# Foundations of Constructive Mathematics

### by Michael J. Beeson

Introduction
Part One. Practice and Philosophy of Constructive Mathematics
Part Two. Formal Systems of the Seventies
Part Three. Metamathematical Studies
Part Four. Metaphilosophical Studies.
Historical Appendix

## Part One - Practice and Philosophy of Constructive Mathematics

Chapter I - Examples of Constructive Mathematics
Chapter II - Informal Foundations of Constructive Mathematics
Chapter III - Some Different Philosophies of Constructive Mathematics
Having in common the two principles:
1. "there exists an x" means we can find x explicitly
2. "truth" has no a priori meaning; a proposition is true just in case we can find a proof of it.
Chapter IV - Recursive Mathematics: Living with Church's Thesis
Chapter V - The Role of Formal Systems in Foundational Studies

## Historical Appendix

From Gauss to Zermelo: the origins of non-constructive mathematics
From Kant to Hilbert: logic and philosophy
Brouwer and the Dutch Intuitionists
Early Formal Systems for Intuitionism
Kleene: the marriage of recursion theory and intuitionism
The Russian constructivists and recursive analysis
Model theory of Intuitionistic systems
Logical studies of intuitionistic systems
Bishop and his followers

## Part One - Practice and Philosophy of Constructive Mathematics

### Chapter III - Some Different Philosophies of Constructive Mathematics

1. The Russian Constructivists
Every mathematical object is (or is given by) a word in a finite alphabet. These are manipulated by Markov algorithms. Reasoning uses the intuitionistic predicate calculus + Markov's principle: for all x in R, x is not less than or equal to zero implies x is greater than zero.
2. Recursive Analysis
Similar to Russian constructivist but using Classical logic (a variant is intuitionist)
3. Bishops Constructivism
4. Objective Intuitionism
5. Sets in Intuitionism
6. Brouwerian Intuitionism
7. Martin-Löf's Philosophy
8. Church's Thesis