From Foundations to Philosophy of Mathematics : An Historical Account of Their Development in the XX Century and Beyond

238 Pages·2012·1.994 MB·English

by  Joan Roselló

From Foundations to Philosophy of Mathematics

From Foundations to Philosophy of Mathematics:
An Historical Account of their Development
in the XX Century and Beyond
By
Joan Roselló
From Foundations to Philosophy of Mathematics:
An Historical Account of their Development in the XX Century and Beyond,
by Joan Roselló
This book first published 2012
Cambridge Scholars Publishing
12 Back Chapman Street, Newcastle upon Tyne, NE6 2XX, UK
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
Copyright © 2012 by Joan Roselló
All rights for this book reserved. No part of this book may be reproduced, stored in a retrieval system,
or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or
otherwise, without the prior permission of the copyright owner.
ISBN (10): 1-4438-3459-9, ISBN (13): 978-1-4438-3459-9
TO MY FATHER

CONTENTS
Introduction.................................................................................................1
Chapter One.................................................................................................7
Frege’s Logic and Logicism
1.1 Fregean Logicism.............................................................................7
1.2 Frege’s Logic...................................................................................9
1.3 The Reduction of the Concept of Order in a Series.......................12
1.4 The Definition of the Concept of Number.....................................14
1.5 The Infinitude of the Natural Number Series.................................18
1.6 The Basic Laws of Arithmetic.......................................................20
1.7 Russell’s Paradox...........................................................................24
1.8 Conclusion.....................................................................................26
Chapter Two..............................................................................................29
Russell’s Logicism and Type Theory
2.1 Russell, Logicism and the Paradoxes.............................................29
2.2 The Transfinite Paradoxes and the No Classes Theory..................32
2.3 The Theory of Descriptions and the Vicious Circle Principle.......34
2.4 The Theory of Logical Types.........................................................38
2.5 Realism versus Constructivism......................................................41
2.6 The Notion of Predicativity and the Axiom of Reducibility..........45
2.7 The Extensional Hierarchy: The Multiplicative and Infinity
Axioms...........................................................................................48
2.8 Conclusion.....................................................................................50
Chapter Three............................................................................................55
Zermelo and the Origins of Axiomatic Set Theory
3.1 Cantor, Dedekind and the Origins of Set Theory..........................55
3.2 Cantor and Transfinite Number Theory.........................................58
3.3 Dedekind’s Logicism and Set Theory............................................60
3.4 The Origins of Zermelo’s Axiomatic Set Theory..........................62
3.5 First Definitions and Axioms.........................................................65
3.6 The Axiom of Separation and Russell’s Paradox...........................67
3.7 The Axioms of Choice and of Infinity...........................................70
3.8 The Cumulative Hierarchy of Sets.................................................72
3.9 Conclusion.....................................................................................74
viii Contents
Chapter Four..............................................................................................79
Brouwer’s Intuitionism and the Foundational Debate
4.1 Intuitionism and Brouwerian Intuitionism.....................................79
4.2 The Primordial Intuition of Time...................................................81
4.3 Mathematics, Language and Logic................................................83
4.4 The Unreliability of the Principle of Excluded Middle..................86
4.5 The Continuum..............................................................................89
4.6 Intuitionist Mathematics................................................................92
4.7 Intuitionist Set Theory...................................................................95
4.8 The Grundlagenstreit: Intuitionism versus Formalism..................98
4.9 Conclusion...................................................................................103
Chapter Five............................................................................................107
Hilbert’s Formalism and Finitism
5.1 Hilbert’s Early Career.................................................................107
5.2 The Geometrical Context............................................................109
5.3 Hilbert’s Grundlagen der Geometrie..........................................111
5.4 Formalism...................................................................................113
5.5 The Foundations of Arithmetic...................................................116
5.6 Logic and Logicism....................................................................119
5.7 Finitism.......................................................................................123
5.8 Proof Theory...............................................................................126
5.9 Conclusion..................................................................................130
Chapter Six..............................................................................................133
Poincaré, Weyl and Predicativism
6.1 Poincaré’s Intuitionism and the Predicativist Constraint.............133
6.2 Russell, Poincaré and the Emergence of Predicativism...............136
6.3 Poincaré against the Logicists, Formalists and Cantorians..........140
6.4 Weyl’s Contributions to the Foundations of Mathematics...........145
6.5 Das Kontinuum (I): Foundational Issues......................................148
6.6 Das Kontinuum (II): The Reconstruction of Analysis..................151
6.7 Conclusion...................................................................................153
From Foundations to Philosophy of Mathematics ix
Chapter Seven..........................................................................................155
Gödel’s Incompleteness Theorems and Platonism
7.1 Gödel and Hilbert’s Program.......................................................155
7.2 Completeness and Incompleteness...............................................157
7.3 Peano’s Arithmetic and Sentences of Goldbach’s Type..............160
7.4 The First Incompleteness Theorem..............................................165
7.5 The Second Incompleteness Theorem and the Incompletability
of Mathematics............................................................................169
7.6 The Consistency of the Axiom of Choice and the Continuum
Hypothesis....................................................................................173
7.7 Platonism or Conceptual Realism................................................176
7.8 Conclusion...................................................................................179
Chapter Eight...........................................................................................183
New Perspectives in the Philosophy of Mathematics: The Foundational
Programs Revisited
8.1 Neologicism.................................................................................183
8.2 Constructivism.............................................................................188
8.3 Second-Order Arithmetic.............................................................192
8.4 Subsystems of PA and Reverse Mathematics.............................194
2
8.5 Finitist Reductionism...................................................................197
8.6 Predicativist Reductionism..........................................................199
8.7 Coda.............................................................................................201
References...............................................................................................207
Subject Index...........................................................................................221