THE LIMITS OF MATHEMATICS
Greg Chaitin
IBM Research Division
P. O. Box 704
Yorktown Heights, NY 10598
chaitin@watson.ibm.com
These are the abstract & foils of a talk given Monday 22 September
1997 at the International Conference on Complex Systems, Nashua, New Hampshire.
Abstract
I'll outline the final version of my course on information theory and the
limits of formal reasoning. It's currently available online
and also as a book from Springer-Verlag. This course uses algorithmic information
theory to show that mathematics has serious limitations, and features a
new more didactic approach to algorithmic information theory using LISP
and Mathematica software. The thesis of the course is that the incompleteness
phenomenon discovered by Gödel is much more widespread and serious
than hitherto suspected. Also Gödel and Einstein's views on the foundations
of mathematics are discussed, and it is suggested that mathematics is quasi-empirical
and that experimental mathematics should be used more freely. The software
for this course can be downloaded from MathSource at Wolfram Research at
http://www.wolfram.com by querying
for ``0208-820''.
Foil A---BOOK ANNOUNCEMENT!
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Greg Chaitin (IBM)
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The Limits of Mathematics---
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A course on information theory
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& the limits of formal reasoning
Foil B---BOOK ANNOUNCEMENT!
Foil 1
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Is metamathematics complex systems theory?
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Complexity of axioms + logic vs. complexity of theorem
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Algorithmic complexity aka. algorithmic information theory aka. program-size
complexity
Foil 2
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Physics ---> math
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Randomness versus Reason
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True for no reason
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True by accident!
Foil 3---INNOVATION I
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Now size of real computer programs
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Real universal Turing machine!
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Run programs
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LISP & Mathematica
Foil 4---INNOVATION II
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Simpler proofs
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NO MEASURE THEORY
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Now it's just computer programming
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Now idea is just Berry paradox
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``First + integer that can't be defined in less than a billion words''
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Berry paradox is due to Bertrand Russell!
Foil 5---MORAL OF STORY
Foil 6---QUESTIONS/COMMENTS