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This month's topics:
Seki Takakazu (Seki Kowa). Image from Tensai no Eiko to Zasetsu by Masahiko Fujiwara (Shincho-sha, Tokyo, 2002), used with permission. "Samurai Mathematician Set Japan Ablaze With Brief, Bright Light" is the title of a History of Science NewsFocus piece in the October 10 2008 Science. Dennis Normile reports from Tokyo, site of a history of mathematics conference last August dedicated to the memory of Seki Takakazu. Seki (c. 1642-1708, also Seki Kowa) was in fact born into a samurai family; his portrait includes the the two swords (daisho) attesting his warrior status. That he "set Japan ablaze" may be an overstatement, but the "bright light" is appropriate: Seki (an almost exact contemporary of Leibniz, but working in complete isolation from European mathematics) "devised new notation for handling equations with several variables and developed solutions for equations with an unknown raised to the fifth power." Furthermore, according to Normile, "His most significant work focused on determinants ... , a field he pioneered a year or two ahead of ... Leibniz." The word "brief" is, unfortunately, also correct. Seki was ahead of his time, and soon after his death, even though his works were gathered and preserved by his students, "the Japanese mathematical tradition hit a dead end." Normile quotes the science historian Hikosaburo Komatsu: Seki's more erudite work "was too difficult for people to pick up and carry forward." Only now are his most innovative contributions being recognized. Namely, his "discovery around the year 1680 of a general theory of elimination, a method of solving simultaneous equations by whittling down the number of unknown quantities one by one." According to Komatsu, the work had been overlooked because it led to calculations "almost beyond human capabilities." The diagonalization of physics Cantor's diagonal argument occurs in his (second, 1891) proof of the uncountability of the real numbers. As Wikipedia tells us, "it demonstrates a powerful and general technique, which has since been reused many times in a wide range of proofs, also known as diagonal arguments ... The most famous examples are perhaps Russell's paradox, the first of Gödel's incompleteness theorems, and Turing's answer to the [Halting Problem]." Now this technique has been extended to the real world, or at least to our understanding of it. Philippe Binder reports in a Philosophy of Science News & Views piece in Nature (October 16, 2008) on work of the physicist/generalist David Wolpert published earlier this year (Physica D 237 1257-1281). According to Binder, Wolpert has demonstrated "that the entire physical Universe cannot be fully understood by any single inference system that exists within it." How does he get there? In Binder's telling, Wolpert "introduces the idea of inference machines -- physical devices that may or may not involve human input -- that can measure data and perform computations, and that model how we come to understand and predict nature." These machines process U, "the space of all world-lines (sequences of events) in the Universe that are consistent with the laws of physics." Wolpert defines strong inference as "the ability of one machine to predict the total conclusion function of another machine for all possible set-ups." And then he uses diagonalization to prove:
"The two statements together imply that, at best, there can be only a 'theory of almost everything'." Binder goes on to give some smaller-scale possible instantiations of the phenomenon. MathFest 2008 was held from July 31 to August 2 in Madison, Wisconsin. Barry Cipra was on hand and reported on items of general interest in the September 5 2008 issue of Science. Among his selections:
 The densest arrangement of 5 equal circles on the triangular torus covers 71.1% of the area. Images courtesy of William Dickinson. Another student (Sandi Xhumari) solved the problem for 6 circles this summer. "Together, the ideas they developed enabled Dickinson to nail down the densest packing for five circles on the square torus."  On the square torus, the densest packing of 5 equal circles covers π/4 = 78.5% of the area.
Math meets the campaign ad on YouTube  Click to watch "The Thinking Man," created by R. Larkin Clarke. This may amuse or infuriate you, but it does seem to involve some kind of mathematics. Thanks to Jonathan Farley for the timely link. Tony Phillips
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 Math DigestSummaries of media coverage of math Recent Math Digest Summaries: Amid the finger-pointing that follows from the recent financial crisis, some have suggested that the mathematicians and physicists who created some of the models used in the financial industry are to blame. Several mathematicians argue, however, that the real problem lies with financial analysts who used the models. First, the analysts didn't build the models to include the possibility of extreme risk and real-world catastrophe. Second, some analysts exploited flaws in the credit-rating agencies' models in order to develop securities that would earn undeservedly high ratings. One of the mathematicians also points out that the "fancy math" of today was not around during previous financial tumbles like the Great Depression, a situation just as dire but without the scientific scapegoat. This article was prompted in part by a segment on 60 Minutes, "A Look at Wall Street's Shadow Market," (aired on October 5, 2008). --- Lisa DeKeukelaere
Citations for reviews of books, plays, movies and television shows that are related to mathematics (but are not aimed solely at the professional mathematician). The alphabetical list includes links to the sources of reviews posted online, and covers reviews published in magazines, science journals and newspapers since 1996.
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