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Mathematics for Humanists Mathematics for Humanists Mathematics for Humanists Herbert Gintis xxxxxxxxx xxxxxxxxxx Press xxxxxxxxx and xxxxxx Copyright c 2021 by ... Published by ... All Rights Reserved Library of Congress Cataloging-in-Publication Data Gintis, Herbert Mathematics for Humanists/ Herbert Gintis p. cm. Includes bibliographical references and index. ISBN ...(hardcover: alk. paper) HB... xxxxxxxxxx British Library Cataloging-in-Publication Data is available The publisher would like to acknowledge the author of this volume for providing the camera-ready copy from which this book was printed This book has been composed in Times and Mathtime by the author Printed on acid-free paper. Printed in the United States of America 10987654321 This book is dedicated to my mathematics teachers: Pincus Shub, Walter Gottschalk, Abram Besikovitch, and Oskar Zariski Contents Preface xii 1 ReadingMath 1 1.1 Reading Math 1 2 The LanguageofLogic 2 2.1 TheLanguageofLogic 2 2.2 FormalPropositionalLogic 4 2.3 Truth Tables 5 2.4 ExercisesinPropositionalLogic 7 2.5 Predicate Logic 8 2.6 ProvingPropositionsinPredicateLogic 9 2.7 ThePerilsofLogic 10 3 Sets 11 3.1 Set Theory 11 3.2 PropertiesandPredicates 12 3.3 OperationsonSets 14 3.4 Russell’s Paradox 15 3.5 Ordered Pairs 17 3.6 MathematicalInduction 18 3.7 SetProducts 19 3.8 RelationsandFunctions 20 3.9 PropertiesofRelations 21 3.10 Injections,Surjections,andBijections 22 3.11 CountingandCardinality 23 3.12 The Cantor-Bernstein Theorem 24 3.13 InequalityinCardinalNumbers 25 3.14 Power Sets 26 3.15 TheFoundationsofMathematics 27 viii Contents 4 Numbers 30 4.1 TheNaturalNumbers 30 4.2 RepresentingNumbers 30 4.3 TheNaturalNumbersandSetTheory 31 4.4 ProvingtheObvious 33 4.5 MultiplyingNaturalNumbers 35 4.6 The Integers 36 4.7 TheRationalNumbers 38 4.8 TheAlgebraicNumbers 39 4.9 ProofoftheFundamentalTheoremofAlgebra 41 4.10 TheRealNumbers 47 4.11 DenumerabilityandtheReals 53 5 Number Theory 56 5.1 ModularArithmetic 56 5.2 PrimeNumbers 57 5.3 RelativelyPrimeNumbers 57 5.4 The Chinese Remainder Theorem 59 6 Probability Theory 60 6.1 Introduction 60 6.2 ProbabilitySpaces 60 6.3 De Morgan’s Laws 61 6.4 Interocitors 61 6.5 TheDirectEvaluationofProbabilities 61 6.6 ProbabilityasFrequency 62 6.7 Craps 63 6.8 A Marksman Contest 63 6.9 Sampling 63 6.10 AcesUp 64 6.11 Permutations 64 6.12 CombinationsandSampling 65 6.13 Mechanical Defects 65 6.14 Mass Defection 65 6.15 House Rules 65 6.16 TheAdditionRuleforProbabilities 66 6.17 AGuessingGame 66 6.18 NorthIsland,SouthIsland 66 Contents ix 6.19 ConditionalProbability 67 6.20 Bayes’ Rule 67 6.21 ExtrasensoryPerception 68 6.22 Les Cinq Tiroirs 68 6.23 Drug Testing 68 6.24 Color Blindness 69 6.25 Urns 69 6.26 TheMontyHallGame 69 6.27 TheLogicofMurderandAbuse 69 6.28 ThePrincipleofInsufficientReason 70 6.29 TheGreensandtheBlacks 70 6.30 TheBrainandKidneyProblem 70 6.31 The Value of Eyewitness Testimony 71 6.32 WhenWeaknessIsStrength 71 6.33 TheUniformDistribution 74 6.34 Laplace’s Law of Succession 75 6.35 FromUniformtoExponential 75 7 Calculus 76 7.1 InfinitesimalNumbers 76 7.2 TheDerivativeofaFunction 77 8 Vector Spaces 80 8.1 TheOriginsofVectorSpaceTheory 80 8.2 TheVectorSpaceAxioms 82 8.3 NormsonVectorSpaces 83 8.4 PropertiesofNormandInnerProduct 83 8.5 TheDimensionofaVectorSpace 84 8.6 Vector Subspaces 86 8.7 RevisitingtheAlgebraicNumbers 87 9 Groups 90 9.1 Groups 90 9.2 PermutationGroups 91 9.3 GroupsofUnits 93 9.4 Groups:BasicProperties 93 9.5 ProductGroups 95 9.6 Subgroups 96 x Contents 9.7 Factor Groups 97 9.8 TheFundamentalTheoremofAbelianGroups 98 10 Real Analysis 101 10.1 LimitsofSequences 101 10.2 Compactness and Continuity in R 103 10.3 PowersandLogarithms 105 11 Table of Symbols 108 References 110 Index 110 Preface The eternal mystery of the world is its comprehensibility. Albert Einstein 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 un- derstand. 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 probabilityand 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. Often mathematics is justified by its being useful, and it is. But useful things can be beautiful, and things can be beautiful without be useful at all. This is something the humanists have taught us. I stress in this book that mathematics is beautiful even if we don’t care a whit about its usefulness. Mathematics is beautiful in the same way that music, dance, and literature are beautiful. Mathematics may even be more beautiful because it takes a lot of work to appreciate it. A lot of work. If you get through this book, you will have done a lot of work. But, on the other hand, you will have acquired the capacity to envision worlds you never even dreamed were there. Worlds of logical and algebraic structures. More times than I care to remember I have heard intelligent people pro- claim that they are awful at math, that they hate it, and that they never use it. In so doing, they project an air of imperfectly concealed self-satisfaction. I usually smile and let it pass. I know that these people expect to be admired for their capacity to avoid the less refined of life’s activities. In fact, I do feel sympathy for these friends. I let their judgments on the topic pass without comment, rather than inform them of the multitudinous pleasurable insights their condition precludes their enjoying. Thinking that xii Preface xiii mathematics is calculating is about as silly as thinking that painting a land- scape is like painting a woodshed, that ballet is aerobic exercise, or that singing is simply setting the air in vibration. This book treats mathematics as a language that fosters certain forms of truthful communication that are difficult to express without specialized symbols. Some mathematicians are fond of saying that anything worth ex- pressing can be expressed in words. Perhaps. But that does not mean any- thing can be understood in words. Even the simplest mathematical state- ment would take many thousands of words to write out in full. I assume that if you are reading this book, your distaste for the math you learned is school is due to the fact that school math stresses mindless com- putation and rote learning of how to do a fixed set of traditional problems. I assume your distaste is not because you are lacking in intelligence or will- ingness to work hard. This book is very challenging and it will make you sweat, especially if you take it seriously. If you just want bedtime reading, go somewhere else. 1 Reading Math 1.1 Reading Math Reading math is not like reading English. In reading a novel, a history book, or the newspaper, you can read a sentence, not understand it perfectly (perhaps there’s an ambiguous word, or you’re not sure what a pronoun refers to), and yet move on to the next sentence. In reading math, you must understand every expression perfectly or you do not understand it at all. Thus, you either understand something like equation (??) perfectly, or you don’t understand it at all. If the latter is the case, read the expression symbol by symbol until you come to the one that doesn’t make sense. Then find out exactly what it means before you go on. 1 2 The Language of Logic People are not logical. They are psychological. Unknown 2.1 The Languageof Logic The term true is a primitive of logic; i.e., we do not define the term ‘true’ in terms of more basic terms. The logic used in mathematical discourse has only three primitive terms in addition to ‘true’. These are ‘not,’ ‘and,’ and ‘for all.’ We define “false” as “not true,” and we define a propositional variable as an entity that can have the value either true or false, but not both. By the way, “true” is a propositional constant because, unlike a variable, its value cannot change. The only other propositional constant is “false.” In this chapter, we will use p, q, r, s and so on to represent propositional variables. In general, if p is a propositional variable, we define p (read “not p”) to be a propositional variable that is true exactly when p:is false. When we assert p, we are saying that p has the value true. From these, we can define all sorts of other logical terms. We when we assert “p and q”, we are asserting that both p and q are true. We write this logically as p q. The four terms true, false, not, and and are used in logic almost exactly^ as in natural language communication. Many other logical terms can be defined in terms of and and not. We define “p or q”, which we write as p q, to be a true when either p or q is true, and is false otherwise. We can define_ this in terms of “not” and “and” as . p q/. we call the inclusive or because in natural language “or” can: be: either^: inclusive or_ exclusive.
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