March 20, 2013 2:32:14 PM

Ian Stewart is a professional mathematician who may be a professor emeritus, but continues his researches into such things as oscillators and the patterns of animal movement. Along the way he has written the "Mathematical Recreations" column of *Scientific American*, and has also written over twenty books explaining aspects of math to lay readers. *Visions of Infinity: The Great Mathematical Problems* (Basic Books) is his most ambitious such book, as it attempts to explain mathematical research that is, or has been recently, on the cutting edge, big problems that have finally been solved or which might be tantalizingly close to a solution. As you can imagine, it is not his easiest book for non-mathematicians to enjoy, but it does represent his best attempts at explanation of specialized and important mathematical arcana. Even under Stewart's friendly, humorous tutelage, you aren't going to understand this stuff as well as a mathematician does. The subjects here are well above the heads of most of us, but Stewart reminds us that "a basic aim of mathematics is to uncover the underlying simplicity of apparently complicated questions." Mathematicians may find some simplicities in these big subjects, but their simplicities are not our simplicities. Stewart often says something like, "The discussion is going to get a bit technical, but there is a sensible story behind the ideas and all we'll need is a broad outline. Bear with me." A great problem in mathematics, he says, "has to be *hard*," and the emphasis is Stewart's.

The amount of work mathematicians do, however, makes what they do worth looking at. Stewart writes, "Mathematics is newer, and more diverse, than most of us imagine. At a rough estimate, the world's research mathematicians number about a hundred thousand, and they produce more than *two million* pages of new mathematics every year." It is a wonderful, optimistic effort - after all, only some of that mathematics has obvious practical benefit, and most mathematicians are busy seeking their patterns without much interest in any application; few of the problems described on these pages are going to affect the way the world goes on about its business. This isn't always the case. Stewart reminds us that for centuries, number theorists, the ones who study one, two, three, and all the following whole numbers, worked in their ivory towers, among other things finding ways to factor numbers into primes. Then in the last couple of decades, as computer systems used prime numbers as keys to encryption, suddenly factoring huge numbers into primes became a high security endeavor; if the number theorists make certain advances in factoring, people in charge of banking, military security, credit cards, and a whole lot more are going to have to change everything they do with computers.

One of the most famous problems in number theory covered here is Fermat's Last Theorem. This one is solved, and Stewart introduces it by reminding us that there was a fine BBC documentary about its solution, a documentary that focused on the fascinating human interest story of the man who solved it, Andrew Wiles. Wiles had been enchanted by the simplicity of the problem, and its lack of a proof, as a kid, and went into mathematics specifically to prove it. There's little to the problem itself; over three centuries ago, Fermat said that although there are infinite whole number solutions for x^{2} + y^{2} = z^{2}, all to powers of two, there were none for powers of three, four, or any number higher than two. Fermat said he had proved it, and his reputation is unsurpassed among mathematicians, but that counts for nothing because he didn't tell what his proof was. The communal effort among mathematicians is often a theme on these pages, but when Wiles got to working on the proof (seven years of pure thought in his office just under the roof of his house), he did so in secret. This isn't usual, but the prize (nonmonetary) of finding the long-sought solution was so great that he did not risk that someone would hear about part of his work and complete it before he did.

Another famous problem covered here is the Four Color Theorem, and it is also one that has been proved and also has little application beyond pure mathematics. It was first proposed in the mid-nineteenth century, and is again a simple-sounding problem: you can draw four regions on a map each of which touches the others, but not five. One mathematician after another thought he had solved the problem, only to be caught out by other mathematicians checking his work. The complexity of the problem was huge, and was eventually only handled by computer in 1976. The computer essentially looked at all the ways regions might touch each other, and spat out 700 pages of calculations to show that if you were coloring a map, four colors sufficed. This brought out a philosophical issue: how do you know the computer got it right? Twenty years after the computer proof was announced, a new team of mathematicians with new computer programs had a go at it, simplified the configurations, and re-proved it successfully. It's still a computer proof, and though mathematicians are more accepting of computer-aided proofs than they were twenty years ago, they will be very happy when someone comes up with a proof that can be written down, studied, and checked with no computer assistance. In a similar proof covered here, for a problem about packing spheres, a proof consisted of three gigabytes of computer files, and a panel of twelve referees declared that they were "99 per cent certain" that the proof was correct. So there is still work to do.

The problem of proof has taken a toll on the man who has proved the Poincaré Conjecture, one of the most difficult problems in topology, proposed in 1904. It was again pure mathematics, but it was so foundationally important that it was declared a Millennial Problem, one of seven with a million dollar prize attached. Grigori Perelman presented his proof in an electronic forum, without all the usual detail in published proofs. He went on to lecture on his proof, and to answer questions about it, and to fill any gaps that anyone found. It was a long process; he had used diverse areas of mathematics and mathematical physics, realms he understood well but which few potential checkers knew so well in total. Because of this, recognition was long in coming, and Perelman became increasingly irritated over what was to him obviously a completed job. The proof has been accepted, and the prize offered, but Perelman has rejected it, not because he was offended about the slowness of his proof's acceptance, but just because he wasn't interested in awards or prizes.

Stewart guides readers as best he can through some very difficult territory, difficult even though mathematicians have settled at least some of these problems forever. In his final chapter, he briefly presents twelve problems for the future, problems which mathematicians are working on now, and whoever proves any will be famous among his peers. I like especially the Collatz Conjecture. Pick a positive whole number. If it is even, divide it by two; and if it is odd, multiply it by three and add one. Repeat. What happens is you always get to the sequence 4, 2, 1, 4, 2, 1 repeating forever. It is usually easy to see that this is so with smaller numbers (though do not try starting at 31 without a calculator), and it is easy to program a computer to accept huge numbers and check them out: every gargantuan number winds up at 4, 2, 1. The problem is that you can't check every number that way, and there may be some gigantic number you insert that rides the "multiply by three and add one" even higher than the series goes when you start at 31, and forever gets larger; or it may be that some cycle other than 4, 2, 1 gets reached (although, and this humorously shows how tough a proof is going to be, it has been proved that no cycle other than the three-step 4, 2, 1 exists which has fewer than 35,400 steps).

No one knows about exceptions to the Collatz Conjecture. No one knows if there is a proof out there. If an exception is found, it is likely to have little effect on anyone but a small coterie of experts in number theory, and it will undoubtedly not mean better cars or cheaper electronics. This is part of the charm of the problems here, with eggheads worldwide going at them for no particular reason except that we do not know the answer and to know the answer is better than not knowing. It is an admirable and lovely effort, one of the best things that humans do, and getting a glimpse of it, even if you don't have a mathematician's understanding of these problems, is challenging fun under Stewart's expert guidance.

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