[extropy-chat] what is probability?

[gts]

What is a correct sample space? 

Clearly something like a correct sample space exists
in the case of the wine/water paradox (I agree with
Ben's referenced paper on this point) but I don't see
the same as true in the case of the Bertrand class of
paradoxes.

Setting aside the cube factory paradox for the moment,
consider this more famous example:

"Consider a fixed circle and select a chord at random.
What is the probability that this random chord is
longer than the side of the equilateral triangle
inscribed in the circle?" (as given by Gillies, 2000)

At least three very plausible solutions exist to this
problem, and I think it can be shown that an infinity
of them exist, each which gives a different answer to
the question (depending of course on our choice of
sample space). 

ET Jaynes tried to solve this paradox, and may have
done so to his own way of thinking. I think Jaynes
would call himself an objective (or perhaps logical or
empirical) Bayesian. He invented (on his own rational
authority, I suppose) some additional properties of
randomness which he believes should apply to random
chords, and then tossed broom straws onto a circle
drawn on the floor to show his definition of "random
chord" to be superior to the others. 

But Gillies points out that Jaynes' proposed solution
in no way shows the PI to be a logical rather than a
heuristic principle. He writes of a hypothetical
researcher who uses the PI in the same (heuristic) way
as Jaynes to arrive initially at a different but
mistaken result.  
 
Note that, (unless I missed something), aside from the
wine/water paradox, the paper by Durack(?) does not
actually *answer* any other paradoxes such as the one
above, except to explain that not enough information
is given. 

So what should we do in real-world cases in which not
enough information is given? I think we can and should
usually still apply the PI, but, as I suggested in my
message before last, I think we should agree with the
subjectivists on this question and not imagine we are
doing something grounded firmly in objective logic.
The PI should be considered a handy rule of thumb, a
heuristic tool and nothing else. 

At least this is my present thinking on this subject,
and it's likely to change as I continue reading and
discussing these issues with informed people. I 
have not yet personally decided which
theory of probability is most acceptable to me.

-gts

(This message is a modified reply to what I thought
was a public message from Ben)