Editing Nonstandard analysis (section)

== Other applications ==
Other results were received along the line of reinterpreting or reproving previously known results. Of particular interest is Teturo Kamae's proof<ref>T. Kamae: ''[https://link.springer.com/article/10.1007/BF02761408 A simple proof of the ergodic theorem using nonstandard analysis]'', [[Israel Journal of Mathematics]] vol. 42, Number 4, 1982.</ref> of the [[individual ergodic theorem]] or L. van den Dries and [[Alex Wilkie]]'s treatment<ref>L. van den Dries and A. J. Wilkie: ''[http://www.math.uni-muenster.de/u/linus.kramer/vandenDriesWilkie.pdf Gromov's Theorem on Groups of Polynomial Growth and Elementary Logic]'', Journal of Algebra, Vol 89, 1984.</ref> of [[Gromov's theorem on groups of polynomial growth]]. Nonstandard analysis was used by Larry Manevitz and [[Shmuel Weinberger]] to prove a result in algebraic topology.<ref>Manevitz, Larry M.; Weinberger, Shmuel: [https://link.springer.com/article/10.1007/BF02762701 Discrete circle actions: a note using nonstandard analysis]. [[Israel Journal of Mathematics]] 94 (1996), 147--155.</ref>

The real contributions of nonstandard analysis lie however in the concepts and theorems that utilize the new extended language of nonstandard set theory. Among the list of new applications in mathematics there are new approaches to probability,<ref name="Ele"/>
hydrodynamics,<ref>Capinski&nbsp;M., [[Nigel Cutland|Cutland]]&nbsp;N.&nbsp;J. ''[https://books.google.com/books?id=QHoVHTRkcIkC&dq=%22Nonstandard+Methods+for+Stochastic+Fluid+Mechanics%22&pg=PR7 Nonstandard Methods for Stochastic Fluid Mechanics].''
Singapore&nbsp;etc., World Scientific Publishers (1995)</ref> measure theory,<ref>Cutland N. ''[[Peter A. Loeb|Loeb]] Measures in Practice: Recent Advances.'' Berlin&nbsp;etc.: Springer (2001)</ref> nonsmooth and harmonic analysis,<ref>{{citation|url=https://www.springer.com/mathematics/analysis/book/978-1-4020-0738-5|last1=Gordon|first1= E. I.|last3=Kutateladze|first3=S. S.|author3-link= Semën Samsonovich Kutateladze|last2= Kusraev|first2= A. G.|title=Infinitesimal Analysis|publisher=Springer Dordrecht|year=2002}}</ref> etc.

There are also applications of nonstandard analysis to the theory of stochastic processes, particularly constructions of [[Brownian motion]] as [[random walk]]s. Albeverio et al.<ref name="Alb" /> have an introduction to this area of research.

In terms of axiomatics, Boffa’s superuniversality axiom has found application as a basis for axiomatic nonstandard analysis.{{sfnp|Kanovei|Reeken|2004|p=303}}

=== Applications to calculus ===
As an application to [[mathematical education]], [[H. Jerome Keisler]] wrote ''[[Elementary Calculus: An Infinitesimal Approach]]''.<ref name="EC" /> Covering [[nonstandard calculus]], it develops differential and integral calculus using the hyperreal numbers, which include infinitesimal elements. These applications of nonstandard analysis depend on the existence of the ''[[standard part function|standard part]]'' of a finite hyperreal {{mvar|r}}. The standard part of {{mvar|r}}, denoted {{math|st(''r'')}}, is a standard real number infinitely close to {{mvar|r}}. One of the visualization devices Keisler uses is that of an imaginary infinite-magnification microscope to distinguish points infinitely close together.