What's It Do? Nothing, but Mathematicians Relish the Quest

Although proving theorems usually doesn't add up to anything practical, the intellectual allure is powerful

by Fernando Q. Gouvêa

Originally published in the Los Angeles Times on December 29, 2003

It took hundreds of thousands of computers and several years of work, but they got it.

"They" are the participants in the Great Internet Mersenne Prime Search. "It" is one more very large prime number, a monster with 6 million digits, part of a sequence of numbers known as "Mersenne primes" that is expected (but not known) to go on forever.

As mathematical achievements go, this one was fairly minor. It required no theoretical innovation, no conceptual leap; time, persistence, the Internet and lots of computers were enough.

Finding a new Mersenne prime confirms the expectation that it was there to find, but does not give us much more than that. As one of the people involved said this month when the discovery was announced, "It's a neat accomplishment, but it really doesn't have any applicability."

Many great mathematical quests are like this. They are exciting adventures of the mind whose completion takes years of effort by whole communities of mathematicians but whose results are not usually of immediate practical use. This may come as a surprise, since our teachers spent a lot of time telling us that mathematics is important because it is useful. But that wasn't the whole story.

Perhaps the best-known example is Fermat's Last Theorem, scribbled in the margin of an old book around 1636 and finally proved by Andrew Wiles in 1994. Fermat wrote that he had found a "marvelous proof" of a negative statement: If you take a whole number and raise it to some power greater than two, he said, it is not possible to write that number as the sum of two nonzero numbers raised to the same power. So, say, 20,736, which is 12 to the fourth power, cannot be written as a sum of two (nonzero) numbers to the fourth power.

A nonmathematician might wonder why anyone would want to prove that. But not only did people want to, they spent 350 years trying. During those years, many mathematicians put together a vast theoretical edifice dealing with such exotic beasts as "elliptic curves," "modular forms" and "Galois representations." The theory served as the base camp from which Wiles set out to get to the peak. It was an impressive conquest, and the methods are proving to be fruitful indeed, but as far as we know the whole thing has no practical use.

In the 20th century, much effort was directed at solving the "Hilbert Problems." German mathematician David Hilbert listed these at the International Congress of Mathematicians in 1900 as prime targets for mathematical research. Most of the problems, which were more like broad questions than like the problems one finds in textbooks, had no direct applicability. Can arithmetic contradict itself? Can one find a general method to figure out whether it is possible to find whole-number solutions to equations? Is every even number the sum of two prime numbers? Hilbert's personal prestige guaranteed that solving one of his problems would establish one's fame.

The Clay Mathematics Institute has followed Hilbert's lead in the 21st century, even putting up some prize money: \$1 million for the solution of each of seven mathematical problems. Of the seven, two are related to physics, one comes from computer science and the other four are purely mathematical. One of the latter four (asking for a proof of the "Riemann Hypothesis," which, if true, will help in the understanding of prime numbers) was also on Hilbert's list, reminding us that sometimes a century of work isn't enough.

Many other such projects, past and present, have guided mathematical research. Mathematicians usually call them "conjectures." Based on known results, special cases, numerical computations and a general sense of how things are bound to be, conjectures are visions of the future and challenges to current scholars. Some turn out to be false (one of mine just did), but even the false ones prove their worth if they guide people to the right questions and to powerful new methods.

Much of this work has no practical application whatsoever, or at least none is known. But appearances can be deceiving. Non-Euclidean geometries had been studied and developed for half a century with no known application when Einstein arrived and put them to use (by recognizing that the right way to understand gravitation was to postulate a curved space-time. In the 1930s, G.H. Hardy famously boasted that none of his work would ever have practical applications; today, ideas closely related to Hardy's are used in cryptography.

Still, what motivates mathematicians is not necessarily the potential for applications. When asked why they do what they do, some resort to George Mallory's answer: "Because it's there." Perhaps John F. Kennedy's explanation is even closer: "We ... do these things ... because they are hard." At the core, however, is the desire to know and understand, to track the monsters to their lairs, to see how ideas fit together and why.

It may be of no practical use to know the answers to the questions that drive mathematical research. In fact, some of the questions may even be beyond our capacity to answer. But most mathematicians would agree with British mathematician E.C. Titchmarsh: "If we can know, it surely would be intolerable not to know."

Copyright 2003 The Times Mirror Company; Los Angeles Times