Devlin's Angle

February 2003

Squaring the circle

Mathematicians who speak on the radio or write for a local or national newspaper generally find their brief moment of fame results in a small postbag of letters from individuals who claim to have solved some major mathematical problem. In my own experience, Fermat's Last Theorem and Squaring the Circle easily top the list. Having just completed a small blitz of media appearances surrounding the publication last fall of my new book The Millennium Problems, I have received a number of Fermat proofs and circle squaring attempts. (Never mind that neither of these is a Millennium Problem.)

Word certainly seems to have gotten around that someone claimed to have proved Fermat's Last Theorem a few years ago, though few of my correspondents at least could name the solver as Princeton's Andrew Wiles or the year as 1994. (Please don't write and say it was 1993. That was the year of the initial announcement of the result, but Wiles subsequently withdrew his claim after a major flaw was found in the proof. He corrected it the following year, and hence 1994 is the year it was solved.) Unfortunately for the poor mail carrier at my university, however, word also got around that the 1994 proof is long and complicated, and not everyone heard the news that the mathematical community did eventually sign off on the correctness of Wiles' 1994 proof. That left open the door for an enthusiastic amateur to seek a proof using elementary (i.e., high school) methods.

I long ago stopped responding. As someone who loves mathematics, I do not want to prevent others from gaining similar enjoyment from the field, and if an individual gains pleasure from battling away trying to find an elementary proof of Fermat's Last Theorem or any other result, then they have my blessing. Their enjoyment in mathematics is surely not unlike mine. But my earlier approach -- twenty years ago, I should add, when I first forayed into the pages of national newspapers -- of writing back with what I thought was a helpful response, soon had to be abandoned. Finding an error in the proposed proof and gently pointing it out to the correspondent invariably led to a small deluge of attempted corrections, or sometimes to indignant letters insisting that what I had suggested was a mistake was nothing of the kind. Eventually, I learned that I could not win. Sooner or later I would have to stop responding, and at that point I would unavoidably come across as aloof, a member of a cabal-like Establishment that would do anything to prevent a genius outsider from gaining entrance to the Mathematical Club.

With Fermat's Last Theorem, there is of course always a tiny chance that someone does find a proof simpler than the one Wiles came up with -- although it will surely involve more than a variation on the classical proof of Pythagoras' Theorem, which is the favored approach of most amateurs who write to me. With the ancient Greek challenge of Squaring the Circle, however, I can be completely certain that any proposed solution is incorrect. Lindemann's 1882 proof that pi is transcendental implied that it is impossible to construct a square with area equal to that of a given circle using only a ruler (more precisely, an unmarked straightedge) and compass (of a type which loses its separation when taken off the paper). Sadly, word of Lindemann's result does not seem to have gotten around sufficiently well.

I added in those two parenthetical comments when I stated the Squaring the Circle problem just now because many people who send in proofs do not realize that "Squaring the Circle" is a Greek intellectual game with highly constrained rules. Relax those rules -- for instance by allowing the ruler to have two fixed marks on its straightedge -- and it is indeed possible to square a circle. At least, I seem to recall reading that somewhere. I have to confess that except for a brief period at high school (in the US it would have been middle school) when I first encountered ruler-and-compass constructions and found them an enjoyable challenge, I have never had much enthusiasm for the topic. It always struck me as so very artificial and contrived. Which it is.

"Useless" topics like ruler and compass constructions also tend to help perpetuate the myth that mathematicians do not care about applications of the subject. True, I know a few who do not -- or at least that is what they say -- but my sense is they are very much a minority. What may well be the case is that many mathematicians -- at least those who work in universities -- are not primarily motivated by applications. But that is probably true for the majority of scientists and engineers as well. Being truly successful in any field requires such dedication that it can only come from a love of the discipline itself, not the applications. In mathematics, this is particularly acute, since applications of a new discovery or technique often come many years later, perhaps after the individuals who did the mathematics are long dead. But being motivated by the internal mechanics of a discipline does not mean a lack of interest in what others can do with it. On the contrary, mathematicians are like anyone else: we are usually fascinated to see what our efforts lead to.

Much if the blame for the "mathematicians don't care about applications" myth comes, I believe, from the remarks made by G. H. Hardy in his book A Mathematician's Apology. Hardy's views reflect the snobbery prevalent at Cambridge University (and to a lesser extent elsewhere in England) in the early decades of the twentieth century, when academics went to great lengths to portray themselves as "gentlemen" (there were hardly any women there at the time), whose intellectual abilities put them above having to worry about the issues of the real world. I have no idea how such views appeared to outsiders at the time, but they seem completely out of place today. It's time that myth was put to rest once and for all.

Of course, it can sometimes be very difficult providing an answer to the question "What is this good for?" I faced precisely that question many times when I was being interviewed about The Millennium Problems. For some of the problems, I was able to give an answer of sorts. For example, a proof of the Riemann Hypothesis might well have implications for Internet security. The RSA algorithm, the current industry standard, depends on the difficulty of factoring large numbers into prime factors, and a proof of the Riemann Hypothesis would likely have an impact on that issue. But it has to be added that it's not the solution of the problem per se that would make the difference. Number theorists have been investigating the consequences of the hypothesis for years. Rather, it's the new methods that we assume would be required to solve the problem that are likely to impact the RSA method.

The fact is, it's virtually impossible to predict how and when a major advance in mathematics will affect the lives of everyday people. The Millennium Problems lie at the very pinnacles of mathematical mountains. They are the Mount Everests of mathematics. The air is thin up there, and only the most able should attempt to scale those peaks. But their very height is what makes their potential impact so great. As every climber and skier knows full well, if you make a sudden move at the top of a mountain, the small movement in the snowpack that results can lead to a major avalanche thousands of feet down below. It's hard to predict exactly what direction the snow will move, and where the main force of the avalanche will hit when it reaches the bottom. But for the people down at sea level, the impact can be huge. Solving a Millennium Problem would cause a ripple effect all the way down the mountain of mathematics. We can't know in advance where the effects will be felt first. But we can be pretty sure they will be big.

Finally, Scrabble players are likely to enjoy the following "Squaring the Circle" puzzle I came across some years ago. Fill in the blanks in the following square so that every row and column is a word of the English language. This one can be solved. In fact there are several solutions.


Devlin's Angle is updated at the beginning of each month.


Mathematician Keith Devlin ( devlin@csli.stanford.edu) 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 published last fall by Basic Books.