Is Leibnizian Calculus Embeddable in First Order Logic?


To explore the extent of embeddability of Leibnizian infinitesimal calculus in first-order logic (FOL) and modern frameworks, we propose to set aside ontological issues and focus on procedural questions. This would enable an account of Leibnizian procedures in a framework limited to FOL with a small number of additional ingredients such as the relation of infinite proximity. If, as we argue here, first order logic is indeed suitable for developing modern proxies for the inferential moves found in Leibnizian infinitesimal calculus, then modern infinitesimal frameworks are more appropriate to interpreting Leibnizian infinitesimal calculus than modern Weierstrassian ones.

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


  1. 1.

    The adjective non-Archimedean is used in modern mathematics to refer to certain modern theories of ordered number systems properly extending the real numbers, namely various successors of Hahn (1907). In modern mathematics, this adjective tends to evoke associations unrelated to seventeenth century mathematics. Furthermore, defining infinitesimal mathematics by a negation, i.e., as non-Archimedean, is a surrender to the Cantor–Dedekind–Weierstrass (CDW) view. Meanwhile, true infinitesimal calculus as practiced by Leibniz, Bernoulli, and others is the base of reference as far as seventeenth century mathematics is concerned. The CDW system could be referred to as non-Bernoullian, though the latter term has not yet gained currency.

  2. 2.

    On occasion Leibniz used the notation “\({{\mathrm{\,{}_{\ulcorner\!\urcorner }\,}}}\)” for the relation of equality. Note that Leibniz also used our “\(=\)” and other signs for equality, and did not distinguish between “\(=\)” and “\({{\mathrm{\,{}_{\ulcorner\!\urcorner }\,}}}\)” in this regard. To emphasize the special meaning equality had for Leibniz, it may be helpful to use the symbol \({{\mathrm{\,{}_{\ulcorner\!\urcorner }\,}}}\) so as to distinguish Leibniz’s equality in a generalized sense of “up to” from the modern notion of equality “on the nose.”

  3. 3.

    In support of this claim, Cassirer refers here in particular to Gottfried Wilhelm Leibniz, Die philosophischen Schriften, hrg. von Carl Immanuel Gerhardt, 7 Bde., Berlin 1875–1890, Bd. VII, S. 542. (Cassirer 1902, p. xi)

  4. 4.

    In the original: “Leibniz selbst hat es ausgesprochen, daß die neue Analysis aus dem innersten Quell der Philosophie geflossen ist, und beiden Gebieten die Aufgabe zugewiesen, sich wechselseitig zu bestätigen und zu erhellen.”


  1. Alling, N. (1985). Conway’s field of surreal numbers. Transactions of the American Mathematical Society, 287(1), 365–386.

    Google Scholar 

  2. Bair, J., Błaszczyk, P., Ely, R., Henry, V., Kanovei, V., Katz, K., et al. (2013). Is mathematical history written by the victors? Notices of the American Mathematical Society, 60(7), 886–904., arxiv: 1306.5973.

  3. Bair, J., Błaszczyk, P., Ely, R., Henry, V.; Kanovei, V., Katz, K., et al. (2016). Interpreting the infinitesimal mathematics of Leibniz and Euler. Journal for general philosophy of science (to appear). doi:10.1007/s10838-016-9334-z, arxiv:1605.00455

  4. Barreau, H. (1989). Lazare Carnot et la conception leibnizienne de l’infini mathématique. In La mathématique non standard (pp. 43–82). Paris: Fondem. Sci. CNRS.

  5. Bascelli, T., Bottazzi, E., Herzberg, F., Kanovei, V., Katz, K., Katz, M., et al. (2014). Fermat, Leibniz, Euler, and the gang: The true history of the concepts of limit and shadow. Notices of the American Mathematical Society, 61(8), 848–864.

    Article  Google Scholar 

  6. Bascelli, T., Błaszczyk, P., Kanovei, V., Katz, K., Katz, M., Schaps, D., et al. (2016). Leibniz vs Ishiguro: Closing a quarter-century of syncategoremania. HOPOS (Journal of the Internatonal Society for the History of Philosophy of Science), 6(1), 117–147. doi:10.1086/685645, arxiv: 1603.07209.

    Article  Google Scholar 

  7. Benacerraf, P. (1965). What numbers could not be. Philosophical Review, 74, 47–73.

    Article  Google Scholar 

  8. Błaszczyk, P. (2015). A purely algebraic proof of the fundamental theorem of algebra. arxiv: 1504.05609.

  9. Borovik, A., & Katz, M. (2012). Who gave you the Cauchy-Weierstrass tale? The dual history of rigorous calculus. Foundations of Science, 17(3), 245–276. doi:10.1007/s10699-011-9235-x.

    Article  Google Scholar 

  10. Bos, H. (1974). Differentials, higher-order differentials and the derivative in the Leibnizian calculus. Archive for History of Exact Sciences, 14, 1–90.

    Article  Google Scholar 

  11. Carnot, L. (1797). Réflexions sur la métaphysique du calcul infinitésimal. Paris.

  12. Cassirer, E. (1902). Leibniz’ System in seinen wissenschaftlichen Grundlagen. Gesammelte Werke, Hamburger Ausgabe, ECW 1, Hamburg, Felix Meiner Verlag, 1998.

  13. Child, J. (Ed.) (1920). The early mathematical manuscripts of Leibniz. Translated from the Latin texts published by Carl Immanuel Gerhardt with critical and historical notes by J. M. Child. The Open Court Publishing, Chicago-London. Reprinted by Dover in 2005.

  14. Conway, J. (2001). On numbers and games (2nd ed.). Natick, MA: A K Peters.

    Google Scholar 

  15. Euclid.(1660). Euclide’s Elements; The whole Fifteen Books, compendiously Demonstrated. By Mr. Isaac Barrow Fellow of Trinity College in Cambridge. And Translated out of the Latin. London.

  16. Gerhardt, C. (Ed.). (1846). Historia et Origo calculi differentialis a G. G. Leibnitio conscripta. Hahn: Hannover.

    Google Scholar 

  17. Gerhardt, C. (Ed.). (1850–1863). Leibnizens mathematische Schriften. Berlin and Halle: Eidmann.

  18. Guillaume, M. (2014). Review of “Katz, M., & Sherry, D. Leibniz’s infinitesimals: Their fictionality, their modern implementations, and their foes from Berkeley to Russell and beyond. Erkenntnis, 78 (2013), no. 3, 571–625.” Mathematical Reviews.

  19. Hahn, H. (1907). Über die nichtarchimedischen Grössensysteme. Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften, Wien, Mathematisch—Naturwissenschaftliche Klasse 116 (Abteilung IIa), pp. 601–655.

  20. Hewitt, E. (1948). Rings of real-valued continuous functions. I. Transactions of the American Mathematical Society, 64, 45–99.

    Article  Google Scholar 

  21. Ishiguro, H. (1990). Leibniz’s philosophy of logic and language (2nd ed.). Cambridge: Cambridge University Press.

    Google Scholar 

  22. Kanovei, V., Katz, M., & Mormann, T. (2013). Tools, Objects, and Chimeras: Connes on the Role of Hyperreals in Mathematics. Foundations of Science, 18(2), 259–296. doi:10.1007/s10699-012-9316-5, arxiv: 1211.0244.

  23. Kanovei, V., Katz, K., Katz, M., & Sherry, D. (2015). Euler’s lute and Edwards’ oud. The Mathematical Intelligencer, 37(4), 48–51. doi:10.1007/s00283-015-9565-6, arxiv: 1506.02586.

  24. Katz, K., & Katz, M. (2011). Cauchy’s continuum. Perspectives on Science, 19(4), 426–452. doi:10.1162/POSC_a_00047, arxiv: 1108.4201.

  25. Katz, K., & Katz, M. (2012). A Burgessian critique of nominalistic tendencies in contemporary mathematics and its historiography. Foundations of Science, 17(1), 51–89. doi:10.1007/s10699-011-9223-1, arxiv: 104.0375.

  26. Katz, M., & Kutateladze, S. (2015). Edward Nelson (1932–2014). The Review of Symbolic Logic, 8(3), 607–610. doi:10.1017/S1755020315000015, arxiv: 1506.01570.

  27. Katz, M., & Sherry, D. (2012). Leibniz’s laws of continuity and homogeneity. Notices of the American Mathematical Society, 59(11), 1550–1558., arxiv: 1211.7188.

  28. Katz, M., & Sherry, D. (2013). Leibniz’s infinitesimals: Their fictionality, their modern implementations, and their foes from Berkeley to Russell and beyond. Erkenntnis, 78(3), 571–625. doi:10.1007/s10670-012-9370-y, arxiv: 1205.0174.

  29. Knobloch, E. (2002). Leibniz’s rigorous foundation of infinitesimal geometry by means of Riemannian sums. Foundations of the formal sciences 1 (Berlin, 1999). Synthese, 133(1–2), 59–73.

    Article  Google Scholar 

  30. Laugwitz, D. (1992). Leibniz’ principle and omega calculus. [A] Le labyrinthe du continu. Colloq. Cerisy-la-Salle/Fr. 1990, 144–154.

  31. Leibniz, G. (1684). Nova methodus pro maximis et minimis\(\ldots \) Acta Eruditorum, Oct. 1684. See (Gerhardt 1850–1863), V, pp. 220–226. English translation at

  32. Leibniz, G. (1701). “Cum Prodiisset\(\ldots \)” mss “Cum prodiisset atque increbuisset Analysis mea infinitesimalis\(\ldots \)”. In (Gerhardt 1846, pp. 39–50).

  33. Leibniz, G. (1702). To Varignon, 2 Feb. 1702. In (Gerhardt 1850–1863], vol. IV, pp. 91–95.

  34. Leibniz, G. (1710) Symbolismus memorabilis calculi algebraici et infinitesimalis in comparatione potentiarum et differentiarum, et de lege homogeneorum transcendentali. In [Gerhardt 1850–1863], vol. V, pp. 377–382.

  35. Leibniz, G. (1965). Responsio ad nonnullas difficultates a Dn. Bernardo Niewentiit circa methodum differentialem seu infinitesimalem motas. Acta Eruditorum Lipsiae. In (Gerhardt 1850–1863), vol. V, pp. 320–328. A French translation is in (Leibniz 1989, p. 316–334).

  36. Leibniz, G.(1989). La naissance du calcul différentiel. 26 articles des Acta Eruditorum. Translated from the Latin and with an introduction and notes by Marc Parmentier. With a preface by Michel Serres. Mathesis. Librairie Philosophique J. Vrin, Paris.

  37. Lenzen, W. (1987). Leibniz on how to derive set-theory from elementary arithmetics. In Proceedings of the 8th International Congress of Logic, Methodology, and Philosophy of Science, (vol. 3, pp. 176–179) Moscow.

  38. Lenzen, W. (2004). Leibniz’s logic. In The rise of modern logic: From Leibniz to Frege, Handbook of the History of Logic (vol. 3, pp. 1–83), Amsterdam: Elsevier/North-Holland.

  39. Łoś, J. (1955). Quelques remarques, théorèmes et problèmes sur les classes définissables d’algèbres. In Mathematical interpretation of formal systems (pp. 98–113). Amsterdam: North-Holland.

  40. Mormann, T., & Katz, M. (2013). Infinitesimals as an issue of neo-Kantian philosophy of science. HOPOS: The Journal of the International Society for the History of Philosophy of Science, 3(2), 236–280., arxiv: 1304.1027.

  41. Nelson, E. (1977). Internal set theory: A new approach to nonstandard analysis. Bulletin of the American Mathematical Society, 83(6), 1165–1198.

    Article  Google Scholar 

  42. Nowik, T., & Katz, M. (2015). Differential geometry via infinitesimal displacements. Journal of Logic and Analysis, 7(5), 1–44.

  43. Quine, W. (1968). Ontological relativity. The Journal of Philosophy, 65(7), 185–212.

    Article  Google Scholar 

  44. Robinson, A. (1961). Non-standard analysis. Nederl. Akad. Wetensch. Proc. Ser. A 64 = Indag. Math. 23 (1961), 432–440. Reprinted in Selected Works, see item Robinson (1979), pp. 3–11.

  45. Robinson, A. (1966). Non-standard analysis. Amsterdam: North-Holland Publishing Co.

    Google Scholar 

  46. Robinson, A. (1979). Selected papers of Abraham Robinson. Vol. II. Nonstandard analysis and philosophy. In W. A. J. Luxemburg & S. Körner. New Haven, CT: Yale University Press.

  47. Sherry, D., & Katz, M. (2014). Infinitesimals, imaginaries, ideals, and fictions. Studia Leibnitiana, 44(2) (2012), 166–192. (The article was published in 2014 even though the journal issue lists the year 2012.) arxiv: 1304.2137.

  48. Skolem, T. (1933). Über die Unmöglichkeit einer vollständigen Charakterisierung der Zahlenreihe mittels eines endlichen Axiomensystems. Norsk Mat. Forenings Skr., II. Ser. No. 1/12, 73–82.

  49. Skolem, T. (1934). Über die Nicht-charakterisierbarkeit der Zahlenreihe mittels endlich oder abzählbar unendlich vieler Aussagen mit ausschliesslich Zahlenvariablen. Fundamenta Mathematicae, 23, 150–161.

    Article  Google Scholar 

  50. Skolem, T. (1955). Peano’s axioms and models of arithmetic. In Mathematical interpretation of formal systems (pp. 1–14). Amsterdam: North-Holland Publishing.

  51. Stolz, O. (1883). Zur Geometrie der Alten, insbesondere über ein Axiom des Archimedes. Mathematische Annalen, 22(4), 504–519.

    Article  Google Scholar 

  52. Unguru, S. (1976). Fermat revivified, explained, and regained. Francia, 4, 774–789.

    Google Scholar 

Download references


M. Katz was partially funded by the Israel Science Foundation Grant Number 1517/12.

Author information



Corresponding author

Correspondence to Mikhail G. Katz.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Błaszczyk, P., Kanovei, V., Katz, K.U. et al. Is Leibnizian Calculus Embeddable in First Order Logic?. Found Sci 22, 717–731 (2017).

Download citation


  • First order logic
  • Infinitesimal calculus
  • Ontology
  • Procedures
  • Leibniz
  • Weierstrass
  • Abraham Robinson