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.

### 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).

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

• Georges Reeb Net (much information on NSA, including some papers, mostly in French)
• Helsinki Logic Group
• FermiVista (a search engine for math papers, including several in NSA).

"...there are good reasons to believe that nonstandard analysis, in some version or other, will be the analysis of the future." Kurt Gödel.

"Nonstandard analysis has attracted a number of particularly adventurous mathematicians with wide ranging interests." Franklin Wattenberg.

When Abraham Robinson turned fifty, Jon Barwise asked him to what he attributed his unusual productivity as a mathematician. He replied:

I've made it a policy to move every five years, either physically or in my research.

If you have suggestions, papers you would like me to add, or a homepage, you can email me at PhilipApps@hotmail.com