More about Kepler's Problem. "In Math, Computers Don't Lie. Or Do They?" was the headline on Kenneth Chang's exhaustive treatment of the brouhaha surrounding the publication of Thomas Hales' 1998 computer-assisted proof of the Kepler sphere-packing conjecture (New York Times, April 6, 2004). [Synopsis: Robert MacPherson, Editor of the Annals of Mathematics where Hales' proof was submitted, assigned the checking to a large group of referees who spent several years at the task and gave up. Everything they examined was OK, but there was always more. The "Solomon-like" decision of the Annals editors: publish the "theoretical underpinnings," and leave the computer programs, and their output, to be published elsewhere.] Chang describes the problem ("In the Produce Aisle, a Math Puzzle") and some of its history, but his main focus is computers, as used in mathematical proof. He interviewed John Conway ("I don't like them, because you sort of don't feel you understand what's going on") and Larry Wos, who claims that the advantage of computers is their lack of preconceptions: "They can follow paths that are totally counterintuitive." He also did some research on the natural history of mathematical proof. "Even in traditional proofs, reviewers rarely check every step, instead focusing mostly on the major points. In the end, they either believe the proof or not." An exemplary piece of journalism about mathematics.
Math is hard! This isn't Barbie speaking, it's Keith Devlin, NPR's "The Math Guy," and he was delivering the keynote address to 15,000 members of the National Council of Teachers of Mathematics at their annual meeting last month in Philadelphia. His remarks were picked up and disseminated by Joann Loviglio of the Associated Press (April 21, 2004). She paraphrases Devlin: "Our brains aren't well equipped to grasp those kinds of advanced mathematics" (those kinds include adding fractions and calculus). What the brain does naturally is "counting, algebra, geometry and simple arithmetic." This "natural mathematics" is contrasted with the "formal mathematics" that many NCTM members are condemned to teach, stuff that "seems counter common sense to our brains." How did Devlin himself become a math professor at Stanford? "Devlin said it was not until he was a graduate student that he really understood what he was doing. 'I learned to play the game first ... to manipulate the symbols to get the right answer, and the understanding came later,' he said." Like Pascal's method for attaining faith through prayer. More of Loviglio's paraphrase: "Maybe formalized math should be taught in a manner similar to the immersion method used for teaching language, in which a teacher just starts speaking in a foreign tongue and students eventually start figuring out what's being said. But not all students learn language that way - and not all students will master formal mathematics." The AP feed was posted on the webpage of the State College, PA Centre Daily Times. A webcast of the entire opening session, including Devlin's address, is available on the NCTM's website.
Recent math history in the Chronicle. "Math with a Moral" is the title of Robert Osserman's contribution to "The Chronicle Review" in the April 23 2004 Chronicle of Higher Education. Osserman sets the intellectual stage for the Poincaré conjecture and leads us through the main steps in its resolution. This is large-scale and coarse-grained mathematical history for a general audience, but very skillfully done. Osserman leaps from shoulder to shoulder (Riemann, Poincaré, Thurston) in sketching the flow of ideas from geometry through topology and back to geometry. He has a nice metaphor for Thurston's geometrization conjecture: "William Thurston's great contribution was to see a way to systematize all those shapes -- to provide a kind of periodic table with which to classify and organize all possibilities, as built up out of components based on the original positively and negatively shaped geometries of Riemann, together with a few other basic types." Then the more recent developments (Hamilton, Perelman) and the news that Perelman's published and accepted work has been shown, by Perelman himself, and by Toby Colding (NYU) and William Minicozzi (Johns Hopkins), to be already sufficient to settle the Poincaré conjecture. [According to my sources this may be premature: Perelman's second paper, necessary for this proof, has still not fully been digested. TP] Perelman's full proof of the geometrization conjecture is still under examination. The story has two morals: "When faced with a problem that seems intractable, the best strategy is sometimes to formulate what appears to be an even harder problem. By expanding one's horizons, one may find an unanticipated route that leads to the goal. Second, ... usually mathematics is a highly social activity, with collaboration between two or more individuals the rule rather than the exception. ... Even when an individual takes the last step in solving a problem, the solution invariably depends on elaborate groundwork laid by others ..."
Inside gifted brains. An item in John O'Neil's "Vital Signs" column (New York Times Science, April 13, 2004) has the heading "Peering into a Math Whiz's Brain." It is based on an article (available online) in the current issue of Neuropsychology, by Harnam Singh (US Army Research Institute for the Behavioral and Social Sciences) and Michael O'Boyle (Melbourne) which shows, as O'Neil puts it, "that innate talent at mathematics is a result of a difference in brain functioning." The title of the article sets up the story: "Interhemispheric Interaction During Global-Local Processing in Mathematically Gifted Adolescents, Average-Ability Youth, and College Students." Singh and O'Boyle recruited 60 right-handed males (18 MG, 18 AA, 24 CS) and, as Neal explains it, "asked them to determine if a pair of letters flashing on a screen were the same or different." The letters were hierarchical - made up themselves of smaller letters, and the subjects were tested separately on local and global discrimination. Three sets of tests were run. In the first (LVF-RH) the two letters were presented to the Left Visual Field (which feeds to the Right Hemisphere); in the second (RVF-LH), to the Right Visual Field and, consequently, the Left Hemishpere; and in the third (Cooperative) one letter appeared in the LVF, and one in the RVF. The most striking result can be presented in the following table. Here the mean reaction times for the AA and CS are compared with that of the gifted group (MG) taken as 1.
As the authors put it: "The findings of the current study reinforce our hypothesis that superior coordination of cortical resources is a unique processing characteristic of the MG brain."
The Bayesian revolution. In April's Nature Reviews Genetics is an article by Mark A. Beaumont and Bruce Rannala reviewing three facets of what they call "The Bayesian revolution in genetics." Bayesian inference is a way of working back from a data distribution to the likelihood distribution for the parameters which are supposed to "explain" the data. When there are many interdependent parameters this problem can be intractable, since it involves integration over huge-dimensional spaces, but with recent increases in processing power it has proved amenable to attack with computation-intensive Markov Chain Monte Carlo techniques. Our authors survey three areas, in each of which they examine several applications.
Chaos in Nature. "they have developed a powerful new method to determine from experimental observation of a system whether it is chaotic, and, if it is, what the precise quantitative nature of that chaos is." Thomas Halsey (ExxonMobil Research) and Mogens Jensen (Niels Bohr Institute) are commenting on recent work of Sam Gratrix and John N. Elgin (Physical Review Letters 92 014101), in a "News and Views" piece in the March 11, 2004 Nature. Halsey and Jensen briefly review the methods currently available for determining if a set is or is not a strange attractor. The criterion is multifractality, but box-counting ("simply reconstruct its trajectory through phase space, cover that trajectory with boxes, measure the amount of time spent in each box, and then determine whether or not the multifractal structure you have computed is consistent with chaos") is unreliable. A safer method involves periodic trajectories. "Mathematicians know that the strange attractor can actually be constructed from the union of all periodic trajectories of a system, provided that trajectories of arbitrarily long periods are included ..." This method can be applied to an analytically given dynamical system, for example the Lorenz attractor: "Using an ingenious method to categorize these long trajectories, Gratrix and Elgin have reconstructed in great detail both the Lorenz attractor and its multifractal properties." For systems in nature, there is rarely time for finding enough trajectories to apply this method. But Grantz and Elgin have developed "a much simpler approach, based on recurrence times" and have shown, by applying it to the Lorenz attractor, that it matches the periodic-trajectory method, and should give reliable diagnoses of chaos. "Because calculations based on recurrence times should be relatively straightforward for experimentalists, and as we now have reason to believe that they will be more reliable than box-counting results, we can confidently await a new series of experimental demonstrations of the chaotic properties of a variety of natural systems." The title of the piece is "Hurricanes and butterflies."
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