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10
AndrzejPelczar
privatethattheythinktheycreatesomething,agoodexampleoragoodidea.
Initsmostextremeform,thishumanistorconstructivistoutlookrejectsthe
ideaofmathematiciansaspassivediscoverersofaremoterealityandinstead
assertsthatmathematicalphenomenaarecreatedbyhumansaloneanddonot
otherwiseexist.
[...]Atheoremmaybediscoveredbutitsproofisusuallyinvented.[...]
Einsteinisreportedtohavefeltthatheinventedtheconceptofrelativistic
spacetime,afterthefact,butthathefelthewasdiscoveringanaspectofreality
whilehewasdoingthework.Itisafeelingfamiliartomanymathematicians:
“weareallPlatonistsinthetrenches”,asoneputit.Butthedebatecontinues
wherevermathematiciansgather.“Doyouinventordoyoudiscover?”
Letusadd,thatthedisputeamongmathematiciansandphilosophersthat
hasbeenonforcenturies,andthatisregardedwiththeissueofdiscovering
vs.inventingmathematicalentities2,wasrecentlyboostedbyanumberof
statementspublishedbyprominentmathematicianswiththeNewslettersof
EuropeanMathematicalSociety(see[2,5,6,7]),whosedifferencesinopinions
(quiteoftenfarfrombeingdramatic)canbealreadyseeninthetitlesoftheir
papers(compare,forexample,[2]and[6]).
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Puttingthecruxofthematteraside,letusconsiderthelanguageusedto
describethe(present)mathematics.Onesays(or,shallwesay,onewrites)
quiteuniversally3abouttheconstructionoftherealnumbers,orthecon-
structionofabasisofsomespace.Whilepresentingproofs,oneoftensays:
somebodygaveaproof.Isthethought(perhapssubconscious?)hidingbehind
thesephraseologicalpatternsthatsomebodyactuallyinventedtheproof,or
ratherfound(hence–discovered)theproof?Perhapsourfeelingisthat–that
somebody–found(discovered)apossibilityofaproof(thatis–discovered
thepossibilityofinventingtheproof)?
Probablymattersarequitesimilarwhenitcomestopresentingexamples
or,socalled,counterexamples.Oneoftensaysthatsomebodyfound(hence
discovered)anexampleoracounterexample,butalsothatsomebodyinvented
suchanexampleorcounterexample.Onealsoveryoftensaysthatsomebody
gaveanexample(acounterexample).Inthislastcaseonemay,perhaps,
admitbothintuitiveinterpretations:foundsinceonefirstdiscovered,orfound
sinceinvented.
TheseremarksapplydirectlytotheaforementionedwordsbySullivanand
2ThisisthetermIletmyselfusehere,fullyrealizingitmaybetakenasanabuseof
thewidelyacceptednomenclature,yethopingthatIshallsucceedinavoidingconfusion,
andphilosophersamongthereadersshallshowenoughunderstanding.
3Perhapsoneshouldsayfrequentlyratherthanuniversally,orusebothterms
simultaneously?
