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8
AndrzejPelczar
B.Ifweagreetotheabove,weinevitablyfacetheHeternal”questionaboutthe
natureofmathematicsinitsHphilosophicaldimension”–namelythequestion,
whetherthisinductivedevelopmentofmathematicsreliesmoreondiscovery
orinvention.Itouchonthisquestionwithmuchhesitation.
Thequestionwhethermathematiciansdiscoverorinventhasbeenincon-
siderationforalongtime–Hsinceforever”.Itseemsthatmostmathematicians
wouldfavortheformer,butitishardtojudgeifthatfeelinghasaquantitative
basis.
Theauthorofthistextdarestostatehisown,highlysubjective,opinions.
Hewouldliketoavoidstatementssuchas:Hitisso”,because,infact,heusually
doesnotknowifHitisso”.However,hedoesnotavoidsharingwiththereader
hisownfeelingsandquiteoftenelaboratesonwhatseemstohimtobetrue.
Hecountsonthereader’spolitenesstoforgivehimthelackofphilosophical
accuracyinnumerousstatementsaswellasthenegligenceincitingthevast
literatureonthesubject,and–heredeliberate–informalityinmathematical
considerations.Itseemsthattheideabehindthisexpositioncanbepresented
withoutmuchrigor.
∗
∗
∗
Ageniussculptor(Michelangelo?)wassupposedtosay,whilestaringat
ablockofmarble:Thereisalreadyastatueinside,Iamonlyremovingthe
extraneousmaterialanditwillcomeoutonitsown.
Doesitactuallymean,thathediscoveredthatthesculptureisthere(that
is,discoveredthestatue),orratherthathediscoveredthepossibilityof“in-
venting”(byHremovingthesuperfluousmaterial”)?
∗
Ifinsteadofsculptinginmarble,hewouldbeusingclay,sothatrather
thanHremovingtheextraneousmaterial”hewouldbeHadding”it(or,atleast,
processing),wouldhealsosaythatthestatueisalreadyinside,butneedsto
beformed?Probablynot.Hewouldprobablysay,thatheisinventingthe
statue,but–otherthanbyHremovingtheextraneousmaterial”–byinventing
it“outofnothing”.TheHnothing”heredoesnotmeantheclay,butthefact
thatatthebeginningoftheprocesshestartedfromHgroundzero”,without
discovering(thatthestatueisalreadythere,anditremainstoletitcomeout
byHremovingtheextraneousmaterial”).
∗
Ishallrisktoproposetakingthesoundnessofthefollowingconstatation
underconsideration:inbothinstancesdescribedaboveonecanthinkofdis-
coveringthepossibilityofinventing.
