- Contemporary messages sorted: [ by date ] [ by thread ] [ by subject ] [ by author ] [ by messages with attachments ]

From: Nick Bostrom <bostrom.domain.name.hidden>

Date: Sat, 28 Feb 1998 15:24:03 +0000

Wei Dai's first paradox was as follows:

*> At time 0 the experiment starts. At time 1 a coin is flipped and the
*

*> result observed by the experimenter. At time 2 a second coin is flipped
*

*> and the result observed by the experimenter. At time 3 the experimenter is
*

*> duplicated if and only if both coins show heads. Now suppose the coins are
*

*> fair so that the experimenter would predict at time 0 that at time 1 he
*

*> will observe heads with probability 1/2. Applying Tegmark's method, he
*

*> would also predict that at time 3 he will observe two heads with
*

*> probability 2/5. But suppose he has already observed heads at time 1, then
*

*> he would predict that with probability 2/3 he will observe another head at
*

*> time 3. So he is left with the following beliefs at time 0:
*

*>
*

*> 1. At time 1 I will observe heads with probability 1/2.
*

*>
*

*> 2. If I observe heads at time 1, I will observe another head at time 3
*

*> with probability 2/3.
*

*>
*

*> 3. I will observe two heads at time 3 with probability 2/5.
*

*>
*

*> But these three statements contradict each other since 1 and 2 together
*

*> implies
*

*>
*

*> 3a. I will observe two heads at time 3 with probability 1/3.
*

*>
*

*> I don't know what to make of this... anyone have any ideas?
*

Here are some comments:

1.

I think premiss 1is false. In order to see this, we have to consider

observer-moments, rather than persons. Let's assume the world is

empty except for the things refered to in the paradox. Then, finding

that your present observer-moment is at time 0 gives you reason

(because of Bayes' theorem) to prefer a hypothesis according to which

a larger fraction of all observer-moments are at time 0 to a

hypothesis according to which a smaller fraction of all

observer-moments are at that time. In the present example, that means

that finding yourself at t=0, you should conclude that the chance

that both coins will land heads is less than 1/4. This also means

that the chance of the first coin landing heads is less than 1/2.

If the world cointains a lot of other observes (outsiders), then the

very fact that your present observer-moment is in this experiment in

the first place indicates that the experiment contains many

observer-moments, i.e. that the chance of both coins landing heads is

greater than 1/4. If you then in addition find that your present

obserever-moment is at time 0, then that gives you reason (as

explained above) to adjust your probability estimate of getting two

heads downwards again. (As the number of outsiders (observer-moments

outside the experiment) goes to infinity, I think you will end up

with a probability asymptotically approaching the one you have

assumed.

2.

Your paradox can be simplified to only one coin toss.

3.

If all branches exist (if there is one real world in which the coins

both land heads, and other real worlds in which they land the other

ways) then I might say the following: If I find myself at time 1 and

observing tails, there is a greater than zero chance that I will find

myself later at time 3 observing two heads! For there will be future

observer-moments observing two heads and other future

observer-moments observing at least one tail, and there doesn't seem

to be any fact of the matter as to which one of these

observer-moments is "really" the future me. I think this might be the

solution to the paradox. It is, metaphorically speaking, possible for

"me" to jump from one branch to another, since there is no fact of

the matter as to which which of the several future me:s I should say

is the true continuation of my present me.

(If only one branch exists, then I would reject premiss 3.)

_____________________________________________________

Nick Bostrom

Department of Philosophy, Logic and Scientific Method

London School of Economics

n.bostrom.domain.name.hidden

http://www.hedweb.com/nickb

Received on Sat Feb 28 1998 - 07:30:02 PST

Date: Sat, 28 Feb 1998 15:24:03 +0000

Wei Dai's first paradox was as follows:

Here are some comments:

1.

I think premiss 1is false. In order to see this, we have to consider

observer-moments, rather than persons. Let's assume the world is

empty except for the things refered to in the paradox. Then, finding

that your present observer-moment is at time 0 gives you reason

(because of Bayes' theorem) to prefer a hypothesis according to which

a larger fraction of all observer-moments are at time 0 to a

hypothesis according to which a smaller fraction of all

observer-moments are at that time. In the present example, that means

that finding yourself at t=0, you should conclude that the chance

that both coins will land heads is less than 1/4. This also means

that the chance of the first coin landing heads is less than 1/2.

If the world cointains a lot of other observes (outsiders), then the

very fact that your present observer-moment is in this experiment in

the first place indicates that the experiment contains many

observer-moments, i.e. that the chance of both coins landing heads is

greater than 1/4. If you then in addition find that your present

obserever-moment is at time 0, then that gives you reason (as

explained above) to adjust your probability estimate of getting two

heads downwards again. (As the number of outsiders (observer-moments

outside the experiment) goes to infinity, I think you will end up

with a probability asymptotically approaching the one you have

assumed.

2.

Your paradox can be simplified to only one coin toss.

3.

If all branches exist (if there is one real world in which the coins

both land heads, and other real worlds in which they land the other

ways) then I might say the following: If I find myself at time 1 and

observing tails, there is a greater than zero chance that I will find

myself later at time 3 observing two heads! For there will be future

observer-moments observing two heads and other future

observer-moments observing at least one tail, and there doesn't seem

to be any fact of the matter as to which one of these

observer-moments is "really" the future me. I think this might be the

solution to the paradox. It is, metaphorically speaking, possible for

"me" to jump from one branch to another, since there is no fact of

the matter as to which which of the several future me:s I should say

is the true continuation of my present me.

(If only one branch exists, then I would reject premiss 3.)

_____________________________________________________

Nick Bostrom

Department of Philosophy, Logic and Scientific Method

London School of Economics

n.bostrom.domain.name.hidden

http://www.hedweb.com/nickb

Received on Sat Feb 28 1998 - 07:30:02 PST

*
This archive was generated by hypermail 2.3.0
: Fri Feb 16 2018 - 13:20:06 PST
*