When something completely unexpected happens, I just love it. Really, that’s what makes math so cool: you’re working along and *ka-POW*, something that you didn’t expect at all just jumps and pops you in the nose. Great fun.
The other day, I got some great wisdom from my algebra professor, John Sullivan. My buddy Barry had a few questions for him after lecture, so I decided to hang around and listen in. This class, Galois Theory, is the first one I’ve taken from Sullivan—so far, I’ve enjoyed his lectures a lot. He works hard to communicate the ideas and thoughts behind the principles, instead of spending time on the mechanisms and contraptions used in the proofs of the princpiles. The downshot of that, having to figure out the mechanics for myself, is a bit annoying, but most likely good for the mind. Anywho, I should stop yammering before I break my “short and sweet” promise.
After talking about a few specific problems, Sullivan started to explain to us some of his philosophy about homework, and math problems in general. I’ll try to get as close as I can to what he said, but it’s not an exact quote.
Once every few weeks, he hands out a fact sheet, with all the important theorems and statements we’ll need for the homework, and he was just starting to explain those to us…
“See, I’m not giving you guys those fact sheets as the keys to solving the problems, not at all. They’re tools to add to your collection, to use along the way to your solution; they’re not the solution itself, not at all. Don’t think about the problems as obstacles, look at them as opportunities—that’s really what they are, they aren’t obstacles.
When you look at one of these problems—when I first look at one of these problems, I have no idea what’s going on, what I really actually am trying to do, what the real underlying idea is. You have to play around with it, try some different things, look at what tools you have to work with. After doing that, after you come to an understanding the problem…that’s when you get a spark, an idea. That’s where the solution has to come from, an inspiration. You need that spark—that’s what these problems are, they’re opportunities for you to work through the problem, and have that spark. That’s why you can’t just use the fact sheet to crank out a solution—you miss the whole point, the experience, and really, what it’s all about. ”
So, yeah. That Sullivan guy, I think he knows a thing or two. He tacked down a part of what math is about, right there. But then I kept thinking about what he said, and then the unexpected happened: it hit me, this idea doesn’t seem like a principle of mathematics as much as it is an outlook on life. I think most can agree—life is all about the journey, the story.
Sullivan’s advice about Galois theory suddenly seemed to connect with that idea. By not looking a problem full on, but grinding through it with what I know how to do–as with math problems, could it be possible to miss a bit of what life is all about? When I work a math problem, I can cobble together a handful of propositions and theorems, call it done, and it’ll probably look OK. I’m safe. Heck, maybe it’ll be original, or even elegant. But, tragically, I will have missed what it’s all about.
I think some of what life is about—just like the beauty of that little spark of inspiration—comes from if, and how, we square with the hard times.
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