by
on
Foundations of Constructive Mathematics
 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:
 "there exists an x" means we can find x explicitly
 "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 nonconstructive 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. MartinLöf's Philosophy

 8. Church's Thesis

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created 1995318 modified 199723