On mathematical realism and
applicability of hyperreals

E. Bottazzi; V. Kanovei; M. Katz; T. Mormann; D. Sherry

doi:10.15330/ms.51.1.200-224

We argue that Robinson’s hyperreals have just as much claim to applicability as the garden variety reals. In a recent text, Easwaran and Towsner (ET) analyze the applicability of mathematical techniques in the sciences, and introduce a distinction between techniques that are applicable and those that are merely instrumental. Unfortunately the authors have not shown that their distinction is a clear and fruitful one, as the examples they provide are superficial and unconvincing. Moreover, their analysis is vitiated by a reliance on a naive version of object realism which has long been abandoned by most philosophical realists in favor of truth-value realism. ET’s argument against the applicability of hyperreals based on automorphisms of hyperreal models involves massaging the evidence and is similarly unconvincing. The purpose of the ET text is to argue that Robinson’s infinitesimal analysis is merely instrumental rather than applicable. Yet in spite of Robinson’s techniques being applied in physics, probability, and economics (see e.g., [70, Chapter IX], [1], [76], [60]), ET don’t bother to provide a meaningful analysis of even a single case in which these techniques are used. Instead, ET produce page after page of speculations mainly imitating Connesian chimera-type arguments ‘from first principles’ against Robinson. In an earlier paper Easwaran endorsed real applicability of the $\sigma$-additivity of measures, whereas the ET text rejects real applicability of the axiom of choice, voicing a preference for ZF. Since it is consistent with ZF that the Lebesgue measure is not $\sigma$-additive, Easwaran is thereby walking back his earlier endorsement. We note a related inaccuracy in the textbook Measure Theory by Paul Halmos. ET’s arguments are unacceptable to mathematicians because they ignore a large body of applications of infinitesimals in science, and massage the evidence of some crucial mathematical details to conform with their philosophical conclusions.

1. S. Albeverio, R. Hoegh-Krohn, J. Fenstad, T. Lindstrom, Nonstandard Methods in Stochastic Analysis and Mathematical Physics, Pure and Applied Mathematics, 122, Academic Press, Orlando, FL, 1986.

2. L. Arkeryd, N. Cutland, C.W. Henson, (Eds.) Nonstandard analysis. Theory and applications, Proceedings of the NATO Advanced Study Institute on Nonstandard Analysis and its Applications held in Edinburgh, 1996. NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, 493. Kluwer Academic Publishers Group, Dordrecht, 1997 (reprinted by Springer in 2012).

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

4. T. Bascelli, E. Bottazzi, F. Herzberg, V. Kanovei, K. Katz, M. Katz, T. Nowik, D. Sherry, S. Shnider, Fermat, Leibniz, Euler, and the gang: The true history of the concepts of limit and shadow, Notices of the American Mathematical Society, 61 (2014), №8, 848-864.

5. P. Benacerraf, What numbers could not be, Philos. Rev., 74 (1965), 47-73.

6. V. Benci, E. Bottazzi, M. Di Nasso, Elementary numerosity and measures, Journal of Logic & Analysis, 6 (2014), paper 3, 14 pages.

7. V. Benci, L. Horsten, S. Wenmackers, Non-archimedean probability, Milan Journal of Mathematics, 81 (2013), №1, 121-151.

8. V. Benci, L. Horsten, S. Wenmackers, Infinitesimal probabilities, The British Journal for the Philosophy of Science, 69 (2018), №2, 509-552.

9. K. Bhaskara Rao, M. Bhaskara Rao, Theory of charges. A study of finitely additive measures. With a foreword by D.M. Stone, Pure and Applied Mathematics, 109, Academic Press [Harcourt Brace Jovanovich, Publishers], New York, 1983.

10. E. Bishop, Foundations of Constructive Analysis, McGrawHill, New York, 1967.

11. P. B laszczyk, A. Borovik, V. Kanovei, M. Katz, T. Kudryk, S. Kutateladze, D. Sherry, A non-standard analysis of a cultural icon: The case of Paul Halmos, Logica Universalis, 10 (2016), №4, 393-405.

12. E. Bottazzi, Nonstandard models in measure theory and in functional analysis, Doctoral dissertation, University of Trento, 2017.

13. E. Bottazzi, A grid function formulation of a class of ill-posed parabolic equations, preprint, 2017, See https://arxiv.org/abs/1704.00472

14. E. Bottazzi, Grid functions of nonstandard analysis in the theory of distributions and in partial differential equations, Advances in Mathematics, 345 (2019), 429-482.

15. D. Bridges, L. Vita, Techniques of Constructive Analysis, Universitext, Springer, New York, 2006.

16. D. Bridges, M. McKubre-Jordens, Solving the Dirichlet problem constructively, Journal of Logic and Analysis, 5 (2013), article 3, 1-22.

17. N. Brunner, K. Svozil, M. Baaz, The axiom of choice in quantum theory, Math. Logic Quart., 42 (1996), №3, 319-340.

18. F. Campillo, C. Lobry, Effect of population size in a predator.prey model, Ecological Modelling, 246 (2012), 1-10.

19. M. Capinski, N. Cutland, Nonstandard methods for stochastic fluid mechanics, Series on Advances in Mathematics for Applied Sciences, 27, World Scientific, River Edge, NJ, 1995.

20. E. Cassirer, Substanzbegriff und Funktionsbegriff, Untersuchungen uber die Grundfragen der Erkenntniskritik, Berlin, Bruno Cassirer, 1910, Translated as Substance and Function, Chicago, Open Court, 1923.

21. A. Chakravartty, Scientific realism, Stanford Encyclopedia of Philosophy, 2017. See https://plato. stanford.edu/entries/scientific-realism

22. J.A. Coffa, The semantic tradition from Kant to Carnap: To the Vienna station, Cambridge University Press, Cambridge, 1991.

23. M. Colyvan, In defence of indispensability, Philos. Math. (3), 6 (1998), №1, 39-62.

24. A. Connes, Noncommutative geometry, Academic Press, San Diego, CA, 1994.

25. A. Connes, Cyclic cohomology, noncommutative geometry and quantum group symmetries, In Noncommutative geometry, 1.71, Lecture Notes in Math., 1831, Fond. CIME/CIME Found. Subser., Springer, Berlin, 2004.

26. A. Connes, Geometry and the quantum, In Foundations of mathematics and physics one century after Hilbert, 159.196, Springer, Cham, 2018.

27. B. de Finetti, Theory of Probability, V. 1 & 2 (A. Mach & A. Smith, Trans.). New York, Wiley, 1974.

28. W. Dean, Strict finitism, feasibility, and the sorites, Rev. Symb. Log., 11 (2018), №2, 295–346.

29. M. Dummett, Frege: Philosophy of Language, 2nd edn, Harvard University Press, Cambridge, MA, 1981.

30. K. Easwaran, Why countable additivity? Thought: A Journal of Philosophy, 2 (2013), 53–61.

31. K. Easwaran, Regularity and hyperreal credences, Philosophical Review, 123 (2014), №1, 1–41.

32. K. Easwaran, Rebutting and undercutting in mathematics, Epistemology, 146–162, Philos. Perspect., 29, Wiley-Blackwell, Malden, MA, 2015.

33. K. Easwaran, H. Towsner, Realism in mathematics: The case of the hyperreals, (2018), preprint. See https://www.dropbox.com/s/vmgevvgmy9bhl2c/Hyperreals.pdf?raw=1

34. K. Easwaran, H. Towsner, Realism in mathematics: The case of the hyperreals, 9 February 2019 version. See http://u.math.biu.ac.il/~katzmik/easwaran19.pdf

35. A. Ehrenfeucht, Discernible elements in models for Peano arithmetic, J. Symbolic Logic, 38 (1973), 291–292.

36. L. Evans, Partial Differential Equations, Providence (RI), American Mathematical Society, 1998.

37. W. Faris, (Ed.) Diffusion, quantum theory, and radically elementary mathematics, Edited by William G. Faris. Mathematical Notes, 47, Princeton University Press, Princeton, NJ, 2006.

38. P. Fletcher, K. Hrbacek, V. Kanovei, M. Katz, C. Lobry, S. Sanders, Approaches to analysis with infinitesimals following Robinson, Nelson, and others, Real Anal. Exchange, 42 (2017), №2, 193–252.

39. M. Foreman, F.Wehrung, The Hahn-Banach theorem implies the existence of a non-Lebesgue measurable set, Fund. Math., 138 (1991), 13–19.

40. D. Fremlin, Measure theory, V.5., Set-theoretic Measure Theory, Part II, Torres Fremlin, Colchester, 2008.

41. M. Friedman, Reconsidering Logical Positivism, Cambridge University Press, Cambridge, 1999.

42. M. Friedman, Dynamics of Reason, CSLI Publications, Stanford, Ca., 2001.

43. A. Gutman, M. Katz, T. Kudryk, S. Kutateladze, The Mathematical Intelligencer flunks the Olympics, Found. Sci. 22 (2017), №3, 539–555.

44. S. Haack, Realisms and their rivals: Recovering our innocence, Facta Philosophica, 4 (2002), 67–88.

45. P. Halmos, Measure Theory, Graduate Texts in Mathematics, 18, Springer-Verlag, New York, 1974 (reprint of the edition published by Van Nostrand, New York, 1950).

46. T. Hawkins, Lebesgue’s theory of integration. Its origins and development, Reprint of the 1979 corrected second ed. AMS Chelsea, Providence, RI, 2001.

47. C.W. Henson, On the nonstandard representation of measures, Transactions of the American Mathematical Society, 172 (1972), 437–446.

48. H. Herrlich, Axiom of choice, Springer, Berlin, 2006.

49. V. Kanovei, M. Katz, A positive function with vanishing Lebesgue integral in Zermelo–Fraenkel set theory, Real Anal. Exchange, 42 (2017), №2, 385–390.

50. V. Kanovei, M. Katz, T. Mormann, Tools, objects, and chimeras: Connes on the role of hyperreals in mathematics, Found. Sci., 18 (2013), №2, 259–296.

51. M. Katz, Review of Sergeyev, Y. Numerical infinities and infinitesimals: methodology, applications, and repercussions on two Hilbert problems. EMS Surv. Math. Sci. 4 (2017), № 2, 219–320, Mathematical Reviews, 2018. See https://mathscinet.ams.org/mathscinet-getitem?mr=3725242 See also http://u.math.biu.ac.il/~katzmik/pirahodimetz.html

52. M. Katz, S. Kutateladze, Edward Nelson (1932-2014), The Review of Symbolic Logic, 8 (2015), №3, 607–610.

53. M. Katz, E. Leichtnam, Commuting and noncommuting infinitesimals, American Mathematical Monthly, 120 (2013), №7, 631–641.

54. M. Katz, L. Polev, From Pythagoreans and Weierstrassians to true infinitesimal calculus, Journal of Humanistic Mathematics, 7 (2017), №1, 87–104.

55. M. Katz, D. Sherry, Leibniz’s infinitesimals: Their fictionality, their modern implementations, and their foes from Berkeley to Russell and beyond, Erkenntnis, 78 (2013), №3, 571–625.

56. L. Kauffman, Review of Sergeyev, Y. Numerical infinities and infinitesimals: methodology, applications, and repercussions on two Hilbert problems, EMS Surv. Math. Sci., 4 (2017), №2, 219–320, Zentralblatt (2018), See https://zbmath.org/?q=an3A1390.03048

57. A. Kock, Synthetic differential geometry, Second ed. London Mathematical Society Lecture Note Series, 333. Cambridge University Press, Camb., 2006.

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64. E. Nelson, Internal set theory: a new approach to nonstandard analysis, Bulletin of the American Mathematical Society, 83 (1977), №6, 1165-1198.

65. E. Nelson, Radically elementary probability theory, Annals of Mathematics Studies, 117, Princeton University Press, Princeton, NJ, 1987.

66. Y. Pinchover, J. Rubinstein, An introduction to partial differential equations, Cambridge University Press, Cambridge, 2005.

67. H. Putnam, What is Mathematical Truth? Reprinted in his Mathematics, Matter, and Method, Philosophical Papers, V.1, Cambridge University Press, Cambridge, 1975.

68. H. Putnam, Indispensability Arguments in the Philosophy of Mathematics, In H. Putnam, Philosophy in an Age of Science: Physics, Mathematics and Skepticism, edited by Mario de Caro and David MacArthur, Harvard, Harvard University Press, 2012.

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	On mathematical realism and applicability of hyperreals
Author	E. Bottazzi¹, V. Kanovei², M. Katz³, T. Mormann⁴, D. Sherry⁵ emanuele.bottazzi@unipv.it¹, kanovei@googlemail.com², katzmik@macs.biu.ac.il³, thomasarnold.mormann@ehu.eus⁴, David.Sherry@nau.edu⁵ 1) Department of Civil Engineering and Architecture, the University of Pavia, Italy; 2) IPPI RAS, Moscow, Russia; 3) Department of Mathematics, Bar Ilan University, Ramat Gan, Israel; 4) Univ. of the Basque Country UPV/EHU, Donostia San Sebastian, Spain; 5) Department of Philosophy, Northern Arizona University, Flagstaff, USA
Abstract	We argue that Robinson’s hyperreals have just as much claim to applicability as the garden variety reals. In a recent text, Easwaran and Towsner (ET) analyze the applicability of mathematical techniques in the sciences, and introduce a distinction between techniques that are applicable and those that are merely instrumental. Unfortunately the authors have not shown that their distinction is a clear and fruitful one, as the examples they provide are superficial and unconvincing. Moreover, their analysis is vitiated by a reliance on a naive version of object realism which has long been abandoned by most philosophical realists in favor of truth-value realism. ET’s argument against the applicability of hyperreals based on automorphisms of hyperreal models involves massaging the evidence and is similarly unconvincing. The purpose of the ET text is to argue that Robinson’s infinitesimal analysis is merely instrumental rather than applicable. Yet in spite of Robinson’s techniques being applied in physics, probability, and economics (see e.g., [70, Chapter IX], [1], [76], [60]), ET don’t bother to provide a meaningful analysis of even a single case in which these techniques are used. Instead, ET produce page after page of speculations mainly imitating Connesian chimera-type arguments ‘from first principles’ against Robinson. In an earlier paper Easwaran endorsed real applicability of the $\sigma$-additivity of measures, whereas the ET text rejects real applicability of the axiom of choice, voicing a preference for ZF. Since it is consistent with ZF that the Lebesgue measure is not $\sigma$-additive, Easwaran is thereby walking back his earlier endorsement. We note a related inaccuracy in the textbook Measure Theory by Paul Halmos. ET’s arguments are unacceptable to mathematicians because they ignore a large body of applications of infinitesimals in science, and massage the evidence of some crucial mathematical details to conform with their philosophical conclusions.
Keywords	object realism; truth-value realism; applicability; hyperreals; infinitesimals; instrumentalism; rigidity; automorphism; Lotka–Volterra model; Lebesgue measure; $\sigma$-additivity
DOI	doi:10.15330/ms.51.2.200-224
Reference	1. S. Albeverio, R. Hoegh-Krohn, J. Fenstad, T. Lindstrom, Nonstandard Methods in Stochastic Analysis and Mathematical Physics, Pure and Applied Mathematics, 122, Academic Press, Orlando, FL, 1986. 2. L. Arkeryd, N. Cutland, C.W. Henson, (Eds.) Nonstandard analysis. Theory and applications, Proceedings of the NATO Advanced Study Institute on Nonstandard Analysis and its Applications held in Edinburgh, 1996. NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, 493. Kluwer Academic Publishers Group, Dordrecht, 1997 (reprinted by Springer in 2012). 3. J. Bair, P. Bliaszczyk, R. Ely, V. Henry, V. Kanovei, K. Katz, M. Katz, S. Kutateladze, T. McGaffey, P. Reeder, D. Schaps, D. Sherry, S. Shnider, Interpreting the infinitesimal mathematics of Leibniz and Euler, Journal for General Philosophy of Science, 48 (2017), №2, 195-238. 4. T. Bascelli, E. Bottazzi, F. Herzberg, V. Kanovei, K. Katz, M. Katz, T. Nowik, D. Sherry, S. Shnider, Fermat, Leibniz, Euler, and the gang: The true history of the concepts of limit and shadow, Notices of the American Mathematical Society, 61 (2014), №8, 848-864. 5. P. Benacerraf, What numbers could not be, Philos. Rev., 74 (1965), 47-73. 6. V. Benci, E. Bottazzi, M. Di Nasso, Elementary numerosity and measures, Journal of Logic & Analysis, 6 (2014), paper 3, 14 pages. 7. V. Benci, L. Horsten, S. Wenmackers, Non-archimedean probability, Milan Journal of Mathematics, 81 (2013), №1, 121-151. 8. V. Benci, L. Horsten, S. Wenmackers, Infinitesimal probabilities, The British Journal for the Philosophy of Science, 69 (2018), №2, 509-552. 9. K. Bhaskara Rao, M. Bhaskara Rao, Theory of charges. A study of finitely additive measures. With a foreword by D.M. Stone, Pure and Applied Mathematics, 109, Academic Press [Harcourt Brace Jovanovich, Publishers], New York, 1983. 10. E. Bishop, Foundations of Constructive Analysis, McGrawHill, New York, 1967. 11. P. B laszczyk, A. Borovik, V. Kanovei, M. Katz, T. Kudryk, S. Kutateladze, D. Sherry, A non-standard analysis of a cultural icon: The case of Paul Halmos, Logica Universalis, 10 (2016), №4, 393-405. 12. E. Bottazzi, Nonstandard models in measure theory and in functional analysis, Doctoral dissertation, University of Trento, 2017. 13. E. Bottazzi, A grid function formulation of a class of ill-posed parabolic equations, preprint, 2017, See https://arxiv.org/abs/1704.00472 14. E. Bottazzi, Grid functions of nonstandard analysis in the theory of distributions and in partial differential equations, Advances in Mathematics, 345 (2019), 429-482. 15. D. Bridges, L. Vita, Techniques of Constructive Analysis, Universitext, Springer, New York, 2006. 16. D. Bridges, M. McKubre-Jordens, Solving the Dirichlet problem constructively, Journal of Logic and Analysis, 5 (2013), article 3, 1-22. 17. N. Brunner, K. Svozil, M. Baaz, The axiom of choice in quantum theory, Math. Logic Quart., 42 (1996), №3, 319-340. 18. F. Campillo, C. Lobry, Effect of population size in a predator.prey model, Ecological Modelling, 246 (2012), 1-10. 19. M. Capinski, N. Cutland, Nonstandard methods for stochastic fluid mechanics, Series on Advances in Mathematics for Applied Sciences, 27, World Scientific, River Edge, NJ, 1995. 20. E. Cassirer, Substanzbegriff und Funktionsbegriff, Untersuchungen uber die Grundfragen der Erkenntniskritik, Berlin, Bruno Cassirer, 1910, Translated as Substance and Function, Chicago, Open Court, 1923. 21. A. Chakravartty, Scientific realism, Stanford Encyclopedia of Philosophy, 2017. See https://plato. stanford.edu/entries/scientific-realism 22. J.A. Coffa, The semantic tradition from Kant to Carnap: To the Vienna station, Cambridge University Press, Cambridge, 1991. 23. M. Colyvan, In defence of indispensability, Philos. Math. (3), 6 (1998), №1, 39-62. 24. A. Connes, Noncommutative geometry, Academic Press, San Diego, CA, 1994. 25. A. Connes, Cyclic cohomology, noncommutative geometry and quantum group symmetries, In Noncommutative geometry, 1.71, Lecture Notes in Math., 1831, Fond. CIME/CIME Found. Subser., Springer, Berlin, 2004. 26. A. Connes, Geometry and the quantum, In Foundations of mathematics and physics one century after Hilbert, 159.196, Springer, Cham, 2018. 27. B. de Finetti, Theory of Probability, V. 1 & 2 (A. Mach & A. Smith, Trans.). New York, Wiley, 1974. 28. W. Dean, Strict finitism, feasibility, and the sorites, Rev. Symb. Log., 11 (2018), №2, 295–346. 29. M. Dummett, Frege: Philosophy of Language, 2nd edn, Harvard University Press, Cambridge, MA, 1981. 30. K. Easwaran, Why countable additivity? Thought: A Journal of Philosophy, 2 (2013), 53–61. 31. K. Easwaran, Regularity and hyperreal credences, Philosophical Review, 123 (2014), №1, 1–41. 32. K. Easwaran, Rebutting and undercutting in mathematics, Epistemology, 146–162, Philos. Perspect., 29, Wiley-Blackwell, Malden, MA, 2015. 33. K. Easwaran, H. Towsner, Realism in mathematics: The case of the hyperreals, (2018), preprint. See https://www.dropbox.com/s/vmgevvgmy9bhl2c/Hyperreals.pdf?raw=1 34. K. Easwaran, H. Towsner, Realism in mathematics: The case of the hyperreals, 9 February 2019 version. See http://u.math.biu.ac.il/~katzmik/easwaran19.pdf 35. A. Ehrenfeucht, Discernible elements in models for Peano arithmetic, J. Symbolic Logic, 38 (1973), 291–292. 36. L. Evans, Partial Differential Equations, Providence (RI), American Mathematical Society, 1998. 37. W. Faris, (Ed.) Diffusion, quantum theory, and radically elementary mathematics, Edited by William G. Faris. Mathematical Notes, 47, Princeton University Press, Princeton, NJ, 2006. 38. P. Fletcher, K. Hrbacek, V. Kanovei, M. Katz, C. Lobry, S. Sanders, Approaches to analysis with infinitesimals following Robinson, Nelson, and others, Real Anal. Exchange, 42 (2017), №2, 193–252. 39. M. Foreman, F.Wehrung, The Hahn-Banach theorem implies the existence of a non-Lebesgue measurable set, Fund. Math., 138 (1991), 13–19. 40. D. Fremlin, Measure theory, V.5., Set-theoretic Measure Theory, Part II, Torres Fremlin, Colchester, 2008. 41. M. Friedman, Reconsidering Logical Positivism, Cambridge University Press, Cambridge, 1999. 42. M. Friedman, Dynamics of Reason, CSLI Publications, Stanford, Ca., 2001. 43. A. Gutman, M. Katz, T. Kudryk, S. Kutateladze, The Mathematical Intelligencer flunks the Olympics, Found. Sci. 22 (2017), №3, 539–555. 44. S. Haack, Realisms and their rivals: Recovering our innocence, Facta Philosophica, 4 (2002), 67–88. 45. P. Halmos, Measure Theory, Graduate Texts in Mathematics, 18, Springer-Verlag, New York, 1974 (reprint of the edition published by Van Nostrand, New York, 1950). 46. T. Hawkins, Lebesgue’s theory of integration. Its origins and development, Reprint of the 1979 corrected second ed. AMS Chelsea, Providence, RI, 2001. 47. C.W. Henson, On the nonstandard representation of measures, Transactions of the American Mathematical Society, 172 (1972), 437–446. 48. H. Herrlich, Axiom of choice, Springer, Berlin, 2006. 49. V. Kanovei, M. Katz, A positive function with vanishing Lebesgue integral in Zermelo–Fraenkel set theory, Real Anal. Exchange, 42 (2017), №2, 385–390. 50. V. Kanovei, M. Katz, T. Mormann, Tools, objects, and chimeras: Connes on the role of hyperreals in mathematics, Found. Sci., 18 (2013), №2, 259–296. 51. M. Katz, Review of Sergeyev, Y. Numerical infinities and infinitesimals: methodology, applications, and repercussions on two Hilbert problems. EMS Surv. Math. Sci. 4 (2017), № 2, 219–320, Mathematical Reviews, 2018. See https://mathscinet.ams.org/mathscinet-getitem?mr=3725242 See also http://u.math.biu.ac.il/~katzmik/pirahodimetz.html 52. M. Katz, S. Kutateladze, Edward Nelson (1932-2014), The Review of Symbolic Logic, 8 (2015), №3, 607–610. 53. M. Katz, E. Leichtnam, Commuting and noncommuting infinitesimals, American Mathematical Monthly, 120 (2013), №7, 631–641. 54. M. Katz, L. Polev, From Pythagoreans and Weierstrassians to true infinitesimal calculus, Journal of Humanistic Mathematics, 7 (2017), №1, 87–104. 55. M. Katz, D. Sherry, Leibniz’s infinitesimals: Their fictionality, their modern implementations, and their foes from Berkeley to Russell and beyond, Erkenntnis, 78 (2013), №3, 571–625. 56. L. Kauffman, Review of Sergeyev, Y. Numerical infinities and infinitesimals: methodology, applications, and repercussions on two Hilbert problems, EMS Surv. Math. Sci., 4 (2017), №2, 219–320, Zentralblatt (2018), See https://zbmath.org/?q=an3A1390.03048 57. A. Kock, Synthetic differential geometry, Second ed. London Mathematical Society Lecture Note Series, 333. Cambridge University Press, Camb., 2006. 58. G. 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Pages	200-224
Volume	51
Issue	2
Year	2019
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

On mathematical realism and applicability of hyperreals