By Reuben Hersh

Collection of the main attention-grabbing fresh writings at the philosophy of arithmetic written by way of hugely revered researchers from philosophy, arithmetic, physics, and chemistry

Interdisciplinary e-book that would be invaluable in numerous fields—with a cross-disciplinary topic sector, and contributions from researchers of varied disciplines

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But I am sure this shows only that our simile fails at this point. Certainly there is an answer, only I do not know how to find it. SOCRATES Your guess is correct that the paradox arose because we kept too close to the simile of the reflected image. A simile is like a bow-if you stretch it too far, it snaps. Let us drop it and choose another one. You certainly know that travellers and sailors make good use of maps. HIPPOCRATES I have experienced that myself. Do you mean that mathematics furnishes a map of the real world?

Furthermore, to claim that the logic of mathematics is deductive logic clashes with the results of the neurosciences, which show that the human brain is very inefficient even in moderately long chains of deductive inferences. Similarly, to claim that, when it comes to explaining the remarkable phenomenon that work on a mathematical problem may end in a result that everyone finds definitive and conclusive, the notion of deduction is a central one, overlooks the fact that, according to the dominant view, several Euclid’s proofs are flawed.

This problem arises because “the point of view of common sense is perhaps that, if a proposition is true, it is because there are entities existing independently of the proposition which have the properties or stand in the relations which the proposition asserts of them”8. This “suggests that since mathematical propositions are true, that there are entities in virtue of which the propositions are true. The ontological issue is whether there are such entitities and if so what their nature is”9. This problem supplements “the epistemological question” of the justification of mathematics, namely, “how mathematical beliefs come to be completely justified”10.

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