperreal was described by A. H. Lightstone [95]. See also

77An ultrafilter on N can be thought of as a way of making a systematic choice,between each pair of complementary infinite subsets of N, so as to prescribe whichone is “dominant” and which one is “negligible”. Such choices have to be madein a coherent manner, e.g., if a subset A ⊂ N is negligible then any subset of Ais negligible, as well. The existence of ultrafilters was proved by Tarski [131], seeKeisler [85, Theorem 2.2]. See a related remark about P-points in footnote 69.


heuristic concept mathematical implementation

adequality standard part function

law of continuity transfer principle

infinitesimal-enriched continuum hyperreal number line

Table 3. Modern mathematical implementation of 17th

century heuristic concepts.

P. Roquette [113] for infinitesimal reminiscences. A discussion of infin-itesimal optics is in K. Stroyan [127], H. J. Keisler [83], D. Tall [128],L. Magnani & R. Dossena [99, 36], and Bair & Henry [7].

Applications of the B-continuum range from aid in teaching calculus[41, 73, 74, 129, 130] to the Bolzmann equation (see L. Arkeryd [4, 5]);modeling of timed systems in computer science (see H. Rust [115]);mathematical economics (see R. Anderson [3]); mathematical physics(see Albeverio et al. [1]); etc.

An interesting elementary example of a B-continuum is the ring ofdual numbers, that is, numbers of the form a + bδ, where δ2 = 0. It isuseful, for example, in the theory of linear algebraic groups and linearLie groups, and enable a quick and transparent computation of Liealgebras of groups like SO3(R).

Recently, Giordano [55, 56, 57] introduced a systematic way of en-riching the reals by nilpotent infinitesimals, referring to the resultingstructure as the ring of Fermat reals .

The transfer principle of the modern theory of infinitesimals is amathematical implementation of Leibniz’s heuristic law of continuity,see Robinson [112, p. 266], and Laugwitz [92]. The standard partfunction is a mathematical implementation of Fermat’s concept of ad-equality, see Table 3.

References

[1] Albeverio, S.; Høegh-Krohn, R.; Fenstad, J.; Lindstrøm, T.: Nonstandardmethods in stochastic analysis and mathematical physics. Pure and AppliedMathematics , 122. Academic Press, Inc., Orlando, FL, 1986.

[2] Anderson, M.; Katz, V.; Wilson, R. (Eds.): Who Gave You the Epsilon?& Other Tales of Mathematical History. The Mathematical Association ofAmerica, 2009.

[3] Anderson, R. M.: Infinitesimal Methods in Mathematical Economics,Preprint, 2000. See http://www.econ.berkeley.edu/∼anderson/Book.pdf.

http://www.econ.berkeley.edu/~anderson/Book.pdf


[4] Arkeryd, L.: Intermolecular forces of infinite range and the Boltzmann equa-tion. Arch. Rational Mech. Anal. 77 (1981), no. 1, 11–21.

[5] Arkeryd, L.: Nonstandard analysis. American Mathematical Monthly 112

(2005), no. 10, 926-928.[6] Artin, E.; Schreier, O.: 1926, Algebraische Konstruktion reeller Korper.

Abhandlungen aus dem Mathematischen Seminar der Hamburgischen Uni-veristat, Leipzig 5, pp. 85-99. Reprinted in Serge Lang and John T. Tate(editors), The Collected Papers of Emil Artin, Addison-Wesley PublishingCompany, Reeding, MA., 1965, pp. 258-272.

[7] Bair, J.; Henry, V.: Implicit Differentiation with Microscopes. The Mathe-matical Intelligencer 32 (2010), no. 1, 53–55.

[8] Benci, V.; Di Nasso, M.: Alpha-theory: an elementary axiomatics for non-standard analysis. Expo. Math. 21 (2003), no. 4, 355–386.

[9] Benci, V.; Di Nasso, M.: A purely algebraic characterization of the hyperrealnumbers. Proc. Amer. Math. Soc. 133 (2005), no. 9, 2501–2505.

[10] Bishop, E.: The crisis in contemporary mathematics. Proceedings of theAmerican Academy Workshop on the Evolution of Modern Mathematics(Boston, Mass., 1974). Historia Math. 2 (1975), no. 4, 507-517.

[11] Bishop, E.: Schizophrenia in contemporary mathematics. In Errett Bishop:reflections on him and his research (San Diego, Calif., 1983), 1–32, Contemp.Math. 39, Amer. Math. Soc., Providence, RI, 1985 [published posthumously;originally distributed in 1973].

[12] Bjorling, E.G.: Sur une classe remarquable des series infinies, J. math. puresappl., (1) 17 (1852), 454–472.

[13] Bjorling, E.G.: Om det Cauchyska kriteriet pade fall, dafunctioner af envariabel lata utveckla sig i serie, fortgaende efter de stigande digniteterna afvariabeln, Kongl. Vetens. Akad. Forh. Stockholm series 1b (1852), 166–228.

[14] B laszczyk, P.: Nonstandard analysis from a philosophical point of view. InNon-Classical Mathematics 2009 (Hejnice, 18-22 june 2009), pp. 21-24.

[15] B laszczyk, P.; Katz, M.; Sherry, D.: Ten misconceptions from the historyof analysis and their debunking. Foundations of Science, 2012. Seehttp://dx.doi.org/10.1007/s10699-012-9285-8and http://arxiv.org/abs/1202.4153

[16] Borel, E.: Lecons sur les series a termes positifs, ed. Robert d’Adhemar,Paris (Gauthier-Villars), 1902.

[17] Borovik, A.; Jin, R.; Katz, M.: An integer construction of infinitesimals:Toward a theory of Eudoxus hyperreals. Notre Dame Journal of FormalLogic 53 (2012), no. 4, 557-570.

[18] Bos, H. J. M.: Differentials, higher-order differentials and the derivative inthe Leibnizian calculus. Arch. History Exact Sci. 14 (1974), 1–90.

[19] Boyer, C.: The concepts of the calculus. Hafner Publishing Company, 1949.[20] Boyer, C.: The history of the calculus and its conceptual development.

Dover Publications, Inc., New York, 1959.[21] Bradley, R.; Sandifer, C.: Cauchy’s cours d’analyse. An annotated transla-

tion. Sources and Studies in the History of Mathematics and Physical Sci-ences . Berlin, Springer, 2009.

[22] Brating, K.: A new look at E. G. Bjorling and the Cauchy sum theorem.Arch. Hist. Exact Sci. 61 (2007), no. 5, 519–535.

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

http://arxiv.org/abs/1202.4153


[23] Brillouet-Belluot, N.: review of Felscher [46] for MathSciNet, online athttp://www.ams.org/mathscinet-getitem?mr=1792418

[24] Cauchy, A. L.: Cours d’Analyse de L’Ecole Royale Polytechnique. PremierePartie. Analyse algebrique. Paris: Imprimerie Royale, 1821. Online athttp://books.google.com/books?id= mYVAAAAQAAJ&dq=cauchy&lr=&source=gbs navlinks s

[25] Cauchy, A. L.: Resume des Lecons donnees a l’Ecole Royale Polytechniquesur le Calcul Infinitesimal. Paris: Imprimerie Royale, 1823.

[26] Cauchy, A. L.: Leons sur les applications du calcul infinitesimal a lageometrie. Paris: Imprimerie Royale, 1826.

[27] Cauchy, A. L. (1829): Lecons sur le calcul differentiel. Paris: Debures. InOeuvres completes, Series 2, Vol. 4, pp. 263-609. Paris: Gauthier-Villars,1899.

[28] Cauchy, A. L. (1853) Note sur les series convergentes dont les divers termessont des fonctions continues d’une variable reelle ou imaginaire, entre deslimites donnees. In Oeuvres completes , Series 1, Vol. 12, pp. 30-36. Paris:Gauthier–Villars, 1900.

[29] Chwistek, L.: The Limits of Science. London: Kegan Paul, 1948.[30] Cleave, J.: The concept of “variable” in nineteenth century analysis. British

J. Philos. Sci. 30 (1979), no. 3, 266–278.[31] Cooke, R. L.: Review of Medvedev [101] for MathSciNet,

see http://www.ams.org/mathscinet-getitem?mr=923796[32] Cutland, N.; Kessler, C.; Kopp, E.; Ross, D.: On Cauchy’s notion of infini-

tesimal. British J. Philos. Sci. 39 (1988), no. 3, 375–378.[33] Dauben, J.: Abraham Robinson. The creation of nonstandard analysis. A

personal and mathematical odyssey. With a foreword by Benoit B. Mandel-brot. Princeton University Press, Princeton, NJ, 1995.

[34] Dauben, J.: Arguments, logic and proof: mathematics, logic and the infi-nite. History of mathematics and education: ideas and experiences (Essen,1992), 113–148, Stud. Wiss. Soz. Bildungsgesch. Math., 11, Vandenhoeck &Ruprecht, Gottingen, 1996.

[35] Davis, M.: Applied nonstandard analysis. Pure and Applied Mathematics.Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977.Reprinted by Dover, NY, 2005, seehttp://store.doverpublications.com/0486442292.html

[36] Dossena, R.; Magnani, L.: Mathematics through diagrams: microscopes innon-standard and smooth analysis. Studies in Computational Intelligence(SCI) 64 (2007), 193–213.

[37] du Bois-Reymond, P.: Die allgemeine Funktionentheorie. Erster Teil. Meta-physik und Theorie der mathematischen Grundbegriffe: Grosse, Grenze,Argument. Laupp, Tubingen, 1882.

[38] Dubinsky, E.: Reflective abstraction in advanced mathematical thinking.Advanced mathematical thinking (see item [129]), Springer, 1991.

[39] Ehrlich, P.: The rise of non-Archimedean mathematics and the roots ofa misconception. I. The emergence of non-Archimedean systems of magni-tudes. Arch. Hist. Exact Sci. 60 (2006), no. 1, 1–121.

[40] Ehrlich, P.: The absolute arithmetic continuum and the unification of allnumbers great and small. Bulletin of Symbolic Logic 18 (2012), no. 1.

http://www.ams.org/mathscinet-getitem?mr=1792418

http://books.google.com/books?id=_mYVAAAAQAAJ

http://www.ams.org/mathscinet-getitem?mr=923796

http://store.doverpublications.com/0486442292.html


[41] Ely, R.: Nonstandard student conceptions about infinitesimals. Journal forResearch in Mathematics Education 41 (2010), no. 2, 117-146.

[42] Feferman, S.: Conceptions of the continuum [Le continu mathematique.Nouvelles conceptions, nouveaux enjeux]. Intellectica 51 (2009) 169-189.See also http://math.stanford.edu/∼feferman/papers/ConceptContin.pdf

[43] Felscher, W.: Naive Mengen und abstrakte Zahlen. III. [Naive sets andabstract numbers. III] Transfinite Methoden. With an erratum to Vol. II.Bibliographisches Institut, Mannheim, 1979.

[44] Felscher, W.: Dialogues, strategies, and intuitionistic provability. Ann. PureAppl. Logic 28 (1985), no. 3, 217–254.

[45] Felscher, W.: Dialogues as a Foundation for Intuitionistic Logic, in Hand-book of philosophical logic, Volume 5. ed. Dov M. Gabbay and Franz Guen-thner. Kluwer, 1986 (second edition in 2001).

[46] Felscher, W.: Bolzano, Cauchy, epsilon, delta. American MathematicalMonthly 107 (2000), no. 9, 844–862.

[47] Fisher, G.: Cauchy’s Variables and Orders of the Infinitely Small. TheBritish Journal for the Philosophy of Science, 30 (1979), no. 3, 261–265.

[48] Fisher, G.: The infinite and infinitesimal quantities of du Bois-Reymondand their reception. Arch. Hist. Exact Sci. 24 (1981), no. 2, 101–163.

[49] Fraenkel, A.: Einleitung in die Mengenlehre. Dover Publications, New York,N. Y., 1946 [originally published by Springer, Berlin, 1928].

[50] Frayne, T.; Morel, A. C.; Scott, D. S.: Reduced Direct Products, Funda-menta Mathematica, 51 (1962-3), 195–228.

[51] Freudenthal, H.: Cauchy, Augustin-Louis. In Dictionary of Scientific Biog-raphy, ed. by C. C. Gillispie, vol. 3 (New York: Charles Scribner’s sons,1971), 131-148.

[52] Freudenthal, H.: Did Cauchy plagiarise Bolzano? Arch. Hist. Exact Sci. 7(1971), 375-392.

[53] Gelfand, I.; Kolmogoroff, A.: On rings of continuous functions on topologicalspaces. [J] C. R. (Dokl.) Acad. Sci. URSS, n. Ser. 22 (1939), 11–15.

[54] Gilain, C.: Cauchy et le cours d’analyse de l’Ecole polytechnique. With aneditorial preface by Emmanuel Grison. Bull. Soc. Amis Bibl. Ecole Polytech.1989, no. 5, 145 pp. See http://www.sabix.org/bulletin/b5/cauchy.html

[55] Giordano, P.: The ring of Fermat reals. Advances in Mathematics 225(2010), no 4, 2050–2075.

[56] Giordano, P.: Infinitesimals without logic. Russian Journal of MathematicalPhysics, 17 (2010), no, 2, 159–191.

[57] Giordano P.: Fermat-Reyes method in the ring of Fermat reals. Advancesin Mathematics 228 (2011), 862-893.

[58] Giordano, P.; Katz, M.: Two ways of obtaining infinitesimals by refiningCantor’s completion of the reals, 2011, see http://arxiv.org/abs/1109.3553

[59] Goldblatt, R.: Lectures on the hyperreals. An introduction to nonstandardanalysis. Graduate Texts in Mathematics, 188. Springer-Verlag, New York,1998.

[60] Gordon, E. I.; Kusraev, A. G.; Kutateladze, S. S.: Infinitesimal analysis.Updated and revised translation of the 2001 Russian original. Translatedby Kutateladze. Mathematics and its Applications , 544. Kluwer AcademicPublishers, Dordrecht, 2002.

http://math.stanford.edu/~feferman/papers/ConceptContin.pdf

http://www.sabix.org/bulletin/b5/cauchy.html



[61] Grabiner, J.: Who gave you the epsilon? Cauchy and the origins of rigorouscalculus. American Mathematical Monthly 90 (1983), no. 3, 185–194.

[62] Grattan-Guinness, I.: Letter. Math. Intelligencer 1 (1978/79), no. 4, 247–248.

[63] Grattan-Guinness, I.: The Cauchy-Stokes-Seidel story on uniform conver-gence again: was there a fourth man? Bull. Soc. Math. Belg. Sr. A 38

(1986), 225–235 (1987).[64] Grattan-Guinness, I.: The mathematics of the past: distinguishing its his-

tory from our heritage. Historia Mathematica 31 (2004), 163-185.[65] Gray, E.; Tall, D.: Duality, Ambiguity and Flexibility: A Proceptual View

of Simple Arithmetic. The Journal for Research in Mathematics Education26 (1994), no. 2, 115–141.

[66] Gray, J.: Plato’s ghost. The modernist transformation of mathematics.Princeton University Press, Princeton, NJ, 2008.

[67] Hardy, G. H.: 1910, Orders of Infinity, The “Infinitarcalcul” of Paul DuBois-Reymond. Cambridge University Press, Cambridge. Second Edition,1924. First Edition reprinted by Hafner, New York, 1971.

[68] Hawking, S., Ed.: The mathematical breakthroughs that changed history.Running Press, Philadelphia, PA, 2007.

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

[70] Hewitt, E.: So far, so good: my life up to now. The Mathematical Intelli-gencer 12 (1990), no. 3, 58–63.See http://at.yorku.ca/t/o/p/c/86.dir/hewitt.htm

[71] Isbell, J. R.: More on the continuity of the real roots of an algebraic equa-tion. Proc. Amer. Math. Soc. 5 (1954), 439.

[72] Kanovei, V.; Katz, M.; Mormann, T.: Tools, Objects, and Chimeras:Connes on the Role of Hyperreals in Mathematics. Foundations of Science(to appear). See http://dx.doi.org/s10699-012-9316-5

[73] Katz, K.; Katz, M.: Zooming in on infinitesimal 1−.9.. in a post-triumvirateera. Educational Studies in Mathematics 74 (2010), no. 3, 259-273. SeearXiv:1003.1501.

[74] Katz, K.; Katz, M.: When is .999 . . . less than 1? The Montana MathematicsEnthusiast 7 (2010), No. 1, 3–30.

[75] Katz, K; Katz, M.: Cauchy’s continuum. Perspectives on Science 19 (2011),no. 4, 426-452. See http://dx.doi.org/10.1162/POSC a 00047and http://arxiv.org/abs/1108.4201

[76] Katz, K.; Katz, M.: Meaning in classical mathematics: is it at odds withIntuitionism? Intellectica 56 (2011), no. 2, 223–302. Seehttp://arxiv.org/abs/1110.5456

[77] Katz, K.; Katz, M.: A Burgessian critique of nominalistic tendencies incontemporary mathematics and its historiography. Foundations of Science17 (2012), no. 1, 51–89. See http://dx.doi.org/10.1007/s10699-011-9223-1and http://arxiv.org/abs/1104.0375

[78] Katz, K.; Katz, M.: Stevin numbers and reality, Foundations of Science 17

(2012), no. 2, 109-123. See http://arxiv.org/abs/1107.3688 andhttp://dx.doi.org/10.1007/s10699-011-9228-9

http://at.yorku.ca/t/o/p/c/86.dir/hewitt.htm

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


http://dx.doi.org/10.1162/POSC_a_00047



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



http://dx.doi.org/10.1007/s10699-011-9228-9


[79] Katz, M.; Schaps, D.; Shnider, S.: Almost Equal: The Method of Adequalityfrom Diophantus to Fermat and Beyond. Perspectives on Science 21 (2013),no. 3, to appear.

[80] Katz, M.; Sherry, D.: Leibniz’s infinitesimals: Their fictionality, their mod-ern implementations, and their foes from Berkeley to Russell and beyond.Erkenntnis (online first), see http://dx.doi.org/10.1007/s10670-012-9370-yand http://arxiv.org/abs/1205.0174

[81] —, Leibniz’s laws of continuity and homogeneity. Notices of the AmericanMathematical Society 59 (2012), no. 11, to appear.

[82] Katz, M.; Tall, D.: The tension between intuitive infinitesimals and formalmathematical analysis. Chapter in: Bharath Sriraman, Editor. Crossroadsin the History of Mathematics and Mathematics Education. The MontanaMathematics Enthusiast Monographs in Mathematics Education 12, Infor-mation Age Publishing, Inc., Charlotte, NC, 2011, pp. 71-89. Seehttp://arxiv.org/abs/1110.5747 andhttp://www.infoagepub.com/products/Crossroads-in-the-History-of-Mathematics

[83] Keisler, H. J.: Elementary Calculus: An Infinitesimal Approach. SecondEdition. Prindle, Weber & Schimidt, Boston, 1986.

[84] Keisler, H. J.: The hyperreal line. In Real numbers, generalizations of thereals, and theories of continua, 207–237, Synthese Lib., 242, Kluwer Acad.Publ., Dordrecht, 1994.

[85] Keisler, H. J.: The ultraproduct construction. Proceedings of the UltramathConference, Pisa, Italy, 2008.

[86] Klein, F.: Elementary Mathematics from an Advanced Standpoint. Vol. I.Arithmetic, Algebra, Analysis. Translation by E. R. Hedrick and C. A. No-ble [Macmillan, New York, 1932] from the third German edition [Springer,Berlin, 1924]. Originally published as Elementarmathematik vom hoherenStandpunkte aus (Leipzig, 1908).

[87] Kurepa, D.: Around Bolzano’s approach to real numbers. CzechoslovakMath. J. 32 (107) (1982), no. 4, 655–666.

[88] Lakatos, I.: Proofs and refutations. The logic of mathematical discov-ery. Edited by John Worrall and Elie Zahar. Cambridge University Press,Cambridge-New York-Melbourne, 1976.

[89] Lakatos, I.: Cauchy and the continuum: the significance of nonstandardanalysis for the history and philosophy of mathematics. Math. Intelligencer1 (1978), no. 3, 151–161 (paper originally presented in 1966).

[90] Laugwitz, D.: Infinitely small quantities in Cauchy’s textbooks. HistoriaMath. 14 (1987), no. 3, 258–274.

[91] Laugwitz, D.: Definite values of infinite sums: aspects of the foundations ofinfinitesimal analysis around 1820. Arch. Hist. Exact Sci. 39 (1989), no. 3,195–245.

[92] Laugwitz, D.: Early delta functions and the use of infinitesimals in research.Revue d’histoire des sciences 45 (1992), no. 1, 115–128.

[93] Laugwitz, D.: On the historical developement of infinitesimal mathematics.Part II. The conceptual thinking of Cauchy. Translated from the Germanby Abe Shenitzer with the editorial assistance of Hardy Grant. AmericanMathematical Monthly 104 (1997), no. 7, 654–663.

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



http://www.infoagepub.com/products/Crossroads-in-the-History-of-Mathematics


[94] Lawvere, F. W.: Toward the description in a smooth topos of the dynam-ically possible motions and deformations of a continuous body. Third Col-loquium on Categories (Amiens, 1980), Part I. Cahiers Topologie Geom.Differentielle 21 (1980), no. 4, 377–392.

[95] Lightstone, A. H.: Infinitesimals. American Mathematical Monthly 79

(1972), 242–251.[96] Los, J.: Quelques remarques, theoremes et problemes sur les classes defi-

nissables d’algebres. In Mathematical interpretation of formal systems, 98–113, North-Holland Publishing Co., Amsterdam, 1955.

[97] Lutzen, J.: The foundation of analysis in the 19th century. A history ofanalysis, 155–195, Hist. Math., 24, Amer. Math. Soc., Providence, RI, 2003.

[98] Luxemburg, W.: Nonstandard analysis. Lectures on A. Robinson’s The-ory of infinitesimals and infinitely large numbers. Pasadena: MathematicsDepartment, California Institute of Technology’ second corrected ed., 1964.

[99] Magnani, L.; Dossena, R.: Perceiving the infinite and the infinitesimalworld: unveiling and optical diagrams in mathematics. Foundations of Sci-ence 10 (2005), no. 1, 7–23.

[100] Mal’tsev, A.I. [Malcev, Mal’cev]: Untersuchungen aus dem Gebiete dermathematischen Logik. [J] Rec. Math. Moscou (Matematicheskii Sbornik),1 (43) (1936), 323–335.

[101] Medvedev, F. A.: Nonstandard analysis and the history of classical analy-sis. Patterns in the development of modern mathematics (Russian), 75–84,“Nauka”, Moscow, 1987.

[102] Medvedev, F. A.: N. N. Luzin on non-Archimedean time. (Russian) Istor.-Mat. Issled. 34 (1993), 103–128.

[103] Medvedev, F. A.: Nonstandard analysis and the history of classical analysis.Translated by Abe Shenitzer. American Mathematical Monthly 105 (1998),no. 7, 659–664.

[104] Meschkowski, H.: Aus den Briefbuchern Georg Cantors. Archive for Historyof Exact Sciences 2 (1965), 503-519.

[105] Nelson, E.: Internal set theory: a new approach to nonstandard analysis.Bull. Amer. Math. Soc. 83 (1977), no. 6, 1165–1198.

[106] Newton, I.: Sir Isaac Newton’s mathematical principles of natural philoso-phy and his system of the world, a revision by F. Cajori of A. Motte’s 1729translation. Berkeley: Univ. of California Press, 1946.

[107] Newton, I.: The Principia: Mathematical principles of natural philosophy,translated by I. B. Cohen & A. Whitman, preceded by A guide to Newton’sPrincipia by I. B. Cohen. Berkeley: Univ. of California Press, 1999.

[108] Nunez, R.; Edwards, L.; Matos, J.: Embodied cognition as grounding forsituatedness and context in mathematics education. Educational Studies inMathematics 39 (1999), no. 1-3, 45-65, DOI: 10.1023/A:1003759711966.

[109] Pourciau, B.: Newton and the notion of limit. Historia Math. 28 (2001),no. 1, 18–30.

[110] Pringsheim, A.: On two theorems of Abel pertaining to the continuity ofsums of series. (Ueber zwei Abel’sche Satze, die Stetigkeit von Reihensum-men betreffend.) (German) [J] Munch. Ber. [Sitz.-Ber. Konigl. Akad. Wiss.Munchen] 27 (1897), 343-356.


[111] Robert, A. M.: Nonstandard analysis. Mineola, NY: Dover Publications,2003. ISBN 0-486-43279-3.

[112] Robinson, A.: Non-standard analysis. North-Holland Publishing Co., Ams-terdam 1966.

[113] Roquette, P.: Numbers and models, standard and nonstandard. MathSemesterber 57 (2010), 185–199.

[114] Russell, B.: The Principles of Mathematics. Routledge. London 1903.[115] Rust, H.: Operational Semantics for Timed Systems. Lecture Notes in Com-

puter Science 3456 (2005), 23-29, DOI: 10.1007/978-3-540-32008-1 4.[116] Sad, L. A.; Teixeira, M. V.; Baldino, R. B.: Cauchy and the problem of

point-wise convergence. Archives Internationales d’Histoire des Sciences 51

(2001), no. 147, 277–308.[117] Schmieden, C.; Laugwitz, D.: Eine Erweiterung der Infinitesimalrechnung.

Mathematische Zeitschrift 69 (1958), 1–39.[118] Schubring, G.: Conflicts between generalization, rigor, and intuition. Num-

ber concepts underlying the development of analysis in 17–19th CenturyFrance and Germany. Sources and Studies in the History of Mathematicsand Physical Sciences. Springer-Verlag, New York, 2005.

[119] Schwarz, H. A.: Letter to G. Cantor dated 1 april 1870, in Meschkowski[104].

[120] Sfard, A.: On the dual nature of mathematical conceptions: Reflections onprocesses and objects as different sides of the same coin. Educational Studiesin Mathematics 22 (1991), 1–36.

[121] Sherry, D.: The wake of Berkeley’s Analyst: rigor mathematicae? Stud.Hist. Philos. Sci. 18 (1987), no. 4, 455–480.

[122] Siegmund-Schultze, R.: review of Felscher [46] for Zentralblatt MATH(2000). Seehttp://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1016.01009

[123] Sinaceur, H.: Cauchy et Bolzano. Revue d’Histoire des Sciences et de leursApplications 26 (1973), no. 2, 97–112.

[124] Skolem, Th.: Uber die Nicht-charakterisierbarkeit der Zahlenreihe mittelsendlich oder abzahlbar unendlich vieler Aussagen mit ausschliesslich Zahlen-variablen. Fundamenta Mathematicae 23, 150-161 (1934).

[125] Smithies, F.: Cauchy’s conception of rigour in analysis. Arch. Hist. ExactSci. 36 (1986), no. 1, 41–61.

[126] Stolz, O., 1885, Vorlesungen uber Allgemeine Arithmetik, Erster Theil: All-gemeines und Arithhmetik der Reellen Zahlen, Teubner, Leipzig. Leipzig.

[127] Stroyan, K.: Uniform continuity and rates of growth of meromorphicfunctions. Contributions to non-standard analysis (Sympos., Oberwolfach,1970), pp. 47–64. Studies in Logic and Foundations of Math., Vol. 69, North-Holland, Amsterdam, 1972.

[128] Tall, D: Looking at graphs through infinitesimal microscopes, windows andtelescopes, The Mathematical Gazette 64 (1980), 22–49.

[129] Tall, D. : The psychology of advanced mathematical thinking, in Advancedmathematical thinking. Edited by D. O. Tall, Mathematics Education Li-brary, 11. Kluwer Academic Publishers Group, Dordrecht, 1991.

http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1016.01009


[130] Tall, D.: Dynamic mathematics and the blending of knowledge structuresin the calculus. Transforming Mathematics Education through the use ofDynamic Mathematics, ZDM 41 (june 2009), no. 4, 481-492.

[131] Tarski, A.: Une contribution a la theorie de la mesure, Fundamenta Math-ematicae 15 (1930), 42-50.

Alexandre Borovik is Professor of Pure Mathematics at the Univer-sity of Manchester, United Kingdom, where he has been working for thepast 20 years. His principal research lies in algebra, model theory, andcombinatorics—topics on which he published several monographs anda number of papers. Recently he has become interested in studyingalgebraic phenomena inhabiting the murky boundary between finiteand infinite. He also has an interest in cognitive aspects of math-ematical practice and recently published a book Mathematics underthe Microscope: Notes on Cognitive Aspects of Mathematical Practicewhich explains a mathematician’s outlook at psychophysiological andcognitive issues in mathematics. An (almost final) draft of his bookShadows of the Truth: Metamathematics of Elementary Mathematicscan be found at http://www.maths.manchester.ac.uk/∼avb/ST.pdf

Mikhail G. Katz is Professor of Mathematics at Bar Ilan University,Ramat Gan, Israel. Two of his joint studies with Karin Katz were pub-lished in Foundations of Science: “A Burgessian critique of nominal-istic tendencies in contemporary mathematics and its historiography”and “Stevin numbers and reality”, online respectively at

http://dx.doi.org/10.1007/s10699-011-9223-1 and athttp://dx.doi.org/10.1007/s10699-011-9228-9A joint study with Karin Katz entitled “Meaning in classical mathe-

matics: is it at odds with Intuitionism?” is due to appear in Intellectica.His joint study with David Tall, entitled “The tension between intu-

itive infinitesimals and formal mathematical analysis”, is due to appearas a chapter in a book edited by Bharath Sriraman, seehttp://www.infoagepub.com/products/Crossroads-in-the-History-of-Mathematics

School of Mathematics, University of Manchester, Oxford Street,

Manchester, M13 9PL, United Kingdom

E-mail address : [email protected]

Department of Mathematics, Bar Ilan University, Ramat Gan 52900

Israel

E-mail address : [email protected]

http://www.maths.manchester.ac.uk/~avb/ST.pdf

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

http://dx.doi.org/10.1007/s10699-011-9228-9

http://www.infoagepub.com/products/Crossroads-in-the-History-of-Mathematics

arxiv:1108.2885v3 [math.ho] 16 oct 2012 · · 2012-10-17arxiv:1108.2885v3 [math.ho] 16 oct 2012...

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