256
Views
6
CrossRef citations to date
0
Altmetric
Articles

Procedures of Leibnizian infinitesimal calculus: an account in three modern frameworks

, ORCID Icon, ORCID Icon, ORCID Icon & ORCID Icon

References

  • Albeverio, S; Høegh-Krohn, R; Fenstad, J E, and Lindstrøm, T, Nonstandard methods in stochastic analysis and mathematical physics (Pure and Applied Mathematics, 122), Orlando, FL: Academic Press, 1986.
  • Andersen, K, ‘One of Berkeley's arguments on compensating errors in the calculus’, Historia Mathematica, 38(2) (2011), 219–231.
  • Arthur, R (Tr.), The Labyrinth of the Continuum. Writings on the Continuum Problem, 1672–1686. G. W. Leibniz, New Haven: Yale University Press, 2001.
  • Arthur, R, ‘x+dx=x’: Leibniz's Archimedean infinitesimals (2007). See https://www.humanities.mcmaster.ca/rarthur/papers/x+dx=x.pdf.
  • Arthur, R, ‘Leibniz's syncategorematic infinitesimals’, Archive for History of Exact Sciences, 67(5) (2013), 553–593.
  • Arthur, R, Leibniz. Classic Thinkers, Polity Press, 2014.
  • Arthur, R, ‘Leibniz's actual infinite in relation to his analysis of matter’, in GW Leibniz (ed), interrelations between mathematics and philosophy, Archimedes, 41, Dordrecht: Springer, 2015, 137–156.
  • Arthur, R, ‘Leibniz's Syncategorematic Actual Infinite’, in Ohad Nachtomy and Reed Winegar (eds), Infinity in early modern philosophy (The New Synthese Historical Library. Texts and Studies in the History of Philosophy, 76), Springer, 2018, 155–179.
  • Arthur, R, ‘Leibniz in Cantor's Paradise’, in V De Risi (ed), Leibniz and the structure of sciences. Modern perspectives on the history of logic, mathematics, epistemology (Boston Studies in Philosophy and History of Science), Berlin: Springer, 2019, 71–109.
  • Avigad, J, ‘Weak theories of nonstandard arithmetic and analysis’, in Reverse mathematics 2001 (Lect. Notes Log., 21), La Jolla, CA: Assoc. Symbol. Logic, 2005, 19–46.
  • Bair, J; Błaszczyk, P; Ely, R; Heinig, P, and Katz, M, ‘Leibniz's well-founded fictions and their interpretations’, Matematychni Studii, 49(2) (2018), 186–224. See https://doi.org/10.15330/ms.49.2.186-224
  • Bair, J; Błaszczyk, P; Ely, R; Henry, V; Kanovei, V; Katz, K; Katz, M; Kudryk, T; Kutateladze, S; McGaffey, T; Mormann, T; Schaps, D, and Sherry, D, ‘Cauchy, Infinitesimals and ghosts of departed quantifiers’, Matematychni Studii, 47(2) (2017a), 115–144. See https://doi.org/10.15330/ms.47.2.115-144, https://arxiv.org/abs/1712.00226.
  • Bair, J; Błaszczyk, P; Ely, R; Henry, V; Kanovei, V; Katz, K; Katz, M; Kutateladze, S; McGaffey, T; Reeder, P; Schaps, D; Sherry, D, and Shnider, S, ‘Interpreting the infinitesimal mathematics of Leibniz and Euler’, Journal for General Philosophy of Science, 48(2) (2017b), 195–238. See https://doi.org/10.1007/s10838-016-9334-z and https://arxiv.org/abs/1605.00455
  • Bair, J; Błaszczyk, P; Fuentes Guillén, E; Heinig, P; Kanovei, V, and Katz, M, ‘Continuity between Cauchy and Bolzano: Issues of antecedents and priority’, British Journal for the History of Mathematics, 35(3) (2020), 207–224.
  • Bair, J; Błaszczyk, P; Heinig, P; Kanovei, V, and Katz, M, ‘19th century real analysis, forward and backward’, Antiquitates Mathematicae, 13 (2019), 19–49. See http://doi.org/10.14708/am.v13i1.6440 and https://arxiv.org/abs/1907.07451.
  • Bascelli, T; Błaszczyk, P; Kanovei, V; Katz, K; Katz, M; Schaps, D, and 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(1) (2016), 117–147. See https://doi.org/10.1086/685645 and http://arxiv.org/abs/1603.07209
  • Bassler, O, ‘An enticing (im)possibility: Infinitesimals, differentials, and the Leibnizian calculus’, in Goldenbaum–Jesseph (2008), Berlin–New York: Walter de Gruyter, 2008, 135–151.
  • Beeley, P, ‘‘Nova methodus investigandi’. On the concept of analysis in John Wallis's mathematical writings’, Studia Leibnitiana, 45(1) (2013), 42–58.
  • Bell, J, Continuity and Infinitesimals. Stanford Encyclopedia of Philosophy, 2005–2013. See https://plato.stanford.edu/entries/continuity.
  • Bell, J, A primer of infinitesimal analysis, Cambridge: Cambridge University Press, 2nd edition, 2008.
  • Bell, J, The continuous, the discrete and the infinitesimal in philosophy and mathematics (The Western Ontario Series in Philosophy of Science, 82), Cham: Springer, 2019.
  • Blåsjö, V, Transcendental curves in the Leibnizian calculus (Studies in the History of Mathematical Enquiry), London: Elsevier/Academic Press, 2017.
  • Blåsjö, V, Comment on Rabouin & Arthur (2020). 7 April 2020. See http://intellectualmathematics.com/blog/comment-on-rabouin-arthur-2020.
  • Błaszczyk, P; Kanovei, V; Katz, K; Katz, M; Kutateladze, S, and Sherry, D, ‘Toward a history of mathematics focused on procedures’, Foundations of Science, 22(4) (2017), 763–783. See https://doi.org/10.1007/s10699-016-9498-3 and https://arxiv.org/abs/1609.04531.
  • Bos, H, ‘Differentials, higher-order differentials and the derivative in the Leibnizian calculus’, Archive for History of Exact Sciences, 14 (1974), 1–90.
  • Bottazzi, E; Kanovei, V; Katz, M; Mormann, T, and Sherry, D, ‘On mathematical realism and the applicability of hyperreals’, Matematychni Studii, 51(2) (2019), 200–224. See http://doi.org/10.15330/ms.51.2.200-224 and https://arxiv.org/abs/1907.07040.
  • Bottazzi, E, and Katz, M, ‘Infinite lotteries, spinners, and the applicability of hyperreals’, Philosophia Mathematica, 29(1) (2021a). See https://doi.org/10.1093/philmat/nkaa032 and https://arxiv.org/abs/2008.11509.
  • Bottazzi, E, and Katz, M, ‘Internality, transfer, and infinitesimal modeling of infinite processes’, Philosophia Mathematica, 29(2) (2021b). See https://doi.org/10.1093/philmat/nkaa033 and https://arxiv.org/abs/2008.11513.
  • Boyer, C, ‘Cavalieri, limits and discarded infinitesimals’, Scripta Mathematica, 8 (1941), 79–91.
  • Bradley, R; Petrilli, S, and Sandifer, C (eds), L'Hôpital's Analyse des infiniment petits. An annotated translation with source material by Johann Bernoulli (Science Networks Historical Studies, 50), Cham: Birkhäuser/Springer, 2015.
  • Breger, H, ‘Leibniz's Calculation with Compendia’, in Goldenbaum–Jesseph (2008), Berlin–New York: Walter de Gruyter, 2008, 185–198.
  • Breger, H, ‘On the grain of sand and heaven's infinity’, in Wenchao Li (ed), ‘Für unser Glück oder das Glück anderer’ Vorträge des X. Intemationalen Leibniz-Kongresses Hannover, 18.-23. Juli 2016, in collaboration with Ute Beckmann, Sven Erdner, Esther-Maria Errulat, Jürgen Herbst, Helena Iwasinski und Simona Noreik, Band VI, Hildesheim–Zurich–New York: Georg Olms Verlag, 2017, 64–79.
  • Cajori, F, ‘Grafting of the theory of limits on the calculus of Leibniz’, American Mathematical Monthly, 30(5) (1923), 223–234.
  • Carnap, R, The logical structure of the world: and, pseudoproblems in philosophy, Chicago and La Salle: Open Court Publishing, 2003. Based on a 1926 dissertation. First English translation in 1967.
  • Child, J (ed), The early mathematical manuscripts of Leibniz. Translated from the Latin texts published by Carl Immanuel Gerhardt with critical and historical notes by J. M. Child. Chicago–London: The Open Court Publishing, 1920. Reprinted by Dover in 2005.
  • Connes, A, ‘An interview with Alain Connes. Part I: conducted by Catherine Goldstein and Georges Skandalis (Paris)’, European Mathematical Society. Newsletter, 63 (2007), 25–30. See http://www.ems-ph.org/journals/newsletter/pdf/2007-03-63.pdf.
  • Easwaran, K, and Towsner, H, Realism in mathematics: The case of the hyperreals. 8 February 2019 version. See http://u.math.biu.ac.il/katzmik/easwaran19.pdf.
  • Ehrlich, P, ‘Contemporary infinitesimalist theories of continua and their late nineteenth- and early twentieth-century forerunners’, in Shapiro–Hellman (2021), 2021, 502–570. See https://arxiv.org/abs/1808.03345.
  • Ferraro, G, ‘Euler and the structure of mathematics', Historia Mathematica 50 (2020), 2–24.
  • Fletcher, P; Hrbacek, K; Kanovei, V; Katz, M; Lobry, C, and Sanders, S, ‘Approaches to analysis with infinitesimals following Robinson, Nelson, and others’, Real Analysis Exchange, 42(2) (2017), 193–252. See https://arxiv.org/abs/1703.00425 and http://doi.org/10.14321/realanalexch.42.2.0193.
  • Forti, M, ‘A topological interpretation of three Leibnizian principles within the functional extensions’, Logical Methods in Computer Science, 14(3) (2018). Paper No. 5, 11 pp.
  • Gerhardt, C (ed), Historia et Origo calculi differentialis a G. G. Leibnitio conscripta, Hannover: Hahn, 1846.
  • Gerhardt, C (ed), Leibnizens mathematische Schriften, Berlin and Halle: A. Asher, 1850–63.
  • Goldenbaum, U, and Jesseph, D (eds), Infinitesimal Differences: Controversies between Leibniz and his Contemporaries, Berlin–New York: Walter de Gruyter, 2008.
  • Gray, J, ‘A short life of Euler’, BSHM Bulletin, 23(1) (2008a), 1–12.
  • Gray, J, Plato's ghost. The modernist transformation of mathematics, Princeton, NJ: Princeton University Press, 2008b.
  • Gray, J, The real and the complex: a history of analysis in the 19th century (Springer Undergraduate Mathematics Series), Cham: Springer, 2015.
  • Guicciardini, N, Review of Knobloch (1999). Mathematical Reviews MR1705300 (2000). See https://mathscinet.ams.org/mathscinet-getitem?mr=1705300.
  • Guicciardini, N, ‘Review of ‘De quadratura arithmetica circuli ellipseos et hyperbolae cujus corollarium est trigonometria sine tabulis, by Gottfried Wilhelm Leibniz, edited by Eberhard Knobloch’’, The Mathematical Intelligencer, 40(4) (2018), 89–91.
  • Hacking, I, Why is there philosophy of mathematics at all?, Cambridge: Cambridge University Press, 2014.
  • Henle, J, ‘Non-nonstandard analysis: real infinitesimals’, The Mathematical Intelligencer, 21(1) (1999), 67–73.
  • Hermann, J, ‘Responsio ad Clarissimi Viri Bernh. Nieuwentiit Considerationes Secundas circa calculi differentialis principia’, editas Basileae, Literis Johannis Conradi a Mechel, 1700.
  • Hewitt, E, ‘Rings of real-valued continuous functions. I’, Transactions of the American Mathematical Society, 64 (1948), 45–99.
  • Horváth, M, ‘On the attempts made by Leibniz to justify his calculus’, Studia Leibnitiana, 18(1) (1986), 60–71.
  • Hrbacek, K, ‘Axiomatic foundations for nonstandard analysis’, Fundamenta Mathematicae, 98(1) (1978), 1–19.
  • Hrbacek, K, and Katz, M, Infinitesimal analysis without the axiom of choice (2020). See https://arxiv.org/abs/2009.04980.
  • Ishiguro, H, Leibniz's philosophy of logic and language, Cambridge: Cambridge University Press, 2nd edition, 1990.
  • Jesseph, D, ‘Leibniz on the elimination of infinitesimals’, in Norma B Goethe, Philip Beeley, and David Rabouin (eds), G.W. Leibniz, Interrelations between mathematics and philosophy (Archimedes Series, 41), Dordrecht–Heidelberg–New York–London: Springer Verlag, 2015, 189–205.
  • Jesseph, D, ‘The indivisibles of the continuum. Seventeenth-century adventures in infinitesimal mathematics’, in Shapiro–Hellman (2021), 2021, 104–122.
  • Kanovei, V, and Reeken, M, Nonstandard analysis, axiomatically (Springer Monographs in Mathematics), Berlin: Springer-Verlag, 2004.
  • Katz, M, ‘Mathematical conquerors, Unguru polarity, and the task of history’, Journal of Humanistic Mathematics, 10(1) (2020), 475–515. See https://doi.org/10.5642/jhummath.202001.27 and https://arxiv.org/abs/2002.00249.
  • Katz, M, and Sherry, D, ‘Leibniz's laws of continuity and homogeneity’, Notices of the American Mathematical Society, 59(11) (2012), 1550–1558. See http://www.ams.org/notices/201211/rtx121101550p.pdf and https://arxiv.org/abs/1211.7188.
  • Katz, M, and Sherry, D, ‘Leibniz's infinitesimals: Their fictionality, their modern implementations, and their foes from Berkeley to Russell and beyond’, Erkenntnis, 78(3) (2013), 571–625. See https://doi.org/10.1007/s10670-012-9370-y and http://arxiv.org/abs/1205.0174.
  • Keisler, H J, Elementary calculus: an infinitesimal approach, Boston: Prindle, Weber & Schmidt, 2nd edition, 1986. An updated version is online at http://www.math.wisc.edu/keisler/calc.html.
  • Knobloch, E, ‘L'infini dans les mathématiques de Leibniz’, in Lamarra (ed), L'infinito in Leibniz, Rome: Edizioni dell' Ateneo, 1990, 33–51.
  • Knobloch, E, ‘The infinite in Leibniz's mathematics -- The historiographical method of comprehension in context’, in K Gavroglu, J Christianidis, E Nicolaidis (eds), Trends in the historiography of science, Dordrecht: Kluwer, 1994, 265–278.
  • Knobloch, E, ‘Galileo and Leibniz: different approaches to infinity’, Archive for History of Exact Sciences, 54(2) (1999), 87–99.
  • Knobloch, E, ‘Leibniz's rigorous foundation of infinitesimal geometry by means of Riemannian sums’. Foundations of the formal sciences, 1 (Berlin, 1999), Synthese, 133(1–2) (2002), 59–73.
  • Knobloch, E, Review of ‘Arthur, R. Leibniz's syncategorematic infinitesimals. Arch. Hist. Exact Sci. 67, No. 5, 553–593 (2013)’. Zentralblatt, 2013, Zbl 1273.01021. See https://zbmath.org/?q=an%3A1273.01021.
  • Knobloch, E, ‘Letter to the editors of the journal Historia Mathematica. Remarks on the paper by Blåsjö’, Historia Mathematica, 44(3) (2017), 280–282.
  • Knobloch, E, ‘Leibniz and the infinite’, Quaderns d'Història de l'Enginyeria, 14 (2018), 11–31.
  • Knobloch, E, ‘Leibniz's Parisian studies on infinitesimal mathematics’, in Ioannis M Vandoulakis and Dun Liu (eds), Navigating across mathematical cultures and times: exploring the diversity of discoveries and proofs. to appear, 2021?. See summary at https://www.bokus.com/bok/9789814689366.
  • Kock, A, Synthetic differential geometry (London Mathematical Society Lecture Note Series, 333). Cambridge: Cambridge University Press, 2nd edition, 2006.
  • Kunen, K, Set theory. An introduction to independence proofs (Studies in Logic and the Foundations of Mathematics, 102). Amsterdam–New York: North-Holland, 1980.
  • Lawvere, F, ‘Toward the description in a smooth topos of the dynamically possible motions and deformations of a continuous body’. Third Colloquium on Categories (Amiens, 1980), Part I, Cahiers Topologie Géom. Différentielle, 21(4) (1980), 377–392.
  • Leibniz, G, ‘Nova methodus pro maximis et minimis’, Acta Erudit. Lips., Oct. 1684’, 467–473, in Gerhardt (1850–63), vol. V, 220–226.
  • Leibniz, G, ‘To l'Hospital’, 14/24 June 1695a, in Gerhardt (1850–63), vol. I, 287–289.
  • Leibniz, G, ‘Responsio ad nonnullas difficultates a Dn. Bernardo Niewentiit circa methodum differentialem seu infinitesimalem motas’, Acta Erudit. Lips. (1695b), 310–316, in Gerhardt (1850–63), vol. V, p. 320–328. A French translation by Parmentier is in Leibniz (1989a, 316–334).
  • Leibniz, G, ‘Cum Prodiisset… mss Cum prodiisset atque increbuisset Analysis mea infinitesimalis…’, in Gerhardt 1846, 1701, 39–50. See http://books.google.co.il/books?id=UOM3AAAAMAAJ English translation in Child (2005), 145–158.
  • Leibniz, G, ‘Letter to Varignon, 2 February 1702’, in Gerhardt (1850–63), vol. IV, 91–95. Published as ‘Extrait d'une Lettre de M. Leibnitz à M. Varignon, contenant l'explication de ce qu'on a raporté de luy dans les Memoires de Trevoux des mois de Novembre & Decembre derniers.’ Journal des sçavans, 20 March 1702, 183–186. See also http://www.gwlb.de/Leibniz/Leibnizarchiv/Veroeffentlichungen/III9.pdf. Translation in Leibniz (1989b), 542–546.
  • Leibniz, G, ‘Symbolismus memorabilis calculi algebraici et infinitesimalis in comparatione potentiarum et differentiarum, et de lege homogeneorum transcendentali’, in Gerhardt (1850–63), vol. V, 1710, 377–382.
  • Leibniz, G, Nouveaux Essais sur l'entendement humain, Ernest Flammarion, 1921. Originally composed in 1704; first published in 1765.
  • Leibniz, G, La naissance du calcul différentiel. 26 articles des Acta Eruditorum. Translated from the Latin and with an introduction and notes by Marc Parmentier. With a preface by Michel Serres. Mathesis. Librairie Philosophique J. Vrin, Paris, 1989a. See https://books.google.co.il/books?id=lfEy-OzaWkQC.
  • Leibniz, G, Philosophical essays. Translated and edited by Roger Ariew and Daniel Garber. Indianapolis, IN: Hackett Publishing, 1989b.
  • Leibniz, G, Philosophical papers and letters (Synthese Historical Library, 2), Leroy E. Loemker, Editor and Translator. Dordrecht–Boston–London: Kluwer Academic Publishers, 2nd edition, 1989c.
  • Leibniz, G, Quadrature arithmétique du cercle, de l'ellipse et de l'hyperbole. Marc Parmentier (Trans. and ed) /Latin text by Eberhard Knobloch (ed), J. Vrin, Paris, 2004.
  • Leibniz, G, Mathesis universalis. Écrits sur la mathématique universelle. Translated from the Latin and with an introduction and notes by David Rabouin. Mathesis. Librairie Philosophique J. Vrin, Paris, 2018.
  • Levey, S, ‘Leibniz on mathematics and the actually infinite division of matter’, The Philosophical Review, 107(1) (1998), 49–96.
  • Levey, S, ‘Archimedes, infinitesimals and the law of continuity: on Leibniz's Fictionalism’, in Goldenbaum–Jesseph (2008), 2008, 107–134.
  • Levey, S, ‘The continuum, the infinitely small, and the law of continuity in Leibniz’, in Shapiro–Hellman (2021), 2021, 123–157.
  • Łoś, J, ‘Quelques remarques, théorèmes et problèmes sur les classes définissables d'algèbres’, in Mathematical interpretation of formal systems, Amsterdam: North-Holland Publishing, 1955, 98–113.
  • Mancosu, P, ‘The metaphysics of the calculus: a foundational debate in the Paris Academy of Sciences, 1700–1706’, Historia Mathematica, 16(3) (1989), 224–248.
  • Mancosu, P, Philosophy of mathematics and mathematical practice in the seventeenth century, New York: The Clarendon Press, Oxford University Press, 1996.
  • Mercer, C, ‘Leibniz on mathematics, methodology, and the good: a reconsideration of the place of mathematics in Leibniz's philosophy’, Early Science and Medicine, 11(4) (2006), 424–454.
  • Moerdijk, I, and Reyes, G, Models for smooth infinitesimal analysis, New York: Springer-Verlag, 1991.
  • Mormann, T, and 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(2) (2013), 236–280. See https://doi.org/10.1086/671348 and https://arxiv.org/abs/1304.1027.
  • Nagel, F, ‘Nieuwentijt, Leibniz, and Jacob Hermann on Infinitesimals’, in Goldenbaum–Jesseph (2008), 2008, 199–214.
  • Nelson, E, ‘Internal set theory: a new approach to nonstandard analysis’, Bulletin of the American Mathematical Society, 83(6) (1977), 1165–1198.
  • Nelson, E, Radically elementary probability theory (Annals of Mathematics Studies, 117). Princeton, NJ: Princeton University Press, 1987. 98 pp.
  • Pasini, E, ‘Die private Kontroverse des GW Leibniz mit sich selbst. Handschriften über die Infinitesimalrechnung im Jahre 1702’, in Leibniz. Tradition und Aktualität, Hannover: Leibniz-Gesellschaft, 1988, 695–709.
  • Probst, S, ‘The calculus’, in MR Antognazza (ed), The oxford handbook of Leibniz, Oxford: Oxford University Press, 2018, 211–224.
  • Pruss, A, ‘Underdetermination of infinitesimal probabilities’, Synthese (2018), online first at https://doi.org/10.1007/s11229-018-02064-x.
  • Rabouin, D, ‘Leibniz's rigorous foundations of the method of indivisibles’, in Vincent Jullien (ed), Seventeenth-century indivisibles revisited (Science Networks. Historical Studies, 49), Basel: Birkhäuser, 2015, 347–364.
  • Rabouin, D, and Arthur, R, ‘Leibniz's syncategorematic infinitesimals II: their existence, their use and their role in the justification of the differential calculus’, Archive for History of Exact Sciences, 74 (2020), 401–443.
  • Robinson, A, ‘Non-standard analysis’, Nederl. Akad. Wetensch. Proc. Ser. A 64 = Indag. Math. 23 (1961), 432–440. (Reprinted in Selected Papers Robinson 1979, 3–11).
  • Robinson, A, Non-standard analysis, Amsterdam: North-Holland Publishing, 1966.
  • Robinson, A, Selected papers of Abraham Robinson. Vol. II. Nonstandard analysis and philosophy. Edited and with introductions by W. A. J. Luxemburg and S. Körner. New Haven, Conn: Yale University Press, 1979.
  • Sanders, S, ‘The unreasonable effectiveness of Nonstandard Analysis’, Journal of Logic and Computation, 30(1) (2020), 459–524. See https://doi.org/10.1093/logcom/exaa019 and http://arxiv.org/abs/1508.07434.
  • Shapiro, S, and Hellman, J (eds), The History of Continua: Philosophical and Mathematical Perspectives, Oxford University Press, 2021. See https://doi.org/10.1093/OSO/9780198809647.003.0018 and https://global.oup.com/academic/product/the-history-of-continua-9780198809647.
  • Sherry, D, ‘The wake of Berkeley's Analyst: rigor mathematicae?’, Studies in History and Philosophy of Science, 18(4) (1987), 455–480.
  • Sherry, D, and Katz, M, ‘Infinitesimals, imaginaries, ideals, and fictions’, Studia Leibnitiana, 44(2) (2012), 166–192. See http://www.jstor.org/stable/43695539 and https://arxiv.org/abs/1304.2137 (Article was published in 2014 even though the journal issue lists the year as 2012).
  • Skolem, T, ‘Über die Unmöglichkeit einer vollständigen Charakterisierung der Zahlenreihe mittels eines endlichen Axiomensystems’, Norsk Mat. Forenings Skr., II. Ser. No. 1/12 (1933), 73–82.
  • Skolem, T, ‘Über die Nicht-charakterisierbarkeit der Zahlenreihe mittels endlich oder abzählbar unendlich vieler Aussagen mit ausschliesslich Zahlenvariablen’, Fundamenta Mathematicae, 23 (1934), 150–161.
  • Sommer, R, and Suppes, P, ‘Finite models of elementary recursive nonstandard analysis’, Notas de la Sociedad Matematica de Chile, 15 (1996), 73–95.
  • Spalt, D, Die Analysis im Wandel und im Widerstreit. Eine Formierungsgeschichte ihrer Grundbegriffe. [On title page: Eine Formierungsgeschichte ihrer Grundgeschichte]. Freiburg: Verlag Karl Alber, 2015.
  • Strømholm, P, ‘Fermat's methods of maxima and minima and of tangents. A reconstruction’, Archive for History Exact Sciences, 5(1) (1968), 47–69.
  • Tarski, A, and Givant, S, A formalization of set theory without variables (American Mathematical Society Colloquium Publications, 41), Providence RI: American Mathematical Society, 1987.
  • van den Berg, B, and Sanders, S, ‘Reverse mathematics and parameter-free transfer’, Annals of Pure and Applied Logic, 170(3) (2019), 273–296.
  • Väth, M, Nonstandard analysis, Basel: Birkhäuser Verlag, 2007.
  • Yokoyama, K, ‘Formalizing non-standard arguments in second-order arithmetic’, Journal of Symbolic Logic, 75(4) (2010), 1199–1210.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.