Recent publications on infinitesimals



Provided below are links to recent publications on infinitesimals and related subjects by Jacques Bair, Tiziana Bascelli, Piotr Błaszczyk, Alexandre Borovik, Emanuele Bottazzi, Robert Ely, Peter Fletcher, Valérie Henry, Frederik Herzberg, Renling Jin, Vladimir Kanovei, Karin Katz, Taras Kudryk, Semen Samsonovich Kutateladze, Eric Leichtnam, Thomas McGaffey, Thomas Mormann, Tahl Nowik, Luie Polev, Patrick Reeder, David Schaps, Mary Schaps, David Sherry, Steven Shnider, and David Tall.

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.


year '17

17a. 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. 1. See http://dx.doi.org/10.1007/s10838-016-9334-z and http://arxiv.org/abs/1605.00455 Here we examine Giovanni Ferraro's flawed historical scholarship and propose a sounder alternative.

17b. 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, to appear. See http://dx.doi.org/10.1007/s10699-016-9512-9

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, online first. http://dx.doi.org/10.1007/s10699-016-9495-6 and http://arxiv.org/abs/1605.03501

17d. Błaszczyk, P.; Kanovei, V.; Katz, K.; Katz, M.; Kutateladze, S.; Sherry. D. "Toward a history of mathematics focused on procedures." Foundations of Science. See http://dx.doi.org/10.1007/s10699-016-9498-3 and http://arxiv.org/abs/1609.04531 Here we 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. See http://dx.doi.org/10.1007/s10699-015-9473-4 and http://arxiv.org/abs/1601.00059 See also Reception

17f. Fletcher, P. et al. "Approaches to analysis with infinitesimals following Robinson, Nelson, and others." Real Analysis Exchange 42 (2017), no. 1, to appear.

17g. Gutman, A.; Katz, M.; Kudryk, T.; Kutateladze, S. "The Mathematical Intelligencer Flunks the Olympics." Foundations of Science. See http://dx.doi.org/10.1007/s10699-016-9485-8 and http://arxiv.org/abs/1606.00160 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, to appear.


year '16

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. See http://dx.doi.org/10.1086/685645 and http://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://dx.doi.org/10.1007/s11787-016-0153-0 and http://arxiv.org/abs/1607.00149 and 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. See http://dx.doi.org/10.1007/s40509-016-0074-x and http://arxiv.org/abs/1604.06663 and http://www.ams.org/mathscinet-getitem?mr=3531864


year '15

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. See http://dx.doi.org/10.1007/s10699-013-9340-0 and http://www.ams.org/mathscinet-getitem?mr=3312498 Here we examine Errett Bishop's criticisms of Robinson's framework

15b. Kanovei, V.; Katz, K.; Katz, M.; Sherry, D. "Euler's lute and Edwards' oud." The Mathematical Intelligencer 37 (2015), 48-51. See http://dx.doi.org/10.1007/s00283-015-9565-6 and http://arxiv.org/abs/1506.02586 and 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. See http://dx.doi.org/10.1017/S1755020315000015 and http://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. See http://www.logicandanalysis.com/index.php/jla/article/view/237 and http://u.cs.biu.ac.il/~katzmik/dgnsa_arxiv.pdf and http://arxiv.org/abs/1405.0984 and http://www.ams.org/mathscinet-getitem?mr=3457545


year '14

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. See http://www.ams.org/notices/201408/rnoti-p848.pdf and http://arxiv.org/abs/1407.0233

14b. Katz, K.; Katz, M.; Kudryk, T. "Toward a clarity of the extreme value theorem." Logica Universalis 8 (2014), no. 2, 193-214. See http://dx.doi.org/10.1007/s11787-014-0102-8 and http://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. See http://arxiv.org/abs/1304.2137 (Article was published in 2014 even though the journal issue lists the year as 2012)

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. See http://dx.doi.org/10.1007/s10649-014-9531-9 and http://arxiv.org/abs/1401.1468


year '13

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 http://arxiv.org/abs/1306.5973

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. See http://dx.doi.org/10.1007/s10699-012-9285-8, http://www.ams.org/mathscinet-getitem?mr=3031794, http://arxiv.org/abs/1202.4153, and Reception

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. See http://dx.doi.org/10.1007/s10699-012-9316-5, http://www.ams.org/mathscinet-getitem?mr=3064607, and http://arxiv.org/abs/1211.0244 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. See http://dx.doi.org/10.4169/amer.math.monthly.120.07.631, http://www.ams.org/mathscinet-getitem?mr=3096469, and http://arxiv.org/abs/1304.0583 Here we examine Alain Connes' criticisms of Robinson's framework

13e. Katz, M.; Schaps, D.; Shnider, D. "Almost Equal: The Method of Adequality from Diophantus to Fermat and Beyond." Perspectives on Science 21 (2013), no. 3, 283-324. See http://www.mitpressjournals.org/doi/abs/10.1162/POSC_a_00101, http://www.ams.org/mathscinet-getitem?mr=3114421, and http://arxiv.org/abs/1210.7750 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. See http://dx.doi.org/10.1007/s10670-012-9370-y, http://www.ams.org/mathscinet-getitem?mr=3053644, and http://arxiv.org/abs/1205.0174

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

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. See http://dx.doi.org/10.1086/671348 and http://arxiv.org/abs/1304.1027


year '12

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. See http://arxiv.org/abs/1210.7475, http://dx.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. see http://dx.doi.org/10.1007/s10699-011-9235-x, http://arxiv.org/abs/1108.2885, and http://www.ams.org/mathscinet-getitem?mr=2950620, as well as http://u.cs.biu.ac.il/~katzmik/straw.html Here we examine Judith Grabiner's flawed Cauchy scholarship and propose a sounder alternative.

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

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. See http://dx.doi.org/10.1007/s10699-011-9223-1, http://arxiv.org/abs/1104.0375, and http://www.ams.org/mathscinet-getitem?mr=2896999

12e. Katz, M.; Sherry, D. "Leibniz's laws of continuity and homogeneity." Notices of the American Mathematical Society 59 (2012), no. 11, 1550-1558. See http://www.ams.org/notices/201211/rtx121101550p.pdf, http://arxiv.org/abs/1211.7188, http://www.ams.org/mathscinet-getitem?mr=3027109, and http://u.cs.biu.ac.il/~katzmik/straw2.html

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, Inc., Charlotte, NC, 2012, pp. 71-89. See http://arxiv.org/abs/1110.5747


year '11

11a. Katz, K.; Katz, M. "Meaning in Classical Mathematics: Is it at Odds with Intuitionism?" Intellectica 56 (2011), no. 2, 223-302. See http://arxiv.org/abs/1110.5456 Here we examine Errett Bishop's criticisms of Robinson's framework

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


year '10

10a. Ely, R. "Nonstandard student conceptions about infinitesimal and infinite numbers." Journal for Research in Mathematics Education 41 (2010), no. 2, 117-146. See http://www.nctm.org/publications/article.aspx?id=26196 and http://u.cs.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. See http://dx.doi.org/10.1007/s10649-010-9239-4 and http://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. See http://scholarworks.umt.edu/tme/vol7/iss1/11 and http://arxiv.org/abs/arXiv:1007.3018



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

American Mathematical Monthly 13d
Erkenntnis 13f
Foundations of Science 17b, 17c, 17d, 17e, 17g, 15a, ...
HOPOS (Journal of the International Society for the History of Philosophy of Science) 13h, 16a
Intellectica 11a
Journal for General Philosophy of Science 17a
Journal of Humanistic Mathematics 17h
Journal of Logic and Analysis 15d
Logica Universalis 14b, 16b
Mathematical Intelligencer 15b
Notices of the American Mathematical Society 12e, 13a, 14a
Notre Dame Journal of Formal Logic 12a
Perspectives on Science 11b, 13e
Quantum Studies: Mathematics and Foundations 16c
Real Analysis Exchange 17f
Review of Symbolic Logic 15c
Studia Leibnitiana 14c



List of critics in alphabetical order:


Critic Journal where rebuttal appeared Link to article containing rebuttal
Richard Arthur Erkenntnis 13f
Richard Arthur Foundations of Science 17d
Errett Bishop Foundations of Science 15a
Errett Bishop Intellectica 11a
Alain Connes Foundations of Science 13c
Alain Connes American Mathematical Monthly 13d
John Earman Erkenntnis 13f
Kenny Easwaran Notices of the American Mathematical Society 14a
Harold M. Edwards Mathematical Intelligencer 15b
Harold M. Edwards Journal for General Philosophy of Science 17a, section 4.13
Giovanni Ferraro Journal for General Philosophy of Science 17a
Judith Grabiner Foundations of Science 12b
Jeremy Gray Foundations of Science 17d
Paul Halmos Foundations of Science 16b
Hide Ishiguro HOPOS (Journal of the International Society for the History of Philosophy of Science) 16a
Gert Schubring Foundations of Science 17e
Detlef Spalt Perspectives on Science 11b
Yaroslav Sergeyev Foundations of Science 17g



Other critics of infinitesimals and/or Robinson Journal where rebuttal appeared Link to article containing rebuttal
George Berkeley Erkenntnis 13f
François-Napoléon-Marie Moigno Erkenntnis 13f
Georg Cantor Erkenntnis 13f
Bertrand Russell Erkenntnis 13f
Henk Bos Journal of General Philosophy of Science 17a, section 2.7
Herbert Breger Perspectives on Science 13e



Amos Shalit: An analysis of Halmos's critique of nonstandard analysis
Borovik's blog Infinitesimals: Their Mathematics, Philosophy, History
Arithmetic, Geometry, and Topology (AGT) Seminar: current schedule
Jim Holt "Infinitesimally yours"
Infinitesimal topics
Special session AMS/IMU on the history and philosophy of mathematics
Salvaging Leibniz
Teaching True Infinitesimal Calculus
Terry Tao on hyperreals

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If, as Kurt Goedel said, Robinson's theory is the analysis of the future, then it should be called standard analysis (or infinitesimal analysis), not nonstandard analysis. Hello?