Awesome, I was looking for a good explanation of the Peano axioms!

About six months ago I had a series of arguments with my housemate, who's been doing a philosophy degree at a Catholic university. He argued that I should leave the door open for some way other than observation to gather knowledge, because we had things like maths giving us knowledge in this other way, which meant we couldn't assume we'd come up with some other other way to discover, say, ethical or aesthetic truths.

I couldn't convince him that all we could do in ethics was reason from axioms, because he didn't understand that maths was just reasoning from axioms --- and I didn't actually understand the Peano axioms, so I couldn't explain them.

So, thanks for the post.