Count on numbers to always be there

By Dan Rockmore

The Boston Globe, Tuesday, August 8, 2000

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After someone discovers I'm a mathematician, the first question I'm usually asked is, "How do you do research in mathematics? Don't we already know all the numbers?"

It's certainly an honest enough question. For most people, all the experience that they have with mathematics is via the numbers of everyday life - prices in the grocery store; the size of jeans, or the genome; the population of the Earth; your age. And it's true: We never seem to run out of numbers to describe these things. You'll never see a headline that says: "Shaquille O'Neal signed to new contract; mathematicians still looking for number to describe it!"

What most people do not know is that mathematics is much more than the numbers that describe our daily concerns and like any living and vibrant science, the more we learn, the more we see that we do not understand. Like looking at the Earth from afar, it seems easy to divide things into land and water.

But closer inspection reveals further and further divisions. The more detailed the taxonomy, the more we question both what distinguishes things and how they are related. Our initially primitive natural science evolves into the wild morass of disciplines that live today. The simple division into one or the other is exposed for the naivete that it manifests. The more we know, the less we know, even about numbers.

This week on the UCLA campus, the American Mathematical Society is sponsoring a conference called "Mathematical Challenges of the 21st Century." The title harkens back to the famous meeting held in August 1900 of the International Congress of Mathematicians in Paris. There, on Aug. 8, David Hilbert, professor of mathematics at Gottingen, gave a lecture titled "Mathematical Problems," in which he listed what he then believed to be the 23 most important unsolved problems in mathematics. To this day, only nine of the problems have been solved, and work on Hilbert's program has driven much of mathematics research in the 20th century.

Part of the purpose of the UCLA meeting is to do a bit of the same for the 21st century: Outline the state of the art in the many, many different subdisciplines into which mathematics has evolved, as well as give some indication of what currently seem to be the fruitful and important directions of inquiry and unsolved problems to pursue. As Hilbert said in 1900, "As long as a branch of science offers an abundance of problems, so long is it alive."

Mathematics is alive and well, and, in some sense, the UCLA meeting will try to answer the question of my innocent interlocutor: Don't we already know all the numbers? I can honestly say no, from many points of view we don't.

Even at the simplest level - do we really know the numbers of everyday life, those God-given natural numbers: 1,2,3. .. - the answer is no, we don't really know them! Like an old dear friend who, after many, many years, still seems to possess a never-ending wealth of surprises, the natural numbers are still a source of mystery for mathematicians.

The most outstanding open problem here is a detailed understanding of the frequency with which prime numbers appear as we list all the natural numbers. Prime numbers are those numbers with no divisors other than 1 and themselves.

Any natural number can be obtained in only one way by multiplying together some prime numbers. In this sense, these are the basic building blocks, or atoms, of our natural numbers. But, just like atoms, a more careful inspection of the primes has yielded years and years of unexpected discoveries and compelling conjectures.

But perhaps my friend is asking a more subtle question. From time immemorial, numbers have been a part of the mathematical means by which we describe the world. Seven days and seven nights; the animals arrive two by two. Perhaps I am being asked if we have all the tools at hand necessary to describe nature. Well, clearly we don't, as anyone who has ever relied on a weather prediction will tell you. Mathematicians have barely touched the surface of being to able to describe the phenomenon of turbulence, which both jangles us during air travel or hypnotizes us as we watch the steam drifting above our morning coffee. This is just one part of the world of fluid dynamics.

Moving from the kitchen to the cosmos, the mathematics of string theory is still evolving to help us better understand both the origin and geometric description of the universe. This is mathematics used to describe the world around us, but there are also many challenges posed by the problems of describing the worlds within us. Now that the genome has effectively been sequenced, the problem of modeling genetic regulation and genetic networks in order to use the information given by the genome is a very exciting direction of new mathematical research.

The more we know about the world around us, the more we need new mathematics and mathematical tools to describe it.

Do we know all the numbers? Let me ask the question differently and instead wonder if we know all the ways in which we can compute numbers. Have we found the best possible technique or machine to solve any given problem? Once again, we stumble upon another vibrant and thriving direction of mathematical research. Mathematicians are working hard at understanding the power of quantum computers, computational platforms that depend on the subtleties and mysteries of quantum mechanics.

Even for our regular silicon-based PCs, the most efficient methods for computation are still, in many cases, unknown. The world of computational problems is a crazy quilt of patches labeled by the theoretical efficiency of the fastest known solution. This is called the complexity of the problem. The sticking point is that, just because we don't know a fast procedure, or algorithm, today, it doesn't mean that someone wont discover one tomorrow. Thus, it is not clear whether some of these complexity classes don't, in fact, overlap.

To date, the outstanding open question in complexity theory is whether there is an efficient solution to the Traveling Salesman Problem. In this problem, we imagine a salesman who needs to travel to many cities and wishes to both minimize his expenses while never visiting any city more than once. If there is an efficient algorithm, then it turns out that this algorithm could be transformed into efficient algorithms for many different problems. This is the famous P = NP question, and it is the holy grail of theoretical computer science.

So, really, we do not know all the numbers, but in Los Angeles this week, well collect ourselves to see what sorts of questions are on the horizon, and then well get to work. You can count on it.

Dan Rockmore is professor of mathematics and computer science at Dartmouth College in Hanover, N.H.