I'm learning to think
more like a mathematician, and that's a good thing.
Last year I co-taught the "The Edge of Reason," the paired Math/Science Fiction courses that my friend and colleague, Professor Bill Goldbloom Bloch and I designed (Bill, by the way has an absolutely brilliant book on the mathematics of Borges' "Library of Babel" coming out any day now from Oxford). So every teaching day I was in a classroom with a mathematician, learning more of the very high-end math along with the students and really starting to understand how mathematicians think.
This year Bill and I are teaching an experimental course, Logic and Language, in which we will be taking students (mainly sophomores) through, among other things, Chomsky's Syntactic Structures, Computability and Logic by George Boolos, et. al., and Daniel Dennett's Darwin's Dangerous Idea. Topics to be covered include transformational generative grammar, information theory, computability, Turing machines, and Gödell's incompleteness theorem.
This summer I have been collaborating with Prof. of Computer Science Mark LeBlanc and Prof. of Mathematics Mike Kahn (a specialist in statistics). We have been building some pretty cool software.
In prepping and teaching the courses, and doing the research, I've learned a lot of important reasoning skills, because mathematicians and computer scientists have different ways of thinking that are incredibly useful. Here's just one.
Let's say other people have proven a lot of things about Turing Machines. And let's say you're working on something else, like Wang Tiles. Well, a mathematician thinks, "hmmm... there's all this cool stuff about Turing Machines over here. If I can prove that a set of Wang Tiles can work like a Turing Machine then, boom!, I've just proved a whole ton of stuff about Wang Tiles without having to prove it specifically for Wang Tiles."
This sounds obvious in my summary, but it's the kind of obvious that I really only got after we'd done stuff like it a bunch of times. For example, number theory, long thought to be the most abstruse and useless part of math, turns out to be absolutely essential for doing secure credit card transactions on the internet. That's nice to know, but it's mind-blowing to work through all the underlying math and see how it all falls together.
And this brings me to just one quick opportunity to shill for a liberal arts education that includes a lot of math and science: The only way to make those kinds of connections, to say "Oh, this set of Wang Tiles is really just a Turing Machine," is to have, floating around in your head, a lot of information about different and disparate fields. That's why I read as much evolutionary biology (and now as much math) as I can get my hands on: the more I pack into my skull, I think, the more likely I am to be able to make those kinds of connections and to think like a mathematician.