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 Tony Phillips' Take on Math in the MediaA monthly survey of math news |
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.
Local discrimination: H H S S H H S S H H H H vs. S S S S H H S S H H S S |
|
||||||||||||||||
Global discrimination:
S S S S S S
S S S
S S S S vs. S S S S
S S S
S S S S S S
|
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| In the global discrimination test, the mathematically gifted students did no better than the other two groups using the left hemisphere alone, slightly better using the right hemisphere alone, but significantly better when the two hemispheres could cooperate. Adapted from Singh and O'Boyle, Table 1. |
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."
-Tony Phillips
Stony Brook
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