Why isn’t the fundamental theorem of arithmetic obvious? (2011)
I know I'm going up against a brilliant mathematician and Fields medalist here, but I find this article to be unenlightening. It seems that Gowers has glossed over something about the integers that's built incredibly deeply into our intuition about them when he talks about Z[sqrt(-5)]:> These numbers have various properties in common with the integers: you can add them and multiply them, there are identities for both addition and multiplication, and every number has an additive inverse. And as with integers, if you divide one by another, you don’t always get a third, so the notion of divisibility makes sense too. That means that we could if we wanted try to define a notion of a “prime” number of the form a+b\sqrt{-5}.He skipped over the fact that the positive integers are well-ordered by < (and in fact < also respects the arithmetic operations on the positive integers as well). I just can't take the comparison seriously without this being explicitly discussed, because the ordering of the integers is incredibly fundamental to human intuition about them.