Skip to main content
SpringerLink
Log in

Leibniz’s syncategorematic infinitesimals

  • Published:
Archive for History of Exact Sciences Aims and scope Submit manuscript

Abstract

In contrast with some recent theories of infinitesimals as non-Archimedean entities, Leibniz’s mature interpretation was fully in accord with the Archimedean Axiom: infinitesimals are fictions, whose treatment as entities incomparably smaller than finite quantities is justifiable wholly in terms of variable finite quantities that can be taken as small as desired, i.e. syncategorematically. In this paper I explain this syncategorematic interpretation, and how Leibniz used it to justify the calculus. I then compare it with the approach of Smooth Infinitesimal Analysis, as propounded by John Bell. I find some salient differences, especially with regard to higher-order infinitesimals. I illustrate these differences by a consideration of how each approach might be applied to propositions of Newton’s Principia concerning the derivation of force laws for bodies orbiting in a circle and an ellipse.

“If the Leibnizian calculus needs a rehabilitation because of too severe treatment by historians in the past half century, as Robinson suggests (1966, 250), I feel that the legitimate grounds for such a rehabilitation are to be found in the Leibnizian theory itself.”—(Bos 1974–1975, 82–83).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

Notes

  1. Relevant extracts from Galileo’s Discorsi may be found in (LoC 352-357).

  2. At EN 78, Salviati says: “for I believe that these attributes of greatness, smallness and equality do not befit infinities, of which one cannot be said to be greater than, smaller than, or equal to another”; and again at the end of the ensuing proof: “in final conclusion, the attributes of equal, greater, and less have no place in infinities, but only in bounded quantities” (EN 79; LoC 355-56). For Gregory of St Vincent’s opinion, see his 1647, lib. 8, pr. 1, theorema, 870 ff.

  3. See Arthur (2009) for a discussion of the development of Leibniz’s early views on infinitesimals.

  4. The exposition I give is indebted to Knobloch (2002), but in the symbolization I have followed Leibniz’s own from the De quadratura (Leibniz 1993).

  5. Thus, Henk Bos, in his classic article on Leibniz’s differentials (1974–1975, 55), sees Leibniz as pursuing “two different approaches to the foundations of the calculus; one connected with the classical methods of proof by ‘exhaustion’, the other in connection with a law of continuity.” The quotations given in my text are from Bos (1974–1975), 12, 13. See also Herbert Breger’s similar criticisms of Bos on these points in his (2008, 195–197).

  6. Thus, to give a recent example of this kind of interpretation, Douglas Jesseph writes: “Leibniz often made grand programmatic statements to the effect that derivations which presuppose infinitesimals can always be re-cast as exhaustion proofs in the style of Archimedes. But he never, so far as I know, attempted anything like a general proof of the eliminability of the infinitesimal ...” (Jesseph 2008, 233).

  7. Leibniz adds: “Or, to put it more commonly, when the cases (or given quantities) continually approach one another, so that one finally passes over into the other, the consequences or events (or what is sought) must do so too.” (A VI 4, 371, 2032; translations mine.)

  8. Leibniz (1846), 40; Bos (1974–1975), 56.

  9. Nova methodus pro maximis et minimis, itemque tangentibus, quae nec fractas nec irrationales quantitates moratur, et singulare pro illis calculi genus, Acta Eruditorum Lips. 1684, GM V 220–226

  10. Nova Methodus, GM V 220.

  11. Henk Bos has drawn attention to the importance of Cum prodiisset in his article on Leibniz’s differentials (Bos 1974–1975). As he observes, since it contains a reference to a work by Gouye of 1701, and “deals with the problems which were discussed in 1701–1702, it is probable that it originated in or not much later than 1701” (1974–1975, 56, n. 92). In a marginal note Leibniz mentions “the Parisians” and urges that “All this must be reviewed very carefully so that it can be published, leaving out the sourer things said in contradicting others.” (Leibniz 1846, 39)

  12. Leibniz (1846), 44. Note that Leibniz characteristically draws his diagrams with the abscissas and \(x\)-axis vertical and the ordinates and \(y\)-axis horizontal. I have altered his \(_{1}X\), \(_{2}Y\), etc. to\( X_{1}\), \(Y_{2}\), etc., for ease of reading, and also corrected his dx to \((\text{ d })x\), as he seems to have intended, but otherwise this is his figure.

  13. This is a response to Nieuwentijt’s definition of an infinitesimal as something actually infinitely small: see below.

  14. Here I disagree with Bos, who claims that in taking the secant as the limiting position of the tangent, Leibniz “did not invoke the law of continuity; as will be seen, he used the law later, presupposing that the limiting position of the secant is the tangent.” (Bos 1974–1975, 57). But that is how the law of continuity works: it does presuppose a limiting case, it does not establish its existence.

  15. Leibniz (1846), 46–47; quoted from (Bos 1974–1975, 58). The asterisked phrase is an error for “or \((\text{ d })y:(\text{ d })x = [x(\text{ d })v/(\text{ d })x + v]:a\), but this does not figure in the remaining calculation.

  16. “... restat xdy \(+\) ydx \(+\) dxdy. Sed hic dxdy rejiciendum, ut ipsis xdy \(+\) ydx incomparabiliter minus, et fit d, xy \(=\) xdy \(+\) ydx, ita ut semper manifestum sit, re in ipsis assignabilibus peracta, errorem, qui inde metui queat, esse dato minorem, si quis calculum ad Archimedis stylum traducere velit.” (Leibniz to Wallis, 30th March 1699; GM IV 63).

  17. Here Leibniz appears to be applying a principle he had just formulated: “If a given motion can be resolved into two motions, one of them possible and the other impossible, the given motion will be impossible.” (A VI 3, 492; LoC 72–73) The quoted passage should be compared with the quotation Bertoloni Meli gives from Leibniz’s letter to Claude Perrault, also written in 1676: “I take it as certain that everything moving along a curved line endeavours to escape along the tangent of this curve; the true cause of this is that curves are polygons with an infinite number of sides, and these sides are portions of the tangents...” (quoted from Bertoloni Meli 1993, 75).

  18. See (Bos 1974–1975, 57, 63). Certainly some of the wording of Leibniz’s later justifications seems to show Newton’s influence, as Guicciardini notes in his excellent discussion (Guicciardini 1999, 161), even if he had developed the method of finite surrogates independently.

  19. As Meli reports (Bertoloni Meli 1993, 103), Leibniz’s attitude seems to have undergone a sea change on moving on from Sect. 1 of the Principia on First and Last Ratios, to Sect. 2, on the determination of central forces. Accordingly, in the first set of Excerpts Leibniz took from the Principia the following year, “lemma 9 is transcribed without commentary, and seems to be accepted without difficulty” (242).

  20. Marginalia, M 42 A: translated from the Latin quoted by Meli, 107.

  21. In a sympathetic but exacting analysis of SIA, Geoffrey Hellman observes that the status of infinitesimals in SIA is odd. On the one hand, “the “existence of non-zero nilsquares” is actually refutable”, but on the other, “in the background one assumes their ‘possibility’ in making sense of the whole theory” (Hellman 2006, 10). Certainly there is no constructive definition of them (11), even as fictional entities. Here, I think, Leibniz’s theory is more in keeping with the constructivist spirit.

  22. More precisely, the logic of smooth toposes is free first-order intuitionistic or constructive logic. See Bell (1998), 101–102.

  23. See the lucid account of Nieuwentijt’s theory in Mancosu (1996), 158ff, and also Nagel (2008), 200–205, for a succinct account of his dispute with Leibniz.

  24. Mancosu has very aptly summarized the difference between Nieuwentijt’s approach and Leibniz’s in his (Mancosu 1996, 160). One of these differences is that on Nieuwentijt’s theory, the infinitesimal \(b/m\) cannot be eliminated from computations; but Bell’s Principle of Microcancellation shows how to circumvent this problem. Bell has also provided a useful comparison of the difference between Nieuwentijt’s approach and Leibniz’s in his (2006, 98–100).

  25. On Varignon and Leibniz on central force see (Bertoloni Meli 1993, 81–83, 201ff.).

  26. For the second edition (published in 1713) Newton composed a new Proposition VI, which remains the same in the third (1726), and the first edition text became Corollary 1. Although the first edition can be pieced together from the footnotes in Newton (1999), Part I is usefully presented in its entirety (in a different translation by Mary Ann Rossi) by Brackenridge (1995), 235–267.

  27. This proof is replaced by a new proof (of Corollary 1 to the new proposition 6) in the 2nd and 3rd editions, where QR is said to be “equal to the sagitta of an arc that is twice the length of an arc QP, with \(P\) being in the middle; and twice the triangle SQP (or SP \(\times \) QT) is proportional to the time in which twice that arc is described and, therefore, can stand for the time.” (453–454). Nonetheless, Newton does write after the new proof of Prop. 6: “The proposition is easily proved by lem. 10, coroll. 4.”.

  28. Bertoloni Meli notes how, in his notes on Corollary 1 to the First Law, Leibniz objected to Newton’s use of “force” in stating it—writing after “... vi”, “(malo dicere conatu)”—meaning that vis should be replaced by conatus, that is (directed) infinitesimal tendency to motion. See (Bertoloni Meli 1993, 224, 239, 163).

  29. See Bertoloni Meli (1993), 162.

  30. “La voye est plus simple,” Leibniz wrote to Varignon in October 1706, “qui ne met pas l’acceleration dans les elemens, lorsqu’on n’en a point besoin. Je m’en suis servi depuis de 30 ans.” (GM IV 150–151; Bertoloni Meli 1993, 81).

  31. Tentamen de Motuum Coelestium Causis, Leibniz’s major work on cosmology, in which he re-derives Newton’s results in accordance with his own physico-mathematical principles, was published in the Acta of Leipzig in February 1689; I follow the English translation given by Bertoloni Meli, An Essay on the Causes of Celestial Motions (Bertoloni Meli 1993, 126–142; 131).

  32. Leibniz to De Volder, GP II 154, 156; quoted by Bertoloni Meli 1993, 88.

  33. Here Leibniz makes a trivial slip, as pointed out by (Bertoloni Meli 1993, 118). In order for \(a \theta \) to be twice the area of the elementary triangle, \(a\) needs to be half the latus rectum. Thus, XW \(=\) 2\(a\), and \(b^{2}= 2aq\), introducing an error of a factor of 2 in the second term of Eq. (4.19).

  34. This is a slightly simplified version of Leibniz’s figure, with the abscissas drawn vertically as was his custom, and with his \(_{1}X,\, _{1}Y\), etc., written as \(X_{1}, \,Y_{1}\), etc., as before.

  35. See Leibniz’s reply to Nieuwentijt in his (1695), and Bos’s lucid account: “Obviously, the procedure can be repeated again, by which LEIBNIZ has shown that finite line-variables can be given proportional to differentials of any order” (Bos 1974–1975, 65).

  36. Bell’s definition of second-order differentials here differs from that given in his (1998, 90–91), where they are defined as nilcube infinitesimals, ones such that \(\varepsilon ^{3} = 0\). But as I pointed out in an earlier version of this paper, this makes the decision of what type of infinitesimal to adopt (nilsquare or nilcube, etc.) depend on their applicability to the problem at hand; moreover, all the principles of SIA that we have depended upon above depend critically on the nilsquare property, which fails if the infinitesimals are nilcube. Bell has generously offered the present account (private communication) to circumvent such difficulties.

  37. See Bos (1974–1975), 5–10, 12–35 and Bertoloni Meli (1993), 66–73 for clear expositions of the difference between Leibniz’s understanding of differentiation and integration and the modern conception.

References

  • Arthur, R.T.W. 2006. The remarkable fecundity of Leibniz’s work on infinite series. Annals of Science 63(2): 221–225.

    Google Scholar 

  • Arthur, R.T.W. 2008. Leery Bedfellows: Newton and Leibniz on the Status of Infinitesimals. In Infinitesimal Differences: Controversies between Leibniz and his Contemporaries, ed. Ursula Goldenbaum and Douglas Jesseph, 7–30, Berlin and New York: De Gruyter.

  • Arthur, R.T.W. 2009. Actual Infinitesimals in Leibniz’s Early Thought. In The Philosophy of the Young Leibniz, Studia Leibnitiana Sonderhefte 35, ed. Mark Kulstad, Mogens Laerke and David Snyder, 11–28.

  • Bell, J.L. 1998. A Primer of Infinitesimal Analysis. Cambridge: Cambridge University Press.

  • Bertoloni Meli, Domenico. 1993. Equivalence and Priority: Newton versus Leibniz. Oxford: Clarendon Press.

  • Bos, H.J.M. 1974–1975. Differentials, Higher-Order Differential and the Derivative in the Leibnizian Calculus. Archive for History of Exact Sciences 14: 1–90.

  • Brackenridge, J. Bruce. 1995. The Key to Newton’s Dynamics: the Kepler Problem and the Principia. Berkeley: University of California Press.

  • Breger, Herbert. 2008. Leibniz’s Calculation with Compendia. In Infinitesimal Differences: Controversies between Leibniz and his Contemporaries, ed. Ursula Goldenbaum and Douglas Jesseph, 185–198, Berlin and New York: De Gruyter.

  • Cohen, I. Bernard, and George E. Smith. 2002. The Cambridge Companion to Newton. Cambridge: Cambridge University Press.

  • Gerhardt, C.I. ed. 1849–1863. Leibnizens Mathematische Schriften (Berlin and Halle: Asher and Schmidt; reprint ed. Hildesheim: Olms, 1971), 7 vols; cited by volume and page, e.g. GM II 316.

  • Gerhardt, C.I. ed. 1875–1890. Der Philosophische Schriften von Gottfried Wilhelm Leibniz (Berlin: Weidmann; reprint ed. Hildesheim: Olms, 1960), 7 vols; cited by volume and page, e.g. G II 268.

  • Grassmann, Hermann. 2000. Die Ausdehnungslehre: Vollständig und in strenger Form bearbeitet (1862) [The Theory of Extension, Thoroughly and Rigorously Treated]. English translation, 2000. ed. Lloyd Kannenberg. Extension Theory: American Mathematical Society.

  • Gregory of St Vincent. 1647. Opus Geometricum quadraturae circuli et sectionum coni, 2 volumes. Antwerp.

  • Guicciardini, Niccolò. 1999. Reading the Principia. Cambridge: Cambridge University Press.

  • Hellman, Geoffrey. 2006. Mathematical Pluralism: The Case of Smooth Infinitesimal Analysis. Journal of Philosophical Logic 35: 621–651.

    Google Scholar 

  • Ishiguro, Hidé. 1990. Leibniz’s Philosophy of Logic and Language (2nd ed.) Cambridge: Cambridge University Press.

  • Jesseph, Douglas. 2008. Truth in Fiction: Origins and Consequences of Leibniz’s Doctrine of Infinitesimal Magnitudes. In Infinitesimal Differences: Controversies between Leibniz and his Contemporaries, ed. Ursula Goldenbaum and Douglas Jesseph, 215–233, Berlin and New York: De Gruyter.

  • Knobloch, Eberhard. 2002. Leibniz’s Rigorous Foundation of Infinitesimal Geometry by means of Riemannian Sums. Synthese 133:59–73.

    Google Scholar 

  • Leibniz, G.W. 1684. Nova methodus pro maximis et minimis, itemque tangentibus, quae nec fractas, nec irrationales quantitates moratur, et singulare pro illis calculi genus. Acta Eruditorum, 467–473; GM V 220–226.

  • Leibniz, G.W. 1695. Responsio ad nonnullas difficultates a dn. Bernardo Nieuwentijt circa methodum differentialem seu infinitesimalem motas. Acta Eruditorum 310–316; GM V 320–326.

  • Leibniz, G.W. 1701. Cum prodiisset atque increbuisset Analysis mea infinitesimalis ... In Leibniz 1846, 39–50.

  • Leibniz, G.W. 1846. Historia et Origo calculi differentialis a G. G. Leibnitzio conscripta. ed. C. I. Gerhardt. Hanover.

  • Leibniz, G.W. 1923. Sämtliche Schriften und Briefe, ed. Akademie der Wissenschaften der DDR. Darmstadt and Berlin: Akademie-Verlag; cited by series, volume and page, e.g. A VI 2, 229.

  • Leibniz, G.W. 1993. De quadratura arithmetica circuli ellipseos et hyperbolae cujus corollarium est trigonometria sine tabulis. Ed. and commentary by Eberhard Knobloch. Göttingen: Vandenhoek & Ruprecht.

  • Leibniz, G.W. 2001. The Labyrinth of the Continuum: Writings on the Continuum Problem, 1672–1686. Ed., sel. & transl. R. T. W. Arthur. New Haven: Yale University Press; abbreviated LoC with page number.

  • Levey, S. 2008. Archimedes, Infinitesimals and the Law of Continuity: On Leibniz’s Fictionalism. In Infinitesimal Differences: Controversies between Leibniz and his Contemporaries, ed. Ursula Goldenbaum and Douglas Jesseph, 107–133, Berlin and New York: De Gruyter.

  • Mancosu, P. 1996. Philosophy of Mathematics and Mathematical Practice in the Seventeenth century. Oxford: Oxford University Press.

  • Nagel, Fritz. 2008. Nieuwentijt, Leibniz, and Jacob Hermann on Infinitesimals. In Infinitesimal Differences: Controversies between Leibniz and his Contemporaries, ed. Ursula Goldenbaum and Douglas Jesseph, 199–214, Berlin and New York: De Gruyter.

  • Newton, Isaac 1967. The Mathematical Papers of Isaac Newton, Volume 1, 1664–1666. ed. Whiteside, D. T. Cambridge: Cambridge University Press.

  • Newton, Isaac. 1999. The Principia: Mathematical Principles of Natural Philosophy. Trans. I. Bernard Cohen and Anne Whitman. Berkeley: University of California Press.

  • Nieuwentijt, Bernhard. 1695. Analysis Infinitorum, seu Curvilineorum proprietatates ex polygonuarum natura deductae. Amsterdam: Wolters.

  • Whiteside, D.T. 1966. Newtonian Dynamics. History of Science 5: 104–117.

Download references

Acknowledgments

An earlier version of this paper was originally submitted for publication in September 2005 in a volume on infinitesimals edited by William Harper and Craig Fraser, and remained available on the internet in the meantime. This explains the fact that references have been made to it in articles that have already been published, such as Levey (2008), Arthur (2008, 2009). But this revised version has, I hope, been much improved by the criticisms and constructive feedback of Niccolò Guicciardini and John L. Bell; any remaining errors or misunderstandings are my responsibility alone.

Author information

Authors and Affiliations

  1. McMaster Univeristy, Hamilton, ON, Canada

    Richard T. W. Arthur

Authors
  1. Richard T. W. Arthur
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Richard T. W. Arthur.

Additional information

Communicated by : Niccolò Guicciardini.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Arthur, R.T.W. Leibniz’s syncategorematic infinitesimals. Arch. Hist. Exact Sci. 67, 553–593 (2013). https://doi.org/10.1007/s00407-013-0119-z

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00407-013-0119-z

Keywords

Search

Navigation