A Solution to Plato's Problem - The Latent Semantic Analysis Theory of Acquisition, Induction and Representation of Knowledge

http://lsa.colorado.edu/papers/plato/plato.annote.html

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MATHEMATICS: CATASTROPHE THEORY, STRANGE ATTRACTORS, CHAOS

The following points are made by Nigel Calder (citation below):

1) Go out of Paris on the road towards Chartres and after 25
kilometers you will come to the Institut des Hautes Etudes
Scientifiques at Bures-sur-Yvette. It occupies a quite small building
surrounded by trees. Founded in 1958 in candid imitation of the
Institute for Advanced Study in Princeton, it enables half a dozen
lifetime professors to interact with 30 or more visitors in pondering
new concepts in mathematics and theoretical physics. A former
president, Marcel Boiteux, called it "a monastery where deep-sown
seeds germinate and grow to maturity at their own pace."
2) A recurring theme for the institute at Bures has been complicated
behavior. In the 21st century this extends to describing how
biological molecules -- nucleic acids and proteins -- fold themselves
to perform precise functions. The mathematical monks in earlier days
directed their attention towards physical and engineering systems that
can often perform in complicated and unpredictable ways.
3) Catastrophe theory was invented at Bures-sur-Yvette in 1968. In the
branch of mathematics concerned with flexible shapes, called topology,
Rene Thom found origami-like ways of picturing abrupt changes in a
system, such as the fracture of a girder or the capsizing of a ship.
Changes that were technically catastrophic could be benign, for
instance in the brain's rapid switch from sleeping to waking. As the
modes of sudden change became more numerous, the greater the number of
factors affecting a system.
4) Fascinated colleagues included Christopher Zeeman at Warwick, who
became Thom's chief publicist. He and others also set out to apply
catastrophe theory to an endless range of topics. From shock waves and
the evolution of species, to economic inflation and political
revolution, it seemed that no field of natural or social science would
fail to benefit from its insights.
5) Thom himself blew the whistle to stop the folderol. "Catastrophe
theory is dead," he pronounced in 1997. "For as soon as it became
clear that the theory did not permit quantitative prediction, all good
minds... decided it was of no value."
6) In an age of self-aggrandizement, Thom's dismissal of his own
theory set a refreshing example to others. But the catastrophe that
overtook catastrophe theory has another lesson. Mathematics stands in
relation to the rest of science like an exotic bazaar, full of pretty
things but most of them useless to a visitor. Descriptions of logical
relationships between imagined entities create wonderful worlds that
never were or will be.
7) Mathematical scientists have to find the small selection of
theorems that may describe the real world. Many decades can elapse in
some cases before a particular item turns out to be useful. Then it
becomes a jewel beyond price. Recent examples are the mathematical
descriptions of subatomic particles, and of the motions of pieces of
the Earth's crust that cause earthquakes.
8) Sometimes the customer can carry a piece of mathematics home, only
to find that it looks nice on the sideboard but doesn't actually do
anything useful. This was the failure of catastrophe theory. Thom's
origami undoubtedly provided mathematical metaphors for sudden
changes, but it was not capable of predicting them.
9) When the subject is predictability itself, the relationship of
science and mathematics becomes subtler. The next innovation at the
leafy institute at Bures came in 1971. David Ruelle, a young
Belgian-born permanent professor, and Floris Takens visiting from
Groningen, were studying turbulence. If you watch a fast-moving river,
you'll see eddies and swirls that appear, disappear and come back, yet
are never quite the same twice.
10) For understanding this not-quite-predictable behavior in an
abstract, mathematical way, Ruelle and Takens wanted pictures. They
were not sure what they would look like, but they had a curious name
for them: "strange attractors". Within a few years, many scientists'
computers would be doodling strange attractors on their monitors and
initiating the genre of mathematical science called "chaos theory".
11) To understand what attractors are, and in what sense they might be
strange, you need first to look back to the pictures of Henri Poincare
(1854-1912). He was France's top theorist at the end of the 19th
century. Wanting to visualize changes in a system through time,
without getting mired in the details, he came up with a brilliantly
simple method.
12) Put a dot in the middle of a blank piece of paper. It represents
an unchanging situation. Not necessarily a static one, to be sure,
because Poincare was talking about dynamical systems, but something in
a steady state. It might be, for example, a population where births
and deaths are perfectly balanced. All of the busy drama of courtship,
childbirth, disease, accident, murder and senescence is then summed up
in a geometric point. And around it, like the empty canvas that taunts
any artist, the rest of the paper is an abstract picture of all
possible variations in the behavior of the system. Poincare called it
"phase space". You can set it to work by putting a second dot on the
paper. Because it is not in the middle, the new dot represents an
unstable condition. So it cannot stay put, but must evolve into a
curved line wandering across the paper. The points through which it
passes are a succession of other unstable situations in which the
system finds itself, with the passage of time. In the case of a
population, the track that it follows depends on changes in the birth
rate and death rate.
13) Considering the generality of dynamic systems, Poincare found that
the curve often evolved into a loop that caught its own tail and
continued on, around and around. It is not an actual loop, but a
mathematical impression of a complicated system that has settled down
into an endlessly repetitive cycle. A high birth rate may in theory
increase a population until starvation sets in. That boosts the death
rate and reverses the process. When there's plenty to eat again, the
birth rate recovers -- and so on, ad infinitum.
14) Poincare also realized that systems coming from different starting
conditions could finish up on the same loop in phase space, as if
attracted to it by a latent preference in the type of dynamic
behavior. A hypothetical population might commence with any
combination of low or high rates of birth and death, and still finish
up in the oscillation mentioned. The loop representing such a favored
outcome is called an "attractor".
15) In many cases the ultimate attractor is not a loop but the central
dot representing a steady state. This may mean a state of repose, as
when friction brings the swirling liquid in a stirred teacup to rest,
or it may be the steady-state population where the birth rate and
death rate always match. Whether they are loops or dots, Poincare
attractors are tidy and you can make predictions from them.
16) By a strange attractor, Ruelle and Takens meant an untidy one that
would capture the essence of the not-quite-predictable. Unknown to
them an American meteorologist, Edward Lorenz, had already drawn a
strange attractor in 1963, unaware of what its name should be. In his
example it looked like a figure of eight drawn by a child taking a
pencil around and around the same figure many times, but not at all
accurately. The loop did not coincide from one circuit to the next,
and you could not predict exactly where it would go next time.
17) When mathematicians woke up to this convergence of research in
France and the USA, they proclaimed the advent of "chaos". The strange
attractor was its emblem. An irony is that Poincare himself had
discovered chaos in the late 1880s, when he was shocked to find that
the motions of the planets are not exactly predictable. But as he
didn't use an attention-grabbing name like chaos, or draw any pictures
of strange attractors, the subject remained in obscurity for more than
80 years, nursed mainly by mathematicians in Russia.
18) Chaos in its contemporary mathematical sense acquired its name
from James Yorke of Princeton, in a paper published in 1975. Assisting
in the relaunch of the subject was Robert May, also at Princeton, who
showed that a childishly simple mathematical equation could generate
extremely complicated patterns of behavior. And in the same year,
Mitchell Feigenbaum at the Los Alamos National Laboratory in New
Mexico discovered a magic number. This is delta, 4.669201..., and it
keeps cropping up in chaos, as pi does in elementary geometry.
Rhythmic variations can occur in chaotic systems, and then switch to a
rhythm at twice the rate. The Feigenbaum number helps to define the
change in circumstances -- the speed of a stream for example -- needed
to provoke transitions from one rhythm to the next.
19) Here was evidence of latent orderliness that distinguishes certain
kinds of erratic behavior from mere chance. "Chaos is not random: it
is apparently random behavior resulting from precise rules," explained
lan Stewart of Warwick. "Chaos is a cryptic form of order." During the
next 20 years, the mathematical idea of chaos swept through science
like a tidal wave. It was the smart new way of looking at everything
from fluid dynamics to literary criticism. Yet by the end of the
century the subject was losing some of its glamor.
20) Exhibit A, for anyone wanting to proclaim the importance of chaos,
was the weather. Indeed it set the trend, with Lorenz's unwitting
discovery of the first strange attractor. That was a by-product of his
experiments on weather forecasting by computer at the beginning of the
1960s. As an atmospheric scientist of mathematical bent at the
Massachusetts Institute of Technology, Lorenz used a very simple
simulation of the atmosphere by numbers, and computed changes at a
network of points.
21) He was startled to find that successive runs from the same
starting point gave quite different weather predictions. Lorenz traced
the reason. The starting points were not exactly the same. To launch a
new calculation he was using rounded numbers from a previous
calculation. For example, 654321 became 654000. He had assumed,
wrongly, that such slight differences were inconsequential. After all,
they corresponded to mere millimeters per second in the speed of the
wind.
22) This was the "Butterfly Effect". Lorenz's computer told him that
the flap of a butterfly's wings in Brazil might stir up a tornado in
Texas. A mild interpretation said that you would not be able to
forecast next week's weather very accurately because you couldn't
measure today's weather with sufficient precision. But even if you
could do so, and could lock up all the lepidoptera, the sterner
version of the Butterfly Effect said that there was enough
unpredictable turbulence in the smallest cloud to produce chance
variations of a greater degree.
23) The dramatic inference was that the weather would do what it damn
well pleased. It was inherently chaotic and unpredictable. The
Butterfly Effect was a great comfort to meteorologists trying to use
the primitive computers of the 1960s for long-range weather forecasts.
"We certainly hadn't been successful at doing that anyway," Lorenz
said, "and now we had an excuse."

Adapted from: Nigel Calder: Magic Universe: The Oxford Guide to Modern
Science. Oxford University Press 2003, p.133. More information at:
http://www.amazon.com/exec/obidos/ASIN/0198507925/scienceweek

ScienceWeek http://www.scienceweek.com - (Sited on Jan 13/2012)

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JC Teacher