What is nonstandard analysis?

What is nonstandard analysis?

Nonstandard analysis (NSA) is a technique of mathematics which provides a logical foundation for the idea of an infinitesimal; a number which is less than 1/2, 1/3, 1/4, 1/5 ... and yet greater than 0. Newton and Leibniz used infinitesimal methods in their development of the calculus, but were unable to make them precise, and Weierstrass eventually provided the formal epsilon-delta idea of limits.

Abraham Robinson developed nonstandard analysis in the 1960's, and the theory has since been investigated for its own sake, and has been applied in areas such as Banach spaces, differential equations, probability theory, microeconomic theory and mathematical physics.

Nonstandard analysis is also sometimes referred to as infinitesimal analysis, or Robinsonian analysis.

The simplicity of NSA

How to make infinitesimals

Paul Hertzel's notes on NSA

The hyperreal line

Books, descriptions and order forms online

An unfinished book on NSA, by Edward Nelson (in dvi format)

Nonstandard Analysis: Theory and Applications Description and order form. (from a recent NATO Advanced Study Institute).

A description of the book Nonstandard topology, by Paul Bankston

Description of and order form for Abraham Robinson: The Creation of Nonstandard Analysis, A Personal and Mathematical Odyssey, by Joseph Dauben

Description of and order form for NSA in practice, by Francine and Marc Diener

Description of and order form for Lectures on the Hyperreals: An Introduction to Nonstandard Analysis, by Rob Goldblatt

Description of and order form for Nonstandard Methods in Analysis, by A. G. Kusraev and S.S. Kutateladze

Description of and order form for Non-standard Analysis by Abraham Robinson

Further reading

H.J. Keisler : Elementary calculus (An introductory calculus textbook written using nonstandard analysis. Suitable for college freshmen. Now out of print.)

T. Lindstrøm : An invitation to nonstandard analysis (in Nonstandard analysis and its applications, edited by N. Cutland, Cambridge Univ. Press 1988) (A good introduction to the subject suitable for graduate students or advanced math undergrads. It was the basis for much of these pages).

Davis and Hersh's book The Mathematical Experience has a short section on nonstandard analysis.

Robert Anderson: Nonstandard analysis with applications to economics, in Handbook of Mathematical Economics Vol. 4 (A good introduction with several references)

Albeverio S, Fenstad J, Hoegh-Krohn R and Lindstrøm T (1986): Nonstandard methods in stochastic analysis and mathematical physics. Academic Press

Hurd and Loeb: An introduction to nonstandard real analysis

More references by subject area

(Note: This division is somewhat arbitrary. If you are interested in mathematical physics, for example, you will probably find items of interest in the analysis and probability lists.)

Mechanizing Nonstandard Analysis

Jacques Fleuriot has done some work on mechanizing NSA in a generic theorem prover, Isabelle. The mechanized construction can be found here and details can be found in his PhD thesis.

See also J. D. Fleuriot and L. C. Paulson. Mechanizing Nonstandard Real Analysis. LMS J. Computation and Mathematics 3 (2000), 140-190.

Homepages of people interested in nonstandard analysis

Other links