[extropy-chat] Paradox? What paradox?
gts_2000 at yahoo.com
Sun Jan 7 18:25:14 UTC 2007
Here is the explanation of the cube paradox as given by the "Stanford
Encyclopedia of Philosophy":
"The paradox [about choosing a random cube] arises because the principle
of indifference can be used in incompatible ways. We have no evidence that
favors the side-length lying in the interval [0, 1/2] over its lying in
[1/2, 1], or vice versa, so the principle requires us to give probability
1/2 to each. Unfortunately, we also have no evidence that favors the
face-area lying in any of the four intervals [0, 1/4], [1/4, 1/2], [1/2,
3/4], and [3/4, 1] over any of the others, so we must give probability 1/4
to each. The event ‘the side-length lies in [0, 1/2]’, receives a
different probability when merely redescribed. And so it goes, for all the
other reformulations of the problem. We cannot meet any pair of these
constraints simultaneously, let alone all of them."
This explanation fits well with Gillies' assessment that the principle of
indifference is best considered a heuristic principle and not a logical
principle. Logical principles should be true in some absolute sense, while
heuristic principles may sometimes lead us astray.
However I am encouraged by the paper Ben cited...
> Von Mises' wine/water paradox is fairly convincingly addressed in
> where it is shown that the paradox goes away if one assumes a 2D
> sample space covering the set of pair of values
I agree these paradoxes seem "fairly convincingly addressed". Thanks Ben!
I think Burock's main insight here, (applicable not only to the wine/water
paradox but to the others as well), is that the paradoxes may be resolved
when one considers that "the probability of an outcome is fundamentally
dependent upon the specific sample space in which the outcome is
contained". I wonder if his paper has created any stir in academia.
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