And another anecdote:

I've got two friends who are talented maths graduates, and somehow both of them had managed to get through their first degrees without ever writing programs. Both of them asked me to teach them.

The first one I've made several attempts with. He sort-of gets it, but he doesn't see why you'd want to. A couple of times he's said 'Oh yes, I get it, sort of like experimental mathematics'. But any time he gets a problem about numbers he tries to solve it with pen and paper, even when it looks obvious to me that a computer will be a profitable attack.

The second, I spent about two hours showing him how to get to "hello world" in python and how to fetch a web page. Five days later he shows me a program he's written to screen-scrape betfair and place trades automatically when it spots arbitrage opportunities. I was literally speechless.

So I reckon that whatever-makes-you-a-mathematician and whatever-makes-you-a-programmer might be different things too. Which is actually a bit weird. They feel the same to me.