## Math field ponders hard-to-fathom quandaries

May 11, 2003

I hit a wall halfway through calculus.

Saul Stahl, a Kansas University math professor, isn't surprised. He says, "Lots of people get lost halfway through calculus."

April being Mathematics Awareness Month, I tried to atone for my lack of math skills. I e-mailed the KU math department and said, "What should I be made aware of about mathematics?"

Professor Judy Roitman reminded me that math serves many other academic disciplines, including industry and even art and music. One of its beauties, she says, is that it lets us talk about ideas that are hard to talk about.

"There are lots of infinities," she says.

Even an infinity of infinities, according to Stahl.

In the book "One, Two, Three ... Infinity: Facts and Speculations of Science," the late theoretical physicist George Gamow writes that one might think the infinity of even numbers would be less than the infinity of the even AND odd numbers put together. But that's wrong. They're the same.

Now here's another twister. A line, even if it's only an inch long, contains more points than all the even and odd numbers put together. And a one-inch line contains just as many points as a line between here and the farthest star -- no more, no less.

On the other hand, there's zero. There's a book about zero by Robert Kaplan, Stahl says. It's got a great title: "The Nothing That Is."

"Do many mathematicians focus on zero?" I ask.

Stahl says no. "There's not enough that's interesting about it."

Well then, how about straight lines versus curved lines -- which are more interesting?

"There's more to be said about curved lines," Stahl says.

Are there other kinds of lines besides those two? An infinity, perhaps?

Stahl and his math colleague, associate professor Tom Creese, say that there are these things called projective lines.

And those are ... ?

The look on Creese's face says "Forget it," so I do.

Creese says, "The trouble with talking about zero, or a straight line or a circle is that they're being considered alone. The interesting problems in math come when you talk about things in relation to each other, not in isolation.

"When you talk about how spheres intersect, now that's a lovely problem."

I'm still at sea. I say, "If you're a novelist looking around a room, you notice details that tell you about the character of the person who lives there. If you're an anthropologist, the details subtly reveal the values of the culture that built the room.

"So what do you see if you're a mathematician?"

Creese remembers a professor who left the KU math department years ago.

"People said, 'He's not going to write a lot of papers, but wherever he goes, other people will start writing a lot of papers.'"

Mathematicians stimulate each other's searches but often don't understand the details of those searches, Stahl says. Mathematical specialties are extremely technical and detailed.

Creese says, "Sometimes, mathematicians are strongly affected by the separateness between them and sometimes by the closeness between them."

Suddenly, I wonder whether the number of feelings that people can have about each other is infinite, and I realize I'm having a mathematical moment.

-- Roger Martin is a research writer and editor for the Kansas University Center for Research and editor of Explore, KU's research magazine Web site, which can be found at www.research.ku.edu. Martin's e-mail address is martin@kucr.ku.edu.

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