Mathematics for Humanists

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