Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics
Plume
448 pages
$14.40
The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics
Farrar, Straus and Giroux
342 pages
$21.60
Editor's note: Science in Focus is on vacation in August, resuming our regular schedule in September. Meanwhile, we're going to the archives for science-related pieces from the pages of Books & Culture. This week we're featuring a piece by Karl-Dieter Crisman from the January/February 2009 issue.
Ask the next person you meet what the greatest unsolved problem in mathematics is, and you'll be lucky if the worst you get in return is a quizzical stare. A few might mention the search for a grand unification of modern physics, but that's not really right, since even the most abstract mathematical physics is (in theory, at least) connected to the real world.
But most people probably will say, "I don't care," if they respond at all. So it may be more than a little surprising to discover that in roughly the last five years no fewer than seven books, all (ostensibly) aimed squarely at the layman, treat precisely this question.
One, by veteran math popularizer Keith Devlin, attempts to describe all seven of the Millennium Prize problems [1], for the solution of which the Clay Institute for Mathematics (www.claymath.org) has recently offered million-dollar prizes. Two more describe the intense human drama of the recent solution of one of these, a century-old problem known as the Poincaré Conjecture. [2]
Given the compelling math, and the story's culmination in the explicit rejection of the most prestigious prize in mathematics by Grigory Perelman, the Russian genius behind the solution, [3] the choice of this topic for a book makes sense. Yet it is telling of the consensus in mathematics as to the answer to our little question that all four of the other books tackle a different Millennium Prize problem, known as the Riemann Hypothesis (or RH, for short). We here review the two of these of greatest interest to Books & Culture readers.
With the history of post-Napoleonic Europe as a lush backdrop, John Derbyshire's Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics attempts to unpack the problem itself; on the other hand, Karl Sabbagh's The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics has a far lighter treatment of the math, focusing on "the humanity of mathematicians," in particular those mathematicians working on the RH on the cusp of the 21st century. Both achieve their major goals, though each book has certain deficiencies I'll outline later. [4]
Even at the risk of pedantry, it seems strange to go on without briefly describing what the Riemann Hypothesis is. Here is the Clay Institute's own description of the RH:
Some numbers have the special property that they cannot be expressed as the product of two smaller numbers, e.g., 2, 3, 5, 7, etc. Such numbers are called prime numbers, and they play an important role, both in pure mathematics and its applications. The distribution of such prime numbers among all natural numbers does not follow any regular pattern, however the German mathematician G.F.B. Riemann (1826–1866) observed that the frequency of prime numbers is very closely related to the behavior of an elaborate function "?(s)" called the Riemann Zeta function. The Riemann hypothesis asserts that all interesting solutions of the equation ? (s) = 0 lie on a straight line. This has been checked for the first 1,500,000,000 solutions. A proof that it is true for every interesting solution would shed light on many of the mysteries surrounding the distribution of primes.
Even most professional mathematicians could only add the definition of the Riemann Zeta function and what the word "interesting" means; the "close relation" is the heart of this most difficult question.
Given the spate of popular math books of the last two decades, and the relative inaccessibility of the problem, why would anyone write or read about it? In Sabbagh's book, the interest is to some extent in TV-style voyeurism. Lest anyone think I mean a reality-show race to wealth, keep in mind that the problem is 145 years old, has been seriously worked on for over a century, and will almost certainly not be solved anytime soon. [5] Instead of Survivor, the researchers in The Riemann Hypothesis are on All My Children. We hear shouts of controversy over two claims on the same proof of an important theorem by Erdös and Selberg; we witness a chance conversation at Princeton that yields one of the main means of attacking the RH; we see the late entrance of the "lovely" Fields medalist Alain Connes as a knight to (possibly) save the day. There is even a black sheep in the family, Louis de Branges; his story begins like the rest, but becomes a tragic portrait of a man "for whom there is nothing in the world but mathematics and persecution," for even if his latest proof is correct, he has cried wolf too often for anyone to listen any longer.
Sabbagh is interested in what drives mathematicians (not the money, in case you were wondering). He seeks to describe flashes of mathematical insight and give a feel for the RH itself at a basic level, and he succeeds—but at a cost. Despite his intermittent attempts to portray high-level mathematicians as folks very much like you and me, his deep-seated attitude toward his subjects is captured well in the prologue: "It is given to them to see truths." Sabbagh's mathematicians are people, yes, but people who operate on a totally different plane when it comes to math, and often don't operate so well outside that rarefied realm. This picture isn't entirely wrong—Sabbagh's description of a typical math conference rings so true that, while reading the book at such a gathering, I thought maybe he was hiding behind the chalkboard, taking notes—but it's exaggerated. For example, his claim that mathematicians find thinking abstractly "a more satisfying activity than any of the other pleasures the world has to offer" needs a correction of "any" to "many."
Derbyshire's goals are entirely different, and so are his mathematicians. Prime Obsession is loosely written as a two-track book, one which goes into some detail about the math of the RH, the other following the history behind each advance along the way. As both tracks are following paper trails to some degree, the book is more grounded in reality. That's not to say Derbyshire shies from hyperbole; in the chapter "Turning the Golden Key," he refers to the mathematical step his prose has built up to for half the book by saying, "I simply cannot tell you how wonderful this result is."
But the historical view places such comments in context. We meet Bernhard Riemann, who proposed the now-famous hypothesis in a then little-noticed 1859 paper, in his milieu of the still-fragmented pre-Bismarck Germany. Like Sabbagh's de Branges (who appears in a single endnote here), Riemann seems "a rather sad and slightly pathetic character." Yet we also hear about his "very pious" Lutheranism and his devotion to his family. Each player in the narrative is thus portrayed, sympathetically and within the overall saga of the RH. If Sabbagh's book is a high-toned soap opera, this one is at least on the History Channel, maybe even PBS. [6] On the other hand, the story never grows boring, and it starts to prepare the reader for the in-depth mathematical treatment.
The mathematics does not totally drive the book (Derbyshire even suggests that the reader skip every other chapter once it gets too hairy). Still, it clearly excites him, and rightly so. I have never seen a treatment of the Riemann Hypothesis that builds up so clearly and with a writer's flair for suspense (Derbyshire is a novelist as well a popular historian of mathematics). That is the great strength of this "remarkable book," as Nobel laureate John Nash's jacket blurb suggests, and it is also the great weakness of an otherwise exceptional math popularization. Another blurb puts the dilemma best; the book "explains the hypothesis in ways understandable by ordinary mathematicians and even by laymen." [7] That locution—"ordinary mathematicians"—is nicely judged.
The early going is manageable, and some of the graphics are truly spectacular. On the other hand, even advanced undergraduates have difficulty understanding big-Oh notation and finite fields on the first try (and later), and ascent to the mathematical high point alluded to above requires considerable training and painstaking care. No one will go away without some level of understanding, but especially the last few math chapters would be well suited for a classroom setting, as is perhaps appropriate for a book published indirectly via the National Academy of Science.
The Riemann Hypothesis also tries to give some idea of what the RH is all about, but Sabbagh's tendency to over-elevate the (math-related) thoughts of mathematicians also infects his treatment of the math itself. He describes the mechanics of the statement at a level many readers will be able to understand; however, this leaves mysterious why Riemann should have made such a statement in the first place. This would be fine—that's not his goal, after all—if Sabbagh didn't at the same time leave behind some other readers completely. The "Toolkit" appendices will help readers who have forgotten the concepts involved, but they are not sufficient to teach a neophyte what a logarithm or an eigenvalue is.
The nice thing is that these mathematical caveats don't obscure why someone would want to read the books. Every math book ever written has a specific level of sophistication in mind—no less so here. But these books are different from the usual crop of math popularization, in which the stories about people are merely window dressing to entice the reader into exploring a number of interesting concepts like infinity or the golden mean. Here the focus really is on people. Especially in Prime Obsession, the daunting problem itself is also of key significance, but we delve into the lives of those who either are developing it now, or who brought it to its current prominence. Though I consider Prime Obsession a quite strong book, The Riemann Hypothesis is also worth a look for different reasons; neither is right for everyone.
One final aspect of these books will be of particular interest to readers of this journal. The past century has seen some great debates about whether mathematics is created or invented, and how much of what is out there will ever be discovered; both authors take rather strong positions for non-experts. Sabbagh makes it clear he believes math exists to be discovered, but includes some interesting anecdotes by various mathematicians working on the RH regarding what they think, from the strong realism of Connes to one mathematician's comparing his Serbian Orthodoxy to believing the RH is true (in a very weak sense); even those who disagree seem so involved in the quest that their very obsession argues for the independent existence of mathematical ideas. Unfortunately, Sabbagh doesn't dig deeper, and talks of such abstract truth-seeking as a curious property of his researchers, "apart from, say, meditation or religion."
Derbyshire spends less time on such issues, but his statement is even stronger: he ends the book by quoting the eternal progress theme of the great German mathematician David Hilbert, that "We must know, we will know," whether the RH is true. [8] It is not clear if this is warranted. [9] Still, thanks to Derbyshire writing his people in their Sitz im Leben, we find humility as well as hubris among the mathematicians themselves.
Derbyshire attributes mathematician Leonhard Euler's "serenity and inner strength" to his "rock-solid religious faith"; even more so, Riemann's "daily self-examination before the face of God" comes up several times, including in the epilogue, where Riemann dies, his wife reciting the Lord's Prayer at his side. His epitaph? "All things work together for good to them that love God" (Rom. 8:28). Neither the solution of the greatest problem in mathematics nor its absence can change that.
Karl-Dieter Crisman is assistant professor of mathematics at Gordon College, where he studies math as it relates to voting systems and music theory (among other things).
1. Unfortunately, as far as non-experts are concerned (including other mathematicians), Devlin doesn't pass the intelligibility test on the two most thorny of the seven problems (the Hodge Conjecture and the Birch and Swinnerton-Dyer Conjecture), but the same can be said of the official publication on the prizes, and at least Devlin acknowledges that his errand is doomed to failure.
2. These are George Szpiro's Poincaré's Prize and Donal O'Shea's The Poincaré Conjecture.
3. Apparently for reasons even more enigmatic than for any Nobel rejection, certainly more so than Sartre's or Pasternak's.
4. Neither of the two books actually written by mathematicians (highly respected ones, in fact) treats the contemporary human element as well as Sabbagh, or the mathematics and history as deeply as Derbyshire; nonetheless, Marcus du Sautoy's The Music of the Primes and Dan Rockmore's Stalking the Riemann Hypothesis also generated reasonable accolades upon publication.
5. For example, see the article "The Riemann Hypothesis" in the March 2003 issue of Notices of the American Mathematical Society. Aimed at experts and written by someone who directs an institute with solving the RH as a primary objective, the view taken is long-term.
6. For instance, the treatment of the Dreyfus Affair is quite detailed—and even adds to the overall exposition; this is not your typical math popularization.
7. By the great mathematics-as-recreation advocate Martin Gardner.
8. "Wir müssen wissen, wir werden wissen."
9. In the article mentioned in the first footnote, the author gives as an argument for the validity of the RH, "It seems unlikely that nature is that perverse!" Possibly true, but not exactly inspiring confidence.
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