Husserl on

Formal Mathematics and

how it relates to intuition Mirja Hartimo

University of Jyväskylä, Finland FPMW12

November 5, 2020

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.

1.   HusserlonformalmathematicsFormaleundtranszendentaleLogik,1929- His”mostmature,iftooconcentrated”work;containsa“definitiveclarificationofthesenseofpureformalmathematics…,accordingtotheprevailingintentionofmathematicians”(FTL,11).- The“prevailingintentionofmathematicians”>“mathematicsofmathematicians”vs.“mathematicswithalogicalinterest”- Zermelovs.Hilbert,Weyl

•  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)

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.”

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.

•  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.

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.

Transitionallink:Husserlinsistsonthetraditionalformofjudgement

SispTheformsofjudgmentleaveitindeterminatewhetherthetermsarecomplexornot(i.e.,inHusserl’sterms,whetherthesubjectorthepredicatearesyntacticallystructured).

”Butitcanbeseenapriorithatanyactualorpossiblejudgmentleadsbacktoultimatecoreswhenwefollowupitssyntaxes;accordinglythatitisasyntacticalstructurebuiltultimately,thoughperhapsfarfromimmediately,outofelementarycores,whichnolongercontainanysyntaxes”(§82).

”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”.

•  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.

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)

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.

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).

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).

• Husserlseemstobesayingthatinformalmathematicstheinformationaboutthe”determination”ofindividualsisabstractedaway.

• However,itisultimatelyneededtoshowthattheindividualscan”materially”relatetoeachother,intheharmoniousunityofexperientialbasis.

• Coherenceisnotamerelysyntacticmatter(asopposedtoHilbert).•  Semantic?Intermsoftheexistenceofmodels(asIhavesuggestedbefore)?–notquite,butperhapsintermsofmeaningexplanations?Syntacticalstuffsasthe”typification”or”computablecontent”tobeincludedinthemeaninggenesis?

•  But,thisseemstocontradictHusserl’sclaimthatthepuremathematicianneedsnottocareaboutsuchfoundationalmatters.

•  EitherHusserlholdsthatevenpuremathematicshastoberestrictedtobesofounded(Husserlwouldbeunawareofanyrestrictions),orelseheintroducesanothersenseinwhichmathematicsmayhavefoundations,indistinctevidence.

•  Myview:Husserlisopentonewformsofevidence;ifhelivednow,heshouldandwouldacknowledgethatthisisnottheevidenceaccordingtotheprevailingintentionsofpuremathematicians.Itdoescarveouta”distinctlyevident”partofmathematics.

•  Phenomenology,ingeneral,isametaphysicallyneutralmethodwithwhichanyexperiencecanbeexamined.Noreasontocutout,say,descriptivesettheory.

•  Foundationalinterestsofmathematiciansmaysuggesttheexistenceoffurtherkindsofevidencethatcarveoutsomeotherpartsofmathematics.

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.

Thank you!

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