I think most people, if you asked them, would say that teaching math in school is a good thing. If you ask why, the usual answer is something like, it allows you to figure out how much carpet and wallpaper you need to buy to redecorate your living room, to determine whether the 12-can pack of ravioli at Costco is cheaper than what you can find at Safeway, and so forth.

But that’s all elementary and High School level stuff: arithmetic, geometry, a dash of algebra. I’ve rarely used trigonometry after college, and I know one person who used calculus in quilting. I think it’s safe to say that most people never use calculus, differential equations, prepositional logic, etc. after college. But I still say there’s value in studying math beyond the practical.

Let me ask a contrasting question: why should kids play sports in school? Only a tiny minority of them will make a living playing or coaching sports, and less than half will even play on the office softball team or the like.

The answer I get most often is that sports teach teamwork: how to subsume your immediate desires for the greater good, make sure you do your part and trust your teammates to do theirs, and communicate effectively to make sure you’re not working at cross-purposes. You learn to win graciously and learn from your failures.

In other words, it’s not so much that playing sports develops muscle tone and general fitness. But rather, it’s an indirect way of teaching other skills like cooperation. They may not be taught explicitly, and there are other ways of teaching them, but this works.

The same thing, I think, applies to math. A few years after learning it, you probably won’t be able to prove that there is an infinite number of primes, or how to integrate a function. But that’s okay, because math teaches skills other than the purely pragmatic.

Proving theorems, for instance, forces you to distinguish between what you think is true, and what you can demonstrate; what looks right, and what *is* right. Geometry teachers always admonish students not to reason from the diagram because concrete examples are often misleading: just because line AB is perpendicular to line CD in this drawing doesn’t mean that that’s always the case. The point is to figure out what’s universally true, not just what’s true about this particular instance.

These are skills that apply to non-mathematical professions: judges and lawyers often deal with people who have clearly done something wrong, and need to distinguish between “I know that ain’t right” and whether the action in question is legal or not. Police officers likewise need to know the difference between “I know Jimmy’s been selling crack to students” and “I can prove to a jury that Jimmy’s been selling crack”. Programmers will find that they write code that works in situations that they didn’t imagine, because it’s a truism that end users will do weird things with your software that you never dreamed of. Financiers should be able to put together a portfolio that can survive unexpected catastrophic changes in the market. (And as much as it pains me to defend the Bush administration, Donald Rumsfeld was quite right in distinguishing between “known unknowns” and “unknown unknowns”, and trying to plan for both.)

Or take the discussion about defining information in the “I Get Email” thread; specifically, the exchange between Tom and Troublesome Frog on whether all living beings contain information. It seems that Tom doesn’t quite get the difference between ∀ *x* ∀ *y* *p* (no matter how you define information, all living beings contain information) and ∃ *x* ∀ *y* *p* (we can come up with a definition of “information” such that all living beings contain information).

Math is very big on abstraction. This starts in algebra, which is all about figuring things out about things that you know you don’t know much about. And it just gets more abstract from there. The first time you demonstrate that there is no solution for some equation, or better yet, that there can be no proof of a given proposition, can be quite a thrill.

And abstract thought is one of those things we humans are good at. It’s what allows us to formulate moral rules that apply to everyone, and see things like “the market” instead of a bunch of people trading stuff. It seems that we ought to learn to do it well.

One of my favorite types of SAT question is the one that presents a math problem, and one of the possible answers is “not enough information to solve the problem”. The lesson here isn’t just “know your limitations”. It also shows that just because something *looks* solvable doesn’t mean that it is; that just because something is printed in an official-looking book doesn’t make it kosher. And also, the sooner you figure out that a problem has no solution, the less time you’ll waste looking for one.

Finally, one thing that everyone should get out of any math class is that you can figure stuff out on your own, without looking the answers up in the back of the book. Yes, the same is true of science and other classes, but math is one of those branches where you don’t need fancy equipment to work on a problem and figure out the solution.

One problem I see is that a lot of people seem to think that knowledge is something that is handed down from on high, rather than something that can be created. This seems to be at the root of the claim that evolution is just another religion: “I have my priests who tell me that God created humans, and you have your priests who tell you humans evolved. The only difference between creation and evolution is which team you’re on.” There’s no arguing with someone who simply repeats what they were taught; but if you’re in an argument with someone who thinks that mere mortals can work out answers on their own, you might get somewhere. And that’s something we could use more of.

Two words: Number theory. I haven’t found a single subject in any discipline that more does a better job of developing that type of reasoning skill. I wish more people had the opportunity to enjoy it. Number theory just makes you smarter. I think that there are a lot of reasons for it:

1) Everything familiar to you is stripped away. You don’t get rules of thumb, “tricks” taught by your math teachers, or anything else. What does it *really* mean for a number to be prime? Really? Is A divisible by B? How do you know?

2) You can do a huge amount of it with nothing more than an elementary understanding of arithmetic and very basic algebra.

3) Even though (2) makes the operations easy, even people who are really good at more advanced math are initially helpless. It takes a really clever person to figure out how to prove that the square root of 2 is irrational. It doesn’t take much skill or brain power to understand the proof once it’s done. You’re learning a whole new way to think, even if you know “practical” mathematics well.

4) A lot of the results can be seen in geometry and the practial world. As most programmers know, modular arithmetic is the key to solving all sorts of real world problems.

If you haven’t read it, I highly recommend Elementary Number Theory in Nine Chapters. It’s as great for an introductory class as it is for reading on your couch without pen or paper in hand. I’m on my third copy because people who borrow it tend to keep it.

Troublesome Frog:

If number theory is what I think it is, then I seem to recall that Martin Gardner had a lot of that in his Mathematical Recreations column in Scientific American. And yeah, there’s a lot of cool stuff in there.

I remember once wondering why it is that you can tell whether a number is divisible by 3 (or 9) by adding up its digits and seeing if that sum is divisible by 3 (or 9). I wound up proving that in any base N, that rule works for factors of N-1.

arensb,

That’s the type of thing number theory covers. A good number theory book can take you from playing with triangular numbers to understanding serious cryptography. Even better, since you’re often working with operators that are defined as part of the subject itself, there really aren’t many prerequisites to learn. I’m always amazed by how problems that deal with such simple operations can be simultaneously incredibly hard to solve and incredibly easy to grasp once you’ve seen the solution. It’s a great puzzle subject.

On a totally unrelated note: Have you seen this? I rooted for Kasparov against Deep Blue, but I’m rooting for the computer this time around. If they even make a good showing, it will be a pretty phenomenal achievement.

Huh. It turns out that I wrote earlier that “math is no more about equations than music is about staves and sharps”. I should’ve used that in this post. Also, this post about Pascal’s triangle seems somewhat apropos.

Troublesome Frog:

I saw a passing reference to IBM’s plan to have a computer compete on Jeopardy, but haven’t been following the story.

arensb wrote:

http://www.youtube.com/watch?v=FC3IryWr4c8 is an enjoyable 4 minute overview. A co-worker asked jokingly if they’d rigged up a robotic arm for Watson to actuate the Jeopardy trigger and it turns out yes, they’d rigged up a robotic arm (FVVOA) for Watson to actuate the Jeopardy trigger.

Wait a second, that sounds familiar. The synthesized voice, the inhuman intelligence, the single arm sticking out of a squat lump of machinery…

Oh, no! IBM is building a Dalek!

arensb Says:

And here I thought you were going to say Stephen Hawking.

[I’ll get my coat – no, no, I know where the door is, thanks.]