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

The latest decade

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

---

© created 1995-3-18 modified 1997-2-3