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Computers,mathematicalproof,andthenatureofthehumanmind
POMSIGMAAKeynoteAddressPhilosophyofMathematicsAmericanMathematicalSocietyandMathematicalAssociationofAmericaJoint
MeetingsJeffBuechner
RutgersUniversity-NewarkandTheSaulKripke Center,CUNY,TheGraduateCenter
January6th,2017
1
1976AppelandHakenprovethefour-colortheorem
• June21,1976WolfgangHakenandKennethAppel,withtheaidofJohnKoch,completedtheirproofoftheFour-ColorTheorem(4CT).(Hakenturned48yearsoldonthatday.)
• Theirproofwaspublishedin1977:“Everyplanarmapisfourcolorable,”Parts1andII,andSupplementsIandII,IllinoisJournalofMathematics,XXI,84,September1977
2
1976AppelandHakenprovethefour-colortheorem
• Atoneplaceintheproofofthe4CT,AppelandHakenneedtofindafinitelistofreducibleconfigurationswiththepropertythateverygraphcontainsatleastoneconfigurationinthelist.Todothis,alemmamustbeproved:thateveryconfigurationinanunavoidablesetisreducible.Acomputerisneededtoprovethatalloftheconfigurationsarereducible.Forinstance,toshowthatonekindofconfigurationinthesetisreduciblerequires1,000,000steps.
3
1976AppelandHakenprovethefour-colortheorem
• AlthoughcomputershadalreadybeenusedtoprovetheoremsinmathematicsbeforetheAppel-Hakenproof,theimportanceofthe4CTbroughttheuseofacomputerinprovingittotheforefrontofattentionofmathematicians,aswellasalaypublic.Moreover,somemathematiciansdidnotbelievethatthetheoremhadbeenproved,sincethecomputerproofofpartofthe4CTistoolongforahumanbeingtosurvey.
4
1976AppelandHakenprovethefour-colortheorem:discordintheranks
• “Intheanalysisofeachcasethecomputeronlyannouncedwhetherornottheprocedureterminatedsuccessfully.Theentireoutputfromthemachinewasasequenceofyeses.Thismustbedistinguishedfromaprogramwhichproducesaquantityasoutputwhichcansubsequentlybeverifiedbyhumansasbeingthecorrectanswer…Therealthrillofmathematicsistoshowthatasafeatofpurereasoningitcanbeunderstoodwhyfourcolorssuffice.AdmittingthecomputershenanigansofAppelandHakentotherealmofmathematicswouldonlyleaveusintellectuallyunfulfilled.”DanielCohen“Thesuperfluousparadigm,”1991
5
1976AppelandHakenprovethefour-colortheorem:discordintheranks
• “Nowhereintheirlongandoftenirrelevantaccountdotheyprovidetheevidencethatwouldenablethereadertocheckwhattheysay.Itmayormaynotbe‘possible’toprovethecolortheoremthewaytheyclaim.Whatismorecertainisthattheydidnotdoso…notonlyisnoprooftobefoundinwhattheypublished,butthereisnotanythingthatevenbeginstolooklikeaproof.Itisthemostridiculouscaseof‘TheKing’sNewClothes’thathaseverdisgracedthehistoryofmathematics.”GeorgeSpencer-Brown,appendixtoGermaneditionofhisLawsofForm
6
TheirproofisimplicitlyrecognizedasvalidbytheUnitedStatesPostal
Authority
7
TheirproofisimplicitlyrecognizedasvalidbytheUnitedStatesPostal
Authority• Themarking‘FOURCOLORSSUFFICE’wasmadebyaUniversityofIllinois-Urbanapostalmeter,notataUnitedStatesPostOffice.Butrecognitionisimplicit,sinceanythingwhichisillegalcannotbemarkedonastampbyauniversitypostalmeter.Sofar,however,theUnitedStatesPostalAuthoritydoesnottakemistakenmathematicalproofstobeillegal.
8
Ashorterandimprovedproof
• NeilRobertson,DanielSanders,PaulSeymour,andRobinThomasprovideanewproofofthefour-colortheoremin1994.AnoutlineoftheirproofispublishedinProceedingsoftheInternationalCongressofmathematiciansin1995.
9
1979Tymoczko ontheFour-ColorTheorem
• Thefirstpaperinthephilosophyofmathematicsonthephilosophicalimportanceofthefour-colortheoremappearedinFebruary,1979.
• ThomasTymoczko “TheFour-ColorProblemandItsPhilosophicalSignificance,”JournalofPhilosophyVol.76,No.21,pp.57-83
10
1979Tymoczko ontheFour-ColorTheorem
• “Whatreasonisthereforsayingthatthe4CTisnotreallyatheoremorthatmathematicianshavenotreallyproducedaproofofit?Justthis:nomathematicianhasseenaproofofthe4CT,norhasanyseenaproofthatithasaproof.Moreover,itisveryunlikelythatanymathematicianwilleverseeaproofofthe4CT.”Tymoczko,op.cit.p.58
• Elementaryinference:AppelandHakenaremathematicians.Soneitherhaseverseenaproofofthe4CT.
11
Tymoczko onwhatthefour-colortheoremshows
• “Ifweacceptthe4CTasatheoremwearecommittedtochangingthesenseoftheunderlyingconceptof‘proof.’”
• “Theuseofcomputersinmathematics,asinthe4CT,introducesempiricalexperimentsintomathematics,andraisesagainforphilosophytheproblemsofdistinguishingmathematicsfromthenaturalsciences.”Tymoczko op.cit.p.58
12
Tymoczko onthefour-colortheorem
• “Theanswerastowhetherthe4CThasbeenprovedturnsonanaccountoftheroleofcomputersinmathematics.”op.cit.p.58
• “The4CTissubstantialpieceofpuremathematicswhichcanbeknownbymathematiciansonlyaposteriori.Ourknowledgemustbequalifiedbytheuncertaintyofourinstruments,computerandprogram...Thedemonstrationofthe4CYincludesnotonlysymbolmanipulation,butthemanipulationofsophisticatedexperimentalequipmentaswell:thefour-colorproblemisnotaformalquestion.”Tymoczko,op.cit.pp.77-78
13
Tymoczko onthefour-colortheorem
• “Theideathatapropositionofpuremathematicscanbeestablishedbyappealingtoempiricalevidenceisquitesurprising.Itentailsthatmanycommonlyheldbeliefsaboutmathematicsmustbeabandonedormodified.Consider:
• 1.Allmathematicaltheoremsareknownapriori• 2.Mathematics,asopposedtonaturalscience,hasno
empiricalcontent.• 3.Mathematics,asopposedtonaturalscience,reliesonly
onproofs,whereasnaturalsciencemakesuseofexperiments.
• 4.Mathematicaltheoremsarecertaintoadegreethatnotheoremofnaturalsciencecanmatch.Tymoczko,p.63
14
Tymoczko onmathematicalproof• “Proofsareconvincing...[In]Wittgenstein’sRemarksontheFoundationsofMathematics,thisisallthereistoproofs:theyareconvincingtomathematicians.Thisistobetakenasabrutefact,somethingforwhichnoexplanationcanbegivenandnoneisnecessary.Mostphilosophersareunhappywiththispositionandinsteadfeelthattheremustbesomedeepercharacterizationsofmathematicalproofswhichexplains,atleasttosomeextent,whytheyareconvincing.”Tymoczko,op.cit.p.59
15
Tymoczko onmathematicalproof
• Whyaremathematicalproofsconvincing?• “Thatproofsaresurveyable andthattheyareformalizable aretwosuchcharacterizations[J.B.ofwhymathematicalproofsareconvincing].”Tymoczko,op.cit.p.59
16
Tymoczko onmathematicalproof
• “Weoftensaythataproofmustbeperspicuous,orcapableofbeingcheckedbyhand.Itisanexhibition,aderivationoftheconclusion,anditneedsnothingoutsideofitselftobeconvincing.”Tymockzo,op.cit.p.59
17
Tymoczko’s circle• Unfortunately,Tymoczko’s definitionofsurveyabilityusestheideathatamathematicalproofneedsnothingoutsideitselftobeconvincing.Soonemustalreadyknowwhatamathematicalproofisbeforeoneknowswhatsurveyability consistsin;butsurveyability isonecriterionofbeingamathematicalproof.
• “Themathematiciansurveys theproofinitsentiretyandtherebycomestoknowtheconclusion.”Tymoczko,op.cit.,p.59
• “Theconstructionthatwesurveyedleavesnoroomfordoubt.”Tymoczko,op.cit.p.60
18
PaulTelleronTymoczko
• “Surveyability isneeded,notbecausewithoutitaproofisinanysensenotaproof,butbecausewithoutsurveyability weseemnottobeabletoverifythataproofiscorrect.Sosurveyability isnotpartofwhatitistobeaproofinouraccustomedsense.”PaulTeller“ComputerProof,”JournalofPhilosophy,December1980,pp.797-803
19
PaulTelleronTymoczko
• “…wemaytakeadvantageofnewmethodsofsurveyingaslongastheseenableustomeetsensibledemandsoncheckingproofs,andashiftinthemeansofsurveyingactuallyusedmeansonlyashiftinmethodsofcheckingproofs,notashiftinourconceptionsofthethingschecked.”Teller,op.cit.,p.798
• Notashiftinourconceptionsofthethingschecked=notashiftinourconceptofproof
20
ThedisputebetweenTellerandTymoczko:theconceptof
mathematicalproof
• Tymoczko:surveyability isanessentialfeatureoftheconceptofamathematicalproof.
• Teller:surveyability isnotanessentialfeatureoftheconceptofamathematicalproof.
• Whoisright?Onwhatgroundsaretheyright?
21
Quine’sproblemforconcepts
• W.V.Quinearguedinhisepochalpaper“TwoDogmasofEmpiricism,”thatthereisnohardandfastdistinctionbetweenmeaning-constitutingbeliefsandauxillary beliefs(beliefsthatarenotmeaning-constituting).
• Thismeansthatitisimpossibletodrawahard-and-fastlinebetweenessential(ornecessary)featuresofaconceptandnon-essential(contingent)featuresofaconcept.
22
Quine’sproblemforconcepts
• IfAproposesthatXisanessentialfeatureoftheconceptofmathematicalproof,andBproposesthatitisanaccidentalfeatureoftheconceptofmathematicalproof,thereisnoprincipledwayofadjudicatingbetweenthetwoproposals.
• Adjudicationshouldgobywayofcanonsofrationalityandcanonsofscientificinquiry,suchasconservatism—upholdingasmanycurrentlyestablishedbeliefsaspossible.Howwouldthatworkfortheconceptofmathematicalproof?
23
Quine’sproblemforconcepts
• ThereisnoconsensusviewastowhetherQuineiscorrectornotonthis,butmostphilosopherstakeQuinetobecorrect.
• Teller’sclaimthatsurveyability isnotanessentialfeatureoftheconceptofmathematicalproofcouldbeupheldifitsatisfiedmorecanonsofrationalityandofscientificinquirythandoesTymoczko’s claim.
24
Quine’sproblemforconcepts
• However,theideaofsurveyability surfacesinthecontextofusingcomputersinmathematicalproofs.
• Therewasnotmuchdata(i.e.,featuresonwhichthereiscommonagreement)concerningtheuseofcomputersinmathematicalproofintheperiod1976-1980.
• ThedisagreementbetweenTellerandTymoczko isastalemate.
25
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
26
Detlefsen onTymoczko• Detlefsen providesseveralexamplesofmathematicalproofswhicharesurveyable andinwhichcomputationsaremade.Hearguesthatsuchcomputationsnecessarilyutilizeempiricalpremises(suchas:thecomputingagentcorrectlyexecutestheprogramrequiredtomakethecomputation).
• Ifhisargumentissound,Detlefsen hasshownthatunsurveyability isnotnecessaryforthepresenceofanempiricalelementinmathematicalproofs.ThisrefutesamajorclaiminTymoczko’s paper.
27
Detlefsen onTymoczko• “ThiscreatesadilemmaforTymoczko.Foreitheronerejectshisreasoning,inwhichcaseheisleftwithoutanargumentfortheempiricalcharacteroftheproofofthe4CToroneacceptshisreasoning,butisthenforcedtoviewthepresenceofcalculationorcomputationinaproofasinjectinganempiricalelementintothatproof.TheconsequenceofsuchaviewisthatempiricalproofsaremorewidespreadthanTymoczkohimselfindicates.”MichaelDetlefsen,op.cit.,p.809
28
Detlefsen onTymoczko• “ThiscreatesadilemmaforTymoczko.Foreitheronerejectshisreasoning,inwhichcaseheisleftwithoutanargumentfortheempiricalcharacteroftheproofofthe4CToroneacceptshisreasoning,butisthenforcedtoviewthepresenceofcalculationorcomputationinaproofasinjectinganempiricalelementintothatproof.TheconsequenceofsuchaviewisthatempiricalproofsaremorewidespreadthanTymoczkohimselfindicates.”MichaelDetlefsen,op.cit.,p.809
29
BackgroundtoBurge:Descartesonmathematicalproofs
• [Inlongdeductions]“thelastlinkisconnectedwiththefirst,eventhoughwedonottakeinbymeansofoneandthesameactofvisionalltheintermediatelinksonwhichthatconnectiondepends,butonlyrememberthatwehavetakenthemsuccessivelyunderreview…”Descartes,RulesfortheDirectionofMind
• ForDescartes,“ifthatknowledgeisdeducedfromevidentmathematicalpremises,itiscertainanddemonstrative.”TylerBurge,ContentPreservation,PhilosophicalIssues,1995,p.271
30
BackgroundtoBurge:Chisholmonmathematicalproofs
• “WhatifSderivesapropositionfromasetofaxioms,notbymeansofoneortwosimplesteps,butasaresultofacomplexproof,involvingaseriesininterrelatedsteps?Iftheproofisformallyvalid,thenshouldn’twesaythatSknowsthepropositionapriori?Ithinkthattheanswerisno.”RoderickChisholm,TheoryofKnowledge,2nd edition
31
BackgroundtoBurge:Chisholmonmathematicalproofs
• “[I]f,inthecourseofademonstration,wemustrelyuponmemoryatvariousstages,thususingaspremisescontingentpropositionsaboutwhatwehappentoremember,then,althoughwemightbesaidtohave‘demonstrativeknowledge’ofourconclusion,inasomewhatbroadsenseoftheexpression‘demonstrativeknowledge,’wecannotbesaidtohaveaa prioridemonstrationoftheconclusion.”RoderickChisholm,op.cit.,
32
WhyisChisholm’spointimportant?
• a posterioriknowledge:knowledgewhichisarrivedatonthebasisofsenseexperiencesorperceptualbeliefs.
• aprioriknowledge:knowledgewhichisarrivedatonthebasisofintellectualprocesseswhichdonotinvolvereferencetoorrelianceuponsenseexperiences.
• aposteriorijustification:justificationwhichreliesuponsenseexperiences.
• apriorijustification:justificationwhichemploysintellectualprocesseswhichdonotinvolvereferencetoorrelianceuponsenseexperiences.
33
WhyisChisholm’spointimportant?
• IfChisholmisrightthatlongmathematicalproofsrequireapremiseaboutwhatwehappentoremember—andthusarenoteitherknownorjustifiedapriori—theniteasilyfollowsthatthoseproofsareknownorjustifiedempirically.Theyrelyuponorrefertosenseexperiences.
• Tymoczko isrightabouttheepistemologicalstatusofthe4CTifweacceptChisholm’spoint.Butheiswrongthatonlyunsurveyablemathematicalproofsrequire(inwholeorinpart)empiricaljustification.
34
BackgroundtoBurge:Fallis ondispensingwithempiricalevidence
• “…thereisasenseinwhichtheproofofthefour-colortheoremisanapriorijustification.Itneednotappealtoanyparticularempiricaldataandinprincipleneednotappealtoempiricaldataatall.Forinstance,therelevantcomputationcouldbeperformedbyadeviceotherthanadigitalcomputerandinprinciplecouldbeperformedinthemathematician’smind.”DonFallis,MathematicalProofandtheReliabilityofDNAEvidence,AmericanMathematicalMonthly,June-July,1996,p.496
35
BackgroundtoBurge:Fallis ondispensingwithempiricalevidence
• Fallis thinksthat,unlessitisnecessarythataphysicalmachineofsomekindperformsomecomputation,thenthecomputationisapriori,becauseitispossiblethatahumanmindcouldperformthecomputation.
• Itispossiblethatahumanmindcouldcompleteaninfinitecomputationalprocess(in,say,aMalament-Hogarthuniverse).Shouldwethensaythatsuchcomputationsareaprioriknowable?
36
Burgeontheuseofcomputersinmathematicalproofs
• TylerBurge,in1998,inhisimportantpaper“ComputerProof,APrioriKnowledge,andOtherMinds,”TheSixthPhilosophicalPerspectivesLecture(pp.1-38),sidestepstheneedtounderstandwhatamathematicalproofisinaskingwhethertheuseofcomputersinmathematicalproofsaddsanempiricalelementtosuchproofs.Burgewillarguethatnoempiricalelementneedbeaddedwhencomputersareusedinmathematicalproofs,suchastheproofofthe4CT.The4CTcanbeknownapriori(tobetrue).
37
Burgeonmathematicalproofs
• Burgeusestheterms‘epistemicentitlement,’‘epistemicwarrant,’and‘epistemicjustification’(sometimeswithoutthequalifier‘epistemic’).
• Unlessyouareaphilosopherworkinginepistemology,itisbesttotreatallofthemasmeaningthesame—namely,justification.
38
Burgeonmathematicalproofs
• “Iconclude,givenourassumptions,onecanbeinaposition,fromthethirdpersonpointofview,tobeaprioriwarrantedinbelieving,infact,knowing,ondefeasibleinductivegrounds,thatthe[4CT]hasbeenproved.Onecanknowthisevenifonecannotreplicatetheproof.”TylerBurgeComputerProof,APrioriKnowledge,andOtherMinds,p.23
39
Burgeonmathematicalproofs• “Theentitlementforrelyingonthesourcedoesspecifythesource[thecomputer].Butitneednotspecifytheobject’sempiricallydiscerniblecharacteristics,ortheempiricalbackgroundconditionsthatenabletherecipienttoaccessandrelyonthesource.Itcanspecifythesourceinthenon-empiricalwaythattheapriorijustificationdoes.”Burge,op.cit.,p.29
• Workonproofassistants(e.g.,byHarveyFriedmanandbyJeremyAvigad)providesanentitlementforrelyingonthesource(theIBM370-160AusedintheAppel-Hakenproofofthe4CT).
40
Burgeonmathematicalproofs• “Perceptualexperienceofthewordsorofthebodyofthesourceneedplaynoroleinjustifyingone’sunderstandingof,orintellectualusesof,thecontentofthewordsorthepresentationsofthesource.”Noticethatthispointwould,ifcorrect,refuteChisholm.
• “Perceptionisonlythemodeofaccess,anenablingconditionwhichmakesnocontributiontotheepistemicforceofthewarrant.”
• Perceptionismerelyaconditionthatenablesonetomakeuseofaresourceforreasonandunderstanding.”Burge,op.cit.,p.30
41
WhathasBurgeshown?
• Ifhisargumentsaresound,Burgehasshownthattheunsurveyability ofmathematicalproofsisnotsufficientfortheexistenceofanempiricalelementinsuchproofs.
42
WhathaveDetlefsen andBurgeshown?
• RecallthatDetlefsen hasshownthattheunsurveyability ofmathematicalproofsisnotnecessaryfortheexistenceofanempiricalelementinsuchproofs.Burgehasshownthatitisnotsufficient.
• TheresultsofBurgeandDetlefsen,ifbotharecorrect,showthereisnoconceptualconnectionbetweentheunsurveyability ofmathematicalproofsandtheexistenceofanempiricalelementinsuchproofs.However,bothcannotbecorrect.
43
WhathaveDetlefsen andBurgeshown?
• Ifthereisnoconceptualconnectionbetweentheunsurveyability ofmathematicalproofsandtheexistenceofanempiricalelementinsuchproofs,iteasilyfollowsthatunsurveyability hasnothingtodowiththeexistenceofanempiricalelementinmathematicalproofs.
• ExaminingtheargumentsofbothDetlefsen andBurge,thisisnotshocking,norevensurprising.
44
WhathaveDetlefsen andBurgeshown?
• Detlefsen arguedthatwhetheramathematicalproofissurveyable orunsurveyable,theremightbeempiricalelementsinit.
• Burgearguedthatwhetheramathematicalproofissurveyable orunsurveyable,thereneednotbeempiricalelementsinit.
• Ofcourse,neitherruleoutthattheremightbe,northattheremightnotbe,empiricalelementsinamathematicalproof.Butwhetherthereareorarenotisnotamatterofwhattheconceptofamathematicalproofconsistsin.Itis,rather,anentirelycontingentmatter.
45
FourProblemsforBurge
• TherearethreeproblemsforBurge’saccountofhowwecanhaveaprioriknowledgeoftheoutputofacomputer.
• Thefirstisthathisaccountmakesittooeasytohavegettiered knowledge.GettiercounterexamplesarecasesinwhichasubjectShasatrue,justifiedbeliefthatp,butinwhichSdoesnotknowthatp.
46
FourProblemsforBurge
• Hereishow,followingBurge’saccountofepistemicjustificationinthecontextofmathematicaltruths,onecanhaveatrue,justifiedbeliefinthe4CT,butnotknowthe4CT.
• Supposethatthe4CTistrue,butthatthecomputerprogramforresolvingthecasesisfallacious.OnBurge’saccount,asubjectSwillbejustifiedinbelievingthe4CTtobetrue.Sinceitistrue(byassumption),Shasatrue,justifiedbeliefinthe4CT.ButSdoesnotknowthe4CT.
47
FourProblemsforBurge
• WeshouldtakeGettier counterexamplesveryseriously.(DavidLewishasremarkedthatthereareonlytworesultsthatallphilosopherstaketobedefinitive:GödelandGettier.)
• IfanaccountofepistemicjustificationmakesittooeasyforGettier counterexamples(andnotjustpossibleforthemtoarise)toarise,thatisareasontorejectsuchanaccount.
48
FourProblemsforBurge
• ThesecondproblemforBurgeconcernshisclaimthatthemodeofaccesstosomeepistemicallywarrantedsetofpropositionsisnotnecessaryforbeingepistemicallyjustifiedinbelievingthosepropositions:“[t]heentitlementforrelyingonthesource…neednotspecifytheempiricalbackgroundconditionsthatenabletherecipienttoaccessandrelyonthesource.”Burge,op.cit.,p.29
49
FourProblemsforBurge• Thatiswhywecandiscounttheroleofmemoryindeterminingwhetherweareepistemicallyjustifiedinbelievingatheoremofmathematicsonthebasisoftheproofofthattheorem.Memoryisameremodeofaccesstotheproof.
• ForBurge,thesameistrueofcomputerproofs—wecandiscountthemodeofaccesstothetheoremanditsproof(thecomputerprogram)whichisthecomputer.
• Ifmemoryisfaulty,thatdoesnotshowthattheproofisfaulty.Indeed,afaultymemoryhasnothingtodowithaproof—whichisanabstractobject.Canwesaythesameofacomputer?Burgethinkswecan.
50
FourProblemsforBurge• Isaywecannot.Hereiswhy.Unlikememory,acomputerisnotanorganicintrinsicpartofahumanbeing.Itistheuniquemodeofaccesstothecomputerprogramandthecomputationsofthatprogram—allofwhichareabstractobjects.
• Butitismore.Itisthemeansbywhichtheabstractobjectsarephysicallyrealized.Memory,ontheotherhand,neednotbethemeansbywhichaproofisphysicallyrealized.Rather,aproofcanbephysicallyrealizedonapieceofpaperusinginktomakeinscriptions.
51
FourProblemsforBurge• Imagineacultureinwhichthereisnopaper,nowriting
instruments,andnoconceptsofwriting(onpaper,usingwritinginstruments).However,thereistheconceptofaproof.Allproofsareinhumanmemory.
• Insuchacase,weshouldsaythataproblemwithhumanmemorywouldcreateaprobleminepistemicentitlementtotheproof.Why?Becausehumanmemoryistheonlymeansbywhichtheproofisphysicallyrealized,aswellasthemodeofaccesstotheproof.
• Withoutmemory,wedonothaveaproof,sincewedonothaveanymodeofaccesstotheabstractobjectwhichistheproof.(Comparewithaproofwhichissodifficultthatnomathematicalconceptsavailabletothehumanmindareadequateforrepresentingtheproof.Insuchacase,eventhoughtheproofhasanabstractexistence,weshouldsaythatwecannotbeepistemicallyentitledtoitsincewehavenomeansbywhichtoaccessit.)
52
FourProblemsforBurge• ThethirdproblemforBurgeconcernsfourassumptionsthatBurgemakesinhisargument.Oneassumptionisthat“individual’sknowledgeofpuremathematics,restingonspecificallymathematicalunderstandingorreasoning,isordinarilyapriori.”(Burge,op.cit.,p.4)
• ThiscontradictsDetlefsen’s claim—whichdependsonTymoczko’s definitionofmathematicalproof—thatempiricalpremisesareusedinmathematicalproofsthataresurveyable (aswellasthosewhichareunsurveyable).
• WedeferourexpositionofthefourthproblemforBurge.
53
Thedialecticsofhowthingsstand• Tymoczko andTeller:stalemate• Tymoczko andDetlefsen:IfDetlefsen iscorrect,empiricalpremisesoccurinmathematicalproofsthatarebothsurveyable andunsurveyable.Thisputspressureongettingclearonwhatwemeanbyamathematicalproof.
• Tymoczko andBurge:IfBurgeiscorrect,thentheuseofcomputersinmathematicalproofsdoesnotintroduceanempiricalelementintothoseproofs(nordoestheuseofcomputationsinmathematicalproofs).Tymoczko andDetlefsen arebothrefuted.
54
Thedialecticsofhowthingsstand• ButwehavepresentedreasonsforthinkingthatBurge’sargumentfails.
• Itisclearthatmuchphilosophicalworkstillneedstobedoneinexplicatingtheconceptofamathematicalproof.
• Butnomatterwhatthatexplicationeventuallyconsistsin,itmustbecompatiblewithourviewsaboutthenatureofcomputersandthenatureofthehumanmind.Thatthis(perhapsstartlingview)issowillbearguedintheremainderofthistalk.
55
Alineofthoughtnottaken• WewillnowdiscussalineofthoughtthatisbroachedbyTeller,Detlefsen,Davis,andTymoczko,butnottakenupbyanyofthem.
• Inhispaper,Tellerwrites:“Theallegednonsurveyability alsounderliesTymoczko’s secondconclusion:thecomputerproofofthecombinatoriallemmaissubjecttoerror—computerscanmakemistakes.Wecannotguardagainstthispossibilityofmechanicalfailureorerrorinprogramminginthetraditionalwaybecausewecannotsurveytheproof.”Teller,op.cit.,p.798
56
Alineofthoughtnottaken
• “Whatiftheprogrammingwaserroneous?Whatiftheinitialdatawerefake?Whatiftherewasamachinemalfunction?”
• “Theseconsiderationsleadustoaposition—whichisrarelydiscussedinworksonthephilosophyofmathematicsandwhichisveryunpopular—thatamathematicalproofhasmuchincommonwithaphysicalexperiment.”P.Davis,“Formac MeetsPappus,”American
mathematicalMonthly,1969,pp.903-904.
57
JohnHortonConwayoncomputers
• Awell-knownmathematician,JohnH.Conway,hasbeenquotedassaying:“Idon’tlikethem[computers],becauseyousortofdon’tfeelyouunderstandwhat’sgoingon.”NewYorkTimes,April6,2004KennethChang“Inmath,computersdon’tlie.Ordothey?”anarticleontheuseofcomputersinmathematicalproofs
58
Wittgensteinonmachinecomputation
• “Ifweknowthemachine,everythingelse…seem[s]tobealreadycompletelydetermined.Wetalkasifthesepartscouldonlymoveinthisway,asiftheycouldnotdoanythingelse.Isthishowitis?Doweforgetthepossibilityoftheirbending,breakingoff,melting,andsoon?Yes,inmanycaseswedon’tthinkofthatatall.Weuseamachine,orapictureofamachine,asasymbolofaparticularmodeofoperation.Forinstance,wegivesomeonesuchapicture,andassumethathewillderivethesuccessivemovementsofthepartsfromit.”LudwigWittgensteinPhilosophicalInvestigations,§ 193
59
Kripke onWittgenstein
• “Wittgensteinhimselfdrawsthedistinctionbetweenthemachineasanabstractprogram(‘derMachineals Symbol,’PI193)andtheactualphysicalmachine,whichissubjecttobreakdown(‘Doweforgetthepossibilityoftheirbending,breakingoff,melting,andsoon?’PI193)”SaulKripke,WittgensteinonRulesandPrivateLanguage,p.35,fn.24
60
Naivecomputerviewofthemind
• “Amachinecanfollowthisrule;whencedoesahumanbeinggainafreedomofchoiceinthismatterwhichamachinedoesnotpossess?”
SirMichaelDummett “Wittgenstein’sPhilosophyofMathematics,”PhilosophicalReviewVol.68(1959),pp.324-348,atp.351
61
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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Kripke’s argumentisnotanunderdetermination argument
• AsmoreandmorevaluesofeitherForofthedigitsofF(n)arecomputed,moreandmorepossiblefunctionsarisethatthecomputingmachinemightbecomputing.Thisishowthephenomenonofmachinemalfunctionisimportantlydifferentfromthephenomenonoftheunderdetermination oftheorybyevidence.
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Kripke’s argumentisnotanunderdetermination argument
• Inunderdetermination arguments,wecanmeaningfullyspeakofhowlikelyitisthatsomefunctionFhasbeencomputed,sincewehavedataconcerningallofthefunctionswhichthecomputermighthavecomputed.
• InKripke’s argumentagainstfunctionalism,wecannotmeaningfullyspeakoflikelihoods.
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Whylikelihoodsareruledout
• WhichfunctionFagivencomputercomputesmightbeanyoneof2n differentfunctions.
• ButunlessoneidealizesastowhichfunctionFagivencomputercomputes,itwon’tbeanyofthose2n functions.
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Whylikelihoodsareruledout
• Itwouldappeartomakesensetoask:“HowlikelyisitthatFiscomputed?”
• Giventhat2n functionscouldbecomputed,weanswer:“Itis1/2n likelythatFiscomputed.”
• Butthismakessenseonlyifthereisafact-of-the-matterastowhichFiscomputed.
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Whylikelihoodsareruledout
• 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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Whatcouldweconcludefromanunderdetermination argument?
• NomathematicianiseverjustifiedthatacomputingmachineiscomputingFbecausetheprobabilitythatthemachineiscomputingFislessthanorequalto.5
• Indeed,forallcomputationsofanyfunction,theprobabilitythemachineiscomputingthatfunctionislessthanorequalto.5
• WehavenomorereasontobelievethecomputingmachineiscomputingFthanwehavereasontobelievethatafairflipofafaircoinwillcomeupheads.
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Whatcouldweconcludefromanunderdetermination argument?
• Sincethereare2n differentfunctionscomputedthatarecompatiblewithagivenoutputofacomputingmachine,theprobabilitythatthecomputingmachinecomputesFis1/2n.
• ThemoredigitsinF(n)thatarecomputed,themorelikelyitisthatF(n)hasbeencomputedbythecomputingmachine.
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Whatcouldweconcludefromanunderdetermination argument?
• Supposethatacomputingmachineoutputsm,whichhappenstobethevalueoftherecursivefunctionF(n).ItalsohappenstobethevalueofG(k),…
• Supposethatthereareinfinitelymanyrecursivefunctionsthatoutputmforagiveninputvaluen.(ThereareinfinitelymanyrecursivefunctionsthatagreewithF(n)fordomainvaluen.)
• ItwouldthenfollowthattheprobabilitythatthecomputingmachinecomputesFis1/∞=0(asalimit,butofwhatfunction)orindeterminate
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Whatcouldweconcludefromanunderdetermination argument?
• 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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Beingrightandsayingacomputerisrightinwhatitcomputes
• IfWandaidealizesagivencomputingmachineascomputingF,thenthatiswhatitcomputes—viz.,F.
• IfGregidealizestheverysamecomputingmachineascomputingG,thenthatiswhatiscomputes—viz.,G.
• Thereisnofact-of-the-matterastowhichidealizationiscorrect.Sowecannotspeakinthesecasesof‘correctness.’
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Beingrightandsayingacomputerisrightinwhatitcomputes
• 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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FourthProblemforBurge
• Inordertotrackthestateofthesystemmakingthecomputations,therecipientwillneedtoidealizethebehaviorofthatsystem.Why?Becauseintheabsenceofanidealization,therecipientcannotsaywhatthesystemiscomputing:whetheritiscomputingthefunctiontherecipienttakesittobecomputing,orwhetheritiscomputinganotherfunction
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FourthProblemforBurge
• Withoutmakingtheidealization,therecipientcannotknowwhetherthemachineiscomputingthefunctionshetakesittobecomputing,undernormalconditionsofoperation,orcomputinganotherfunctionshedoesnottakeittobecomputing,underabnormalconditionsofoperation.
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FourthProblemforBurge
• Withoutmakingsuchanidealization,therecipientcannotknowwhetherthemachineisoperatingundernormalconditions,oroperatingunderabnormalconditions.
• Ifthemachineisidealizedasoperatingundernormalconditions,anditoutputswhattherecipientthinksitshouldoutput,thenthemachineiscomputingthefunctiontherecipienttakesittobecomputing.
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FourthProblemforBurge
• Makingsuchanidealizationisanecessarypartofunderstandingwhatfunctionthemachineiscomputing.Noticethateveniftherecipienthasestablishedthemathematicalpowersofthemachinebyapriorireasoning,thatdoesnotestablishherepistemicentitlementthatsheiswarrantedinbelievingthemachinewillcorrectlycomputethefunctionsshetakesittobecomputing.
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FourthProblemforBurge
• 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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