Download 11 L unreasonable math

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About Math
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1
The Unreasonable Effectiveness
of Mathematics in the Natural
Sciences
by
Eugene Wigner
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The Unreasonable Effectiveness
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Papers on Google Drive
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• Mathematical concepts turn up in entirely
unexpected settings
• Often permit very close and accurate
descriptions of phenomena
• We are limited because we don’t know why
mathematics works and whether it is
unique
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Effectiveness of Mathematics
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Mysterious Usefulness
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• Usefulness borders on the uncanny
• We have no rational explanation
• Raises the question of uniqueness of
physical theories
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• Elementary concepts clearly evolved in
connection with entities in the actual
world.
• counting, arithmetic
• calculus (?)
• More abstract concepts were devised to
be interesting to mathematicians, not to
mimic reality.
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What is Mathematics
• Yet they very often do.
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What is Mathematics
• abstract geometry: General Relativity
• analysis: calculus
• group theory: Eightfold Way
• Amazing that Darwin’s natural selection
brought human reasoning power to it’s
current high level.
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• Examples
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Consider complex numbers:
• No analog in reality
• Basis for a large field in theory of
equations, power series, and analytic
functions.
• Very valuable in classical physics
• Essential to quantum mechanics
• Crucial in engineering
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What is Mathematics…
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Do mathematical entities have some
sort of reality separate from the
human mind?
Would an alien species have the same
mathematics that we have?
Are mathematical concepts somehow
linked to human evolution?
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What is Mathematics…
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What is Physics?
• Schrodinger: a miracle that we can
discover regularities in real events
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• Galileo’s dropped weights
• Invariance in space
• Results are true in Pisa and Tokyo
• Invariance in time
• Results are true in 1596 A.D. and 6000 B.C.
• Invariance under other externalities
• Doesn’t matter if it’s sunny or cloudy
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If there were no phenomena which
were independent of all but a
manageably small set of conditions,
physics would be impossible.
It is not at all natural that “laws of
nature” exist, much less that man is
able to discover them.
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What is Physics? (cont.)
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Laws of nature incorporate only a small part
of our knowledge of reality. They do not
include information about the existence of
bodies, initial conditions, etc.
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What is Physics? (cont.)
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• Evaluating the consequences of
established theories is not the most
important function of math – this is
applied math serving as a tool.
• The laws of nature are written in the
language of mathematics.
• Sometimes the appropriate mathematics
is independently (re)discovered by the
physicist.
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Mathematics in Physical Theories
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• Important to note that mathematics is
developed to please the mathematician,
not, usually, to fill a perceived need.
• Miraculous that abstract mathematics
keeps cropping up in physics and that the
human mind can follow long chains of
complex reasoning.
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Math in Physics (cont.)
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• Expressing crude experience in
mathematical form remarkably often
leads to an amazingly accurate description
of a large class of phenomena.
• Suggests that mathematics is more than
just a human convenience.
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Math in Physics (cont.)
• Consider the example of planetary motion.
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Newton considered freely falling objects
and the Moon. Arrived at equations of
motion and gravity.
• Newton verified gravity to about 4%.
• Theory has proved accurate to better
than 0.0001%.
• Laws of motion are not obvious
• Second derivative is not a product of
common sense
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Planetary Motion
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• Laws of nature based on abstract
mathematical objects
• Proposed on the basis of crude
measurements
• Lead to predictions of amazing accuracy
• Are very limited in scope
• Wigner calls these observations “the
empirical law of epistemology.”
• an article of faith of the physicist
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Summary of Math in Physics
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Theories of Everything
• Will disparate physical laws prove to be
approximations of deeper underlying
rules?
• Will some physical laws never be brought
into a greater structure?
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• Will new theories include such
externalities as initial conditions?
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General Relativity and Quantum
Mechanics
• Much effort, arguable progress, and wide
belief that such a connection is possible.
• We should entertain the possibility that
the connection will not, possibly cannot,
ever be made.
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• Currently no connection between these
two important theories.
19
We have experience with false theories
that functioned well for a while, sometimes
a very long while.
• Aristotelian mechanics:
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False Theories
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False Theories (cont.)
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• Ptolemy’s epicycles
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So long as we don’t know why
mathematics works so wonderfully
with physics, we cannot be certain
that accuracy proves truth and
consistency.
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Caution
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The miracle of the
appropriateness of the language
of mathematics for the
formulation of the laws of physics
is a wonderful gift which we
neither understand nor deserve.
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Wigner’s Coda
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Finis
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How might Galileo study falling bodies
without doing an experiment?
• Think about when two bodies compose a
single body
• Think about when one body comprises two
bodies
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We See What We Look For
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