[extropy-chat] Paradox? What paradox?
Jef Allbright
jef at jefallbright.net
Sat Jan 6 16:47:04 UTC 2007
I don't see the origins of this thread in my email nor in the archives
at
http://lists.extropy.org/pipermail/extropy-chat/2007-January/date.html.
What am I missing?
- Jef
> -----Original Message-----
> From: extropy-chat-bounces at lists.extropy.org
> [mailto:extropy-chat-bounces at lists.extropy.org] On Behalf Of gts
> Sent: Saturday, January 06, 2007 7:43 AM
> To: rafal at smigrodzki.org; ExI chat list
> Subject: Re: [extropy-chat] Paradox? What paradox?
>
> On Sat, 06 Jan 2007 00:52:03 -0500, Rafal Smigrodzki
> <rafal.smigrodzki at gmail.com> wrote:
>
> > I'd rather say the problem is a trick question rather than
> a paradox
> > :)
>
> It's a real paradox, in the class of paradoxes known
> generically as Bertrand' Paradox(es). However they are
> paradoxes iff we assume the principle of indifference.
>
> Here is a different version of the same paradox:
>
> http://www.cut-the-knot.org/bertrand.shtml
>
> (This is the version of Bertrand's Paradox to which Jaynes
> offered a possible resolution. But as I mentioned to Ben,
> it's my understanding that Jaynes' apparent solution does not
> apply to every version of Bertrand's Paradox and so cannot be
> considered a real solution.)
>
> > The problem may seem vexing at first glance, until one notices that
> > the procedure used by the factory to choose which cube to
> make is not
> > defined in the formulation of the problem.
>
> I might say the procedure *is* defined in the problem; that
> it is defined as a "random" procedure.
>
> > therefore the event of "randomly choosing" a cube is not defined
> > sufficiently
>
> Yes. The paradox goes away, as I think you see, when we
> realize we have no idea what we mean we speak of selecting a
> "random" cube or of performing a "random selection procedure".
>
> More generally, the principle of indifference seems not to
> apply to continuous variables. (I mentioned this paradox in
> the first place by way of criticising the principle of
> indifference upon which some epistemic theories of
> probability depend.)
>
> We can say "with respect to a random coin-flip, if we have no
> evidence to favor either heads or tails then we should assume
> the two possible outcomes to have equal probabilities". We
> are indifferent to the two outcomes and so the two outcomes
> receive the same subjective probability.
> This is the (supposed) principle of indifference in action.
>
> However we run into trouble when we say things like "with
> respect to a random cube selection (as per the paradox) we
> should assume the two possible outcomes have equal
> probabilities." Here again we are indifferent to the two
> possible outcomes, just as in the coin-flip example above,
> but the principle of indifference fails miserably.
>
> -gts
>
>
>
>
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