L. E. J. Brouwer, the founder of mathematical intuitionism, believed that mathematics and its objects must be humanly graspable. He initiated a program rebuilding modern mathematics according to that principle. This book introduces the reader to the mathematical core of intuitionism – from elementary number theory through to Brouwer's uniform continuity theorem – and to the two central topics of 'formalized intuitionism': formal intuitionistic logic, and formal systems for intuitionistic analysis. Building on that, the book proposes a systematic, philosophical foundation for intuitionism that weaves together doctrines about human grasp, mathematical objects and mathematical truth.
Author(s): Carl J. Posy
Publisher: Cambridge University Press
Year: 2020
Language: English
Pages: 116
Cover
Mathematical Intuitionism
Contents
1 Introduction: Three Faces of Intuitionism
1.1 The Mathematical Face of Intuitionism
1.2 The Logical Face
1.3 Philosophy
1.4 Preview
2 The Mathematical Face of Intuitionism
2.1 Classical Foil and Formalist Foe
2.1.1 The Cantorian Continuum
Natural Numbers
Integers and Rationals
ℝ and its Fine Structure
Order
Topology
Functions
2.1.2 Consequences
Universal Ontology
2.1.3 Conceptual Problems
Problem One: A Brittle Continuum
Problem Two: Infinity
Hilbert’s Programme
Brouwer
The Negative Programme
The Positive Programme
2.2 A Primer of Intuitionistic Mathematics
2.2.1 Finite Numbers
Natural Numbers
Integers and Rational Numbers
2.2.2 Real Numbers and the Structure of the Continuum
Species
Infinite Sequences and Real Numbers
Real-Number Arithmetic
The Measurable Order Relation <ℝ
Virtual Order
Topology
2.2.3 Non-determinate Sequences
Choice Sequences
The Creating Subject
Step 1
Step 2
Step 3
Step 4
2.2.4 Functions
Spreads
The Binary Spread MB
The Spread Mℝ
The Spread M½0;1
The Spread Mð0;1Þ
Functions and Continuity
Uniform Continuity
Three Consequences of the Fan Theorem
Two Final Remarks about Brouwer and Continuity
Bar Induction and the Bar Theorem
Brouwer’s Mathematical Coup
3 Formalised Intuitionism
3.1 Formal Logic and Number Theory
3.1.1 Heyting’s System IL
Axioms
Rules
3.1.2 Formal Intuitionistic Number Theory
Axioms for Identity
Number Theoretic Axioms
3.1.3 Syntactic Comparisons
3.1.4 Semantics for ISL
Topological Semantics
Model Theoretic Semantics for ISL
3.1.5 Quantificational Kripke Models
3.2 Formalising Analysis
3.2.1 The System for Lawlike Sequences
Lawlike Sequences
The System EL
3.2.2 Finite-Input Functionals
The Formal System IDB
3.2.3 Systems for Choice Sequences
Lawless Sequences
The System LS
Extending K’s Reach
Intermediate Notions of Choice Sequence
The System CS
The Elimination Theorem
3.2.4 The Creating Subject
Kripke’s Schema
Refutation of the ‘Intuitionistic Church’s Thesis’
Refutation of the 8α9β Continuity Principle
3.2.5 Real Analysis and Recursion
Recursive Realisability
Recursive Analysis
4 The Intuitionistic Standpoint
4.1 Experience and Knowledge
4.1.1 Phenomenology of Intuitive Grasp
Empirical Grasp
Mathematical Grasp
The First Act of Intuitionism: Grasping Numbers and Other Finite
Objects
The Second Act of Intuitionism: Grasping IPS’s and Species
IPS’s and Spreads
Species
The Intuition of the Continuum
4.1.2 Reflection on Intuition
Nature and Limits of Knowledge
A Priority
4.1.3 Hilbert and Husserl on Mathematical Intuition
Hilbert
Husserl
4.2 Ontology
4.2.1 Brouwer versus Heyting on Ontology
4.2.2 Intuitionistic Proto-Ontology
Finite Objects
Processes
Species
4.3 Truth and Logic
4.3.1 Heyting’s Proof Interpretation
4.3.2 Dummett
4.3.3 Brouwer’s Semantic Insight
4.4 The Standpoint of Intuitionism
4.4.1 Finitude and the Inner Perspective
Finitude
4.4.2 Indeterminacy
Varieties of Indeterminacy
Brouwer’s Negative Continuity Theorem
4.5 In Sum
Afterword
Overall Surveys
Brouwer-Focused Works
Intuitionism and Physics
Afterword
Bibliography
Acknowledgements
