 December 2002

Why are equations important?

Given the hours that mathematics teachers spend instructing students how to solve equations, it would be easy to assume that the most important thing to do with an equation is to find a solution. But that is rarely the case. Most of the equations that arise in real world contexts cannot be solved. Even if they can, it is often simpler and faster to use a computational method to find a numerical solution. The real power of equations is that they provide a very precise way to describe various features of the world. (That is why a solution to an equation can be useful, when one can be found. )

Before I find myself inundated with hundreds of angry emails from teachers who don't want their students getting the idea that learning how to solve equations is not important, I should say that it is indeed an important exercise. But the reason is not that the student is likely to find her or himself actually solving equations - outside the math class, that is. Rather, mastering the processes required to solve equations is arguably the best way to become adept at understanding what equations tell us. This, of course, is the same reason why English teachers ask their students to write essays. Few of those students are likely to go on to become novelists or journalists, but writing essays is the best way to learn how to use written language.

One of the most dramatic illustrations of the unimportance - outside of mathematics itself - of solving equations is provided by modern physics. The fundamental theory of matter that physicists work with today is the most accurate scientific theory the world has ever known. Predictions made on the basis of the fundamental equations of matter have been experimentally verified to many places of decimals. And yet, none of those equations has been solved. You have to go back to the 1920s to find equations of matter that anyone has been able to solve.

Living as we do in a world filled with high tech gadgets that depend upon modern physics - the computer I am writing this on and the CD player that is keeping me entertained as I do being just two such - it is obvious that the lack of a solution has hardly held the physicists back, or the engineers who take modern physical theory and turn it into products. Without the precise understanding provided by the equations, the world would not have silicon chips, compact disk players, MRI medical examinations, or many of the other things we now take for granted. But none of those applications required that those equations be solved in the strict mathematical sense.

Physicists have spent the past eighty years trying to find a single framework that explains what are now believed to be the (only) four fundamental forces of nature: electromagnetism, gravity, the strong nuclear force, and the weak nuclear force. Most of the effort has been directed toward developing an extension of quantum theory of a kind that physicists call "quantum field theory" (QFT). The picture of matter that QFT has given us, which represents our best current knowledge of the nature of the material that makes up universe, is generally referred to as the "standard model of particle physics."

Edward Witten of the Institute for Advanced Study in Princeton, New Jersey, one of the present leaders in this ongoing research, has described the current version of QFT as "a twentieth century scientific theory that uses twenty-first century mathematics." By that, he means that much of the mathematics remains to be worked out - in other words, mathematicians have yet to solve the equations!

It may seem that Witten is being hard on the mathematicians for being so tardy, but in fact he is simply being realistic. Scientists and mathematicians have been in this position before. Much of Newton's science depended on the methods of calculus, which he invented for the purpose, but the details of calculus were not fully worked out as a mathematical theory until two hundred years later!

A specific unsolved mathematical problem arising from QFT research was chosen as one of the seven Millennium Problems that the Clay Mathematics Institute announced in the year 2000, offering a prize of \$1 million to the first person to solve each problem. (I describe these seven problems in as close to layperson's terms as I can in my recent book The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time.) This particular Millennium Problem, the only one on the list that comes from modern physics, asks for a solution (under certain specified conditions) to the Yang-Mills equations (a quantum field analog of Maxwell's equations for electromagnetism), together with a subsequent explanation, based on that solution, of the so-called "mass gap" (the conjectured, and hitherto observed, minimum level to the mass that any matter may have).

Despite its origins in physics, the problem as stated is essentially a mathematical one. Indeed, many physicists regard the problem as in large part already solved. MIT's Frank Wilczek, one of the leading figures in QFT, and one of the initiators of quantum chromodynamics (QCD), the most comprehensive theory within the overall QFT framework, comments:

"Specifically, one has direct evidence for the existence of the basic elements of the theory [QCD] - quarks and gluons - and for the basic interactions the theory postulates. Most of the evidence is from studies of jets in high-energy processes, and comparison of their observed properties with very precise and unambiguous calculations in QCD ... Another sort of evidence is from actually integrating the full equations directly, using powerful computers. This work directly addresses, and to me effectively solves, the Clay problem. We not only know that there is a mass gap, but have calculated it, and successfully compared it with reality. Of course I understand that numerical results, however convincing and well controlled, are not traditionally considered mathematical proofs."

If physicists like Wilczek regard the Clay problem as already solved, why did the Clay Institute include it in their list of the seven most difficult and important unsolved mathematical problems at the start of the third millennium? The answer is provided by Arthur Jaffe of Harvard University, an expert in the mathematics of quantum field theory and until recently the director of the Clay Institute. He said, "The problem of Yang-Mills Theory and the Mass Gap Hypothesis was chosen as a Millennium Problem because its solution would mark the beginning of a major new area of mathematics, having deep and profound connections to our current understanding of the universe." In other words, solving equations is an important goal within mathematics.

Jaffe's remark does not, however, imply that a solution to the Yang-Mills Theory and Mass Gap problem, if one were discovered next week, would not have major consequences for physics. On the contrary, it would, almost certainly, lead in time to an increased understanding of matter, and from there to who-knows-what new technological gadgets to improve and enhance our world. But - and this is why Wilczek and Jaffe are not in conflict here - applications will almost certainly not start with the solution, as such, rather will come from the methods used to find that solution. As is so often the case in mathematics, in the long run, the method is likely to be more important than the answer.

Remember, all you students out there, I am not saying that solving equations is not important. It is important for several reasons. Rather, my point is that it is generally not the most important thing to do with an equation, whose real power is as a formal and precise description of our world.

We should also remind ourselves that any new mathematical result has the potential to change the world. Many have.

Devlin's Angle is updated at the beginning of each month.
Mathematician Keith Devlin ( [email protected]) is the Executive Director of the Center for the Study of Language and Information at Stanford University and "The Math Guy" on NPR's Weekend Edition. His book The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time was just published by Basic Books.