More Precisely: The Math You Need to Do Philosophy

250 Pages·2017·7.51 MB·English

by  Eric Steinhart

MORE
PRECISELY
The Math You Need
to Do Philosophy

More Precisely
The Math You Need to Do Philosophy
Second Edition
Eric Steinhart
broadview press
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Library and Archives Canada Cataloguing in Publication
Steinhart, Eric, author
More precisely : the math you need to do philosophy / Eric
Steinhart. - Second edition.
Includes bibliographical references and index.
ISBN 978-1-55481-345-2 (softcover)
I. Mathematics-Philosophy. 2. Philosophy. I. Title.
QA8.4.S84 2017 5IO.l C2017-906154-2
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Edited by Robert M. Martin
Cover design by Aldo Fierro
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Contents
Prefa ce ............................................................................................. x
1. SETS ............................................................................................ 1
1. Collections of Things .................................................................................... l
2. Sets and Members ......................................................................................... .2
3. Set Builder Notation ...................................................................................... 3
4. Subsets .......................................................................................................... .4
5. Small Sets ...................................................................................................... 6
6. Unions of Sets ................................................................................................ 7
7. Intersections of Sets ....................................................................................... 8
8. Difference of Sets ······••.•···············································································lO
9. Set Algebra .................................................................................................. 10
10. Sets of Sets ............................................................................................... .11
I 11. Union of a Set of Sets ............................................................................... .12
12. Power Sets ................................................................................................. 13
13. Sets and Selections ................................................................................. :. . 15
14. Pure Sets .................................................................................................... 16
i
15. Sets and Numbers ...................................................................................... 19
16. Sums of Sets of Numbers ......................................................................... .20
17. Ordered Pairs ............................................................................................. 21
18. Ordered Tuples ................. :. ....................................................................... 22
19. Cartesian Products ..................................................................................... 23
2. RELATIONS ............................................................................2 5
',_·, 1. Relations ...................................................................................................... 25
2. Some Features of Relations ......................................................................... 26
3. Equivalence Relations and Classes ............................................................. 27
I 4. Closures of Relations .................................................................................. .29
5. Recursive Definitions and Ancestrals ......................................................... 32
. I 6. Personal Persistence ................................................................................... .33
j 6.1 The Diachronic Sameness Relation ...................................................... .33
I 6.2 The Memory Relation ............................................................................ 34
6.3 Symmetric then Transitive Closure ....................................................... 35
6.4 The Fission Problem .............................................................................. 38
6.5 Transitive then Symmetric Closure ...................................................... .40
7. Closure under an Operation ........................................................................ .42
8. Closure under Physical Relations ............................................................... .42
9. Order Relations ........................................................................................... .44
IO. Degrees of Perfection ............................................................................... .45
I I . Parts of Sets .............................................................................................. .48
vi Contents
12. Functions ...................................................................................................4 9
13. Some Examples of Functions .................................................................... 52
14. Isomorphisms ............................................................................................ 55
15. Functions and Sums ................................................................................... 59
16. Sequences and Operations on Sequences .................................................. 60
17. Cardinality ................................................................................................. 61
18. Sets and Classes ......................................................................................... 62
3. MACHINES ..............................................................................6 5
1 . Machines ...................................................................................................... 65
2. Finite State Machines .................................................................................. 65
2.1 Rules for Machines ................................................................................6 5
2.2 The Careers of Machines ....................................................................... 68
2.3 Utilities of States and Careers ............................................................... 70
3. The Game of Life ........................................................................................ 71
3.1 A Universe Made from Machines ......................................................... 71
3.2 The Causal Law in the Game of Life .................................................... 72
3.3 Regularities in the Causal Flow .............................................................7 3
3 .4 Constructing the Game of Life from Pure Sets ..................................... 7 5
4. Turing Machines ........................................................................................ :.78
5. Lifelike Worlds ............................................................................................ 81
4. SEMANTICS ........................................................................... .85
,- I
1. Extensional Semantics ............................................................. ,. .................. 85
1.1 Words and Referents ............................................................................. 85 F·1 '
1.2 A Sample Vocabulary and Model ......................................................... 88
l
1.3 Sentences and Truth-Conditions .......... , ................................................. 89 ,
..
2. Simple Modal Semantics ............................................................................. 92
2.1 Possible Worlds ..................................................................................... 92
2.2 A Sample Modal Structure .................................................................... 95
2.3 Sentences and Truth at Possible Worlds ............................................... 95
2.4 Modalities .............................................................................................. 97
2.5 lntensions ............................................................................................... 99
2.6 Propositions ........................................................................................... 99
3. Modal Semantics with Counterparts ........................................................ .101
3.1 The Counterpart Relation ................................................................... .101
3.2 A Sample Model for Counterpart Theoretic Semantics ..................... .102
3.3 Truth-Conditions for Non-Modal Statements ..................................... 103
3 .4 Truth-Conditions for Modal Statements ............................................. .105
5. PROBABILITY ..................................................................... 108
l. Sample Spaces ........................................................................................... 108
Contents vii
2. Simple Probability ..................................................................................... 109
3. Combined Probabilities ............................................................................ .111
4. Probability Distributions ........................................................................... 113
5. Conditional Probabilities .......................................................................... .115
5.1 Restricting the Sample Space .............................................................. 115
5 .2 The Definition of Conditional Probability .......................................... 116
5.3 An Example Involving Marbles .......................................................... 117
5.4 Independent Events ............................................................................. 118
6. Bayes Theorem .......................................................................................... 119
6 .1 The First Form of Bayes Theorem ..................................................... .119
6.2 An Example Involving Medical Diagnosis ........................................ .119
6.3 The Second Form of Bayes Theorem ................................................. .121
6.4 An Example Involving Envelopes with Prizes ................................... .123
7. Degrees of Belief ....................................................................................... 124
7 .1 Sets and Sentences ............................................................................... 124
7 .2 Subjective Probability Functions ........................................................ .125
8. Bayesian Confirmation Theory ................................................................ .127
8.1 Confirmation and Disconfirmation ...................................................... 127
8.2 Bayesian Conditionalization ................................................................ 128
9. Know ledge and the Flow of Information ................................................. .129
6. INFORMATION THEORY ................................................. 131
l . Communication ......................................................................................... 131
2. Exponents and Logarithms ....................................................................... .132
3. The Probabilities of Messages ................................................................... 134
4. Efficient Codes for Communication .......................................................... 134
4.1 A Method for Making Binary Codes .................................................. .134
4.2 The Weight Moving across a Bridge .................................................. .137
4.3 The Information Flowing through a Channel ..................................... .139
4.4 Messages with Variable Probabilities ................................................. 140
4.5 Compression ........................................................................................ 141
4.6 Compression Using Huffman Codes .................................................. .142
5. Entropy ...................................................................................................... 144
5.1 Probability and the Flow oflnformation ............................................ .144
5.2 Shannon Entropy ................................................................................. 145
5.3 Entropy in Aesthetics ......................................................................... .147
5.4 Joint Probability ................................................................................... 147
5.5 Joint Entropy ......................................................................, . ................ 148
6. Mutual Information ................................................................................... 150
6.1 From Joint Entropy to Mutual Information ........................................ .150
6.2 From Joint to Conditional Probabilities ............................................. .151
6.3 Conditional Entropy ............................................................................ 152
6.4 From Conditional Entropy to Mutual Information .............................. 153
6.5 An Illustration of Entropies and Codes .............................................. .154
7. Information and Mentality ........................................................................ .160
7. l Mutual Information and Mental Representation ................................. 160
Vlll Contents
7.2 Integrated Information Theory and Consciousness ............................ .160
7. DECISIONS AND GAMES .................................................. 163
1. Act Utilitarianism ...................................................................................... 163
1.1 Agents and Actions ............................................................................. .163
1.2 Actions and Their Consequences ....................................................... .164
1.3 Utility and Moral Quality ................................................................... .165
2. From Decisions to Games ........................................................................ .165
2.1 Expected Utility ................................................................................... 165
2.2 Game Theory ....................................................................................... 167
2.3 Static Games ........................................................................................ 168
3. Multi-Player Games ................................................................................... 170
3 .1 The Prisoner's Dilemma ...................................................................... 170
3 .2 Philosophical Issues in the Prisoner's Dilemma ................................. 171
3.3 Dominant Strategies ............................................................................ 172
3.4 The Stag Hunt ...................................................................................... 175
3.5 Nash Equilibria .................................................................................... 176
4. The Evolution of Cooperation .................................................................. .177
4 .1 The Iterated Prisoner's Dilemma ........................................................ .177
4.2 The Spatialized Iterated Prisoner's Dilemma ..................................... .180
4.3 Public Goods Games ........................................................................... 181
4.4 Games and Cooperation ..................................................................... .183
8. FROM THE FINITE TO THE INFINITE .......................... 186
1. Recursively Defined Series ...................................................................... .186
2. Limits of Recursively Defined Series ....................................................... .188
2.1 Counting through All the Numbers ..... ). ............................................. .188
2.2 Cantor's Three Number Generating Rules ......................................... .190
2.3 The Series of Von Neumann Numbers ................................................ 190
3. Some Examples of Series with Limits ...................................................... .191
3.1 Achilles Runs on Zeno's Racetrack ................................................... .191
3 .2 The Royce Map ................................................................................... 192
3 .3 The Hilbert Paper ............................................................................... .193
3 .4 An Endless Series of Degrees of Perfection ....................................... .194
4. Infinity ....................................................................................................... 195
4.1 Infinity and Infinite Complexity ......................................................... .195
4.2 The Hilbert Hotel ................................................................................. 197
4.3 Operations on Infinite Sequences ....................................................... .198
5. Supertasks .................................................................................................. 198
5.1 Reading the Borges Book .................................................................... 198
5 .2 The Thomson Lamp ............................................................................ 199
5 .3 Zeus Performs a Super-Computation ................................................. .200
5 .4 Accelerating Turing Machines ........................................................... .200
Contents ix
9. BIGGER INFINITIES ...........................................................2 02
1. Some Transfinite Ordinal Numbers .......................................................... .202
2. Comparing the Sizes of Sets ......................................................................2 03
3. Ordinal and Cardinal Numbers .................................................................. 205
4. Cantor's Diagonal Argument .................................................................... 207
5. Cantor's Power Set Argument ................................................................... 211
5 .1 Sketch of the Power Set Argument ..................................................... 211
5.2 The Power Set Argument in Detail ..................................................... 213
5.3 The Beth Numbers ............................................................................... 214
6. The Aleph Numbers .................................................................................. 216
7. Transfinite Recursion ............................................................................... .217
7 .1 Rules for the Long Line ....................................................................... 217
7 .2 The Sequence of Universes ................................................................. 218
7 .3 Degrees of Di vine Perfection .............................................................. 219
Further Study .............................................................................2 21
Glossary of Symbols ...................................................................2 23
References ...................................................................................2 26
Index ............................................................................................2 31
Preface
Anyone doing philosophy today needs to have a sound understanding of a wide
range of basic mathematical concepts. Unfortunately, most applied mathematics
texts are designed to meet the needs of scientists. And much of the math used in
the sciences is not used in philosophy. You're looking at a mathematics book
that's designed to meet the needs of philosophers. More Precisely introduces
the mathematical concepts you need in order to do philosophy today. As we
introduce these concepts, we illustrate them with many classical and recent
philosophical examples. This is math for philosophers.
It's important to understand what More Precisely is and what it isn't. More
Precisely is not a philosophy of mathematics book. It's a mathematics book.
We're not going to talk philosophically about mathematics. We are going to
teach you the mathematics you need to do philosophy. We won't enter into any
of the debates that rage in the current philosophy of math. Do abstract objects
exist? How do we have mathematical knowledge? We don't deal with those
issues here. More Precisely is not a logic book. We're not introducing you to
any logical calculus. We won't prove any theorems. You can do a lot of math
with very little logic. If you know some propositional or predicate logic, it
won't hurt. But even if you've never heard of them, you'll do just fine here.
More Precisely is an introductory book. It is not an advanced text. It aims to
cover the basics so that you're prepared to go into the depths on the topics that
are of special interest to you. To follow what we're doing here, you don't need
anything beyond high school mathematics. We introduce all the technical
notations and concepts gently and with many examples. We'll draw our
examples from many branches of philosophy - including metaphysics,
philosophy of mind, philosophy of language, epistemology, ethics, and
philosophy of religion.
It's natural to start with some set theory. All branches of philosophy today
.make some use of set theory. If you want to ht: able to follow what's going on
in philosophy today, you need to master at least the basic language of set theory.
You need to understand the specialized notation and vocabulary used to talk
about sets. For example, you need to understand the concept of the intersection
of two sets, and to know how it is written in the specialized notation of set
theory. Since we're not doing philosophy of math, we aren't going to get into
any debates about whether or not sets exist. Before getting into such debates,
you need to have a clear understanding of the objects you're arguing about. Our
purpose in Chapter 1 is to introduce you to the language of sets and the basic
ideas of set theory. Chapter 2 introduces relations and functions. Basic set
theoretic notions, especially relations and functions, are used extensively in the
later chapters. So if you're not familiar with those notions, you've got to start
with Chapters 1 and 2. Make sure you've really mastered the ideas in Chapters
1 and 2 before going on.
After we discuss basic set-theoretic concepts, we go into concepts that are
used in various branches of philosophy. Chapter 3 introduces machines. A
machine (in the special sense used in philosophy, computer science, and
mathematics) isn't an industrial device. It's a formal structure used to describe