I don’t believe in the existence of infinitely many primes because of some philosophical reason; I believe it because Euclid proved it. And I don’t believe in the existence of this coffee-cup for metaphysical reasons; I believe it on the practical and by no means logically conclusive grounds, that it has so far done a good job of holding my coffee. Should it turn out I’m only a brain in a vat, worse luck: no coffee-cup after all. But the number of primes will still be infinite.
Thus far the thoughts of a simple man, after a tasty cup of coffee. Then come the objections: And why do you trust your sense-perceptions of that cup? Simple answer: Because that’s what creatures do; I’m like the bunny, trusting the carrot. Surrejoinder: a years-long onslaught of the findings of philosophy and physics. Somewhat chastened, I revise the periphery of my ontology in complicated ways, demoting a number of entities to various levels of Meinongian subsistence. But the coffee-cup stays just as it was. There is still a frenziedly skeptical attack on that (the brain-in-a-vat bit), which I cannot logically refute; but I meet it with faith, which in this case requires no great leap.
As for the integers…Well I know of no attack that even needs meeting. There are attacks on various characterizations of what numbers “are” (rather than, so to speak, what they “do” – divide evenly into twelve, and so forth); these are harmless, perhaps even true. They do not strike at the heart of numbers, any more than a bad statue or overidealized painting strike at the reality of Napoleon.
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