Equality and Near-Equality in a Nonstandard World
DOI:
https://doi.org/10.12775/LLP.2022.018Keywords
equality, nonstandard analysis, paradoxes of identityAbstract
In the context of nonstandard analysis, the somewhat vague equality relation of near-equality allows us to relate objects that are indistinguishable but not necessarily equal. This relation appears to enable us to better understand certain paradoxes, such as the paradox of Theseus’s ship, by identifying identity at a time with identity over a short period of time. With this view in mind, I propose and discuss two mathematical models for this paradox.
References
Bair, J., P. Błaszczyk, R. Ely, P. Heinig, and M. Katz, 2018, “ Leibniz’s wellfounded fictions and their interpetations”, Mat. Stud., 49 (2): 186–224.
Bair, J., P. Błaszczyk, P. Heinig, V. Kanovei, and M. Katz, 2020, “Cauchy’s work on integral geometry, centers of curvature, and other applications of infinitesimals”, Real Analysis Exchange, 45 (1): 1–23.
Barnes, J., 2002, The Presocratic Philosophers, Arguments of the Philosophers, Routledge.
Bascelli, T., E. Bottazzi, F. Herzberg, V. Kanovei, K. Katz, M. Katz, T. Nowik, D. Sherry, and S. Shnider, 2014, “Fermat, Leibniz, Euler, and the gang: the true history of the concepts of limit and shadow”, Notices Amer. Math. Soc., 61 (8): 848–864. DOI: http://dx.doi.org/10.1090/noti1149
Bedürftig, T., and R. Murawski, 2018, Philosophy of Mathematics, De Gruyter.
Bender, S., 2019, “Is Leibniz’s principle of the identity of indiscernibles necessary or contingent?”, Philosophers’ Imprint, 19 (42).
Berkeley, G., 2005, “The analyst, or a discourse addressed to an infidel mathematician”, pages 60–92 in W. Ewald (ed.), From Kant to Hilbert Volume 1: A Source Book in the Foundations of Mathematics, Oxford University Press.
Black, M., 1952, “The identity of indiscernibles”, Mind, 61: 153–164.
Bos, H., 1974, “Differentials, higher-order differentials and the derivative in the Leibnizian calculus”, Arch. History Exact Sci., 14: 1–90. DOI: http://dx.doi.org/10.1007/BF00327456
Boursin, J.-L., 1983, Dicionário elementar de matemáticas modernas, Publicações Dom Quixote.
Callot, J.-L., 1992, “Trois leçons d’analyse infinitésimale”, pages 369–381 in J. M. Salanskis and H. Sinaceur (eds.), Le labyrinthe du continu, Springer-Verlag, Paris.
Clark, M., 2012, Paradoxes from A to Z, Routledge.
Cortes, A., 1976, “Leibniz’s principle of the identity of indiscernibles: A false principle”, Philosophy of Science, 43 (4): 491–505. DOI: http://dx.doi.org/10.1086/288707
Deutsch, H. 2008, “Relative identity”, in Stanford Encyclopedia of Philosophy.
di Nasso. M., 1999, “On the foundations of nonstandard mathematics”, Math. Japon., 50 (1): 131–160.
Diener, F., and M. Diener (eds.), 1995, Nonstandard Analysis in Practice, Universitext, Springer-Verlag, Berlin. DOI: http://dx.doi.org/10.1007/978-3-642-57758-1
Diener, F., and G. Reeb, 1989, Analyse non standard, volume 40 of Collection Enseignement des Sciences, Hermann, Paris.
Dinis, B., and I. van den Berg, 2017, “Axiomatics for the external numbers of nonstandard analysis”, J. Logic & Analysis, 9(7): 1–47. DOI: http://dx.doi.org/10.4115/jla.2017.9.7
Dinis, B., and I. van den Berg, 2019, Neutrices and External Numbers: A Flexible Number System, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL. DOI: http://dx.doi.org/10.1201/9780429291456
Fletcher, P., K. Hrbacek, V. Kanovei, M. Katz, C. Lobry, and S. Sanders, 2017, “Approaches to analysis with infinitesimals following Robinson, Nelson, and others”, Real Anal. Exchange, 42 (2): 193–251. DOI: http://dx.doi.org/10.14321/realanalexch.42.2.0193
French, S., 2019, “Identity and individuality in quantum theory”, in E. N. Zalta (ed.), The Stanford Encyclopedia of Philosophy, winter 2019 edition.
Katz, K., and M. Katz, 2010a, “Zooming in on infinitesimal 1 – .9.. in a posttriumvirate era”, Educational Studies in Mathematics, 74 (3): 259–273.
Katz, K., and M. Katz, 2010b, “When is .999... less than 1?”, The Montana Mathematics Enthusiast, 7 (1): 3–30.
Katz, M., and D. Sherry, 2013, “Leibniz’s infinitesimals: their fictionality, their modern implementations, and their foes from Berkeley to Russell and beyond”, Erkenntnis, 78 (3): 571–625. DOI: http://dx.doi.org/10.1007/s10670-012-9370-y
Kunen, K., 1980, Set Theory, vol. 102 of “Studies in Logic and the Foundations of Mathematics”, North-Holland Publishing Co., Amsterdam–New York.
Kusraev, A., and S. Kutateladze, 1994, Nonstandard Methods of Analysis, vol. 291 of “Mathematics and its Applications”, Kluwer Academic Publishers Group, Dordrecht. Translated from the 1990 Russian original by L. Iouzina Tarvainen. DOI: http://dx.doi.org/10.1007/978-94-011-1136-2
Leibniz, G., 1989, Philosophical Essays, edited and translated by R. Ariew and D. Garber, Hackett Classics Series, Hackett Publishing Company.
Locke, J., 2015, An Essay Concerning Human Understanding: With Thoughts On the Conduct of the Understanding, Sagwan Press.
Lutz, R., 1987, Rêveries infinitésimales, Gazette des mathématiciens, 34: 79–87.
McGraw-Hill Dictionary of Athematics, 2003, McGraw-Hill, New York, 2nd edition.
Nelson, E., 1987, Radically Elementary Probability Theory, Annals of Mathematical Studies, vol. 117, Princeton University Press, Princeton, N. J. Noonan, H., 2017, “Relative identity”, pages 1013–1032 in A Companion to the Philosophy of Language, John Wiley and Sons, Ltd. DOI: http://dx.doi.org/10.1002/9781118972090.ch40
Potter, M., 2004, Set Theory and its Philosophy, Oxford University Press, New York. DOI: http://dx.doi.org/10.1093/acprof:oso/9780199269730.001.0001
Quine, W., 1951, “Two dogmas of empiricism”, Philosophical Review, 60 (1): 20–43.
Robinson, A., 1961, “Non-standard analysis”, Nederl. Akad. Wetensch. Proc. Ser. A 64, 23: 432–440.
Robinson, A., 1966, Non-Standard Analysis, North-Holland Publishing Co., Amsterdam.
Teixeira, J., 1999, “Elliptic differential equations and their discretizations”, ProQuest LLC, Ann Arbor, MI.
van den Berg, I., 1987, Nonstandard Asymptotic Analysis, Lecture notes in mathematics 1249, Springer-Verlag. DOI: http://dx.doi.org/10.1007/bfb0077577
van der Corput, J., 1959/1960, “Introduction to the neutrix calculus”, J. Analyse Math., 7: 281–399. DOI: http://dx.doi.org/10.1007/bf02787689
Wittgenstein, L., 1953, Philosophical Investigations, translated by G. E. M. Anscombe, The Macmillan Co., New York.
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