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From: Hal Finney <hal.domain.name.hidden>

Date: Wed, 15 Jun 2005 10:56:51 -0700 (PDT)

I wanted to add a few points to my earlier posting about how to derive

OM measure in a Schmidhuberian multiverse model.

The method is basically to take all the universes where the OM appears

and to sum up the contribution they make to the OM measure. However,

the key idea is that this contribution has two components. One is the

measure of the universe. The greater the measure of the universe, the

greater the contribution to OM measure. But the other is the fraction

of the universe that is involved in the OM. This means that a smaller

universe that contains an OM gives a greater fraction of its measure

as its contribution to the OM measure. Smaller universes make more

contribution than larger ones.

This last step may seem ad hoc but in fact it can be seen in a very

natural way. It can be thought of as a two step way to output the

description of a given OM: first write a program to output a universe

with the OM in it, then write a program to take that universe and output

the OM. We can think of combining these two programs into one: write a

program that outputs the OM. Then, the sum of the measure of all such

programs is the measure of the OM.

That last sentence is merely the definition of measure in a Schmidhuberian

context - the measure of anything is the fraction of all programs that

output that thing. It IMPLIES the formula I described for downgrading a

universe's contribution to an OM by virtue of the relative size of the OM

compared to the universe. It can be said that we have derived and proven

that relationship by assuming this fundamental definition of measure.

Note that we could also write a program to output an OM without regard

to creating a universe first. However, I believe that at least for

observers like us, it will always be a much simpler program to first

create a universe and then find the OM in it. This lets evolution work

and everything is simple. Ultimately, this allows the AUH (all universe

hypothesis, ie the multiverse exists) to JUSTIFY the belief that we are

not brains (or OMs) in vats, that the universe is probably real.

Okay, so that's just restating what I had before in different words,

explaining it from a different perspective that might be more obvious.

Here are a couple of interesting additions.

First, what about our universe? Why is it so damn big? If the measure

of an OM is smaller in a big universe, the AUH should predict that

the universe is no bigger than it needs to be. Yet, looking around,

our universe looks a lot bigger than necessary. There's a lot of

wasted space.

I conclude that it is likely that the universe is not in fact much

bigger than it needs to be. It actually needs to be as big as it is.

This might imply that intelligent life is extremely rare in universes

like ours. Only by creating a truly enormous universe can we have a

good chance of creating observers.

Let me expand on this a little. All universes exist. Some have

complex laws of physics and some are simple. Some have complex initial

conditions and some are simple. Physicists believe that our universe

is relatively simple by both measures. The laws of physics are not

completely understood but the ones we know have a very simple mathematical

formulation. And the initial conditions also appear to represent a very

smooth and uniform condition immediately after the Big Bang. The bottom

line is that you would not have to write a very big program to simulate

our universe.

Yet, even with these simple laws, our universe supports life that can

evolve into consciousness. That's pretty amazing, maybe. What are

the odds that another universe with equally simple laws could do so?

We know that our own physical laws appear to be relatively "fine tuned"

such that even a tiny change in various properties would cause life as

we know it to be impossible. That suggests that maybe it is not so

easy to have life. Maybe almost no universes with laws as simple as

ours create life.

And, maybe life is not all that easy to create even in our universe.

What if life, at least intelligent life, is overwhelmingly unlikely,

even in a universe as well suited as our own? Maybe we need ten billion

light years' worth of galaxies, stars and planets in order to have a

decent chance of evolving life. Maybe, in short, our universe is as

big as it needs to be, given our laws of physics, to allow life to evolve.

There may be other sets of laws of physics that would be more fecund,

where life could evolve more easily. Those might get by with smaller

universes. But if so, the AUH would predict that such universes would

have much more complicated laws of physics and/or initial conditions

than our own. Otherwise we would live there.

Given that the universe is as big as we see, and given the AUH, we can

predict that it is not full of intelligent life. We can predict that

there should be almost no other intelligent civilizations within the

universe. This then solves the Fermi paradox - where are the aliens.

There are no aliens, not for cosmological distances.

This leads to my second point: what about infinities? In some models,

our own universe is infinite in size. Tegmark's level 1 multiverse

postulates a physically infinite space. His level 2 sees our universe

as an infinite bubble inside a larger chaotic inflation region, with

infinite numbers of other bubbles. How could my proposed formulation

of OM measure work if the universe is infinite in size? How can we

determine the fraction of universe measure dedicated to an OM if the

OM is infinitesimal in size compared to the universe?

I have two thoughts about this. One is that if the universe is truly

spatially infinite, any OM should be repeated infinite times, as Tegmark

predicts. Therefore the OM still occupies a finite fraction of the

universe resources, and we can calculate that fraction by taking the

limit as spatial size goes to infinity. I know that some people don't

like this limit approach, they get upset by trying to divide one infinity

by another, but in practice this method seems sensible and produces a

reasonable value.

The other approach focuses on a paradox between level 1 and level 2.

If the level 1 universe is infinite in extent, where are the level 2

universes? Other dimensions? That doesn't really work, physically.

Here is how I understand it. In physics, time and space are relative.

*>From the level-2 perspective, our universe is finite in size. However,
*

it is constantly and eternally growing. From within the universe,

the level-1 perspective (our point of view!) time and space have gotten

shifted so that the finite-size but infinite-time outside view becomes

infinite-size from the inside view. I've seen drawings of this. It is

consistent to have a universe that looks spatially infinite but from the

outside is finite in size but constantly growing. It's all a matter of

event horizons and such.

This provides a method to at least partially solve the infinite-size

paradox. From the level 2 perspective, the universe is no longer infinite

in size, however it is infinite in time. This means that it takes only

a finite amount of computation to simulate the universe up to any given

point in space and time. We can simulate the past light cone of any point

in spacetime within our universe using a finite amount of computation,

even though the universe looks spatially infinite from each point.

This allows us to again apply our program to find OMs within the output

of the program creating a universe. Each OM appears after only a finite

length of output. And the farther out we go, the longer the OM-finding

program is going to need to be, because it has to localize the OM within

the output of the universe program. That means at some point we can

ignore further OM instances appearing in the tape as having negligible

measure (he says, waving his hands furiously) because the program to

find them would have to be so big.

In effect what we are saying is that there is a nonuniform measure

over spacetime in our own universe, one that tails off to zero in space

and time. Now, I'm not 100% sure how that works spatially, it seems to

suggest that there is a point in space that has more measure than all

others, which doesn't seem very physical. I'll have to think about it.

But it is basically consistent with the general Schmidhuber principle

that measure of X is 1/2^KC(X), applied to OMs. If we live in a level

2 multiverse, as our best physical theories suggest, then the measure

of the OMs in that multiverse have to work in a method similar to what

I have outlined here.

Hal Finney

Received on Wed Jun 15 2005 - 14:48:30 PDT

Date: Wed, 15 Jun 2005 10:56:51 -0700 (PDT)

I wanted to add a few points to my earlier posting about how to derive

OM measure in a Schmidhuberian multiverse model.

The method is basically to take all the universes where the OM appears

and to sum up the contribution they make to the OM measure. However,

the key idea is that this contribution has two components. One is the

measure of the universe. The greater the measure of the universe, the

greater the contribution to OM measure. But the other is the fraction

of the universe that is involved in the OM. This means that a smaller

universe that contains an OM gives a greater fraction of its measure

as its contribution to the OM measure. Smaller universes make more

contribution than larger ones.

This last step may seem ad hoc but in fact it can be seen in a very

natural way. It can be thought of as a two step way to output the

description of a given OM: first write a program to output a universe

with the OM in it, then write a program to take that universe and output

the OM. We can think of combining these two programs into one: write a

program that outputs the OM. Then, the sum of the measure of all such

programs is the measure of the OM.

That last sentence is merely the definition of measure in a Schmidhuberian

context - the measure of anything is the fraction of all programs that

output that thing. It IMPLIES the formula I described for downgrading a

universe's contribution to an OM by virtue of the relative size of the OM

compared to the universe. It can be said that we have derived and proven

that relationship by assuming this fundamental definition of measure.

Note that we could also write a program to output an OM without regard

to creating a universe first. However, I believe that at least for

observers like us, it will always be a much simpler program to first

create a universe and then find the OM in it. This lets evolution work

and everything is simple. Ultimately, this allows the AUH (all universe

hypothesis, ie the multiverse exists) to JUSTIFY the belief that we are

not brains (or OMs) in vats, that the universe is probably real.

Okay, so that's just restating what I had before in different words,

explaining it from a different perspective that might be more obvious.

Here are a couple of interesting additions.

First, what about our universe? Why is it so damn big? If the measure

of an OM is smaller in a big universe, the AUH should predict that

the universe is no bigger than it needs to be. Yet, looking around,

our universe looks a lot bigger than necessary. There's a lot of

wasted space.

I conclude that it is likely that the universe is not in fact much

bigger than it needs to be. It actually needs to be as big as it is.

This might imply that intelligent life is extremely rare in universes

like ours. Only by creating a truly enormous universe can we have a

good chance of creating observers.

Let me expand on this a little. All universes exist. Some have

complex laws of physics and some are simple. Some have complex initial

conditions and some are simple. Physicists believe that our universe

is relatively simple by both measures. The laws of physics are not

completely understood but the ones we know have a very simple mathematical

formulation. And the initial conditions also appear to represent a very

smooth and uniform condition immediately after the Big Bang. The bottom

line is that you would not have to write a very big program to simulate

our universe.

Yet, even with these simple laws, our universe supports life that can

evolve into consciousness. That's pretty amazing, maybe. What are

the odds that another universe with equally simple laws could do so?

We know that our own physical laws appear to be relatively "fine tuned"

such that even a tiny change in various properties would cause life as

we know it to be impossible. That suggests that maybe it is not so

easy to have life. Maybe almost no universes with laws as simple as

ours create life.

And, maybe life is not all that easy to create even in our universe.

What if life, at least intelligent life, is overwhelmingly unlikely,

even in a universe as well suited as our own? Maybe we need ten billion

light years' worth of galaxies, stars and planets in order to have a

decent chance of evolving life. Maybe, in short, our universe is as

big as it needs to be, given our laws of physics, to allow life to evolve.

There may be other sets of laws of physics that would be more fecund,

where life could evolve more easily. Those might get by with smaller

universes. But if so, the AUH would predict that such universes would

have much more complicated laws of physics and/or initial conditions

than our own. Otherwise we would live there.

Given that the universe is as big as we see, and given the AUH, we can

predict that it is not full of intelligent life. We can predict that

there should be almost no other intelligent civilizations within the

universe. This then solves the Fermi paradox - where are the aliens.

There are no aliens, not for cosmological distances.

This leads to my second point: what about infinities? In some models,

our own universe is infinite in size. Tegmark's level 1 multiverse

postulates a physically infinite space. His level 2 sees our universe

as an infinite bubble inside a larger chaotic inflation region, with

infinite numbers of other bubbles. How could my proposed formulation

of OM measure work if the universe is infinite in size? How can we

determine the fraction of universe measure dedicated to an OM if the

OM is infinitesimal in size compared to the universe?

I have two thoughts about this. One is that if the universe is truly

spatially infinite, any OM should be repeated infinite times, as Tegmark

predicts. Therefore the OM still occupies a finite fraction of the

universe resources, and we can calculate that fraction by taking the

limit as spatial size goes to infinity. I know that some people don't

like this limit approach, they get upset by trying to divide one infinity

by another, but in practice this method seems sensible and produces a

reasonable value.

The other approach focuses on a paradox between level 1 and level 2.

If the level 1 universe is infinite in extent, where are the level 2

universes? Other dimensions? That doesn't really work, physically.

Here is how I understand it. In physics, time and space are relative.

it is constantly and eternally growing. From within the universe,

the level-1 perspective (our point of view!) time and space have gotten

shifted so that the finite-size but infinite-time outside view becomes

infinite-size from the inside view. I've seen drawings of this. It is

consistent to have a universe that looks spatially infinite but from the

outside is finite in size but constantly growing. It's all a matter of

event horizons and such.

This provides a method to at least partially solve the infinite-size

paradox. From the level 2 perspective, the universe is no longer infinite

in size, however it is infinite in time. This means that it takes only

a finite amount of computation to simulate the universe up to any given

point in space and time. We can simulate the past light cone of any point

in spacetime within our universe using a finite amount of computation,

even though the universe looks spatially infinite from each point.

This allows us to again apply our program to find OMs within the output

of the program creating a universe. Each OM appears after only a finite

length of output. And the farther out we go, the longer the OM-finding

program is going to need to be, because it has to localize the OM within

the output of the universe program. That means at some point we can

ignore further OM instances appearing in the tape as having negligible

measure (he says, waving his hands furiously) because the program to

find them would have to be so big.

In effect what we are saying is that there is a nonuniform measure

over spacetime in our own universe, one that tails off to zero in space

and time. Now, I'm not 100% sure how that works spatially, it seems to

suggest that there is a point in space that has more measure than all

others, which doesn't seem very physical. I'll have to think about it.

But it is basically consistent with the general Schmidhuber principle

that measure of X is 1/2^KC(X), applied to OMs. If we live in a level

2 multiverse, as our best physical theories suggest, then the measure

of the OMs in that multiverse have to work in a method similar to what

I have outlined here.

Hal Finney

Received on Wed Jun 15 2005 - 14:48:30 PDT

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