Certainty, Beauty, Usefulness



PRINCETON, NEW JERSEY — Princeton, New Jersey. -- The front pages of many of the world's newspapers were even more startling than usual the other day. Somebody seems to have gotten the right answer to one of the world's vexing problems.

Andrew Wiles of Princeton University announced from England that he had solved ''Fermat's last theorem,'' a problem that has bedeviled mathematicians for more than three centuries. News of the solution, understandable to only a few in the world today, flashed globally across E-mails and faxes in minutes and got to the front pages within hours. That such a problem appears to be solved is stunning -- for its certainty, its beauty, its usefulness. And maybe for how few appreciated it.

Certainty is rare and not often enough pursued. Instead there is ''partial credit.'' When I was teaching at Harvard, students in a calculus course expected partial credit for getting a problem started or arriving at a reasonable -- but wrong -- answer. I asked them, ''Suppose you end up being a doctor. Are your patients going to be satisfied if your diagnosis is partially correct?'' The point is a serious one: In mathematics there are definite answers, and a student can take real satisfaction in ''getting it right.''

It is a well-kept secret that mathematics is really fun -- at least for mathematicians -- and I am amazed at how often we use the word ''beautiful'' to describe work that satisfies us. A great mathematician was once speculating with some anthropologists about early experiments with fire. One anthropologist suggested that humans were motivated by a desire for cooking; another thought they were after a dependable source of heat. The mathematician said he believed that fire came under human control because of fascination with flame -- for that beautiful abstraction that yields the good cooking and reliable heat we all need.

Few understand that the most advanced math is useful. A famous example is part of the folklore of Albert Einstein. He had spent years brooding over the possibility that gravitation is really a symptom of the curvature of spacetime, but he lacked the language to express it. He turned to his friend Marcel Grossman, who told him that an entire subfield of mathematics, the study of curved space, had been elaborated decades before by the great mathematician Bernhard Reimann. A vast body of research was already available. Einstein, a physicist first and a mathematician only by necessity, breathed a sigh of relief and went ahead with the general theory of relativity.

Several decades ago, the engineer Alan M. Cormack was searching for a non-invasive way to pinpoint the location and density of objects in the body. At the time, only X-rays were available, and they gave information in only two dimensions. Cormack discovered that a mathematical solution had been around for many years, dating from the early-20th-century work of Johan Radon. Using Radon's solution, Cormack could take X-rays from many angles and determine the location and shape of objects in the body, such as organs and tumors. Thus was born the CAT scan, followed by magnetic resonance imaging and position emission tomography. Radon's solution has also been applied in oceanography, astronomy and even anthropology.

Math is today a constant partner of useful science. The complex Navier-Stokes equations have enabled work in fluid dynamics, including the study of hurricanes swirling through the atmosphere, the rush of blood through the heart, and the seepage of petroleum through porous ground.

Fluid dynamics has led to computational models of the kidney, pancreas and ear; to improved design of artificial heart valves. It is applied to chaotic behavior, where small changes can produce large effects: Relatively few molecules of chlorofluorocarbons precipitate a large ozone hole in our atmosphere.

Mathematics brings new efficiencies to almost everything we do. High-performance aircraft, which used to be designed primarily by wind-tunnel experiments and risky test flights, are now shaped by mathematically designed modeling. Integration of design and performance is more efficient, and more test pilots live to become grandparents. Mathematicians have joined biologists to explore the mechanics of DNA replication. Math contains a subfield called knot theory, which along with probability theory and combinatorics, is helping us understand the complexities of genetic architecture.

There is also a major effort to model mathematically the AIDS epidemic, which is showing that HIV does not spread like the agents of most other diseases.

And one of the deepest interactions between mathematics and science today is string theory, which gets its name from the notion that the fundamental units of matter may be shaped like tiny, vibrating strings. This is the first theory to incorporate gravity into a broad description of matter on a microscopic scale -- a theory of everything.

Yet mathematicians have dwelled so long in splendid isolation that the public poorly understands what we do. The mathematical community has yet to show the public that though we do not design widgets or cure diseases, our impact on engineering and medicine is enabling and significant. Our role in sustaining science and the economy is critical. If we are to hope for better and more support, then we mathematicians have to communicate better and produce better math teachers.

I can honestly say that the most important person in my own career in forming my decision to become a mathematician was Lottie Wilson, my high school mathematics teacher. She had two essential qualities for getting her subject across. She understood the majesty and mystery of mathematics. She also knew there is no substitute for getting the right answer.

Phillip A. Griffiths is a mathematician and director of the Institute for Advanced Study in Princeton.

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