Cauchy, infinitesimals and ghosts of departed quantifiers

J. Bair, P. B laszczyk, R. Ely, V. Henry, V. Kanovei, K. U. Katz, M. G. Katz, T. Kudryk, S. S.
Kutateladze, T. McGaffey, T. Mormann, D. M. Schaps, D. Sherry

doi:10.15330/ms.47.2.115-144

Procedures relying on infinitesimals in Leibniz, Euler and Cauchy have been interpreted in both a Weierstrassian and Robinson’s frameworks. The latter provides closer proxies for the procedures of the classical masters. Thus, Leibniz’s distinction between assignable and inassignable numbers finds a proxy in the distinction between standard and nonstandard numbers in Robinson’s framework, while Leibniz’s law of homogeneity with the implied notion of equality up to negligible terms finds a mathematical formalisation in terms of standard part. It is hard to provide parallel formalisations in a Weierstrassian framework but scholars since Ishiguro have engaged in a quest for ghosts of departed quantifiers to provide a Weierstrassian account for Leibniz’s infinitesimals. Euler similarly had notions of equality up to negligible terms, of which he distinguished two types: geometric and arithmetic. Euler routinely used product decompositions into a specific infinite number of factors, and used the binomial formula with an infinite exponent. Such procedures have immediate hyperfinite analogues in Robinson’s framework, while in a Weierstrassian framework they can only be reinterpreted by means of paraphrases departing significantly from Euler’s own presentation. Cauchy gives lucid definitions of continuity in terms of infinitesimals that find ready formalisations in Robinson’s framework but scholars working in a Weierstrassian framework bend over backwards either to claim that Cauchy was vague or to engage in a quest for ghosts of departed quantifiers in his work. Cauchy’s procedures in the context of his 1853 sum theorem (for series of continuous functions) are more readily understood from the viewpoint of Robinson’s framework, where one can exploit tools such as the pointwise definition of the concept of uniform convergence. As case studies, we analyze the approaches of Craig Fraser and Jesper LЁutzen to Cauchy’s contributions to infinitesimal analysis, as well as Fraser’s approach toward Leibniz’s theoretical strategy in dealing with infinitesimals. The insights by philosophers Ian Hacking and others into the important roles of contextuality and contingency tend to undermine Fraser’s interpretive framework.

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3. Bair J., Blaszczyk P., Ely R., Henry V., Kanovei V., Katz K., Katz M., Kutateladze S., McGaffey T., Schaps D., Sherry D., Shnider S., Is mathematical history written by the victors? Notices of the American Mathematical Society, 60, no.7, 886-904.

4. Bair J., Blaszczyk P., Ely R., Henry V., Kanovei V., Katz K., Katz M., Kutateladze S., McGaffey T., Reeder P., Schaps D., Sherry D., Shnider S., Interpreting the infinitesimal mathematics of Leibniz and Euler, Journal for General Philosophy of Science, 48 (2017), no.2, 195-238.

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8. Bascelli T., Blaszczyk P., Kanovei V., Katz K., Katz M., Schaps D., Sherry D., Leibniz vs Ishiguro: Closing a quarter-century of syncategoremania HOPOS: Journal of the Internatonal Society for the History of Philosophy of Science, 6 (2016), no.1, 117-147.

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10. Bascelli T., Blaszczyk P., Borovik A., Kanovei V., Katz K., Katz M., Kutateladze S., McGaffey T., Schaps D., Sherry D., Cauchy's infinitesimals, his sum theorem, and foundational paradigms, Foundations of Science (2018), see https://arxiv.org/abs/1704.07723

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	Cauchy, infinitesimals and ghosts of departed quantifiers
Author	J. Bair, P. Blaszczyk, R. Ely, V. Henry, V. Kanovei, K. U. Katz, M. G. Katz, T. Kudryk, S. S. Kutateladze, T. McGaffey, T. Mormann, D. M. Schaps, D. Sherry j.bair@ulg.ac.be; pb@up.krakow.pl; ely@uidaho.edu; vhen@math.fundp.ac.be; kanovei@rambler.ru; katzmik@math.biu.ac.il; katzmik@macs.biu.ac.il; kudryk@mail.lviv.ua; sskut@math.nsc.ru; thomasmcgaffey@sbcglobal.net; ylxmomot@sf.ehu.es; dschaps@mail.biu.ac.il; david.sherry@nau.edu HEC-ULG, University of Liege, 4000 Belgium; Institute of Mathematics, Pedagogical University of Cracow, Poland; Department of Mathematics, University of Idaho, Moscow, Russia; Department of Mathematics, University of Namur, Belgium; IPPI, Moscow, and MIIT, Moscow, Russia; Department of Mathematics, Bar Ilan University, Ramat Gan, Israel; Department of Mathematics, Bar Ilan University, Ramat Gan, Israel; Department of Mathematics, Lviv National University, Ukraine; Sobolev Institute of Mathematics, Novosibirsk State University, Russia; Rice University, US; Department of Logic and Philosophy of Science, University of the Basque Country UPV/EHU, Donostia San Sebastian, Spain; Department of Classical Studies, Bar Ilan University, Ramat Gan, Israel; Department of Philosophy, Northern Arizona University, Flagstaff, US
Abstract	Procedures relying on infinitesimals in Leibniz, Euler and Cauchy have been interpreted in both a Weierstrassian and Robinson’s frameworks. The latter provides closer proxies for the procedures of the classical masters. Thus, Leibniz’s distinction between assignable and inassignable numbers finds a proxy in the distinction between standard and nonstandard numbers in Robinson’s framework, while Leibniz’s law of homogeneity with the implied notion of equality up to negligible terms finds a mathematical formalisation in terms of standard part. It is hard to provide parallel formalisations in a Weierstrassian framework but scholars since Ishiguro have engaged in a quest for ghosts of departed quantifiers to provide a Weierstrassian account for Leibniz’s infinitesimals. Euler similarly had notions of equality up to negligible terms, of which he distinguished two types: geometric and arithmetic. Euler routinely used product decompositions into a specific infinite number of factors, and used the binomial formula with an infinite exponent. Such procedures have immediate hyperfinite analogues in Robinson’s framework, while in a Weierstrassian framework they can only be reinterpreted by means of paraphrases departing significantly from Euler’s own presentation. Cauchy gives lucid definitions of continuity in terms of infinitesimals that find ready formalisations in Robinson’s framework but scholars working in a Weierstrassian framework bend over backwards either to claim that Cauchy was vague or to engage in a quest for ghosts of departed quantifiers in his work. Cauchy’s procedures in the context of his 1853 sum theorem (for series of continuous functions) are more readily understood from the viewpoint of Robinson’s framework, where one can exploit tools such as the pointwise definition of the concept of uniform convergence. As case studies, we analyze the approaches of Craig Fraser and Jesper LЁutzen to Cauchy’s contributions to infinitesimal analysis, as well as Fraser’s approach toward Leibniz’s theoretical strategy in dealing with infinitesimals. The insights by philosophers Ian Hacking and others into the important roles of contextuality and contingency tend to undermine Fraser’s interpretive framework.
Keywords	historiography; infinitesimal; Latin model; butterfly model; law of continuity; ontology; practice; Cauchy; Leibniz
DOI	doi:10.15330/ms.47.2.115-144
Reference	1. Alling N., Conway's field of surreal numbers, Trans. Amer. Math. Soc., 287 (1985), no.1, 365-386. 2. Artin E., Schreier O., Algebraische Konstruktion reeller Korper, Abhandlungen aus dem Mathematischen Seminar der Hamburgischen Univeristat, Leipzig 5 (1926), p. 85-99. Reprinted in Serge Lang and John T. Tate (editors), The Collected Papers of Emil Artin, Addison-Wesley Publishing Company, Reeding, MA., 1965, p. 258-272. 3. Bair J., Blaszczyk P., Ely R., Henry V., Kanovei V., Katz K., Katz M., Kutateladze S., McGaffey T., Schaps D., Sherry D., Shnider S., Is mathematical history written by the victors? Notices of the American Mathematical Society, 60, no.7, 886-904. 4. Bair J., Blaszczyk P., Ely R., Henry V., Kanovei V., Katz K., Katz M., Kutateladze S., McGaffey T., Reeder P., Schaps D., Sherry D., Shnider S., Interpreting the infinitesimal mathematics of Leibniz and Euler, Journal for General Philosophy of Science, 48 (2017), no.2, 195-238. 5. Bair J., Katz M., Sherry D., Fermat's dilemma: Why did he keep mum on infinitesimals? and the European theological context, Foundations of Science (2018), to appear. 6. Barabashev A., In support of significant modernization of original mathematical texts (in defense of presentism), Philos. Math. (3), 5 (1997), no.1, 21-41. 7. Bascelli T., Bottazzi E., Herzberg F., Kanovei V., Katz K., Katz M., Nowik T., Sherry D., Shnider S., Fermat, Leibniz, Euler, and the gang: The true history of the concepts of limit and shadow, Notices of the American Mathematical Society, 61 (2014), no.8, 848-864. 8. Bascelli T., Blaszczyk P., Kanovei V., Katz K., Katz M., Schaps D., Sherry D., Leibniz vs Ishiguro: Closing a quarter-century of syncategoremania HOPOS: Journal of the Internatonal Society for the History of Philosophy of Science, 6 (2016), no.1, 117-147. 9. Bascelli T., Blaszczyk P., Kanovei V., Katz K., Katz M., Kutateladze S., Nowik T., Schaps D., Sherry D., Gregory's sixth operation Foundations of Science, see https://arxiv.org/abs/1612.05944 10. Bascelli T., Blaszczyk P., Borovik A., Kanovei V., Katz K., Katz M., Kutateladze S., McGaffey T., Schaps D., Sherry D., Cauchy's infinitesimals, his sum theorem, and foundational paradigms, Foundations of Science (2018), see https://arxiv.org/abs/1704.07723 11. Benacerraf P., What numbers could not be The Philosophical Review, 74 (1965), 47-73. 12. Blasjo V., On what has been called Leibniz's rigorous foundation of infinitesimal geometry by means of Riemannian sum, Historia Math., 44 (2017), no.2, 134-149. 13. Blaszczyk P., A note on Otto Holder's memoir; Die Axiome der Quantitat und die Lehre vom Mass, Annales Universitatis Paedagogicae Cracoviensis Studia ad Didacticam Mathematicae Pertinentia, V (2013), 129-142. 14. 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Pages	115-144
Volume	47
Issue	2
Year	2017
Journal	Matematychni Studii
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