Mathematical Mountaintops: The Five Most Famous Problems of All Time

  • John Casti
Oxford University Press: 2001. 288 pp. £19.95, $25

The recent boom in mathematics bestsellers has contributed a great deal towards raising the public profile of the subject. But such books ignore a significant section of potential readers, namely those who have more of a mathematical background than the general reader but who are not professional mathematicians. Such mathematical enthusiasts have no doubt enjoyed some of the popular books, but would really prefer a more technical treatment. This is exactly what John Casti provides in Mathematical Mountaintops. It is neither a textbook nor a pop maths book, rather it is a serious in-depth look at the great problems of mathematics.

Casti has picked “the five most famous problems of all time”, and spends 30 to 40 pages describing each one. The problems are Hilbert's tenth problem, the four-colour problem, the continuum hypothesis, the Kepler conjecture and Fermat's last theorem. Each of these has now been solved, so, in addition to outlining the problem, the author is able to explain the solution and recount the story behind it. Four of the problems have been written about extensively elsewhere, but perhaps not with Casti's balance of technical explanation and background narrative.

Casti's remaining problem, the Kepler conjecture, has (to my knowledge) not been written about since the recent announcement that it has been proved, and provides perhaps the most interesting chapter. The problem dates back to 1606, when Johannes Kepler posed a question in a paper for his patron Johann Matthäus Wacker of Wackenfels, Knight Bachelor. Kepler asked, what is the most efficient way to stack spheres so as to minimize the spaces between them? Alternatively, what is the best way to pack oranges in an infinite box? Kepler proposed that the best arrangement was the face-centred cubic lattice, in which every sphere in the first layer is surrounded by six others, and each subsequent layer is built by putting spheres in the dimples of the layer below. This arrangement has a packing efficiency of 74.048%. Grocers, who traditionally stack oranges in this way, suspected that Kepler was right, but it took mathematicians almost four centuries to prove it.

Credit: DAVID NEWTON

There were some notable milestones along the way. In 1694, Isaac Newton and the Scottish astronomer James Gregory argued about the sphere-kissing problem: what is the maximum number of spheres you can place simultaneously in contact with a central sphere? Newton said that the answer was twelve, which is easily achievable, but Gregory was convinced that it was possible to squeeze in a thirteenth sphere. Newton turned out to be right, but this took 180 years to prove.

The Kepler conjecture was eventually proved in 1998 by Thomas Hales of the University of Michigan, following an approach developed in the 1950s by the Hungarian mathematician Lázlo Fejes Tóth. Hales showed that it was possible to determine the maximum packing efficiency by analysing a cluster of just 50 spheres. Each sphere has a position in three-dimensional space, so the packing efficiency depends on an equation containing 150 variables. Maximize the equation and you have the maximum packing efficiency, although this is not a trivial problem.

Hales and his graduate student Samuel Ferguson maximized the equation, thanks to some clever mathematical short-cuts and a tremendous computing effort which used a program that relied on three gigabytes of storage. The fact that Hales's proof relies so much on a computer gives rise to one of the most interesting aspects of Casti's book, namely the validity of computer proof.

In fact, although each chapter is about a particular problem, these problems are used to convey broader ideas about the nature of mathematics in general, its motivation and objectives, its culture and rules. Over the past 25 years, the use of computers has changed the nature of mathematics, solving some previously intractable problems, but sowing discord within the mathematical community.

For example, Casti writes about the four-colour problem, which was also solved with the aid of a computer. After Kenneth Appel and Wolfgang Haken proved it in 1976, their lectures were sometimes met with hostility from their colleagues and some professors barred their graduate students from talking to the notorious duo because this was a computer proof, not a traditional proof. A rumour began to spread that there was a bug in the program, but no bug was ever found. In fact, it was the hand-generated part of the proof that contained errors, none of which turned out to be serious.

Casti explains that a good proof should satisfy three criteria. First, it should be convincing — in other words, mathematicians should believe it when they see it. Second, it should be formalizable, which means it can be incorporated within an established logical framework. Third, it should be surveyable, or capable of being understood, studied, communicated and verified by rational analysis. However, a computer proof fails to satisfy the third requirement, at least in any traditional sense. In the past, mathematicians could work through a proof line by line and explain it to one another. In a computer proof, the broad approach can be checked, but the detailed calculations are embedded within computer code and can be performed only by a microprocessor. The proof, to some extent, has to be taken on trust.

For readers who are particularly inspired by Mathematical Mountaintops and who have some spare time, Casti's final chapter briefly discusses the unsolved Clay Institute problems. Last year, the Clay Mathematics Institute held a press conference and identified seven problems that were crucial to mathematics in the new millennium. This resonates with German mathematician David Hilbert's list of outstanding problems announced in 1900. Whoever solves any of the Clay problems will win a prize of $1 million. But more importantly, they will earn a place in mathematical history and perhaps their own chapter in a subsequent edition of Mathematical Mountaintops.