Mathematics, history, and philosophy of infinitesimals



What's new with infinitesimals? Provided below are links to over 70 recent publications on infinitesimals and related subjects by Jacques Bair, Tiziana Bascelli, Piotr Błaszczyk, Alexandre Borovik, Emanuele Bottazzi, Robert Ely, Peter Fletcher, Elías Fuentes Guillén, Peter Heinig, Valérie Henry, Frederik Herzberg, Karel Hrbacek, Renling Jin, Vladimir Kanovei, Boris Katz, Karin Katz, Taras Kudryk, Karl Kuhlemann, Semen Samsonovich Kutateladze, Eric Leichtnam, Claude Lobry, Thomas McGaffey, Thomas Mormann, Tahl Nowik, Luie Polev, Patrick Reeder, Sam Sanders, Jan Peter Schäfermeyer, David Sherry, Monica Ugaglia, Mark van Atten, and others.

A nice introduction to our program can be found in the MathSciNet review by M. Guillaume in pdf
To see where the papers have appeared click on List of periodicals
See also List of critics and Reception
See also Reappraisal of the procedures of the pioneers of infinitesimal analysis


year '24

24a. Katz, M.; Kuhlemann, K.; Sherry, D.; Ugaglia, M. "Leibniz on bodies and infinities: rerum natura and mathematical fictions." Review of Symbolic Logic 17 (2024), no. 1. https://doi.org/10.1017/S1755020321000575, https://arxiv.org/abs/2112.08155


year '23 (7 publications)

23a. Bair, J.; Borovik, A.; Kanovei, V.; Katz, M.; Kutateladze, S.; Sanders, S.; Sherry, D.; Ugaglia, M.; van Atten, M. "Is pluralism in the history of mathematics possible?" The Mathematical Intelligencer 45 (2023), no. 1, 8. http://doi.org/10.1007/s00283-022-10248-0, https://arxiv.org/abs/2212.12422, https://mathscinet.ams.org/mathscinet-getitem?mr=4559464. See also Depictions.

23b. Heinig, P.; Katz, M.; Kuhlemann, K.; Schaefermeyer, J.P.; Sherry, D. "Exploring Felix Klein's contested modernism." Antiquitates Mathematicae 17 (2023), 101-137. https://dx.doi.org/10.14708/am.v17i1.7245, https://arxiv.org/abs/2402.00122

23c. Hrbacek, K.; Katz, M. "Constructing nonstandard hulls and Loeb measures in internal set theories." Bulletin of Symbolic Logic 29 (2023), no. 1, 97-127. https://doi.org/10.1017/bsl.2022.43, https://arxiv.org/abs/2301.00367, https://mathscinet.ams.org/mathscinet-getitem?mr=4560535

23d. Hrbacek, K.; Katz, M. "Effective infinitesimals in ℝ." Real Analysis Exchange 48 (2023), no. 2, 365-380. https://arxiv.org/abs/2305.09672, https://doi.org/10.14321/realanalexch.48.2.1671048854, https://mathscinet.ams.org/mathscinet/article?mr=4668954

23e. Hrbacek, K.; Katz, M. "Peano and Osgood theorems via effective infinitesimals." Journal of Logic and Analysis 15:6 (2023), 1-19. https://doi.org/10.4115/jla.2023.15.6, https://arxiv.org/abs/2311.01374, https://mathscinet.ams.org/mathscinet/article?mr=4673816

23h. Katz, M.; Sherry, D.; Ugaglia, M. "When does a hyperbola meet its asymptote? Bounded infinities, fictions, and contradictions in Leibniz." Revista Latinoamericana de Filosofía 49 (2023), no. 2, 241-258. https://doi.org/10.36446/rlf2023359, https://arxiv.org/abs/2311.06023

23i. Ugaglia, M.; Katz, M. "Evolution of Leibniz’s thought in the matter of fictions and infinitesimals." In: Sriraman, B. (ed.) Handbook of the History and Philosophy of Mathematical Practice. Springer, Cham, 2023. https://doi.org/10.1007/978-3-030-19071-2_149-1, https://arxiv.org/abs/2310.14249


year '22 (2 publications)

22a. Bair, J.; Borovik, A.; Kanovei, V.; Katz, M.; Kutateladze, S.; Sanders, S.; Sherry, D.; Ugaglia, M. "Historical infinitesimalists and modern historiography of infinitesimals." Antiquitates Mathematicae 16 (2022), 189-257. https://doi.org/10.14708/am.v16i1.7169, http://arxiv.org/abs/2210.14504, https://mathscinet.ams.org/mathscinet-getitem?mr=4570174

22b. Katz, M.; Kuhlemann, K.; Sherry, D.; Ugaglia, M.; van Atten, M. "Two-track depictions of Leibniz's fictions." The Mathematical Intelligencer 44 (2022), no. 3, 261-266. https://doi.org/10.1007/s00283-021-10140-3, https://arxiv.org/abs/2111.00922, https://mathscinet.ams.org/mathscinet-getitem?mr=4480193. See also Depictions.


year '21 (7 publications)

21a. Bair, J.; Błaszczyk, P.; Ely, R.; Katz, M.; Kuhlemann, K. "Procedures of Leibnizian infinitesimal calculus: An account in three modern frameworks." British Journal for the History of Mathematics 36 (2021), no. 3, 170-209. https://doi.org/10.1080/26375451.2020.1851120, https://arxiv.org/abs/2011.12628, https://mathscinet.ams.org/mathscinet-getitem?mr=4353153, cited by:

21b. Bottazzi, E; Katz, M. "Infinite lotteries, spinners, and the applicability of hyperreals." Philosophia Mathematica 29 (2021), no. 1, 88-109. https://doi.org/10.1093/philmat/nkaa032, https://arxiv.org/abs/2008.11509, https://mathscinet.ams.org/mathscinet-getitem?mr=4267988

21c. Bottazzi, E; Katz, M. "Internality, transfer, and infinitesimal modeling of infinite processes." Philosophia Mathematica 29 (2021), no. 2, 256-277. https://doi.org/10.1093/philmat/nkaa033, https://arxiv.org/abs/2008.11513, https://mathscinet.ams.org/mathscinet-getitem?mr=4492449

21d. Bottazzi, E; Katz, M. "Infinitesimals via Cauchy sequences: Refining the classical equivalence." Open Mathematics 19 (2021), 477-482. https://doi.org/10.1515/math-2021-0048, https://arxiv.org/abs/2106.00229, https://mathscinet.ams.org/mathscinet-getitem?mr=4267478

21e. Hrbacek, K.; Katz, M. "Infinitesimal analysis without the Axiom of Choice." Annals of Pure and Applied Logic 172 (2021), no. 6, 102959. https://doi.org/10.1016/j.apal.2021.102959, https://arxiv.org/abs/2009.04980, https://mathscinet.ams.org/mathscinet-getitem?mr=4224071. See also Introduction to infinitesimal analysis without the axiom of choice

21f. Katz, M. "A two-track tour of Cauchy's Cours." Mathematics Today 57 (2021), no. 4, 154-158. Reprint in pdf, https://arxiv.org/abs/2107.00207, https://mathscinet.ams.org/mathscinet-getitem?mr=4401322

21g. Katz, M.; Kuhlemann, K.; Sherry, D.; Ugaglia, M. "Three case studies in current Leibniz scholarship." Antiquitates Mathematicae 15 (2021), 147-168. https://dx.doi.org/10.14708/am.v15i1.7087, https://arxiv.org/abs/2201.02047, https://mathscinet.ams.org/mathscinet-getitem?mr=4467506


year '20 (5 publications)

20a. Bair, J.; Błaszczyk, P.; Fuentes Guillén, E.; Heinig, P.; Kanovei, V.; Katz, M. "Continuity between Cauchy and Bolzano: Issues of antecedents and priority." British Journal for the History of Mathematics 35 (2020), no. 3, 207-224. https://doi.org/10.1080/26375451.2020.1770015, https://arxiv.org/abs/2005.13259, https://mathscinet.ams.org/mathscinet-getitem?mr=4154872, cited by:

20b. Bair, J.; Błaszczyk, P.; Heinig, P.; Kanovei, V.; Katz, M. "Cauchy's work on integral geometry, centers of curvature, and other applications of infinitesimals." Real Analysis Exchange 45 (2020), no. 1, 127-150. reprint, https://arxiv.org/abs/2003.00438, https://mathscinet.ams.org/mathscinet-getitem?mr=4196072

20c. Ely, R. "Teaching calculus with infinitesimals and differentials." ZDM 53 (2021), 591-604. https://doi.org/10.1007/s11858-020-01194-2

20d. Kanovei, V.; Katz, M.; Nowik, T. "Metric completions, the Heine-Borel property, and approachability." Open Mathematics 18 (2020), 162-166. https://doi.org/10.1515/math-2020-0017, https://arxiv.org/abs/2002.07536, https://mathscinet.ams.org/mathscinet-getitem?mr=4080273

20e. Katz, M. "Mathematical conquerors, Unguru polarity, and the task of history." Journal of Humanistic Mathematics 10 (2020), no. 1, 475-515. http://doi.org/10.5642/jhummath.202001.27, https://arxiv.org/abs/2002.00249, https://mathscinet.ams.org/mathscinet-getitem?mr=4060619, cited by:


year '19 (4 publications)

19a. Bair, J.; Błaszczyk, P.; Heinig, P.; Kanovei, V.; Katz, M. "19th century real analysis, forward and backward." Antiquitates Mathematicae 13 (2019), 19-49. http://doi.org/10.14708/am.v13i1.6440, https://arxiv.org/abs/1907.07451, https://mathscinet.ams.org/mathscinet-getitem?mr=4075256

19b. Bascelli, T.; Błaszczyk, P.; Kanovei, V.; Katz, K.; Katz, M.; Kutateladze, S.; Nowik, T.; Schaps, D.; Sherry, D. "Gregory's sixth operation." In The Best Writing on Mathematics 2019, 195-207. Edited by Mircea Pitici. Princeton University Press, Princeton, NJ, 2019. https://books.google.co.il/books?id=RcmXDwAAQBAJ, https://mathscinet.ams.org/mathscinet-getitem?mr=4528717

19c. Bottazzi, E. "Homomorphisms between rings with infinitesimals and infinitesimal comparisons." Mat. Stud. 52 (2019), no. 1, 3-9. http://doi.org/10.30970/ms.52.1.3-9, https://arxiv.org/abs/1902.06076

19d. Bottazzi, E.; Kanovei, V.; Katz, M.; Mormann, T.; Sherry, D. "On mathematical realism and the applicability of hyperreals." Mat. Stud. 51 (2019), no. 2, 200-224. http://doi.org/10.15330/ms.51.2.200-224, https://arxiv.org/abs/1907.07040, https://mathscinet.ams.org/mathscinet-getitem?mr=3988243, cited by:


year '18 (12 publications)

18a. Bair, J.; Błaszczyk, P.; Ely, R.; Heinig, P.; Katz, M. "Leibniz's well-founded fictions and their interpretations." Mat. Stud. 49 (2018), no. 2, 186-224. http://doi.org/10.15330/ms.49.2.186-224, https://arxiv.org/abs/1812.00226, https://mathscinet.ams.org/mathscinet-getitem?mr=3882551, cited by:

18b. Bair, J.; Błaszczyk, P.; Heinig, P.; Katz, M.; Schäfermeyer, J.; Sherry, D. "Klein vs Mehrtens: restoring the reputation of a great modern." Mat. Stud. 48 (2017), no. 2, 189-219. https://arxiv.org/abs/1803.02193, http://doi.org/10.15330/ms.48.2.189-219 , https://mathscinet.ams.org/mathscinet-getitem?mr=3819950

18c. Bair, J.; Błaszczyk, P.; Katz, K.; Katz, M.; Kudryk, T.; Sherry, D. "Analyzing Benardete's comment on decimal notation." Philosophy of Mathematics Education Journal no. 33, january 2018. at journal and https://arxiv.org/abs/1706.00191

18d. Bair, J.; Katz, M.; Sherry, D. "Fermat's dilemma: Why did he keep mum on infinitesimals? and the European theological context." Foundations of Science 23 (2018), no. 3, 559-595. http://doi.org/10.1007/s10699-017-9542-y, https://arxiv.org/abs/1801.00427, https://mathscinet.ams.org/mathscinet-getitem?mr=3836239, cited by:

18e. Bascelli, T.; Błaszczyk, 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 23 (2018), no. 2, 267-296. http://doi.org/10.1007/s10699-017-9534-y, https://arxiv.org/abs/1704.07723, https://mathscinet.ams.org/mathscinet-getitem?mr=3803893, cited by:

18f. Bascelli, T.; Błaszczyk, P.; Kanovei, V.; Katz, K.; Katz, M.; Kutateladze, S.; Nowik, T.; Schaps, D.; Sherry, D. "Gregory's sixth operation." Foundations of Science 23 (2018), no. 1, 133-144. http://doi.org/10.1007/s10699-016-9512-9, https://arxiv.org/abs/1612.05944, https://mathscinet.ams.org/mathscinet-getitem?mr=3772065

18g. Błaszczyk, P.; Kanovei, V.; Katz, M.; Nowik, T. "Monotone subsequence via ultrapower." Open Mathematics 16 (2018), 149-153. https://doi.org/10.1515/math-2018-0015, https://arxiv.org/abs/1803.00312, https://mathscinet.ams.org/mathscinet-getitem?mr=3772690

18h. Herzberg, F.; Kanovei, V.; Katz, M.; Lyubetsky, V. "Minimal axiomatic frameworks for definable hyperreals with transfer." Journal of Symbolic Logic 83 Issue 1, march 2018, pp. 385-391. http://doi.org/10.1017/jsl.2017.48, https://arxiv.org/abs/1707.00202, https://mathscinet.ams.org/mathscinet-getitem?mr=3796290

18i. Kanovei, V.; Katz, K.; Katz, M.; Mormann, T. "What makes a theory of infinitesimals useful? A view by Klein and Fraenkel." Journal of Humanistic Mathematics 8 (2018), no. 1, 108-119. http://scholarship.claremont.edu/jhm/vol8/iss1/7, https://arxiv.org/abs/1802.01972, https://mathscinet.ams.org/mathscinet-getitem?mr=3762866

18j. Katz, B.; Katz, M; Sanders, S. "A footnote to The crisis in contemporary mathematics." Historia Mathematica 45 (2018), no. 2, 176-181. https://doi.org/10.1016/j.hm.2018.03.002, https://arxiv.org/abs/1804.02645 , https://mathscinet.ams.org/mathscinet-getitem?mr=3802555 A portrait of Errett Bishop as a young... chicken.

18k. Sherry, D. "The jesuits and the method of indivisibles." Foundations of Science 23 (2018), no. 2, 367-392. http://doi.org/10.1007/s10699-017-9525-z, https://mathscinet.ams.org/mathscinet-getitem?mr=3803897

18l. Sanders, S. "To be or not to be constructive, that is not the question." Indag. Math. (N.S.) 29 (2018), no. 1, 313-381. https://doi.org/10.1016/j.indag.2017.05.005, https://mathscinet.ams.org/mathscinet-getitem?mr=3739620


year '17 (9 publications)

17a. Bair, J.; Błaszczyk, P.; Ely, R.; Henry, V.; Kanovei, V.; Katz, K.; Katz, M.; Kudryk, T.; Kutateladze, S.; McGaffey, T.; Mormann, T.; Schaps, D.; Sherry, D. "Cauchy, infinitesimals and ghosts of departed quantifiers." Mat. Stud. 47 (2017), no. 2, 115-144. http://doi.org/10.15330/ms.47.2.115-144, https://arxiv.org/abs/1712.00226, https://mathscinet.ams.org/mathscinet-getitem?mr=3733080

17b. Bair, J.; Błaszczyk, 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. http://doi.org/10.1007/s10838-016-9334-z and https://arxiv.org/abs/1605.00455 and http://www.ams.org/mathscinet-getitem?mr=3663035 Here we analyze Euler's approach to infinitesimal analysis and his proof of the infinite product decomposition for the sine function. We also examine Giovanni Ferraro's flawed historical scholarship and propose a sounder alternative.

17c. Błaszczyk, P.; Kanovei, V.; Katz, K.; Katz, M.; Kudryk, T.; Mormann, T.; Sherry, D. "Is Leibnizian calculus embeddable in first order logic?" Foundations of Science 22 (2017), no. 4, 717-731. http://doi.org/10.1007/s10699-016-9495-6 and https://arxiv.org/abs/1605.03501 and https://mathscinet.ams.org/mathscinet-getitem?mr=3720412

17d. Błaszczyk, P.; Kanovei, V.; Katz, K.; Katz, M.; Kutateladze, S.; Sherry, D. "Toward a history of mathematics focused on procedures." Foundations of Science 22 (2017), no. 4, 763-783. http://doi.org/10.1007/s10699-016-9498-3 and https://arxiv.org/abs/1609.04531 and https://mathscinet.ams.org/mathscinet-getitem?mr=3720415 Here we propose an approach to the history of mathematics that focuses on the procedures of the historical masters rather than set-theoretic ontology of the entities they use. We also examine Jeremy Gray's flawed historical scholarship and propose a sounder alternative.

17e. Błaszczyk, P.; Kanovei, V.; Katz, M.; Sherry, D. "Controversies in the foundations of analysis: Comments on Schubring's Conflicts." Foundations of Science 22 (2017), no. 1, 125-140. http://doi.org/10.1007/s10699-015-9473-4, https://arxiv.org/abs/1601.00059, http://www.ams.org/mathscinet-getitem?mr=3605125 See also Reception

17f. Fletcher, P.; Hrbacek, K.; Kanovei, V.; Katz, M.; Lobry, C.; Sanders, S. "Approaches to analysis with infinitesimals following Robinson, Nelson, and others." Real Analysis Exchange 42 (2017), no. 2, 193-252. reprint, http://doi.org/10.14321/realanalexch.42.2.0193, https://arxiv.org/abs/1703.00425, https://mathscinet.ams.org/mathscinet-getitem?mr=3721800, cited by over 40 articles:

17g. Gutman, A.; Katz, M.; Kudryk, T.; Kutateladze, S. "The Mathematical Intelligencer flunks the Olympics." Foundations of Science 22 (2017), no. 3, 539-555. http://doi.org/10.1007/s10699-016-9485-8, https://arxiv.org/abs/1606.00160, http://www.ams.org/mathscinet-getitem?mr=3696393 Here we examine Yaroslav Sergeyev's grossbit pathos.

17h. Katz, M.; Polev, L. "From Pythagoreans and Weierstrassians to true infinitesimal calculus." Journal of Humanistic Mathematics 7 (2017), no. 1, 87-104. http://doi.org/10.5642/jhummath.201701.07, https://arxiv.org/abs/1701.05187

17i. Sanders, S. "Reverse Formalism 16." Synthese 197 (2020), no. 2, 497-544. http://doi.org/10.1007/s11229-017-1322-2, https://arxiv.org/abs/1701.05066, https://mathscinet.ams.org/mathscinet-getitem?mr=4072261


year '16 (3 publications)

16a. Bascelli, T.; Błaszczyk, P.; Kanovei, V.; Katz, K.; Katz, M.; Schaps, D.; Sherry, D. "Leibniz versus Ishiguro: Closing a Quarter Century of Syncategoremania." HOPOS: The Journal of the International Society for the History of Philosophy of Science 6 (2016), no. 1, 117-147. http://doi.org/10.1086/685645, https://arxiv.org/abs/1603.07209

16b. Błaszczyk, P.; Borovik, A.; Kanovei, V.; Katz, M.; Kudryk, T.; Kutateladze, S.; Sherry, D. "A non-standard analysis of a cultural icon: The case of Paul Halmos." Logica Universalis 10 (2016), no. 4, 393-405. http://doi.org/10.1007/s11787-016-0153-0, https://arxiv.org/abs/1607.00149, http://www.ams.org/mathscinet-getitem?mr=3566230

16c. Kanovei, V.; Katz, K.; Katz, M.; Nowik, T. "Small oscillations of the pendulum, Euler's method, and adequality." Quantum Studies: Mathematics and Foundations 3 (2016), no. 3, 231-236. http://doi.org/10.1007/s40509-016-0074-x and https://arxiv.org/abs/1604.06663 and http://www.ams.org/mathscinet-getitem?mr=3531864


year '15 (4 publications)

15a. Kanovei, V.; Katz, K.; Katz, M.; Schaps, M. "Proofs and Retributions, Or: Why Sarah Can't Take Limits." Foundations of Science 20 (2015), no. 1, 1-25. http://doi.org/10.1007/s10699-013-9340-0, http://www.ams.org/mathscinet-getitem?mr=3312498 Here we examine Errett Bishop's criticisms of Robinson's framework. We also compare Bishop's attitude with Heyting's.

15b. Kanovei, V.; Katz, K.; Katz, M.; Sherry, D. "Euler's lute and Edwards' oud." The Mathematical Intelligencer 37 (2015), no. 4, 48-51. http://doi.org/10.1007/s00283-015-9565-6, https://arxiv.org/abs/1506.02586, http://www.ams.org/mathscinet-getitem?mr=3435825 see also Reception

15c. Katz, M.; Kutateladze, S. "Edward Nelson (1932-2014)." The Review of Symbolic Logic 8 (2015), no. 3, 607-610. http://doi.org/10.1017/S1755020315000015, https://arxiv.org/abs/1506.01570

15d. Nowik, T; Katz, M. "Differential geometry via infinitesimal displacements." Journal of Logic and Analysis 7:5 (2015), 1-44. http://www.logicandanalysis.com/index.php/jla/article/view/237, http://u.math.biu.ac.il/~katzmik/dgnsa_arxiv.pdf, https://arxiv.org/abs/1405.0984, http://www.ams.org/mathscinet-getitem?mr=3457545


year '14 (4 publications)

14a. 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. http://www.ams.org/notices/201408/rnoti-p848.pdf, https://arxiv.org/abs/1407.0233. cited by over 50 articles:

14b. Katz, K.; Katz, M.; Kudryk, T. "Toward a clarity of the extreme value theorem." Logica Universalis 8 (2014), no. 2, 193-214. http://doi.org/10.1007/s11787-014-0102-8 and https://arxiv.org/abs/1404.5658 and http://www.ams.org/mathscinet-getitem?mr=3210286

14c. Sherry, D.; Katz, M. "Infinitesimals, imaginaries, ideals, and fictions." Studia Leibnitiana 44 (2012), no. 2, 166-192. http://www.jstor.org/stable/43695539, https://arxiv.org/abs/1304.2137 (Article was published in 2014 even though the journal issue lists the year as 2012) cited by 40 articles:

14d. Tall, D.; Katz, M. "A cognitive analysis of Cauchy's conceptions of function, continuity, limit, and infinitesimal, with implications for teaching the calculus." Educational Studies in Mathematics 86 (2014), no. 1, 97-124. http://doi.org/10.1007/s10649-014-9531-9 and https://arxiv.org/abs/1401.1468


year '13 (8 publications)

13a. Bair, J.; Błaszczyk, 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 (2013) no. 7, 886-904. Accessible here, http://www.ams.org/notices/201307/rnoti-p886.pdf, http://www.ams.org/mathscinet-getitem?mr=3086638, and https://arxiv.org/abs/1306.5973. cited by 50 articles:

13b. Błaszczyk, P.; Katz, M.; Sherry, D. "Ten misconceptions from the history of analysis and their debunking." Foundations of Science 18 (2013), no. 1, 43-74. http://doi.org/10.1007/s10699-012-9285-8, http://www.ams.org/mathscinet-getitem?mr=3031794, https://arxiv.org/abs/1202.4153, and Reception. cited by over 60 articles:

13c. Kanovei, V.; Katz, M.; Mormann, T. "Tools, objects, and chimeras: Connes on the role of hyperreals in mathematics." Foundations of Science 18 (2013), no. 2, 259-296. http://doi.org/10.1007/s10699-012-9316-5, https://arxiv.org/abs/1211.0244, http://www.ams.org/mathscinet-getitem?mr=3064607. cited by over 40 articles: Here we examine Alain Connes' criticisms of Robinson's framework

13d. Katz, M.; Leichtnam, E. "Commuting and noncommuting infinitesimals." American Mathematical Monthly 120 (2013), no. 7, 631-641. http://doi.org/10.4169/amer.math.monthly.120.07.631, https://arxiv.org/abs/1304.0583, and http://www.ams.org/mathscinet-getitem?mr=3096469. Here we examine Alain Connes' criticisms of Robinson's framework

13e. Katz, M.; Schaps, D.; Shnider, S. "Almost equal: The method of adequality from Diophantus to Fermat and beyond." Perspectives on Science 21 (2013), no. 3, 283-324. http://doi.org/10.1162/POSC_a_00101, https://arxiv.org/abs/1210.7750, http://www.ams.org/mathscinet-getitem?mr=3114421. cited by over 40 articles: Here we refute Herbert Breger's interpretation of Fermat and propose a sounder alternative.

13f. Katz, M.; Sherry, D. "Leibniz's infinitesimals: Their fictionality, their modern implementations, and their foes from Berkeley to Russell and beyond." Erkenntnis 78 (2013), no. 3, 571-625. http://doi.org/10.1007/s10670-012-9370-y, https://arxiv.org/abs/1205.0174, and http://www.ams.org/mathscinet-getitem?mr=3053644, cited by over 110 articles:

13g. Katz, M.; Tall, D. "A Cauchy-Dirac delta function." Foundations of Science 18 (2013), no. 1, 107-123. http://doi.org/10.1007/s10699-012-9289-4, https://arxiv.org/abs/1206.0119, and http://www.ams.org/mathscinet-getitem?mr=3031797

13h. Mormann, T.; Katz, M. "Infinitesimals as an issue of neo-Kantian philosophy of science." HOPOS: The Journal of the International Society for the History of Philosophy of Science 3 (2013), no. 2, 236-280. http://doi.org/10.1086/671348 and https://arxiv.org/abs/1304.1027


year '12 (6 publications)

12a. Borovik, A.; Jin, R.; Katz, M. "An integer construction of infinitesimals: Toward a theory of Eudoxus hyperreals." Notre Dame Journal of Formal Logic 53 (2012), no. 4, 557-570. https://arxiv.org/abs/1210.7475, http://doi.org/10.1215/00294527-1722755, and http://www.ams.org/mathscinet-getitem?mr=2995420

12b. Borovik, A.; Katz, M. "Who gave you the Cauchy-Weierstrass tale? The dual history of rigorous calculus." Foundations of Science 17 (2012), no. 3, 245-276. http://doi.org/10.1007/s10699-011-9235-x, https://arxiv.org/abs/1108.2885, and http://www.ams.org/mathscinet-getitem?mr=2950620, as well as http://u.math.biu.ac.il/~katzmik/straw.html Here we examine Judith Grabiner's flawed Cauchy scholarship and propose a sounder alternative. cited by over 70 articles:

12c. Katz, K.; Katz, M. "Stevin numbers and reality." Foundations of Science 17 (2012), no. 2, 109-123. http://doi.org/10.1007/s10699-011-9228-9 and https://arxiv.org/abs/1107.3688 and http://www.ams.org/mathscinet-getitem?mr=2935194. cited by over 40 articles:

12d. Katz, K.; Katz, M. "A Burgessian critique of nominalistic tendencies in contemporary mathematics and its historiography." Foundations of Science 17 (2012), no. 1, 51-89. http://doi.org/10.1007/s10699-011-9223-1, https://arxiv.org/abs/1104.0375, and http://www.ams.org/mathscinet-getitem?mr=2896999 cited by 40 articles:

12e. Katz, M.; Sherry, D. "Leibniz's laws of continuity and homogeneity." Notices of the American Mathematical Society 59 (2012), no. 11, 1550-1558. https://doi.org/10.1090/noti921, https://arxiv.org/abs/1211.7188, http://www.ams.org/mathscinet-getitem?mr=3027109, and http://u.math.biu.ac.il/~katzmik/straw2.html. cited by 60 articles:

12f. Katz, M.; Tall, D. "Tension between intuitive infinitesimals and formal mathematical analysis." Chapter in: Bharath Sriraman, Editor. Crossroads in the History of Mathematics and Mathematics Education. The Montana Mathematics Enthusiast Monographs in Mathematics Education 12, Information Age Publishing, Charlotte, NC, 2012, pp. 71-89. https://arxiv.org/abs/1110.5747


year '11 (2 publications)

11a. Katz, K.; Katz, M. "Meaning in classical mathematics: Is it at odds with Intuitionism?" Intellectica 56 (2011), no. 2, 223-302. https://arxiv.org/abs/1110.5456 and https://www.persee.fr/doc/intel_0769-4113_2011_num_56_2_1154 Here we examine Errett Bishop's criticisms of Robinson's framework. We also compare Bishop's attitude with Heyting's.

11b. Katz, K.; Katz, M. "Cauchy's continuum." Perspectives on Science 19 (2011), no. 4, 426-452. http://doi.org/10.1162/POSC_a_00047, https://arxiv.org/abs/1108.4201, http://www.ams.org/mathscinet-getitem?mr=2884218. cited by 40 articles:


year '10 (3 publications)

10a. Ely, R. "Nonstandard student conceptions about infinitesimal and infinite numbers." Journal for Research in Mathematics Education 41 (2010), no. 2, 117-146. http://www.nctm.org/publications/article.aspx?id=26196 and http://u.math.biu.ac.il/~katzmik/ely10.pdf

10b. Katz, K.; Katz, M. "Zooming in on infinitesimal 1-.9.. in a post-triumvirate era." Educational Studies in Mathematics 74 (2010), no. 3, 259-273. http://doi.org/10.1007/s10649-010-9239-4 and https://arxiv.org/abs/arXiv:1003.1501

10c. Katz, K.; Katz, M. "When is .999... less than 1?" The Montana Mathematics Enthusiast 7 (2010), No. 1, 3-30. http://scholarworks.umt.edu/tme/vol7/iss1/11, https://arxiv.org/abs/arXiv:1007.3018



List of 30 periodicals where the articles have appeared, in alphabetical order:

1. American Mathematical Monthly 13d
2. Annals of Pure and Applied Logic 21e
3. Antiquitates Mathematicae 19a, 21g, 22a, 23b
4. British Journal for the History of Mathematics 20a, 21a
5. Bulletin of Symbolic Logic 23c
6. Erkenntnis 13f
7. Foundations of Science 18d, 18e, 18f, 17c, 17d, 17e, 17g, 15a, 13b, 13c, 13g, 12b, 12c, 12d
8. Historia Mathematica 18j
9. Handbook of the History and Philosophy of Mathematical Practice 23i
10. HOPOS (Journal of the International Society for the History of Philosophy of Science) 13h, 16a
11. Indag. Math. 18l
12. Intellectica 11a
13. Journal for General Philosophy of Science 17b
14. Journal of Humanistic Mathematics 20e, 18i, 17h
15. Journal of Logic and Analysis 23e, 15d
16. Journal of Symbolic Logic 18h
17. Logica Universalis 14b, 16b
18. Mat. Stud. 17a, 18a, 18b, 19c, 19d
19. The Mathematical Intelligencer 15b, 22b 23a
20. Mathematics Today 21f
21. Notices of the American Mathematical Society 12e, 13a, 14a
22. Notre Dame Journal of Formal Logic 12a
23. Open Mathematics 21d, 20d, 18g
24. Perspectives on Science 11b, 13e
25. Philosophia Mathematica 21b, 21c
26. Quantum Studies: Mathematics and Foundations 16c
27. Real Analysis Exchange 23d, 20b, 17f
28. Review of Symbolic Logic 15c, 24a
29. Revista Latinoamericana de Filosofía 23h
30. Studia Leibnitiana 14c
31. Synthese 17i
32. ZDM 20c



Reappraisal of the procedures of the pioneers of infinitesimal analysis from Stevin to Cauchy


Pioneer Journal where reappraisal appeared Link to article containing reappraisal
Simon Stevin Foundations of Science 12c
Pierre Fermat Perspectives on Science 13e
Pierre Fermat Foundations of Science 18d
Pierre Fermat Journal of Humanistic Mathematics 20e
Pierre Fermat Dedicated page Fermat dedicated page
James Gregory Foundations of Science 18f
James Gregory The Best Writings on Mathematics 2019, M. Pitici, Ed. 19b
Gottfried Leibniz Notices AMS 12e
Gottfried Leibniz Erkenntnis 13f
Gottfried Leibniz Studia Leibnitiana 14c
Gottfried Leibniz HOPOS (Journal of the International Society for the History of Philosophy of Science) 16a
Gottfried Leibniz Mat. Stud. 18a
Gottfried Leibniz British Journal for the History of Mathematics 21a
Gottfried Leibniz Antiquitates Mathematicae 21g
Gottfried Leibniz Review of Symbolic Logic 24a
Gottfried Leibniz The Mathematical Intelligencer 22b
Gottfried Leibniz Dedicated page Leibniz Dedicated page
Leonhard Euler The Mathematical Intelligencer 15b
Leonhard Euler Journal for General Philosophy of Science 17b
Leonhard Euler Dedicated page Euler's infinitesimal analysis
A. L. Cauchy Perspectives on Science 11b
A. L. Cauchy Foundations of Science 12b
A. L. Cauchy Foundations of Science 13g
A. L. Cauchy Mat. Stud. 17a
A. L. Cauchy Foundations of Science 17e
A. L. Cauchy Foundations of Science 18e
A. L. Cauchy Antiquitates Mathematicae 19a
A. L. Cauchy Real Analysis Exchange 20b
A. L. Cauchy British Journal for the History of Mathematics 20a
A. L. Cauchy Mathematics Today 21f
A. L. Cauchy Dedicated page Cauchy's infinitesimal analysis
Felix Klein Dedicated page Felix Klein
Thoralf Skolem Dedicated page Thoralf Skolem


List of critics in alphabetical order:


Critic Venue where rebuttal appeared Link to article/venue containing rebuttal
Tom Archibald Antiquitates Mathematicae 22a, Section 5. See also Depictions
Richard Arthur Erkenntnis 13f
Richard Arthur Foundations of Science 17d
Richard Arthur British Journal for the History of Mathematics 21a See also Depictions
Errett Bishop Intellectica 11a
Errett Bishop Foundations of Science 15a
Errett Bishop Historia Mathematica 18j
Bishop-Connes Synthese 17i
Errett Bishop Dedicated page Dedicated page
Umberto Bottazzini Math Overflow Q&A thread
Herbert Breger Perspectives on Science 13e
Herbert Breger Foundations of Science 18d
Rudopf Carnap British Journal for the History of Mathematics 21a, Section 1.7
Alain Connes Foundations of Science 13c
Alain Connes American Mathematical Monthly 13d
Alain Connes Math Overflow Q&A thread
Alain Connes Annals of Pure and Applied Logic 21e
Alain Connes Dedicated page Dedicated page
John Earman Erkenntnis 13f
Kenny Easwaran Notices of the American Mathematical Society 14a
Kenny Easwaran Mat. Stud. 19d
Harold M. Edwards Mathematical Intelligencer 15b
Harold M. Edwards Journal for General Philosophy of Science 17b, section 4.13
Giovanni Ferraro Journal for General Philosophy of Science 17b
Giovanni Ferraro Foundations of Science 18f. See also Depictions
Craig Fraser Foundations of Science 18e
Craig Fraser Mat. Stud. 17a, Section 4
Craig Fraser Math Overflow Q&A thread
C. Gilain Antiquitates Mathematicae 19a
Judith Grabiner Foundations of Science 12b
Judith Grabiner Foundations of Science 18e
Jeremy Gray Foundations of Science 17d. See also Depictions
Jeremy Gray Stack Exchange Q&A thread
Paul Halmos Logica Universalis 16b
Hide Ishiguro Studia Leibnitiana 14c
Hide Ishiguro HOPOS (Journal of the International Society for the History of Philosophy of Science) 16a
Douglas Jesseph Antiquitates Mathematicae 22a, Section 5. See also Depictions
Jesper Lützen Mat. Stud. 17a, Section 3. See also Depictions
Ohad Nachtomy Mat. Stud. 18a, Section 1.7
Marco Panza Journal for General Philosophy of Science 17b, Section 2.8, pp. 204-205. See also Depictions
Matthew W. Parker TBA TBA.
Alexander Pruss Philosophia Mathematica 21b
Alexander Pruss Philosophia Mathematica 21c
David Rabouin Mat. Stud. 18a, Sections 4.4, 4.6
David Rabouin British Journal for the History of Mathematics 21a See also Depictions
Gert Schubring Foundations of Science 17e See also Reception
Yaroslav Sergeyev Foundations of Science 17g
Yaroslav Sergeyev EMS Surveys in Mathematical Sciences "Both [EICs] have assumed responsibility for [the mistake of publishing Sergeyev's paper] and resigned from their position."
Yaroslav Sergeyev Retraction Watch Editors-in-chief of math journal resign over controversial paper
Yaroslav Sergeyev Zentralblatt Review by Louis Kauffman
Yaroslav Sergeyev Mathematical Reviews Review by Mikhail Katz
Yaroslav Sergeyev dedicated page Dedicated page
Reinhard Siegmund-Schultze Antiquitates Mathematicae 19a
Detlef Spalt Perspectives on Science 11b
Detlef Spalt Antiquitates Mathematicae 22a, Sections 3.2, 3.3
Henry Towsner Mat. Stud. 19d
Klaus Viertel Foundations of Science 18e, Section 4.5


tallberkeley12


Other critics:


Other critics of infinitesimals and/or Robinson Journal where rebuttal appeared Link to article containing rebuttal
George Berkeley (1685-1753) Erkenntnis 13f
George Berkeley (1685-1753) Dedicated page
François-Napoléon-Marie Moigno (1804-1884) Erkenntnis 13f
Georg Cantor (1845-1918) Erkenntnis 13f
Bertrand Russell (1872-1970) Erkenntnis 13f, section 11.1
Henk Bos (1940- ) Erkenntnis 13f, section 11.3
Henk Bos (1940- ) Journal for General Philosophy of Science 17b, section 2.7



Alok Singh's talk on NSA
Nonstandard analysis-based software development
Introduction to infinitesimal analysis without the axiom of choice (SPOT and SCOT)
Kathleen Sullivan's '76 study of teaching calculus with infinitesimals based on Keisler's book
Amos Shalit: An analysis of Halmos's critique of nonstandard analysis
Arithmetic, Geometry, and Topology (AGT) Seminar: current schedule
Jim Holt, "Infinitesimally yours"
Infinitesimal topics
Special session AMS/IMU (Israel Mathematical Union) on the history and philosophy of mathematics
Stevin
Fermat
Leibniz
Berkeley
Euler
Cauchy
Riemann
Cantor
Klein
Skolem
Heyting
Robinson
Nelson
Hrbacek
Teaching True Infinitesimal Calculus
Terry Tao on ultrafilters, nonstandard analysis, and epsilon management (june '07)
Terry Tao on a cheap version of nonstandard analysis
Terry Tao: there is more to mathematics than rigor and proofs
Cauchy's sum theorem
Hyperreals and surreals
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