Are these abstractions "real"? That depends, of course, on how you define "real". Are the rules of chess "real"? How do we perceive them? I agree it's not wi

Are these abstractions "real"? That depends, of course, on how you define "real". Are the rules of chess "real"? How do we perceive them? I agree it's not with our eyes and ears. And here we're back to Kant. They are intelligible, not sensible. We perceive them with our mind/intellect. (Interestingly, in Buddhist philosophy, there are six senses-- the mind joining the other five, so they push the intelligible/sensible distinction down a level.) "That's just the way it is" isn't an explanation of anything, though. How do we know, once we've proven 17 prime, using these rules, that we will never discover a factorization of 17? How do we know that our reasoning about these abstractions is sound? I'm not saying "That's just the way it is"-- I'm saying "That's the way it has to be because we defined it that way." How do we know that in some particular game of chess, the bishop won't suddenly be allowed to move sideways? Anyway: if you are really interested in the reduction of arithmetic to pure logic, check out Russell and Whitehead's Principia Mathematica-- or better yet, read the graphic novel "Logicomix" which does a wonderful job of covering the domain for non-specialists.