LEIBNIZ ON BODIES AND INFINITIES: RERUM NATURA AND MATHEMATICAL FICTIONS

The way Leibniz applied his philosophy to mathematics has been the subject of longstanding debates. A key piece of evidence is his letter to Masson on bodies. We offer an interpretation of this often misunderstood text, dealing with the status of infinite divisibility in nature, rather than in mathematics. In line with this distinction, we offer a reading of the fictionality of infinitesimals. The letter has been claimed to support a reading of infinitesimals according to which they are logical fictions, contradictory in their definition, and thus absolutely impossible. The advocates of such a reading have lumped infinitesimals with infinite wholes, which are rejected by Leibniz as contradicting the partwhole principle. Far from supporting this reading, the letter is arguably consistent with the view that infinitesimals, as inassignable quantities, are mentis fictiones, i.e., (well-founded) fictions usable in mathematics, but possibly contrary to the Leibnizian principle of the harmony of things and not necessarily idealizing anything in rerum natura. Unlike infinite wholes, infinitesimals – as well as imaginary roots and other well-founded fictions – may involve accidental (as opposed to absolute) impossibilities, in accordance with the Leibnizian theories of knowledge and modality.

[1]  G. Leibniz,et al.  Philosophical papers and letters. , 2011 .

[2]  David Sherry,et al.  Two-Track Depictions of Leibniz’s Fictions , 2021, The Mathematical Intelligencer.

[3]  Patrick Riley,et al.  Leibniz's Philosophy of Logic and Language , 1973 .

[4]  G. Leibniz,et al.  The Early Mathematical Manuscripts of Leibniz: Translated from the Latin Texts Published by Carl Immanuel Gerhardt with Critical and Historical Notes , 2012 .

[5]  M. Antognazza.  The Leibniz-Des Bosses Correspondence , 2009 .

[6]  Philip Ehrlich,et al.  The Rise of non-Archimedean Mathematics and the Roots of a Misconception I: The Emergence of non-Archimedean Systems of Magnitudes , 2006 .

[7]  G. Leibniz,et al.  Quadrature arithmétique du cercle, de l'ellipse et de l'hyperbole et la trigonométrie sans tables trigonométriques qui en est le corollaire , 2004 .

[8]  Samuel Levey.  Leibniz on Mathematics and the Actually Infinite Division of Matter , 1998 .

[9]  E. Knobloch.  Galileo and Leibniz: Different Approaches to Infinity , 1999 .

[10]  M. Kulstad,et al.  The Philosophy of the Young Leibniz , 2009 .

[11]  Vladimir Kanovei,et al.  Controversies in the Foundations of Analysis: Comments on Schubring’s Conflicts , 2016, 1601.00059.

[12]  Henk J. M. Bos,et al.  Differentials, higher-order differentials and the derivative in the Leibnizian calculus , 1974 .

[13]  R. Sleigh,et al.  Metaphysics: The Early Period to the Discourse on Metaphysics , 1994 .

[14]  The Difficulty of Being Simple: On Some Interactions Between Mathematics and Philosophy in Leibniz’s Analysis of Notions , 2015 .

[15]  R. Hissette.  The Cambridge History of Later Medieval Philosophy. From the Rediscovery of Aristotle to the Disintegration of Scholasticism 1100-1600. Editors Norman Kretzmann, Anthony Kenny, Jan Pinborg. Associate editor Eleonore Stump , 1985 .

[16]  Philosophical Essays , 1997, Nature.

[17]  E. Knobloch.  The Infinite in Leibniz’s Mathematics — The Historiographical Method of Comprehension in Context , 1994 .

[18]  S. Shapiro,et al.  Varieties of Continua: From Regions to Points and Back , 2018 .

[19]  D. Rabouin.  Leibniz’s Rigorous Foundations of the Method of Indivisibles , 2015 .

[20]  P. Beeley.  Leibniz, Philosopher Mathematician and Mathematical Philosopher , 2015 .

[21]  Richard T. W. Arthur,et al.  The labyrinth of the continuum : writings on the continuum problem, 1672-1686 , 2001 .

[22]  Mikhail G. Katz,et al.  Infinitesimals, Imaginaries, Ideals, and Fictions , 2012 .

[23]  G. Leibniz,et al.  Confessio Philosophi: Papers Concerning the Problem of Evil, 1671-1678 , 2004 .

[24]  Vladimir Kanovei,et al.  Interpreting the Infinitesimal Mathematics of Leibniz and Euler , 2016, 1605.00455.

[25]  DAS KONTINUUM BEI LEIBNIZ , 2012 .

[26]  Oscar M. Esquisabel,et al.  Fiction, possibility and impossibility: three kinds of mathematical fictions in Leibniz’s work , 2021, Archive for History of Exact Sciences.

[27]  Akademie der Wissenschaften in Göttingen,et al.  Sämtliche Schriften und Briefe , 1923 .

[28]  R. Arthur.  Leibniz’s Syncategorematic Actual Infinite , 2018 .

[29]  Mikhail G. Katz,et al.  Procedures of Leibnizian infinitesimal calculus: an account in three modern frameworks , 2020, British Journal for the History of Mathematics.

[30]  Piotr Blaszczyk,et al.  Leibniz versus Ishiguro: Closing a Quarter Century of Syncategoremania , 2016, HOPOS: The Journal of the International Society for the History of Philosophy of Science.

[31]  Mikhail G. Katz,et al.  Leibniz’s Infinitesimals: Their Fictionality, Their Modern Implementations, and Their Foes from Berkeley to Russell and Beyond , 2012, 1205.0174.

[32]  D. Rutherford.  Leibniz’s ‘New System’ and Associated Contemporary Texts , 1999 .

[33]  Richard T. W. Arthur.  Leibniz’s syncategorematic infinitesimals , 2013 .

[34]  R. Arthur,et al.  Leibniz’s syncategorematic infinitesimals II: their existence, their use and their role in the justification of the differential calculus , 2020, Archive for History of Exact Sciences.