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1、MathematicalIntuitionandtheCognitiveRootsofMathematicalConceptsGiuseppeLongo•ArnaudViarougePublishedonline:20January2010SpringerScience+BusinessMediaB.V.2010AbstractThefoundationofMathematicsisbothalogicaformalissueandanepistemologicalone.Bythefirst,wemeantheexplicitationandanalys
2、isofformalproofprinciples,which,largelyaposteriori,groundproofongeneraldeductionrulesandschemata.Bythesecond,wemeantheinvestigationoftheconstitutivegenesisofconceptsandstructures,theaimofthispaper.This‘‘genealogyofconcepts’’,isnecessarybothinordertoenrichthefoundationalanalysiswi
3、thanoftendisregardedaspect(thecognitiveandhistoricalconstitutionofmathematicalstructures)andbecauseoftheprovableincompletenessofproofprinciplesalsointheanalysisofdeduction.Forthepurposesofourinvestigation,wewillhintheretoaphilosophicalframeaswellastosomerecentexperimentalstudieso
4、nnumericalcognitionthatsupportourclaimonthecognitiveoriginandtheconstitutiveroleofmathematicalintuition.Keywords:NumericalcognitionMathematicalintuitionFoundationsofmathematics1FromLogictoCognitionOverthecourseofthetwentiethcentury,therelationshipsbetweenPhilosophyandMathematic
5、shavebeendominatedbyMathematicalLogic.AmostinterestingareaofMathematicswhich,from1931onwards,yearofoneofthemajormathematicalresultsofthecentury(Go¨delianIncompleteness),enjoyedthedoublestatusofadisciplinethatis11bothtechnicallyprofoundandphilosophicallyfundamental.Fromthefoundati
6、onalpointofview,ProofTheoryconstituteditsmainaspect,alsoonaccountofotherremarkableresults(OrdinalAnalysis,TypeTheoryinthemannerofChurch-Go¨del-Girard,variousformsofincompleteness-independenceinSetTheoryandArithmetics),andproducedspin-offswithfar-reachingpracticalconsequences:thef
7、unctionsforthecomputationofproofs(Herbrand,Go¨del,Church),theLogicalComputingMachine(Turing),andhenceourdigitalmachines.Thequestionshavingarisenattheendofthenineteenthcentury,duetothefoundationaldebacleofEuclideancertitudes,motivatedthecentralityoftheanalysisofproofs.Inparticular
8、,theinvestigationoftheformalconsistencyo
