RESUMEN
There are different definitions of axioms, but the one that seems to have general approval is that axioms are statements whose truths are universally accepted but cannot be proven; they are the foundation from which further propositional truths are derived. Previous attempts, led by David Hilbert, to show that all of mathematics can be built into an axiomatic system that is complete and consistent failed when Kurt Gödel proved that there will always be statements which are known to be true but can never be proven within the same axiomatic system. But Gödel and his followers took no account of brain mechanisms that generate and mediate logic. In this largely theoretical paper, but backed by previous experiments and our new ones reported below, we show that in the case of so-called 'optical illusions', there exists a significant and irreconcilable difference between their visual perception and their description according to Euclidean geometry; when participants are asked to adjust, from an initial randomised state, the perceptual geometric axioms to conform to the Euclidean description, the two never match, although the degree of mismatch varies between individuals. These results provide evidence that perceptual axioms, or statements known to be perceptually true, cannot be described mathematically. Thus, the logic of the visual perceptual system is irreconcilable with the cognitive (mathematical) system and cannot be updated even when knowledge of the difference between the two is available. Hence, no one brain reality is more 'objective' than any other.