How/where to start to teach someone MATH from SCRATCH?

[nonequilibrium]

Hello,

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?

 

[nonequilibrium]

Who moved it to here? I don't think it's serious enough for this board, it's meant to be more light-hearted...

 

[AlephZero]

You could try some Euclidean geometry.

He might able to read Euclid in the original greek (it loses quite a bit in translation). Point him at the introduction to T L Heath's edition of the Elements (available online) which has a long section on Greek ideas about mathematical logic etc.

Then point him at Hilbert's "Foundations of Geometry" (also online) to see the difference between the classical and modern (well, 100-year-old!) approach to the "same" mathematics.

If none of that interests a classicist and/or philosopher, maybe should should just forget about the whole thing...

 

[Poopsilon]

I'm sure that he can't read ancient greek. Considering the enormous scope of mathematics it's probably best to narrow down the choices based on what he hopes to get out of this mathematical excursion. If he just wants to learn more about how mathematicians think and learn to sharpen his analytical skills then I would recommend classical logic (derivations, quantifiers, UI EI & EG, that whole spiel) as it's not as dry a subject as set theory. Learning about why logicians set up the rules the way they did and how ignoring certain rules leads to particular fallacies in argument, fallacies which likely have quasi-formal analogues in the philosophy he studies.

If on the other hand he's more interested in appreciating the beauty of mathematics then I would maybe first teach him the minimum amount of prerequisite material required and then introduce him to infinite series, as this is a fascinating and counter-intuitive subject with lots of colorful examples; you could mention Zeno's paradox and its mathematical resolution. Or if not that then probably something with a geometric flavor although I'm not sure what.

I personally would probably shy away from abstract algebra, since without a lot of previous exposure to elementary algebra and various sets of numbers it's likely to come off as overly formal and unmotivated.

Either way it's going to require a considerable investment of time and effort on his part to get anything out of it, mathematics is not a spectator sport.

 

[nonequilibrium]

Why do you think he can't read ancient greek? He assures me he can.

As for the advice so far, I like them.

 

[mathwonk]

euler wrote his algebra book for his butler. it is free online.

http://www.archive.org/stream/elementsalgebra00lagrgoogEuclid's geometry in english is also excellent from green lion press. i also put the notes from a summer course in euclid for 8-10 year olds at "epsilon camp" on my web site.