MORE PRECISELY The Math You Need to Do Philosophy More Precisely The MathYou Need to Do Philosophy Second Edition Eric Steinhart B broadview press BROADVIEWPRESS—www.broadviewpress.com Peterborough,Ontario,Canada Foundedin 1985,BroadviewPressremainsawhollyindependentpublishinghouse.Broadview’sfocus ison academicpublishing;ourtitlesareaccessibletouniversityandcollegestudentsaswellasscholarsandgeneral readers.Withover600titlesinprint,Broadviewhasbecomealeadinginternationalpublisherinthehumanities, withworld-widedistribution.Broadviewiscommittedtoenvironmentallyresponsiblepublishingandfairbusi- nesspractices. Theinteriorofthisbookisprinted (stunt Ancient on |00%recycledpaper. ∙16" :oresct“ ’ �∕∕�∕ rien y'“ ©20'8 EricSteinhart PERMANENT 100% 31:9:nGovAs km Allrightsreserved.Nopartofthisbookmaybereproduced,keptinaninformationstorageandretrievalsystem, ortransmitted inanyformorbyanymeans,electronicormechanical,includingphotocopying,recording,or otherwise,exceptasexpresslypermittedbytheapplicablecopyrightlawsorthroughwrittenpermission from thepublisher. LibraryandArchivesCanadaCataloguinginPublication Steinhart.Eric,author Moreprecisely:themathyouneedtodophilosophy/ Eric Steinhart.—Secondedition. Includesbibliographicalreferencesandindex. ISBN978-1-55481-345-2(softcover) l.Mathematics—Philosophy. 2.Philosophy. I.Title. 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BroadviewPressacknowledgesthefinancialsupportoftheGovernmentofCanadathroughtheCanadaBook Fundforourpublishingactivities. �∙� EditedbyRobertM.Martin Canada CoverdesignbyAldoFierro InteriortypesetbyEricSteinhart PRINTEDINCANADA Contents ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙ Preface x 1.CollectionsofThings ....................................................................................1 2.SetsandMembers..........................................................................................2 3.SetBuilder Notation............................3 4.Subsets...........................................................................................................4 5.SmallSets∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙6 6.UnionsofSets................................................................................................7 7.IntersectionsofSets.......................................................................................8 8.DifferenceofSets........�∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙l0 9.SetAlgebra..................................................................................................10 10.SetsofSets................................................................................................11 11.UnionofaSetofSets................................................................................12 12.PowerSets.................................................................................................13 13.SetsandSelections..................................................................................'..15 14.PureSets....................................................................................................16 15.SetsandNumbers......................................................................................19 16.SumsofSetsofNumbers..........................................................................20 17.OrderedPairs.............................................................................21 18.OrderedTuples..........................................................................................22 - 19.CartesianProducts.....................................................................................23 2.RELATIONS ∙∙∙∙∙∙ ∙∙∙∙∙∙25 1.Relations......................................................................................................25 2.SomeFeaturesofRelations.........................................................................26 3.EquivalenceRelationsandClasses .............................................................27 4.ClosuresofRelations...................................................................................29 5.RecursiveDefinitionsandAncestrals .........................................................32 6.PersonalPersistence ....................................................................................33 6.1TheDiachronicSamenessRelation.......................................................33 6.2TheMemoryRelation............................................................................34 6.3SymmetricthenTransitiveClosure.......................................................35 6.4TheFissionProblem..............................................................................38 6.5Transitivethen SymmetricClosure.......................................................40 7.ClosureunderanOperation.........................................................................42 8.ClosureunderPhysicalRelations ............................................42 9.OrderRelations............................................................................................44 10.DegreesofPerfection ................................................................................45 11.Partsof Sets...............................................................................................48 vi Contents 12.Functions...................................................................................................49 13.SomeExamplesofFunctions....................................................................52 14.Isomorphisms ............................................................................................55 15.FunctionsandSums...................................................................................59 16.SequencesandOperationsonSequences..................................................60 17.Cardinality.................................................................................................61 18.SetsandClasses.........................................................................................62 1.Machines......................................................................................................65 2.FiniteStateMachines..................................................................................65 2.1RulesforMachines................................................................................65 2.2TheCareersofMachines.......................................................................68 2.3UtilitiesofStatesandCareers...............................................................70 3.TheGameofLife........................................................................................71 3.1AUniverseMadefromMachines .........................................................71 3.2TheCausalLawintheGameof Life ....................................................72 3.3RegularitiesintheCausalFlow.............................................................73 � 3.4ConstructingtheGameofLifefromPureSets.....................................75 4.TuringMachines........................................................................................1.78 5.LifelikeWorlds............................................................................................81 1.ExtensionalSemantics...................85 1.1WordsandReferents .............................................................................85 1.2A SampleVocabulary andModel .........................................................88 1.3SentencesandTruth-Conditions..........t.................................................89 2.SimpleModalSemantics.............................................................................92 ∙ 2.1PossibleWorlds.....................................................................................92 2.2ASampleModalStructure....................................................................95 2.3SentencesandTruthatPossibleWorlds ...............................................95 2.4Modalities..............................................................................................97 2.5Intensions...............................................................................................99 2.6Propositions...........................................................................................99 3.ModalSemanticswithCounterparts .........................................................101 3.1TheCounterpartRelation ....................................................................101 3.2A SampleModelforCounterpartTheoreticSemantics......................102 3.3Truth-ConditionsforNon-Modal Statements.....................................103 3.4Truth-ConditionsforModalStatements..............................................105 5.PROBABILIT1Y 08 1.SampleSpaces...........................................................................................108 Contents vii ��∙������∙� .SimpleProbability.....................................................................................109 .CombinedProbabilities.............................................................................111 .ProbabilityDistributions ...........................................................................113 .ConditionalProbabilities...........................................................................115 5.1RestrictingtheSampleSpace..............................................................115 5.2TheDefinitionofConditionalProbability ..........................................116 5.3AnExampleInvolvingMarbles..........................................................117 5.4IndependentEvents .............................................................................118 .BayesTheorem..........................................................................................119 6.1TheFirstFormofBayesTheorem ......................................................119 6.2AnExampleInvolvingMedicalDiagnosis .........................................119 6.3TheSecondFormofBayesTheorem..................................................121 6.4An ExampleInvolvingEnvelopeswith Prizes....................................123 .DegreesofBelief.......................................................................................124 7.1SetsandSentences...............................................................................124 7.2SubjectiveProbabilityFunctions.........................................................125 .BayesianConfirmationTheory .................................................................127 8.1ConfirmationandDisconfirmation......................................................127 8.2BayesianConditionalization................................................................128 .KnowledgeandtheFlow ofInformation ..................................................129 6.INFORMATIONTHEOR1Y 31 ��������−� ∙Communication .........................................................................................l31 .ExponentsandLogarithms........................................................................132 .TheProbabilitiesofMessages...................................................................134 .EfficientCodesforCommunication..........................................................134 4.1AMethod forMakingBinaryCodes...................................................134 4.2TheWeightMovingacrossaBridge...................................................137 4.3TheInformationFlowingthroughaChannel......................................139 4.4MessageswithVariableProbabilities .................................................140 4.5Compression........................................................................................141 4.6CompressionUsing HuffmanCodes...................................................142 .Entropy ......................................................................................................144 5.1ProbabilityandtheFlowofInformation.............................................144 5.2ShannonEntropy .................................................................................145 5.3EntropyinAesthetics ..........................................................................147 5.4JointProbability...................................................................................147 5.5JointEntropy ......................................................................�.................148 .MutualInformation ...................................................................................150 6.1FromJointEntropy toMutual Information.........................................150 6.2FromJointtoConditionalProbabilities ................................151 6.3ConditionalEntropy ............................................................................152 6.4FromConditionalEntropytoMutualInformation..............................153 6.5AnIllustrationofEntropiesandCodes...............................................154 .Information andMentality.........................................................................160 7.1Mutual Information andMentalRepresentation .................................160 viii Contents 7.2IntegratedInformationTheoryandConsciousness.............................160 .............................. 7.DECISIONSAND GAMES 163 1.ActUtilitarianism......................................................................................163 1.1AgentsandActions..............................................................................163 1.2ActionsandTheirConsequences........................................................164 1.3Utility andMoralQuality....................................................................165 2.FromDecisionstoGames .........................................................................165 2.] ExpectedUtility...................................................................................165 2.2GameTheory.......................................................................................167 2.3StaticGames...........................................168 3.Multi-PlayerGames...................................................................................170 3.1ThePrisoner’sDilemma......................................................................170 3.2PhilosophicalIssuesinthePrisoner’sDilemma .................................171 3.3DominantStrategies............................................................................172 3.4TheStagHunt......................................................................................175 3.5Nash Equilibria....................................................................................176 4.TheEvolutionofCooperation...................................................................177 4.1TheIteratedPrisoner’sDilemma.........................................................177 4.2TheSpatializedIteratedPrisoner’sDilemma......................................180 . 4.3PublicGoodsGames...........................................................................181 4.4GamesandCooperation ......................................................................183 8.FROM THEFINITETOTHEINFINITE..........................186 .Recursively Defined Series.......................................................................186 .LimitsofRecursivelyDefinedSeries........................................................188 2.1CountingthroughAll theNumbers.....J...............................................188 2.2Cantor’sThreeNumber GeneratingRules..........................................190 2.3TheSeriesofVonNeumann Numbers................................................190 .SomeExamplesof Serieswith Limits.......................................................191 3.1AchillesRunsonZeno’sRacetrack ....................................................191 3.2TheRoyceMap...................................................................................192 3.3TheHilbertPaper ................................................................................193 3.4AnEndlessSeriesof Degreesof Perfection........................................194 .Infinity.......................................................................................................195 4.1InfinityandInfiniteComplexity..........................................................195 4.2TheHilbertHotel.................................................................................197 4.3OperationsonInfiniteSequences........................................................198 .Supertasks..................................................................................................198 5.1ReadingtheBorgesBook.....................∙...............................................198 5.2TheThomsonLamp ............................................................................199 5.3ZeusPerformsaSuper-Computation ..................................................200 5.4AcceleratingTuringMachines............................................................200 Contents ix 9.BIGGER INFINITIES∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙ ∙∙∙∙∙202 1.SomeTransfiniteOrdinalNumbers...........................................................202 2.Comparingthe Sizesof Sets......................................................................203 3.Ordinal andCardinalNumbers..................................................................205 4.Cantor’sDiagonalArgument ....................................................................207 5.Cantor’sPowerSetArgument...................................................................2]l 5.1Sketchof thePowerSetArgument .....................................................211 5.2ThePowerSetArgument inDetail .....................................................213 5.3TheBeth Numbers...............................................................................214 6.TheAlephNumbers ..................................................................................216 7.TransfiniteRecursion ................................................................................217 7.] RulesfortheLongLine.......................................................................217 7.2TheSequenceofUniverses.................................................................218 7.3Degreesof DivinePerfection ..............................................................219 FurtherStudy0.0.0...OOOOOOOOOOOOOOOOO0.000000IOOOOOOOOOOOOOOOOO...OOOOOOOOOOOOOOOOO0.0.0.221 ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙ Glossary Of symbOIS OO..0...0.0.00.0.0.00.00.00.000000000223 ................................................................................... References 226 ......... Index 231 Preface Anyone doingphilosophy today needs to have a sound understanding of a wide rangeofbasicmathematicalconcepts. Unfortunately,mostapplied mathematics textsaredesignedtomeettheneedsof scientists. And much of themath 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 philosophicalexamples. Thisismath forphilosophers. It’simportanttounderstand what MorePrecisely isandwhat itisn’t. More Precisely is not a philosophy of mathematics book. It’s a mathematics book. We’re not going to talk philos0phically about mathematics. We are going to teachyou themathematicsyou need todophilosophy. We won’t enterinto 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 dojust fine here. More Precisely is an introductory book. It is not anadvanced text. It aims to coverthe basics sothat you’re prepared to go into the depths on the topics that are of specialinterestto you. Tofollow what we’re doinghere,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 ofreligion. It’snatural tostartwith somesettheory. All branches of philosophy today .makesomeuse of settheory. If you want tobe ableto follow what’s going on inphilosophy today,you need tomasteratleastthebasic languageof settheory. You need to understand the specialized notation and vocabulary used to talk about sets. Forexample,you need tounderstand the conceptof the intersection of two sets, and to know how it is written in the specialized notation of set theory. Sincewe’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 tohaveaclearunderstanding of theobjectsyou’re arguingabout. 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. Soif you’re not familiar with those notions, you’ve got to start with Chapters 1and 2. Make sureyou’ve really mastered the ideas in Chapters 1and2beforegoingon. 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 industrialdevice. It’s a formal structureused to describe Preface xi somelawful pattern of activity. Machines areoften used in philosophy of mind — many philosophers model minds as machines. Machines are sometimesused in metaphysics — simple universes can be modeled as networks of interacting machines. You can use these models to study space, time, and causality. Chapter4introducessomeof themath used inthephilosophy of language. Sets, relations, and functions are extensively used in formal semantic theories — especiallypossibleworldssemantics. Chapter5introducesbasic probability theory. Probability theory isused in epistemology and the philosophy of science (e.g., Bayesian epistemology, Bayesianconfirmationtheory). Building onthetheory of probability,Chapter6 discusses information theory. Information theory is used inlepistemology, philosophy of science,philosophy of mind, aesthetics,and in other branches of philosophy. A firm understanding of informationtheory isincreasingly relevant ascomputersplay evermore importantroles in human life. Chapter7discusses some of the uses of mathematics in ethics. It introduces decision theory and game theory. Finally, the topic of infinity comes up in many philosophical discussions. Is the mind finitely or infinitely complex? Can infinitely many tasks be done in finite time? What does it mean to say that God is infinite? Chapter 8 introduces the notion of recursion and countable infinity. Chapter 9 shows that there is an endless progression of bigger and bigger infinities. It introducestransfiniterecursion. . We illustrate the mathematical concepts with philosophical examples. We aren’t interested in the philosophical soundness of these examples. Do sets really exist? Was Platoright? It’sirrelevant. Whetherthey really existornot, it’s extremely important to understand set theory to do philosophy today. Chapter 3discusses mechanical theories of mentality. Are minds machines? It doesn’tmatter. What matters isthat mechanical theories of mentality use math, and that you need to know that math before you can really understand the philosophical issues. As another example,we’ll spend many pages explaining the mathematical apparatus behind various versions of possible worlds semantics. Isthisbecause possible worlds really exist? We don’tcare. We do care that possible worlds semantics makes heavy use of sets, relations, and functions. As we develop the mathematics used in philosophy, we obviously talk about lots and lots of mathematical objects. We talk about sets,numbers, functions,and soon. Our attitudeto these objects is entirely uncritical. We’re engaged in exposition, not evaluation. We leave the interpretations and evaluations up to you. Although we aim to avoid philosophical controversies, More Precisely is not a miscellaneous assortment of mathematical tools and techniques. If you look closely, you’ll see that the ideas unfold in an orderly andconnectedway. MorePreciselyisaconceptualnarrative. Ourhopeisthatlearningthemathematicswepresent inMorePreciselywill help you to do philosophy. You’ll be better equipped to read technical philosophical articles. Articles and ideas that once might have seemed far too formal will become easy to understand. And you’ll be able to apply these concepts in your own philosophical thinking and writing. Of course, some philosophers might object: why should philosophy use mathematics at all? Shouldn’t philosophy avoid technicalities? We agree that technicality for its own sake ought to be avoided. As Ansel Adams once said, “There’s nothing