Abraham Robinson Abraham Robinson (born Robinsohn; October 6, 1918 – April 11, 1974) was a mathematician who is most widely known for development of nonstandard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were reincorp ...

's theory of

nonstandard analysis 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–delt ...

has been applied in a number of fields.

Probability theory

"Radically elementary probability theory" of

Edward Nelson Edward Nelson (May 4, 1932 – September 10, 2014) was an American mathematician. He was professor in the Mathematics Department at Princeton University. He was known for his work on mathematical physics and mathematical logic. In mathematic ...

combines the discrete and the continuous theory through the infinitesimal approach. The model-theoretical approach of nonstandard analysis together with Loeb measure theory allows one to define

Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...

as a hyperfinite random walk, obviating the need for cumbersome measure-theoretic developments.

Jerome Keisler Howard Jerome Keisler (born 3 December 1936) is an American mathematician, currently professor emeritus at University of Wisconsin–Madison. His research has included model theory and non-standard analysis. His Ph.D. advisor was Alfred Tarski ...

used this classical approach of nonstandard analysis to characterize general

stochastic processes In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that a ...

as hyperfinite ones.

Economics

Economists have used nonstandard analysis to model markets with large numbers of agents (see

Robert M. Anderson (economist) Robert Murdoch Anderson (born 1951) is Professor of Economics and of Mathematics at the University of California, Berkeley. He is director of the Center for Risk Management Research, University of California, Berkeley and he was chair of the U ...

Education

An article by

Michèle Artigue Michèle Artigue (born 1946) is a French expert in mathematics education, a professor emeritus at Paris Diderot University and the former president of the International Commission on Mathematical Instruction. Early life and education Artigue was b ...

concerns the teaching of analysis. Artigue devotes a section, "The non standard analysis and its weak impact on education" on page 172, to non-standard analysis. She writes: :The non-standard analysis revival and its weak impact on education. The publication in 1966 of Robinson's book NSA constituted in some sense a rehabilitation of infinitesimals which had fallen into disrepute .. obinson's proposalwas met with suspicion, even hostility, by many mathematicians ..Nevertheless, despite the obscurity of this first work, NSA developed rapidly ..The attempts at simplification were often conducted with the aim of producing an elementary way of teaching NSA. This was the case with the work of Keisler and Henle-Kleinberg .. Artigue continues specifically with reference to the calculus textbook: : eisler's workserved as a reference text for a teaching experiment in the first year in university in the Chicago area in 1973-74. Sullivan used 2 questionnaires to evaluate the effects of the course, one for teachers, the other for students. The 11 teachers involved gave a very positive opinion of the experience. The student questionnaire revealed no significant difference in technical performance ..but showed that those following the NSA course were better able to interpret the sense of the mathematical formalism of calculus ..The appearance of the 2nd book of Keisler led to a virulent criticism by Bishop, accusing Keisler of seeking ..to convince students that mathematics is only "an esoteric and meaningless exercise in technique", detached from any reality. These criticisms were in opposition to the declarations of the partisans of NSA who affirmed with great passion its simplicity and intuitive character. ..However, it is necessary to emphasize the weak impact of NSA on contemporary education. The small number of reported instances of this approach are often accompanied with passionate advocacy, but this rarely rises above the level of personal conviction.

Authors of books on hyperreals

Sergio Albeverio Sergio Albeverio (born 17 January 1939) is a Swiss mathematician and mathematical physicist working in numerous fields of mathematics and its applications. In particular he is known for his work in probability theory, analysis (including infini ...

Leif Arkeryd Leif O. Arkeryd (born 24 August 1940) is professor emeritus of mathematics at Chalmers University of Technology. He is a specialist on the theory of the Boltzmann equation. Arkeryd earned his doctorate from Lund University in 1966, under the su ...

Nigel Cutland Nigel J. Cutland is Professor of Mathematics at the University of York , mottoeng = On the threshold of wisdom , established = , type = Public research university , endowment = £8.0 million , budget = £403.6 million , chancel ...

Martin Davis Martin Davis may refer to: * Martin Davis (Australian footballer) (born 1936), Australian rules footballer * Martin Davis (Jamaican footballer) (born 1996), Jamaican footballer * Martin Davis (mathematician) Martin David Davis (March 8, 1928 � ...

Jens Erik Fenstad Jens Erik Fenstad (15 April 1935 – 13 April 2020) was a Norwegian mathematician. Fenstad graduated as ''mag.scient.'' from the University of Oslo in 1959, and worked as a research fellow there and at UC Berkeley. He was a professor at the U ...

Robert Goldblatt __notoc__ Robert Ian Goldblatt (born 1949) is a mathematical logician who is Emeritus Professor in the School of Mathematics and Statistics at Victoria University, Wellington, New Zealand. His most popular books are ''Logics of Time and Computatio ...

Reuben Hersh Reuben Hersh (December 9, 1927 – January 3, 2020) was an American mathematician and academic, best known for his writings on the nature, practice, and social impact of mathematics. Although he was generally known as Reuben Hersh, late in life h ...

Raphael Høegh-Krohn Jan Raphael Høegh-Krohn (10 February 1938 – 24 January 1988) was a Norwegian mathematician. He finished his Ph.D. in 1966, titled ''On Partly Gentle Perturbation with Application to Perturbation by Annihilation-Creation Operator,'' under the ...

Karel Hrbáček Karel Hrbáček (born 1944) is professor emeritus of mathematics at City College of New York. He specializes in mathematical logic, set theory, and non-standard analysis. Early life and education Karel studied at Charles University with Petr Vo ...

Vladimir Kanovei Vladimir G. Kanovei (born 1951) is a Russian mathematician working at the Institute for Information Transmission Problems in Moscow, Russia. His interests include mathematical logic and foundations, as well as mathematical history The histo ...

* H. Jerome Keisler *

Semen Samsonovich Kutateladze Semen, also known as seminal fluid, is an organic bodily fluid created to contain spermatozoa. It is secreted by the gonads (sexual glands) and other sexual organs of male or hermaphroditic animals and can fertilize the female ovum. Semen is ...

Peter Loeb Peter Albert Loeb (born July 3, 1937) is a mathematician at the University of Illinois at Urbana–Champaign. He co-authored a basic reference text on nonstandard analysis (Hurd–Loeb 1985). Reviewer Perry Smith for MathSciNet wrote: :This book is ...

Wilhelmus Luxemburg Wilhelmus Anthonius Josephus Luxemburg (Delft, 11 April 1929 – 2 October 2018) was a Dutch American mathematician who was a professor of mathematics at the California Institute of Technology. He received his B.A. from the University of Leiden ...

Alain M. Robert Alain M. Robert is Honorary Professor at University of Neuchâtel. Robert received his PhD from the University of Neuchâtel in 1967, where he studied under Roger Bader. His dissertation ''Quelques Questions d'Espaces Vectoriels Topologiques'' conc ...

Keith Stroyan Keith D. Stroyan is Professor of Mathematics at the University of Iowa. His main research interests are in analysis and visual depth perception. Publications *Stroyan, K. D.; Luxemburg, W. A. J. ''Introduction to the theory of infinitesimals''. P ...

Terence Tao Terence Chi-Shen Tao (; born 17 July 1975) is an Australian-American mathematician. He is a professor of mathematics at the University of California, Los Angeles (UCLA), where he holds the James and Carol Collins chair. His research includes ...

Notes

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Bibliography

* Zdzislaw Pawlak: Rough set approach to knowledge-based decision support. European Journal of Operational Research Volume 99, Issue 1, 16 May 1997, Pages 48–57. *

Melvin Fitting Melvin Fitting (born January 24, 1942) is a logician with special interests in philosophical logic and tableau proof systems. He was a professor at City University of New York, Lehman College and the Graduate Center. from 1968 to 2013. At the ...

: First-order logic and automated theorem-proving. Springer, 1996. *Daniel Lehmann,

Menachem Magidor Menachem Magidor (Hebrew: מנחם מגידור; born January 24, 1946) is an Israeli mathematician who specializes in mathematical logic, in particular set theory. He served as president of the Hebrew University of Jerusalem, was president of ...