...Prove Their Worth...

"Problems worthy of attack
prove their worth
by hitting back." - Piet Hein

A kind of running diary and rambling pieces on my struggles with assorted books, classes, and other things, as they happen. You must be pretty bored to be reading this...

Wednesday, July 31, 2002

This is the posting that's been sitting on my hard drive that I'd referred to last time, with a bit of a trim for sanity. Without much further ado...



Most people meet math in school. They are taught to add, to subtract, to multiply, to divide. They are forced to memorize tables of facts. No one denies this is useful - I suspect even the kids know it. But kids aren't stupid, and they know about calculators and computers, and they know that a calculator can do arithmetic faster and better than they can. Arithmetic is boring. It's just a bunch of arbitrary algorithms (as far as kids are concerned).



It doesn't get much better as kids advance through the system. Algebra, then trig, then calculus - all are presented as a bunch of techinques to memorize, formulas to use for 'plug and chug', and always, in the end, to reduce things to arithmetics and get some 'numbers' out of things. Any proofs given (and this happens rarely) are turgid, hairy things, with the teacher more or less quoting the book, not really understanding what each step is for, where it fits in the larger picture, where the approach of the proof fits...



What's even worse is that most teachers sound bored while teaching. It's genuinely rare to find a teacher who is so excited about both teaching and his material that it shows in his eyes, in the way he talks about it, etc. This is not surprising to me, because most teachers have to repeat the same material several times a day, year after year after year. How anyone manages to stay excited in such circumstances (and a few do!) is beyond my understanding. So I sympathize. But the problem is that it's hard to get someone to care about what you're talking about if you don't really care about it yourself anymore.



Math education sucks. People grow up to hate math, and they are right to do so - math, as they have seen it, is dead boring.



I disliked math in high school and college. To me, math was mostly a tool that was handy for some things, but it was a pretty boring tool, and one that often savaged me when I least expected it to (ex.: doing things like 2+2=5 for large values of 2 on SATs...). Occasionally, I'd see something 'neat' in math, but that was as far as it went. I didn't like proofs, and I sucked at constructing them. I just wanted the cookbook recipie, damn it.



But the core reason I didn't like math was deeper. It didn't seem real, and it didn't seem accesible. That is, I could never see where most of the math I was taught 'came from'! I knew, intellectually, that lots of dead white men (and a few women, and a few non-Europeans*) spent centuries developing it. I didn't see how. As far as I was concerned, it was all pulled out of someone's ass. That it worked and was handy was the only reason to learn it, as far as I was concerned. But it's hard to get excited about what you think is the output of someone's rear, and so I didn't.



I suspect this is one of the reasons so many people don't like math, apart from the tedium and the bad instruction.



As I've mentioned in a long-past entry, my opinion of math has since changed to 'strikingly engrossing and beautiful, oh, and also useful'. (Blame Needham's Complex Analysis book for that change of heart.) The big difference is that now I can feel (and not just know, in a vague intellectual sort of way) why, exactly, it's not not just pulled out of someone's ass. I can do some of the proofs, and when I'm lucky, I know why each part of the proof is needed, and what happens when you take it out, and how else you can set up the proof, and what it means**. I can feel how various ideas are put together. I know where they come from, and when I don't (which is most of the time), I've some degree of confidence that I could figure it out, given time and some references.



Learning math (and related things) is to me a way of learning about how the world works. It's an amazingly elegant way of doing so, too. Here's a for-instance (one, sadly, that I suspect won't make any sense to anyone who doesn't already know about cross products and wedge products and wedgies and other things already, but such is life). Everyone knows about cross products of vectors. They're fine and dandy, very nice for a lot of practical things, and so on. However, they a) have an ass-nasty definition involving crap with determinants of 3x3 matrices (whose ass was it pulled out of?) and b) are limited to being defined on R^3 and c) require a right hand rule convention. The second and third problems are the more serious ones. It'd be nice to figure out some sort of generalization of a cross-product, one that isn't just limited to R^3, but which does behave like a cross-product in R^3.



To do this, we need to figure out the essential features of the cross-product. It turns out there's really only one: antisymmetry. That is, v x w = - w x v, where v and w are vectors in R^3. So, let's define a 'wedge product' (aka 'exterior product'), and say that it has this property. That is, v /\ w = - w /\ v, where v, w are elements of a vector space V. An additional piece of info that we'll need to tuck into the definition is that v /\ w will lie in a new vector space, denoted /\V. Actually, we'll say that /\V is an algebra (giving it a bit more structure than a vector space, but whatever).



That is pretty much the only thing we need to figure out how to take wedge products. Say V is three-dimensional, and is spanned by dx, dy, dz. Then the wedge product of two vectors in V is: v /\ w = (v_x w_y - v_y w_x) dx /\ dy + (v_y w_z - v_z w_y) dy /\ dz + (v_z w_x - v_x w_z) dz /\ dx. The only thing needed to derive this is the idea that /\V is an algebra together with the anticommutativity of the wedge product! Also, notice that this already bears a suspicious resemblance to the cross product!



What I think is cool here is just how little information was needed to get this fancy complicated thing (well, it only looks complicated - it's really not.) And reading on in Baez in Munian, I think I now grok where cross products come from. With more reading, I hope to be able to say that I grok stuff about exterior algebra with differential forms, and I'll know how to do 'cross products' on manifolds.



Anyways, I obviously think this stuff is interesting, and I've totally lost my thread of thought, and, err, I might as well post this shit now -- I don't think I'll be able to get it any more presentable.




* - Ok, preemptive strike. Yes, I know about the arabs doing some things with algebra, and so on, and so forth. However, whether anyone likes it or not, the Europeans are responsible for almost all of 'modern' mathematics. That doesn't mean they are The Smartest People Ever, and Everyone Else is a Retard.


** - This is sort a description of the 'nirvana state' - I don't actually grok things that deeply most of the time. Instead, most of the time I curse and despair of understanding anything, and then try again later, and sometimes I get it, and sometimes (most of the time) I don't. What keeps me going is that I do hope to eventually 'get it', and it's entertaining enough to keep my interest in the meantime.

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