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Husserl on
Formal Mathematics and
how it relates to intuition Mirja Hartimo
University of Jyväskylä, Finland FPMW12
November 5, 2020
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Outline
1. Husserlonformalmathematics2. ”Mathematicsofmathematicians”andmathematicswithlogical
interest3. Kindsofevidence:distinctness[Deutlichkeit]vs.clarity[Klarheit]4. ”ATransitionalLink”showshowformalmathematicsisrelatedto
judgmentsaboutindividualobjects.5. Abstractionprinciplesorintuitionistictypetheory?6. Distinctevidence,furtherdemands.7.Conclusion:”Mathematicsofmathematicians”andmathematics
withtwokindsoffoundationalinterests.
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1. HusserlonformalmathematicsFormaleundtranszendentaleLogik,1929- His”mostmature,iftooconcentrated”work;containsa“definitiveclarificationofthesenseofpureformalmathematics…,accordingtotheprevailingintentionofmathematicians”(FTL,11).- The“prevailingintentionofmathematicians”>“mathematicsofmathematicians”vs.“mathematicswithalogicalinterest”- Zermelovs.Hilbert,Weyl
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• InProlegomena(1900)Leibnizianmathesisunversalis,partiallyrealizedinRiemann’stheoryofmanifolds,Grassmann’stheoryofextensions,Hamilton’squaternions,andCantor’ssettheory(e.g.,§§60,68,70)
• InFTL,thecoreidearemainsunchanged:”Thegreatadvanceofmodernmathematics,particularlyasdevelopedbyRiemannandhissuccessors,consists…initshavinggoneontoviewsuchsystem-formsthemselvesasmathematicalObjects,toalterthemfreely,universalizethemmathematically,andparticularizetheuniversalities–…inconformitywiththesuperordinationsandsubordinationsthatpresentthemselvesintheprovinceoftheformal”(§30).
• I.e.,modernstructuralmathematics.• Combinationof”formalapophantics”(theoryofjudgments)and”formalontology”(studyofformalobjects,i.e.,mathematics).
• Actsofjudgmentsincludenotonlyactsofpredicatingsomethingoftheobjects,butalsoactsofcollecting,counting,orderingandcombiningmathematically(FTL,§§39;100)
• TheoryofjudgmentsembracesallofpuremathematicsHusserlexplicitlypointsoutthatitincludestraditionalanalysis[dietraditionelleformale“Analysis”derMathematiker],settheory[dieMathematikderMengen],theoriesofcombinationsandpermutations,cardinalsorordinalsbelongingtovariouslevels,ofmanifolds[Mannigfaltigkeiten],etc.(FTL,§24)
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2.Mathematicsofmathematiciansandmathematicswithlogicalinterest§51:“themathematicianassuchneednotbeatallconcernedwiththefact
thatthereactuallyaremultiplicitiesinconcrete‘actuality’.“§52:“Mathesispura”asproperlylogicalandasextra-logical.The
“mathematicsofmathematicians”“onecansetupawholesciencethat,freedfromthespecificallylogicalaim,neitherexploresnorintendstoexploreanythingbeyondtheuniversalrealmofpureapophanticsenses.Itbecomesapparentthat,whenquestionsaboutpossibletruthareconsistentlyexcludedinthismanner,andthetruth-conceptitselfissimilarlyexcluded,onehasnotactuallylostanyofthislogicalmathesis;onestillhasthewholeofit:as‘purely’formalmathematics’.”“Onemustseethataformalmathematics,reducedtotheabovedescribedpurity,hasitsownlegitimacyandthat,formathematicsthereisinanycasenonecessitytogobeyondthatpurity.…inthismannerthepropersenseof‘formalmathematics’,themathematicstowhicheveryproperlylogicalintention(thatis:everyintentiontoatheoryofscience)remainsalien–themathematicsofmathematicians–atlastbecomesfundamentallyclarified.”
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3.Kindsofevidence:distinctness[Deutlichkeit]vs.clarity[Klarheit]• Differencebetweenthetwoisultimatelythatmathematicsofmathematiciansandwithlogicalinterestaimatdifferentkindsofevidence
Mathematics(ofmathematicians)>distinctness[Deutlichkeit]–logicofnon-contradictionMathematicswithlogicalinterest>distinctnessandclarity[Klarheit]–logicoftruth
(Inaddition,bothseekgrammaticalrigor.Husserlalsotalksaboutevidencerelatedto“anewsortofcategorialformation,”namely,constructionalinfinities>opennesstofurtherkindsofevidencetosurfaceinmathematics.)
• §53,Husserl’sexample:Euclidiangeometryasapossiblesystemoftruepropositionsvs.orasoneamonganopeninfinityofpossibledeductivescienceshavingthissamecategorialform,i.e.,asasystemofpropositionspurelyassenses,indistinctevidence,asasystematicwhole,“unifiablewithoutcontradiction”
• ThedistinctionbetweenthetwokindsofevidenceisfirstpresentedinthelecturecourseErstePhilosophie(1923-24)
• Purelymathematicaltheorieshave“unityofaninternallycoherentvalidity”(Husserl1956,19/20).
• Whetherajudgmentbelongstothistheoreticalunityisindependentofthequestionofwhetherthejudgmentsaretrueorfalse.
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• Distinctnessderivesfromtheexperienced“theoreticalunity”;whereasclarityisparadigmaticallyobtainedbyseeingthatwhatisjudgedisindeedthecase.I.e.,“originarily”,“inperson”.“Judgingwith‘clarity’hasatonceclarityoftheaffairs,intheperformanceofthejudgment-steps,andclarityofthepredicativelyformedaffair-complexinthewholejudging”…“itisanewevidence,pertainingtoagivennessoriginaliteroftheaffairsthemselves,ofthepredicativelyformedaffair-complexitself,atwhichoneaimsinthejudgingthatstrivestowardcognition”…(§16b).
• Takingstock:thereispurelyformalmathematics,characterizedbytheevidenceofdistinctness,andappliedmathematics(logicoftruth)thatisdeterminedbytheevidenceofclarity(i.e.,whatisactuallytrue).i.e.,twokindsof“intuition”,whereinthecaseofclarity,thenotionofintuitionisparadigmaticallyperceptionoftheobjectsintheworld.
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4.TheTransitionalLink“werequirehereanimportantsupplementationofthepurelogicofnon-contradiction,asupplementationthat,tobesure,goesbeyondformalmathematicsproper,butstilldoesnotbelongtotruth-logic.Itisamatter,sotospeak,ofatransitionallink[Übergangsglied]betweenthem”(§82).
• Aimistoshowhowjudgmentsofformalmathematicsultimatelyrelatetojudgmentsaboutindividuals.“Formathesisuniversalis,asformalmathematics,theseultimateshavenoparticularinterest.Quitethecontraryfortruth-logic:becauseultimatesubstrate-objectsareindividuals,aboutwhichverymuchcanbesaidinformaltruth,andbacktowhichalltruthultimatelyrelates“(§82).
• Whereaspuremathematiciansneednotcareaboutit,themathematicianwithalogical(=foundational)interestwantstoknowhowmathematicsrelatestojudgmentsaboutindividualobjects,andthisiswhatthe“transitionallink”shows.
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Transitionallink:Husserlinsistsonthetraditionalformofjudgement
SispTheformsofjudgmentleaveitindeterminatewhetherthetermsarecomplexornot(i.e.,inHusserl’sterms,whetherthesubjectorthepredicatearesyntacticallystructured).
”Butitcanbeseenapriorithatanyactualorpossiblejudgmentleadsbacktoultimatecoreswhenwefollowupitssyntaxes;accordinglythatitisasyntacticalstructurebuiltultimately,thoughperhapsfarfromimmediately,outofelementarycores,whichnolongercontainanysyntaxes”(§82).
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”byreduction,wereachacorrespondingultimate,thatis:ultimatesubstrates-…absolutesubjects…,ultimatepredicates,…ultimateuniversalities,ultimaterelations”…”thereductionsignifiesthat,purelybyfollowingupthemeanings,wereachultimatesomething-meanings;firstofall,then,asregardsthemeantorsupposedjudgment-objects,supposedabsoluteobjects-about-which.”Ifweoperateonlywiththedistinctjudgment-senses,onecanonlyreachaclaimabout”sense-elements”astheultimate”core-stuffs.”§83:Inlogicoftruth,thereisacorrespondingreductiontotruths,tojudgmentsthatareaboutindividualobjects,
”objectsthatthereforecontainwithinthemselvesnojudgment-syntaxesandthat,intheirexperienceablefactualbeing,arepriortoalljudging”…”reductivedeliberationteaches,asanApriori,thateveryconceivablejudgmentultimately…hasrelationtoindividualobjects(inanextremeblybroadsense,realobjects),andthereforehasrelationtoarealuniverse,a’world’ora’world-province,’forwhichitholdsgood”.
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• The”transitionallink”issupposedtoreveal”hiddenintentionalimplications”injudging.Ituncovers”thesense-genesis”ofjudgments.(§85)
• Itshowsthatthelowestlevelisjudgementsaboutindividuals;”andconsequently,inthecaseofevidentjudgments,inthesenseofseeingsofthepredicativelyformedaffair-complexesthemselves,itbringsustothoseevidencesofsomethingindividualthatbelongtothesimplesttype.Thesearethepureandsimpleexperientialjudgments,judgmentsaboutdataofpossibleperceptionandmemory,whichgivenormsforthecorrectnessofcategoricaljudicialmeaningsatthelowestlevelconcerningindividuals”(§86)
So,likeforHilbertandforWeyl,forHusserl,logicpresupposessomepre-existingindividualobjects,henceHusserl’s”foundations”closertothemthantoBrouwer.
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5.Abstractionprinciplesorintuitionistictypetheory?• Recently,Husserl’stransitionallinkhasbeendescribedintermsof(Fregean)abstractionprinciples(Costantini).
• Dynamicabstractionism(Linnebo)tointroducenewobjects.• Theaccountregardsthereductioninthetransitionallinkasa”denominalization”,where”nominalization”isalinguisticcounterpartoftheprocessofabstraction(thatgeneratesnewobjects),sothatthetruthofcomplexjudgementsisgroundedontheobjectsofourexperience.
• Asymmetricalabstractionprinciplestoexpanddomains;PFOLwithamodalizedquantifier□∀toexpressgeneralstatementsoveranypossibleexpansion.(E.g.,inIdeasI,§119,Husserltalksabout”pluralconsciousness”andthelawofnominalization)
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Aninterestingsuggestion,but…• Theapproachisuntyped,whereasHusserl’suniverseistyped,asindicatedbyhisinsistenceonthetraditionalformofjudgment;
• ThewayHusserl’stheoryofjudgmentandmathematicsare”entangled”showsatighterconnectionbetween”logic”andmathematics.
• Husserlseemstosuggestthatthereductionismechanical,i.e.,itiscomputable(thiswouldensurethemediationofevidence).
• Insum,thetransitionallinkhaspropertiesofintuitionistictypetheory.(Typeduniverse,theformofjudgment,entanglementduetoCurry-Howardisomorphism,andduetoit,decidability.)
• Bethatasitmay,inHusserl’sviewthefoundationalintereststrivesustoshowhowmathematicsisrelatedtoaperceptionofordinaryobjectsandjudgmentsaboutthem.
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Whataboutthemathematicsofmathematicians?- HusserlexplainshowwhenweascendfromgivenindividualobjectstotheformalApriori,”each’individual’mustbeemptiedtobecomeanythingwhatever”(§86).I.e.,”Inplaceofsomethingindividual,thereenterseverywherethepositingof’acertainsubstrate(ofwhateversort)aboutwhichonecanjudge”(§86).
- ”TheevidenceoflawspertainingtotheanalyticApriorineedsnosuchintuitionsofdeterminateindividuals”(§86).
- ”Sisp”>”p”;lossof”determinateness”ofindividuals,i.e.,thetypification.- Theevidenceofdistinctness,i.e.,non-contradictorinessoftheunityofjudgment,ashetalkedaboutitinErstePhilosophielecturesin1923-4;>”existenceofamodel”?
But,here:- ”Nevertheless,thesense-relationofallcategorialmeaningstosomethingindividual,…surelycannotbeinsignificantforthesenseandthepossibleevidenceofthelawsofanalytics,includingthehighestones,theprinciplesoflogic.Otherwise,howcouldthoselawsclaimformal-ontologicalvalidity:validityforeverythingconceivablyexisting?”(§86).
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Theprecisesourceofdistinctness:• ”theunitaryeffectibilityofthejudgment-content”,ortheideal’existence’ofthejudgment-content”.
• Husserl:”thepossibilityofajudgment(asameaning)isrootednotonlyinthesyntacticalformsbutalsointhesyntacticalstuffs.”(§89b)
• Theintentionalgenesis:”Everyjudgmentassuchhasitsintentionalgenesisor,aswecanalsosay,itsessentiallynecessarymotivationalfoundations,withoutwhichitcouldnotatfirstexistintheprimitivemode,certainty,norbemodalizedthereafter.Thesefoundationsincludethenecessitythatthesyntacticalstuffsoccuringintheunityofajudgmenthavesomethingtodowithoneanother”(§89b).
• ”Priortoalljudging,thereisauniversalexperientialbasis.Itisalwayspresupposedasaharmoniousunityofpossibleexperience.Inthisharmony,everythinghas’todo’materiallywitheverythingelse….Thus,inrespectofitscontent,everyoriginaljudgingandeveryjudgingthatproceedscoherently,hascoherencebyvirtueofthecoherenceofthemattersinthesyntheticunityoftheexperience,whichisthebasisonwhichthejudgingstands.Wedonotintendtosayinadvancethattherecanbeonlyoneuniverseofpossibleexperienceasthebasisforjudgment,andthatthereforeeveryintuitivejudgmenthasthesamebasisandalljudgmentsbelongtoasinglemateriallycoherentwhole.Toreachadecisionaboutthatwouldrequireaseparateinvestigation.”(§89b).
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• Husserlseemstobesayingthatinformalmathematicstheinformationaboutthe”determination”ofindividualsisabstractedaway.
• However,itisultimatelyneededtoshowthattheindividualscan”materially”relatetoeachother,intheharmoniousunityofexperientialbasis.
• Coherenceisnotamerelysyntacticmatter(asopposedtoHilbert).• Semantic?Intermsoftheexistenceofmodels(asIhavesuggestedbefore)?–notquite,butperhapsintermsofmeaningexplanations?Syntacticalstuffsasthe”typification”or”computablecontent”tobeincludedinthemeaninggenesis?
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• But,thisseemstocontradictHusserl’sclaimthatthepuremathematicianneedsnottocareaboutsuchfoundationalmatters.
• EitherHusserlholdsthatevenpuremathematicshastoberestrictedtobesofounded(Husserlwouldbeunawareofanyrestrictions),orelseheintroducesanothersenseinwhichmathematicsmayhavefoundations,indistinctevidence.
• Myview:Husserlisopentonewformsofevidence;ifhelivednow,heshouldandwouldacknowledgethatthisisnottheevidenceaccordingtotheprevailingintentionsofpuremathematicians.Itdoescarveouta”distinctlyevident”partofmathematics.
• Phenomenology,ingeneral,isametaphysicallyneutralmethodwithwhichanyexperiencecanbeexamined.Noreasontocutout,say,descriptivesettheory.
• Foundationalinterestsofmathematiciansmaysuggesttheexistenceoffurtherkindsofevidencethatcarveoutsomeotherpartsofmathematics.
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7.Conclusion:”Mathematicsofmathematicians”andmathematicswithtwokindsoffoundational interests:• Formalmathematicscanbeapproachedintwoways:purelymathematicallyand”logically,”withafoundationalinterest.
• Husserlclaimsthattheformerseeksevidenceofdistinctnessthatultimatelyderivesfromtheharmoniousunityofexperience,inwhichthesomethings-whateverforma”material”unity.
• Thelatterseeksevidenceofclarity,whichisparadigmaticallyobtainedbyperceivinganobject.• Thesekindsofevidencearereachedwiththe”transitionallink”• Husserlclaimsthateventhoughthepuremathematicianneedsnottocareaboutit,evenhersubjectmatterisultimatelyrelatedtotheharmoniousunityofexperienceviathetransitionallink.
• Husserl’sviewofdistinctevidencecannotcoverallof”mathematicsofthemathematicians”.Asuggestion:Today,phenomenologistshouldclarifynew,contemporarykindsofevidencethatdeterminemathematiciansintentionstoday.
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Thank you!
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