Is Leibnizian Calculus Embeddable in First Order Logic?

Abstract

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.

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Notes

  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.”

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Acknowledgments

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

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Błaszczyk, P., Kanovei, V., Katz, K.U. et al. Is Leibnizian Calculus Embeddable in First Order Logic?. Found Sci 22, 717–731 (2017). https://doi.org/10.1007/s10699-016-9495-6

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Keywords

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