Philosophical Perspectives on Infinity

✍ Scribed by Graham Oppy

• Publisher: Cambridge University Press
• Year: 2006
• Tongue: English
• Leaves: 336
• Category: Library

✦ Synopsis

This book is an exploration of philosophical questions about infinity. Graham Oppy examines how the infinite lurks everywhere, both in science and in our ordinary thoughts about the world. He also analyses the many puzzles and paradoxes that follow in the train of the infinite. Even simple notions, such as counting, adding and maximising present serious difficulties. Other topics examined include the nature of space and time, infinities in physical science, infinities in theories of probability and decision, the nature of part/whole relations, mathematical theories of the infinite, and infinite regression and principles of sufficient reason.

✦ Table of Contents

Cover
Half-title
Title
Copyright
Contents
Preface
Acknowledgments
Introduction
1 Beginnings and Puzzles
2 Mathematical Preliminaries
2.1 Set Theory
2.2 Numbers
(a) Ordinals
(b) Cardinals
2.3 Cantor's Paradise
(a) Infinite Ordinals and Infinite Cardinals
(b) Cantorian Arguments
(c) Arithmetic for Infinite Ordinals and Infinite Cardinals
(d) Additional Axioms?
2.4 Standard Analysis
(a) Rational Numbers and Real Numbers
(b) Continuity and Convergence
(c) Differentiation
(d) Integration
2.5 Nonstandard Numbers
(a) Model Theory
(b) Internal Set Theory
(c) Conway Numbers
(d) Finite Mathematics
3 Some Cases Discussed
3.1 Al-ghazali’s Problem
3.2 Hilbert’s Hotel
3.3 craig’s library
3.4 tristram shandy
3.5 counting from infinity
3.6 infinite paralysis
3.7 stick
3.8 spaceship
3.9 thomson’s lamp
3.10 black’s marble shifter
3.11 pi machine
3.12 goldbach machine
3.13 ross urn
3.14 deafening peals
3.15 invisibility
3.16 infinity mob
3.17 string
3.18 some concluding remarks
4 Space, Time, and Spacetime
4.1 zeno’s paradoxes
4.2 grunbaum’s metrical puzzle
(a) Dimension
(b) Measure
(c) A Zeno-Style Argument
4.3 skyrms’s measure puzzle
4.4 points, regions, and finite lattices
(a) Regions as Primitives
(b) Relations as Primitive
(c) Fictionalism
(d) Supervenience
(e) Lattices and Graphs
4.5 first kantian antinomy
(a) The world has a beginning in time
(b) The World is Spatially Finite
(c) The World has no Beginning in Time
(d) The World is Spatially Infinite
Conclusion
4.6 infinity machines in relativistic spacetimes
(a) Pitowsky Spacetimes
(b) Malament-Hogarth Spacetimes
4.7 singularities
4.8 concluding remarks
5 Physical Infinities
5.1 hotter than infinite temperatures
5.2 extensive magnitudes
5.3 infinite extensive magnitudes?
5.4 renormalisation
5.5 the dark night sky
5.6 some more general considerations
5.7 possibilities
5.8 physical possibilities
6 Probability and Decision Theory
6.1 probabilities
6.2 additivity principles
6.3 Decision Theory
(a) Decisions under Uncertainty
(b) Decisions under Risk
6.4 Approaching Infinite Decision Theory
6.5 infinite utility streams
6.6 infinite decision rules
6.7 Two Envelopes
6.8 st. petersburg game
6.9 heaven and hell
6.10 concluding remarks
7 Mereology
7.1 second kantian antinomy
(a) On Behalf of Simples
(b) Against Simples
7.2 some postulates
7.3 forrest’s challenge
7.4 taking stock
7.5 atoms and indiscernible particles
7.6 vagueness and infinite divisibility
7.7 continuity
7.8 our universe
8 Some Philosophical Considerations
8.1 distinctions
8.2 philosophies of pure mathematics
8.3 knowing the infinite
8.4 putting classical mathematics first
8.5 extrapolation from finite mathematics
8.6 some notes about set theory
8.7 philosophies of applied mathematics
8.8 skolem’s paradox
8.9 infinitesimals
8.10 concluding remarks
9 Infinite Regress and Sufficient Reason
9.1 strong principles of sufficient reason
9.2 self-explanation and sufficient reason
9.3 necessitation and sufficient reason
9.4 problems for strong principles of sufficient reason
9.5 weaker principles of sufficient reason
9.6 infinite regresses
9.7 concluding remarks
Conclusion
References
Index