FOM: intuition
Vladik Kreinovich
vladik at cs.utep.edu
Sun Jan 7 21:04:29 EST 2001
There is a formal justification of this "physics-type" induction in math using
Kolmogorov complexity; see, e.g., M. Li and P. Vitanyi, An Introduction to
Kolmogorov Complexity and its Applications, Springer-Verlag, 1997 (especially
Chpater 5).
Crudely speaking, Kolmogorov complexity of a word is the shortest length of a
program (in some universal language like C) which computes this word. There are
quite a few results justifying induction. The main result is that, crudely
speaking, if for every n, we select the simplest (in this sense) hypothesis
consistent with the first n observations, then at some point, we will find the
correct hypothesis. In this sense, thge correct answer minus 10^{-something}
is, of course, also consistent, but it is NOT the simplest hypothesis.
Vladik
P.S. To add my two cents worth: I have my own paper dealing with induction in
math:
V. Kreinovich,
"Coincidences are Not Accidental: a Theorem",
Cybernetics and Systems: An International Journal,
1999, Vol. 30, No. 5, pp. 429-440.
Can be downloaded from http://www.cs.utep.edu/vladik/1997/tr97-36.ps
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