Which beliefs of the Pythagoreans were invalidated by the discovery of irrationals?
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I think it was their intuitive conception that all lengths should be "commensurable". I.e. there should be some conversion formula between the two...98 of this unit equals 79 of this unit. But such is not the case with $1$ and $\sqrt{2}$.
answered Jan 18, 2014 at 5:39