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A monthly survey of math news
Everything and More is the title of David Foster Wallace's new book. The subtitle, "a compact history of ∞" begins to explain the joke in the title with another joke, a nice topological twist on Stephen Hawking's "Short History of Time." The book has been extensively (and favorably) reviewed in the press. One of DFW's fans has a webpage listing the reviews, with links to most of them. Many reviewers plunge right into the math. Jim Holt, in the November 3 2003 New Yorker, gives his own take on Wallace's witty definition of the Groucho Club: "a sort of salon des refusés: it consists of all the people who are paired with clubs that won't have them as members" and goes on to reproduce the proof that a one-one match of people with clubs is impossible, and that "the infinity of sets of things is bigger than the infinity of things." And to point out that "this principle, which has come to be known as Cantor's theorem, ... can be applied over and over again" to yield "a never-ending tower of infinities." The Boston Globe interviewed Wallace in NYC (Caleb Crain, 10/26/2003). "CRAIN: In 'Everything and More,' once some of the technical questions about infinity are answered, a new abyss opens up with Godel's incompleteness theorem. WALLACE:... After Godel, the idea that mathematics was not just a language of God but a language we could decode to understand the universe and understand everything -- that doesn't work any more. It's part of the great postmodern uncertainty that we live in." And also: "WALLACE: The idea is, if the book works right, you finish it with a better idea of not just how Cantorian transfinite set theory works but actually why it was a big deal and why it's beautiful and amazing."
The Most Important Problem. On Tuesday, November 11 2003 the Science Times was 25 years old. To celebrate, the New York Times, under the general rubric "Does Science Matter?" ran 25 short pieces "asking 25 of the most compelling questions facing science today." Edging out "Does the Paranormal Exist?" for 24th place was "What Is the Most Important Problem in Math Today?". The actual question is ... the Riemann Hypothesis. Gina Kolata was borrowed back from her new beat (Biology and Medecine) to contribute a few quick paragraphs: primes, zeroes, computer simulations ("The record is the 10 to the 23rd zero, which, as Dr. Andrew Odlyzko of the University of Minnesota discovered, falls on the line, just as 20 billion of its neighbors do.") and the link to Physics ("The distribution of energy levels in highly excited atomic nuclei looks similar to the distribution of zeroes of the zeta function."). A very rapid but expert job.
Who invented the electronic digital computer? It was a professor of math and physics at Iowa State University, and it happened between 1937 and 1941. An article in the Ames, Iowa Tribune for November 1, 2003 reports on a meeting of "computer experts from around the world" to celebrate what would have been the 100th birthday of John Vincent Atanasoff who, together with his graduate student Clifford Berry, developed and built the first modern computer. What about ENIAC, you might ask. The Army Research Laboratory webpage is still running The ENIAC Story (Martin Weik, 1961) which states "The world's first electronic digital computer was developed by Army Ordnance to compute World War II ballistic firing tables," crediting Dr. John W. Mauchly and J. P. Eckert, Jr., of the University of Pennsylvania, for the original design and quoting from the patent (No 3,120,606) they filed on June 26, 1947. But 12 years after that story was written, a Federal Judge ruled that the ENIAC patents were invalid, and that "Eckert and Mauchly did not themselves first invent the automatic digital electronic computer, but instead derived that subject matter from one Dr. John Vincent Atanasoff." It's not a pretty story, and it's all told on the ISU Dept of Computer Science website.
Far East Math Flap. It happened with the New York State Regents Math A Exam, it can happen anywhere. The November 11 2003 China Daily reports that "Controversy has arisen in East China's Jiangsu Province concerning the validity of a math question in this year's college entrance examination." It turns out that the very first question on the test, administered last June, was a multiple choice question that could be interpreted in such a way that none of the four proposed answers fit. The controversy has involved the Ministry of Education and the Chinese Academy of Sciences, but the case "remains unresolved after five months of repeated evaluations by high-profile experts." The author of the article (available online) concludes: "an efficient rectification mechanism should be established to promptly make up for the possible repercussions for the students who wrote the exam."
Hilbert Hotel Onstage. This summer in Milan, "an esoteric play about mathematics was the talk of the town." This is the start of Marcus de Sautoy's piece in the November 4 2003 Guardian about "Infinities," the fruit of a collaboration between "Cambridge-based cosmologist John Barrow and Italian theatre director Luca Ronconi." Apparently Barrow and Ronconi want to avoid the use of science as a context for the exploration of human character. "They wanted to create a piece of theatre about scientific ideas, not about people: to put the science centre stage." The first scene is based on Hilbert's Hotel. "Its infinitely many rooms can accommodate new guests even if every room is occupied. The owner just asks each guest to move into the next room along. The guest in room 1 moves to room 2, the guest in room 2 moves to room 3 and so on. Everyone can still be accommodated but now room 1 is free for the new guest." De Sautoy does not tell us what the other four scenes are about, but he does say that the production "has more in common with installation art than with the drawing-room dramas served up in London's West End." De Sautoy's essay is available online.
Auxin and the Fibonacci Numbers. A European team led by Didier Reinhart and Eva-Rachele Pesce of the University of Bern has made great progress towards an understanding of the biochemical basis of phyllotaxis, the regular arrangement of leaves around a plant's stem that leads to spirals with charcteristic mathematical properties. Namely, the numbers of left and right-turning spirals are almost always two consecutive Fibonacci numbers. The team performed an ingenious set of experiments using recently developed mutant strains of Arabidopsis to show that the concentration of the plant hormone auxin and the distribution of primordia (leaf buds) participate in a positive-negative feedback system analogous to "the short-range activator and long-range inhibitor in reaction-diffusion mechanisms." They conclude: "Our model accounts for the reiterativity and the stability of organ positioning." But the way the precise divergence angles are determined (which is where the "golden angle" 137.5° and the Fibonacci numbers enter the picture) is only addressed speculatively. The research is reported in an article in Nature, November 20, 2003.