Book Review The New York Times Book of Mathematics Reviewed by Ian Stewart an order that helps to The New York Times Book of Mathematics tell a story—which is, Edited by Gina Kolata after all, what journal- Sterling, 2013 ism is about. US$24.95, 496 pages This choice makes ISBN-13: 978-1402793226 the book far more ac- cessible to the general Mathematicians often complain that their subject reader, though I’m is neglected in the mass media, although when it tempted to photocopy does get reported there always seem to be a few the contents list and of us who find it impossible to resist the urge to shuffle the articles complain bitterly about inaccuracies and “hype”, back into the order which usually seems to mean the promotion of the of publication. Read- area concerned instead of their own. As this sump- ing them in that order tuous volume demonstrates, neither complaint would give a dramatic can sensibly be directed at the New York Times perception of how our subject has changed over [NYT ]. For more than a century this high-quality the last 120 years; however, this comes over any- newspaper has done sterling work in the service way. Paul Hoffman’s foreword and Gina Kolata’s in- of our profession, the public, and the cause of sci- troduction provide concise and helpful overviews ence journalism. of such questions, and the first group of topics This collection contains more than a hundred addresses general issues of what mathematics is NYT articles on mathematics and its applications and what it’s for. That said, this is clearly a book of all kinds, dating back to 1892 and written by that you can dip into fairly randomly, and I’m over thirty authors—some journalists, some math- going to spend much of this review doing just ematicians, some neither. Together they provide that. Along the way, I’ll say a little about how the an unusual overview of our subject and how it is selected articles illuminate broader “philosophical” perceived by everyone else. It is an eclectic col- issues—just as the whim takes me. lection, ranging from an anonymous piece on life Roughly speaking, the seven sections deal with insurance at the end of the nineteenth century to the nature of mathematics, chance, famous prob- Grigori Perelman’s proof of the Poincaré Conjec- lems, chaos, cryptography, computers, and biogra- ture. The editor has resisted the temptation to phy. Immediately we get a glimpse of which areas place the articles in chronological order, grouping of mathematics are deemed newsworthy or, per- them into seven major areas and then following haps, just appealed to the editor. Large regions of the mathematical landscape are untouched—only Ian Stewart is a science writer and professor emeritus of passing reference to Fourier transforms (though mathematics at the University of Warwick. His email ad- the use of wavelets for image compression is men- dress is [email protected]. tioned), nothing on symmetry except the E8 “size DOI: http://dx.doi.org/10.1090/noti1117 of Manhattan” calculation. The only mathematical MAY 2014 NOTICES OF THE AMS 495 biology is a short piece on the flocking of birds help them tell the story accurately as it unfolds. from 1987. I’m not complaining about what’s not In every other field of human activity, this kind been included; just posting a mild warning notice of thing happens on a daily basis. I’ve lost count for potential readers. What has been included is of how many times a cosmologist has announced worthy, valuable, and fascinating, fully justifying that dark matter has been detected definitively, the book’s existence. only for another one to shoot the whole thing I’ll jump in with the third section, on famous down in flames. problems, solved and unsolved. This covers what If keeping the lid on Miyaoka’s ideas and Wiles’s by now is familiar ground to most mathemati- discovery was difficult two decades ago, it’s totally cians: the Poincaré conjecture, the four-color impossible in this age of Twitter and Facebook. problem, the Goldbach conjecture, Fermat’s Last We may privately wish the world were otherwise, Theorem [FLT], and a flurry of less celebrated but with discoveries being made public only when still important questions such as optimal packing they have become definitive—but it’s not. Indeed, of tetrahedra and Steiner trees. In its treatment in most areas outside mathematics something of Fermat’s Last Theorem, the NYT offered its can be definitive for decades and then turn out readers glimpses of mathematics in the making. to be totally wrong. The payoff from this kind of “Mathematics expert may soon resolve a 350-year warts-and-all disclosure is almost entirely positive: problem…” You’d be justified in assuming this the public experiences the excitement of serious was advance notice of Andrew Wiles’s work, but new mathematics, and sees new things happening actually the piece is about an abortive attempt by as they occur. They discover that we are actually Yoichi Miyaoka in 1988. It reminds us that math- doing things. It may be “unfortunate” when some- ematical research does not consist of triumphs times the excitement fizzles out, but a week later alone; it involves many unsuccessful attempts. The the only people fretting about it are a few anally article is suitably cautious: “Nobody has the right retentive specialists. Get a life. to have an opinion yet.” The subsequent article, James Gleick features prominently in the book, “Fermat’s theorem solved? Not this time” provides because he writes so much and so well about our the dénouement. subject, so it is hardly surprising that chaos and A month later, the NYT is reporting what ex- related topics get a section to themselves. The area perts in number theory had been quietly saying for is, actually, rather important. Remarkably, the NYT about a decade: the road to a solution might well spotted this coming in 1938—in a way. It reported be opened up by the theory of elliptic curves. Five Norbert Wiener’s work on Brownian motion, reveal- years later “At last, shout of Eureka!” told the world ing statistical regularities in random movements. of Wiles’s Cambridge announcement, confirming But it also mentioned George Birkhoff’s approach that belief. This was followed up by a longer de- to chaotic events, which later led Stephen Smale scription of what Wiles did—“you use Hilbert ir- to the horseshoe, a key step in understanding reducibility and the Cebotarevˇ density theorem … that deterministic chaotic dynamics has its own to produce a noncuspidal rational point”—well patterns. This section is an eclectic mix of various done, James Gleick! Handled so deftly that he topics from nonlinear dynamics—chaos, catas- gets away with it, while readers gain a subliminal trophes, complexity, stochastics—ending with an insight that this stuff is difficult and advanced, obituary of Benoît Mandelbrot and a summary of yet follows some kind of coherent mathematical the development of fractals. narrative. Except… “Flaw is found in math proof, Much of this area was a cause of controversy but repairs are under way.” Then “Fermat’s puzzle when the media got hold of it; some still is. “Here is still not quite Q.E.D.” And finally, “How a gap in is a mathematician’s nightmare that I heard in the the Fermat proof was bridged.” 1980s,” Gleick writes. “[A] mathematician dreamed This is what real research is like. A few steps that Mandelbrot died and God spoke: ‘You know, forward, a step or two back. With luck and a fol- there really was something to that Mandelbrot.’” lowing wind, you may actually reach your destina- Today’s mathematicians mostly think that God had tion. But should any of this have been reported? Is a point, though a few reckon we are still living the it “premature” to inform the press of major new nightmare. Perhaps if they read Science and Na- developments as they are happening, and before ture regularly they would realize how thoroughly the community has peer-reviewed them? Well, fractals now pervade numerous areas of science. Wiles kept everything secret until he was confident A third major theme, one that future historians he was right, and his announcement was low-key: of mathematics will have to come to grips with, is nothing terribly wrong there. But, gosh, well… the remarkable interplay between mathematics there was a gap. These things happen. Wiles was and computing. Once again the NYT is on the case lucky, Miyaoka not. The plain fact is that, even in early; once again, the story is not what we might the early 1990s, there is no way that the media expect. In 1927 it carried a short article on Van- would not have got wind of a possible solution nevar Bush’s “thinking machine”—more properly, of FLT. The best we can do, as a profession, is to the “product integraph”, an analog device for 496 NOTICES OF THE AMS VOLUME 61, NUMBER 5 solving differential equations. By 1947 we are once brought a live tiger into a lecture room full hearing of two “giant electronic brains” being es- of children. tablished by the National Bureau of Standards, suc- You’ll guess that both of them are the same cessors to ENIAC and similar beasts. “A third com- person—me. Even the second one pales into in- puting machine might be built in the near future to significance compared to the remarkable math- speed up the calculations of the Census Bureau.” In ematicians that actually have been featured in the the same year, ENIAC is being “converted so that it NYT.