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The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics
The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics
Karl Sabbagh
Farrar, Straus and Giroux, 2004
342 pp., $23.00

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Karl-Dieter Crisman


Three Prime Numbers Go into a Bar

The curious history of the Riemann Hypothesis.

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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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