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That’s a nice definition of mathematical constants: Numbers whose calculation specifications have a Kolmogorov complexity that is low enough so that they reappe
by sawwit 11y ago
That’s a nice definition of mathematical constants: Numbers whose calculation specifications have a Kolmogorov complexity that is low enough so that they reappear in different contexts with high probability.
- murbard2 11y agoMore precisely it would be the algorithmic probability. Not only are there short programs that compute mathematical constants, but there are many such programs, so the total mass is even higher.
- sawwit 11y agoI'm not sure what you mean. Do you mean that a mathematical constant has multiple different shortest programs?
- murbard2 11y agoNot necessarily shortest, but it has many short programs. The algorithmic probability of a sequence x is the sum over the set of all prefix-free programs u that calculate x of 2^-len(u). Think of the shortest description length as a MAP, while the algorithmic probability integrates over the full prior. A constant which has many short programs can thus have a greater algorithmic probability than another constant with a slightly lower Kolmogorov complexity. To put it back in context, it's possible that a very short program computes some constant, but it's unlikely to be an important mathematical constants. What's particular about mathematical constants is that they keep appearing in many different situations.