Mathematics for Humanists

✍ Scribed by Herbert Gintis

• Publisher: University of Massachusetts
• Year: 2021
• Tongue: English
• Leaves: 123
• Category: Library

✦ Synopsis

This book is for people who believe that those who study the world using formal mathematical models might have something to say to them, but for whom mathematical formalism is a foreign language that they do not understand. More generally it is for anyone who is curious about the nature of mathematical and logical thought and their relationship to society and to the universe. I say relationship to society because we will eventually use what we have learned to understand how probability and game theory can be used to gain insight into human behavior, to understand why we have been so successful as a species, and to assess how likely our success in spreading across the world with also be our undoing. I say relationship to the universe because you will see how a few simple equations have a stunning ability to explain the physical processes that have mystified our ancestors for ages.

✦ Table of Contents

Preface xii
1 Reading Math 1
1.1 Reading Math 1
2 The Language of Logic 2
2.1 The Language of Logic 2
2.2 Formal Propositional Logic 4
2.3 Truth Tables 5
2.4 Exercises in Propositional Logic 7
2.5 Predicate Logic 8
2.6 Proving Propositions in Predicate Logic 9
2.7 The Perils of Logic 10
3 Sets 11
3.1 Set Theory 11
3.2 Properties and Predicates 12
3.3 Operations on Sets 14
3.4 Russell’s Paradox 15
3.5 Ordered Pairs 17
3.6 Mathematical Induction 18
3.7 Set Products 19
3.8 Relations and Functions 20
3.9 Properties of Relations 21
3.10 Injections, Surjections, and Bijections 22
3.11 Counting and Cardinality 23
3.12 The Cantor-Bernstein Theorem 24
3.13 Inequality in Cardinal Numbers 25
3.14 Power Sets 26
3.15 The Foundations of Mathematics 27
4 Numbers 30
4.1 The Natural Numbers 30
4.2 Representing Numbers 30
4.3 The Natural Numbers and Set Theory 31
4.4 Proving the Obvious 33
4.5 Multiplying Natural Numbers 35
4.6 The Integers 36
4.7 The Rational Numbers 38
4.8 The Algebraic Numbers 39
4.9 Proof of the Fundamental Theorem of Algebra 41
4.10 The Real Numbers 47
4.11 Denumerability and the Reals 53
5 Number Theory 56
5.1 Modular Arithmetic 56
5.2 Prime Numbers 57
5.3 Relatively Prime Numbers 57
5.4 The Chinese Remainder Theorem 59
6 Probability Theory 60
6.1 Introduction 60
6.2 Probability Spaces 60
6.3 De Morgan’s Laws 61
6.4 Interocitors 61
6.5 The Direct Evaluation of Probabilities 61
6.6 Probability as Frequency 62
6.7 Craps 63
6.8 A Marksman Contest 63
6.9 Sampling 63
6.10 Aces Up 64
6.11 Permutations 64
6.12 Combinations and Sampling 65
6.13 Mechanical Defects 65
6.14 Mass Defection 65
6.15 House Rules 65
6.16 The Addition Rule for Probabilities 66
6.17 A Guessing Game 66
6.18 North Island, South Island 66
6.19 Conditional Probability 67
6.20 Bayes’ Rule 67
6.21 Extrasensory Perception 68
6.22 Les Cinq Tiroirs 68
6.23 Drug Testing 68
6.24 Color Blindness 69
6.25 Urns 69
6.26 The Monty Hall Game 69
6.27 The Logic of Murder and Abuse 69
6.28 The Principle of Insufficient Reason 70
6.29 The Greens and the Blacks 70
6.30 The Brain and Kidney Problem 70
6.31 The Value of Eyewitness Testimony 71
6.32 When Weakness Is Strength 71
6.33 The Uniform Distribution 74
6.34 Laplace’s Law of Succession 75
6.35 From Uniform to Exponential 75
7 Calculus 76
7.1 Infinitesimal Numbers 76
7.2 The Derivative of a Function 77
8 Vector Spaces 80
8.1 The Origins of Vector Space Theory 80
8.2 The Vector Space Axioms 82
8.3 Norms on Vector Spaces 83
8.4 Properties of Norm and Inner Product 83
8.5 The Dimension of a Vector Space 84
8.6 Vector Subspaces 86
8.7 Revisiting the Algebraic Numbers 87
9 Groups 90
9.1 Groups 90
9.2 Permutation Groups 91
9.3 Groups of Units 93
9.4 Groups: Basic Properties 93
9.5 Product Groups 95
9.6 Subgroups 96
9.7 Factor Groups 97
9.8 The Fundamental Theorem of Abelian Groups 98
10 Real Analysis 101
10.1 Limits of Sequences 101
10.2 Compactness and Continuity in R 103
10.3 Powers and Logarithms 105
11 Table of Symbols 108
References 110
Index 110