I'm in college, and my roomy is a very close friend who does Classical Philology and Philosophy and I, on the other hand, am completely passionate about physics and mathematics and we've agreed that I'd teach him about mathematics this year (for fun). He knows close to nothing, he's had the minimum of hours of math in high school, so nothing much to build on (except some intuition, hopefully). He is, however, certainly very interested and a very smart guy.
My question is: what do I focus on? And more particularly: where do I start? (he has expressed interest in logic and probability; of course this is to be taken with a grain of salt: he has little idea of what is out there)
Some options: really fundamentally with set theory and the foundations of logic; I know close to nothing about this (in my last year of bachelor in math), but I'm very interested so I think I could look up and grasp the basics, anyway it seems like the most "genuine" place to start, especially since he has a deep interest in philosophy (albeit continental philosophy, but I'll cure that)
Another option: the way they start in a real analysis course: defining the real numbers, concepts like order, completeness, those things; this would be interesting to see the gap between intuition and rigour.
Or: algebra. Don't talk about numbers, but groups and rings and fields and algebras and matrices. This would be of interest to show how mathematics succeeds in talking about structures themselves and not just concrete realizations, a jump into the abstract.
Or, well, maybe geometry, although that seems like a weird place to start nowadays, it is after all the way math began and comes with a load of intuition, intuition that can be shattered by the interesting non-euclidean spaces or projective spaces.
And one other way I can think of: to not spend too much on the basics, but just jump in with calculus, to get to complex analysis quickly: a piece of beauty I don't want to deny him!
Any suggestions or comments?