Recent publications on infinitesimals



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, Karel Hrbacek, Renling Jin, Vladimir Kanovei, Karin Katz, Taras Kudryk, Semen Samsonovich Kutateladze, Eric Leichtnam, Claude Lobry, Thomas McGaffey, Thomas Mormann, Tahl Nowik, Luie Polev, Patrick Reeder, David Schaps, Mary Schaps, David Sherry, Steven Shnider, and David Tall can be found below.

A nice introduction to our program can be found in the following review by M. Guillaume: Guillaume's review in pdf


year '16

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 (2016), to appear. See http://dx.doi.org/10.1007/s10838-016-9334-z and http://arxiv.org/abs/1605.00455

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

Błaszczyk, P.; Borovik, A.; Kanovei, V.; Katz, K.; Katz, M.; Kudryk, T.; Kutateladze, S.; Sherry, D. "A non-standard analysis of a cultural icon: The case of Paul Halmos." Logica Universalis. http://dx.doi.org/10.1007/s11787-016-0153-0

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

Błaszczyk, P.; Kanovei, V.; Katz, K.; Katz, M.; Kutateladze, S.; Sherry. D. "Toward a history of mathematics focused on procedures." Foundations of Science, online first.

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

Gutman, A.; Katz, M.; Kudryk, T.; Kutateladze, S. "The Mathematical Intelligencer Flunks the Olympics." Foundations of Science (2016). See http://dx.doi.org/10.1007/s10699-016-9485-8 and http://arxiv.org/abs/1606.00160

Kanovei, V.; Katz, K.; Katz, M.; Nowik, T. "Small oscillations of the pendulum, Euler's method, and adequality." Quantum Studies: Mathematics and Foundations, online first. See http://dx.doi.org/10.1007/s40509-016-0074-x and http://arxiv.org/abs/1604.06663


year '15

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

Kanovei, V.; Katz, K.; Katz, M.; Sherry, D. "Euler's lute and Edwards' oud." The Mathematical Intelligencer 37 (2015), 48-51. See http://arxiv.org/abs/1506.02586 and http://dx.doi.org/10.1007/s00283-015-9565-6 and http://www.ams.org/mathscinet-getitem?mr=3435825 see also Reception

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

Nowik, T; Katz, M. "Differential geometry via infinitesimal displacements." Journal of Logic and Analysis 7:5 (2015), 1-44. See http://www.logicandanalysis.org/index.php/jla/article/view/237/106 and http://arxiv.org/abs/1405.0984 and http://www.ams.org/mathscinet-getitem?mr=3457545


year '14

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

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

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)

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


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

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

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

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

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

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

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

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://www.jstor.org/stable/10.1086/671348 and http://arxiv.org/abs/1304.1027


year '12


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

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

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

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

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

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


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

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


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

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://arxiv.org/abs/arXiv:1003.1501

Katz, K.; Katz, M. "When is .999... less than 1?" The Montana Mathematics Enthusiast 7 (2010), No. 1, 3--30. See http://www.math.umt.edu/tmme/vol7no1/TMME_vol7no1_2010_article1_pp.3_30.pdf and http://arxiv.org/abs/arXiv:1007.3018







A few questions on nonstandard analysis (MSE)
Terry Tao's book on Hilbert's fifth problem
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

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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, not nonstandard analysis. Hello?