(普特南) Putnam - 1979 - 'Philosophy of Mathematics - Why Nothing Works'

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1、28.PhilosophyofMathematics:WhyNothingWorksInsteadofdescribingthemaintrendsinthephilosophyofmathemat-ics,Ihavedecidedtobeginwiththetheme"whynothingworks."ButIamnotsimplybeingperverse:explainingwhyandhowitseemsthat"nothingworks,"thatis,whyandhowitisthateveryphilosophyseemstofailwhenitcomestoexplai

2、ningthephenomenonofmathe-maticalknowledge,willinvolvesayingsomethingabouteachofthe"maintrends."Anditwillalso,paradoxically,servetopointoutwhyphilosophyofmathematicsissuchacrucialfield.Ifthereissuchathingas"philosophicalprogressn-andIconfesstoanunregeneratefaiththatthisisnotachimera-thenitcomesfr

3、omfocusingtheatten-tionofphilosophersonareasandproblemswheretheirpetideasrunintotrouble,areaswhere"nothingworks."Wewillnotgetbeyondthepresentphilosophicaldiscussionbyarguingonemoretimewhetherornotthereare"canonsofscientificmethod,"orbydrag-gingoutthefamiliarexamplesfromthehistoryofphysics;wemayg

4、ettoanewplateaubytakingseriouslytheideathattherearerealprob-lemsforallthestandardviews-andnoareaismorelikelytomakeusawareofthisthanthephilosophyofmathematics.LogicismLogicism-theviewthat"mathematicsislogicindisguise"andthatthatiswhataccountsforitscertainty-appearstobedefunctasfarashavingpresenta

5、dherentsgoes.However,itisoftenoverlookedthatsomethingofpermanentvaluecameoutoflogicism.Ishallnotrepeattheobjectionstologicismhere-theyaretoowellknowntoneedrep-etition,Ibelieve-butinsteadIshallmakethefollowingobservation:TheDiversityoftheSciencesSincetheworkofFregeandRussell,'weareallmuchmoreawar

6、eofhowmuchmathematicscanbedoneinlogisticsystemswhicharesetuptocodifydeductivelogic.Thosephilosopherswhowouldcountsecond-orderlogicaslogic-andtherearesome-wouldholdthatallofstandardmathematicscanbeformalizedwithinlogic(foritcanbeformalizedwithinsecond-orderlogic):eventhoughtheywouldnotIclaimthatt

7、hismakestheepistemologyofmathematicsanyeasier(perhapsitmakesitharder!);whileevenifwefollowwhatseemstobethemajorityfashion,andlimittheterm"logic"tofirst-orderlogic,we1stillhavetorecognizethatagooddealofwhatanymathematicianwou