ABSTRACT
ThediscoveryofirrationalnumbersdestroyedthePythagoreanprogrammeofderivingcosmology(andgeometry)fromthearithmeticof naturalnumbers.Platorealizedthisfact,andsoughttoreplacethearithmeticaltheoryoftheworldbyageometricaltheoryoftheworld.18The famousinscriptionoverthegatesofhisAcademymeantexactlywhatit said:thatarithmeticwasnotenough,andthatgeometrywasthefundamentalscience.HisTimaeuscontains,asopposedtothepreviousarithmeticalatomism,ageometricalatomictheoryinwhichthefundamental particleswereallconstructedoutoftwotriangleswhichhadassidesthe (irrational)squarerootsoftwoandthree.Platobequeathedhisproblemto hissuccessors,andtheysolvedit.Euclid'sElementsfulfilledPlato'sprogramme,sinceinitgeometryisdevelopedautonomously,thatis,without the'arithmetical'assumptionofcommensurabilityorrationality.Plato's largelycosmologicalproblemsweresolvedsosuccessfulybyEuclidthat theywereforgotten:theElementsisregardedasthefirsttextbookofpure deductivemathematics,insteadofthecosmologicaltreatisewhichIbelieve ittohavebeen.
