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Notes by RBJ on

Constructive Analysis

by Errett Bishop and Douglas Bridges

Preface
Originally intended as a second edition of The Foundations of Constructive Analysis, to be prepared by Bridges under the supervision of Bishop, the volume was not completed until after Bishop's death and was given a new title because of the extent of the changes.
Prolog
The Prolog is Bishop's original Preface. In it he draws the distinction between two kinds of mathematical statement:

and describes the principles on which the book is written, and its purposes

Chapter 1 - A Constructivist Manifesto
In this Chapter Bishop describes, first, the descriptive basis of mathematics, which is not entirely dissimilar to what others might call its constructive content. Next he describes its idealistic component, by which he may mean the non constructive parts. Finally he talks about the constructivisation of mathematics, which is the process of extracting from classical mathematics those constructive results which underpin it, and discarding the rest.
Chapter 2 - Calculus and the Real Numbers
Chapter 3 - Set Theory
Chapter 4 - Metric Spaces
Chapter 5 - Complex Analysis
Chapter 6 - Integration
Chapter 7 - Normed Linear Spaces
Chapter 8 - Locally Compact Abelian Groups
Chapter 9 - Commutative Banach Algebras

Prolog

This book is "a piece of constructivist propaganda", which is intended to present a satisfactory alternative to classical mathematics, and "to give a numerical meaning to as much as possible of classical abstract analysis".
Three principles are enunciated:
  1. make every concept affirmative
  2. avoid definitions that are not relevant
  3. avoid pseudogenerality
and three purposes:
  1. to present the constructive point of view
  2. to show that the constructive program can succeed
  3. to lay the foundation for further work

Chapter 1 - A Constructivist Manifesto

1. The Descriptive Basis of Mathematics
"The primary concern of mathematics is number, and this means the positive integers."
"We are not interested in properties of the positive integers which have no descriptive meaning for finite man. When a man proves a positive integer to exist, he should show how to find it."
"A set is not an entity which has an ideal existence: a set exists only when it has been defined. To define a set we prescribe, at least implicitly, what we [..] must do in order to construct an element of the set, and what we must do in order to show that two elements of the set are equal."
Similarly, to define a function from set A to set B we must "prescribe a finite routine" leading from elements of A to those of B, which respects equality on these sets.
By thus building sets and functions from the integers all the structures required by mathematics are obtained.
".. every mathematical statement ultimately expresses the fact that if we perform certain computations within the set of positive integers, we shall get certain results." This does not exclude abstract mathematics, which can be justified in terms of its applicability in the more concrete branches.
2. The Idealist Component of Mathematics
Examples are given of non-constructive results in classical analysis, viz:
  • "that every non-void bounded set A of real numbers has a least upper bound"
  • "[that] every continuous real valued function f on the closed interval [0,1], with f(0)<0 and f(1)>0, vanishes as some point"
A discussion of Brouwer's contribution to the fight against idealism in mathematics follows.
3. The Constructivisation of Mathematics


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