Wednesday, November 14, 2007

Math is Hard... and Then You Die: Chapter 5

(Author's note: I'm going to try to keep this one short, but it's going to be an effort. When we get to the moment when make the big change, I could easily give into my inner Marcel Proust. You know, eat a cookie, think about the past, write a seven volume novel. I'll see if I can in touch with my inner Richard Brautigan. You know, one short sentence can be a paragraph, or even an entire chapter.)

So now I'm a junior in college and a computer science major. I've taken a lot of math and some computer science classes. I've gone through the calculus series and taken some linear algebra. I've had some good teachers, including Ken Rebman, Ed Keller and Russ Merris in math, and Dan Jurca in computer science, but the math topics have been interesting, but not particularly compelling. It's still more fun for me to write a computer program than to solve a math problem.

Then I take Modern Algebra from Ted Tracewell.

I don't have a picture of Ted Tracewell, so I put up two pictures of celebrities he kind of looked like.

The guy in color is Adolph Green, actor and screenwriter. He was in My Favorite Year as King Kaiser's friend and lawyer. He also wrote Singin' In The Rain with Betty Comden.

The guy in black and white is Isaac Asimov.

I could have just said Ted Tracewell looked like Asimov without the ridiculous mutton chops, but he also had Green's energy when he got excited about a topic. Sometimes class was like a musical comedy without the music.

Modern Algebra is now called Abstract Algebra at most schools. Since it was developed in the 19th Century, it's more modern than the rest of algebra, but the name is a little misleading. There are many ways to teach the class, but the standard way is to concentrate on group theory.

Oh.. My.. God... group theory!

Here's where I could go all Marcel Proust. Don't get me started. Let me try to contain my explanation of group theory to three short paragraphs.

The Big Idea: Group theory is the mathematical study of symmetry. Since almost everything that is beautiful has symmetry, you could say that group theory is the mathematics of beauty.

Definition of a group (sort of): You have a set of things called elements, and you have an operation that takes two elements and combines them into a third element in the set. (There are some other rules. That's why I said sort of.)

Example of a group: Take all the integers, from negative infinity to infinity. Writing them as a list, we use the ellipses as follows: ... -3, -2, -1, 0, 1, 2, 3,...
Use addition as the operation.
What it means to be a group is that if you add two integers, you get another integer.
Also, the number 0 is important because a + 0 = a for all a, and a + (-a) = 0.
These are the ideas of an identity element and a unique inverse element for every element.

While the initial rules for defining a group are simple for math, there's a lot to group theory. Mathematicians get lost in it, so much so that it created a decades long rift between mathematicians and physicists. Symmetry is incredibly useful for simplifying and solving differential equations that physicists need, but mathematicians would yap, yap yap about all the wonderful things in group theory, all the symmetries within symmetries, when physicists just wanted the answer to a single problem.

(Sorry, that story is definitely my inner Proust talking. Long story short: the most recent generation of physicists decided not to listen to their physics professors and became good at group theory. Some of the most important advances in the 20th Century in the subject were done by physicists, not mathematicians.)

So it's my first week in class, and Tracewell has defined a group, shown some examples, and defined a subgroup, and shown some examples of subgroups in groups we have already learned about. It's a Friday, and I go up to ask a question. "All the subgroups and groups you have shown so far have the property that the size of the group can be divided evenly by the size of every one of its subgroups. Is that just coincidence or is it always true?"

Tracewell lit up like a Christmas tree. "It's always true! Prove it yourself this weekend! Don't look it up!"

So I did. And I was a math major.

There are lots of pretty and fun parts of math, but it's absolutely astounding how often group theory is lurking around under the surface of the pretty stuff. It's really fantastic. There were a lot of good teachers at Cal State back in the 1970's and a lot of good ones now as well, but I consider myself fortunate that I got to learn the basics of group theory from Ted Tracewell.

I called this little memoir Math is Hard... and Then You Die as a play on words on the famous thing Barbie said and the pessimistic motto "Life is Hard, and Then You Die.", but I now have to interject an actual death. Before I took classes from Tracewell, he was a butt-end smoker. He had two packs of cigarettes in his pockets, two different brands, and he would light his next one with the butt of the one he just finished. He taught with chalk in one hand and a cigarette in the other.

In the 1970s, he quit smoking. He started walking more and dedicated himself to living a more healthy life.

But in the 1980s, after I left Cal State, he died in class, at the chalkboard. If some want to say it was a fitting end, I won't disagree.

For me, how he lived is more important than how he died. He counted me as his friend as well as his student, and that is one of the great blessings of my life.

Chapter 6: The incomparable Dr. Stuart Smith.

Now playing: Smokey Robinson & The Miracles - I'll Try Something New
via FoxyTunes


Karla said...

What a cool dude he was!

Don't you wish all teachers were like that?

dguzman said...

Neat, Matty. I often wish I were more math-inclined (or even interested) but I just missed out on that gene. My whole family has it, but I don't. But then, for a long time I was the only one who could swim. So there.

Matty Boy said...

I had a lot of good professors at Cal State Hayward. I'm still close friends with one from whom I never took a class, Victor Manjarrez. There were five different professors who introduced me to strangers as their son: Manjarrez, Tracewell, Summers, Rebman and Simon.

Maybe they just didn't want to explain why they were hanging out with a pretty youth, though as I said before, it was completely innocent because I'm not gay and I never have been gay.


CDP said...

That was a good story, and a good explanation of group theory for a former math-phobe.