Nonstandard Analysis

Nonstandard analysis is a branch of mathematical logic which introduces hyperreal numbers to allow for the existence of "genuine infinitesimals," which are numbers that are less than 1/2, 1/3, 1/4, 1/5, ..., but greater than 0. Abraham Robinson developed nonstandard analysis in the 1960s. The theory has since been investigated for its own sake and has been applied in areas such as Banach spaces, differential equations, probability theory, mathematical economics, and mathematical physics.

The axioms used in nonstandard analysis are first-order set theoretical axioms, but many of the topics studied in classical analysis that are axiomatized with higher-order axioms can be reformulated in set theoretical terms in a first-order axiomatization. As an example, consider the notion of a measure on a set. In classical analysis, one studies measure spaces. A measure space consists of a set , together with a measure , which is a function from some sigma-algebra  of subsets of  into the reals. This way of looking at measure spaces is a way that uses higher-order logic, and the measure  is a sort of "higher order object", since it is not an element of . But if one forms the superstructure  that has as its individuals the members of  and the real numbers, and is constructed as described in typical texts on nonstandard analysis, as the union of (roughly) a tower of iterated power sets, with the only fundamental relation being the membership relation, then in the first-order theory of this superstructure, one may refer to the measure  as an element, for it is in fact an element of .

Loosely, nonstandard methods replace higher-order concepts with first-order analogs. It looks at them from a different angle. Crucially, however, the angle at which the nonstandard analyst looks at the axioms of analysis provides for an average case reduction in complexity that provides shorter proofs of various results, and will one day lead to the proof of a result which is not accessible to classical mathematics without nonstandard methods, precisely because its classical proof is too long to write down in the length of time humans will reside on Earth.

In addition, in the nonstandard analysis community, there is a growing number of results that are not being translated into standard results, because the intuitive content of certain theorems is greater and/or clearer when left in nonstandard terminology. Examples include the use of nonstandard analysis in mathematical economics to describe the behavior of large economies and the use of nonstandard methods to give meaning to concepts that do not classically make sense, such as certain products of infinitely many independent, equally weighted random variables.

REFERENCES

Albeverio, S.; Fenstad, J.; Hoegh-Krohn, R.; and Lindstrøom, T. Nonstandard Methods in Stochastic Analysis and Mathematical Physics. New York: Academic Press, 1986.

Anderson, R. M. "Nonstandard Analysis with Applications to Economics." Ch. 39 in Handbook of Mathematical Economics, Vol. 4 (Ed. W. Hildenbrand and H. Sonnenschein). New York: Elsevier, pp. 2145-2208, 1991.

Apps, P. "What is Nonstandard Analysis?" http://members.tripod.com/PhilipApps/nonstandard.html.Dauben, J. W. Abraham Robinson: The Creation of Nonstandard Analysis, A Personal and Mathematical Odyssey. Princeton, NJ: Princeton University Press, 1998.

Davis, P. J. and Hersch, R. The Mathematical Experience. Boston, MA: Birkhäuser, 1981.Hurd, A. E. and Loeb, P. A. An Introduction to Nonstandard Real Analysis. New York: Academic Press, 1985.

Keisler, H. J. Elementary Calculus: An Infinitesimal Approach. Boston, MA: PWS, 1986.

 http://www.math.wisc.edu/~keisler/calc.html.Lindstrøom, T. "An Invitation to Nonstandard Analysis." In Nonstandard Analysis and Its Applications (Ed. N. Cutland). New York: Cambridge University Press, 1988.

Loeb, P. A. and Wolff, M. Nonstandard Analysis for the Working Mathematician. Dordrecht, Netherlands: Kluwer, 2000.

Robinson, A. Non-Standard Analysis. Princeton, NJ: Princeton University Press, 1996.Stewart, I. "Non-Standard Analysis." In From Here to Infinity: A Guide to Today's Mathematics. Oxford, England: Oxford University Press, pp. 80-81, 1996.

Sun, Y. "The Almost Equivalence of Pairwise and Mutual Independence and the Duality with Exchangeability." Probab. Th. Related Fields 112, 425-456, 1998.