- 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.
- The Prolog is Bishop's original Preface.
In it he draws the distinction between two kinds of mathematical statement:
- Those which are "merely evocative, that make assertions without empirical validity."
- and those which are "of immediate empirical validity, which say that certain performable operations will produce certain observable results..".
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
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:
and three purposes:
- make every concept affirmative
- avoid definitions that are not relevant
- avoid pseudogenerality
- to present the constructive point of view
- to show that the constructive program can succeed
- 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:
A discussion of Brouwer's contribution to the fight against idealism in mathematics follows.
- "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"
- 3. The Constructivisation of Mathematics
created 95/4/8 modified 97/2/3