(updated
February 2, 2005)
a new course of
analysis: the main ideas
gabriel stolzenberg
This
new course of analysis—in reality, it is nearly thirty-five years
old—was initially informed by Errett Bishop's Foundations of
Constructive Analysis
(1967) but soon took on a life of its own. And then another life. And yet another. Five in all, each born
from a critique of the preceding one.
This
site, which itself is a work in progress, is dedicated to an exposition of
ideas that could be used by other mathematicians (especially ones who share my
sense of wonder at the origin of analysis in the emergence of the continuous
from the discrete) to create new courses of this kind—ones that might
differ in interesting ways according to the tastes, interests and bright ideas
of their creators.
The
exposition will emerge in stages, starting here with The Real Number System (wherein we construct a machine)
and An Inverse Function Theorem and Some Applications (wherein we operate it): PDF. These will be followed, at the next
stage, by ideas about continuity, integration and the derivative—in short, by ideas for a
theory of the calculus, albeit a nontraditional one. (Much of this is based on Uniform
Calculus and the Law of Bounded Change by Mark Bridger and Gabriel Stolzenberg, American
Mathematical Monthly, August-September, 1999.)
Among possible topics for later stages are complex
analysis, differential equations, metric spaces (and some unfamiliar generalizations
of them) and, my current favorite, vegetarian Lebesgue theory for Fourier
analysis. For an
explanation of what I mean by ‘vegetarian,’ see What I Mean by ‘Vegetarian’: HTML.