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The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta procedures rather than infinitesimals. Non-standard analysis instead reformulates the calculus using a logically rigorous notion of infinitesimal numbers. Non-standard analysis was originated in the early 1960s by the mathematician Abraham Robinson. He wrote: "... the idea of infinitely small or infinitesimal quantities seems to appeal naturally to our intuition. At any rate, the use of infinitesimals was widespread during the formative stages of the Differential and Integral Calculus. As for the objection ... that the distance between two distinct real numbers cannot be infinitely small, Gottfried Wilhelm Leibniz argued that the theory of infinitesimals implies the introduction of ideal numbers which might be infinitely small or infinitely large compared with the real numbers but which were to possess the same properties as the latter". Robinson argued that this law of continuity of Leibniz's is a precursor of the transfer principle. Robinson continued: "However, neither he nor his disciples and successors were able to give a rational development leading up to a system of this sort. As a result, the theory of infinitesimals gradually fell into disrepute and was replaced eventually by the classical theory of limits." "It is shown in this book that Leibniz's ideas can be fully vindicated and that they lead to a novel and fruitful approach to classical Analysis and to many other branches of mathematics. The key to our method is provided by the detailed analysis of the relation between mathematical languages and mathematical structures which lies at the bottom of contemporary model theory." In 1973, intuitionist Arend Heyting praised non-standard analysis as "a standard model of important mathematical research".
References
- Robinson, Abraham (1996). Non-standard analysis (Revised ed.). Princeton University Press. ISBN 0-691-04490-2.
- Robinson, A.: Non-standard analysis. North-Holland Publishing Co., Amsterdam 1966.
- Curt Schmieden and Detlef Laugwitz: Eine Erweiterung der Infinitesimalrechnung, Mathematische Zeitschrift 69 (1958), 1-39
- H. Jerome Keisler, An Infinitesimal Approach. First edition 1976; 2nd edition 1986: full text of 2nd edition http://www.math.wisc.edu/~keisler/calc.html
- Edward Nelson: Radically Elementary Probability Theory, Princeton University Press, 1987, full text http://www.math.princeton.edu/~nelson/books/rept.pdf
- Sergio Albeverio, Jans Erik Fenstad, Raphael Høegh-Krohn, Tom Lindstrøm: Nonstandard Methods in Stochastic Analysis and Mathematical Physics, Academic Press 1986. https://books.google.com/books?hl=en&lr=&id=M_5RB-YGjZsC&oi=fnd&pg=PP2&dq=%22+Nonstandard+Methods+in+Stochastic+Analysis+and+Mathematical+Physics%22&ots=eaaZNfWOVe&sig=IN3-HoUHgNpwbPI0q4R_iAtf07I#v=onepage&q=%22%20Nonstandard%20Methods%20in%20Stochastic%20Analysis%20and%20Mathematical%20Physics%22&f=false
- Edward Nelson: Internal Set Theory: A New Approach to Nonstandard Analysis, Bulletin of the American Mathematical Society, Vol. 83, Number 6, November 1977. A chapter on internal set theory is available at http://www.math.princeton.edu/~nelson/books/1.pdf http://www.math.princeton.edu/~nelson/books/1.pdf
- Vopěnka, P. Mathematics in the Alternative Set Theory. Teubner, Leipzig, 1979.
- Robinson, Abraham: 'Non-Standard Analysis', Kon. Nederl. Akad. Wetensch. Amsterdam Proc. AM (=Indag. Math. 23), 1961, 432-440.
- Allen Bernstein and Abraham Robinson, Solution of an invariant subspace problem of K. T. Smith and P. R. Halmos, Pacific Journal of Mathematics 16:3 (1966) 421-431 http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.pjm/1102994835
- P. Halmos, Invariant subspaces for Polynomially Compact Operators, Pacific Journal of Mathematics, 16:3 (1966) 433-437. http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.pjm/1102994836
- T. Kamae: A simple proof of the ergodic theorem using nonstandard analysis, Israel Journal of Mathematics vol. 42, Number 4, 1982. https://link.springer.com/article/10.1007/BF02761408
- L. van den Dries and A. J. Wilkie: Gromov's Theorem on Groups of Polynomial Growth and Elementary Logic, Journal of Algebra, Vol 89, 1984. http://www.math.uni-muenster.de/u/linus.kramer/vandenDriesWilkie.pdf
- Manevitz, Larry M.; Weinberger, Shmuel: Discrete circle actions: a note using non-standard analysis. Israel J. Math. 94 (1996), 147--155. https://link.springer.com/article/10.1007/BF02762701
- Capinski M., Cutland N. J. Nonstandard Methods for Stochastic Fluid Mechanics. Singapore etc., World Scientific Publishers (1995) https://books.google.com/books?hl=en&lr=&id=QHoVHTRkcIkC&oi=fnd&pg=PR7&dq=%22Nonstandard+Methods+for+Stochastic+Fluid+Mechanics%22&ots=vOSE89HiWB&sig=B18KGBW8jYj9ucP9SshYerNu-Ws#v=onepage&q=%22Nonstandard%20Methods%20for%20Stochastic%20Fluid%20Mechanics%22&f=false
- Cutland N. Loeb Measures in Practice: Recent Advances. Berlin etc.: Springer (2001)
- Gordon E. I., Kutateladze S. S., and Kusraev A. G. Infinitesimal Analysis Dordrecht, Kluwer Academic Publishers (2002)
- Salbany, S.; Todorov, T. Nonstandard Analysis in Point-Set Topology. Erwing Schrodinger Institute for Mathematical Physics. http://www.esi.ac.at/static/esiprpr/esi666.pdf
- Chang, C. C.; Keisler, H. J. Model theory. Third edition. Studies in Logic and the Foundations of Mathematics, 73. North-Holland Publishing Co., Amsterdam, 1990. xvi+650 pp. ISBN 0-444-88054-2
