Leibniz versus Ishiguro: Closing a Quarter Century of Syncategoremania

Did Leibniz exploit infinitesimals and infinities à la rigueur or only as shorthand for quantified propositions that refer to ordinary Archimedean magnitudes? Hidé Ishiguro defends the latter position, which she reformulates in terms of Russellian logical fictions. Ishiguro does not explain how to reconcile this interpretation with Leibniz’s repeated assertions that infinitesimals violate the Archimedean property (i.e., Euclid’s Elements, V.4). We present textual evidence from Leibniz, as well as historical evidence from the early decades of the calculus, to undermine Ishiguro’s interpretation. Leibniz frequently writes that his infinitesimals are useful fictions, and we agree, but we show that it is best not to understand them as logical fictions; instead, they are better understood as pure fictions.

[1]  H. Jerome Keisler,et al.  On the strength of nonstandard analysis , 1986, Journal of Symbolic Logic.

[2]  W. Luxemburg.  Non-Standard Analysis , 1977 .

[3]  T. Tho.  EQUIVOCATION IN THE FOUNDATIONS OF LEIBNIZ’S INFINITESIMAL FICTIONS , 2012 .

[4]  Carl B. Boyer,et al.  The Concepts of the Calculus , 1940 .

[5]  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.

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

[7]  Don't take me half the way: On Berkeley on mathematical reasoning , 1993 .

[8]  P. A. Macmahon,et al.  A Treatise on Algebra , 2010, Nature.

[9]  Daniel J. Cook,et al.  Leibniz's Philosophy of Logic and Language. , 1974 .

[10]  Abraham Robinson.  Concerning Progress In The Philosophy Of Mathematics , 1975 .

[11]  S. Unguru.  Fermat revivified, explained, and regained , 1977 .

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

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

[14]  Patrick Reeder.  Internal Set Theory and Euler's Introductio in Analysin Infinitorum , 2013 .

[15]  Alexandre Borovik,et al.  Who Gave You the Cauchy–Weierstrass Tale? The Dual History of Rigorous Calculus , 2011, 1108.2885.

[16]  Eberhard Knobloch,et al.  Leibniz's Rigorous Foundation Of Infinitesimal Geometry By Means Of Riemannian Sums , 2002, Synthese.

[17]  Paul Benacerraf,et al.  What the numbers could not be , 1983 .

[18]  Ian Mueller,et al.  Philosophy of mathematics and deductive structure in Euclid's Elements , 1981 .

[19]  A. Cauchy.  Cours d'analyse de l'École royale polytechnique , 1821 .

[20]  F. B.,et al.  The Concepts of the Calculus , 1939, Nature.

[21]  B. Russell,et al.  Introduction to Mathematical Philosophy , 1920, The Mathematical Gazette.

[22]  J. M. Child.  The Early Mathematical Manuscripts Of Leibniz , 1921, The Mathematical Gazette.

[23]  Karin U. Katz,et al.  Euler’s Lute and Edwards’s Oud , 2015, 1506.02586.

[24]  Emanuele Bottazzi,et al.  Fermat, Leibniz, Euler, and the gang: The true history of the concepts of limit and shadow , 2014, 1407.0233.

[25]  D. Jesseph,et al.  Archimedes, Infinitesimals and the Law of Continuity: On Leibniz’s Fictionalism , 2008 .

[26]  D. Jesseph.  Truth in Fiction: Origins and Consequences of Leibniz's Doctrine of Infinitesimal Magnitudes , 2006 .

[27]  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 .

[28]  D. Jesseph,et al.  Leery Bedfellows: Newton and Leibniz on the Status of Infinitesimals , 2008 .

[29]  J. Tandy.  Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World , 2015 .

[30]  David Sherry,et al.  The wake of Berkeley's analyst: Rigor mathematicae? , 1987 .

[31]  Douglas M. Jesseph,et al.  Leibniz on The Elimination of Infinitesimals , 2015 .

[32]  Mikhail G. Katz,et al.  EDWARD NELSON (1932–2014) , 2015, The Review of Symbolic Logic.

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

[34]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

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

[36]  Vladimir Kanovei,et al.  Proofs and Retributions, Or: Why Sarah Can’t Take Limits , 2015 .

[37]  Piotr Blaszczyk,et al.  Is mathematical history written by the victors , 2013, 1306.5973.

[38]  Giovanni Ferraro,et al.  The rise and development of the theory of series up to the early 1820s , 2007 .

[39]  A. Alexander.  Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World , 2014 .

[40]  Hide Ishiguro,et al.  Leibniz's Philosophy of Logic and Language , 1972 .

[41]  Mikhail G. Katz,et al.  Almost Equal: the Method of Adequality from Diophantus to Fermat and Beyond , 2012, Perspectives on Science.