gts wrote:
LEADING TO THE ONLY THING REALLY INTERESTING ABOUT THIS DISCUSSION:
What interests me is that the Principle of Indifference is taken for
granted by so many people as a "logical truth" when in reality it is
fraught with logical difficulties.
I think it's been a pretty long time since the PI was taken by any
serious thinkers as a "logical truth", though...
What it is, is a heuristic principle, which can be applied in a number
of ways to any given situation
The connection of the PI with entropy is interesting, in that it
highlights the subjectivity of entropy. To calculate the entropy
information-theoretically, one needs to partition the state space of the
system being measured. Different partitions could lead to different
answers. So, entropy exists subjectively relative to a certain
observer, who takes a certain coarse-grained view of the state space.
This is consistent with how assuming PI with respect to different
partitions of the state space (a vs. ~a, b vs. ~b) can lead to different
answers --- the PI being a special case of entropy maximization.
Philosophically, this is similar to how the Occam prior depends on the
model of computation under assumption.
So, in Zurek's formulation
Physical Entropy = Statistical Entropy + Algorithmic Entropy
the first term is subjective due to dependence on a partition of state
space, and the second term is subjective
due to dependence on a choice of universal computer.
And that's just the way it is.... But, this is all basically old stuff,
and I'm not sure why it requires so much discussion
at this point!
-- Ben
Gillies (2000) makes an analogy between the situation in probability
theory concerning the Principle of Indifference and the situation that
once existed in set theory concerning the Axiom of Comprehension.
Like the Principle of Indifference, the Axiom of Comprehension seemed
logical and intuitively obvious. That axiom states that all things
which share a property form a set. What could be more logical and
intuitively obvious? But the Axiom of Comprehension led to the Russell
Paradox, and a crisis in set theory.
Similarly the Principle of Indifference (and its predecessor the
Principle of Insufficient Reason) led to numerous difficulties, (e.g.,
the Bertrand Paradoxes, and arguments such as Cox's). Subsequently we
saw a schism in probability theory. The classical theory was
discredited, including the classical interpretation of Bayes' Theorem,
and replaced with at least four different alternative interpretations.
Among bayesians, one might say De Finetti and Ramsey and the
subjectivists helped rescue bayesianism from the jaws of
(philosophical) death, by separating bayesianism from that albatross
around its neck which is the Principle of Indifference.
-gts
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