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 Tony Phillips' Take on Math in the MediaA monthly survey of math news |
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. |
Stephen Wolfram and the New York Times.
This image accompanied George Johnson's review of "A New Kind of Science"
in the New York Times Book Review, June 9, 2002. Johnson begins with
the book as a physical object: "1,263 pages ... and 583,313 words," intimidating
perhaps but with marvelous pictures. "Certainly no one has worked so hard
to produce such a beautiful book." He then contrasts Wolfram's publishing
style (everything, all at once) with "the normal thing," i.e. regularly
posted unreadable papers in "fashionable zines" like Physical Review Letters
or Physica D. Johnson presents a cogent digest of Wolfram's main tenet:
"the algorithm is the pure, elemental expression of nature; the equation
is an artifice." And several examples. "One idea after another comes spewing
from the automata in Wolfram's brain."
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.
-Tony Phillips
Stony Brook
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