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Books > Science & Mathematics > Mathematics > Topology > Analytic topology
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Mathematics and the Unexpected (Paperback, New edition)
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Mathematics and the Unexpected (Paperback, New edition)
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Mathematicians have always stressed elegance and economy in
formulating proofs. Ekeland fulfills both these criteria in this
exposition of themes drawn from contemporary mathematics - even
providing anidiomatic English translation from his own French
original. His purpose is to describe how mathematicians have dealt
with time; in so doing, he draws upon the celebrated 19th-century
mathematician Henri Poincare, and several contemporaries: Rene Thom
of catastrophe theory fame, and others associated with the dynamics
of chaos. As background, Ekeland describes the historic attempts to
predict the positions of the planets, which culminated in Kepler's
laws of planetary motion. Later, Newton was able to derive the laws
from his own development of the law of universal gravitation and
calculus. In Kepler's and Newton's formulations, time can be read
forward and backward: the laws are symmetric with respect to time;
the universe is a deterministic clockwork. By the 19th century,
however, it was clear that the values derived from Kepler's laws
were only crude approximations. It took Poincare to show that even
with the better and better approximations, certain orbits could not
be computed. Thus, determinism was gone. In its wake contemporary
mathematicians have seen that "initial conditions" can lead to
extraordinary future states - the repeated "patterns" of chaos, for
example, and curves with "strange attractors" and self-replicating
geometries. In other instances, the conditions that define
"dissipative structures" can lead to values that give rise to the
"cusps" of catastrophe theory. Ekeland believes catastrophe theory
to be important but limited, unwisely generalized by social and
behavioral sciences, in both these mathematical developments, time
loses its eternal character and becomes a one-way street - time
gains an arrowhead. To conclude these heady observations, Ekeland
contrasts the concept of time in the Iliad with the Odyssey,
compares Thorn's uses of time with Proust, and discourses on
evolution, Stephen Jay Gould, and Hieronymous Bosch. For readers of
philosophical and mathematical bent, an elequent expression of new
ideas. (Kirkus Reviews)
In this brief treatise, Ekelund explains some philosophical implications of recent mathematics. He examines randomness, the geometry involved in making predictions, and why general trends are easy to project, but particulars are practically impossible.
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