I can understand the people who dislike math. Most of the mathematics I had to study until the last year of high school or so was, frankly, for the most part just boring repetition of easily memorized rules. Euclidean geometry had some element of fun, but apart from it... well, there really
is nothing all that interesting in memorizing and executing algorithms.
I was actually quite interested in Cartesian geometry, mostly because I liked to dabble a little in computer programming and it is a fundamental tool for that, but yes, math was boring.
"Advanced" math is really quite different. You don't have rules to follow, or not that much: you have problems, and you have a lot of tricks that sometimes happen and sometimes don't, and you have to use a mixture of intuition and creativity in order to get to the solution.
Then there is nonclassical mathematical logic, of course, my own area of interest, in which things become just crazy. In this subject, you
make the rules. You have to be consistent about it, of course*
, and certain rulesets lead to more interesting consequences than others; but really, the main problem here is to define new formalisms and see how they compare with one another — you know, if expressions can be translated from Logic A to Logic B preserving the meaning, this sort of stuff.
Where most mathematicians work with complex constructions deriving from chains of definitions, we tinker with the basic toolbox, messing with it just to see what happens
It is really, really awesome. Do you want non-commutative conjunction? You have them. Do you want infinitely many truth values? You have them too. Do you want infinitely long formulas? Knock yourself out! Do you want to deny the principle of non-contradiction? Eh, join the ranks of all the logics that do that, and see how yours compares to them.
Really, the sky is the limit there. Except when you say that it isn't.
And then there is Abstract Logic
and Abstract Model Theory
, which takes all of this one step further and looks at families
of nonstandard logics from outside and tries to characterize their properties. Funny stuff, really.
edited 6th Jun '11 8:25:14 PM by Carciofus