1/28/2004 11:09:00 PM|||Andrew|||It is a constant source of amazement to me how relatively little mathematics majors know about Math, taken as a whole and in general, and how much I've been able to learn about it by studying philosophy. Granted that my primary area of interest is logic, the area where philosophy is most like mathematics, but in talking to people who have been studying math for four years, I know all about a variety of concepts that they have never (or very briefly) been exposed to.

I've learned fascinating things: the axiom of choice in set theory and the controversial nature of it, non-Euclidian geometries, the theory of transfinite numbers, questions about the reality or antireality of mathematical proofs, Godel's incompleteness theoram, and some seriously deep studies into the very basic units of mathematics, numbers, operations, functions. Just today our my professor spent about 45 minutes talking about the evolving set theories, starting with Frege's, killed before it was born by Russell's paradox, Russell's own type theories (simple and ramified), and a modern theory, ZF.

I'm not terribly surprised that we've covered this kind of stuff in the various classes I've taken, given that I've focused heavily on logic and language, and mathematics seems to be one the very favorite examples used to illustrate various principles. What confuses and/or concerns me is that mathematics students don't seem to be studying this stuff (at least not a lot of it), even at the highest levels. Is it saved until grad school or what? Perhaps the teaching of mathematics is simply too caught up in technique and the established theories to really go into metamathematical theory and alternative systems. Ah well. At least it makes me increasingly glad that I chose the path I did.|||107536019141357654|||