Recent publications on infinitesimals



Links to recent publications on infinitesimals (and related subjects) by Bair, Bascelli, Błaszczyk, Borovik, Bottazzi, Ely, Henry, Herzberg, Jin, Kanovei, Katz, Kudryk, Kutateladze, Leichtnam, T. McGaffey, Mormann, Nowik, D. Schaps, M. Schaps, Sherry, Shnider, and Tall can be found below.


year '15

Vladimir Kanovei Karin U. Katz Mikhail G. Katz Mary Schaps. Proofs and Retributions, Or: Why Sarah Can't Take Limits. Foundations of Science 20 (2015), no. ?, ??-??. See http://dx.doi.org/10.1007/s10699-013-9340-0

year '14

Nowik, T; Katz, M. Differential geometry via infinitesimal displacements. See http://arxiv.org/abs/1405.0984

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

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, and http://arxiv.org/abs/1202.4153

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/, 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 and google scholar.

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







Some interesting questions:
Was the early calculus inconsistent? (MO)
Can Euler's mathematics be reformulated in terms of modern theories?
Did Connes find a catch in the theory?
In what ways did Leibniz's philosophy anticipate modern mathematics?
A few questions on nonstandard analysis (SE)
Is mathematical history written by the victors? (SE)
What did Newton and Leibniz actually discover? (SE)
Who gave you the epsilon? (SE)

Infinitesimal topics

Special session AMS/IMU on the history and philosophy of mathematics
Salvaging Leibniz

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