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The number line is real. Psychologically speaking. That's the conclusion reached by a team of psychologists (Marco Zorzi, Konstantinos Priftis, Carlo Umiltà) at the University of Padua. In "Neglect disrupts the mental number line" (Nature, May 9 2002) they examine right-brain-damaged patients with persistent left neglect: these patients "show a spatial defecit for left-side stimuli. ... When asked to mark the midpoint of a line, they miss the midpoint and place it to the right. The misplacement increases as a function of line length, with a crossover effect (leftward displacement) for very short lines." The team showed that exactly the same systematic errors occurred in mental operations when the patients were asked to name the midpoint of an integral segment [a,b] given its endpoints a and b. The errors occur in the same direction whether the endpoints were given in increasing or in decreasing order, e.g. 1-9 or 9-1, leading them to observe that "the number line is canonically orientated in a left-to-right manner." They conclude: "Although most people focus on symbolic aspects of numbers, ... thinking of numbers in spatial terms (as has been reported by great mathematicians) may be more efficient because it is grounded in the actual neural representation of numbers." The reference is to Hadamard's "The Mathematician's Mind" (Princeton, 1996) which describes his own use of mental imagery but in coordinate-free terms: "a confused mass, ..., a point rather remote from the confused mass, ..., a second point a little beyond the first, ..." etc. (his visualization of Euclid's infinity-of-primes theorem). He also quotes Einstein: "The words or the language, as they are written or spoken, do not seem to play any role in my mechanism of thought."
Click for full-size image. The end of mathematics as we know it? Illustration by Thomas Escher for the review of "A New Kind of Science" in the New York Times Book Review. Used with permission.
The publication of Wolfram's treatise was also covered in the Science Times for June 11. In "Did This Man Just Rewrite Science?" Dennis Overbye relays opinions from several scientists who have worked the same turf. Here is Edward Fredkin, a BU physicist and longtime proponent of viewing nature as a computer: "For me this is a great event. Wolfram is the first significant person to believe in this stuff. I've been very lonely." Fredkin goes on: "An equation is just a thing you write down on a piece of paper. E=mc2 can't keep you warm." But programs are different. "Put them in the computer and they run."
George Johnson is at bat again in "What's So New in a Newfangled Science?" (The Week in Review, June 16). "Interesting ideas rarely spring up in isolation" is the theme of this article, making up for Johnson's neglect of that topic in the Book Review. He surveys some of the current work on the algorithmic universe, including MIT's Seth Lloyd, the author of `Lloyd's hypothesis' (Everything that's worth understanding about a complex system can be understood in terms of how it processes information), and BU's Fredkin. He concludes: "That is how an idea progresses. But sometimes it takes a bombshell to bring it to center stage." and in fact, as Johnson tells us at the start of the piece, " 'A New Kind of Science' was holding its own last week atop Amazon's best-seller chart, along with 'Divine Secrets of the Ya-Ya Sisterhood' and 'The Nanny Diaries.' "
Math in the USSR. That was then. Now it's under the auspices of the Russian Academy of Sciences (RAS) where a struggle is on between the old-style pattern of uniform (and now tiny) support for scientists, and a new effort to "funnel more money to the elite and eliminate redundant projects" according to RAS Vice-president Gennady Mesyats, as cited in a News of the Week report in the May 24 2002 Science. Impetus for the overhaul comes from the very top: Vladimir Putin reportedly decreed in March that "the state would no longer distribute research funding as a kind of welfare." The idea is "to strengthen areas such as mathematics that once commanded respect worldwide but have since lost scores of top minds to emigration." The authors, Vladimir Pokrovsky and Andrei Allakhverdov, speculate on whether the structural reorganization that is supposed to bring about this change is "a genuine transformation of Soviet-style management at the country's research behemoth or a cynical attempt to thwart real reform."
A tough math problem in internet routing is described in "Guessing secrets: applying mathematics to the efficient delivery of Internet content" by Ivars Peterson in the April 6, 2002 Science News. Internet route optimizers need to determine the geographical source of a webpage request in order to connect that "client" with the nearest server holding the webpage. The request comes via an intermediate computer called a nameserver, but only the nameserver's address is immediately available. The client's address must be ascertained by a kind of "20 questions" game with the nameserver. E.g. "is the first digit a '1'?" The problem becomes interesting when, as is often the case, the client has two or more addresses, because then the nameserver still gives a yes-or-no answer. Peterson presents an worst-case example with three addresses and an honest but inscrutable answering algorithm that makes it impossible to guess any digit of any of the addresses. In general, when the information is available, how should one ask the questions to obtain it most efficiently? The matter, which is related to "list decoding" of ambiguous messages, is treated by Tom Leighton, Ron Graham and Fan Chung in the Electronic Journal of Combinatorics.
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