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Leibniz’s syncategorematic infinitesimals II: their existence, their use and their role in the justification of the differential calculus

Abstract

In this paper, we endeavour to give a historically accurate presentation of how Leibniz understood his infinitesimals, and how he justified their use. Some authors claim that when Leibniz called them “fictions” in response to the criticisms of the calculus by Rolle and others at the turn of the century, he had in mind a different meaning of “fiction” than in his earlier work, involving a commitment to their existence as non-Archimedean elements of the continuum. Against this, we show that by 1676 Leibniz had already developed an interpretation from which he never wavered, according to which infinitesimals, like infinite wholes, cannot be regarded as existing because their concepts entail contradictions, even though they may be used as if they exist under certain specified conditions—a conception he later characterized as “syncategorematic”. Thus, one cannot infer the existence of infinitesimals from their successful use. By a detailed analysis of Leibniz’s arguments in his De quadratura of 1675–1676, we show that Leibniz had already presented there two strategies for presenting infinitesimalist methods, one in which one uses finite quantities that can be made as small as necessary in order for the error to be smaller than can be assigned, and thus zero; and another “direct” method in which the infinite and infinitely small are introduced by a fiction analogous to imaginary roots in algebra, and to points at infinity in projective geometry. We then show how in his mature papers the latter strategy, now articulated as based on the Law of Continuity, is presented to critics of the calculus as being equally constitutive for the foundations of algebra and geometry and also as being provably rigorous according to the accepted standards in keeping with the Archimedean axiom.

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Notes

  1. The question of the “existence” of mathematical entities is a tricky one since it can relate to various questions, including ontological problems concerning their existence “in nature” or the like. In this paper, we take mathematical existence in its strictest sense of non-contradiction and we say that an entity does not exist when its positing in the theory involves introducing a contradiction.

  2. That this is not accurate can already be seen in the case of Robinson (1966), who gives a justification of infinitesimals using finitist means (using his compactness theorem), without it following that they are therefore replaceable by finite terms.

  3. For a clear and comprehensive explanation of the development of pre-Robinsonian non-Archimedean theories, see Ehrlich (2006), who gives the lie to Russell’s claim that infinitesimals had been banished from mathematics.

  4. A particularly clear example of this tendency to dichotomize is provided by Mikhail Katz, David Sherry and various co-workers in a spate of recent articles. According to their analysis, there are two rival approaches in the history of infinitesimal mathematics, which they call the “A-track” and the “B-track”. The first is the Weierstrassian approach, with “A” standing for the fact that the elements of the continuum obey the Archimedean axiom, whereas “B” stands for the “Bernoullian continuum”, in which there are elements violating that axiom (non-Archimedean infinitesimals). See, for instance, Katz and Sherry (2013), 575–577, Bair et al. (2013), 888–889).

  5. This must not be confused with the question of whether infinitely small things exist in nature; here also Leibniz denies that they do, holding that there are just arbitrarily small finite things. But this latter fact, Leibniz believes, is what allows the applicability of the ideal notions of mathematics to nature. More on this below.

  6. As we will detail below, these criticisms are made in Blåsjö (2017) and Jesseph (2015).

  7. “There are actually parts in the continuum … and these are actually infinite (Sunt actu partes in continuo …et sunt actu infinita)” (A VI 2, 264/LLC 339). Similarly in his De minimo et maximo (Nov 1672–Jan 1673), Leibniz asserted that “There are in the continuum infinitely small things (Sunt aliqua in continuo infinite parva)” (A VI 3, 98/LLC 12).

  8. For example, Douglas Jesseph writes: “[Leibniz’s] view of infinitesimals as ‘useful fictions’ seems to have taken shape in the mid 1690s, although there are certainly traces of it as early as the 1670s, and a forthright statement of the fictionalist position seems to have come from Leibniz’s pen only in the aftermath of the dispute in the Académie des Sciences over the foundations of the calculus.” (Jesseph 2015, 198). On the contrary, a forthright statement of the fictionalist position from April 1676 is given at the head of this article; and the position is defended in the De Quadratura that Leibniz finished the same year.

  9. See also Leibniz’s remark to Varignon, still in 1713, concerning Grandi’s series: “But one must not rely on reasonings about infinite series, unless one can demonstrate their truth with finite quantities by the methods of Archimedes. [Mais il ne faut se fier aux raisonnemens sur les series infinies, que lorsqu'on en peut demontrer la vérité par les finis à la façons d’Archimède].” (GM IV 191).

  10. See, for example, Historia et Origo Calculi Differentialis, ca. 1713 (Leibniz 1846, 4; Child 1920, 29–30).

  11. Observationes quod rationes seu proportiones, non habeamt locum circa quantitates nihilo minores et de vero sensu methodi infinitesimalis (GM V 389). This paper, which is presented as providing the “true meaning of infinitesimal methods” was published in the Acta Eruditorum in 1712.

  12. “Our thinking about composites is for the most part only symbolic” (Meditations on Knowledge, Truth and Ideas, A VI 4, 588).

  13. “For we often understand the individual words in one way or another, or remember having understood them before, but since we are content with this blind thought and do not pursue the resolution of notions far enough, it happens that a contradiction involved in a very complex notion is concealed from us” (A VI 4, 588). This is why nominal definition is not enough and should be accompanied by a proof of possibility (we’ll encounter this question with the definition of “incomparable” quantities defined as violating Archimedes’ axiom). Cf. Arthur’s discussion of real versus nominal definitions in the context of how Leibniz might have responded to Cantor’s definition of the transfinite (Arthur 2019).

  14. This is also what happens in reductio argument, except that in this case we focus on the part containing the contradiction to exhibit the impossibility involved in the fiction (the connexion with “fictions” and reductio is made explicitly in the correspondence with Clarke, see Correspondance Leibniz-Clarke (Robinet 1991), IV, §16–17, 374). The parallel with a point at infinity may be recalled here since this is a notion which produces a contradiction when inserted in some proofs of Euclid’s Elements (such as I, 27, where we assume that parallel lines meet), but which is also useful (when accompanied with suitable demonstrations) in order to produce general geometrical truths, such as the ones promoted by Desargues and Pascal. In this case, we can see how the very same notion can lead to a contradiction or not depending on the way it is used in the proof.

  15. Cf. Leibniz’s remark in a letter to Foucher: “It is true that from truths one only infers truths; but there are certain falsities useful for finding the truth.” (GP I 406).

  16. See, for example, the De Conditionibus (1665), § 164/307 (A VI 1, 143).

  17. See, for example, the Specimen inventorum de admirandis naturae Generalis arcanis (1688; A VI 4, 1628/LLC 327-9), or the De abstracto et concreto (1688; A VI 4, 991).

  18. See the passage from Observationes quod rationes quoted above, or his 1713 letter to Wolff: “And this is in agreement with the Law of Continuity that I once proposed in Bayle’s Nouvelles Lettres, and applied to the Laws of motion: whence it happens that in continua an exclusive extreme point can be treated as inclusive, so that the ultimate case, even though of a wholly different nature, is subsumed under the general law of the other cases, and at the same time, by a certain paradoxical reasoning and, so to speak, Philosophico-rhetorical Figure, we can understand the point to be included in the line, rest in motion, the special case in the contradistinguished general case, as if the point were an infinitely small or evanescent line, or rest an evanescent motion. And it is the same for other things of that kind, which that most profound gentleman Joachim Jung called true within a tolerance [tolerantur vera], and which are of the greatest service to the art of discovery; even though in my opinion they encompass something of the fictive and imaginary, this can nevertheless be rectified by a reduction to ordinary expressions so readily that no error can intervene.” (GM V 385).

  19. Another important aspect of this parallel is that there exist non-standard set theories, provably equiconsistent with the standard one, in which one introduces a two-tiered ontology with classes as genuine entities. The standard set theorist can henceforth claim at the same time that proper classes do not exist (in her axiomatic system), and that her surface language remains neutral as regard this question of existence, since there are other ways of interpreting it (as recalled by Kunen on that very same page). As we will see, this is exactly the position endorsed by Leibniz in his public declarations regarding infinitesimals.

  20. It is worth emphasizing that the premise of the whole discussion of this important paper, and of the immediately preceding “On Motion and Matter”, is Leibniz’s “recent discovery” that entities he had previously taken as actually infinitely small, such as horn angles and endeavours, are not so after all, and must be classified as fictions. See (A VI 3, 492; LLC 75, 394, 396).

  21. Cf. Leibniz’s comment in his 1675–1676 treatise De Quadratura that he prefers a justification “which simply shows that the difference between two quantities is nothing, so that they are then equal (whereas it is otherwise usually proved by a double reductio that one is neither greater nor smaller than the other)” (Leibniz 1993, 35). More on this below.

  22. For an introduction of this principle and the comparison with Newton, see Arthur (2008). See also Levey’s paper in the same volume: he calls a similar principle Leibniz’s “Principle of Equality” (Levey 2008, 113).

  23. Interestingly enough, in the exchange with Bernoulli, Leibniz refers to the fact that he proved, long ago, that an infinite number is a contradictory notion: “Sane ante multos annos demonstravi numerum seu multitudinem omnium numerorum contradictionem implicare…[I certainly demonstrated many years ago that the number or multiplicity of all numbers implies a contradiction]” (A III 7, 884) This exchange also includes, as we shall see, an important reference to the DQA of which Leibniz copies the argument on the area under the hyperbola and concludes by the fact that one could use infinitely small quantities as fictions as long as one can demonstrate them by using reductio ad absurdum and his Lemmata incomparabilium (A III 7, 857 and Sect. 4).

  24. In the seventeenth century, it was customary to refer to “the syncategorematic infinite” as an abbreviation for “the infinite understood syncategorematically”. For a thorough treatment of the Scholastics’ discussions of the syncategorematic, see Uckelman’s (2015). Leibniz also recognized a third species of the infinite, namely the hypercategorematic infinite. See Antognazza (2015) for discussion.

  25. Compare with what Leibniz wrote to Des Bosses in 1706: “I hold that matter is actually fragmented into parts smaller than any given, that is to say, that there is no part that is not actually subdivided into other parts undergoing different motions. This is demanded by the nature of matter and motion, and by the whole frame of the universe, for physical, mathematical and metaphysical reasons” (2 March 1706; LDB 32–35).

  26. Cf. what Leibniz wrote to Samuel Masson in the last year of his life: “Notwithstanding my Infinitesimal Calculus, I do not at all admit a genuine infinite number, although I confess that the multiplicity of things surpasses every finite number, or rather, every number.” (GP VI 629).

  27. For instance, as Leibniz wrote in a text from about 1698: “In this way, when I say that the infinite series of fractions 1, 1/2, 1/4, 1/8 etc., is equal to 2, I mean that if each of these fractions is assumed and none besides, then neither more nor less is assumed than what is in 2. And in this sense it is understood that the whole infinite series is equal to 2, so that what is called a collective whole is really a distributive one.” (Gerhardt 1876, 605).

  28. The best that could be done, and was done by mathematicians such as Pascal and Torricelli, was to show on examples that the same results were obtained either by using indivisibles or by relying on the “method of the Ancients”.

  29. This is typically the case in Pascal and Wallis, who both believed “indivisibles” to stand for infinitely small homogeneous quantities (Pascal is explicit about the fact that one takes “des petites portions égales”).

  30. As we will see below, this qualification was explicit when Leibniz presented a variant of the proof to Bodenhausen in 1690.

  31. One just has to fix a point as origin and take triangles under the first curve from this point, each triangle being transformed by the general construction into a rectangle of which one can estimate the difference with the second curve. This is crucial in the demonstration for two reasons: first, it enables one to give a proof which works in general (there are, of course, some conditions on the curve, the first one being that it has a tangent at every point, but Leibniz makes these constraints explicit, see Rabouin 2015, 355–356); second, it allows one to transform the “difference” between the triangulation and the curve comparable with a well known figure, of which one can estimate the magnitude (in this case a rectangle, one brilliant idea of Leibniz being the use of an upper bound for the width of this rectangle).

  32. See Knobloch (2002), 65 and Rabouin (2015), 356 for details.

  33. Leibniz mentions the need for “rigour” in several places in the DQA. Thus, at the beginning of Prop. 6 he writes “The reading of this proposition can be omitted if in demonstrating Prop. 7 one does not desire the utmost rigour [Hujus propositionis lectio omitti potest, si quis in demonstranda prop. 7. summum rigorem non desideret].” Elsewhere he writes of severas demonstrationes, and severe demonstrare.

  34. This idea already appears in Torricelli and in Pascal. For the latter, see Whiteside (1968), 341 and Cortese and Rabouin (2019).

  35. The validity of this general result on sum of “differences” has been demonstrated by Leibniz in Prop. 5.

  36. As we will see later, Leibniz still insists on this in the Compendium of 1690, when commenting on the DQA and warning about the danger of reasoning with the infinite: “In Hyperbola Conica, quia zona aequalis zonae conjugatae, fiet totum aequali partem. Unde patet rem reducendam ad demonstrationes apagogicas [In the hyperbolic conic, since a zone is equal to a conjugate zone, this makes the whole equal to a part. Whence it is clear that the matter should be reduced to apagogical demonstrations.]” (GM VI 106).

  37. Interestingly enough, when Leibniz rewrites the proof of Prop. 7 for the Compendium (to which we shall return in Sect. 4), he formulates it in a mixed way: starting as in a reductio by supposing that the intended result does not hold (i.e. that the Quadrilineum is not twice the Trilineum), and then by supposing that there is a difference Z between the two quantities, he demonstrates that the difference can be made smaller than Z. But instead of directly concluding quod est absurdum, he continues: “erit per prop. 5 diff. inter Q et T minor quam 3/4Z, adeoque minor quam Z, adeoque minor data quacunque quantitate, adeoque nulla est haec differentia [By prop. 5 the difference between Q and T will be smaller than 3/4Z and so smaller than Z, and so smaller than any given quantity whatever, and so this difference will be zero].” (GM V 101). We see here very clearly the way in which any reductio of this type can be transformed into a direct argument using PUD and vice versa.

  38. For some hypotheses in this direction, see Rabouin (2015).

  39. According to Whiteside (see note 34 above), Pascal is the first author to have the idea of reasoning directly on the difference (which can be made smaller than any given quantity). In the Lettres d’A. Dettonville, he applies this reasoning to the case of the division of the basis into a “sum of lines” (i.e. the basic techniques of the “method of indivisibles”), (Pascal 1659, 10–11).

  40. This aspect should not be underestimated. According to Whiteside, it was one of the major obstacles which prevented early modern authors from correctly assessing the power of exhaustion methods, when taken in their logical form (Whiteside 1968, 331).

  41. As regards this proposition itself, I hold that it is one of the most general and useful in geometry, inasmuch as it is so general that it applies to all curves, even in the case where they are drawn at will without a specific law; and given any figure it exhibits infinitely many others, the dimension of each of which depends on the former and vice versa. But it can also be reckoned among the most fruitful theorems in geometry; for from it are demonstrated the quadratures of all parabolas and hyperbolas to infinity (…) and not to mention that we have transformed infinitely many other absolute or hypothetical quadratures—certainly the circle and by its aid any conic having a centre—into a rational figure, and from this we derive a rational quadrature of the whole circle and any portion of it, and a true and perfect analytic expression for the arc from a given tangent, all of which it is the business of this treatise to demonstrate” (Scholium to Prop. 7, Leibniz 1993, 35–46, Leibniz 2016, 38). In the following paragraph, Leibniz insists on the fact that he can use triangles or rectangles to perform quadratures.

  42. The idea of fiction is mentioned a first time for designating the “point at infinity” introduced by geometers developing projective considerations, such as Desargues and Pascal (schol. VII). We shall return to that example later.

  43. In the Compendium Quadraturae Arithmeticae Leibniz defines a segment as “the space comprised between two lines, a curve, and another straight line”, and a sector as “a trilineum comprised between two straight lines and a curve” (GM V 101).

  44. On this geneaology, see (Grosholz 2007, chap. 8.2, “Leibniz on Transcendental Curves”).

  45. “To understand the Parabola and the straight line it is necessary to make use of infinite and infinitely small lines. Thus, if we suppose that the line q, or the latus transversum of the Parabola, is of an infinite length, it is evident that the equation 2axq ± ax2 = qy2 will equal to this one: 2axq = qy2, or 2ax = y2 (which is that of the parabola), because the term of the equation ax2 is infinitely small with respect to the others, 2axq, and qy2, for since there are as many letters or dimensions of one term as the other, those one of which one letter is infinite will be infinitely greater than those whose letters are only ordinary: which, consequently, can be neglected, since the error which will result from them will be infinitely small, or less than any given error, that is to say, null.” (chap. XLIV. Notations modernized. A VII 7, 103–104).

  46. (A VII 7, 88). The example of the coincidence taken as infinitely small distance is mentioned in the letter to Varignon from Feb. 2 1702 as a typical case of application of the Law of continuity (A III 9, 14).

  47. We’ll see that this point was at the core of the discussion with Wallis in 1699.

  48. This proposition is crucial for dealing with the quadrature of the simple hyperbola, a case which Leibniz deals with in prop. 12.

  49. This presupposes that the curve is, a least piecewise, convex—a condition which is stated by Leibniz when describing the curve in Prop. 6.

  50. Knobloch (2002) has given other examples, which can be generalized from prop. 20.

  51. See Mancosu and Vailati (1990). Leibniz also mentions other examples of the same kind in Gregory of Saint Vincent and Huygens.

  52. See the beginning of the Cum Prodiisset (Leibniz 1701 in Leibniz 1846, 41). The parallel between the introduction of point at infinite distance and infinitesimals, which appears in the DQA and was already present in Desargues, plays also a prominent role in the Elementa nova matheseos universalis (circa 1683; A VI 4, 521) and the Matheseos Universalis pars prior (1699; GM VII 75, 76).

  53. This consideration will become even more significant when Leibniz realizes that the main supporters of the Differential Calculus in the Académie Royale (the Bernoullis, L’Hôpital and Fontenelle) all believe in the existence of infinitesimal quantities.

  54. Especially when it comes to interpreting the “incomparables”, since what Leibniz expressly says is the following: « C’est ce qui m’a fait parler autres fois des incomparables, par ce que ce que j’en dis a lieu soit qu’on entende des grandeurs infiniment petites ou qu’on employe des grandeurs d’une petitesse inconsiderable et suffisante pour faire l’erreur moindre que celle qui est donnée » [“This is what makes me speak on other occasions of incomparables, because what I said there holds whether one means infinitely small magnitudes or one employs magnitudes of a smallness that is inconsiderable and sufficient to make the error less than that which is given”] (Defense du calcul in Pasini (1988), 708; our emphasis).

  55. A similar point was made in by Tzuchien Tho in his (2012), where he distinguished the question of the elimination of reference to infinitary terms by syncategorematic paraphrase from that of how Leibniz conducted his proofs in the DQA.

  56. Although Leibniz did not use this language in props. 6 and 7, this is what he alludes to when introducing the fiction of “infinitely small” at the beginning of Prop. 8: “Hoc uno verbo confici potest, ex eo quod quae p r o p o s i t i o n e 7. Demonstravimus generalia sunt, et locum habent, utcunque parvae sint rectae, ac proinde etsi sint infinite parvae. [This can be gathered from what was in Prop. 7. We have demonstrated general things, and they hold however small the straight lines may be, and therefore even if they are infinitely small.]”.

  57. Amongst other problems, one difficulty is that it is not easy to find an equivalent of a generic case on which to conduct such a justification.

  58. Victor Blåsjö writes: “it is well known that Leibniz was desperate to fashion a career for himself in intellectual circles at this time. The fact that he wanted to submit his work to the French Academy could very well be a reflection of this desire more than an assessment of the quality of the work, so this in itself proves nothing.” (2017, 136). Doug Jesseph “suspect[s] that he set aside the Arithmetical Quadrature without publishing it because he had turned his attention to more powerful methods that he would introduce in the 1680s…” (Jesseph 2015, 200).

  59. Thus, Jesseph writes that Leibniz’s procedure in the DQA, in its dependence on the construction of auxiliary curves, “requires that we have a tangent construction that will apply to the original curve”, and that although “this is readily available in the case of the circle, and tangents to conic sections and other well-behaved curves are also constructible with classical methods” (Jesseph 2015, 199), this would not extend to more general curves for which no geometrical tangent construction was available.

  60. Blåsjö (2017, 136). We will return to this below.

  61. Many of these remarks denigrating the treatise depend on the bias we noted above concerning the concentration on Proposition 6 to the exclusion of what else is contained in it. When considered in its entirety, the DQA contains many beautiful results which Leibniz had not published at the time and will still praise in later writings: an easy way to find the quadrature of the cycloid, his famous series for the quadrature of the circle, a unified and analytic treatment for trigonometric functions, a study of the logarithmic curve, the presentation of his “harmonic triangle”, a proof of the impossibility of an algebraic quadrature of the circle. By the time he wrote to Bernoulli, it is true, these would not have carried the same weight, since all of this was then published in other places. But as we shall see, this did not derogate from the importance of Proposition 6 for Leibniz.

  62. See in particular A III 4, 520 which Leibniz entitles: Responsio mea ad dubium hujus Epistolae. Among the difficulties Bodenhausen raises is a case where Leibniz concludes from the equation eg = m (where e is extension and m is mass) that d(e)g = d(m), even though g is not a constant. In a lengthy and intriguing reply, Leibniz writes that while e and m are sometimes assignabiles and sometimes inassignabiles, g is introduced by “a kind of fiction” (A III 4, 523).

  63. The passage begins with the following sentence: “Ad proferendum aliquod plausibile specimen nostrorum inventorum Geometricorum, quod ad captum sit eorum, qui veterum Methodis unice assueti sunt, non incommodum erit Theorema generale… [In order to proffer a plausible specimen of our discoveries in geometry, framed for those who are accustomed to the methods of the ancients, a general theorem will not be inconvenient…]”(A III 4, 624).

  64. The proof of Prop. 6 introduces an interesting variant by considering not the chords to the curve, but the triangles. This allows Leibniz to recover a technique by inscribed and circumscribed figures without having the defect of relying on an explicit construction of polygons, which he presents as “Archimedean” (A III 4, 635).

  65. Fateor autem me Theorematis hujusmodi opus non habere, nam quicquid ex illo duci potest, jam in calculo meo comprehenditur [I admit, though, that I do not need a theorem of this kind, for what can be inferred from it can already be understood in my calculus]” (and this is what he will intend to show by rephrasing prop. 8 in the Differential calculus). Notice, however, that Leibniz adds that he still enjoys this method, for it gives some representation in imagination corresponding to the operations of the Calculus (libenter tamen iis utor, quia calculum imaginationi quodammodo conciliant).

  66. De Tetragonismo meo placet difficultatem quae viro eximio occurrerat, sublatam esse, sed quia demonstrationem desiderare videtur, fundamenta hic apponam, ex quibus eam ipse facile absolvet, nam cuncta nunc prolixe explicare non vacat. Habeo quidem plures demonstrandi idem vias, sed haec maxime elegans visa est”. This passage begins the second half of the enclosure Leibniz sent to Otto Mencke for Sturm in November–December 1695, omitted from previous editions that had included the first half, such as Dutens (1768) and Erdmann (1840).

  67. Leibniz is referring to Prestet (1675).

  68. Quadratrix autem si esset parabola, daretur sectio anguli in data ratione, per certi gradus aequationem, quod est impossibile, cum altior sit aequatio prout arcus vel angulus in plures partes secari debet [“But if the quadratrix were a parabola, a section of the angle in the given ratio would be given by an equation of a certain degree, which is impossible, since the equation would be higher [in degree] in proportion as the arc or angle would have to be cut into several parts.”] (A II 3, 104).

  69. Recall that this is also the context in which he mentions his syncategorematic view on the infinite as grounding his conception of “fictions”.

  70. See Leibniz reply in Sect. 2.

  71. The expression is used by Varignon in a letter to Bernoulli, giving us a terminus a quo for these debates: (to Jean Bernoulli, 6 August 1697, Der Briefwechsel von Johann Bernoulli, Band 2, op. cit., p. 124).

  72. Clüver was in fact criticizing Archimedes on a par with the new calculus, see Mancosu and Vailati (1990) and Mancosu (1996), 157.

  73. To Mersenne, 9 January 1639 (AT II, 490). Leibniz is familiar with this letter and recalls it strategically at the beginning of the Cum Prodiisset, adding that Descartes did indeed rely on an Archimedean argument in his Metaphysics (Leibniz 1846, 42).

  74. Letter to Hardy (AT I, 490).

  75. See Méthode de l’Universalité, chap. VI: “Cavalieri, Mr Fermat, Mr Wallis, et autres supposent des certaines lettres, ou lignes infinement petites ou egales a rien. J’ay mis la mesme chose en usage, et j’ay adjousté des lettres qui representent une grandeur infinie, ou des lignes egales à des rectangles, comme sont les asymptotes de l’Hyperbole [Cavalieri, Mr. Fermat, Mr Wallis, and others suppose certain letters to be either infinitely small lines or equal to nothing. I have put the same thing into use, and have adjusted the letters which represent an infinite magnitude, or lines equal to rectangles, as are the asymptotes of the hyperbola]” (VII, 7, 79).

  76. Leibniz sent a first text as a letter to Pinsson, which was published almost in extenso in the Mémoire de Trevoux. This text, a reaction to critiques raised by Father Gouye, raised a lot of perplexity, even amongst his supporters, since it contained a comparison between the various orders of differentials and fixed and finite entities such as a grain of sand and the sun (GM V 96). Being asked about these comparisons by Varignon, Leibniz replied in the famous letter from Feb 2 1702, that it was a coarse way of speaking and that infinitesimals should not be seen as fixed entities. This letter was then published in the Journal des Savants (Extrait d’une lettre à M. Varignon, contenant l’explication de ce qu’on a raporté de luy dans les Memoires de Trevoux des mois de Novembre et Decembre derniers). The Justification was conceived as a follow up to this letter aimed specifically at critique coming from Rolle. It was sent to Pinsson for Varignon for publication in the Journal des Savants, but the project did not succeed.

  77. This is the interpretation in terms of “incomparables” to which we will come back later.

  78. We follow the translation given by Jesseph in his (2015, 201).

  79. un moyen fort palpable de justifier nostre maniere de calculer par le calcul ordinaire d’Algebre”.

  80. This parallel has led some scholars to read the Defense as a version of the Justification. This has had the unfortunate consequence of hiding another new strategy announced in the Defense (and absent from the Justification) in which Leibniz proposes an interpretation devoid of infinitesimals. We’ll give our hypothesis about the proper status of this text below.

  81. In one draft Leibniz thought about the possibility of positing the Law of Continuity as an axiom (LH XXXV, 29, 8 fol. 1–2).

  82. Let us note, for now, that when doing so, Leibniz will be very explicit about the general context in which he takes the Law of Continuity to hold in mathematics mentioning the examples dating back from the Parisian stay: parallel lines seen as meeting at infinity or ellipses transforming into parabolas when one of the focus goes at infinity.

  83. Thus in his 2015 paper, D. Jesseph distinguishes two strategies that might reconcile Leibniz[’s] requirements for rigorous demonstration”, a “syntactic” or “proof-theoretic” one, and a “semantic” or “model-theoretic” approach (2015, 196–7). The former is roughly the idea that “a symbol like ‘dx’ is simply a placeholder for a much more elaborate line of reasoning that makes reference only to finite differences of finite quantities”, while the latter is the idea that “use of infinitesimals would never lead from truth to falsehood”, which would require “something like the proof of a principle that adding infinitesimals to the standard geometry yields a model-theoretic conservative extension of standard geometry” (197). On that view the infinitesimal would be “something like a Hilbertian ideal element” (202). Similarly, Katz and Sherry identify two different approaches, one relying on the Method of Exhaustion, the second embracing infinitesimals as fictions in the sense of “modern formalist positions such as Hilbert’s and Robinson’s” (Katz and Sherry 2012, 1553). They refer this dichotomy to Bos (Katz and Sherry 2012, 1551; 2013, 575), although Bos (1974–5) seems to have been more cautious on this issue.

  84. “I call those magnitudes incomparable of which one multiplied by any finite number whatsoever cannot exceed the other, in the same manner that Euclid takes it in the fifth definition of the fifth book.” (Leibniz to L’Hôpital, GM II 287–289).

  85. Similarly, Karin and Mikhail Katz comment “Leibniz repeatedly asserted that his infinitesimals, when compared to other quantities, violate the Archimedean property, viz., Euclid’s Elements, V.4… This appears directly to contradict Ishiguro’s claim that Leibniz was working with an Archimedean continuum.” (Katz and Katz 2010, 3–4).

  86. Cf. Leibniz to Luigi Grandi, September 1713: “Interea infinite parva concipimus non ut nihila simpliciter et absolute, sed, utnihila respective(ut ipse bene notas), id est ut evanescentia quidem in nihilum, retinentia tamen characterem ejus quod evanescit.…” [Meanwhile we conceive infinitely small [quantities] not as simply and absolutely nothings, but as relative nothings (as you yourself well observe), that is, as indeed vanishing into nothing, yet retaining the character of that which is vanishing.]” (GM IV 218) (More on this below).

  87. Leibniz makes this point to Grandi in the same letter (GM IV 219).

  88. That this transfers to the use of the differential algorithm is immediate since we just have to express the relationship in the symbolism of this calculus: this is what Leibniz does in his letter to Bodenhausen from 1690 and in the Compendium of the Quadratura written at the same period.

  89. Letter to J. Bernoulli, 29 July 1698 (see Leibniz reply in Sect. 2). The same statement may be found in the letter to Bodenhausen A III 5, 149, or at the beginning of the Defense du calcul  « On leur peut tousjour monstrer que tout ce qui se conclut par ce calcul peut estre prouvé par une reduction ad absurdum à la façon d’Archimede: et en se servant des Lemmes des incomparables proposées dans les Actes de Leipzic. » (see second quotation in this section for a translation). See also, in addition to the letter to Wallis of 1699, the Responsio (Leibniz (1695); GM V 322).

  90. For a general presentation, see Jesseph (1998).

  91. This argument plays a crucial role in Jesseph’s idea that Leibniz was fluctuating in his acceptance of infinitesimals as eliminable or not (Jesseph 2008, 215).

  92. We saw in our study of Prop. 8 that this is not a trivial matter.

  93. See (Bos 1974–5) and (Arthur 2013) for details.

  94. Leibniz uses the comma to play a role similar to our parentheses. Here, it indicates that the whole expression xx + 2xdx + dxdx has to be divided by a, and not just the last term.

  95. Hence, it does not seem possible to claim that “The assignable quantity (d)x passes via infinitesimal dx on its way to absolute 0” (Katz and Sherry 2013, 581).

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Acknowledgements

The authors would like to thank the members of the Mathesis group for their comments when this paper was presented to them in Paris on September 25, 2019, and also Sandra Bella and Jeremy Gray for their helpful feedback on earlier versions of the paper. DR wishes to acknowledge the help of a grant from ANR: Mathesis, Édition et commentaires de manuscrits mathématiques inédits de Leibniz (2017–2021), No. ANR-17-CE27-0018-01 AAP GENERIQUE 2017.

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Rabouin, D., Arthur, R.T.W. Leibniz’s syncategorematic infinitesimals II: their existence, their use and their role in the justification of the differential calculus. Arch. Hist. Exact Sci. 74, 401–443 (2020). https://doi.org/10.1007/s00407-020-00249-w

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