A Brief Introduction to Infinitesimal Calculus

Section 1: Intuitive Proofs with "Small" Quantities

Abraham Robinson discovered a rigorous approach to calculus with infinitesimals in 1960 and published it in

Non-standard Analysis, Proceedings of the Royal Academy of Sciences, Amsterdam, ser A, 64, 1961, p.432-440

This solved a 300 year old problem dating to Leibniz and Newton.  Extending the ordered field of (Dedekind) "real" numbers to include infinitesimals is not difficult algebraically, but calculus depends on approximations with transcendental functions.  Robinson used mathematical logic to show how to extend all real functions in a way that preserves their properties in a precise sense.  These properties can be used to develop calculus with infinitesimals.

Infinitesimal numbers have always fit basic intuitive approximation when certain quantities are "small enough," but Leibniz, Euler, and many others could not make the approach free of contradiction.  Robinson's discovery offers the possibility of making rigorous foudations of calculus more accessible.  

Section 1 of this article uses some intuitive approximations to derive a few fundamental results of analysis.  We use approximate equality, , only in an intuitive sense that " is sufficiently close to ".  Intutive approximation gives compelling arguments for the results, but not technically complete proofs.  

H. Jerome Keisler developed simpler approaches to Robinson's logic and began using infinitesimals in beginning U. S. calculus courses in 1969.  The experimental and first edition of his book were used widely in the 1970's.  Section 2 of this article completes the proofs of Section 1 using Keisler's approach to the logic of infinitesimals from

Elementary Calculus: An Infinitesimal Approach, Edition, PWS Publishers, 1986, now available free at http://www.math.wisc.edu/~keisler/calc.html

Created by Mathematica  (September 22, 2004)