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Analysis, nonstandard

DOI
10.4324/9780415249126-Y070-1
DOI: 10.4324/9780415249126-Y070-1
Version: v1,  Published online: 1998
Retrieved March 29, 2024, from https://www.rep.routledge.com/articles/thematic/analysis-nonstandard/v-1

Article Summary

Nonstandard analysis is an important application of mathematical logic to the rest of mathematics. Invented in 1960, it provided a long-sought-for rigorous justification for the use of infinitely large and infinitely small (infinitesimal) quantities in the differential and integral calculus, and the first sound canon for manipulating such quantities.

Consider the structure of real numbers, that is, the set of real numbers together with operations and relations on them. We specify a formal language L , which has names for all individual real numbers as well as for the operations and relations of . We can then obtain an enlargement (a special kind of extension) * (‘pseudo- ‘) of such that * and have exactly the same formal properties: that is, properties expressible in L . But, far from being merely a copy of , * has convenient properties not expressible in L , which are not shared by . In particular, * has additional ‘nonstandard’ objects (which have no names in L ), some of which behave as infinite or infinitesimal quantities. If, using these novel objects and properties of * , we prove a proposition about * that can be expressed in L , then this proposition holds automatically also for . Such ‘nonstandard’ proofs of results about are often easier and more intuitive than conventional proofs, which operate wholly within .

More generally, this method is applied to other structures, in virtually every branch of mathematics – including algebraic number theory, various branches of classical and modern analysis, probability theory and mathematical physics, to yield highly intuitive characterizations of various infinitary concepts, to simplify proofs and to provide new, efficient ways of generating mathematical constructs.

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Citing this article:
Machover, Moshe. Analysis, nonstandard, 1998, doi:10.4324/9780415249126-Y070-1. Routledge Encyclopedia of Philosophy, Taylor and Francis, https://www.rep.routledge.com/articles/thematic/analysis-nonstandard/v-1.
Copyright © 1998-2024 Routledge.

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