Michael Sipser, a 30-year-old rising star in the math department at the Massachusetts Institute of Technology, still remembers the first time he experienced mathematical beauty. "I was very young, about 5 years old, and I remember my father folded down the flaps of a cardboard box so that each flap was holding down the next. And I remember looking at it and being amazed, it was so perfect in a way. I traced it in my mind, and I remember it as my first experience of joy in abstract thought."
Research mathematicians such as Sipser live in a world that is largely unknown to the rest of us, even to other scientists. It is a world of the mind and the imagination, and what they do is create or discover objects that to them are indescribably beautiful.
But, the mathematicians lament, they have no words to tell of such beauty. It is more difficult, they say, to translate mathematics into English than to translate Chinese poetry.
So some mathematicians have given up. Fritz John, for example, of the Courant Institute of New York University, once said his goals are not public understanding of his work, nor are they fame or fortune. All he wants, he remarked, is "the grudging admiration of a few close friends."
But others say they are willing to try to describe what it is they do and why they find it beautiful. What is remarkable is that their descriptions are hauntingly similar to those of musicians or artists or other creative people. The main difference is that when mathematicians have created a work, only highly trained colleagues can truly appreciate it.
The first question usually asked, they say, is, what do you do when you do mathematics research? Do you just go off and think, or do you use a computer to do calculations for you, or do you scribble on a legal pad?
The answer seems to vary with the mathematician. Hungarian Paul Erdos likes to scribble formulas on a pad of paper. Ronald Graham, director of math and statistics research at AT&T Bell Laboratories, says he likes to stare at the walls of his shower stall, which have a grid pattern. Or he draws a small dot, then a larger one far from it and a huge one farther away still on a pad of paper as sort of a mental reminder that the mathematical objects he studies tend to get very big. Sipser keeps a pad of paper in front of him and fills it with scrawled phrases such as, "Let's try doing it like this . . . but that doesn't work for this reason . . ." The phrases, he says, help him focus his thinking. "I fill reams of paper, but I never look at the papers again," he says.
Somehow, out of all this, ideas spring forth. But frequently, the ideas come when they are least expected. French mathematician Jacques Hadmard, who wrote about such experiences in his book, "The Psychology of Invention in the Field of Mathematics," cites what is now considered a classic description by French mathematician Henri Poincare, who did his work around the turn of the century.
Poincare had spent weeks thinking about a kind of abstract object called fuchsian functions, but had gotten nowhere. He had even tried to prove they did not exist, but to no avail. Then, he took a bus trip as a tourist. "The incidents of travel made me forget my mathematical work," he wrote. "Having reached Coutances, we entered an omnibus to go some place or another. At the moment when I put my foot on the step, the idea came to me, without anything in my former thoughts seeming to have paved the way for it . . . I went on with a conversation already commenced, but I felt a perfect certainty. On my return to Caen, for conscience's sake, I verified the result at my leisure."
Persi Diaconis, a statistician at Stanford University, says that when ideas first began to spring into his mind, they would make him nervous. But now, he says, he has come to have more trust in them. "Many times, quite complicated patterns jump into my mind, fully formed, after I have been thinking about something for several weeks. Big chunks of the solution to a problem are revealed to me.
"It's amazing when it happens, and the most amazing thing is that you develop faith in that underlying structure that you see. I've gotten so I say, 'Gee, this must be true.' "
The experiences of artists are strikingly similar. "Sometimes I have observed this moment when a sensation arrives at the mind; it is as a gleam of light, not so much illuminating as dazzling," French poet Paul Valery wrote in an essay, quoted in Hadmard's book. "This arrival calls for attention, points, rather than illuminates, and in fine, is itself an enigma which carries with it the assurance that it can be postponed. You say, 'I see, and then tomorrow I shall see more.' "
But this still leaves the question of what it is that mathematicians see, what it is they create or discover. Sipser says he finds beauty in the interconnections between mathematical objects. "Sometimes when you see links between two things that never seemed related before, there's a sense of joy associated with it," he explains.
Diaconis agrees. "Part of mathematical beauty is seeing that things that look completely different really are manifestations of the same things. It is like a set of roots just under the surface of the earth. Here and there you see something pop up, but underneath there is this very deep set of interconnections." SS ometimes, mathematical insight can arise from the most unlikely experiences. Henry S Pollack, assistant vice president for mathematical communications and computer sciences at Bell Communications Research, tells of a discovery he made that was instigated by figuring out an expense account.
He had traveled from Bell Laboratories in New Jersey to Bell Laboratories in Columbus, Ohio, for a business meeting. From there, he went to Raleigh-Durham, N.C., to visit a high school that specialized in science and math. Then he went on to Tampa to visit his mother-in-law. Finally, he returned to New Jersey. The problem was, how should his expenses be allocated? Who pays for what?
Bell was willing to pay the equivalent of a round trip fare between New Jersey and Columbus. The North Carolina high school was willing to pay a round trip fare between New Jersey and Raleigh-Durham. He was willing to pay his own round trip fare between New Jersey and Tampa. "Of course, the total cost of the ticket was much less than the cost of four round trips," Pollack says.
He recognized that this problem was not just a personal one of figuring out an expense account. It occurs in numerous guises in all sorts of business and industries. For example, people use home telephones for long distance and local calls. How should the costs be divided?
"Or suppose you are trying to do an honest accounting of how much it costs to run a hospital," Pollack says. "You would charge the costs of the operating rooms to surgery, but who should be charged for the corridors? What about the stockroom, the secretaries and the janitors?
"You might say that you could just charge each department in proportion to the amount they use each service, but what do you do about the fact that people who run the stockroom eat in the cafeteria? The service departments service each other."
After looking in the mathematics literature for a way to solve the relatively simple cost allocation problem that arose with his airplane trip, Pollack learned that "there is a literature saying there aren't any good solutions." So he and his colleague Peter Fishburn tried to develop their own.
They began by writing down rules that would have to be obeyed if the cost were to be divided fairly. None of the three parties involved should pay more than the round-trip cost of that portion of the journey. Everyone should pay something. And the total amount payed should equal the cost of Pollack's ticket.
The first thing Pollack and Fishburn proved was that there really is an answer to this problem. The next question was how to find it. After trying a few blind alleys, Pollack hit on a crucial insight: The solution to a very simple variant of the problem could be generalized to fit more complicated problems as well.
All he had to do was solve the problem as if he were going to only two places -- point A on one side of his home, and then point B on the the other. He would be reimbursed for round trips between his home and A and his home and B, but not between points A and B. Solve that, and you've solved the tougher problem, too.
Pollack was amazed that it was all so simple. "I looked at the proof and said, 'Hey, this can't be. I didn't work hard enough.' I checked it again and again and finally I decided it was right. My main feelings were elation tempered with incredulity. I realized that in mathematics you're never so wrong as when you're quite sure you're right."
Although not every cost allocation problem was solved, the two mathematicians did find a way to solve many of them. It is the sort of work that mathematicians say gives them great joy. It is moments like these, Diaconis says, that "make me very happy."