×

Is Leibnizian calculus embeddable in first order logic? (English) Zbl 1398.03223

Summary: 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.

MSC:

03H05 Nonstandard models in mathematics
03B30 Foundations of classical theories (including reverse mathematics)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Alling, N, Conway’s field of surreal numbers, Transactions of the American Mathematical Society, 287, 365-386, (1985) · Zbl 0524.12013
[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. http://www.ams.org/notices/201307/rnoti-p886pdf, arxiv: 1306.5973. · Zbl 1334.01010
[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. · Zbl 0788.01025
[5] Bascelli, T; Bottazzi, E; Herzberg, F; Kanovei, V; Katz, K; Katz, M; etal., Fermat, Leibniz, Euler, and the gang: the true history of the concepts of limit and shadow, Notices of the American Mathematical Society, 61, 848-864, (2014) · Zbl 1338.26001 · doi:10.1090/noti1149
[6] Bascelli, T; Błaszczyk, P; Kanovei, V; Katz, K; Katz, M; Schaps, D; etal., Leibniz vs ishiguro: closing a quarter-century of syncategoremania, HOPOS (Journal of the Internatonal Society for the History of Philosophy of Science), 6, 117-147, (2016) · doi:10.1086/685645
[7] Benacerraf, P, What numbers could not be, Philosophical Review, 74, 47-73, (1965) · doi:10.2307/2183530
[8] Błaszczyk, P. (2015). A purely algebraic proof of the fundamental theorem of algebra. arxiv: 1504.05609.
[9] Borovik, A; Katz, M, Who gave you the Cauchy-Weierstrass tale? the dual history of rigorous calculus, Foundations of Science, 17, 245-276, (2012) · Zbl 1279.01017 · doi:10.1007/s10699-011-9235-x
[10] Bos, H, Differentials, higher-order differentials and the derivative in the Leibnizian calculus, Archive for History of Exact Sciences, 14, 1-90, (1974) · Zbl 0291.01016 · doi:10.1007/BF00327456
[11] Carnot, L. (1797). Réflexions sur la métaphysique du calcul infinitésimal. Paris. · JFM 48.0008.03
[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. · JFM 47.0035.09
[14] Conway, J. (2001). On numbers and games (2nd ed.). Natick, MA: A K Peters. · Zbl 0972.11002
[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.
[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. http://www.ams.org/mathscinet-getitem?mr=3053644. · Zbl 1303.01012
[19] Hahn, H. (1907). Über die nichtarchimedischen Grössensysteme. Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften, Wien, Mathematisch—Naturwissenschaftliche Klasse 116 (Abteilung IIa), pp. 601-655. · JFM 38.0501.01
[20] Hewitt, E, Rings of real-valued continuous functions. I, Transactions of the American Mathematical Society, 64, 45-99, (1948) · Zbl 0032.28603 · doi:10.1090/S0002-9947-1948-0026239-9
[21] Ishiguro, H. (1990). Leibniz’s philosophy of logic and language (2nd ed.). Cambridge: Cambridge University Press.
[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. · Zbl 1392.03012
[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. · Zbl 1342.01017
[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. · Zbl 1292.01028
[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. · Zbl 1283.03006
[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. · Zbl 1323.01037
[27] Katz, M., & Sherry, D. (2012). Leibniz’s laws of continuity and homogeneity. Notices of the American Mathematical Society, 59(11), 1550-1558. http://www.ams.org/notices/201211/rtx121101550p, arxiv: 1211.7188. · Zbl 1284.03064
[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. · Zbl 1303.01012
[29] Knobloch, E, Leibniz’s rigorous foundation of infinitesimal geometry by means of Riemannian sums. foundations of the formal sciences 1 (Berlin, 1999), Synthese, 133, 59-73, (2002) · Zbl 1032.01011 · doi:10.1023/A:1020859101830
[30] Laugwitz, D. (1992). Leibniz’ principle and omega calculus. [A] Le labyrinthe du continu. Colloq. Cerisy-la-Salle/Fr. 1990, 144-154. · Zbl 0769.01005
[31] Leibniz, G. (1684). Nova methodus pro maximis et minimis\(… \)Acta Eruditorum, Oct. 1684. See (Gerhardt 1850-1863), V, pp. 220-226. English translation at http://17centurymaths.com/contents/Leibniz/nova1.
[32] Leibniz, G. (1701). “Cum Prodiisset\(… \)“ mss “Cum prodiisset atque increbuisset Analysis mea infinitesimalis\(… \)”. In (Gerhardt 1846, pp. 39-50). http://books.google.co.il/books?id=UOM3AAAAMAAJ&source=gbs_navlinks_s.
[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. · Zbl 1315.01067
[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. · Zbl 1066.03007
[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. · Zbl 0068.24401
[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. http://www.jstor.org/stable/10.1086/671348, arxiv: 1304.1027.
[41] Nelson, E, Internal set theory: A new approach to nonstandard analysis, Bulletin of the American Mathematical Society, 83, 1165-1198, (1977) · Zbl 0373.02040 · doi:10.1090/S0002-9904-1977-14398-X
[42] Nowik, T., & Katz, M. (2015). Differential geometry via infinitesimal displacements. Journal of Logic and Analysis, \(7\)(5), 1-44. http://www.logicandanalysis.org/index.php/jla/article/view/237/106. · Zbl 1331.26052
[43] Quine, W, Ontological relativity, The Journal of Philosophy, 65, 185-212, (1968) · doi:10.2307/2024305
[44] Robinson, A. (1961). Non-standard analysis. Nederl. Akad. Wetensch. Proc. Ser. A64 = Indag. Math.23 (1961), 432-440. Reprinted in Selected Works, see item Robinson (1979), pp. 3-11. · Zbl 0102.00708
[45] Robinson, A. (1966). Non-standard analysis. Amsterdam: North-Holland Publishing Co. · Zbl 0151.00803
[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. · Zbl 0424.01031
[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. · Zbl 0007.19305
[49] Skolem, T, Über die nicht-charakterisierbarkeit der zahlenreihe mittels endlich oder abzählbar unendlich vieler aussagen mit ausschliesslich zahlenvariablen, Fundamenta Mathematicae, 23, 150-161, (1934) · Zbl 0010.04902 · doi:10.4064/fm-23-1-150-161
[50] Skolem, T. (1955). Peano’s axioms and models of arithmetic. In Mathematical interpretation of formal systems (pp. 1-14). Amsterdam: North-Holland Publishing. · Zbl 0068.24603
[51] Stolz, O, Zur geometrie der alten, insbesondere über ein axiom des archimedes, Mathematische Annalen, 22, 504-519, (1883) · JFM 15.0022.02 · doi:10.1007/BF01443264
[52] Unguru, S, Fermat revivified, explained, and regained, Francia, 4, 774-789, (1976)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.