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Descriptionofmathematics
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9
IntheextremelyinterestingbookMathematicstoday:twelveinformales-
sayseditedbyL.A.SteenandAllenHammond[4]inhisessayquotesan
excerptfromthediscussionbetweenLipmanBersandDennisSullivan:
LipmanBers:Doyouinventordoyoudiscover?Whatisyourgutfeeling?
DennisSullivan:Sometimesyoucomeuponsomethingsortofnatural,it’s
likeyou’rediscoveringit.Butsometimesyoujustmakesomethingupoutof
thinair,sotospeak;maybeforceitalittlebit.
Hammondhimselfpresentshisviewsinabroadercontext,alongwith
referring,inahighlygeneralway,theviewsof–inhisopinion–mostmath-
ematicians(oreven:thevastmajorityofmathematicians).Herearethe
excerptsfromthistext:
Theargumentamongmathematiciansoverwhethermathematicaltruths
areinventedordiscoveredhasbeengoingonalongtime.Itisnotanargu-
mentthatiseasilyresolvedbutitisrevealingofhowmathematiciansthink
abouttheirwork.Thetwopointsofviewareatfirstglancequitedistinct.
Oneholdsthatmathematiciansdiscoverapieceofreality[...],atruthnot
oftheirownmakingbutratheraninherentpartoftheuniverse.“Godmade
theintegers”,asonenineteenthcenturymathematicianputit[...].Hencethe
propertiesoftheintegersencompassedinsimplearithmeticandinthemore
sophisticatedtheoremsofnumbertheoryareviewedbymostmathematicians
inmuchthesamewayasastronomersthinkoftheplanets–discoveredele-
mentsoftheheavens.ThisabsolutistorPlatonistviewpoint–mathematics
asrealityrevealed–extendstootherareasofmathematicsaswellandisin
factthedominantdogmainthemathematicalcommunity.[...]
Thesecondpointofviewemphasizestheroleofhumancreativityininvent-
ingmathematicalstructures.Clearlythereisanelementofhumancreativity
involved,buthowmuch,wheretodrawtheline?Anysystemofmathematics
restsultimatelyonaseriesofaxioms,forexample,andthereisinmanyin-
stancesanelementofchoiceastowhichaxiomstouse.Euclideangeometry
wasbasedonfivesupposedlybasicandself-evidentaxiomsaboutthenatureof
space,butjusthowarbitrarysuchchoicescanbewasshownbythediscovery
(inphysics,notinmathematics)thatspaceisnotEuclideanafterallbutrather
Riemannian-thatanalternatesetofaxiomsduetoRiemannprovidedage-
ometrythatcorrespondsmoretophysicalrealitythanthatofEuclid’s.Other
mathematicianshavesinceinventedandexploredthepropertiesofstillother
geometries,allgoodmathematics,butwithanelementofchoice.Noristhis
elementofhumaninventivenessconfinedtogeometry;thereexistalternate,
“nonstandard”modelsoftherealnumbersystemtoo.Eventhelawsoflogic
onwhichallofmathematicalreasoningdependsarenotuniversallyregarded
asabsolute.Somemathematicianshavearguedthattheretoothereisanel-
ementofchoiceandconvention.Infactmanymathematicianswilladmitin
