A young man strolls along a wooded path, stopping every so often to stare off into space. Onlookers at the park can see him talking to himself, and every so often gesturing into the air with his hands. Finding a sunny spot near a tree, the man lies back and gazes off into the great blue yonder. The clouds dance by, time passes, and still he lies there. Then, without warning, he leaps to his feet, almost stumbling in the process. A wild look crosses his face, followed by an expression of near euphoria. With a scream reminiscent of the winning pitcher throwing the last strike in the World Series, he tears across the park and runs out of sight. Mothers with young children instinctively hold them near, muttering under their breaths about the loonies out and about today.

Somewhere a little child tugs on his mother’s arm and asks, "Mommy, what was that man doing?"

"Who knows, darling... who knows."

This could have been the story of a recent escapee from a local mental hospital. It could have been an absent-minded young man that realized he left the oven on. It could also have been a quite standard procedure for doing some groundbreaking mathematical research.

If you asked someone with no scientific background what a physicist does, he could picture people walking around big machines with Einstein-type hair. He might tell you that chemists walk around all day in lab coats mixing things and watching them foam and react. Ask about a biologist and you might get images of doctors doing molecular research, or looking at animals. For almost every branch of scientific research, the average person on the street could give you a mildly accurate description of what a research scientist in that area might do. In mathematics, however, this is not the case. What does a mathematician do? How can someone stand to sit around and do long division all day?

Perhaps the greatest barrier to public appreciation of research in pure mathematics is that most people, even the scientifically inclined, have no idea what pure mathematics is. To most people, math is what you can do on a calculator. Numbers collide in various ways and make new numbers, and therein lies the use and extent of mathematics. The more scientifically inclined have a notion or even a mastery of trigonometry and some of the finer elements of calculus. Even those, however, have little more than an inkling of what "true mathematics," if such a thing could be defined, is.

A brief glimpse can be seen of mathematics at its most pure with little or no mathematical training. Number theory is unique among areas of mathematics, perhaps among all sciences, in that many of its more complicated problems can be understood and even worked on by those with no more than a high school education. Familiarity with multiplication tables and some notion of what a prime number is will suffice for a large number of problems. The average passerby on a city street can understand one of its most difficult puzzles.

Most of us are familiar with Pythagorean triples. To remind the nonmathematical reader, think of the sides of a right triangle. A squared plus B squared equals C squared. Sound familiar? A Pythagorean triple is a set of three whole numbers that can be put in for A, B, and C. For example, 3, 4, and 5 work. 5, 12, and 13 also work. Now what happens if we cube the numbers? Raise them to the fourth? What if we raise them to any power n, where n is greater than 2? Can there be any whole number answers for these formulas?

"Cuius rei demonstrationem mirabilem sane detexi hanc marginis exiguitas non caperet."

"I have discovered a truly marvelous demonstration of this proposition which this margin is too narrow to contain." - Pierre de Fermat.

Work all you want, but I doubt if you'll be able to find an answer. Those words, scrawled in the margin of Fermat's copy of Diophantus' Arithmetica, would confound mathematicians for over 350 years (Singh, 62). For the curious reader, the answer is no. The proof, however, of this answer would be the crowning point of a lifetime of work by one of the most famous mathematicians of the twentieth century, Andrew Wiles. The work spans novels, and involves mathematics understood by only a handful of people. This could not be the proof that Fermat had if indeed he had one, as it uses mathematical tools unheard of in Fermat's time.

What is the point, though? What does this mean? Who cares whether or not some equation has a whole number solution? The question of "Who cares?" plagues many mathematicians today. Why would anybody care about their work, whatever "math research" is? Interestingly enough, some mathematicians are about as focused on the world as it is on them. G.H. Hardy once bragged of his research (in number theory), "I have never done anything 'useful'. No discovery of mine has made, or is likely to make, directly or indirectly, for good or for ill, the least bit of difference to the amenity of the world," (Hardy, 150). Despite his attempts, Hardy's work was useful after all, and in a way he would have deplored. Very recently prime numbers have found a home in the Pentagon as the basis for the military's most secure codes.

John Tierney summed up the relationship between mathematicians and the world around them. "This is the remarkable paradox of mathematics. No matter how determinedly its practitioners ignore the world, they consistently produce the best tools for understanding it," (Hoffman 178). Many of the tools found most useful in describing the world were sought out by mathematicians for no reason other than that the mathematics was interesting. Perhaps it is fitting that the world cares little for the current work of pure mathematicians, since they focus so little on the world.

The main difference between mathematics and the more familiar sciences is that the sciences find ways to describe the world that we have. Mathematics defines worlds to describe. The geometer offers the physicist a variety of universes to choose from, and the physicist picks the one that seems to best resemble ours. It is important to have this multitude of universes from which to choose, because we are never sure if we have the right one. One great example of this necessity arose when Bernhard Riemann decided to find out what a geometry would look like if you threw out the parallel postulate. That is, suppose that no two lines could be parallel. They would always intersect somewhere, maybe at infinity. This seems absurd, but later Einstein would announce that this is the best way to describe our universe. Riemannian geometry is a rich field today.

A mathematical line of inquiry differs then from a scientific one in the questions it asks and the tools it uses. A chemist might ask, "Why do these reagents react in this way?" A mathematician, when presented the same data might ask, "Suppose they don't react this way. Then what do we have?" That is indeed the greatest tool in mathematical research, the supposition. It is the tool I find most enjoyable, for in what other field do you have such godlike power over what you work with? If you are curious about the consequences of a certain action, suppose that it happens and then work from there. What would the laws of physics be if gravity worked by a worked according to a different formula? Is there a compelling reason for it to have the mathematical structure that it has?

Perhaps it is important then that research in pure mathematics has little to no concern for the real world; we never know when we may realize that the way we are looking at the "real world" is wrong. It would be helpful in such a time to already have an alternate universe described. If we concern ourselves only with what we think is real, then how will we know where to turn when what is real changes? The seeking of a good and interesting challenge by mathematicians everywhere, then, protects us from this. Why climb Mt. Everest? Because it is there. Why throw out the parallel postulate? Just to see what happens. That is the joy and motivation of working with a subject that is in a way all a figment of our collective imaginations.

I hope that I have managed to at least illuminate to you some of the motivation for working with "long division" for a lifetime. The subject matter of mathematics is limited only by the imagination of its practitioners. So, next time you see a math professor, don't ask him how his day has been. Ask him, "Suppose it was raining today. Then how would your day have gone?"

Hardy, G. H. A Mathematician's Apology. 1969. Cambridge University Press.

Hoffman, Paul. The Man Who Loved Only Numbers. 1998. Hyperion. New York.