8/9/2019 Abstract Shahidrahman

1/2

CenterforPhilosophyofScienceoftheUniversityofLisbon

InternationalColloquiumThePhilosophersandMathematics

Abstract

TheIntensionalTakeontheAxiomofChoice:

ADialogicalPerspectiveonItsProof

ShahidRahman1

UniversitdeLille,UMR8163:STL

The present talk studies the interplay between a Philosophers reflections onmathematics, namely,

JaakkoHintikkaandaMathematicianreflectingonthephilosophicalandepistemologicalfoundationsof

his science.Despiteourcriticismof the formerwedo think thathisworkopenedanewpath for the

interchangebetweenphilosophyandmathematicsinawaythatwasalreadyprefiguredbyHenriPoincar

thoughwewillnotdiscussherePoincarsviewontheissue.Thetalk,Ithink,nicelyfitswiththebeautiful

titleofthismeetingthathonoursRoshdiRashedwhoseworkisalandmarkinthefield.

Ithasbeensaid ,andrightlyso,thattheprincipleofsettheoryknownastheAxiomofChoice(AC)is

probablythe

most

interesting

and

in

spite

of

its

late

appearance,

the

most

discussed

axiom

of

mathematics,

secondonly toEuclidsAxiomofParallelswhichwas introducedmore than two thousandyearsago

(Fraenkel/BarHillelandLevy[1973]).

AccordingtoErnstZermelosformulationof1904ACamountstotheclaim;that,givenanyfamilyAof

nonemptysets,itispossibletoselectasingleelementfromeachmemberofA. Theselectionprocessis

carriedoutbyafunctionfwithdomaininM,suchthatforanynonemptysetMinA,thenf(M)isanelement

ofM.Theaxiomhasbeenresistedfromitsverybeginningsandtriggeredheatedfoundationaldiscussions

concerningamongothers,mathematicalexistenceandthenotionofmathematicalobjectingeneraland

offunctioninparticular.However,withthetime,thefoundationalandphilosophicalreticencefadedaway

and

was

replaced

by

a

kind

of

praxis

driven

view

by

the

means

of

which

AC

is

accepted

as

a

kind

of

postulate (rather than as an axiom the truth of which is manifest) necessary for the practice and

developmentofmathematics.

RecentlythefoundationaldiscussionsaroundACexperiencedanunexpectedrevivalwhenPerMartinLf,

showed(around1980)thatinconstructivelogic(thatdoesnotpresupposetheexcludedmiddle)theaxiom

ofchoiceislogicallyvalid(howeverinitsintensionalversion)andthatthislogicaltruthnaturally(almost

trivially) follows from theconstructivemeaningof thequantifiers involved it is this evidence, that

makesitanaxiomratherthanapostulate.Extensionalitycanalsobeprovedbutthen,eitherthirdexcluded

or unicity of the functionmust be assumed.MartinLfs proof, forwhich hewas awardedwith the

prestigiousKolmogorov price,showedthatattherootoftheolddiscussionsanoldconceptualproblem

wasatstake,namelythetensionbetweenintensionandextension.

1ThepresenttalkisbasedonClerbout/RahmanLinkingGamesandConstructiveTypeTheory:Dialogical

Strategies,CTTDemonstrationsandtheAxiomofchoice.(inprint).

8/9/2019 Abstract Shahidrahman

2/2

AnevenmorerecentdevelopmentstudiesthegametheoreticalinterpretationofACbroughtforwardby

JaakkoHintikkaby19962,thoughhedidnotconsiderMartinLfsproofpresumablysobecauseHintikka

isnot favorable toconstructivistapproaches.Theaimof thepaper is todevelopanew constructivist

approachtothegametheoreticalinterpretationofACbasedontheCTTproof.Moreprecisely,Clerbout

andRahmanshowedthattheCTT understandingofAC,thatstressesthetypedependenceinvolvedby

thefunction

that

constitutes

the

proof

object

of

the

antecedent,

can

be

seen

as

the

result

of

an

outside

insideapproachtomeaning3. Itisthisapproachtomeaning,soweclaim,thatprovidesanaturaldialogical

interpretationtoAC,wherethe(intensional)functioninvolvedunderstoodasrulesofcorrespondence

producedbytheplayersinteraction constitutesaplayobjectforthe(firstorder)universalquantifierthat

occurs in the antecedent of the formal expression of this axiom.Different toHintikkas own game

theoreticalapproachthedialogicaltakeonACdoesnotrequireanotaxiomatisablelanguagesuchasthe

oneunderlying IndependentFriendlyLogic(IFlogic).AspointedoutbyJovanovic(2014)thedialogical

approachtoCTTsupportsHintikkaclaimsthatagametheoreticaljustifiesZermelo'saxiomofchoiceina

firstorder way perfectly acceptable for the constructivists, however, no underlying IFsemantics is

required.Moreover,HintikkasownformulationofAC,whenspelledout,yieldstheCTTformulationof

MartinLf,

that

is

constructivist

after

all.

Summing

up,

though

Hintikka

is

right

in

stressing

the

perspicuity

ofthegametheoreticalinterpretationofACheiswronginrelationtothetheoryofmeaningrequiredfor

thisinterpretation.OneofthemainreasonsbehindHintikkascriticismoftheconstructivistapproachis

thatheassumesthattherejectionoftheclassicalunderstandingoftheACbytheconstructivistshasits

rootsintherejectionofafunctionthatisnotarecursiveone.However,asthoroughlydiscussedbyThierry

Coquand (2014), already ArendHeyting (1960) pointed out that recursive functions cannot (without

circularity)beusedtodefineconstructivityandfinallyErretBishop(1967)showedthatrecursivefunctions

are not at all needed to develop constructivemathematics. The very point of the rejection by the

constructivistsoftheclassicaltakeonACistheir(theclassical)assumptionthatthefunctionatstakeisan

extensionalone.Forshort,theCTTproofofACisbasedontheintensionaltakeonfunctionsandthisis

whatthe

dialogical

interpretation

displays.

We

will

conclude

with

some

reflections

on

the

conceptual

link

betweentheconstructivistnotionoffunctionasruleofcorrespondenceanddialogicalinteractionwithin

humanplayablegames,thatmightrestatesomeofHintikkasremarksalbeitinadifferentframe.

Iwill finishbynoticing that, asmentioned above, it isHenryPoincar (1905)who already compared

mathematicalknowledgewiththehabilitytounderstandthedevelopmentofplayofchess.According,to

thisview,andifwejoinandPoincarswithWittgenstein Hintikkasremarksinthecontextofthenotion

ofproofaswinningstrategy,theknowledgeofaproofamountstograspingitsmeaningandthisamounts

onbeingabletoshowhowtoconstructthisproof:aproofbeyondourabilitiestoconstructitisnotproof

atall:humanplayableorreachableproofprovidesthefoundationtothenotionofinferenceandtruth.

2SeeforexampleHintikka(1996,2001).3ForathoroughdiscussionontheissueseeJovanovich(2014,2015).