Procedures of Leibnizian infinitesimal calculus: an account in three modern frameworks

Abstract

Recent Leibniz scholarship has sought to gauge which foundational framework provides the most successful account of the procedures of the Leibnizian calculus (LC). While many scholars (e.g. Ishiguro, Levey) opt for a default Weierstrassian framework, Arthur compares LC to a non-Archimedean framework SIA (Smooth Infinitesimal Analysis) of Lawvere–Kock–Bell. We analyze Arthur's comparison and find it rife with equivocations and misunderstandings on issues including the non-punctiform nature of the continuum, infinite-sided polygons, and the fictionality of infinitesimals. Rabouin and Arthur claim that Leibniz considers infinities as contradictory, and that Leibniz' definition of incomparables should be understood as nominal rather than as semantic. However, such claims hinge upon a conflation of Leibnizian notions of bounded infinity and unbounded infinity, a distinction emphasized by early Knobloch. The most faithful account of LC is arguably provided by Robinson's framework for infinitesimal analysis. We exploit an axiomatic framework for infinitesimal analysis SPOT to formalize LC.

Acknowledgments

We are grateful to Viktor Blåsjö, Elías Fuentes Guillén, Karel Hrbacek, Vladimir Kanovei, Eberhard Knobloch, Sam Sanders, David Schaps, and David Sherry for helpful comments. The influence of Hilton Kramer (1928–2012) is obvious.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Correction Statement

This article has been corrected with minor changes. These changes do not impact the academic content of the article.

Notes

1 See note 3 for Boyer's summary of the procedure of dropping negligible terms, and Section 4.5 for a formalization.

2 Possibly Johann Bernoulli did, but there may have been significant differences between Leibniz and Bernoulli; see Nagel (2008).

3 As observed by Boyer, ‘The general principle that in an equation involving infinitesimals those of higher order are to be discarded inasmuch they have no effect on the final result is sometimes regarded as the basic principle of the differential calculus. … this type of doctrine constituted the central theme in the developments leading to the calculus of Newton and Leibniz … It was, indeed, precisely upon this general premise that Leibniz sought to establish the method of differentials’ (Boyer 1941, 79). For a possible formalization of the procedure of discarding terms see Section 4.5.

4 The view of infinitesimals as limits is no late stratagem for Guicciardini. Nearly two decades earlier, he already claimed that ‘Leibniz carefully defines the infinitesimal and the infinite in terms of limit procedures’ (Guicciardini 2000). The claim occurs in a review of (Knobloch 1999). Note that the article under review does not mention the word ‘limit’ in connection with the Leibnizian calculus.

5 Similar remarks apply to the question of continuity between Leibniz and other Leibnizians such as Jacob Hermann; see note 16. Jesseph analyzes the use of infinitesimals by Torricelli and Roberval, and concludes: ‘By taking indivisibles as infinitely small magnitudes of the same dimension as the lines, figures, or solids they compose, [Roberval] could avoid the paradoxes that seemed to threaten Cavalieri's methods, just as Torricelli had done’ (Jesseph 2021, 117). Thus Torricelli and Roberval viewed infinitesimals as homogeneous with ordinary quantities but incomparable with them. In Section 1.5 we argue that this was Leibniz's view, as well. Guicciardini's limit-centered interpretation of Leibnizian infinitesimals anachronistically postulates a discontinuity between Leibniz with both the preceding and the following generations of mathematicians, and is symptomatic of butterfly-model thinking (see Section 1.7).

6 Thus Ishiguro's Chapter 5 fits in a modern tradition of scholarship in search of, to paraphrase Berkeley, the ghosts of departed quantifiers that targets not only Leibniz but also Cauchy (see e.g. Bair et al. 2017a).

7 Among the authors who have pursued such an Archimedean line is Levey. Levey is of the opinion that ‘Leibniz has abandoned any ontology of actual infinitesimals and adopted the syncategorematic view of both the infinite and the infinitely small …. The interpretation is worth stating in some detail, both for propaganda purposes and for the clarity it lends to some questions that should be raised concerning Leibniz's fictionalism’ (Levey 2008, 107; emphasis on ‘syncategorematic’ in the original; emphasis on ‘propaganda’ added). In conclusion, ‘Leibniz will emerge at key points to be something of an Archimedean’ (ibid.). Levey does not elaborate what ‘propaganda purposes’ he may have in mind here, but apparently they don't include considering Robinson's interpretation as a possibility. Most recently, Levey claims that ‘Particularly critical for grasping the conceptual foundation of his method … is the fact that for Leibniz the idea of ‘infinitely small’ is itself understood in terms of a variable finite quantity that can be made arbitrarily small’ (Levey (2021, 142); emphasis in the original), but provides no evidence.

8 Levey quotes this passage in (2021, 146) and claims that ‘Leibniz's view in DQA of the infinitely small and its correlative idea of the infinitely large as useful fictions, whose underlying truth is understood in terms of relations among variable finite quantities that could be spelled out painstakingly if necessary, will be his longstanding position’ (ibid.), without however providing any evidence. As we argued, rather than support the syncategorematic reading, the 1676 passage actually furnishes evidence against it.

9 Here and in the following we preserve the spelling as found in Gerhardt.

10 Loemker's English translation: ‘It follows from this that even if someone refuses to admit infinite and infinitesimal lines in a rigorous metaphysical sense and as real things, he can still use them with confidence as ideal concepts which shorten his reasoning, similar to what we call imaginary roots in the ordinary algebra’ (Leibniz 1989c, 543).

11 In their note 8, RA attribute such a view to Jesseph. However, Jesseph observes that ‘there are certainly traces of [Leibniz's fictionalist position] as early as the 1670s’ (Jesseph 2015, 195). RA give an erroneous page for Jesseph's observation.

12 Knobloch quotes only the first sentence ‘I have left my manuscript on arithmetical quadratures at Paris so that it may some day be printed there’ and infers that ‘Leibniz did want to publish it’ (Knobloch 2017, 282). The rest of the quotation makes such an inference less certain.

13 The incomparables in this sense are sometimes described by Leibniz as common. Thus, they are described as incomparables communs in the 1702 letter to Varignon quoted in Section 3.2. The distinction has often been overlooked by commentators; see note 34.

14 Leibnizian conciliatory methodology (see Mercer 2006) could account for his desire to include a mention of the A-method (favored by some contemporaries on traditionalist grounds) alongside the B-method.

15 As mentioned earlier, this interpretation is untenable since in the preceding sentences – not quoted by Levey – Leibniz defines his generalized equality where the difference may be incomparably small rather than necessarily absolutely zero.

16 Hermann defines infinitesimals as follows: ‘Quantitas vero infinite parva est, quae omni assignabili minor est: & talis Infinitesima vel Differentiale vocatur’ (Hermann 1700, 56). Thus an infinitesimal is smaller than every assignable quantity, just as in Leibniz.

17 Not to be confused with common incomparables; see notes 13 and 34.

18 Additional difficulties with Levey's reading are signalled in notes 7, 8, 37, 40 and 51.

19 Apart from its failure to either cite or name scholars being criticized, Gray's position endorsing historical authenticity seems reasonable–until one examines Gray's own historical work, where Euler's foundations are claimed to be ‘dreadfully weak’ (Gray 2008a, 6) and ‘cannot be said to be more than a gesture’ (Gray 2015, 3), whereas Cauchy's continuity is claimed to be among concepts Cauchy defined using ‘limiting arguments’ (Gray 2008b, 62). Such claims are arguably symptomatic of butterfly-model thinking (see Section 1.7). For a more balanced approach to Euler and Cauchy see respectively (Bair et al. 2017b) and (Bair et al. 2020).

20 It is commendable that in a recent text, Rabouin and Arthur (2020) engage their opponents in a more open fashion than has been the case until now, even though we raise many questions concerning the quality of their engagement in Sections 1.4 and 3.

21 Rabouin and Arthur (2020) seem to acknowledge the presence of two methods, but then go on to claim that method B exploiting infinitesimals dx and dy is easily transcribed in terms of assignable quantities $\left(d\right)x$ and $\left(d\right)y$. The mathematical coherence (or otherwise) of such a claim is analyzed in Section 6.2.

22 For a case study in Cauchy scholarship see Bair et al. (2019). Opposition to Marburg neo-Kantians' interest in infinitesimals is documented in Mormann and Katz (2013).

23 Arthur's finessing comment is analyzed in Section 5.1.

24 The definition of the part Δ appears in Section 5.3.

25 It is worth noting that bounded/unbounded is not the same distinction as potential/actual infinity. Bounded infinity is a term Leibniz reserves mainly to characterize the quantities used in his infinitesimal calculus, namely the infinitely small and infinitely large.

26 Furthermore, RA misinterpret Leibnizian bounded infinities when they compare them to compactifications in modern mathematics; see Section 3.8.

27 The definition of the part Δ appears in Section 5.3 below.

28 More precisely, an internal polygon with an infinite hyperinteger number of vertices. In Nelson's framework or in the theory SPOT one considers a polygon with μ sides for a nonstandard $\mu \in \mathbb{N}$; see Section 4.6.

29 See Section 1.4 and note 34 on the distinction between incomparables and common incomparables.

30 The Leibnizian passage is also quoted by RA (Rabouin and Arthur 2020, 437) who similarly fail to account for the fact that talking about a translation into the style of Archimedes entails the existence of a separate method exploiting infinitesimals à la rigueur.

31 Furthermore, Arthur's attempt to account for the Leibnizian derivation of the law $d\left(\mathrm{xy}\right)=\mathrm{xdy}+\mathrm{ydx}$ in Archimedean terms, by means of quantified variables representing dx, dy, $d\left(\mathrm{xy}\right)$ taking assignable values and tending to zero, would run into technical difficulties. Since all three tend to zero, the statement to the effect that ‘the error that could accrue from this would always be smaller than any given’ understood literally is true but vacuous: 0 = 0 + 0. To assign non-vacuous meaning to such an Archimedean translation of Leibniz, one would have to rewrite the formula, for instance, in the form $\frac{d\left(\mathrm{xy}\right)}{\mathrm{dx}}=x\frac{\mathrm{dy}}{\mathrm{dx}}+y$. While such a paraphrase works in the relatively simple case of the product rule, it becomes more problematic for calculations involving, e.g. transcendental functions.

32 We add a comma for clarity.

33 Here and in the following we preserve the original spelling.

34 Such common incomparables are not to be confused with the inassignable ones; see Section 1.4. Loemker gives the following translation of this sentence: ‘[T]hese incomparable magnitudes themselves, as commonly understood, are not at all fixed or determined but can be taken to be as small as we wish in our geometrical reasoning and so have the effect of the infinitely small in the rigorous sense’ (Leibniz 1989c, 543). The translation is not successful as it obscures the fact that communs modifies incomparables. Leibniz is not speaking of common understanding of incomparable quantities, but rather of common incomparables as opposed to bona fide ones. Horváth in (1986, 66) provides both the original French and an English translation, but omits the adjective communs in his translation.

35 To buttress such a claim, RA offer what they describe as ‘a short digression on Leibniz's theory of knowledge’ (Rabouin and Arthur 2020, 406). The evidence they provide is the following Leibnizian passage: ‘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’ (Leibniz as translated by RA in Rabouin and Arthur 2020, note 13). Readers may judge for themselves how compelling this evidence is for RA's claim.

36 Similarly, Jesseph notes that ‘a central task for Leibniz's fictionalism about infinitesimal magnitudes is to show that their introduction is essentially harmless in the sense that it does not yield a contradiction’ (Jesseph 2015, 196; emphasis added).

37 Levey claims that ‘For Leibniz there is no genuine infinite magnitude, great or small: no infinite line, no infinite quantity, no infinite number’ (Levey 2021, 147). He seeks support for his claim in the following Leibnizian passage from (Leibniz 1921): ‘It is perfectly correct to say that there is an infinity of things, i.e. that there are always more of them than one can specify. But it is easy to demonstrate that there is no infinite number, nor any infinite line or other infinite quantity, if these are taken to be genuine wholes’ (emphasis added). Levey appears not to have noticed that the quoted claim is conditional upon taking infinity to be a whole. Unbounded infinity taken as a whole contradicts the part-whole axiom; bounded infinity doesn't (see Section 2.2).

38 Arthur's conjured-up Leibniz specifically compares the violation of the part-whole axiom by unbounded infinities, to the set-theoretic paradox of the set of all ordinals, in Arthur (2019, 104).

39 Thus, Väth comments: ‘Without the existence of δ-free ultrafilters … we were not able to construct nonstandard embeddings. … [I]n the author's opinion … nonstandard analysis is not a good model for ‘real-world’ phenomena’ (Väth 2007, 85). Easwaran and Towsner claim to ‘point out serious problems for the use of the hyperreals (and other entities whose existence is proven only using the Axiom of Choice) in describing the physical world in a real way’ (Easwaran and Towsner 2019, 1). Sanders catalogues many such comments in (2020, 459). A rebuttal of the claims by Easwaran and Towsner and Väth appears in Bottazzi et al. (2019). A rebuttal of related criticisms by Pruss (2018) appears in Bottazzi and Katz (2021a, 2021b). The theories developed in Hrbacek and Katz (2020) expose as factually incorrect a claim by Alain Connes to the effect that ‘as soon as you have a non-standard number, you get a non-measurable set’ (Connes 2007, 26).

40 In connection with the law of continuity, Levey observes: ‘It is not hard to think of counterexamples to the law [‘the rules of the finite are found to succeed in the infinite’] and at a minimum the law of continuity requires additional sharpening and guidance to be used correctly in specific cases’ (Levey 2021, 153). However, Levey fails to point out that such sharpening and guidance are indeed provided in Robinson's classic framework for infinitesimal analysis and its modern axiomatic versions (see the beginning of the current Section 4). Unlike Probst (2018), Levey provides no hint of the existence of an alternative to the syncategorematic reading.

41 Bassler claims the following: ‘while it is difficult if not impossible to imagine a line starting at a particular point, going on forever, and then terminating at another point, the idea that two points on a line could be infinitely close to each other seems considerably more palatable. … At any rate, this is much different from Leibniz's conception that the model for the mathematical infinite is the sequence of natural numbers, which has a beginning but no end’ (Bassler 2008, 145). Arguably the two conceptions are not ‘much different’ since the interval $[0,\mu ]\subseteq \mathbb{R}$ would formalize Leibniz's bounded infinite line. It is therefore not difficult to imagine a line starting at a particular point, terminating at another point, and having infinite length.

42 Fragments of nonstandard arithmetic are studied by Avigad (2005), Sommer and Suppes (1996), Nelson (1987), Sanders (2020), van den Berg and Sanders (2019), Yokoyama (2010), and others.

43 Thus, speaking of Leibnizian infinite-sided polygons, Arthur claims that ‘this means that a curve can be construed as an ideal limit of a sequence of such polygons, so that its length L will be the limit of a sequence of sums ns of their sides s as their number $n\to \mathrm{\infty }$' (Arthur 2001, note 4, 393).

44 For an analysis of the procedures/ontology distinction see Błaszczyk et al. (2017).

45 See Section 2.1 for Arthur's translation of Leibniz's endorsement of the law of excluded middle.

46 In the original: ‘Proposito quocunque transitu continuo in aliquem terminum desinente, liceat raciocinationem communem instituere, qua ultimus terminus comprehendatur’ (Leibniz 1701, 40).

47 Spalt uses the German term Grenze (Spalt 2015, 111) which is more successful because it differs from the technical term Grenzwert for limit.

48 Leibniz goes on to write: ‘Truly it is very likely that Archimedes, and one who seems so have surpassed him, Conon, found out their wonderfully elegant theorems by the help of such ideas; these theorems they completed with reductio ad absurdum proofs, by which they at the same time provided rigorous demonstrations and also concealed their methods’ (ibid.). Leibniz's reference to the concealment of the direct methods (similar to Leibniz's own) by Archimedes indicates the existence of an alternative (infinitesimal) method to be concealed, contrary to the IA thesis. See related comments on translation into Archimedean terms in note 30.

49 Arthur's translation is rather unsatisfactory. Child translated this passage as follows: ‘Hence, it may be seen that there is no need in the whole of our differential calculus to say that those things are equal which have a difference that is infinitely small, but that those things can be taken as equal that have not any difference at all, provided that the calculation is supposed to be general, including both the cases in which there is a difference and in which the difference is zero; and provided that the difference is not assumed to be zero until the calculation is purged as far as is possible by legitimate omissions, and reduced to ratios of non-evanescent quantities, and we finally come to the point where we apply our result to the ultimate case’ (Leibniz as translated by Child in (2005, 151–152).

50 See note 3 on discarding terms.

51 Levey's page-and-a-half discussion of this Leibnizian derivation manages to avoid mentioning the crucial Leibnizian distinction between dx and $\left(d\right)x$. He concludes that ‘in fact dx stands for a variable finite quantity, and its behavior reflects precisely the fact – ensured by the continuity of the curve AY – that the difference between the abscissas can be taken as small as one wishes, all the way to zero’ (Levey 2021, 152). However, the conclusion applies only to the $\left(d\right)x$, not the dx as per Levey.