That would be helluva loaded dice. Get you banned from the cosmic casino for sure. It just wouldn’t fit the description of being random. — apokrisis

It's exactly as likely as any other outcome: 1/∞.

Jeez, you were serious?

The odds of landing on the face marked Earth might be 1/∞. The odds of landing on the face marked Earth an infinite number of times in a row is another matter. It could only be 1/∞ in relation to an infinite ensemble of multiverse creations. So in a multiverse of multiverses, you would almost surely get your one multiverse in which every planet wound up being replica Earth, faithful down to us speaking in Korean about flower arranging, or whatever other modal possibility we could imagine.

I agree that presuming infinity entails any madness you care to suggest. But first, you would have to motivate this new tale of yours about multiverses of multiverses where a random process can then turn out its most unlikely possible result with certainty.

But first, you would have to motivate this new tale of yours about multiverses of multiverses where a random process can then turn out its most unlikely possible result with certainty. — apokrisis

I didn't say that. I said that any particular outcome (whether infinite Mars or infinite everything) is almost surely never going to be the case.

The odds of landing on the face marked Earth might be 1/∞. The odds of landing on the face marked Earth an infinite number of times in a row is another matter. — apokrisis

Also 1/∞.

So in a multiverse of multiverses, you would almost surely get your one multiverse in which every planet wound up being replica Earth, faithful down to us speaking in Korean about flower arranging, or whatever other modal possibility we could imagine. — apokrisis

I think you almost surely won't.

You are just ignoring the fact that your scenario demands an infinite creation of multiverses. That is different from figuring the odds of repeated configurations within just the one multiverse with “a fair die”.

There is nothing to motivate your assumption that the one multiverse would be so atypical. A multiverse with a die that produced your selective outcome could not be believed to be random. There would be no proper basis for such a presumption. It would be mad not to believe the die was loaded.

So nice try, but no dice.

A multiverse with a die that produced your selective outcome could not be believed to be random. There would be no proper basis for such a presumption. It would be mad not to believe the die was loaded. — apokrisis

The same is true of any particular outcome, as any particular outcome is as (un)likely as any other.

Having an infinite number of every planet is no more likely than having an infinite number of just one planet.

And whether the coin (or die) is biased makes no difference when we're dealing with infinity. Both a fair coin and a head-heavy coin have the same probability of always landing heads (so long as landing tails is always possible).

Even in Euclidean space, as soon as you introduce something to break the symmetry, you already have some kind of "preference." For example, in a universe that is a flat space with one black hole there is an obvious "center." — SophistiCat

That one makes a bit of a hash of the Copernican principle at least. Ossipoff's initial post on the prior page was such a violation, but there is no such principle in the view he was supporting there.

I agree they are not other universes. — apokrisis

The type-3 ones are also not other universes, for more or less the same reasons.

So spatial infinity would seem to guarantee that there should be an infinity of Earths where you and me are having this exact discussion - plus every other even faintly similar or utterly different interactions. We could be discussing hair-do's, speaking in Korean, typing random sequences. And the fact any of those might be the case would mean that all those varieties of cloned Earths would have to be infinite in number themselves. There would be an infinite number of replica planets with us speaking Korean, etc.

You mean there is a pile of near-replicas to go with each actual replica. Yes. Those aren't so far away, depending on how loose you allow your definition of 'near replica' to be.

There just is no end to the madness once you let actual infinity run riot in your ontology.

You seem to be apeirophobic *. I followed the argument until it was suddenly labelled madness.
* The word seems to mean more fear of eternity, not infinities of the non-temporal sort. So fear of realities that involve infinities. Could find no better word for that.

Anyway, even in a spatially infinite universe, we would presume that it all expands and cools in the same way. And cooling steadily - or in fact, exponentially - removes material possibilities. If every portion of the universe is losing energy density at a shared rate, that means there is only a tiny time window for replica earths to actually form.

Of course. Any replica of Earth would be the exact same age. A replica cannot begin to form by chance for example, centuries from now on the other side of our galaxy.

Of course. Any replica of Earth would be the exact same age. A replica cannot begin to form by chance for example, centuries from now on the other side of our galaxy. — noAxioms

I'm not sure about this. I wonder if the principle behind the Boltzmann brain hypothesis can also apply here. It is more likely that an Earth-like planet spontaneously forms than for a Hubble volume to grow and develop as ours is believed to have done.

I'm not sure about this. I wonder if the principle behind the Boltzmann brain hypothesis can also apply here. It is more likely that an Earth-like planet spontaneously forms than for a Hubble volume to grow and develop as ours is believed to have done. — Michael

Interesting. Perhaps we could define a duplicate as not just a state, but one that persists for a second or so as a natural duplicate should. A Boltzmann Earth duplicate ceases to be a duplicate immediately just like the brain ceases to be a brain in a moment

It's exactly as likely as any other outcome: 1/∞. — Michael

Can you explain exactly what you mean by that? Are you using your own private system of mathematics? Making an argument based on a vague misunderstanding of nonstandard analysis? Something else?

Do you know what countable additivity is? Did you know that there's no uniform probability measure on the natural numbers? Does that information affect your use of this notation?

Having an infinite number of every planet is no more likely than having an infinite number of just one planet. — Michael

If the odds of earth being the case on any one roll are 1 in infinity, then the odds of earth being the case every time in an infinite series of rolls are 1 in an infinity of infinities.

If you just reduce this to a consideration of the statistics of a one off event - as is the case with the multiverse argument - then you must take a propensity view of the statistics. We should presume typicality for the outcome.

We are talking about a known outcome - the visible universe and the variety of planets if produces. We know what typicality looks like to a reasonable degree. You get gas giants, you get Mercuries, you get Earth-like planets. We are busy counting that variety around other stars now.

We also then have at least some grasp on the propensity for intelligent life arising on other planets. And we can keep tacking on propensity estimates for the history of an earth repeating such that it produces humans who are exactly, atom-for-atom, thought-for-though, like you and me, doing replica actions for as long into the future as this multiverse calculation requires. (Is it still a real multiverse if all the other replica Earths do a Boltzmann Brain disappearing or disintegrating act in the next split second? How long must that exact continuity of a history persist?)

So the actual situation for a multiverse "just one throw of a die" propensity calculations is that being "Earth-like" in the vague way astrobiologists have in mind is reasonably typical. There are many ways to be Earth-like. So it happens a lot. Even inside our visible universe.

Then harbouring Earth-like life is way less typical. How typical the biology of the Earth is - as an outcome of the physics and constraints of the universe - is an open question. Recent work - like Nick Lane's The Vital Question - is arguing that the ways life can biochemically develop are surprisingly limited. So the odds of Earth-like biology now seem much higher - if life develops on other planets at all.

Then we have the question of the typicality of a re-run of the Homo sapiens story down to the level of historic accident that produces you, me, and the rest of PF. The level of accident, the level of information discarded, argues for some extremely low propensity. I would say "almost never". Or probability = 0.

You thus run into a collision between two notions of the infinite. The combinatorial one says every possible combination simply gets realised. The constraints based one says the steady shrinking towards an infinite unlikelihood means you are headed towards almost never, a probability that is actually zero (fine print: for all practical purposes). That is, the principle of indifference kicks in to allow the state of infinite constraint to be achieved.

One notion of infinity operates on an already closed and bounded set. The other has to achieve that claimed closure.

So as I tried to point out, the simple minded combinatorial notion of infinity employed to motivate multiverse arguments is itself in question. It depends on the assumption of a bounded space with no internal correlations. That gives you one picture of what "infinity" means.

And then the more appropriate notion is a constraints-based infinity where the correlations get counted too. Restrictions on what is typical arise due to histories. Material accidents and formal necessities go into making those histories. The story in irreducibly complex and non-linear.

The propensities of Earth-like biology might be much higher than naive combinatrics would predict, if we buy Nick Lane's arguments about the biophysical constraints on life forming. But then the propensity of Homo sapien history being exactly repeated to the point of producing doppleganger you and me, is way less than naive combinatrics would predict.

But even if we put aside the difference between a combinatorial statistics and a constraints-based, negentropy-including, one, you are still only dealing with a one-off propensity story with the multiverse argument. Unless you can motivate the further idea of a multiverse of multiverses, we are only talking about a one time "roll of the dice" so far as there was a Big Bang that produce an infinite amount of spatial regions, all with the same propensity for star and planet formation.

The typicality is wired in just by observation. We have a sample size of the solar system, and now the solar systems of other stars. Already that is a constraint we can't just ignore (as good Bayesians).

But the very description of the die to be tossed - this spherical die with its infinity of marked faces - demands we assign a Bayesian propensity to the typicality of its outcomes. We already must "know" that it is going to generate the statistically typical, not the statistically infinitely unlikely. The only way we could think different is if we imagine an infinity of infinite throws. Then our propensity switches to thinking it almost sure that the 1 in infinity outcome would be among all the combinations that happen. Now, it couldn't not ... for all practical purposes.

To sum up, the multiverse scenario I was addressing only permits a one-shot propensity view. It was about the likelihoods within a single infinite Big Bang space.

Then a simple combinatrics view - the one that counts only entropy or degrees of freedom - would give you a naive number for how many times some exact combination of atoms and thoughts could appear in just such an infinite space.

But I argued that simple combinatrics is simply going to be wrong. A realistic calculation of the odds has to include correlations and the emergent constraints on combinations that will result.

If that propensity calculation could be done correctly, my gut feeling is that the probability of that propensity would shrink towards zero, or almost surely not, even with the infinity of a multiverse to play with. The magic of infinity would lose its power to conjure up every possibility.

The big "if" is doing that particular calculation. But I think Scott Aaronson provides some of the right conceptual tools here - https://www.scottaaronson.com/papers/npcomplete.pdf

If the odds of earth being the case on any one roll are 1 in infinity, — apokrisis

Why are you doubling down on this mathematical nonsense?

Nonsense? I think it really gets us to the heart of some really telling confusion.

This is a philosophy discussion group. When things seem unarguably right, that's when you know there must be the whole flip-side to the story. Dialectics always rules. So someone's nonsense is always the start of someone else's sense.

This is a philosophy discussion group. When things seem unarguably right, that's when you know there must be the whole flip-side to the story. Dialectics always rules. So someone's nonsense is always the start of someone else's sense. — apokrisis

When you replied to me earlier spouting buzzwords about the Hubble radius and such, I tended to believe you since I don't know much physics.

But when you write the expression 1/infinity with a straight face; and when I call you on it, and you reply as you did in the quoted paragraph; I have to assume you have no idea what you're talking about. So which is it?

There is no such thing as a probability of 1/infinity. Not in standard analysis and not in nonstandard analysis, which I took the trouble to learn something about a while back.

The problem I have in this thread is that whenever someone lays on the physics jargon, I can't respond or even form an opinion, since I lack the background. But whenever those same individuals (not just you) wander into a domain in which I have technical expertise, they're often spouting bullshit. [In the Frankfurt sense, of course]. And this leads me to think they're doing the same with their physics jargon.

So I ask you again: When you claim an event has probability 1/∞, exactly what do you mean? Is that a claim of standard probability theory? A misunderstanding of nonstandard probability theory? A vague, meaningless intuition that cannot be properly grounded in logic? Or what, exactly?

And if you can't speak sense to me about things I know about, why should I believe you when you speak to me about things I don't know about?

Of course it was also posted that contemporary physics puts a finite size on the universe — noAxioms

That fact it was posted doesn't mean that it's so.

I too noticed a post that made that claim that physics says the universe is finite, or that most physicists think so. He didn't say where he got that. Maybe there are some physicists, cosmologists, &/or astronomers who say that. I don't know what their arguments are, because I haven't run-across or sought-out their articles.

But, according to Tegmark, the evidence, more and more, points to the universe being infinite. Tegmark also says that that's probably the more widely-held position among physicists.

I refer you to that quote from Tegmark that I posted.

Secondly, the level-1 multiverse only requires a finite universe sufficiently large that light hasn't had time to get from one point to some other point in the age of the universe.

...depending on the definition of a level-1 multiverse. In articles that i've seen, Tegmark referred to an infinite big-bang universe (BBU) when he spoke of a level-1 multiverse.

I'd said:

By the way, I'd expect that if an infinite universe means that there are other civilizations in the universe, then the nearest one is so far away that, for all practical purposes, including communication or transportation, it's the same, for us, as if it weren't there. — Michael Ossipoff

You replied:

How do you get this?

Fermi's paradox.

Astronomers say that this galaxy is old enough for there to have been early civilizations that have had time to thoroughly explore and document its every star and solar-system, even with space-transportion no faster than what we now have. ....but with the help of self-replicating robots.

This planet's potential for life would have been noticed, and a monitor-device could have been left somewhere in the solar-system. Maybe in a distant solar orbit. Maybe closer, if it could be made unnoticeable to us. (Arthur Clarke pointed out that a sufficiently advanced technology would be indistinguishable from magic).

But we haven't had any communication from space, either by robot or radio, etc.

Maybe the super-advanced societies aren't interested in space-exploration. So maybe I should amend what I said, by replacing "civilizations" with "spacefaring civilizations". Still, if there are lots of super-advanced civilizations, that might decrease the likelihood that none will be spacefaring.

Non-intervention prime-directive? I doubt it. I'd expect that technological super-advancement would go
along with corresponding moral/ethical advancement, and some compassion. It's difficult to believe that such beings would observe events on our planet without instituting the policing that would protect us from eachother. ...as in Clarke's Childhood's End.

So, the fact that they haven't helped us means that they aren't there.

I'd said:

Could there not be any other civilizations in this universe, if the universe is infinite?

You replied:

You just got finished saying there is an exact copy of us out there, given infinite space.

Yes, I quoted Tegmark and others about that, and they seem right. But that statement assumes that this universe is natural, not artificial. ...not specifically-designed by some advanced alien technology, to not have any life other than us (I'll call that the high-tech quarantine theory)..

I should clarify that, based on an assumption about compassion, I don't think quarantine without help is likely, and so I don't really believe the high-tech quarantine theory.

I'd said:

Maybe, if, as a form of high-tech quarantine, our belligerent and aggressive species, along with its planet, has been re-located into a universe that was specifically designed, by an advanced technology, to not have any life other than us.

This statement is quite a break from the usual stance I've seen from you. You gone all ID on us?

ID isn't about creation by advanced aliens.. And my high-tech quarantine suggestion (which I don't really believe), wasn't that advanced aliens created us and the Earth. It was just that they've relocated us to an artificial universe made by them, designed to have no life other than us.

Physicists and cosmologists have spoken about the possibility maybe that a physicist working in a laboratory could create a new universe. And, by the way, that wouldn't make him a god.

There is at least one group who believe that advanced aliens created the human race, but even that isn't ID.

Tegmark for instance described a universe not in need of creation, not designed, nor one where we are special.

Of course. Ontic Sructural Realism. Tegmark's External Reality Hypothesis is at the basis of MUH, a Realism.

I don't agree with Realism. But I also don't believe in absolute Anti-Realism.

But there's a sense in which we're special. You're the center of your world. And so the natural way to speak of the world is in terms of the individual's experience...an individual life-experience possibility-story. That system of abstract facts is as valid in its own context as any.

I didn't mean to imply anything about religious issues. In religion discussions, I often mention that the word "create" is too anthropomorphic.

But the fact that Tegmark didn't suggest that aliens created the universe that we're in (and relocated us to it), that doesn't disprove that suggestion (which, as I said, I don't believe, because I assume that an advanced society would be compassionate).

Michael Ossipoff

Just a lazy way to write $$\lim_{n \to ∞} f(\frac{1}{n}) = 0$$

Isn't this how to describe the probability that an infinite number of coin tosses always lands heads, or always lands tails, or lands once on heads and every other time on tails, or [insert any particular outcome]?

Isn't this how to describe the probability that an infinite number of coin tosses always lands heads, or always lands tails, or lands once on heads and every other time on tails, or [insert any particular outcome]? — Michael

No. The probability of any particular sequence of infinitely many coin tosses is zero. Exactly zero.

The question at issue with the "infinitely sided die" is that you toss a set of pingpong balls into a hat. The balls are labelled 1, 2, 3, 4, 5, 6, ... and so forth. The question is, what it the probability of pulling out, say, 6. And the answer is that there is no way to assign that event a probability. That's because of a thing called countable additivity. If you say the probability is zero, then the probability of picking any number at all must be zero. But that's false, because you will pick out SOME pingpong ball.

https://en.wikipedia.org/wiki/Sigma_additivity

On the other hand, if the probability is some tiny real number greater than zero, then the total of all the probabilities for individual balls adds up to infinity. That's impossible too, because in a probability space, the probabilities of all the possible events must add up to 1.

The conclusion is that there is no uniform probability distribution on the natural numbers. That's why the notation 1/∞ makes no sense. It's an attempt to express an intuition that turns out to be logically impossible.

Now you may ask, why is the probability of getting some particular sequence of coin tosses exactly zero then? The answer is that there are uncountably many possible sequences, so countable additivity doesn't apply. The probability of any particular sequence of coin tosses is zero, and the probability that you'll get SOME sequence is 1 (since you must get some result), and the math works out.

Here are the mathematical rules for probabilities.

https://en.wikipedia.org/wiki/Probability_axioms

No. The probability of any particular sequence of infinitely many coin tosses is zero. Exactly zero. — fishfry

Which is why I wrote $$\lim_{n \to ∞} f(\frac{1}{n}) = 0$$

On the other hand, if the probability is some tiny real number greater than zero, then the total of all the probabilities for individual balls adds up to infinity. — fishfry

Is this the same with infinitesimals?

Which is why I wrote
limn→∞f(1n)=0 — Michael

But that number is exactly zero. There is no such thing as 1/∞.

As I said, that was me being lazy. I didn't want to figure out how to do MathJax.

Is this the same with infinitesimals? — Michael

There are no infinitesimals in the real number system. Using nonstandard analysis, in which there are infinitesimals, doesn't help your argument.

↪fishfry As I said, that was me being lazy. I didn't want to figure out how to do MathJax. — Michael

Doesn't help your argument. The limit is exactly zero. But if the probability of one event in a countable event space is zero, your probabilities don't add up to 1. And if the probability is nonzero, it also doesn't add up to 1. There is no uniform probability distribution on a countable event space. Your notation wasn't lazy, it was mathematically incorrect no matter how you notate it.

ps -- To make this a little more clear, there is a probability distribution on the natural numbers, but it's not uniform. If you assign probability 1/2 to 1, 1/4 to 2, 1/8 to 3, and so forth, the sum of the probabilities is indeed 1 (by countable additivity) so this is a probability distribution. But it is not uniform.

My point is that applying bad math to multiverse theories makes skeptics out of people who know math but don't know physics. Tom's mention of ergodicity makes sense to me because if you assume the distribution of states to regions is ergodic, then there are indeed infinitely many earths. When people say things that make sense to me mathematically, I tend to believe their physics. That's all I'm saying.

I don't see how my argument is any different to your argument here:

In infinite probability spaces, probability zero events may still happen. Suppose you flip infinitely many coins and they come up in any sequence whatsoever: hthhthththththhthttthhthththt... say. A completely random sequence. What's the probability? Well, the prob that flip 1 is h is 1/2. The prob that flip 2 is t is 1/2. Etc. The prob of the first n flips being exactly what they are is 1/2^n, and that goes to zero as n goes to infinity. Every particular sequence has probability zero. Do you follow that point? All heads is just as likely as alternating heads and tails which is just as likely as the random sequence above. The probability is zero. Yet SOME probability zero sequence must occur. — fishfry

I don't see how my argument is any different to your argument here: — Michael

The space of coin flips is uncountable. So countable additivity still allows individual events to have probability zero.

When people are talking about an infinitely-sided die, I assume they mean a countable infinity, which has no uniform probability distribution.

But the talk of an infinitely-sided die is quite incorrect in this context anyway. The entire "duplicate earth" idea depends on their being a FINITE number of possible states in any bounded region of space. If the set of states is infinite, then there is no reason at all that any state should be duplicated.

The introduction of the infinitely-sided die is the moment this thread went completely off the rails. The entire foundation of the duplicate earth theory is that there are only finitely many states possible in a given region of space. That's essential to the argument. Drop that assumption and there is no argument at all.

Perhaps you were talking about infinite sequences of coin flips and not infinite-sided dice. I may have misunderstood you in the general confusion of the last few posts. Once someone mention an infinite-sided die to represent possible states, the entire thread went hopelessly off the rails.

The introduction of the infinitely-sided die is the moment this thread went completely off the rails. The entire foundation of the duplicate earth theory is that there are only finitely many states possible in a given region of space. That's essential to the argument. Drop that assumption and there is no argument at all. — fishfry

Even with finite states does it work? Given an ordinary coin and an infinite series, is it almost certain that there will be an infinite number of heads and an infinite number of tails?

Even with finite states does it work? Given an ordinary coin and an infinite series, is it almost certain that there will be an infinite number of heads and an infinite number of tails? — Michael

Yes. Needed to think about that a little. There's only one way there can be zero heads. There are countably many ways there can be one head. (Could be on flip 1, or flip 2, etc.). There are countably many ways there can be exactly two heads: Flips 1-2, 1-3, 1-4, ..., 2-3, 2-4, ..., 3-4, etc., and there are only countably many such combinations. (Countable times countable is countable). Continuing, there are only countably ways to get 3 heads, countably many ways to get 4 heads, etc. Adding up all those possibilities is still countable, since a countable union of countable sets is countable. Each sequence has probability zero, so by countable additivity the probability of finitely many heads is zero. Same analysis for tails.

That's why I said that the duplicate earth theory may fail, but only with probability zero (which can still happen). If you throw in the assumption of ergodicity, you are guaranteed a duplicate earth to the best of my understanding.

limn→∞f(1/n)=0 — Michael

The limit that you wrote doesn't come through very well in ordinary characters. I'm referring to the limit, as n goes to infinity, of some function of 1/n.

Well, if f(n) is the reciprocal function, then that limit certainly wouldn't be equal to zero.


Michael Ossipoff

Had a hard time picking out a consistent point being made in that long post, so I picked this little bit out.

It's difficult to believe that such beings would observe events on our planet without instituting the policing that would protect us from eachother. ...as in Clarke's Childhood's End. — Michael Ossipoff

This sort of makes the assumption that we're worth saving. How can a species that has the collective maturity of an ebola outbreak be the thing they want to save? If there's a test, we certainly have yet to pass it.

Had a hard time picking out a consistent point being made in that long post — noAxioms

If I contradicted myself, or was in some way inconsistent, then feel free to specify a particular instance.

t, so I picked this little bit out:

It's difficult to believe that such beings would observe events on our planet without instituting the policing that would protect us from eachother. ...as in Clarke's Childhood's End

. — Michael Ossipoff

This sort of makes the assumption that we're worth saving. How can a species that has the collective maturity of an ebola outbreak be the thing they want to save? If there's a test, we certainly have yet to pass it.

It isn't a question of saving a society. It's a matter of protecting some individuals from other individuals.

Some individuals are relatively innocent and deserving of protection and a chance to live

Michael Ossipoff