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Computers,mathematicalproof,andthenatureofthehumanmind

POMSIGMAAKeynoteAddressPhilosophyofMathematicsAmericanMathematicalSocietyandMathematicalAssociationofAmericaJoint

MeetingsJeffBuechner

RutgersUniversity-NewarkandTheSaulKripke Center,CUNY,TheGraduateCenter

January6th,2017

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1976AppelandHakenprovethefour-colortheorem

• June21,1976WolfgangHakenandKennethAppel,withtheaidofJohnKoch,completedtheirproofoftheFour-ColorTheorem(4CT).(Hakenturned48yearsoldonthatday.)

• Theirproofwaspublishedin1977:“Everyplanarmapisfourcolorable,”Parts1andII,andSupplementsIandII,IllinoisJournalofMathematics,XXI,84,September1977

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• Atoneplaceintheproofofthe4CT,AppelandHakenneedtofindafinitelistofreducibleconfigurationswiththepropertythateverygraphcontainsatleastoneconfigurationinthelist.Todothis,alemmamustbeproved:thateveryconfigurationinanunavoidablesetisreducible.Acomputerisneededtoprovethatalloftheconfigurationsarereducible.Forinstance,toshowthatonekindofconfigurationinthesetisreduciblerequires1,000,000steps.

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• AlthoughcomputershadalreadybeenusedtoprovetheoremsinmathematicsbeforetheAppel-Hakenproof,theimportanceofthe4CTbroughttheuseofacomputerinprovingittotheforefrontofattentionofmathematicians,aswellasalaypublic.Moreover,somemathematiciansdidnotbelievethatthetheoremhadbeenproved,sincethecomputerproofofpartofthe4CTistoolongforahumanbeingtosurvey.

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1976AppelandHakenprovethefour-colortheorem:discordintheranks

• “Intheanalysisofeachcasethecomputeronlyannouncedwhetherornottheprocedureterminatedsuccessfully.Theentireoutputfromthemachinewasasequenceofyeses.Thismustbedistinguishedfromaprogramwhichproducesaquantityasoutputwhichcansubsequentlybeverifiedbyhumansasbeingthecorrectanswer…Therealthrillofmathematicsistoshowthatasafeatofpurereasoningitcanbeunderstoodwhyfourcolorssuffice.AdmittingthecomputershenanigansofAppelandHakentotherealmofmathematicswouldonlyleaveusintellectuallyunfulfilled.”DanielCohen“Thesuperfluousparadigm,”1991

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1976AppelandHakenprovethefour-colortheorem:discordintheranks

• “Nowhereintheirlongandoftenirrelevantaccountdotheyprovidetheevidencethatwouldenablethereadertocheckwhattheysay.Itmayormaynotbe‘possible’toprovethecolortheoremthewaytheyclaim.Whatismorecertainisthattheydidnotdoso…notonlyisnoprooftobefoundinwhattheypublished,butthereisnotanythingthatevenbeginstolooklikeaproof.Itisthemostridiculouscaseof‘TheKing’sNewClothes’thathaseverdisgracedthehistoryofmathematics.”GeorgeSpencer-Brown,appendixtoGermaneditionofhisLawsofForm

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TheirproofisimplicitlyrecognizedasvalidbytheUnitedStatesPostal

Authority

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TheirproofisimplicitlyrecognizedasvalidbytheUnitedStatesPostal

Authority• Themarking‘FOURCOLORSSUFFICE’wasmadebyaUniversityofIllinois-Urbanapostalmeter,notataUnitedStatesPostOffice.Butrecognitionisimplicit,sinceanythingwhichisillegalcannotbemarkedonastampbyauniversitypostalmeter.Sofar,however,theUnitedStatesPostalAuthoritydoesnottakemistakenmathematicalproofstobeillegal.

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Ashorterandimprovedproof

• NeilRobertson,DanielSanders,PaulSeymour,andRobinThomasprovideanewproofofthefour-colortheoremin1994.AnoutlineoftheirproofispublishedinProceedingsoftheInternationalCongressofmathematiciansin1995.

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1979Tymoczko ontheFour-ColorTheorem

• Thefirstpaperinthephilosophyofmathematicsonthephilosophicalimportanceofthefour-colortheoremappearedinFebruary,1979.

• ThomasTymoczko “TheFour-ColorProblemandItsPhilosophicalSignificance,”JournalofPhilosophyVol.76,No.21,pp.57-83

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1979Tymoczko ontheFour-ColorTheorem

• “Whatreasonisthereforsayingthatthe4CTisnotreallyatheoremorthatmathematicianshavenotreallyproducedaproofofit?Justthis:nomathematicianhasseenaproofofthe4CT,norhasanyseenaproofthatithasaproof.Moreover,itisveryunlikelythatanymathematicianwilleverseeaproofofthe4CT.”Tymoczko,op.cit.p.58

• Elementaryinference:AppelandHakenaremathematicians.Soneitherhaseverseenaproofofthe4CT.

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Tymoczko onwhatthefour-colortheoremshows

• “Ifweacceptthe4CTasatheoremwearecommittedtochangingthesenseoftheunderlyingconceptof‘proof.’”

• “Theuseofcomputersinmathematics,asinthe4CT,introducesempiricalexperimentsintomathematics,andraisesagainforphilosophytheproblemsofdistinguishingmathematicsfromthenaturalsciences.”Tymoczko op.cit.p.58

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Tymoczko onthefour-colortheorem

• “Theanswerastowhetherthe4CThasbeenprovedturnsonanaccountoftheroleofcomputersinmathematics.”op.cit.p.58

• “The4CTissubstantialpieceofpuremathematicswhichcanbeknownbymathematiciansonlyaposteriori.Ourknowledgemustbequalifiedbytheuncertaintyofourinstruments,computerandprogram...Thedemonstrationofthe4CYincludesnotonlysymbolmanipulation,butthemanipulationofsophisticatedexperimentalequipmentaswell:thefour-colorproblemisnotaformalquestion.”Tymoczko,op.cit.pp.77-78

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Tymoczko onthefour-colortheorem

• “Theideathatapropositionofpuremathematicscanbeestablishedbyappealingtoempiricalevidenceisquitesurprising.Itentailsthatmanycommonlyheldbeliefsaboutmathematicsmustbeabandonedormodified.Consider:

• 1.Allmathematicaltheoremsareknownapriori• 2.Mathematics,asopposedtonaturalscience,hasno

empiricalcontent.• 3.Mathematics,asopposedtonaturalscience,reliesonly

onproofs,whereasnaturalsciencemakesuseofexperiments.

• 4.Mathematicaltheoremsarecertaintoadegreethatnotheoremofnaturalsciencecanmatch.Tymoczko,p.63

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Tymoczko onmathematicalproof• “Proofsareconvincing...[In]Wittgenstein’sRemarksontheFoundationsofMathematics,thisisallthereistoproofs:theyareconvincingtomathematicians.Thisistobetakenasabrutefact,somethingforwhichnoexplanationcanbegivenandnoneisnecessary.Mostphilosophersareunhappywiththispositionandinsteadfeelthattheremustbesomedeepercharacterizationsofmathematicalproofswhichexplains,atleasttosomeextent,whytheyareconvincing.”Tymoczko,op.cit.p.59

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Tymoczko onmathematicalproof

• Whyaremathematicalproofsconvincing?• “Thatproofsaresurveyable andthattheyareformalizable aretwosuchcharacterizations[J.B.ofwhymathematicalproofsareconvincing].”Tymoczko,op.cit.p.59

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Tymoczko onmathematicalproof

• “Weoftensaythataproofmustbeperspicuous,orcapableofbeingcheckedbyhand.Itisanexhibition,aderivationoftheconclusion,anditneedsnothingoutsideofitselftobeconvincing.”Tymockzo,op.cit.p.59

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Tymoczko’s circle• Unfortunately,Tymoczko’s definitionofsurveyabilityusestheideathatamathematicalproofneedsnothingoutsideitselftobeconvincing.Soonemustalreadyknowwhatamathematicalproofisbeforeoneknowswhatsurveyability consistsin;butsurveyability isonecriterionofbeingamathematicalproof.

• “Themathematiciansurveys theproofinitsentiretyandtherebycomestoknowtheconclusion.”Tymoczko,op.cit.,p.59

• “Theconstructionthatwesurveyedleavesnoroomfordoubt.”Tymoczko,op.cit.p.60

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PaulTelleronTymoczko

• “Surveyability isneeded,notbecausewithoutitaproofisinanysensenotaproof,butbecausewithoutsurveyability weseemnottobeabletoverifythataproofiscorrect.Sosurveyability isnotpartofwhatitistobeaproofinouraccustomedsense.”PaulTeller“ComputerProof,”JournalofPhilosophy,December1980,pp.797-803

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PaulTelleronTymoczko

• “…wemaytakeadvantageofnewmethodsofsurveyingaslongastheseenableustomeetsensibledemandsoncheckingproofs,andashiftinthemeansofsurveyingactuallyusedmeansonlyashiftinmethodsofcheckingproofs,notashiftinourconceptionsofthethingschecked.”Teller,op.cit.,p.798

• Notashiftinourconceptionsofthethingschecked=notashiftinourconceptofproof

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ThedisputebetweenTellerandTymoczko:theconceptof

mathematicalproof

• Tymoczko:surveyability isanessentialfeatureoftheconceptofamathematicalproof.

• Teller:surveyability isnotanessentialfeatureoftheconceptofamathematicalproof.

• Whoisright?Onwhatgroundsaretheyright?

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Quine’sproblemforconcepts

• W.V.Quinearguedinhisepochalpaper“TwoDogmasofEmpiricism,”thatthereisnohardandfastdistinctionbetweenmeaning-constitutingbeliefsandauxillary beliefs(beliefsthatarenotmeaning-constituting).

• Thismeansthatitisimpossibletodrawahard-and-fastlinebetweenessential(ornecessary)featuresofaconceptandnon-essential(contingent)featuresofaconcept.

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• IfAproposesthatXisanessentialfeatureoftheconceptofmathematicalproof,andBproposesthatitisanaccidentalfeatureoftheconceptofmathematicalproof,thereisnoprincipledwayofadjudicatingbetweenthetwoproposals.

• Adjudicationshouldgobywayofcanonsofrationalityandcanonsofscientificinquiry,suchasconservatism—upholdingasmanycurrentlyestablishedbeliefsaspossible.Howwouldthatworkfortheconceptofmathematicalproof?

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• ThereisnoconsensusviewastowhetherQuineiscorrectornotonthis,butmostphilosopherstakeQuinetobecorrect.

• Teller’sclaimthatsurveyability isnotanessentialfeatureoftheconceptofmathematicalproofcouldbeupheldifitsatisfiedmorecanonsofrationalityandofscientificinquirythandoesTymoczko’s claim.

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• However,theideaofsurveyability surfacesinthecontextofusingcomputersinmathematicalproofs.

• Therewasnotmuchdata(i.e.,featuresonwhichthereiscommonagreement)concerningtheuseofcomputersinmathematicalproofintheperiod1976-1980.

• ThedisagreementbetweenTellerandTymoczko isastalemate.

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Detlefsen onTymoczko• “Theneedfortheappealtoempiricalevidenceisbrought

about,inTymoczko’s view,bythefactthatthecalculationperformedbyanIBM370-160Ainordertodeterminethereducibilityofcertainconfigurationsistoolongtobe‘checked’or‘surveyed’byhumanmathematicians.Becauseofthis,Tymoczko reasons,whateverevidencewehaveforthereliabilityoftheIBM370-160Aindeterminingreducibilityofconfigurationscannottaketheformofa‘surveyable’proofofitsreliability.Andso,itisconcluded,theevidencemustbeempiricalincharacter.”MichaelDetlefsen andMarkLuker“The FourColorTheoremandmathematicalProof,”JournalofPhilosophy,1980,pp.803-820

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Detlefsen onTymoczko• Detlefsen providesseveralexamplesofmathematicalproofswhicharesurveyable andinwhichcomputationsaremade.Hearguesthatsuchcomputationsnecessarilyutilizeempiricalpremises(suchas:thecomputingagentcorrectlyexecutestheprogramrequiredtomakethecomputation).

• Ifhisargumentissound,Detlefsen hasshownthatunsurveyability isnotnecessaryforthepresenceofanempiricalelementinmathematicalproofs.ThisrefutesamajorclaiminTymoczko’s paper.

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Detlefsen onTymoczko• “ThiscreatesadilemmaforTymoczko.Foreitheronerejectshisreasoning,inwhichcaseheisleftwithoutanargumentfortheempiricalcharacteroftheproofofthe4CToroneacceptshisreasoning,butisthenforcedtoviewthepresenceofcalculationorcomputationinaproofasinjectinganempiricalelementintothatproof.TheconsequenceofsuchaviewisthatempiricalproofsaremorewidespreadthanTymoczkohimselfindicates.”MichaelDetlefsen,op.cit.,p.809

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Detlefsen onTymoczko• “ThiscreatesadilemmaforTymoczko.Foreitheronerejectshisreasoning,inwhichcaseheisleftwithoutanargumentfortheempiricalcharacteroftheproofofthe4CToroneacceptshisreasoning,butisthenforcedtoviewthepresenceofcalculationorcomputationinaproofasinjectinganempiricalelementintothatproof.TheconsequenceofsuchaviewisthatempiricalproofsaremorewidespreadthanTymoczkohimselfindicates.”MichaelDetlefsen,op.cit.,p.809

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BackgroundtoBurge:Descartesonmathematicalproofs

• [Inlongdeductions]“thelastlinkisconnectedwiththefirst,eventhoughwedonottakeinbymeansofoneandthesameactofvisionalltheintermediatelinksonwhichthatconnectiondepends,butonlyrememberthatwehavetakenthemsuccessivelyunderreview…”Descartes,RulesfortheDirectionofMind

• ForDescartes,“ifthatknowledgeisdeducedfromevidentmathematicalpremises,itiscertainanddemonstrative.”TylerBurge,ContentPreservation,PhilosophicalIssues,1995,p.271

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BackgroundtoBurge:Chisholmonmathematicalproofs

• “WhatifSderivesapropositionfromasetofaxioms,notbymeansofoneortwosimplesteps,butasaresultofacomplexproof,involvingaseriesininterrelatedsteps?Iftheproofisformallyvalid,thenshouldn’twesaythatSknowsthepropositionapriori?Ithinkthattheanswerisno.”RoderickChisholm,TheoryofKnowledge,2nd edition

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BackgroundtoBurge:Chisholmonmathematicalproofs

• “[I]f,inthecourseofademonstration,wemustrelyuponmemoryatvariousstages,thususingaspremisescontingentpropositionsaboutwhatwehappentoremember,then,althoughwemightbesaidtohave‘demonstrativeknowledge’ofourconclusion,inasomewhatbroadsenseoftheexpression‘demonstrativeknowledge,’wecannotbesaidtohaveaa prioridemonstrationoftheconclusion.”RoderickChisholm,op.cit.,

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WhyisChisholm’spointimportant?

• a posterioriknowledge:knowledgewhichisarrivedatonthebasisofsenseexperiencesorperceptualbeliefs.

• aprioriknowledge:knowledgewhichisarrivedatonthebasisofintellectualprocesseswhichdonotinvolvereferencetoorrelianceuponsenseexperiences.

• aposteriorijustification:justificationwhichreliesuponsenseexperiences.

• apriorijustification:justificationwhichemploysintellectualprocesseswhichdonotinvolvereferencetoorrelianceuponsenseexperiences.

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WhyisChisholm’spointimportant?

• IfChisholmisrightthatlongmathematicalproofsrequireapremiseaboutwhatwehappentoremember—andthusarenoteitherknownorjustifiedapriori—theniteasilyfollowsthatthoseproofsareknownorjustifiedempirically.Theyrelyuponorrefertosenseexperiences.

• Tymoczko isrightabouttheepistemologicalstatusofthe4CTifweacceptChisholm’spoint.Butheiswrongthatonlyunsurveyablemathematicalproofsrequire(inwholeorinpart)empiricaljustification.

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BackgroundtoBurge:Fallis ondispensingwithempiricalevidence

• “…thereisasenseinwhichtheproofofthefour-colortheoremisanapriorijustification.Itneednotappealtoanyparticularempiricaldataandinprincipleneednotappealtoempiricaldataatall.Forinstance,therelevantcomputationcouldbeperformedbyadeviceotherthanadigitalcomputerandinprinciplecouldbeperformedinthemathematician’smind.”DonFallis,MathematicalProofandtheReliabilityofDNAEvidence,AmericanMathematicalMonthly,June-July,1996,p.496

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BackgroundtoBurge:Fallis ondispensingwithempiricalevidence

• Fallis thinksthat,unlessitisnecessarythataphysicalmachineofsomekindperformsomecomputation,thenthecomputationisapriori,becauseitispossiblethatahumanmindcouldperformthecomputation.

• Itispossiblethatahumanmindcouldcompleteaninfinitecomputationalprocess(in,say,aMalament-Hogarthuniverse).Shouldwethensaythatsuchcomputationsareaprioriknowable?

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Burgeontheuseofcomputersinmathematicalproofs

• TylerBurge,in1998,inhisimportantpaper“ComputerProof,APrioriKnowledge,andOtherMinds,”TheSixthPhilosophicalPerspectivesLecture(pp.1-38),sidestepstheneedtounderstandwhatamathematicalproofisinaskingwhethertheuseofcomputersinmathematicalproofsaddsanempiricalelementtosuchproofs.Burgewillarguethatnoempiricalelementneedbeaddedwhencomputersareusedinmathematicalproofs,suchastheproofofthe4CT.The4CTcanbeknownapriori(tobetrue).

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Burgeonmathematicalproofs

• Burgeusestheterms‘epistemicentitlement,’‘epistemicwarrant,’and‘epistemicjustification’(sometimeswithoutthequalifier‘epistemic’).

• Unlessyouareaphilosopherworkinginepistemology,itisbesttotreatallofthemasmeaningthesame—namely,justification.

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Burgeonmathematicalproofs

• “Iconclude,givenourassumptions,onecanbeinaposition,fromthethirdpersonpointofview,tobeaprioriwarrantedinbelieving,infact,knowing,ondefeasibleinductivegrounds,thatthe[4CT]hasbeenproved.Onecanknowthisevenifonecannotreplicatetheproof.”TylerBurgeComputerProof,APrioriKnowledge,andOtherMinds,p.23

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Burgeonmathematicalproofs• “Theentitlementforrelyingonthesourcedoesspecifythesource[thecomputer].Butitneednotspecifytheobject’sempiricallydiscerniblecharacteristics,ortheempiricalbackgroundconditionsthatenabletherecipienttoaccessandrelyonthesource.Itcanspecifythesourceinthenon-empiricalwaythattheapriorijustificationdoes.”Burge,op.cit.,p.29

• Workonproofassistants(e.g.,byHarveyFriedmanandbyJeremyAvigad)providesanentitlementforrelyingonthesource(theIBM370-160AusedintheAppel-Hakenproofofthe4CT).

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Burgeonmathematicalproofs• “Perceptualexperienceofthewordsorofthebodyofthesourceneedplaynoroleinjustifyingone’sunderstandingof,orintellectualusesof,thecontentofthewordsorthepresentationsofthesource.”Noticethatthispointwould,ifcorrect,refuteChisholm.

• “Perceptionisonlythemodeofaccess,anenablingconditionwhichmakesnocontributiontotheepistemicforceofthewarrant.”

• Perceptionismerelyaconditionthatenablesonetomakeuseofaresourceforreasonandunderstanding.”Burge,op.cit.,p.30

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WhathasBurgeshown?

• Ifhisargumentsaresound,Burgehasshownthattheunsurveyability ofmathematicalproofsisnotsufficientfortheexistenceofanempiricalelementinsuchproofs.

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WhathaveDetlefsen andBurgeshown?

• RecallthatDetlefsen hasshownthattheunsurveyability ofmathematicalproofsisnotnecessaryfortheexistenceofanempiricalelementinsuchproofs.Burgehasshownthatitisnotsufficient.

• TheresultsofBurgeandDetlefsen,ifbotharecorrect,showthereisnoconceptualconnectionbetweentheunsurveyability ofmathematicalproofsandtheexistenceofanempiricalelementinsuchproofs.However,bothcannotbecorrect.

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• Ifthereisnoconceptualconnectionbetweentheunsurveyability ofmathematicalproofsandtheexistenceofanempiricalelementinsuchproofs,iteasilyfollowsthatunsurveyability hasnothingtodowiththeexistenceofanempiricalelementinmathematicalproofs.

• ExaminingtheargumentsofbothDetlefsen andBurge,thisisnotshocking,norevensurprising.

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• Detlefsen arguedthatwhetheramathematicalproofissurveyable orunsurveyable,theremightbeempiricalelementsinit.

• Burgearguedthatwhetheramathematicalproofissurveyable orunsurveyable,thereneednotbeempiricalelementsinit.

• Ofcourse,neitherruleoutthattheremightbe,northattheremightnotbe,empiricalelementsinamathematicalproof.Butwhetherthereareorarenotisnotamatterofwhattheconceptofamathematicalproofconsistsin.Itis,rather,anentirelycontingentmatter.

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FourProblemsforBurge

• TherearethreeproblemsforBurge’saccountofhowwecanhaveaprioriknowledgeoftheoutputofacomputer.

• Thefirstisthathisaccountmakesittooeasytohavegettiered knowledge.GettiercounterexamplesarecasesinwhichasubjectShasatrue,justifiedbeliefthatp,butinwhichSdoesnotknowthatp.

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• Hereishow,followingBurge’saccountofepistemicjustificationinthecontextofmathematicaltruths,onecanhaveatrue,justifiedbeliefinthe4CT,butnotknowthe4CT.

• Supposethatthe4CTistrue,butthatthecomputerprogramforresolvingthecasesisfallacious.OnBurge’saccount,asubjectSwillbejustifiedinbelievingthe4CTtobetrue.Sinceitistrue(byassumption),Shasatrue,justifiedbeliefinthe4CT.ButSdoesnotknowthe4CT.

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• WeshouldtakeGettier counterexamplesveryseriously.(DavidLewishasremarkedthatthereareonlytworesultsthatallphilosopherstaketobedefinitive:GödelandGettier.)

• IfanaccountofepistemicjustificationmakesittooeasyforGettier counterexamples(andnotjustpossibleforthemtoarise)toarise,thatisareasontorejectsuchanaccount.

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• ThesecondproblemforBurgeconcernshisclaimthatthemodeofaccesstosomeepistemicallywarrantedsetofpropositionsisnotnecessaryforbeingepistemicallyjustifiedinbelievingthosepropositions:“[t]heentitlementforrelyingonthesource…neednotspecifytheempiricalbackgroundconditionsthatenabletherecipienttoaccessandrelyonthesource.”Burge,op.cit.,p.29

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FourProblemsforBurge• Thatiswhywecandiscounttheroleofmemoryindeterminingwhetherweareepistemicallyjustifiedinbelievingatheoremofmathematicsonthebasisoftheproofofthattheorem.Memoryisameremodeofaccesstotheproof.

• ForBurge,thesameistrueofcomputerproofs—wecandiscountthemodeofaccesstothetheoremanditsproof(thecomputerprogram)whichisthecomputer.

• Ifmemoryisfaulty,thatdoesnotshowthattheproofisfaulty.Indeed,afaultymemoryhasnothingtodowithaproof—whichisanabstractobject.Canwesaythesameofacomputer?Burgethinkswecan.

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FourProblemsforBurge• Isaywecannot.Hereiswhy.Unlikememory,acomputerisnotanorganicintrinsicpartofahumanbeing.Itistheuniquemodeofaccesstothecomputerprogramandthecomputationsofthatprogram—allofwhichareabstractobjects.

• Butitismore.Itisthemeansbywhichtheabstractobjectsarephysicallyrealized.Memory,ontheotherhand,neednotbethemeansbywhichaproofisphysicallyrealized.Rather,aproofcanbephysicallyrealizedonapieceofpaperusinginktomakeinscriptions.

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FourProblemsforBurge• Imagineacultureinwhichthereisnopaper,nowriting

instruments,andnoconceptsofwriting(onpaper,usingwritinginstruments).However,thereistheconceptofaproof.Allproofsareinhumanmemory.

• Insuchacase,weshouldsaythataproblemwithhumanmemorywouldcreateaprobleminepistemicentitlementtotheproof.Why?Becausehumanmemoryistheonlymeansbywhichtheproofisphysicallyrealized,aswellasthemodeofaccesstotheproof.

• Withoutmemory,wedonothaveaproof,sincewedonothaveanymodeofaccesstotheabstractobjectwhichistheproof.(Comparewithaproofwhichissodifficultthatnomathematicalconceptsavailabletothehumanmindareadequateforrepresentingtheproof.Insuchacase,eventhoughtheproofhasanabstractexistence,weshouldsaythatwecannotbeepistemicallyentitledtoitsincewehavenomeansbywhichtoaccessit.)

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FourProblemsforBurge• ThethirdproblemforBurgeconcernsfourassumptionsthatBurgemakesinhisargument.Oneassumptionisthat“individual’sknowledgeofpuremathematics,restingonspecificallymathematicalunderstandingorreasoning,isordinarilyapriori.”(Burge,op.cit.,p.4)

• ThiscontradictsDetlefsen’s claim—whichdependsonTymoczko’s definitionofmathematicalproof—thatempiricalpremisesareusedinmathematicalproofsthataresurveyable (aswellasthosewhichareunsurveyable).

• WedeferourexpositionofthefourthproblemforBurge.

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Thedialecticsofhowthingsstand• Tymoczko andTeller:stalemate• Tymoczko andDetlefsen:IfDetlefsen iscorrect,empiricalpremisesoccurinmathematicalproofsthatarebothsurveyable andunsurveyable.Thisputspressureongettingclearonwhatwemeanbyamathematicalproof.

• Tymoczko andBurge:IfBurgeiscorrect,thentheuseofcomputersinmathematicalproofsdoesnotintroduceanempiricalelementintothoseproofs(nordoestheuseofcomputationsinmathematicalproofs).Tymoczko andDetlefsen arebothrefuted.

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Thedialecticsofhowthingsstand• ButwehavepresentedreasonsforthinkingthatBurge’sargumentfails.

• Itisclearthatmuchphilosophicalworkstillneedstobedoneinexplicatingtheconceptofamathematicalproof.

• Butnomatterwhatthatexplicationeventuallyconsistsin,itmustbecompatiblewithourviewsaboutthenatureofcomputersandthenatureofthehumanmind.Thatthis(perhapsstartlingview)issowillbearguedintheremainderofthistalk.

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Alineofthoughtnottaken• WewillnowdiscussalineofthoughtthatisbroachedbyTeller,Detlefsen,Davis,andTymoczko,butnottakenupbyanyofthem.

• Inhispaper,Tellerwrites:“Theallegednonsurveyability alsounderliesTymoczko’s secondconclusion:thecomputerproofofthecombinatoriallemmaissubjecttoerror—computerscanmakemistakes.Wecannotguardagainstthispossibilityofmechanicalfailureorerrorinprogramminginthetraditionalwaybecausewecannotsurveytheproof.”Teller,op.cit.,p.798

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Alineofthoughtnottaken

• “Whatiftheprogrammingwaserroneous?Whatiftheinitialdatawerefake?Whatiftherewasamachinemalfunction?”

• “Theseconsiderationsleadustoaposition—whichisrarelydiscussedinworksonthephilosophyofmathematicsandwhichisveryunpopular—thatamathematicalproofhasmuchincommonwithaphysicalexperiment.”P.Davis,“Formac MeetsPappus,”American

mathematicalMonthly,1969,pp.903-904.

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JohnHortonConwayoncomputers

• Awell-knownmathematician,JohnH.Conway,hasbeenquotedassaying:“Idon’tlikethem[computers],becauseyousortofdon’tfeelyouunderstandwhat’sgoingon.”NewYorkTimes,April6,2004KennethChang“Inmath,computersdon’tlie.Ordothey?”anarticleontheuseofcomputersinmathematicalproofs

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Wittgensteinonmachinecomputation

• “Ifweknowthemachine,everythingelse…seem[s]tobealreadycompletelydetermined.Wetalkasifthesepartscouldonlymoveinthisway,asiftheycouldnotdoanythingelse.Isthishowitis?Doweforgetthepossibilityoftheirbending,breakingoff,melting,andsoon?Yes,inmanycaseswedon’tthinkofthatatall.Weuseamachine,orapictureofamachine,asasymbolofaparticularmodeofoperation.Forinstance,wegivesomeonesuchapicture,andassumethathewillderivethesuccessivemovementsofthepartsfromit.”LudwigWittgensteinPhilosophicalInvestigations,§ 193

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Kripke onWittgenstein

• “Wittgensteinhimselfdrawsthedistinctionbetweenthemachineasanabstractprogram(‘derMachineals Symbol,’PI193)andtheactualphysicalmachine,whichissubjecttobreakdown(‘Doweforgetthepossibilityoftheirbending,breakingoff,melting,andsoon?’PI193)”SaulKripke,WittgensteinonRulesandPrivateLanguage,p.35,fn.24

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Naivecomputerviewofthemind

• “Amachinecanfollowthisrule;whencedoesahumanbeinggainafreedomofchoiceinthismatterwhichamachinedoesnotpossess?”

SirMichaelDummett “Wittgenstein’sPhilosophyofMathematics,”PhilosophicalReviewVol.68(1959),pp.324-348,atp.351

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Thebasicidea

• Sincephysicalcomputingmachinescanbreakdowninvariousways,howdowereallyknowwhatfunctionFagivenPCMcomputes?

• Onemightthinkthatisnotaseriousproblem.IfFisthesquarefunction,andthePCMcomputesF(2)=4,thePCMisoperatingnormally.IFthePCMcomputesF(2)=8,thenithassufferedabreakdown.

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Thebasicidea• Thebasicideaisnotthatoftheunder-determinationoftheorybydata.Forinstance,boththesquarefunctionandthedoublingfunctionoutput‘4’whentheirinputis‘2.’Indeed,therearemanyinfinitelymanyfunctionswhoseinitialsegmentconsistsoftheinteger‘4.’AsmoreandmorevaluesofFarecomputed(sayn),functionswhoseinitialsegmentconsistofthesequenceofn-1valueswillnolongersharenvalues.

• Butthisisnotamatterofunderdetermination oftheorybydata.Itissomethingquitedifferent.

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Thebasicidea

• Thatviewistoonaïve.Therearemanyotherfunctions(say,G)thatPCMmightbecomputing.Perhapstheoutput‘4’iswhenPCMsuffersabreakdownincomputingG.Perhapstheoutput‘8’iswhenPCMoperatesnormallyincomputingG.

• UnlessitisKNOWNthatthePCMcomputes,say,F,itcannotberuledoutthat,basedonitsbehavior,itiscomputing,say,G.

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Thebasicidea• Inshort,wehavetoidealizethephysicalbehaviorofthePCMascomputing,say,F,ifwearetounderstandjustwhataPCMcomputesandwhatitdoesnotcompute.

• Buttoidealizethephysicalbehaviorofthecomputingmachineascomputing,say,F,wemustalreadyknowthatitdoescomputeF.

• Wheredidweacquirethisknowledge?Certainly,notfromthephysicalbehaviorofthePCM(whichphysicalbehaviorincludeswhatPCMoutputs),sincewehaveidealizedthatphysicalbehaviorontheassumptionthatPCMcomputesF.

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Thebasicidea

• Wecan’tidentifythefunctionaPCMcomputesbyobservingthatitisoperatingnormally,orissufferingabreakdown.

• WecannotdothatbecausewecannotknowwhetherPCMisoperatingnormallyorsufferingabreakdownunlesswealreadyknowwhatfunctionPCMiscomputing.

• ByidealizingthephysicalbehaviorofaPCM,weimplicitlystipulatewhetherconditionsarenormalorbreakdown.

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Thebasicidea

• WecannotappealtotheintentionsofthedesignersofthePCMtodeterminewhatfunctionthePCMcomputesunlesswealreadyknowthattheyintendthePCMtocompute,say,F.

• Ifwemakesuchanappeal,thenwecansaywhetherthePCMisoperatingnormally,orisinbreakdownmode,onlyifwealreadyknowthatthecodeforthePCMisthecodeforcorrectlycomputing,say,F.

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HowmanydistinctdesignersoftheIBM370-160Amighttherebe?

• ConstructaBooleantree,whereforanynode,thetop-mostbranchleavingitrepresentsnormalconditionsandthebottom-mostbranchesleavingitrepresentmalfunctionconditions.

• FeedthesuccessivenodesinthetreesuccessivedigitsinthesequenceofoutputdigitsofsomeF.

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HowmanydistinctdesignersoftheIBM370-160Amighttherebe?

…9677784…..

…677784….

…77784…

…77784…

…677784…

…77784…

…77784…

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Adisturbingconclusion

• AtanygivenstageinthecomputationofF,itismustbeassumedthatthecomputingmachineiscomputingF,andnotsomeotherfunction,suchasG.Evenafterthecomputationends,andonecansee(byobservation)thatthecomputingmachineoutputsthedigitsinthecomputationofF(n),itmuststillbeassumedthatFhasbeencomputed,andnotsomeotherfunction,suchasG(becauseforeachdigitinF(n),itcouldhavebeencomputedbyG,….)

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Kripke’s argumentisnotanunderdetermination argument

• Anunderdetermination argument:givenevidencee,therearenhypothesescompatiblewithe.Whereeistheoutputm=F(n),thereareinfinitelymanyrecursivefunctionswhichagreewiththatoutputforthatdomainvalue.AsothervaluesofFarecomputed,thenumberofhypothesescompatiblewithedecreases.

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• AsmoreandmorevaluesofeitherForofthedigitsofF(n)arecomputed,moreandmorepossiblefunctionsarisethatthecomputingmachinemightbecomputing.Thisishowthephenomenonofmachinemalfunctionisimportantlydifferentfromthephenomenonoftheunderdetermination oftheorybyevidence.

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• Inunderdetermination arguments,wecanmeaningfullyspeakofhowlikelyitisthatsomefunctionFhasbeencomputed,sincewehavedataconcerningallofthefunctionswhichthecomputermighthavecomputed.

• InKripke’s argumentagainstfunctionalism,wecannotmeaningfullyspeakoflikelihoods.

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Whylikelihoodsareruledout

• WhichfunctionFagivencomputercomputesmightbeanyoneof2n differentfunctions.

• ButunlessoneidealizesastowhichfunctionFagivencomputercomputes,itwon’tbeanyofthose2n functions.

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• Itwouldappeartomakesensetoask:“HowlikelyisitthatFiscomputed?”

• Giventhat2n functionscouldbecomputed,weanswer:“Itis1/2n likelythatFiscomputed.”

• Butthismakessenseonlyifthereisafact-of-the-matterastowhichFiscomputed.

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• However,intheabsenceofmakinganidealizationastowhichFacomputercomputes,thereisno fact-of-the-matterastowhichFitcomputes.

• Andoncetheidealizationismade,thefact-of-the-matteristhatonlyonefunctionFiscomputed.SoitiscertainthatFiscomputedundertheidealization.

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Whatcouldweconcludefromanunderdetermination argument?

• Let’sbrieflylookatwhatwewouldsayaboutagivenphysicalcomputerphysicallycomputingsomefunctionFwhereweemployanunderdetermination argument.

• Thisisusefultodo,sinceonemightconfuseKripke’s argumentagainstfunctionalismwithanunderdetrmination argument.

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• NomathematicianiseverjustifiedthatacomputingmachineiscomputingFbecausetheprobabilitythatthemachineiscomputingFislessthanorequalto.5

• Indeed,forallcomputationsofanyfunction,theprobabilitythemachineiscomputingthatfunctionislessthanorequalto.5

• WehavenomorereasontobelievethecomputingmachineiscomputingFthanwehavereasontobelievethatafairflipofafaircoinwillcomeupheads.

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• Sincethereare2n differentfunctionscomputedthatarecompatiblewithagivenoutputofacomputingmachine,theprobabilitythatthecomputingmachinecomputesFis1/2n.

• ThemoredigitsinF(n)thatarecomputed,themorelikelyitisthatF(n)hasbeencomputedbythecomputingmachine.

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• Supposethatacomputingmachineoutputsm,whichhappenstobethevalueoftherecursivefunctionF(n).ItalsohappenstobethevalueofG(k),…

• Supposethatthereareinfinitelymanyrecursivefunctionsthatoutputmforagiveninputvaluen.(ThereareinfinitelymanyrecursivefunctionsthatagreewithF(n)fordomainvaluen.)

• ItwouldthenfollowthattheprobabilitythatthecomputingmachinecomputesFis1/∞=0(asalimit,butofwhatfunction)orindeterminate

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• Underdetermination oftheorybyevidenceisnotwhatisthecasewhereitisthecomputingmachinewhichmayormaynotbeexhibitingabreakdown.Whatfunctionitiscomputingdetermineswhetheritisinbreakdownmodeorisoperatingnormally.Butonecannotknowwhatfunctionitiscomputingwithoutknowingwhetheritisoperatingnormallyorisinbreakdownmode.

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Whycomputersareunreliable

• WehavebelaboredthedifferencebetweenKripke’s argumentagainstfunctionalismandunderdetermination claimssothatonecanseefairlyeasilythatunderdetermination claimsdonotshowcomputersareunreliable.

• However,Kripke’s argumentagainstfunctionalismdoesshowcomputersareunreliable,sinceintheabsenceofmakinganidealizationastowhichFacomputercomputes,thereisno fact-of-the-matterastowhichFitcomputes.

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Whycomputersareunreliable• Thatthereisnofact-of-the-matterastowhichFagivencomputercomputesandthatsomeonewhousesthecomputermuststipulatewhichFitdoescomputeshowstheyareunreliable.

• Reliabilityofaphysicaldeviceisestablishedbyshowingtheextenttowhichtheoutputsofthedevicecorrespondtowhatwetakethedevicetoberegistering,computing,measuring,etc.Butifthereisnofact-of-the-matterattowhatthedeviceregisters,computes,measures,etc.,thenitcannot,bydefinition,bereliable.

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Beingrightandsayingacomputerisrightinwhatitcomputes

• InidealizingacomputerascomputingF,oneisstipulatingthatthecomputercomputesF.Intheabsenceofsuchanidealization—orstipulation—thereisnofact-of-the-matterastowhatthecomputercomputes—indeed,astowhatitdoes.

• Thedistinctionbeingthecomputerbeingrightinwhatitcomputesandoursayingitisrightinwhatitcomputesvanishes.

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• IfWandaidealizesagivencomputingmachineascomputingF,thenthatiswhatitcomputes—viz.,F.

• IfGregidealizestheverysamecomputingmachineascomputingG,thenthatiswhatiscomputes—viz.,G.

• Thereisnofact-of-the-matterastowhichidealizationiscorrect.Sowecannotspeakinthesecasesof‘correctness.’

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• Wherewecannotspeakofafact-of-the-matteraboutwhichoneiseithercorrectornotcorrect,wehaverelativism.

• Truth-relativismisthedoctrinethattruthisrelativetoaspeaker.Itisaninsidiousdoctrinethatphilosophershavedonetheirbesttorefute.

• Computation-relativismisthedoctrinethatwhichcomputationagivencomputermakesisrelativetotheidealizationagivenpersonmakes.ItisaconsequenceofKripke’s argumentagainstfunctionalism.

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Wittgensteinonbeingrightandsayingoneisright

• “AndnowitseemsquiteindifferentwhetherIhaverecognizedthesensationright ornot.LetussupposeIregularlyidentifyitaswrong,itdoesnotmatterintheleast.AndthatalsoshowsthatthehypothesisthatImakeamistakeismereshow.”LudwigWittgenstein,PhilosophicalInvestigations,paragraph270.

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Putnamonbeingrightandsayingoneisright

• “therelativistcannot…makeanysenseofthedistinctionbetweenbeingright andthinkingheisright;andthatmeansthereis…nodifferencebetweenassertingorthinking,ontheonehand,andmakingnoises…ontheother.…Toholdsuchaviewistocommitasortofmentalsuicide.”HilaryPutnam,Reason,Truth,andHistory,p.122

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Truthrelativismandcomputationrelativism

• Computationrelativismappearstobesuchanabsurdview(liketruthrelativism),thatonenaturallytakesittobeareductio ofKripke’sargumentagainstfunctionalism.

• However,althoughtherearecompellingargumentswhichrefutetruth—relativism,therearenocompellingarguments(thusfar)whichrefuteKripke’s argumentagainstfunctionalism.

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FourthProblemforBurge

• “ItisadelicateandunresolvedmatterhowtodistinguishthecasesinwhichwarrantforcontinuingrelianceonasourceQrequiresanempiricalinduction,orevenanempiricalentitlement,fromthecasesinwhichempiricalrecognitioncanbesubmergedintoknowinghowtoaccessarationalresource.”Burge,op.cit.,p.29

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FourthProblemforBurge• “[Casesinwhichempiricalrecognitioncanbesubmerged]requirethattheperceivablepropertiesofacomputerorpersonthatoneusesasarationalresourceberelativelysimple.Ithinkthattheymustbeincorporatedintoanearlyautomaticroutine.Itisimportantthattherecipientneednotengageincontext-dependentempirical(ornon-empirical)trackingexercises,orcomplextheorizing,toreidentify theresource…throughitspossiblychangingphysicalcharacteristics.”Burgeop.cit.,p.29

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• Inordertotrackthestateofthesystemmakingthecomputations,therecipientwillneedtoidealizethebehaviorofthatsystem.Why?Becauseintheabsenceofanidealization,therecipientcannotsaywhatthesystemiscomputing:whetheritiscomputingthefunctiontherecipienttakesittobecomputing,orwhetheritiscomputinganotherfunction

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• Withoutmakingtheidealization,therecipientcannotknowwhetherthemachineiscomputingthefunctionshetakesittobecomputing,undernormalconditionsofoperation,orcomputinganotherfunctionshedoesnottakeittobecomputing,underabnormalconditionsofoperation.

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• Withoutmakingsuchanidealization,therecipientcannotknowwhetherthemachineisoperatingundernormalconditions,oroperatingunderabnormalconditions.

• Ifthemachineisidealizedasoperatingundernormalconditions,anditoutputswhattherecipientthinksitshouldoutput,thenthemachineiscomputingthefunctiontherecipienttakesittobecomputing.

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• Makingsuchanidealizationisanecessarypartofunderstandingwhatfunctionthemachineiscomputing.Noticethateveniftherecipienthasestablishedthemathematicalpowersofthemachinebyapriorireasoning,thatdoesnotestablishherepistemicentitlementthatsheiswarrantedinbelievingthemachinewillcorrectlycomputethefunctionsshetakesittobecomputing.

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• Butmakingsuchanidealizationistoengagein“complextheorizingtore-identifytheresourcethroughitspossiblychangingphysicalcharacteristics.”

• Weneedtorefertoempiricalconstancynotjustforaccess,butalsorefertoitinourwarrant.(Burge:“Werelyonempiricalconstancyforaccess,withouthavingtorefertoitinourwarrant,”[J.B.unlesstherecipientengagesincomplextheorizing.]

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Wheredowegofromhere?

• Thereismuchworktobedoneondevelopingaconceptofmathematicalproofandonproofassistants.Butnomatterwhatthedevelopmentoftheseareaslookslikeinthefuture,unlesswecometotermswiththephilosophicalquestionsconcerningthenatureofthehumanmind,wewillnotbeinapositiontosaywhetheramathematicalproofthatusescomputers(inthewaythe4CTdoes)isagenuinemathematicalproof.

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Wheredowegofromhere?• Boththehumanbrainandaphysicalcomputerarephysicalobjects,subjecttobreakdownandmalfunction.

• Modelingthehumanmindasacomputationaldeviceworksattheabstractlevel,butcomputationaldevicesmustbephysicallyrealized,anditisintheirphysicalrealizationthatproblemsarise.

• Couldwere-thinkhowacomputerworksbyanalogywithanon-computationalmodelofthehumanmind?Wouldthatgetaroundtheproblemsthatarisewithphysicalrealizations?

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Wheredowegofromhere?• Speculation:wewillnothaveanadequateconceptofmachinecomputationsuntilwehaveanadequatesetofconceptsonthenatureofthehumanmind.

• Whethertheseconceptsmustrespectthemathematicalworkoncomputationisanopenquestion.Itmightbethat,e.g.,anewconceptofcomputationalcomplexitywillbeneeded.

• Thisseemsstrange;indeed,itISstrange.ButtheargumentsIhavepresentedheretodayshowthat,althoughstrange,itis(perhaps)necessary.

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TheEnd

• ThankstoBonnieGoldformuchhelpfuleditorialadviceanddiscussion.

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