POST-HUMANMATHEMATICS.
byDavidRuelle†.Abstract: Presentdaymathematicsisahumanconstruct,wherecomputersareused moreandmorebutdonotplayacreativerole.Thissituationmaychangehowever: computersmaybecomecreative,andsincetheyfunctionvery differentlyfromthehuman braintheymayproduceaverydifferentsortofmathematics.Wediscusswhatthisposthumanmathematicsmaylooklike,andthephilosophicalconsequencesthatthismay entail.
† Math.Dept.,RutgersUniversity,andIHES,91440BuressurYvette,France.email: ruelle@ihes.fr
1.Introduction:vitalism.
Thepurposeofthisnoteistolookintowhatmaybethemathematicsoftomorrow,ifandwhencomputerprogramshavegainedamathematicalcreativitythattheyare currentlylacking.
Beforethathoweveritwillbeilluminatingtodiscussbrieflyanaspectofthebiology andchemistryofyesterday,namelyvitalism.Vitalismisthenotionthatlivingmatter containsavitalprinciplewhichisabsentfromnon-livingentities,sothatlivingmatter obeysdifferentlawsfromthosethatrulenon-livingmatter. Thisisanoldidea,anditisby nomeansridiculous.Thisideahasledinchemistrytoadistinctionbetweenorganicand inorganicsubstances.ThegreatSwedishchemistJ.J.Berzelius(1779-1848)suggestedthat organicchemicalcompoundscontainedavitalforce,absent frominorganiccompounds. Indeed,itdidnotappearpossibletosynthesizeorganiccompoundsfrominorganicones. ThatwasuntilthesynthesisofureabyF.W¨ohlerin1828,followedbythesynthesisof aceticacidbyA.Kolbein1845,andthesynthesisofmoreandmoreorganiccompounds afterthat.
Organicsynthesisinthelabhasdifferentfeaturesfromorganicsynthesis invivo,but thereappearstobenolimittowhatcanbesynthesized.Eventhepossibilityofcreating lifeinthelabisnolongerataboosubject;manyscientistsbelievethatitwillbedone, althoughwhatkindoflifeandwhenthiswillbeachievedremaindebatedquestions.
2.Theuniquenessofhumanintelligence.
Althoughvitalismhaslargelybeenabandonedbyscientists,thereisabeliefrelated tovitalismwhichremainsquitepopular,thisconcernsthehumanmindandhumanintelligence.Alotofpeople,andthisincludesveryrespectedscientists,believethatthehuman mindhasuniquecreativeabilitieswhichneitheranimalsnorcomputerscanduplicate.We shallleaveanimalsoutofthepresentdiscussion,andconcentrateononeaspectofcreativity:mathematicalcreativity.Iseetwosignificantargumentsinfavoroftheuniqueness ofhumanmathematicalcreativity.Thefirstargumentisthat wehaveanintrospective feelingofbeingabletothink(inthepresentcasetothinkaboutmathematicalproblems) whichwewoulddenycomputersbecausetheyarejustmachines.Thesecondargumentis thelackofseriousexamplesofmathematicalcreativitybycomputers.Letmenowdiscuss andquestionthesetwoarguments.
2.1Canacomputerthink?
WhenIspeakofacomputerinwhatfollows,Ialwaysmeanhardwareplussoftware, i.e.,acomputerrunningasuitableprogram.Thefeelingthatwecanthinkwhilecomputers cannothasbeenconsideredbyAlanTuring[7].Hediscussedthepossibilityofatestwhich ahumanwouldpass,butwhichamachinewouldfail.Theconsensusatthistimeisthat nosuchtestcanbedevised.Adiscussionofintelligencebasedontheintrospectivefeeling thatonlywecanthinkisthusoflimitedinterest.Wecanhoweversaythatifpresentor futurecomputersthink,theirthinkingmustbeofaverydifferentnaturefromours:this questioncanbeobjectivelyanalyzed,andhasbeendiscussedbyJohannvonNeumann inaremarkablelittlebookcalled ThecomputerandtheBrain [3].Boththecomputer
andthebrainareinformationprocessingsystems,andadetailedcomparisonmadebyvon Neumannshowsthattheyfunctionverydifferently.Thebrain isslow,pronetoerror,has limitedmemory,andisveryhighlyparallel(itisaconnectedmany-channelsystem).By comparison,computersareveryfast,reliable,havelargememory,andareusuallynotvery highlyparallel.Computerintelligence,ifitexists,isthusexpectedtobequitedifferent fromhumanintelligence.
Bothcomputersandhumanshaveseveraltypesofmemories.Forthetaskofputting astringofdigitsinshorttermmemory,acomputerisfarsuperiortoahuman(whocan rememberonlyaboutsevendigits).Butourbrainalsopreservesalifetimeoflongterm memories:anapparentlylimitlessamount.Bycomparison,earlycomputersdidn’tdo thatwell,butthingshavechanged:progressintranslation bycomputersshowsthatthey areprogressivelybecomingabletomasterthehugecorpusof datacorrespondingtothe knowledgeofahumannaturallanguage.
Ineverydaylifewenowfrequentlyhavetointeractwithcomputerprograms,andthis influencestheopinionwemayhaveontheirintelligence.WhenIuseGoogle,Ihavethe feelingofavastintelligencesomehowpresentthere.(This isbecauseGooglehasrapidand intelligentaccesstoahugeamountofdata.)Bycontrast,whenIrecentlytriedtosubmit apaperforpublicationinacoupleofscientificjournals,andhadtofightovertheInternet withtheireditorialprograms,IwasconvincedthatIwasfacingsomethingviciously unintelligent.Youknowhowitis:thethingdemandsthatyou pleaseenterdigital certificateofvirginityofgrandmother,orsomesuchnonsense.Whateveryou answeris,thestupidmachinereplies certificateinvalidpleasetryagain.Letus trytoovercometheirritationproducedbysuchexperiences,andproceedwithaserene discussion.
Ourconclusionforthemomentwillbethis:wecannotexclude thatcomputersthink, butifandwhentheydoitwillprobablybeinamannerverydifferentfromthatofhumans. Thesituationisabitsimilartothatoforganicchemicalsynthesis:artificialsynthesisof organiccompoundsisnotimpossible,farfromthat,butitis generallyachievedquite differentlyfromchemicalsynthesisinlivingorganisms.
2.2Ismathematicalcreativitybycomputerspossible?
Wecomenowtothesecondpointofourdiscussion:thecomputer’scurrentlackof mathematicalcreativity.Computershaveactuallygainedquiteabitofimportancein mathematics.JustthinkthatthegreatBernhardRiemanntestedsomemathematical ideasbylongnumericalcalculations;hismoderncolleaguesoftenworkinasimilarway butdotheircalculationsbycomputerratherthanbyhand.Computersarealsousedinan essentialwaytoprovidepartsofrigorousproofs:theyperformheavylogicalornumerical taskswhicharebeyondhumancapabilities.(Anexamplehere istheproofofthefour colortheorembyKennethAppelandWolfgangHaken[1]).Letmealsomentionthat somedefinitemathematicalcreativityhasbeenobtainedusingWilf-Zeilbergerpairs[4]to producenewidentitiesinvolvinghypergeometricfunctions.
However,atthistime,theclosestthatcomputershavecometoreallydoingmathematicsisincomputer-verifiedproofs(so-called formalproofs).Inbrief,ahumanmathe-
maticiantransformsahumanproofofatheorem(likethe primenumbertheorem)intoa sequenceoflemmasinaformallanguage,andthe(nontrivial)proofthatthelemmasare correctislefttoacomputer.FordetailswerefertoaHales[2]andfurtherpapersina specialissue(December2008)oftheAMSNotices(vol. 55)onthesubject.Letmejust makeafewremarks:
(a)nontrivialtheoremsliketheprimenumbertheoremnowhaveacomputer-verified proof,
(b)computer-verifiedformalproofsarealotmorereliablethanhumanproofs,which alwayscontainsomewhatimpreciseformulations,andsometimesbigmistakes(thishas becomeamajorconcernwithverylongmodernproofs),
(c)partofthecomputer-verifiedformalproofs(theproofof lemmasbyacomputer program)nowescapesintuitivegraspbythehumanmind:theseproofsarenolonger completelyhuman,
(d)nevertheless,thecreativeroleofcomputersincomputer-verifiedproofsisminimal, beinglimitedtoacombinatorialsearchforproofsoflemmas alonglinesprogrammedby humans.
3.Whatismathematicalcreativity?
Thisisnottheplaceforapoeticdiscourseoncreativityingeneral.Rather,Iwantto seewhatcreativityimpliesinthecaseofmathematics,howitisimplementedbyhumans, andhowitmighthaveanon-humanimplementation.
Itisconvenienttoassumethatsomebasisofmathematicshas beenaccepted:logical rulesofdeductionandbasicaxioms.TheaxiomsmaybetheZermelo-Fraenkel-Choice axiomsofsettheory,orsomethingsimilarasimplementedin aprogramforcomputerverifiedproofs.Inbriefweassumethatacommonbasisformathematicsisacceptedby humansandcomputers.Doingmathematicsisthenfindingandprovingtheoremsonthe basisoftheaxioms,usingacceptedrulesoflogic.
Thereisagenerallimitationindoingmathematicsthatappliesbothtohumansand tocomputers:atheoremwithashortformulationmayhaveanextremelylongproof.This fact,notedbyG¨odel,isoflogicalorigin,andrelatedtotheincompletenesstheorem.
Aswehaveseen,thehumanbrainhaslimitedmemoryandispronetoerror.A humanmathematicaltextisthuscomposedofsmallunits(afewlines,whereacomputer couldhandle105 pages).Agreathelpinobtainingsmallunitsistheuseof definitions (for instancethedefinitionofacompactgroup,orthatofcomplex numbers)whichareagreed uponbeforemakingamathematicalstatement.
Letmesaythisagain:thehumanwayofdoingmathematicsisto writeamathematical text,consistingofshortpieceswhichmaybedefinitionsortheorems.Typicallythereisa maintheoremwithalongproof,thelongproofconsistingofdefinitionsandlemmas(the lemmasarelittletheoremswhichfolloweasilyfromwhatisalreadyknown).
MathematicianslikeHadamardandPoincar´ehavenotedthat doingmathematicsis acombinatorialtask:puttingpiecestogethertoobtainaninterestingtheorem.There
aremanychoicesinvolvedinconjecturingan interesting theoremandputtingtogetherthe piecesofaproof.Toguidethesechoiceswehaveabackground ofresultsinthepublished literature,andoftheoreticalideaswhichmaybemoreorlessvagueorprecise.Thepublishedliteratureincreaseswithtime,andthebackgroundoftheoreticalideas(whichdefine whatisaninterestingtheorem)alsochanges.Forinstancemathematiciansareguidedby ideasonthenaturalstructuresofmathematics;suchideashavebeenformulatedprecisely byBourbaki,orhavebeenlaterembodiedinthetheoryofcategoriesandfunctors.Structuralideasnowplayamajorroleincertainareasofmathematics,wheresomequestions willappearnaturalforstructuralreasons,andbesystematicallyaskedandinvestigated. Lessprecisetheoreticalideasconsistofanalogies,suchastheanalogybetweenthetheory ofC∗ -algebrasandthetheoryofcompactspaces(thisanalogycomesfromthefactthatan abelianC∗ -algebraispreciselythealgebraofcomplexcontinuousfunctionsonacompact space–butthisdoesnotsayexactlyhowtoperformthegeneralizationfromtheabelian tothenon-abeliansituation).
Tosummarize,doingmathematicsmaybeviewedasasuccessionofguessesand routineverifications.Theguessesareguidedbytheoreticalideasthatevolvewithtime. Foramoredetaileddiscussionofthesequestionsseemybook TheMathematician’sBrain [5].
Theabovewasadescriptionofhumanmathematics.Inthecase ofcomputer-verified mathematics(formalproofs)apartoftheguessesishuman,buttheroutineverifications aremadebycomputer,andthisinvolvesanontrivialcombinatorialpart,i.e.,makingmany lowlevelguesses.Whatremainsofhumancreativityarethemanyhigherlevelguesses, basedontheoreticalideaswhicharenoteasilyformalizedtopermittheirsystematicuse.
4.Limitstomathematicalcreativity.
Theabilitytodomathematicsisarecentdevelopmentintheevolutionofthehuman brain.Mathematicalabilityisrelated(amongotherthings)totheacquisitionoflanguage,whichispoorlyunderstood.Theabilitytospeakwasclearlyfavoredbyevolution, andthesamemightbesaidoftheabilitytocountfrom1to10.Buttheabilitytodo highermathematics(likestudyingGaloistheory)isanothermatter,andmostpeopleget alongsuccessfullywithoutthisability.Onethingthatstrikesmeisthegreatdisparityof performancesofthebestmathematicians:ifonetriestoassessquantitativelythecontributionsofRiemann,G¨odel,orGrothendiecktomathematics,onecouldsaythatitisten toahundredtimesgreaterthanthatofa“normal”high-level mathematician.Inother words,thecontributionofoneofthe“great”mathematiciansmentionedaboveisworthas muchasthecontributionoftentoahundredmembersofthemathematicalsectionofa goodacademyofsciences(sayFrench,orUS).Ithinkthatmostmathematicalcolleagues wouldagreewiththisquantitativeestimate(althoughperhapswithoutenthusiasm).This isquitedifferentfromthesituationfor100mrunningwheretheperformancesofthebest racersarequitesimilar.Tounderstandthedifference,onemayappealtonaturalselection, whichisclearlynotthesameforrunningandfordoingmathematics,butnaturalselection argumentsaretricky*,andweshallnotgofurtherinthisdirection.
*IamindebtedtoHenriKornforadiscussionofthismatter.
Note,bytheway,thatagreatmathematicianisonewhodoessomethingnew,not onewhoisgoodatdoingagainthingsthathavebeendonebefore.Weexpecttherefore thatgreatmathematiciansarequitedifferentfromeachother,sothattheycantackle problemsindifferentmanners.Thisisindeedthecase;forexamplewhileoneslimvolume containsthecompleteworksofRiemann,theworkofGrothendieckcoversmanythousands ofpages.Anywayonelooksatthings,thegreatestmathematiciansarethusverydissimilar people*:theyarenotclusteredagainstsomenaturallimitofwhatcanbedonebyhumans inmathematics.Wehaveseenearlierthattherearelimitsto humancreativityimposed bylogicandbythestructureofthehumanbrain,butnowitalsoappearsthatindividual mathematiciansarenotclosetoauniversallimit.Thedifficultytoputalimitonhuman mathematicalperformancesuggeststhatitwillalsobedifficulttoputalimittocomputer mathematicalperformance,oncecomputersstarttobecreative.
5.Post-humanmathematics.
Ihavepointedoutthattheintellectualabilitytodomathematicsisarecentdevelopmentfromthepointofviewofthebiologicalevolutionofthe humanbrain.Ifindithardto believethatthisrecentdevelopmenthasproducedsomethingsouniquethatitcannotbe successfullyimitatedbycomputers.Ithinkthatthesituationofmathematicalcreativity todayislikethatoforganicsynthesisbeforeW¨ohler:itis onestageinanevolution,and therewillbelaterstages.Thebigquestionisthen:whatkindofmathematicscouldbe producedbyartificialmathematicalcreativity?Whatifwehaveacomputerwithaccess tosomesortofmathematicalliterature,theabilitytoperformroutineproofs,butalsothe abilitytomakeintelligentguessesbecauseitwouldhavebeentaughtaproperbackground oftheoreticalideas?Whatifitdevelopeditsownnon-human backgroundoftheoretical ideas?
Letmeinterruptmyselfhereforabriefpsychologicaldigression.Iamnoteagerto seecomputersreplacehumanmathematicians.Itwould,orwill,beasadthinginsome respects,butIdon’twanttoshutmyeyestothepossibility. Thinkoftheenormous effectsthatindustrialorganicsynthesishashadonmankind,somehavebeenniceand someterrible,butthereiscertainlynowaybacktothepre-W¨ohlertimes.Similarly, mathematicsisprobablyenteringsoonacompletelynewera, andwemightaswelltryto guesswherethiswillleadus.
Ifweassumethatacomputerhasbeentaughttobemathematicallycreative,we canthenimaginethatitcouldbeathumanmathematiciansattheirowngame.This meansthatitcouldgiveproofsofconjectures,orinterestingnewtheorems,whichhuman mathematicianscouldunderstandandperhapsadmire.Butit ismorelikelythat,once acomputerbecomescreative,itwilldothingsquitedifferentlyfromhumans.Herearea coupleofpossibilities:
(a)Thecomputercouldproveaninterestingresult,butwith aproofimpenetrableto humans,becauseitwoulduselongdevelopmentinsomeformal languagewithnoreason-
*Theintellectualdiversityofpeopleisofcoursenotlimitedtomathematics.Aglimpse intononstandardintellectualabilitiesisprovidedbysomeautists,see[7].
ablybrieftranslationintofamiliarhumanlanguage.(TheAppel-Hakenproofofthefour colortheorem,orthecomputerverificationsusingformalproofs,areexamplesofthis).
(b)Thecomputercouldproveanimportanttheorem,butwitha statementimpenetrabletohumans(againbecauseitwouldhavenoreasonablybrieftranslationtohuman mathematicallanguage).Thecomputermightconvinceusthatthistheoremisimportant, becauseitimpliesanumberofinterestingconjecturesthat wecanunderstand.Butour braincouldnotmakesenseofthetheoremitself.
Theabovepossibilitiesraisebigquestionsonthenatureof mathematics.Wecan seetoday’smathematicsasareasonablywellstructuredlandscape,withbigdomainslike algebraicgeometryorsmoothdynamics,andimportanttheoremsliketheprimenumber theorem.Thisstructuredmathematicallandscapeisrelatedtothepossibilitiesofthe humanbrain.Isthereastructuretomathematicswhichisindependentofthehuman brain?Couldanintelligentcomputerdevelopanewmathematicallandscapesimilarto theoneweknow?Todiscussthesequestions,rememberthelogicalfactthatatheorem withashortformulationmayhaveanextremelylongproof.To developmathematical knowledgeinaneconomicalway,oneavoidsrepeatingsimilarverylongproofs.Onetries insteadtoobtainnewresultsbyarelativelyshortproofusingalreadyknownresults. Thehumanwayofdevelopingmathematicsproducesthusanetworkofresultsrelatedby “understandable”proofs(nottoolong,butalsonot“inhumanly”formal).Thenetwork isconstantlybeingreworkedandimprovedbyhumanmathematiciansto“revealnatural underlyingstructures”.Thisishowthestructurallandscapethatweknowforhuman mathematicshasbeenobtained.Aswehavepointedout,there isalogicalreasonbehind thisstructurallandscape,buttherearealsospecificallyhumanreasons(notanalyzed indetailbut,inbrief:thehumanbrainfavorsshortformulations,“understandable”,and “interesting”arguments).Dowebelievethatlogicalfactorsprevailoverhumanspecificities inproducingthesortofstructuralmathematicallandscape thatweknow?
Ifearthatwemustconsideranotherpossibility:perhapscomputerswilldevelop mathematicalabilitiessothattheycananswerefficientlyquestionsthatweaskthem,but perhapstheirefficientwayofthinkingwillhavenostructuralbasisrecognizablebyhumans. Ifthishappens,thesuperiorityofourhumanintelligencewillbestronglychallenged: weshallwatchanintelligentcomputerdoingmathematicsin muchthesamewayasa chimpanzeecouldwatchahumanscientistreadingabookonGaloistheory.*
Thegutfeelingofmanyloversofmathematicswillbethattheycan’tbelievemy chimpanzeestory.JustasBerzeliuscouldnotbelievethatorganiccompoundswould everbesynthesizedinthelab.Togobeyondsuchvisceralreactions,Iwouldliketo comparemathematicsandmusic.Mathematiciansoftenlovemusic:thereisharmony, beauty,andasenseofinfinityinbothmathematicsandmusic. Thereisalsotheuse offractionstodescribemusicalintervals(approximately)butthisisasomewhatlimited relation.Infact,mathematicsandmusicareconceptuallyverydifferentthings,butitis importantforourdiscussionthattheyevokesimilarestheticreactions,probablybecause
*Inthisrespect,Jean-PierreEckmannremindedmeofFredHoyle’snovel TheBlack Hole [],whichdescribeshumancontactwithasuperiorintelligence.
theyinvolverelatedactivitiesofthebrain.Herewemustrememberthattherearetwosides tomathematics:oneisnon-humanlogicalnecessity,theotherishumanbrainactivity.The non-humanlogichaslittletodowithmusic,butitcouldbeaccessibletocomputers.Asa humanbrainactivity,mathematicsisrelatedtootherbrain activities,andhasapparently aprivilegedrelationwithmusic.Theinterplaybetweenthe humanandnon-humansides ofmathematicsisbeingmodifiedbytheirruptionofcomputersintothegame.Howthis modifiedinterplaywilldevelopintheyearstocomewillbefascinatingtoobserve.
Protagorashassaidthat“manisthemeasureofallthings:of thingswhichare,that theyare,andofthingswhicharenot,thattheyarenot”.This remainstrueforusin thesensethateverythingweknowisknowntousthroughourownhumanbrain.This remainstrueeventhoughweunderstandtodaythattheplanet Earthonwhichweliveisan infinitesimalspeckofdustinthephysicaluniverse.Thiswillremaintruetomorroweven ifwefindthathumanmathematicaldiscoveriesaredwarfedandhumbledbycomputer mathematics.
Thepresenttextisaslightlyreworkedversionofapresentationmadeatthe20-th anniversaryconferenceoftheESIinVienna,29April2013
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