RichardKennaway:
Ariithmetic is complex because it can not be captured in a small set of axioms. More precisely, it cannot be specified by any (small or large) set of axioms, because any set of (true) axioms about arithmetic applies equally well to other structures that are not arithmetic. Your favorite set of axioms fails to specify arithmetic in the same way that the statement "bricks are rectangular" fails to specify bricks; there are lots of other things that are also rectangular.
This is not true, for example, of euclidean geometry, which can be specified by a set of axioms.
Silas Barta's remarks notwithstanding, the question of which truths we can know has nothing to do with this; we can never know all the truths of euclidean geometry, but we can still specify euclidean geometry via a set of axioms. Not so for arithmetic.
Here we go again.
Arithmetic is complex because it can not be captured in a small set of axioms.
Then the universe doesn't use that arithmetic in implementing physics, and it doesn't have the significance you claim it does. Like I said just above, it uses the kind of arithmetic that can be captured in a small set of axioms. And like I said in our many exchanges, it's true that modern computers can't answer every question about the natural numbers, but they don't need to. Neither does the universe.
...Your favorite set of axioms fails to specify arithmet
A monthly thread for posting rationality-related quotes you've seen recently (or had stored in your quotesfile for ages).
ETA: It would seem that rationality quotes are no longer desired. After several days this thread stands voted into the negatives. Wolud whoever chose to to downvote this below 0 would care to express their disapproval of the regular quotes tradition more explicitly? Or perhaps they may like to browse around for some alternative posts that they could downvote instead of this one? Or, since we're in the business of quotation, they could "come on if they think they're hard enough!"