19th century real analysis, forward and backward
Jacques Bair, Piotr Blaszczyk, Peter Heinig, Vladimir Kanovei, Mikhail G. Katz
aa r X i v : . [ m a t h . HO ] J u l JACQUES BAIR, PIOTR B LASZCZYK, PETER HEINIG,VLADIMIR KANOVEI, AND MIKHAIL G. KATZ
Abstract. limite in Cauchy’swork necessarily imply that Cauchy was working with an Archi-medean continuum, whereas infinitesimals were merely a conve-nient figure of speech, for which Cauchy had in mind a completejustification in terms of Archimedean limits. However, there is an-other formalisation of Cauchy’s procedures exploiting his limite ,more consistent with Cauchy’s ubiquitous use of infinitesimals, interms of the standard part principle of modern infinitesimal anal-ysis.We challenge a misconception according to which Cauchy wasallegedly forced to teach infinitesimals at the
Ecole Polytechnique .We show that the debate there concerned mainly the issue of rigor ,a separate one from infinitesimals . A critique of Cauchy’s approachby his contemporary de Prony sheds light on the meaning of rigor toCauchy and his contemporaries. An attentive reading of Cauchy’swork challenges received views on Cauchy’s role in the history ofanalysis, and indicates that he was a pioneer of infinitesimal tech-niques as much as a harbinger of the
Epsilontik .Keywords: butterfly model; continuity; infinitesimals; limite ;standard part; variable quantity; Cauchy; de Prony
Contents
1. Introduction 21.1.
Limites limite ? 41.4. Butterfly model 41.5. Siegmund-Schultze on
Cours d’Analyse limite and infiniment petit Un infiniment petit in Cauchy 112.4. Variable quantities, infinitesimals, and limits 122.5. Assigning a sign to an infinitesimal 132.6. Gilain on omnipresence of limits 142.7.
Limite and infinity 153. Minutes of meetings, Poisson, and de Prony 163.1. Cauchy pressured by Poisson and de Prony 163.2. Reports by de Prony 173.3. Course summaries 183.4. Cauchy taken to task 183.5. Critique by de Prony 193.6. De Prony on small oscillations 203.7. Foundations, limits, and infinitesimals 213.8. Cauchy’s A-track arguments 223.9. Lacroix, Laplace, and Poisson 224. Modern infinitesimals in relation to Cauchy’s procedures 245. Conclusion 25Acknowledgments 25References 25
Since Weierstrass’s time, we haveheld a fairly contemptuous view ofthe infinitesimalists which I regardas unfair. – Ivor Grattan-Guinness Introduction
Cauchy exploited the concepts of variable quantity, limit, and infini-tesimal in his seminal 1821 textbook
Cours d’Analyse (CdA). However,the meaning he attached to those terms is not identical to their modernmeanings. While Cauchy frequently used infinitesimals in CdA, some scholars have argued that Cauchyan infinitesimals are merely short-hand for prototypes of ǫ, δ techniques. Moreover, one can legitimatelyask whether the material found in CdA was actually taught by Cauchyin the classroom of the
Ecole Polytechnique (EP). A valuable resourcethat sheds information on such issues is the archive of summaries ofcourses and various
Conseil meetings at the EP, explored by Guitard([29], 1986), Gilain ([23], 1989), and others. Among the key figuresat EP at the time was Gaspard de Prony, whose critique of Cauchy’steaching will be examined in Sections 3.5 and 3.6. While de Prony wascritical of Cauchy, a careful examination of the criticism indicates thatde Prony’s main target was what he felt was excesssive rigor, ratherthan an alleged absence of infinitesimals. While scholars sometimesclaim that Cauchy avoided infinitesimals in the 1820s, de Prony’s com-ments and other primary documents indicate otherwise.1.1.
Limites . Cauchy defined limits as follows in his
Cours d’Analyse (CdA): On nomme quantit´e variable celle que l’on consid`erecomme devant recevoir successivement plusieurs valeursdiff´erentes les unes des autres. . . . Lorsque les valeurssuccessivement attribu´ees `a une mˆeme variable s’app-rochent ind´efiniment d’une valeur fixe, de mani`ere `a finirpar en diff´erer aussi peu que l’on voudra, cette derni`ereest appel´ee la limite de toutes les autres. (Cauchy [15],1821, p. 4; emphasis in the original)Here Cauchy defines limits in terms of a primitive notion of a variablequantity . As Robinson pointed out, Cauchy “assign[ed] a central roleto the notion of a variable which tends to a limit, in particular to thelimit zero” (Robinson [40], 1966, p. 276).Elsewhere in CdA, Cauchy used what appears to be a somewhatdifferent notion of limit, as for example when the value of the derivativeis extracted from the ratio of infinitesimals ∆ y and ∆ x (see Section 2.1).Two distinct approaches used by Cauchy are analyzed in Section 1.2.1.2. A-track and B-track for the development of analysis.
Thearticle Katz–Sherry [33] introduced a distinction between two types ofprocedures in the writing of the pioneers of infinitesimal calculus: Translation from [14, p. 6]: “We call a quantity variable if it can be considered asable to take on successively many different values. . . . When the values successivelyattributed to a particular variable indefinitely approach a fixed value in such a wayas to end up by differing from it by as little as we wish, this fixed value is calledthe limit of all the other values.”
J. BAIR, P. BLASZCZYK, P. HEINIG, V. KANOVEI, AND M. KATZ (A) procedures in pioneering work in analysis that can be based onan Archimedean continuum (or the A-track approach), cf. [1];and(B) procedures that can be based on a Bernoullian (i.e., infinitesimal-enriched) continuum (the B-track approach), as they appear inLeibniz, Bernoulli, Euler, and others.This is not an exhaustive distinction, but one that helps broaden thelens of a historiography often handicapped by self-imposed limitationsof a Weierstrassian type; see Section 1.4.Here we use the term procedure in a broad sense that encompassesalgorithms but is not limited to them. For instance, Euler’s proof ofthe infinite product formula for the sine function is a rather coherentprocedure though it can hardly be described as an algorithm; see [4]for an analysis of Euler’s proof.Like Leibniz, Cauchy used both A-track and B-track techniques inhis work. The sample discussed in Section 3.8 below illustrates his A-track work. Elsewhere, as we document in this article and in earlierwork (see e.g., [11]), Cauchy used B-track techniques, as well.1.3.
What is Cauchy’s limite ? Scholars who stress Cauchy’s use ofthe limit concept rely on a traditional but flawed dichotomy of infinites-imals vs limits. The dichotomy is flawed because limits are presentwhether one works with an Archimedean or Bernoullian continuum(see Section 1.2). In fact, the definition of derivative found in Cauchy(see Section 2.1) suggests that he works with the B-track version oflimits which is referred to as the standard part function in modern in-finitesimal analysis; see Section 4, formula (4.3). Thus the real issueis whether Cauchy’s continuum was Archimedean or Bernoullian, andthe genuine dichotomy is between A-track ǫ, δ techniques and B-trackinfinitesimal techniques.1.4. Butterfly model.
The articles (Bair et al. [3]), (Bair et al. [4]),and (Fletcher et al. [22]) argued that some historians of mathematicsoperate within a conceptual scheme described in (Hacking [30], 2014)as a butterfly model of development.Inspired in part by (Mancosu [38], 2009), Ian Hacking proposes adistinction between the butterfly model and the
Latin model , namelythe contrast between a model of a deterministic (genetically deter-mined) biological development of animals like butterflies (the egg–larva–cocoon–butterfly development), as opposed to a model of a con-tingent historical evolution of languages like Latin.
Historians working within the butterfly paradigm often assume thatthe evolution of mathematical rigor has a natural direction, leading for-ward to the Archimedean framework as developed by Weierstrass andothers (what Boyer referred to as “the great triumvirate” [13, p. 298]).Such historians also tend to interpret the qualifier rigorous as neces-sarily implying
Archimedean , as we illustrate in Section 1.5.1.5.
Siegmund-Schultze on
Cours d’Analyse . As an illustrationof butterfly model thinking by modern historians, we turn to a reviewby historian Siegmund-Schultze of an English edition of CdA (Bradley–Sandifer [14], 2009). The review illustrates the poignancy of Grattan-Guinness’ comment quoted in our epigraph. The comment appears in(Grattan-Guinness [26], 1970, p. 379) in the context of a discussion ofCdA.Siegmund-Schultze’s Zentralblatt (Zlb) review ([43], 2009) of theEnglish edition of CdA contains two items of interest:(SS1) Siegmund-Schultze quotes part of Cauchy’s definition of con-tinuity via infinitesimals, and asserts that Cauchy’s use of in-finitesimals was a step backward: “There has been . . . an in-tense historical discussion in the last four decades or so howto interpret certain apparent remnants of the past or – as com-pared to J. L. Lagrange’s (1736–1813) rigorous ‘Algebraic Anal-ysis’ – even steps backwards in Cauchy’s book, particularly hisuse of infinitesimals. . . ” ([43]; emphasis added).(SS2) Siegmund-Schultze quotes Cauchy’s comments (in translation)on rigor in geometry, and surmises that the framework for CdAwas Archimedean, similarly to Euclid’s geometry: “a non-Archi-median interpretation of the continuum would clash with theEuclidean theory, which was still the basis of Cauchy’s book. In-deed, Cauchy writes in the ‘introduction’ to the Cours d’Analyse:‘As for methods, I have sought to give them all the rigor thatone demands in geometry , . . . ’ ” (ibid.; emphasis added).Siegmund-Schultze’s Zbl review goes on to continue the quotation fromCauchy:“. . . in such a way as never to revert to reasoning drawnfrom the generality of algebra . Reasoning of this kind,although commonly admitted, particularly in the pas-sage from convergent to divergent series and from realquantities to imaginary expressions, can, it seems to me,only occasionally be considered as inductions suitablefor presenting the truth, since they accord so little with
J. BAIR, P. BLASZCZYK, P. HEINIG, V. KANOVEI, AND M. KATZ the precision so esteemed in the mathematical sciences.”(Cauchy as quoted in [43]; emphasis added).Cauchy’s objections here have to do with the cavalier use of diver-gent series, based on a heuristic principle Cauchy called the gener-ality of algebra , by his illustrious predecessors Euler and Lagrange,rather than with the issue of using or not using infinitesimals, contraryto Siegmund-Schultze’s claim. We will evaluate Siegmund-Schultze’sclaims further in Section 1.6.1.6.
Analysis of a review.
The Zbl review quoted in Section 1.5tends to confirm the diagnosis following Hacking. Namely, the com-ment on infinitesimals quoted in (SS1) leading specifically backward will surely be read by the Zbl audience as indicative of an assumptionof an organic (butterfly model) forward direction (culminating in the great triumvirate ).Similarly, the comment quoted in (SS2) appears to take it for grantedthat Euclid’s framework, being rigorous, was necessarily Archimedean.Yet the facts are as follows:(i) Books I through IV of
The Elements are developed without theArchimedean axiom;(ii) developments around 1900 showed conclusively that the com-pleteness property of R is irrelevant to the development of Eu-clidean geometry, and in fact the latter can be developed in thecontext of non-Archimedean fields.Indeed, Hilbert proved that these parts of Euclidean geometry can bedeveloped in a non-Archimedean plane (modulo some specific assump-tions such as circle–circle intersection and postulation of the congruencetheorems); see further in [5, Section 5].While Euclid relied on the Archimedean axiom to develop his theoryof proportion , Hilbert obtained all the results of Euclidean geometryincluding the theory of proportion and geometric similarity withoutsuch a reliance; see Hartshorne ([31], 2000, Sections 12.3–12.5 and 20–23) or Baldwin ([2], 2017).Furthermore, starting with Descartes’ Geometry , mathematiciansimplicitly relied on ordered field properties rather than the ancienttheory of proportion.
Moreover, it is difficult to understand how Siegmund-Schutze wouldreconcile his two claims. If Cauchy used Euclidean Archimedean math-ematics exclusively, as implied by (SS2), then what exactly were the en-tities that constituted a step backward, as claimed in (SS1)? Siegmund-Schultze’s counterfactual claims are indicative of butterfly-model think-ing as outlined in Section 1.4.Like the Zbl review by Siegmund-Schultze, the Cauchy scholarshipof Gilain tends to be colored by teleological assumptions of the sortdetailed above, as we argue in Sections 2 and 3.A number of historians and mathematicians have sought to chal-lenge the received views on Cauchy’s infinitesimals, as we detail inSections 1.7 through 1.9.1.7.
Robinson on received views.
Abraham Robinson noted thatthe received view of the development of the calculus[would] lead us to expect that, following the rejection ofLeibniz’ theory by Lagrange and D’Alembert, infinitelysmall and infinitely large quantities would have no placeamong the ideas of Cauchy, who is generally regardedas the founder of the modern approach, or that theymight, at most, arise as figures of speech, as in ‘ x tendsto infinity’. However, this expectation is mistaken. [40,p. 269].Robinson described Cauchy’s approach as follows:Cauchy regarded his theory of infinitely small quanti-ties as a satisfactory foundation for the theory of limitsand (d’Alembert’s suggestion notwithstanding) he didnot introduce the latter in order to replace the former .His proof procedures thus involved both infinitely small(and infinitely large) quantities and limits. [40, p. 271](emphasis added)Note Robinson’s focus on Cauchy’s procedures (for a discussion of theprocedure/ontology dichotomy, see B laszczyk et al. [9]). After quotingCauchy’s definition of derivative, Robinson notes:Later generations have overlooked the fact that in thisdefinition ∆ x and ∆ y were explicitly supposed to be in-finitely small. Indeed according to our present standardideas, we take f ′ ( x ) to be the limit [of] ∆ y/ ∆ x as ∆ x tends to zero, whenever that limit exists, without anymention of infinitely small quantities. Thus, as soonas we consider limits, the assumption that ∆ x and ∆ y J. BAIR, P. BLASZCZYK, P. HEINIG, V. KANOVEI, AND M. KATZ are infinitesimal is completely redundant. It is thereforethe more interesting that the assumption is there, and,indeed, appears again and again also in Cauchy’s laterexpositions of the same topic (Cauchy [1829, 1844]). [40,p. 274]Robinson’s conclusion is as follows:We are forced to conclude that Cauchy’s mental pictureof the situation was significantly different from the pic-ture adopted today, in the Weierstrass tradition. (ibid.)It is such received views in what Robinson refers to as the
Weierstrasstradition that we wish to reconsider here.1.8.
Grattan-Guinness on Cauchy’s infinitesimals.
Robinson’s1966 comments on the Weierstrassian tradition cited in Section 1.7were echoed by historians Ivor Grattan-Guinness and Detlef Laugwitz.Thus, fourteen years later, Grattan-Guinness wrote:[Cauchy’s definition of infinitesimal] is in contrast to theview adopted from the Weierstrassians onwards (and oc-casionally earlier), where an infinitesimal is a variablewith limit zero. . . (Grattan-Guinness [27], 1980, p. 110;emphasis added)Concerning the term limit , it is necessary to disassociate the followingtwo issues:(Ca1) the issue of whether or not limits were at the base of Cauchy’sapproach;(Ca2) the issue of Cauchy’s systematic use of infinitesimals as numbersin his textbooks and research articles.1.9.
Laugwitz on Cauchy’s infinitesimals.
As far as item (Ca2)is concerned, Laugwitz acknowledged that Cauchy started using in-finitesimals systematically in the 1820s (whereas his attitude towardthem during the preceding decade was more ambiguous and limits mayhave played a larger role):. . . after 1820, Cauchy developed his analysis by utilizinginfinitesimals in a deliberate and consequent manner. (Laugwitz [36], 1989, p. 196; emphasis in the original)Laugwitz’ position is consistent with Gilain’s observation that infinites-imals first appeared in Cauchy’s course summary during the academicyear 1820–1821:Ann´ee 1820–1821 . . . Notons aussi l’apparition, pour lapremi`ere fois dans les
Mati`eres des le¸cons , des notions de quantit´es infiniment petites et infiniment grandes(le¸con 3). (Gilain [23], §
52, 1989)In 1997, Laugwitz elaborated on the subject (of Cauchy’s endorsementof infinitesimals circa 1820) in the following terms:Cauchy avoided the use of the infinitely small. This pro-voked growing criticism on the part of his colleagues,including the physicist Petit, who emphasized the di-dactical and practical advantages of the use of infin-itely small magnitudes. In 1819 and in 1820, the Con-seil d’Instruction at the Ecole exerted strong pressureon Cauchy, but this alone would not have made thisrather stubborn man change his mind.
Around 1820,he must have realized that infinitesimal considerationswere a powerful research method at a time when he wasin a state of constant rivalry, especially with Poisson.(Laugwitz [37], 1997, p. 657; emphasis added)In the textbook
Cours d’Analyse [15], limite is not the only centralfoundational concept for Cauchy, as we argue in Section 2.We challenge a common misconception according to which Cauchywas forced to teach infinitesimals at the
Ecole Polytechnique allegedlyagainst his will. We show that the debate there concerned mainly theissue of rigor , a separate one from infinitesimals ; see Section 3.2.
Cauchy’s limite and infiniment petit
In this section we will analyze the meaning of Cauchy’s terms limite and infiniment petit .2.1.
Differentials and infinitesimals.
In his work, Cauchy care-fully distinguishes between differentials ds, dt which to Cauchy arenoninfinitesimal variables, on the one hand, and infinitesimal incre-ments ∆ s, ∆ t , on the other:. . . soit s une variable distincte de la variable primitive t .En vertu des d´efinitions adopt´ees, le rapport entre lesdiff´erentielles ds, dt , sera la limite du rapport entre les Translation: “Year 1820–1821 . . . We also note the appearance, for the first timein the
Lesson summaries , of the notions of infinitely small and infinitely largequantities (lesson 3).” accroissements infiniment petits ∆ s, ∆ t . (Cauchy [17],1844, p. 11; emphasis added)Cauchy goes on to express such a relation by means of a formula interms of the infinitesimals ∆ s and ∆ t :On aura donc dsdt = lim. ∆ s ∆ t (2.1)(ibid., equation (1); the period after lim in “lim.” in theoriginal; equation number (2.1) added)Cauchy’s procedure involving the passage from the ratio of infinitesi-mals like ∆ s ∆ t to the value of the derivative dsdt as in equation (2.1) hasa close parallel in Robinson’s infinitesimal analysis, where it is carriedout by the standard part function; see equations (4.1) and (4.2) inSection 4.Paraphrasing this definition in Archimedean terms would necessarilyinvolve elements that are not explicit in Cauchy’s definition. ThusCauchy’s “lim.” finds a closer proxy in the notion of standard part,as in formula (4.3), than in any notion of limit in the context of anArchimedean continuum; see also Bascelli et al. ([6], 2014).2.2. Definite integrals and infinitesimals.
Similar remarks applyto Cauchy’s 1823 definition of the definite integral which exploits apartition of the domain of integration into infinitesimal subintervals.Here Cauchy writes: “D’apr`es ce qui a ´et´e dit dans la derni`ere le¸con, sil’on divise X − x en ´el´emens infiniment petits x − x , x − x . . . X − x n − , la somme(1) S = ( x − x ) f ( x ) + ( x − x ) f ( x ) + . . . + ( X − x n − ) f ( x n − )convergera vers une limite repr´esent´ee par l’int´egrale d´efinie(2) Z Xx f ( x ) dx. Des principes sur lesquels nous avons fond´e cette proposition il r´esulte,etc.” (Cauchy [16], 1823, Le¸con 22, p. 85; emphasis added).Note that there is a misprint in Cauchy’s formula (1): the difference( x − x ) should be ( x − x ). In this passage, Cauchy refers to thesuccessive differences x − x , x − x , X − x n − as infinitely smallelements . Translation: “Let s be a variable different from the primitive variable t . By virtueof the definitions given, the ratio of the differentials ds, dt will be the limit of theratio of the infinitely small increments ∆ s, ∆ t .” We preserved the original spelling.
Analogous partitions into infinitesimal subintervals are exploited inKeisler’s textbook [34] (and throughout the literature on infinitesimalanalysis; see e.g., [24, p. 153]). Cauchy’s use of limite in the pas-sage above is another instance of limit in the context of a Bernoulliancontinuum, which parallels the use of the standard part function (seeSection 4) enabling the transition from a sum of type (1) above to thedefinite integral (2), similar to the definition of the derivative analyzedin Section 2.1.2.3.
Un infiniment petit in Cauchy.
What is the precise meaningof Cauchy’s infiniment petit (infinitely small)? All of Cauchy’s text-books on analysis contain essentially the same definition up to slightchanges in word order:Lorsque les valeurs num´eriques successives d’une mˆemevariable d´ecroissent ind´efiniment, de mani`ere `a s’abaisserau-dessous de tout nombre donn´e, cette variable devientce qu’on nomme un infiniment petit ou une quantit´e in-finiment petite . Une variable de cette esp`ece a z´ero pourlimite. [15, p. 4] (emphasis in the original)An examination of the books [15], [16] reveals that Cauchy typically did not define his infinitely small literally as a variable whose limit iszero. Namely, he rarely wrote “an infinitely small is a variable, etc.”but said, rather, that a variable becomes ( devient ) an infinitely small.Thus, the passage cited above is the first definition of the infinitelysmall in Cours d’Analyse . The next occurrence is on page 26 there,again using devient , and emphasizing infiniment petite by means of ital-ics. On page 27 Cauchy summarizes the definition as follows: “Soit α une quantit´e infiniment petite, c’est-`a-dire, une variable dont la valeurnum´erique d´ecroisse ind´efiniment.” This is a summary of the definitionalready given twice, the expression “infiniment petite” is not italicized,and “is” is used in place of “becomes” as shorthand for the more de-tailed and precise definitions appearing earlier in Cauchy’s textbook.An identical definition with devient appears in his 1823 textbook [16,p. 4]. Translation: “When the successive numerical values of such a variable decreaseindefinitely, in such a way as to fall below any given number, this variable becomeswhat we call infinitesimal , or an infinitely small quantity . A variable of this kindhas zero as its limit” [14, p. 7].
Cauchy’s term becomes implies a change of nature or type . Namely,a variable is not quite an infinitesimal yet, but only serves to generate or represent one, as emphasized by Laugwitz:Cauchy never says what his infinitesimals are ; we aretold only how infinitesimals can be represented . (Laug-witz [35], 1987, p. 271)See also Sad et al. [41]. This indicates that Cauchy considered aninfinitesimal as a separate type of mathematical entity, distinct fromvariable or sequence.2.4. Variable quantities, infinitesimals, and limits.
To commentmore fully on Cauchy’s passage cited in Section 2.3, note that there arethree players here:(A) variable quantity;(B) infinitesimal;(C) limit zero.We observe that the notion of variable quantity is the primitive notionin terms of which both infinitesimals and limits are defined (see Sec-tion 1.1 for Cauchy’s definition of limit in terms of variable quantity).This order of priorities is confirmed by the title of Cauchy’s very firstlesson in his 1823 book:1. re Le¸con. Des variables, de leurs limites, et des quan-tit´es infiniment petites [16, p. ix]Thus, Cauchy is proposing a definition and an observation:(Co1) a variable quantity that diminishes indefinitely becomes an in-finitesimal; and(Co2) such a variable quantity has zero as limit.Here item (Co2) is merely a restatement of the property of diminishingindefinitely in terms of the language of limits. As noted in Section 1,Robinson pointed out that Cauchy assigned a central role to the notionof a variable which tends to a limit. Cauchy’s notion of limit here isclose to the notion of limit of his predecessor Lacroix (see Section 3.9). To illustrate such a change in modern terms, note that in the context of thetraditional construction of the real numbers in terms of Cauchy sequences u =( u n ) ∈ Q N of rational numbers, one never says that a real number is a sequence, butrather that a sequence represents or generates the real number, or to use Cauchy’sterminology, becomes a real number. A related construction of hyperreal numbersout of sequences of real numbers, where a sequence tending to zero generates aninfinitesimal, is summarized in Section 4. Assigning a sign to an infinitesimal.
Cauchy often uses thenotation α for a generic infinitesimal, in both his 1821 and 1823 text-books. In his 1823 textbook Cauchy assumes that α is either positiveor negative:Cherchons maintenant la limite vers laquelle convergel’expression (1 + α ) α , tandis que α s’approche ind´efini-ment de z´ero. Si l’on suppose d’abord la quantit´e α positive et de la forme m , m d´esignant un nombre entiervariable et susceptible d’un accroissement ind´efini, onaura (1 + α ) α = (cid:0) m (cid:1) m . . . Supposons enfin que α devienne une quantit´e n´egative. Si l’on fait dans cettehypoth`ese 1 + α = β , β sera une quantit´e positive,qui convergera elle-mˆeme vers z´ero, etc. [16, pp. 2–4]It is well known that variable quantities or sequences that generateCauchyan infinitesimals are not necessarily monotone. Indeed, Cauchyhimself gives a non-monotone example at the beginning of CdA: , , , , , , &c. . . . [15, p. 27]This poses a problem since it is not obvious how to assign a sign plus orminus to an arbitrary null sequence (i.e., a sequence tending to zero).When Cauchy actually uses infinitesimals in proofs and applications,he assumes that they can be manipulated freely in arithmetic opera-tions and other calculations. While formal order theory is a few decadesaway and is not to be found as such in Cauchy, he does appear to as-sume that a definite sign can be attached to an infinitesimal. Besidesassuming that they have a well-defined sign, Cauchy also routinely ap-plies arithmetic operations to infinitesimals.This creates a difficulty to those who consider that Cauchy merelyused the term “infinitely small” as shorthand for a sequence withlimit 0, since it is unclear how to assign a sign to an arbitrary nullsequence, whereas Cauchy does appear to assign a sign to his infinites-imals.Which process exactly did Cauchy envision when he spoke of a se-quence becoming an infinitesimal? Cauchy does not explain. However,Cauchy’s assumption that each infinitesimal has a sign suggests that asequence is not identical to the infinitesimal it generates.Even monotone sequences are not closed under arithmetic opera-tions. Namely, such operations necessarily lead to non-monotone ones,including ones that change sign.Cauchy routinely assumes in his work, particularly on integrals, thatone can freely add infinitesimals and obtain other infinitesimals, i.e.,that the numbers involved are closed under arithmetic operations. Such an assumption is valid in modern theories of ordered fieldsproperly extending R , but if one is working with sequences, such anassumption leads to a dilemma:(1) either one only works with monotone ones, in which case onegets into a problem of closedness under natural arithmetic op-erations;(2) or one works with arbitrary sequences, in which case the as-sumption that a sequence can be declared to be either positiveor negative becomes problematic.Cauchy was probably not aware of the difficulty that that one can’t both assign a specific sign to α , and also have the freedom of applyingarithmetic operations to infinitesimals. The point however is that theway he uses infinitesimals indicates that both conditions are assumed,even though from the modern standpoint the justification providedis insufficient. In other words, Cauchy’s procedures are those of aninfinitesimal-enriched framework, though the ontology of such a systemis not provided.Cauchy most likely was not aware of the problem, for otherwise hemay have sought to address it in one way or another. He did havesome interest in asymptotic behavior of sequences. Thus, in some ofhis texts from the late 1820s he tried to develop a theory of the orderof growth at infinity of functions. Such investigations were eventuallypicked up by du Bois-Reymond, Borel, and Hardy; see Borovik–Katz([11], 2012) for details.2.6. Gilain on omnipresence of limits.
Gilain refers to Cauchy’scourse in 1817 as acours tr`es important historiquement, o`u les bases de lanouvelle analyse, notamment celle de l’
Analyse alg´ebrique de 1821, sont pos´ees. . . [23, § variable quantity (see beginning of Section 1 as well as Section 2.4). It is therefore difficultto agree with Gilain when he claims to know the following:On sait que Cauchy d´efinissait le concept d’infinimentpetit `a l’aide du concept de limite, qui avait le premierrˆole (voir Analyse alg´ebrique, p. 19; . . . ) [23, note 67]Here Gilain claims that it is the concept of limite that played a pri-mary role in the definition of infinitesimal, with reference to page 19 inthe 1897 Ouevres Compl`etes edition of CdA [15]. The correspondingpage in the 1821 edition is page 4. We quoted Cauchy’s definition inSection 2.3 and analyzed it in Section 2.4. An attentive analysis of thedefinition indicates that it is more accurate to say that it is the conceptof variable quantity (rather than limite ) that “avait le premier rˆole.”Cauchy exploited the notion of limit in [15, Chapter 2, §
3] in theproofs of Theorem 1 and Theorem 2. Theorem 1 compares the con-vergence of the difference f ( x + 1) − f ( x ) and that of the ratio f ( x ) x .Theorem 2 compares the convergence of f ( x +1) f ( x ) and [ f ( x )] x . Theseproofs can be viewed as prototypes of ǫ, δ arguments. On the otherhand, neither of the two proofs mentions infinitesimals. Therefore nei-ther can support Gilain’s claim to the effect that Cauchy allegedly usedlimits as a basis for defining infinitesimals. The proof of Theorem 1 isanalyzed in more detail in Section 3.8.Cauchy’s procedures exploiting infinitesimals have stood the test oftime and proved their applicability in diverse areas of mathematics,physics, and engineering.Gilain and some other historians assume that the appropriate mod-ern proxy for Cauchy’s limite necessarily operates in the context ofan Archimedean continuum (see Section 2.4). Yet the vitality androbustness of Cauchy’s infinitesimal procedures is obvious given theexistence of proxies in modern theories of infinitesimals. What we ar-gue is that modern infinitesimal proxies for Cauchy’s procedures aremore faithful to the original than Archimedean proxies that typicallyinvolve anachronistic paraphrases of Cauchy’s briefer definitions andarguments.This article does not address the historical ontology of infinitesimals(a subject that may require separate study) but rather the procedures ofinfinitesimal calculus and analysis as found in Cauchy’s oeuvre (see [9]for further details on the procedure/ontology dichotomy).2.7. Limite and infinity.
As we noted in Section 1.3, the use ofthe term limite by Cauchy could be misleading to a modern reader.Consider for example its use in the passage cited in Section 2.3. The fact that Cauchy is not referring here to a modern notion of limit isevident from his very next sentence:Lorsque les valeurs num´eriques successives d’une mˆemevariable croissent de plus en plus, de mani`ere `a s’´eleverau-dessus de tout nombre donn´e, on dit que cette vari-able a pour limite l’infini positif indiqu´e par le signe ∞ s’il s’agit d’une variable positive. . . [16, p. 4]In today’s calculus courses, it is customary to give an ( ǫ, δ ) or ( ǫ, N )definition of limit of, say, a sequence, and then introduce infinite ‘limits’in a broader sense when the sequence diverges to infinity. But Cauchydoes not make a distinction between convergent limits and divergentinfinite limits.Scholars ranging from Sinaceur ([44], 1973) to Nakane ([39], 2014)have pointed out that Cauchy’s notion of limit is distinct from theWeierstrassian Epsilontik one (this is particularly clear from Cauchy’sdefinition of the derivative analyzed in Section 2.1); nor did Cauchyever give an ǫ, δ definition of limit, though prototypes of ǫ, δ arguments do occasionally appear in Cauchy; see Section 1.2.3.
Minutes of meetings, Poisson, and de Prony
Here we develop an analysis of the third of the misconceptions diag-nozed in Borovik–Katz ([11], 2012, Section 2.5), namely the idea thatCauchy was forced to teach infinitesimals at the
Ecole Polytechnique allegedly against his will. We show that the debate there concernedmainly the issue of rigor , a separate one from infinitesimals .Minutes of meetings at the
Ecole are a valuable source of informa-tion concerning the scientific and pedagogical interactions there in the1820s.3.1.
Cauchy pressured by Poisson and de Prony.
Gilain providesdetailed evidence of the pressure exerted by Sim´eon Denis Poisson,Gaspard de Prony, and others on Cauchy to simplify his analysis course.Thus, in 1822Poisson et de Prony. . . insistent [sur la] n´ecessit´e. . . desimplifier l’enseignement de l’analyse, en multipliant lesexemples num´eriques et en r´eduisant beaucoup la partieanalyse alg´ebrique plac´ee au d´ebut du cours. [23, § Translation: “When the successive numerical values [i.e., absolute values] of thesame variable grow larger and larger so as to rise above each given number, onesays that this variable has limit positive infinity denoted by the symbol ∞ whenthe variable is positive.” Similarly, in 1823, Cauchy’s course was criticized for being too compli-cated: des voix se sont ´elev´ees pour trouver trop compliqu´eesles feuilles de cours en question et il ´etait d´ecid´e deproposer au Ministre la nomination d’une commissionqui serait charg´ee chaque ann´ee de l’examen des feuillesd’analyse et des modifications ´eventuelles `a y apporter.[23, § § rigoureuses .[23, §
86] (emphasis added)Note however that in these discussions, the issue is mainly that of rigor (i.e., too many proofs) rather than choice of a particular approach tothe foundations of analysis. While Cauchy’s commitment to simplifythe course may have entailed skipping the proofs in the style of the
Epsilontik of Theorems 1 and 2 in [15, Chapter 2, §
3] (see end ofSection 2.4), it may have also entailed skipping the proofs of as manyas eight theorems concerning the properties of infinitesimals of variousorders in [15, Chapter 2, § Reports by de Prony.
Gilain notes that starting in 1826, thereis a new source of information concerning Cauchy’s course, namely thereports by de Prony:de Prony reproche de fa¸con g´en´erale `a Cauchy de ne pasutiliser suffisamment les consid´erations g´eom´etriques etles infiniment petits, tant en analyse qu’en m´ecanique.[23, § rigor but also insufficientuse of infinitesimals was being contested. Even here, the complaintis not an alleged absence of infinitesimals, but merely insufficient use thereof. We will examine de Prony’s views in Section 3.5. Course summaries.
According to course summaries reproducedin [23], Cauchy taught both continuous functions and infinitesimals(and presumably the definition of continuity in terms of infinitesimalsafter 1820) in the premi`ere ann´ee during the academic years 1825–1826,1826–1827, 1827–1828, and 1828–1829 (the summaries for the premi`ereann´ee during the 1829–1830 academic year, Cauchy’s last at the
EcolePolytechnique , are not provided). All these summaries contain identicalcomments on continuity and infinitesimals for those years:Des fonctions en g´en´eral, et des fonctions continues enparticulier. – Repr´esentation g´eom´etrique des fonctions continues d’une seule variable. – Du rapport entre l’accroisse-ment d’une fonction et l’accroissement de la variable. –Valeur que prend ce rapport quand les accroissemens de-viennent infiniment petits . (Cauchy as quoted by Gilain;emphasis added)In 1827 for the first time we find a claim of an actual absence of in-finitesimals from Cauchy’s teaching. Thus, on 12 january 1827,le cours de Cauchy a de nouveau ´et´e mis en cause pour sadifficult´e, (le gouverneur affirmant que des ´el`eves avaientd´eclar´e qu’ils ne le comprenaient pas), et son non-usage de la m´ethode des infiniment petits (voir document C12). [23, § Cauchy taken to task.
Gilain writes that during the 1826–1827academic year, Cauchy was taken to task in the
Conseil de Perfection-nement of the ´Ecole Polytechnique for allegedly not teaching infinites-imals (see [23, § To comment on Gilain’s “document C12” (denoted C in [23]), it is necessary toreproduce what the document actually says: “Un membre demande si le professeurexpose la m´ethode des infiniment petits, ainsi que le voeu en a ´et´e exprim´e.” Whatwas apparently Cauchy’s response to this query is reproduced in the next paragraphof document C12: “On r´epond que le commencement du cours ne pourra ˆetre fond´esur les notions infinit´esimales que l’ann´ee prochaine, parce que le cours de cetteann´ee ´etait commenc´e `a l’´epoque o`u cette disposition a ´et´e arrˆet´ee; que M. Cauchys’occupe de la r´edaction de ses feuilles, en cons´equence, et qu’il a promis de lescommuniquer bientˆot `a la commission de l’enseignement math´ematique.”Thus, the actual contents of document C12 indicate that Gilain’s claim of “ non-usage ” is merely an extrapolation. that Cauchy exploited infinitesimals anyway that year, in developingthe theory of contact of curves:S’il ne fonde pas le calcul diff´erentiel et int´egral sur la‘m´ethode’ des infiniment petits, Cauchy n’en utilise pasmoins de fa¸con importante ces objets (consid´er´es commedes variables dont la limite est z´ero), en liaison no-tamment avec l’exposition de la th´eorie du contact descourbes. [23, note 111]It emerges that Cauchy did use infinitesimals that year in his treatmentof a more advanced topic (theory of contact). Thus Cauchy’s actualscientific practice was not necessarily dependent on his preliminarydefinitions. There is conflicting evidence as to whether Cauchy usedinfinitesimals (as developed in [15] and [16]) in the introductory partof his course that year. As we mentioned in Section 3.2, the coursesummary for 1826–1827 does include both continuity and infinitesimals.3.5. Critique by de Prony.
Gilain describes de Prony’s criticism ofCauchy as follows:[De Prony] critique notamment l’emploi de la m´ethodedes limites par Cauchy au lieu de celle des infinimentpetits, faisant appel ici `a l’autorit´e posthume de Laplace,d´ec´ed´e depuis le 5 mars 1827 (voir document C14).[23, § du mouve-ment vari´e se sont encor trouv´ees mel´ees de considera-tions relatives aux limites ; . . . (de Prony as quoted inGrattan-Guinness [28], 1990, p. 1339; emphasis in theoriginal)Having specified the target of his criticism, namely Cauchy’s conceptof limite , de Prony continues:. . . il me semble qu’en employant, immediatement et ex-clusivement, la methode des infiniment petits, on abregeet on simplifie les raisonnements sans nuire `a la clart´e; Gilain’s parenthetical remark here is an editorial comment for which he providesno evidence. The remark reveals more about Gilain’s own default expectations (seeSection 1) than about Cauchy’s actual foundational stance. The spelling as found in (Gilain [23, Document C ]) is g´en´erales (i.e., the modernFrench spelling). Gilain similarly replaced encor by encore , mel´ees by mˆel´ees , immediatement by imm´ediatement , methode by m´ethode , abrege by abr`ege , and collegue by coll`egue . rappellons nous combien cette methode ´etait recommand´eepar l’illustre collegue [Laplace] que la mort nous a en-lev´e. (ibid.)What is precisely the nature of de Prony’s criticism of Cauchy’s ap-proach to analysis? Does his criticism focus on excessive rigor, or oninfinitesimals, as Gilain claims? The answer depends crucially on un-derstanding de Prony’s own approach, explored in Section 3.6.3.6. De Prony on small oscillations.
In his work
M´ecanique philo-sophique , de Prony considers infinitesimal oscillations of the pendulum(de Prony [19], 1799, p. 86, § π q ag where a is the length of the cord, and g is acceleration under gravity.Limits are not mentioned. In the table on the following page 87, hestates the property of isochronism , meaning that the halfperiod π q ag is independent of the size of the infinitesimal amplitude. This howeveris not true literally but only up to a passage to limits, or taking thestandard part; see Section 4. Thus de Prony’s own solution to theconceptual difficulties involving limits/standard parts in this case ismerely to ignore the difficulties and suppress the limits.In his article “Suite des le¸cons d’analyse,” de Prony lets n = Az ([18], 1796, p. 237). He goes on to write down the formulacos z = (cid:2) cos zn + sin zn √− (cid:3) n + (cid:2) cos zn − sin zn √− (cid:3) n A diminue et n augmente, ces ´equations s’approchent de devenircos z = h z √− n i n + h − z √− n i n Even if literally infinitesimal amplitudes are admitted, there is still a discrepancydisallowing one to claim that the halfperiod is literally π q ag . This difficulty can beovercome in the context of modern infinitesimal analysis; see Kanovei et al. ([32],2016). De Prony’s formula (3.1) is correct only up to taking the standard partof the right-hand side (for infinite n ). Again de Prony handles theconceptual difficulty of dealing with infinite and infinitesimal numbersby suppressing limits or standard parts. Note that both of de Prony’sformulas are taken verbatim from (Euler [21], 1748, §
133 – § It is reasonable to assume that de Prony’s criticism of Cauchy’steaching of prospective engineers had to do with what Prony saw asexcessive fussiness in dealing with what came to be viewed later asconceptual difficulties of passing to the limit, i.e., taking the standardpart. Note that in the comment by de Prony cited at the beginning ofthis section, he does not criticize Cauchy for not using infinitesimals,but merely for excessive emphasis on technical detail involving limites .Therefore Gilain’s claim to the contrary cited at the beginning of Sec-tion 3.5 amounts to massaging the evidence by putting a tendentiousspin on de Prony’s criticism.3.7.
Foundations, limits, and infinitesimals.
Can one claim thatCauchy established the foundations of analysis on the concept of infin-itesimal?The notions of infinitesimal, limit, and variable quantity are all fun-damental for Cauchy. One understands them only by the definitionwhich explains how they interact. If Cauchy established such foun-dations it was on the concept of a variable quantity, as analyzed inSection 2.4.Can one claim that Cauchy conferred upon limite a central role inthe architecture of analysis? The answer is affirmative if one takesnote of the frequency of the occurrence of the term in Cauchy’s oeuvre;similarly, Cauchy conferred upon infinitesimals a central role in thesaid architecture.A more relevant issue, however, is the precise meaning of the term limite as used by Cauchy. As we saw in Section 2.1 he used it in the differential calculus in a sense closer to the standard part function thanto any limit concept in the context of an Archimedean continuum; andas we saw in Section 2.2, he used it in the integral calculus in a sensecloser to the standard part than any Archimedean counterpart. Schubring lodges the following claim concerning de Prony: “The break with pre-vious tradition, which was probably the most visible to his contemporaries, was theexclusion and rejection of infiniment petits by the analytic method. In de Prony theinfiniment petits were excluded from the foundational concepts of his teaching bysimply not being mentioned; only in a heading did they appear in a quotation, as‘so-called analysis of the infinitely small quantities’” (Schubring [42], 2005, p. 289).Schubring’s assessment of de Prony’s attitude toward infinitesimals seems about asapt as his assessment of Cauchy’s; see (B laszczyk et al. [10], 2017).
Did Cauchy ever seek a justification of infinitesimals in terms of lim-its? Hardly so, since he expressed both concepts in terms of a primitivenotion of variable quantity. In applications of analysis, Cauchy makesno effort to justify infinitesimals in terms of limits.3.8.
Cauchy’s A-track arguments.
Let us examine in more detailthe issue of ǫ, δ arguments in Cauchy, as found in [15, Section 2.3,Theorem 1] (already mentioned in Section 2.6). Cauchy seeks to showthat if the difference f ( x +1) − f ( x ) converges towards a certain limit k ,for increasing values of x , then the ratio f ( x ) x converges at the same timetowards the same limit; see [14, p. 35].Cauchy chooses ǫ >
0, and notes that we can give the number h avalue large enough so that, when x is equal to or greater than h , thedifference f ( x + 1) − f ( x ) is always contained between k − ǫ and k + ǫ .Cauchy then arrives at the formula f ( h + n ) − f ( h ) n = k + α, where α is a quantity contained between the limits − ǫ and + ǫ , andeventually obtains that the ratio f ( x ) x has for its limit a quantity con-tained between k − ǫ and k + ǫ .This is a fine sample of a prototype of an ǫ, δ proof in Cauchy. How-ever, as pointed out by Sinkevich, Cauchy’s proofs are all missing thetell-tale sign of a modern proof in the tradition of the Weierstrassian Epsilontik , namely exhibiting an explicit functional dependence of δ (orin this case h ) on ǫ (Sinkevich [45], 2016).One of the first occurrences of a modern definition of continuity inthe style of the Epsilontik can be found in Schwarz’s summaries of 1861lectures by Weierstrass; see (Dugac [20], 1973, p. 64), (Yushkevich [47],1986, pp. 74–75). This definition is a verbal form of a definition fea-turing a correct quantifier order (involving alternations of quantifiers).The salient point here is that this sample of Cauchy’s work has nobearing on Cauchy’s infinitesimals. Nor does it imply that infinitesi-mals are merely variables tending to zero, since the term infinitely small does not occur in this proof at all. Nor does Cauchy’s argument showthat he thought of limits in anything resembling post-Weierstrassianterms since his recurring definition of limit routinely falls back on theprimitive notion of a variable quantity, rather than on any form of analternating quantifier string, whether verbal or not.3.9.
Lacroix, Laplace, and Poisson.
The Bradley–Sandifer editionquotes a revealing comment of Cauchy’s on the importance of infinites-imals. The comment is found in Cauchy’s introduction:
In speaking of the continuity of functions, I could notdispense with a treatment of the principal propertiesof infinitely small quantities, properties which serve asthe foundation of the infinitesimal calculus. (Cauchy astranslated in [14, p. 1])Bradley and Sandifer then go on to note: “It is interesting that Cauchydoes not also mention limits here” (ibid., note 6; emphasis added).The circumstances of the publication of the 1821
Cours d’Analyse indicate that attaching fundamental importance to infinitesimals ratherthan limits (noted by Bradley and Sandifer) was Cauchy’s personalchoice, rather than being dictated by the constraints of his teaching atthe ´Ecole Polytechnique . Indeed, unlike Cauchy’s later textbooks, his1821 book was not commissioned by the ´Ecole but was rather writtenupon the personal request of Laplace and Poisson, as acknowledged in(Gilain [23], 1989, note 139).Sinaceur points out that Cauchy’s definition of limit resembles, notthat of Weierstrass, but rather that of Lacroix dating from 1810(see [44, p. 108–109]). This is acknowledged in (Grabiner [25], 1981,p. 80).Cauchy’s kinematic notion of limit was expressed, like his notion ofinfinitesimal α , in terms of a primitive notion of variable quantity (seeSection 2.4). Thus, Cauchy’s comment that when a variable becomesan infinitesimal α , the limit of such a variable is zero, can be interpretedin two ways. It can be interpreted in the context of an Archimedeancontinuum. Alternatively, it could be interpreted as the statement thatthe assignable part of α is zero, in the context of a Bernoullian (i.e.,infinitesimal-enriched) continuum, or in modern terminology, that the standard part of α is zero; see Section 4. As a student at the
Polytechnique , Cauchy attended Lacroix’s course in analysisin 1805; see (Belhoste [8], 1991, p. 10, 243). Sinaceur explicitly denies Cauchy the honor of having published the first arith-metic definition of limits, by writing: “Or, 1) l’´epsilonisation n’est pas l’œuvre deCauchy, mais celle de Weierstrass ; . . . on ne peut dire qu’il en donne une d´efinitionpurement arithm´etique ou purement analytique. Sa d´efinition . . . n’enveloppe pasmoins d’intuition g´eom´etrique que celle contenue dans le
Trait´e de Lacroix. . . ” Modern infinitesimals in relation to Cauchy’sprocedures
While set-theoretic justifications for either A-track or B-track mod-ern framework are obviously not to be found in Cauchy, Cauchy’s proce-dures exploiting infinitesimals find closer proxies in Robinson’s frame-work for analysis with infinitesimals than in a Weierstrassian frame-work. In this section we outline a set-theoretic construction of a hy-perreal extension R ֒ → ∗ R , and point out specific similarities betweenprocedures using the hyperreals, on the one hand, with Cauchy’s pro-cedures, on the other.Let R N denote the ring of sequences of real numbers, with arithmeticoperations defined termwise. Then we have ∗ R = R N / MAX whereMAX is the maximal ideal consisting of all “negligible” sequences ( u n ).Here a sequence is negligible if it vanishes for a set of indices of fullmeasure ξ , namely, ξ (cid:0) { n ∈ N : u n = 0 } (cid:1) = 1. Here ξ : P ( N ) → { , } is a finitely additive probability measure taking the value 1 on cofinitesets, where P ( N ) is the set of subsets of N . The subset F ξ ⊆ P ( N )consisting of sets of full measure ξ is called a nonprincipal ultrafilter.These originate with (Tarski [46], 1930). The set-theoretic presentationof a Bernoullian continuum (see Section 1.2) outlined here was thereforenot available prior to that date.The field R is embedded in ∗ R by means of constant sequences. Thesubring h R ⊆ ∗ R consisting of the finite elements of ∗ R admits a map st to R , known as standard part st : h R → R , (4.1)which rounds off each finite hyperreal number to its nearest real number(the existence of such a map st is the content of the standard partprinciple ). This enables one, for instance, to define the derivative of s = f ( t ) as f ′ ( t ) = dsdt = st (cid:18) ∆ s ∆ t (cid:19) (4.2)(here ∆ s = 0 is infinitesimal) which parallels Cauchy’s definition ofderivative (see equation (2.1) in Section 2.1) more closely than any Epsilontik definition. Limit is similarly defined in terms of st , e.g., bysetting lim t → f ( t ) = st ( f ( ǫ )) (4.3)where ǫ is a nonzero infinitesimal, in analogy with Cauchy’s limit asanalyzed in Section 1.3. For additional details on Robinson’s frameworksee e.g., [22]. Conclusion
The oft-repeated claim (as documented e.g., in [3]; [7]) that “Cauchy’sinfinitesimal is a variable with limit 0” (see Gilain’s comment cited inSection 3.4) is a reductionist view of Cauchy’s foundational stance, atodds with much compelling evidence in Cauchy’s writings, as we arguedin Sections 2 and 3.Gilain, Siegmund-Schultze, and some other historians tend to adopta butterfly model for the development of analysis, to seek proxies forCauchy’s procedures in a default modern Archimedean framework, andto view his infinitesimal techniques as an evolutionary dead-end inthe history of analysis. Such an attitude was criticized by Grattan-Guinness, as discussed in Section 1. The fact is that, while Cauchydid use an occasional epsilon in an Archimedean sense, his techniquesrelying on infinitesimals find better proxies in a modern frameworkexploiting a Bernoullian continuum.Robinson first proposed an interpretation of Cauchy’s procedures inthe framework of a modern theory of infinitesimals in [40] (see Sec-tion 1.7). A set-theoretic foundation for infinitesimals could not havebeen provided by Cauchy for obvious reasons, but Cauchy’s procedures find closer proxies in modern infinitesimal frameworks than in modernArchimedean ones.
Acknowledgments
We are grateful to Peter Fletcher for helpful suggestions. We thankReinhard Siegmund-Schultze for bringing his review [43] to our atten-tion.
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E-mail address : [email protected] P. B laszczyk, Institute of Mathematics, Pedagogical University ofCracow, Poland
E-mail address : [email protected] P. Heinig
E-mail address : [email protected] V. Kanovei, IPPI RAS, Moscow, Russia
E-mail address : [email protected] M. Katz, Department of Mathematics, Bar Ilan University, RamatGan 52900 Israel
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