Level III Multiverse again.
The Level III Multiverse (Tegmark) gets a lot of stick, despite the fact that it adds zero complexity to our conception of Reality according to known physics.
It is interesting to note some prominent physicists who admit that the multiverse is real, according to this 20 year old TV programme.
We have the usual suspects explaining why the Level 3 Multiverse is real, but what is really interesting is Penrose claiming that the Mutiverse is real according to quantum mechanics, therefore physics is wrong! This is truly bizarre!
Right at the end, we have Hawking declaring the Multiverse is real, plus declaring it was Feynman's idea!
I think it is time to take physics seriously.
It is interesting to note some prominent physicists who admit that the multiverse is real, according to this 20 year old TV programme.
We have the usual suspects explaining why the Level 3 Multiverse is real, but what is really interesting is Penrose claiming that the Mutiverse is real according to quantum mechanics, therefore physics is wrong! This is truly bizarre!
Right at the end, we have Hawking declaring the Multiverse is real, plus declaring it was Feynman's idea!
I think it is time to take physics seriously.
Comments (235)
I think different people have different definitions of what 'is real'.
One can view the universe from within or from without and yield different answers to the same questions, and nobody is wrong or right.
So from within, only what we can sense or what possibly can effect us is real. It is not wrong to say there is no multiverse. From an objective view, there may be multiple worlds, but still one universe. These two different ways of looking at it are completely valid for not just Type 3, but also types 1 and 2.
So it seems there is no multiverse, but only because that is a poor choice of words. Everett proposed one universe with different solutions to measurements, but no 'collapse' that makes one of those solutions the actual one. That's multi-world perhaps, but not multiverse.
Quoting tom
On the contrary, it seems that in the past few years it's becoming time to stop taking physics seriously. Theories that can not be experimentally verified or refuted are not science.
It seems a stretch to say the many worlds interpretation doesn't add an unnecessary amount of complication to our metaphysics.
I still prefer to hope for a physics that fixes the MWI's unbounded world-line branching by incorporating the actuality of a generalised wavefunction collapse.
But with decoherence maths - QM+statistical mechanics - we get a workable collapse anyway. So there's no burning need to embrace the extravagant ontology of MWI. Decoherence delivers a quasi-collapse that makes the theory fit with the world we observe quite adequately while a "proper" theory is still under development.
So while you claim that MWI adds zero complexity to the maths, you have to admit that the same equations produce too many answers. It is only natural to expect the maths therefore misses some constraint on its unbound fecundity. And now just such a thermodynamic limit has been tacked on with decoherence.
So MWI has become more realistic, more in accord with the world as we know it classically. Yet the very fact that QM+limits is an improvement should be evidence that multiverse thinking was always wrong-headed.
MWI has already begun the job of denying itself, even though many people think it is the "decoherence interpretation". We need something extra by way of a world-constraining mechanism so as to actually reduce the complexity of the QM maths.
How does the second sentence follow from the first? Do the universes share the same history? Why should they do that?
The same objection that Apo applies to Type 3 is relevant to type 1. Those distant hubble-spheres are just other (sometimes identical) solutions to the same equations. In both cases, there is no separate ontology to it. There is only the one universe, and not a multiverse of separately existing noninteracting things.
That would not seem to follow, since no distance is infinite.
First, why is the space infinite? Your premise is that the universe is finite but sufficiently large that there are regions inaccessible to each other due to light not having had enough time to get from one place to the other. That doesn't require an infinite universe.
Secondly, even if the universe is infinite AND the possible states in a given region of space are finite, you STILL are not guaranteed a duplicate earth. Say there are two states, and infinitely many universes:
0, 1, 1, 1, 1, 1, 1, ...
So SOME state occurs infinitely often. But the 0 state is never duplicated. In other words if there are infinitely many universes, then SOME rock or bacteria or planet is duplicated for sure. But you personally, or the earth, are not necessarily duplicated.
Again though, you're betraying your original premise. You started with TWO regions of the universe that are causally and informationally isolated from one another. That's two, not infinitely many.
Not guaranteed, no. A coin may flip tails forever. It's just a probability after enough distance.
Edit: I think that mathematically, a coin cannot come up tails forever. There cannot not be a dup Earth given infinite space. The probability of of that is 0.000... which is zero.
Not sure I follow this. If there is a duplicate, there would seem to be an infinite number of them. There is simply a probability as to how far away (measured in non-linear units) the nearest one is. There are infinite type-1 worlds given infinite universe size. In fact, each point in space is centered on such a world, so you are in a different one than I am. Light can in theory reach (immortal) you from your slightly-left-shifted world but never reach me in mine. That means that by the time the light gets to you, the expansion of the universe separates us so far that we're forever isolated from each other.
I think you misunderstood; the point is that there is no actual infinite distance. Even if you traveled away from Earth, for example, forever you would never reach an infinite distance from earth.
On the contrary, I think it is time to acknowledge that physicalism as a natural philosophy has utterly failed, and is now leading us up 10[sup]500[/sup] garden paths.
A 3-manifold, e.g. a 3-sphere, is finite with no edge.
I didn't watch it either, I just fast-forwarded to the parts in which the physicists were talking. David Deutsch is particularly worth listening to.
As an aside, there is a video where DD is being interviewed by Dutch TV in the same scruffy front room of his house. They ask to see his office, so he takes them there. They mention that his office is very spartan, it is empty. He replies that he had never been there before. Hilarious!
Quoting fishfry
Quantum mechanics has been around since 1925, and if you count Einstein's paper on the photo-electric effect as the start (which it was) then since 1905. I have no idea what you mean by "the last few years", but over the last 100 years QM has been the most tested theory ever conceived, and has passed every test. More than that, it has also revealed several astonishing and novel features of reality - interference of matter particles, superposition, entanglement, ...
And by the way, the existence of the other Worlds is experimentally verified.
Quoting noAxioms
It is psychologically interesting that people generally accept the existence of the Type 1 multiverse, when there is absolutely no evidence for it. I'm not sure if evidence for it is even possible?
Accepting the Type 1 multiverse, which is a consequence of an infinite universe subject to the Bekenstein bound, then the Type 2 adds zero complexity, but many people recoil at the idea, even Penrose!
The argument for the existence of the Type 1 multiverse requires an infinite, ergodic, and expanding universe.
True, but I don't understand how this is relevant to what noAxiom was saying.
For those interested, the argument that, as a generic consequence of inflationary cosmology, there almost certainly exist exact duplicates of Earth (among other interesting things) is given here: Many worlds in one, J. Garriga, A. Vilenkin, Phys.Rev. D64 (2001). (This is still within the parameters of "level-I multiverse.")
Just that it would seem there could be no actual infinite number of planets, and hence the argument that a finite number of possible planets would lead to a duplicate earth would seem to fail.
Thanks for the link anyway; I'll take a look.
How does it seem that there can be no infinite number of planets?
Because there is no infinitely large number, just as there can be no infinite distance; the idea is untintelligble.
I see. You are proscribing reality based on your intellectual abilities. Not a good idea!
Correct. Even if the universe is finite there might well be type-1 multiverses, which are simply regions of space too far apart for light to have gotten from one end to the other since the beginning. Which is good, because contemporary physics holds that the universe is finite. There's no sensible theory of an infinite universe, since then set theory would become an experimental science. And there are no grant applications from physics postdocs proposing to investigate the Continuum hypothesis and the axiom of choice. That's because infinity is a mathematical concept and not a physical one. This could change tomorrow, but by contemporary physics, the universe is finite.
Quoting noAxioms
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.
Quoting noAxioms
In your first paragraph you agreed that level-1 multiverses only need a finite universe. And now you are claiming an infinite universe. I ask you (again), what is your evidence for such a claim? Contemporary physics has no such theory other than pure speculation.
There is simply
Quoting noAxioms
Yes but the universe is finite as far as we know. So you're saying, "Well, if Santa Claus is real, he gets to every house in finite time on Christmas eve." That's true, simply by virtue of the premise being false. You say that such and so happens in an infinite universe, but your premise is contradicted by known physics.
Quoting noAxioms
So now there are immortal beings in an infinite universe? This is not the science fiction forum. What is the evidence for your premises that the universe is infinite AND that it contains immortal beings?
You're just making this up.
Quoting tomIsn't just a curvature measurement enough? If flat enough, there are places sufficiently separated to never interact. Yes, expansion is required for that, but not infinite space.
Well then I'm glad I didn't watch it! It's bad form to post a vid you didn't watch IMO. But now that you pointed us to the good parts I'll take a look when I get a chance.
Quoting tom
Multiverse theory, string theory, speculative theories that have zero experimental basis and that can't be experimentally verified even in theory.
Quoting fishfryNo, disagree with this. A finite sequence has a nonzero probability. An infinite one is not a specific one, and has probability zero and does not happen.
Every particular sequence has half the probability of the flip sequence of one-less flip. That is not zero. Some probability-zero sequence is not a particular sequence and thus does not in fact occur.
Well yeah, that's my point. You claimed there's a dup earth, I asked you to support your claim, and you have apparently backtracked. Yes?
Quoting noAxioms
You said all heads has probability zero hence can't happen. I pointed out that ANY specific infinite sequence has probability zero. Are you unclear on this point? By coding H as 1 and T as 0 and putting a binary point in front of the sequence, an infinite sequence of coin flips represents some real number in the unit interval. If you randomly pick one, the probability that you picked it was zero. But SOME number gets picked.
I can see now that an infinitely large number of planets is not needed for the argument, so thanks for correcting me on that.
I remained unconvinced, though, that an infinite number (can there be more than one?) could be specified; because it would seem that any specifiable number must be finite. This is not to deny that an unspecifiable number might be useful for mathematical operations. In any case I see no reason to believe there are infinitely many planets; but admittedly I am no expert on cosmology.
Appreciate that. Just trying to carve out some clarity in a fuzzy discussion.
Quoting noAxioms
Is 1/3 = .3333.... = 3/10 + 3/100 + 3/1000 + ... some number?
There's no difference in principle between decimal and binary notation. And binary notation is just a sequence of H/T choices. A real number between 0 and 1 is an infinite sequence of head/tail choices. If you are picking them randomly then the chance of any particular one is zero. But they all exist. That's the point. In infinite probability spaces, probability zero events may occur.
If you can express it in finite language, it is not a typical number, it is a special one.
You're entirely correct about that, but now you are understanding and agreeing with my point. If you throw a dart at the real line, you'll hit a noncomputable number -- a number whose binary or decimal expansion is random -- with probability 1. And you'll hit some computable number with probability zero. But there are plenty of computable numbers.
You take a stab at a number line with a pointer and you will hit a 'typical number' as I call it. That number cannot be expressed with any amount of digits. It has zero probability of being hit, and yet it was hit.
Is that more what you're after?
Edit: I wrote that before reading your last post. Creepy...
No, I would say it is impossible to hit a computable (or expressible) number, except to say 'this one'.
That's exactly right. The set of computable numbers is said to have "measure zero." https://en.wikipedia.org/wiki/Null_set
Quoting noAxioms
It's not impossible. The computable numbers exist. You might hit one. There are infinitely many of them, after all. In infinite probability spaces, probability zero events may occur; and probability 1 events are not absolutely certain.
Getting back to our original point, this is why even if there are infinitely many universes, SOME state must recur infinitely many times, but not necessarily any particular one. SOME person might have a twin, but probably not you or me.
So at best, the "in an infinite universe there must be two earths" is false in an absolute sense, and is at best a probabilistic argument. Which I think we already agreed on.
Quoting fishfryI would suggest 'possibly not'. If someone has infinite copies, I probably do as well. But I must back off the 'definitely' stance.
Yes
By typical you mean random, or noncomputable. Alan Turing described computable numbers as numbers whose decimal (or binary, same concept) expression can be generated by a program. So numbers like 1/3 are computable. Pi is computable since there are many closed-form expressions that could be programmed into a computer.
The noncomputable, or random reals are the ones whose decimal digits can't be expressed or computed by any program. If you throw a dart at the real number line, the chance that you hit a noncomputable number is 1, and the chance that you hit a computable number is 0. But all the speciic numbers we use in math or science are computable.
I do agree with you that if there are infinitely many universes and only finitely many possible state of matter, it would be very unlikely for only one state to occur infinitely many times. But then again, perhaps there are laws of nature that make some configurations of matter more likely than others, so we don't actually know the true probabilities.
ps -- There's another wrinkle to this. If someone believes that the universe is a computation -- this is the "computable universe hypothesis" -- then the noncomputable numbers don't even exist. Only computable numbers exist. I have a hard time with that because then the real line would be full of holes. But some speculative thinkers believe it.
The quantum multiverse has been known about since 1950s. Schrödinger mentioned it in lectures, several other physicists discovered it independently, most famously Everett in his PhD thesis. The video shows that Hawking accepts it, Feynman discovered it independently, and that Penrose admits it is an inescapable consequence of quantum mechanics.
Many Worlds is at least 60 years old, and has not only passed every test, but has led to the discovery of decoherence and the quantum computer. Every quantum interference experiment is a test of Many Worlds, as are interaction free measurements and many other technological examples.
Perhaps I stand corrected. Are we talking about interpretations of QM? Or actually proven multiverses? My understanding is that Many Worlds is an interpretation, not an experimentally verified fact. But if Feynman and Penrose say so, well ok. I don't know.
Many Worlds is the only known interpretation of quantum mechanics.
Bohmian mechanics - adds hidden variables to QM, thus a different theory. Doesn't work.
GRW - ads stochastic collapse to QM, thus a different theory. Doesn't work.
Transactional - I really don't care.
Copenhagen is the tricky one, that has caused all the confusion. It suffers from a variety of problems, such as the measurement problem, and led many famous physicists to believe consciousness is responsible for wavefunction collapse. It is largely responsible for Deepak Chopra. It is a different theory from MW because it claims reality does not exist (MW claims it does) and it has the Born Rule as an axiom (MW does not).
I can only go by the Wiki article. https://en.wikipedia.org/wiki/Many-worlds_interpretation
They say it's an interpretation. You say it's a done deal. I'm not convinced. Where are all these universes supposed to live? But really that's not the point. The point is that Wiki says MW is an intepretation and you say it's the only known interpretation. I"m in no position to evaluate your claim versus Wiki's except that in this case I tend to believe Wiki.
Why are hidden variables disallowed? I think you're right actually in that it was proved somewhere that there can be no hidden variables, but its supporters obviously don't think the proof carries weight.
These are new ones to me, but again, why does this addition disqualify them?
Copenhagen is not mentioned, and it carries a lot of voting support still.
My personal aversion to most of the other interpretation is non-locality. Bohmian doesn't necessarily have it, but the others do. The ability to alter the past seems a nastier pill to swallow than the (mostly religious) implications of what MWI does to one's biased ideas of personal identity.
Non-locality doesn't necessarily have to violate causality. From what I've read, FTL effects can be accepted without retrocausation provided one give up the principle of relativity and adopt a preferred frame of reference. It can be a price to pay in it's own right for some, since it sort of takes the relativity out of relativity theory but if it takes away the problem of killing your own grandfather then it all comes down to what one thinks is preferable. As someone once said, “Relativity, causality, and FTL: pick any two”.
I sort of wonder whether this is the reason why the Bohmian interpretation is said to require a preferred frame of reference. In a sense, it does seem like a very 19th century theory. Non-locality was never a problem until Einstein showed up (Newtonian gravity I think was non-local), and a deterministic, realistic, and non-relativistic theory sounds like something that would be popular then (though maybe not today).
Interference experiments are a test of Many Worlds. Interaction free measurements test Many Worlds, as does the quantum computer.
Quoting noAxioms
Bohmian mechanics is Many Worlds in a state of chronic denial. A valiant attempt at realism, but by making ad-hoc modifications to a theory, it goes against the scientific method. Non-relativistic QM is not the last word, it's just a handy regime in which to discuss ideas. Bohmian mechanics has not successfully been made relativistic. And let's not forget that both local and non-local hidden variables are ruled out experimentally. Still, there may be loopholes, so the task of looking for them continues.
Quoting noAxioms
With MW you get a standard scientific theory. It explains what is happening in terms of elements of reality that are real and interact locally. There is no measurement problem, and no spooky action at a distance. What is weirder, a reality that consists of many parallel worlds that we can detect by their interactions, and cooperate with in our technologies, or a reality that is not really there and that we are not allowed to question?
I
Unitary Quantum Mechanics is a local theory. It is only when you ad-hoc modify it or burden it with metaphysical baggage like collapse, hidden variables, and unreality that are you forced to appeal to acausal interactions from beyond spacetime.
Copenhagen preserves locality by giving acausal spooky actions at a distance a different name. It calls them "correlations", as if that changes anything.
What are "acausal interactions from beyond spacetime" supposed to mean?
That would be non-locality a.k.a. spooky action at a distance.
Some of them do.
...because the level-1 multiverse notion assumes that this universe is infinite. In an infinite amount of space, with an infinite number of solar-systems and planets, there inevitably, somewhere, will be an identical copy of Earth, with, of course, a copy of you. ...an infinite number of exact Earth copies, in fact.
Tegmark gives a rough estimate of how far away the nearest one is likely to be, and of course it's a very great distance.
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.
Could there not be any other civilizations in this universe, if the universe is infinite? 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.
Though the matter isn't established, there seems to be agreement among various qualified sources, that the evidence is piling-up in favor of this universe being infinite.
Michael Ossipoff
NoAxioms and I just had a lengthy conversation disproving this very point. Could you please review those posts? What you say is simply not true. At best you have a probabilistic argument that falls short of certainty.
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.
It means there will never be enough time. It requires the expansion to be accelerating since steady expansion still allows light to travel from anywhere to anywhere else eventually.
Quoting Michael OssipoffHow do you get this? It seems, especially in the context of this thread, to follow from an assumption that other civilizations must be essentially identical to us to be, well, civilizations.
You just got finished saying there is an exact copy of us out there, given infinite space.
This statement is quite a break from the usual stance I've seen from you. You gone all ID on us? Tegmark for instance described a universe not in need of creation, not designed, nor one where we are special.
We need a thread on the Fermi paradox. Much more relevant to this sort of discussion.
Note that fishfry's source for this claim is, apparently, the bizarre little argument that follows it, and not what scientists actually write, which would be something like the following from a NASA site:
Quoting NASA
Actually, when they say that the near-flatness of the universe "suggests that the Universe is infinite in extent" - that's an oversimplification. A locally flat(tish) universe is compatible with some finite topologies, such as a 3-torus. These guys did some serious number-crunching with WMAP data and came to the conclusion that out of several likely topologies that they considered, a certain compact (finite) topology provided a better fit than the infinite flat topology.
Be that as it may, you would be hard-pressed to find many cosmologists who dismiss the possibility of an infinite universe on such preposterous a priori grounds.
I never dismissed the possibility. I pointed out -- correctly -- that current theory says that the universe is finite. Your own examples support this.
I have alluded to some more exotic mathematics in which an infinite number can actually be specified (and yes, more than one) - that goes all the way to Cantor. But that's really beside the point. All these fanciful cosmologies that are bandied around here are backed by nothing more exotic than standard calculus. And while calculus used be called "infinitesimal calculus," it has long since eschewed any explicit references to the infinitely small or the infinitely large. When we say things like "an infinite number," that's just a shorthand that, in a more rigorous formulation, unpacks into the kind of weaselly formulation that I gave you above, where I offer you to play a game: give me any number and I can give you an even bigger (but still finite) number. It turns out that that's all we need to get going.
False.
Consider the Peano axioms. Given a number n, there's a greater number called the successor of n.
That gives us a sequence 0, 1, 2, 3, ...
However, that is NOT ENOUGH to get calculus off the ground. To do that, you need a completed set of numbers {0, 1, 2, 3, ...} which is given by the axiom of infinity.
In other words Peano + Infinity = Calculus.
Without the axiom of infinity, each number has a successor but there is no set of all the numbers; no infinite set; and no calculus.
To say this another way: ZFC is stronger than Peano. Important to keep this in mind when slinging around ideas like infinite numbers of planets.
Sure, you are right. Though "the axiom of infinity" is just a name for an axiom that posits the existence of a set with certain properties. But those properties do not include being "infinite" in some basic sense. But thanks for the qualification.
This does not change what I was saying though: like it or not, most of modern physics does use calculus. You can claim that most of modern physics is misguided (for what that would be worth), but you cannot deny the facts.
Quoting fishfry
I take back the bit about not denying facts! And this is why I don't usually engage this person.
Stating the totally obvious.
Quoting SophistiCat
I have never said any such thing.
Quoting SophistiCat
If only.
The contemporary view would seem to be more subtle than simply that the Comos is "finitely infinite" like the surface of a sphere.
The spacetime sphere is also a material thing. It both expands and cools in reciprocal measure. So in some sense, the surface is thinning as fast as it is growing. The maths has to represent that fact. And so the area of the sphere is now measured entropically in terms of an event horizon with some ultimate number of bits.
Making it really tricky, to actually arrive at some future finite Heat Death or entropy limit requires the further thing of a cosmological constant or dark energy. The geometry of spacetime can't be either perfectly flat and infinite, nor finite by being a closed hypersphere, but must in fact be made finite by being faintly hyperbolic and open in its curvature.
So contemporary physics knows from observation - unless dark energy can be explained away as observational artefact - that the finitude is looking secondary to an infinitude. Something is faintly accelerating the Universe so that it is generally flexed hyperbolically - it bends away from flat infinity in the other direction. And that is what is actually necessary to fix a future date on when the material contents of an expanding/cooling surface will come to a halt at a fixed temperature.
So finitude is the long-term fate. But for reasons still left open until we can account for the "force" creating the faintly open hyperbolic curvature.
This all has relevance to MWI multiverse hype. The big problem - if you believe in the reality of principles like the conservation of energy, or causal closure, at all - is that MWI violates energy conservation in the most fundamental fashion. That is at the guts of an instinctive objection.
Now if you are not used to taking the materiality of the Universe seriously, then perhaps it is easy just to imagine the free creation of endless worlds, or endless world branches.
But contemporary physics is pretty concerned with an entropic view of reality. Even quantum theory has been "fixed" by welding on statistical mechanics to give us the new and improved decoherence formalism.
The irony is that MWIers latched onto that to peddle their "free lunch" multiverse. But fads come and go. Decoherence is a way to put a thermally coherent limit on a spacetime volume. It brings in the conserving machinery of event horizons.
MWIers still use sleight of tongue to claim infinite branching within the one collective "space". The cost of producing an unlimited number of observers with an unlimited number of points of view is not yet counted by even the expanded quantum formalism, so they can take refuge in that fact.
The maths still only puts numbers on the observables. There is no conservation rule limiting the multiplicity of individual observer. I can be split across as many alternative worlds as you think might be required at "no cost". And so a certain brand of metaphysical nonsense can be promoted as "quantum mechanics with nothing added".
Anyway, the general point is that we do tend to produce simple mathematical frameworks that are open, unbounded and therefore point towards infinity. Then physics comes along and starts to discover the constraints that in fact bound reality and give if some concrete, rather classical-looking, finitude.
The tension between the maths and the physics boils down to issue of how to handle materiality now. Maths is traditionally a view that is spatial. It simplifies by getting rid of time and change and energy within its conceptual metaphysics. But now time and change and energy need to be re-introduced to the mathematical modelling at a fundamental sort of level. They have to indeed emerge from mathematical considerations, not just get tacked on.
Hence string theory, loop quantum gravity, thermal time, anti-de sitter spaces, decoherent QM, holography and other important research projects in contemporary physics.
Multiverse speculation is just the modern equivalent of time travel or "consciousness" based interpretations of QM. A populist sideshow. Metaphysics-lite for the entertainment of the masses.
This is only a minor violation of the principle of relativity, but it galls me enough to discount the significant probability of such a finite topology.
You seem to have a very biased picture of what MWI is. All it states is that any closed system evolves according to Schrodinger's equation.
In math, the idea of a manifold is more abstract than that. Naively when we imagine the 2-torus like a big donut in space, we imagine the donut embedded in Euclidean three-space. We can use the equations of the ambient space to describe and manipulate points on the torus.
But the idea of a manifold is that the donut is there ... and the ambient Euclidean 3-space is not. The donut, or sphere, or whatever shape it is, exists on its own. There is no big rectangular grid in the sky.
In the mathematical discipline called differential geometry they note that for each point on the manifold, there some neighborhood that is diffeomorphic to Euclidean n-space for some n. Diffeomorphic means that there's a little neighborhood around each point that can be smoothly deformed into a Euclidean space.
So a manifold does not live in a Euclidean space. All we can say that at each point we can pretend it's pretty much Euclidean. Just like if you're standing in Kansas it seems like you are at the center of a big plane. [Kansas is a very flat state]. Each point on a sphere is locally stretchable into Euclidean 2-space.
Now if you have two points nearby each other, and each point can be deformed into a Euclidean space, what about any points that are in the overlap? Do they get stretched in a consistent manner? That's part of the basic machinery of differential geometry. Then they use these local mappings to do calculus on the surface of the manifold, without regard to any notion of an ambient space.
Einstein basically discovered differential geometry when he developed relativity, then Minkowski came along and recognized it as something mathematicians had already known about since the days of Riemann in the 1840's.
My understanding is that this is how physicists look at things. They don't need an ambient coordinate system. Which is fortunate, since Einstein noted that there is no preferred frame of reference.
I'm no expert but this is what I know about it.
I have no problem with decoherence as a formalism that describes the time evolution of the probability of sets of observables. As such, it "safely" sidesteps pretty much everything of ontological concern.
But folk want to know if it is really "me" who gets split in a way that "I" can't notice just to make this mathematical account work. So once MWIers start saying yes, we just have to accept any old weirdness the maths implies, then the interpretation bit comes into play.
Tom/Odo is pushing the justification that "simplicity" warrants us making this further interpretive leap. I point out how there is nothing simple about it at all. The simplicity is merely a fact of quantum theory doing things like presuming the existence of time, presuming the definiteness of initiating observables, presuming some entropy-less notion of observers.
Any formalism is going to be simple if you chuck out enough reality. :)
So I am all for decoherence - QM+statistical mechanics. It is the claim that MWI is simply "the maths sans interpretation" which is the sly stunt that I'm objecting to here.
That's pretty mysterious to me. Do you mean that the new universes don't come into existence? Are you saying that they're always there? I should emphasize that my knowledge in this area is limited to Wikipedia. I'm asking on a very simplistic and naive level. My understanding is that MWI says that these universes come into existence at each fork in the road. The cat is dead in one and alive in the other. and those two universes spring out of the one.
If that's not right, can you please explain it to me?
The three axes are orthogonal. So not arbitrary but fixed by this exact relation to each other. They don't need to be fixed in terms of some larger "space" any more than the curvature of 3-sphere surface needs to "flex" or "swell" within an embedding space.
Goedel and Mach explored these issues pretty thoroughly. If the Universe was tied to three constant axes as you seem to imagine, then it could also as a whole have a rotation. And that would be a little awkward.
This seems to counter what you're saying. There can be a continuous finite 2-space that does not exhibit this property? I can't think of how that would work.
Schrodinger's cat is the best example. The cat is both dead and alive, without the creation of a new cat, one of which doesn't survive. Still one cat in superposition. They've done this with a macroscopic object now. They have a box that holds something that can be seen without aid, and put that thing in superposition, not create a second thing, both of which are in the box.
That a manifold is a locally Euclidean object that exists on its own and does not live in an ambient Euclidean space? Most definitely.
That doesn't mean you can't have a torus that IS embedded in an ambient Euclidean space. I didn't say that tori can't live in 3-space. I said they don't HAVE to. The latter is accomplished mathematically via the concept of a manifold.
Apologies. I thought your mention of a torus was a typo. Didn't realise it had been introduced into the thread. If we lived in a 3-torus, we would be able to detect that global alignment as you say. We could get places quicker, with less energy, by going in one direction rather than another.
I meant to ask if there was no grid in the sky if the space was torrid manifold. I then gave the example of the asteroids game which is played on such a manifold, and there is very much a noticeable grid to it, despite the lack of lines painted through the space or the fact that the screen happens to line up with it.
No, there is no grid in the sky on a manifold. There is a distorted grid in the sky at each point. But each point has a different distorted grid, with some regularity conditions to make sure the areas of overlap are consistent.
Quoting noAxioms
Asteroids is a torus represented on a 2-D grid. Not an abstract manifold.
Quoting noAxioms
Oh I see your point. Yes, the torus is orientable. Some manifolds CAN'T have a consistent coordinate system drawn on them. Famous examples being the Mobius strip and the Klein bottle.
https://en.wikipedia.org/wiki/Orientability
But even in the case of a torus, whose points can be described by rotations on two circles, those are not absolute with respect to the ambient space. They move with the torus, if I'm understanding your point correctly.
Agreed. But any kind of topology could be screened off in that fashion with a sufficiently small hubble radius.
Anyway, I see now that SophistiCat rather confused things by talking about a toroid rather than spherical geometry. The simplest compact space that would make the point about the Universe being "finitely infinite" would be a sphere.
The story would then not change no matter how big your hubble factor. Or at least not until all that escaping light came back at you from the opposite direction. :)
A curved spherical space is not orientable. You'd see a copy of Earth in all directions, or at least you would if light speed was infinite or Earth was polite enough to hold still. With a flat 3-torrid manifold, the nearest Earths in the sky would be in 6 directions, hence the orientation suggestion.
I tried it with a 3-klein bottle and saw my backside reflection. Weird. And yes, no suggested way to look to do that. It is beyond my imagination to see the distorted picture of diverging ray tracing in such a space.
If there is an exact copy of Earth way out there, then there is an exact copy of me on it. Is that guy me? There are arguments to both sides of that question, so "depends" is all that can be said for now.
Quoting fishfry
Ahh...well maybe not quite :D
A review of that conversation, and a more recent post from NoAxioms, show that NoAxioms posted a conclusive demonstration that you're mistaken on that.
Yes,that's where I found NoAxioms' initial statement of his argument, which he repeated in a reply to your post that I'm replying to here.
I refer you to NoAxoms' argument. Read it carefully this time.
Obviously if our Big-Bang universe (BBU) is large enough to contain several Hubble-volumes, then it will contain several Hubble volumes.
You're calling that a level -1 multiverse. Define it as you want, but maybe we should let Tegmark define his terms.
In the article that read, his mentions of level-1 all referred to an infinite BBU.
Here are some quotes from Tegmark that might help answer your questions:
(Note that he also answers your question about support for finite vs infinite universe.)
Level I:
A generic prediction of cosmological infla-
tion is an infinite “ergodic” space, which contains
Hubble volumes realizing all initial conditions —
including an identical copy of you about 10
10
29
m
away.
3
I. LEVEL I: REGIONS BEYOND OUR COSMIC
HORIZON
Let us return to your distant twin. If space is infi-
nite and the distribution of matter is sufficiently uniform
on large scales, then even the most unlikely events must
take place somewhere. In particular, there are infinitely
many other inhabited planets, including not just one but
infinitely many with people with the same appearance,
name and memories as you. Indeed, there are infinitely
many other regions the size of our observable universe,
where every possible cosmic history is played out. This
is the Level I multiverse.
A. Evidence for Level I parallel universes
Although the implications may seem crazy and
counter-intuitive, this spatially infinite cosmological
model is in fact the simplest and most popular one on
the market today. It is part of the cosmological concor-
dance model, which agrees with all current observational
evidence and is used as the basis for most calculations
and simulations presented at cosmology conferences. In
contrast, alternatives such as a fractal universe, a closed
universe and a multiply connected universe have been se-
riously challenged by observations.
Space could be finite if it has a convex curva-
ture or an unusual topology (that is, interconnectedness).
A spherical, doughnut-shaped or pretzel-shaped universe
would have a limited volume and no edges. The cosmic
microwave background radiation allows sensitive tests of
such scenarios. So far, however, the evidence is against
them. Infinite models fit the data, and strong limits have
been placed on the alternatives (de Oliveira-Costa
et al.
2003; Cornish
et al.
2003). In addition, a spatially infi-
nite universe is a generic prediction of the cosmological
theory of inflation (Garriga & Vilenkin 2001b), so the
striking successes of inflation listed below therefore lend
further support to the idea that space is after all simple
and infinite just as we learned in school.
Another loophole is that space is infinite but matter is
confined to a finite region around us–the historically pop-
ular ”island universe” model. In a variant on this model,
matter thins out on large scales in a fractal pattern. In
both cases, almost all universes in the Level I multiverse
would be empty and dead. But recent observations of the
three-dimensional galaxy distribution and the microwave
background have shown that the arrangement of matter
gives way to dull uniformity on large scales, with no co-
herent structures larger than about 1024 meters. Assum-
ing that this pattern continues, space beyond our observ-
able universe teems with galaxies, stars and planets.
(quoted from Max Tegmark)
Michael Ossipoff
No. That was indeed necessary to explain how a Big Bang universe could be so remarkably thermally homogenous. And before that, just to resolve Olbers paradox.
Quoting noAxioms
I agree they are not other universes. And even if our known Big Bang universe with its light cone structure were spatially infinite, then I still think important constraints on the "modal realist" version of the multiverse will count.
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.
There just is no end to the madness once you let actual infinity run riot in your ontology.
MWI suffers this because it can't in fact define what causes a branch universe to form. It tries to confine the branching to stuff like simple spin-up/spin-down entanglements. But that is way too ad hoc as it stands. Every photon emission in history could just as well have landed up being absorbed anywhere in the future eternal visible universe. Try extracting the decohered thermal signal from 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.
So the Universe might have infinite space to play with, but a very finite amount of time. Now the madness of infinity means there will still be an infinite number of worlds where we are saying all this in Korean, etc, but infinitely less than there could have been because of a strict time constraint.
If nothing else, there will be infinitely more space between each of these supposed exact replicas. You would have to travel infinitely further to get that kind of big surprise.
Well perhaps not infinitely, literally. But near enough FAPP. :)
If there are two states and infinitely many universes they could be 0, 1, 1, 1, 1, 1, ...
If 0 is the "earth" state, there is no other earth. 1 is maybe Mars. So Mars exists infinitely many times but not earth. If there are a trillion states, same argument. SOME state recurs infinitely many times, but not necessarily any particular state. Maybe there's only one earth even though there are infinitely many copies of Mars. It's perfectly possible.
Yeah. But you get to pick these infinite sequences out of an infinite hat. So you would pick that exact sequence an infinite number of times.
Quoting fishfry
But now you are changing the rules of your own game. Instead of Earth = 0, not-Earth = 1, you are saying reality only has the two options of Earth or Mars. And for some reason, nothing else will get pulled out of the hat.
Furthermore, 0, 1, 1, 1, 1, 1, ... has the same probability as 1, 0, 0, 0, 0, ... . So those exact sequences are equally improbable. The typical sequence will be more like 1, 0, 0, 1, 1, 0, 1, ... And what interpretation are you assigning to that in terms of physical outcomes?
Quoting fishfry
Your maths doesn't give that result as I've argued. You are trying to hardwire in the restriction that Earth or Mars is the binary choice that reality is having to toss a coin on. But this is about a coin with an infinity of faces - one for every possible state of the world. And it gets toss an infinity of times, so lands on every one of those faces an infinite number of times.
The multiverse is pure madness in other words. And maybe that ought to give pause to any Cantorians round about these parts. Maybe the maths version of infinity is not that robust either? Heh, heh.
That doesn't make any sense. You're completely misunderstanding the discussion.
The claim (which I disagree with, but accept only for sake of argument in this instance) is that the universe is infinite. So there are infinitely many finite-sized regions of space. Call them R1, R2, R3, and so forth.
Under this circumstance, we assume that each region can only be in one of finitely many possible configurations.
Under that assumption the claim (which I am showing is incorrect) is that there must be two earths.
Now for simplicity I am saying, let's suppose there are only two states, 0 and 1. We might have that R1 is in state 0, and all the other states are in state 1. In that case, state 0 never gets repeated, contradicting the claim that every state must be repeated.
Now in the more realistic case there are zillions of possible state. Still finite, but very large. But then it's still the case that R1 might be in state 0, and every other region is in some other state. It's still the case that it is possible that some state never repeats.
Quoting apokrisis
But this completely undermines the argument. If there are infinitely many possible states then it is possible that R1 is in state 1, R2 is in state 2, etc., so that NO state ever repeats. It's the assumption that there are only finitely many states that makes the idea even plausible that there are possibly two earths. In the infinite state case the argument totally falls apart.
Quoting apokrisis
If there are zillions of states, it's the same argument. The two state example is only a simplified illustration.
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." This does not violate the principles of relativity, though it may violate some esthetic demand for perfect symmetry. (Of course, the principle of relativity is itself a kind of demand for symmetry.)
The passage you quoted was written by me. I read it very carefully and stand by it.
If there are only finitely many states S1 through Sn, and infinitely many regions R1, R2, ..., there is no reason why some state can't occur only once. That may be statistically unlikely, but it's possible. There is no reason there's necessarily a "duplicate earth."
If the universe is infinite, ergodic, expanding, and subject to the Bekenstein Bound, then it is certain that Hubble volumes exist which are identical to ours.
Quoting fishfry
How do you fit all possible Hubble Volumes instantiating all possible initial conditions into a finite universe? You obviously can't fit in an infinite number of copies of each, which is what you need for a multiverse.
If the universe is infinite and ergodic, it is impossible that state 0 will not repeat.
I'm not the one claiming the universe is infinite. I'm simply pointing out that in an infinite collection of regions, with each region taking on one of a finite number of possible states, there's no reason that any particular state must be necessarily be shared by two regions. It may be that region 1 is in state 1 and all other regions are in some other state. SOME state gets repeated infinitely many times but not necessarily any particular state.
It's called ergodicity. I have mentioned that ergodicity is a requirement for Type 1 multiverse several times.
According to our best theory of cosmology, the universe is infinite and ergodic - both are predictions of inflation.
So you are requiring the assumption that you are then claiming is true? Well by that logic you're certainly right. If you pre-load your desired conclusion into an assumption, your conclusion falls out at the end.
You may be clear in your mind that ergodicity is a requirement, but the other two people arguing for the "duplicate earth" idea certainly never mentioned it. Why do you think that is?
ps -- I had a look at the Wiki page for ergodicity and found no clue to support your argument. Can you be more specific as to how ergodicity rules out the possibility that some state recurs and some state doesn't? https://en.wikipedia.org/wiki/Ergodicity
Ergodicity is a PREDICTION of physics. If you don't like it, you'll need some new physics and some very good arguments and predictions.
It might not. It might land on one face an infinite number of times. It might have been Mars all the way down.
Any particular outcome (whether infinite Mars or infinite everything) is almost surely never going to be the case.
The odds against it are vast, yet finite. So granting an infinite array of such volumes, the configuration would repeat just by accident. Indeed it would occur an unlimited number of times.
Of course, a replica earth with a replica you and me would seem to require even more information to specify it. But even if we toss in another million orders of magnitude to allow for a more negentropic story - one that includes all the information discarded in the course of some evolutionary history - infinity will still ensure that all possible arrangements of finite regions with finite contents must repeat their configurations. It is just a consequence of the atomistic description of the set up.
I’m not a fan of this ergodic analysis. As I’ve indicated, I think it fails to take into account the negentropy of information discarded through interactions. It presumes every degree of freedom is independent. Yet in the real world, particles have dependencies. They interact. And that leads to non linearities that evolve in exponential time rather than polynomial time. In short, the complexity of an actual Hubble volume would grow at a rate that the simple statistical picture cannot even follow.
But if we do treat a Hubble volume as if it is just filled by an ideal gas, then ergodic simplicity applies. The multiverse argument carries if the ontology is atomistic.
Imagine the universe when it just was a gas of radiation, or even a dust of weakly interacting particles. You could go into deep space right now and sample cubic metres of vacuum. Each will average about a dozen hydrogen atoms. Even within our own visible universe, you would think you would get exact repeats - the same configuration of a cubic metre of vacuum.
So the multiverse argument is very straight forward in itself. The flaw would be in its assumptions. Like the willingness to discount interactions and the fact they may screw up any simple statistical extrapolations with their non-polynomial complexity.
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.
And if every planet in the universe could have been replica Mars, then we would be Mars. Venus and Saturn would be Mars. So at least we can rule your hypothesis out observationally.
It's exactly as likely as any other outcome: 1/?.
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.
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.
Quoting apokrisis
Also 1/?.
Quoting apokrisis
I think you almost surely won't.
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.
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).
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.
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.
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.
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.
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?
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
Why are you doubling down on this mathematical nonsense?
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.
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?
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.
...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:
You replied:
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:
You replied:
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:
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.
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
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]?
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
Which is why I wrote $$\lim_{n \to ?} f(\frac{1}{n}) = 0$$
Is this the same with infinitesimals?
But that number is exactly zero. There is no such thing as 1/?.
There are no infinitesimals in the real number system. Using nonstandard analysis, in which there are infinitesimals, doesn't help your argument.
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.
Quoting fishfry
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.
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?
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.
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
If I contradicted myself, or was in some way inconsistent, then feel free to specify a particular instance.
. — Michael Ossipoff
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
You do get your knickers in a twist with great rapidity. As you know, that was Michael's terminology. I went along with it for the sake of discussion.
But also, 1/infinity is the proper definition of the infinitesimal as far as I'm concerned. Now you will get on your high horse and object no doubt. But I went along with Michael's terminology largely because I also like that sly implication. It is another way of getting across that the probability ain't actually zero even if it is almost surely zero when it comes to an infinite spatial universe producing replica earths with replica people doing replica things.
Quoting fishfry
If you remember, it was you who introduced the confusion. I was trying to sort it out for you by pointing out that those would be the kind "coins" you would need to be flipping...
Quoting fishfry
This still reads as nonsense to me. Maybe you agree now as you seem to have discovered ergodicity and moved on to a notion of a universe chopped up into sufficiently large but finite regions - the ensemble of microstates picture that I also have been at pains to criticise.
As I attempted to make clear earlier, I'm ignorant of physics and so I have to take on faith a lot of what's written here. But now and then the discussion wanders into areas I'm familiar with; and more often than not what's said is nonsense. At those moments I do my best to straighten out the math.
As far as your snark, it's not necessary.
Quoting apokrisis
Yes and I corrected it because it's wrong and confuses the subtle probabilistic issues at stake.
Quoting apokrisis
You are entitled to your private definition. The actual definition of an infinitesimal is a quantity x such that 0 < x < 1/n for every natural number n. The gap between your instincts and how math actually works is significant. I understand in general that you do not believe in standard modern math. You made some remark about Cantor earlier along those lines. Philosophically it's perfectly valid that you have alternative ideas about math and don't accept parts of standard modern math. But physics is most definitely based on standard modern math; and to the extent that your outlook diverges from that, you are introducing confusion into the conversation.
Quoting apokrisis
Is that really the best you can do? I'm simply doing my best to explain the viewpoint of standard math, which is the math used in modern physics; and you are arguing from your Peircean viewpoint and claiming that you are entitled to make your own definitions for mathematical terms that already have perfectly clear standard definitions.
Quoting apokrisis
I'm not sure how that supports any point you might be trying to make. I did actually correct his usage first, and I left your first usage alone simply because he was first and you were simply going along. It wasn't till you mentioned it a second time. If that bothers you, I'll apologize if it makes you happy.
For the record I do understand that you have your own private notation and that you reject the standard notation of modern math, on which physics is based. Is that a fair assessment? Since you admit you have your own definition of an infinitesimal, and you assign meaning to the symbol 1/?. And that I was a terrible person for calling your private ideas, which you are perfectly well entitled to hold, nonsense. They're not nonsense. They're merely your personal ideas and notation, totally at odds with modern math.
Quoting apokrisis
Ok. You are trying to express the idea of an infinitesimal probability. We all have these intuitions of infinitesimals, as Leibniz did. I truly get that. But since we're doing physics, it's important to make sure we get the math right. Else I wouldn't bother to bring it up.
Quoting apokrisis
Yes I understand your philosophical point. But your math is wrong. And we're doing physics. If you're using math metaphorically you should say so up front.
Quoting apokrisis
This refers to the infinite die. No that's not true. You introduced it, and the moment you did I realized you didn't have any idea what the conversation is about. The duplicate earth argument depends crucially on there being only a finite set of possible states in any bounded region of space. That's fundamental to the argument. When you brought up an infinite state die, I knew you simply had wandered off into some conversation the rest of us aren't having.
Quoting apokrisis
But that's exactly wrong. You need a finite die with a very large number of faces. One for each admissible state of all the particles in some bounded region of space.
It's ironic that just when you are totally losing track of the thread, you think you're helping me. You've done that before and you were wrong then too.
Quoting apokrisis
I'm perfectly willing to stipulate that much of what I write reads as nonsense to you. You have the same effect on me. Your writing seems extremely learned yet you never make a lick of sense.
Why don't we agree not to interact? I was really surprised earlier that you directly replied to something I said. I don't think our interactions are productive. I find your snark annoying, especially since it generally shows up when your degree of wrongness is at a local maximum.
Quoting apokrisis
As it happens I've spent the afternoon chasing down ergodicity. I do know a little about it relative to the irrational rotations of a circle, which are ergodically dense in the circle. What I've learned today is that by the definition of ergodicity, any set that behaves badly must have measure zero. If that's correct, then my NO-duplicate earth possibility is still alive. Ergodicity is a statistical attribute that describes what happens almost surely. But not absolutely surely.
This is my preliminary understanding. I no longer think ergodicity absolutely guarantees that there is a duplicate earth. If someone knows better and can walk me through the argument, I'd be grateful.
Give snark and you get snark. Fair enough?
Quoting fishfry
Even in philosophy of maths, these are routine debates. And physics doesn't base itself on the "correctness" of maths. So you may feel confused by conversations that veer of the beaten textbook track, but there is no particular reason to think the secrets of reality are already written in those textbooks. The relationship between maths and physics is much more subtle than that.
So look, I am interested in what you have to say about standard views within mathematics. It is good that you can explain the structures of justification upon which certain mathematical positions are constructed. But if you just want to lecture on the correct methods of the academy - the stuff that will get you a pass in class - I don't really call that a conversation in the context of a philosophy forum where the thread is about the material reality of multiverses.
Quoting fishfry
Hardly. I'm quite happy to accept more than one way of doing anything. And happy also that the standard way will have proven itself to be the most pragmatic in finding the simplest path - the one that disposes of the most metaphysical baggage.
But when we are talking about how existence itself comes to be, as we are with multiverses, then that is when the axioms upon which a reductionist simplicity is founded come into metaphysical question. That is when we have to rewind and see what was being left out, or being assumed, when starting down that path.
Quoting fishfry
Err, no. It is important to get the physics right and then find the right maths to make those ideas precise enough to measure.
I'm not denying that physics and maths enjoy a remarkably fruitful relation. But I do deny your understanding of how science works. The maths does not lead, it follows. And the fact that the maths then works as the precise description of the scientific claims merely shows that the maths has been suitably fitted to the task in hand. The maths was convenient.
Quoting fishfry
So why did you introduce the blather about binary coin flips where you either got Earth or Mars in the course of some infinite sequence? That was the misstep I was addressing.
Of course I understood the ergodic argument that motivates the talk of repeating outcomes in a multiverse of infinite spatial extent. You were the one who had to go look it up.
And of course I understood the ridiculousness of the idea of a spherical die that could be said to "land on 1 of its infinite number of sides". It was your coin-flipping nonsense I was lampooning. Exactly which infinitesimal point are we suppose to say a sphere has landed?
Now you are talking about a finite sided die with enough degrees of freedom to represent every thermal microstate in a bounded space. As if this fixes either your blather about Earth/Mars coin-flipping sequences, or deals with my actual objections to the simplicities of the correlation-less ergodic view of nature.
Imagine your finite side die had dependencies between the faces so that all the Earth-like combos are clustered together on one neighbourhood of the die, not represented randomly across the die surface.
Perhaps now you can start to see the mathematical presumptions you are failing to take account of in attempting to construct some physical model of a "multiverse generator"? A fair die would scramble the microstates randomly. But a die more properly representing the material world, with its dependencies and entangled history, would already be built in a way that was pre-loaded. It would either fall somewhere in the neighbour where the many Earth-like outcomes were clustered - or mostly somewhere not anything like the Earth at all.
That should be blindingly obvious. Yet it's not because the idea of modelling probability spaces with correlations is not so "mainstream".
But I have zero expectation you will pick up on such a point and argue it through. You are way too intent on finding an ever higher horse from which to look down your nose at any "non-mathematician".
Quoting fishfry
I'm amused and entertained. I'm finding the dispute productive. What more could I wish for? I'm not complaining.
Quoting fishfry
Hmm. "Ergodic" in this context is really talking about Poincare reccurence -
https://en.wikipedia.org/wiki/Poincar%C3%A9_recurrence_theorem
In an ideal gas, the particles will recross their initial conditions in the long run.
But my counter has been that the universe is more like a box of gas with its lid off. The particles are all escaping as the Universe is expanding/cooling. The locations they might want to recross may well have moved over an event horizon forever.
And then even worse, the gas is far from ideal. All the particles have correlations or dependencies. At the least, gravity and other forces are in play. So a different maths is needed to make any multiverse extrapolations.
Think more of a chaotic attractor where the trajectories could go anywhere within the bounds of the phase space, yet also cluster ... because of the internal dependencies. The system is not random but emergently self-constrained - https://en.wikipedia.org/wiki/Attractor
Inflation generates all possible initial conditions. This means that all possible matter configurations exist in some Hubble Volume. The most likely initial conditions are those which are almost uniform with only slight variations in matter densities, which later become amplified by gravitational clustering. We inhabit a typical Hubble Volume.
The quantum mechanisms that generate the initial conditions do so effectively in the manner of an ergodic random field. What this means, is that given an ensemble of Hubble volumes, each with their own random initial conditions, then the probability distribution you get by sampling different Hubble Volumes is identical to the probability distribution you get if you sample parts of a single Hubble Volume.
It is precisely this property of ergodicity - a generic prediction of Inflation - that guarantees that everything that could have happened here, did in fact occur elsewhere, and that everything that did happen here has multiple copies.
Quoting fishfry
You also need the Bekenstein Bound and infinity.
It's actually John Wallis' definition from his Treatise on the Conic Sections, incidentally where the ? symbol was coined.
With the usual binomial model of fair coin tossing, the event of an infinite sequence of heads is one that 'almost surely' will not occur, which is not the same as saying it cannot occur.
We can be tempted to apply this to an infinite universe and say it will Almost Surely contain duplicate Earths. To do that we need to first assume and specify a probability distribution for the configuration of mass-energy across the universe. I imagine that can be done, but it's too long to get into in this post.
That math might not be right either (I didn't show it), but it seems to hold the probability of hitting a specific countable number in random sample of the uncountable reals.
I am basing a lot of my claims on another thread debating why 0.999... is 1, not just infinitesimally close to it. It was explained by someone who knows their stuff far better than I.
Quoting andrewkNot sure if a model of fair coin tossing applies. In an uncountable sample space, the one you actually hit cannot be represented by any number of coin tosses.
Hah. There certainly is an official position on this. But it is more about what has to be agreed to make the maths come out right than one based on force of metaphysical argument.
And I'm not complaining. Maths needs to secure its constructs. It needs to be axiomatic.
I'm just reminding that this is what happens and so maths isn't in a position to tell metaphysics "what is really going on" due to what if finds works. Maths can act as a powerful constraint on free metaphysical speculation, and also serve as a powerful inspiration to further inquiry. But it isn't how metaphysical truth is discovered.
So the infinite and the infinitesimal speak to the need to establish limits. Constructive actions like counting need to have constraints to bound them as well. It is in fact the same dialectical issue which is at the root of metaphysical reasoning. For something to be the change, something complementary must be made the bit that stands still.
The number line has to be both continuous yet discrete at the same time. It must be a line composed of points. So of course some fancy mathematical machinery must be added to negotiate what must be a tricky change-over going on somewhere. What connects the points? What permits an exact cut?
The official answer works. But it is also pervaded by a spirit of "OK guys, shut your eyes for a moment, don't ask any annoying questions, as we do this bit of tricky surgery".
Am still enjoying the concept. Perhaps a proof that no copy is possible then? The distant Earth might be outside our causal cone now, but it wasn't always. All matter (or whatever it was back then) was at the big bang and is part of the causal history of this planet. Since information is preserved, a perfect copy has it all. Thin ice there. Maybe the info was shipped out of reach, even if preserved.
Anyway, if that works, Earth (or just an apple say), contains the history of its entire causal cone in the past and cannot be recreated without that entire past, which is infinite stuff.
Tegmark assumes Earth can be represented with finite state, and computes the distance needed to get the probability of a repro up to about 1. If if the state is no finite number, then no copy.
I hate discreet anything. How slow can something go? If it moves one Planck length each hour, what does it do while it waits for the next one? A discreet universe would exhibit jaggies. Ewww....
You can see how this can be scaled up by adding more atoms and particles to the system: they each have some finite number of states, and so do their combinations, even allowing for interactions. The number of degrees of freedom rises dramatically as you expand outwards, but the principle remains the same.
I won't vouch that the math actually works out for macroscopic systems, i.e. that the number of possible states increases slower than the size of the system, but that is the argument that Vilenkin and some others make.
Our visible universe may well have been the size of a point at the Big-Bang, but the entire Level 1 Multiverse was not. The Multiverse was as infinite then as it is now.
Inflation guarantees the immediate creation of causally disconnected regions.
Quoting noAxioms
As I've mentioned several times, the Bekenstein bound severely limits the number of states available to any volume of space.
So we have:
Infinite universe+Inflation+all initial conditions+ergodicity+Bekenstein Bound = guaranteed copies of Hubble Volumes.
Quoting SophistiCatOne atom has no position, velocity, or other relations. But a group does, and each atom has innumerable additional states that make up its relationships with the others. Really innumerable??? Maybe not.
Quoting tomMaybe my model is incorrect, but this seems wrong. Since the level-1 spheres overlap, they're all points in the beginning, and all the same point at that, else they'd not overlap. I don't totally grasp eternal inflation theory, where perhaps the inflation stuff rips away as normal space forms in the bubble, but that is not a description of a point except the point where the bubble first began, not necessarily being the point that represents our hubble sphere.
Quoting tomLimits it given finite energy. If the initial infinite universe was actually a point, there is infinite energy/information there. But this actually kills my idea. Earth is a limited space with limited energy. The bound applies. Earth cannot be in a unique state that requires the history of the entire set of material that was once in its causal past. Tegmark was working on a bound such as this, and then just computed how much space was required to make it likely that a good majority of those (valid) states were realized.
Information must be preserved, but like other conserved things, it need not all be conserved here. Most of that information is shipped off elsewhere.
How much expansion is required to produce a literally infinite universe from a point in a mere 13.8 billion years?
Why do you think Hubble Volumes were ever in contact or overlap?
Quoting noAxioms
The initial infinite universe cannot have been finite. No physical process can take something finite and make it infinite in finite time.
The initial energy density of the universe was also finite.
There are also Level 2 multiverse earths.
The way I see it: If the geometry is such that the universe wraps (like the sphere of the balloon analogy), then there are finite Hubble-volumes. Assuming not, then if the expansion rate is increasing, there are infinite such volumes. If the rate is not increasing, light will eventually get from anywhere to anywhere else, and the universe is a single Hubble volume. At no point does "13.8 billions years" play into that.
Why do you think they don't? We are at the exact center of our Hubble volume. Isn't that amazing? From the perspective of a planet 10 BLY away to the left (all this is in comoving coordinates BTW), they are centered on a different volume that encompasses us way to the right. Their volume ends further to the right of us, but not a whole lot further. Some distant galaxy to our right can be seen from here but can never ever be seen by them. It is outside their Hubble Volume. Our volumes overlap else we couldn't see each other.
To say they're all nonoverlapping implies there are discreet chunks of disjoint space with one preferred point in each of them which is their center. My model doesn't look like that.
I would think so, yes. Level 4 as well.
Level 1 is just Schrodinger's box implemented with distance instead of technology.
Do you have any references on what the measure preserving transformation is? I mean, if we're speaking about ergodicity, it has to be the ergodicity of a measure preserving transformation. Another way of putting it is what is a 'step' in the 'trajectory of the universe' defined as? And how can it be established as ergodic?
Another thing - how can ergodicity be used to show not just that the long term probability of set visitation is nonzero, but that its arrival time is finite? There's a distinction in terms of finite Markov chains, having an infinite arrival or 'revisitation' time excludes a state from being ergodic (and thus the chain from being ergodic in terms of all states).
Do those combinations also include how they are arranged together in space? If space were continuous, then there would seem to be an infinite number of possible ways I can put them together. I can imagine two atoms being separated by 1 nm, 2 nm or any 1/n nm, for example. I am not sure if it's physically possible but I can imagine it in my head. Or does that not matter?
So, what rate of expansion do you think might be required to turn a subatomic spec into a literally infinite universe in 13.8 billion years? Have you done the calculation?
Quoting noAxioms
Hubble Volumes are finite. If the "universe wraps" i.e. it is a finite 3-sphere, I think all bets are off for Level 1 (and Level 2) multiverse statistics.
The rate of expansion may be static, increasing, or decreasing. As long as there is a +ve Hubble constant, there will be Hubble Volumes.
Quoting noAxioms
Sure, your Hubble volume and my Hubble volume might be slightly different in 14 billion years. In the mean time, there are an infinite number of Hubble volumes that were never in causal contact with ours.
A Hubble volume is not a type-1 universe. It is just the volume containing the matter whose distance from us is growing at sub-lightspeed. The Type-1 universe is bounded by the event horizon, beyond which things cannot ever have a causal effect here. It is something like 16BLY in radius at this time (comoving coordinates again).
About the Hubble constant, it is not a constant since the expansion rate is increasing. The constant is just what the value is now, and thus can only be measured to so much precision.
I looked at its description on space.com and found text more worthy of the New York Post:
km/sec per megaparsec is not a velocity, so not sure how this could be unimaginably fast. 70km/sec is not much more than the orbital speed of Mercury, and I think I can manage the imagination of it. Sorry. I was hoping for better from a site like that.
The example was about the nearby overlapping ones, not the countless more distant ones.
Maybe you should show your working? Given an initial 1m3 of space-time, what expansion rate is required to turn it into literally infinite volume in any finite time?
We can create a model in which the spacetime contains all times after time zero but does not contain time zero itself. The earlier the time (The smaller its time coordinate), the greater the universe density becomes, so that it increases without limit as t approaches zero. If the universe is spatially infinite now, it will be spatially infinite at all those times too, no matter how small t is.
When I say 'no reasonable model' I mean that a model that included time zero would have to be discontinuous in most significant respects at that time, in which case there's really no point in including time zero in the model since it would have no causal connection with the rest of the spacetime.
These problems do not arise with a spatially finite universe.
In practice, we don't need to worry about a time zero for either a spatially finite or a spatially infinite universe, because the General Theory of Relativity, which is used to do the backwards projection, loses validity as the scale becomes very small, and we have no theory to replace it. We can't use quantum mechanics because it ignores gravitational effects and in a very dense universe those cannot be ignored.
Not going to happen. Universe was never 1m3 it seems.
Suffice it to say that for the purposes of this thread, the declaration that space is infinite implies it was always infinite ever since it was space. The material/energy probably never fully interacted. There is stuff that never was part of our causal history, even going all the way back. No reason to worry about it anyway since the Earth contains finite state and thus has no need of that history to be identical to another Earth.
We do have history though. If an exact duplicate Boltzmann Earth suddenly popped into existence in orbit around a plausible star, I would be on it and that 'me' would notice damn fast that it wasn't the native Earth.
Not according to Inflation, the theory on which the Level 1 and Level 2 Multiverses are premised.
Quoting noAxioms
Precisely! No finite physical process can create something infinite out of something finite in finite time whether you choose to show your workings or not.
Edit: Just to point out that to travel between Level 2 multiverses, you would need to traverse the part of the infinite space-time that our universe emerged from. This would be extremely difficult as these regions of the Master Space-Time are still undergoing inflation. As if travelling between Level 1 multiverses were not difficult enough.
You are passing up a valuable learning opportunity! Go on, give it a try!
That's very interesting. How does it know to do that? In the early moments of the universe it's cranking out all these possible configurations, and it's only got one more left. How does it know that? What if it forgets to do one particular configuration? Can it go back and do it later? Can the universe continue to exist or does this one single imbalance make the universe unstable in some way?
Quoting tom
So my understanding is correct? I'm gratified. "Almost surely" is not the same as "surely." All these statistical arguments leave the possibility of exceptions, unlikely as they may be.
Now the Beckenstein bound I believe has something to do with relating the energy in a bounded region of space to its information carrying capacity. It takes a certain amount of energy to change the state of a system so that the collection of all possible state changes is finite and can be calculated. Is that about right?
So how do we get from there to duplicate earths?
You know I just don't believe this duplicate earth story. Say there's a universe or a multiverse and it's got every possible state represented infinitely many times ... except there is one state that just happens to only occur once, by incredible amazing luck ... one little blue watery planet with bad politics, third from the sun ... and it's the only one like it in the entire multiverse.
I just don't see why such a universe couldn't exist. It would still have all the required statistical properties. But if your universe is infinite, then "almost surely" is NOT the same as "surely" and individual exceptions to the statistical properties may exist. That's one of the limitations of extending probability theory to infinite spaces. You lose certainty.
But we're making progress if you agree that ergodicity by itself is not sufficient.
Your attributing an agency here, an ‘it’ that ‘does’ something. But the same ‘it’ is what you’re trying to explain, so there’s an issue of recursion involved.
Only speaking loosely. Ignorant of the physics. Asking if there's an explanation of why all states must occur. Is this all as in all? Or all as in statistically all, but possibly missing one or two with no harm done to the universe?
I don't understand why you think I'm trying to explain anything. Tom said that during inflation all states are created. I'm asking questions about that.
You know, "How does it know how to do that?" seems pretty clear to me. If I said, "How does a bowling ball know how to fall to the earth?" you wouldn't think I am talking about agency, but rather asking how gravity works. So how does the universe know how to generate all the states? That's a pretty common figure of speech I think.
Much of the attraction of the Multiverse is that it removes the requirement to explain particular initial conditions and physical constants because all are realised somewhere. If you wish to single out a single set of initial conditions for being forgotten or of being uniquely instantiated, then you are going to need a different physical theory than we already have. As I have mentioned a few times, it is a prediction of Inflation that all initial conditions are realised, and that their statistical distribution is ergodic.
Tegmark's view on how these initial conditions obtain is that the early universe, during Inflation (i.e. the inflaton field), is in a superposition of all its possible states. If this is the case (which it must be if physics is unitary) then the same fundamental physical process gives rise to the type 1 and 2 multiverses as type 3. This physical process is decoherence.
Quoting fishfry
So, you prefer "incredible amazing luck" theory to our best theory of the universe? Do you even aspire to rationality?
Quoting fishfry
Now you are straw-manning.
All possible states must occur because of:
1. Infinity, which I have mentioned several times previously. Space must be infinite.
2. Mass fluctuations approach zero on large scales.
3. The mass densities at any set of points has a multivatiate Gaussian probability distribution.
Conditions 2 and 3 may be combined into the weaker condition that the correlation functions of all orders vanish in the limit of infinite spatial separation.
Together the above conditions are called ERGODICITY
The above prediction of Inflation can be traced back to the state of the conjectured scalar field, called the Inflaton, that permeates all of the infinite master space-time that our, and all the other universes and multiverses inflate from. The harmonic oscillator equations that determine the ground state of the Inflaton give a Gaussian wave function.
4. Infinity. But I've already mentioned that.
5. The Bekenstein Bound - the number of possible states scales with surface area, not volume.
Thank you for clarifying. I don't feel that you're engaging with my point.
Your bullet items are probabilistic. And you are applying probability theory to an infinite event space.
It is a fact that when you extend finite probability theory to infinitary realms, you lose certainty. Your event space could be extremely well-behaved statistically, and still allow for many anomolies and exceptions. This is the difference between the phrases "surely" and "almost surely." These are technical terms in probability theory. Almost surely means that something happens everywhere except on a set of measure zero. That means that there can be many individual exceptions to the global statistical properties of the event space. And measure zero sets can be quite large and complicated.
I would certainly agree with the statement that given the premises of the discussion, there is almost surely a duplicate earth.
But that does not mean that it must necessarily happen. And this is the point you're not engaging with. The very infinitary probability theory that you're invoking allows for exceptional sets of measure zero.
You are applying without justification your intuitions and beliefs about finite probabilities to infinite ones. You are not giving sufficient consideration to measure zero sets and the loss of absolute certainty when we pass from finite to infinite probability theory.
This being a philosophy forum, it's worth noting that this is a point of epistemology. When you extend finite probability theory to infinite event spaces, you lose certainty. The theory requires the existence of measure zero sets; and sets of measure zero don't have any effect on the limiting behavior of the probability distributions. You have to take this into account when making metaphysical claims based on probabilistic arguments,
Are they probabilistic? Perhaps you could show what you mean by that, and which points comply with your meaning?
Quoting fishfry
Infinite realms? Loss of certainty?
Quoting fishfry
Intuitions, infinite probabilities?
Quoting fishfry
Metaphysical claims?
Will you inform the Nobel Prize Committee?
Is that related to this point?
DOES THE MULTIVERSE REALLY EXIST? (cover story). By: Ellis, George F. R. Scientific American. Aug2011, Vol. 305 Issue 2, p38-4.
Because, if that is the motivation, then what is behind it? That the suggestion of a 'fine-tuned Universe' is one which most scientists would rather avoid?
The Level 2 Multiverse solves the Fine-Tuning Problem.
Your mistake is in mathematics, not in physics, so if you want to invoke a committee, it would be for something like the Fields medal, not the Nobel prize.
Ask yourself why it’s a problem that needs solving.
Science is about formulating problems and seeking solutions for them in terms of explanations, i.e. accounts of what exists in reality, how it behaves, and why.
So sure, if I look at the constants of nature, a perfectly natural question arises as to why they take particular values. Wouldn't it be nice to find a theory that predicts them? And if I look a little deeper and discover the sensitivity of Reality to slight changes in some of them, the slightly deeper question arises about why the had to take these values for there to be anyone to be questioning them.
In the face of Fine-Tuning it is impossible to maintain the Principle of Mediocrity without the Multiverse.
Perhaps you would like to get in touch with Max Tegmark and give him the bad news?
I suggest you read up on the precise, important mathematical notion of 'almost surely' (also rendered from time to time as 'a.s', 'almost certain' or 'a.c')
Being unaware is not an argument.
I see your suggestion about contacting him as irrelevant, as was the reference to the Nobel committee.
When you have calmed down from your hissy fit, consider this:
An ergodic random field whose harmonic oscillator coefficients span {1,2,3,4,5,6} with a Gaussian distribution centred on 3.5. The field is in a state of superposition and decoheres for infinite time. What is the probability that the decoherence branch with initial condition "4" will be encountered in the multiverse?
Maybe not. Maybe science will never explain the various constants, the 'six numbers' that purportedly underlie everything. Maybe the attempt to explain at that level is forever beyond science, and produces only pseudo-science, 'multiverses' being an example.
Before I compose a detailed reply to each of the points you raised; can you please tell me which part of my post is giving you trouble? Do you not understand the distinction between almost surely and surely? Or do you understand it but think it doesn't apply in the present discussion? Or think that Tegmark doesn't think it applies? I just want to figure out what level of misunderstanding we're having on this point. There's no question that I'm accurately relaying the math of infinitary probabllity theory. So there must be some disconnect between that and the physics.
You mean maybe the six numbers are magical? And of course the usual slur. How predictable!
I look forward to your refutation of our best Cosological theory, right here, on this forum, by a true expert.
Not magical - simply the terminus of explanation.
The short version: the ‘many worlds theory’ is based on avoiding the philosophically unsavoury implications of the observer problem.
The multiverse - ditto for the unsavoury implications of the fine-tuning problem.
They’re both metaphysical issues which physicalists are trying to solve by infinite ad hoc additions to physical theory.
Of course I could never substantiate such an argument, so I won’t pursue it. Consider it a footnote.
I find them elegant solutions. The unsavory feeling you get seems to be a challenge to a religious view of what you are. Yes, I would find that unsavory, and cause for further investigation, not a terminus because it threatens my biases.
That's a pathetic response.
A non-pathetic response to the question I asked you would be one of the following:
* "I don't quite get this measure zero business, can you explain?" or
* I get measure zero ,but it doesn't apply in this case because _______"; or
* I get measure zero and Tegmark gets measure zero but Tegmark says it doesn't apply in this case because _____."
If you make a statistical argument on an infinite probability space and you don't take measure zero events into account, you have to say why they're not relevant in the particular case under discussion. If you can't formulate a coherent reply, you don't understand the ideas you're promoting.
On the contrary - the philosophical problem posed by the ‘observer problem’ and by the ‘fine-tuned Universe’ argument, is in part the motivation for the MWI and multiverse theories (see this quote). You see the same motivation again in respect of whether the Universe can be said to have a beginning: 'While many of us may be OK with the idea of the big bang simply starting everything, physicists, including Hawking, tend to shy away from cosmic genesis. “A point of creation would be a place where science broke down"(seeWhy Physicists Can't Avoid a Creation Event).
My view is 'religious' mainly because it questions materialism - not because I am in the business of defending creation myths. But in today's world, to doubt the dominant narrative is to be regarded as 'religious' - because the dominant narrative now functions as a religion, in the sense of being normative for how educated people are supposed to think.
I understand perfectly that religious creation myths are indeed myths. Whatever is empirically demonstrated by science is an indisputable fact; and I will never dispute the facts. I'm questioning conjecture which I claim is at least partially motivated by philosophical or metaphysical considerations. They influence what kinds of ideas will be considered, and what avoided, on the basis of pre-existing commitment - otherwise known as a bias.
This is really depressing. There are no measure zero events, as explained multiple times in this thread.
Still waiting for your demolition of Cosmic Inflation, you are the self proclaimed expert.
Please explain it to me. If there are no measure zero events, then NO distribution of states to universes is possible. Just like if you flip infinitely many coins. Whatever result comes up, that was a measure zero event.
It's only when you aggregate all the possible results as either "Some state only happens once" versus "All states happen infinitely many times," that the latter has probability 1. That's because you're aggregating uncountably many measure zero events, and comparing that to an aggregate of countably many measure zero events.
If you randomly assign one of finitely many possible events to each of infinitely many universes or regions of an infinite universe, the probability of ANY PARTICULAR result is zero. You can just multiply out the probabilities to see that.
I have not noticed such an explanation. But it's a long thread and I haven't read it all. Can you please point to one such explanation?
Do we have to go back to Kindergarten? If we do, I'm feeling like the much anticipated demolition of Cosmic Inflation has a somewhat uninspiring probability.
What strikes me as particularly strange, is that it seems you think it possible, in a Hubble Volume, to flip a coin infinitely many times, and presumably store the result? I am literally suppressing laughter.
I think maybe you should either accept the physics or try to understand it.
Can you provide a reference to the derivation?
If all you have is insults and you can't coherently explain your point, you've got nothing. You're the one claiming there are infinitely many universes and/or finite regions of an infinite space. My analysis applies.
You earlier claimed that it's been established in this thread that measure zero events can't occur. Two or three people including myself have asked you for the explanation. You have not provided one.
You can go look up the formal definition of ergodicity to learn that the smooth statistical behavior may fail to occur on a set of measure zero. This is a required characteristic of any probabilistic argument, since events on a set of measure zero have no effect on the overall probability.
I repeat. All possible initial states are instantiated over a Gaussian distribution due to the nature of the Inflaton.
There is no member of the initial superposition of all possible states of the Inflaton that is singled out for non-existence.
Physics is Unitary.
After finite time there are a countably infinite number of indistinguishable Hubble Volumes, which, as time progresses, may diverge.
There are measure zero sets in the Gaussian distribution (and in Gaussian Random fields) since their (finite state) distributions are continuous with respect to the Lebesgue measure, tho. :(
What? Are you joking? You need a derivation and a reference? You are giving me a headache.
Yes. If you're going to say a result is established in physics, and is obvious. It should come with either a reference to either the paper or popular science article that establishes it, or a description of the text which suggests it.*
It isn't at all obvious that @andrewk and @fishfry's comments regarding measure zero sets have been addressed, but I am taking it on faith that you know the derivation or where to find it. The alternative is you don't know the derivation or the behaviour, or a reference to either, and your incredulity at our questions is unfounded.
*if requested.
I don't understand how you can say this, yet claim you don't understand the idea of countably many coin flips. They're the same mathematical idea.
You have countably many regions, or universes, or coin flips. Each region or universe or flip is assigned one out of finitely many possible states. One out of a zillion in the case of physics, or one out of 2 in the case of coin flips, but the math is exactly the same either way.
You can't mock the idea of coin flips and then come back with the exact same idea in the guise of countably many universes. There's no mathematical difference between a 2-sided coin or a gazillion-sided coin. If the number of states is finite, then probability theory applies. In the large, it is "almost certain" that all states recur infinitely many times, but it is not absolutely certain. The case of coins or universes are exactly the same. It only depends on there being countably many coins or universes or regions, each taking up one out of at most finitely many states.
The coins are a simplification for the purpose of making the probability theory clear. But I can use a gazillion-sided die if you like, where the number of possible arrangements of particles is a gazillion. The math is absolutely the same.
What? It's a toy example! If you need a reference, you need your head examining. Seriously, I am feeling vicariously embarrassed and ashamed, and somewhat defiled.
Ok then.
Umm, did I ever claim incomprehension?
Quoting fishfry
Umm, no. There are uncountably many Hubble Volumes instantiating countably many initial states, at least that is what the theory says. What is a "zillion"?
Quoting fishfry
You make me laugh.
Quoting tom [My bolding in both quotes]
In one post you said there are countably many Hubble volumes, and in another post you said there were uncountably many. Can you clarify this?
We would all benefit if you would please lay out a clear version of your argument in one place for reference. Assumptions, argument, conclusions. So we can know we're all talking about the same thing.
I perfectly well admit that I'm ignorant of physics. If someone can explain to me why probability theory doesn't apply to this situation I'd be happy to learn.
My understanding of your argument is that at the moment of the creation of the multiverse every possible state gets instantiated.
As I understand it, the argument for that conclusion is probabilistic. It depends on the behavior of statistical distributions at the limit. You used that phrase as one of your bullet items earlier. You are making a probabilistic argument to demonstrate that it's almost surely the case that every state gets instantiated. And you don't understand that your argument allows for the possibility of a large set of exceptions.
I'm asking for a clear explanation for why the well-known rules of probability don't actually apply to the creation of the multiverse, when these very rules are employed to develop the theory. I'm not saying there's no such explanation. I'm asking what the explanation is. Then I'll learn something.
I've heard that Tegmark himself admits he's no mathematician. I wouldn't be surprised if he's just ignored this point and nobody cares much. It's not that big a deal. As I said earlier, what if the multiverse got started and all but one state were instantiated. It would still be ergodic. Can you tell me the exact reason you claim that can't happen? If it happened, would the universe collapse or would we all have three heads? My intuition doesn't see why it couldn't miss a state yet still satisfy all the relevant statistical properties.
Could you just read the quotes again, slowly, and out loud?
Or, try swapping the parts in bold:
There are a countable infinity of INDISTINGUISHABLE Hubble Volumes, which diverge.
There are uncountably many Hubble Volumes, instantiating countably many initial states.
Quoting fishfry
No, the physics is Unitary, all the way back to the Inflaton, and beyond.
You might explain for us hoi polloi how indistinguishable things can be counted, because we would have thought that distinguishing something is a prerequisite for counting it.
Hi Sophisticat. I skimmed that article you linked and was interested to note that Vilenkin makes statements like:
"there are an infinite number of O-regions with identical histories up to the present"
where I think what he means is "there is almost surely an infinite number of .....". That is, I think he over-simplified his statement, presumably because he wanted to make it more accessible to the non-physicist reader, since it is a non-technical article.
I note that in your post you included the crucial qualifier "almost certainly", although it does not occur in the paper. Interestingly, Tegmark also omits the qualifier (bottom of first column on page 4 of this article you linked) but, like Vilenkin, gives no explanation for the omission, and his article is also more pop science than academic.
Do you have a view on why they omitted the 'almost certain' qualifier from their articles?
I didn't say you could count them. You can't count them.
This is truly weird. You think you know better than Garriga and Vilenkin what they actually mean?
In Vilenkin's book, which he published 6yrs later, he repeats that paragraph verbatim. He also writes:
"Yes, dear reader, scores of your duplicates are now holding copies of this book. They live on planets exactly like our earth, with all its mountains, cities, trees and butterflies. The Earths revolve around perfect copies of our Sun, and each Sun belongs to a grand spiral galaxy - an exact copy of our Milky Way."
Also:
"It follows from the theory [of eternal inflation] that island universes are infinite and that the initial conditions at the big bang are set by random quantum processes during inflation. The existence of clones is thus an inevitable consequence of the theory."
Countable means you can assign a number to any of these volumes, and to do that they must be distinguished. If they can't be, they're not countable. So volumes like ours, we're (arbitrarily) 1 and the nearest copy is 2. It is distinguished by being nearest. Of course the copy of me over there considers his volume to be 1 and ours (of which he is unaware) to be 3 because there happens to be another one closer to him. We're both unaware of the actual locations of those copies, so identical state is maintained. It is an objective ordering. Anyway, they're identical in state, but still objectively distinguishable and thus objectively countable.
I think it is a mistake to say this then:
Quoting tomThey must be distinguishable but have at least identical state. If identical state, how can they diverge? You must consider the full set of worlds as the one state, else there is no 'current state' with which another volume can be identical. To do so presumes a QM interpretation like Copenhagen with real chance and action at a distance and a bunch of baggage that muddies the statement that the two volumes are actually identical.
Heck, with that definition, maybe every Hubble Volume is identical state, and yes, that would mean uncountable.
Sorry for this post. Just thinking out loud.
No it doesn't. You can't count your clones. Physics tells us that the cardinality of your clones is Aleph_0.
If you think it is possible to count your clones, I urge you to try.
Quoting noAxioms
I said the Hubble Volumes are INDISTINGUISHABLE not identical. I also mentioned somewhere that the Hubble Volumes would be INDISTINGUISHABLE after FINITE time.
The distinction is that the Volumes have the same history, not necessarily the same future.
If they have the same history, and if determinism is the case, then wouldn't they also have the same future?
Edit: I'm sure I recall you claiming to be a determinist. If not, then I suppose this question can be ignored.
Copenhagen is in theory nondeterministic, but involves faster than light action, and thus destroys the limits of the Hubble volume. It also leaves states in superposition if not measured from an arbitrary observation point. Pull back the camera so to speak and Copenhagen becomes as deterministic as any other interpretation.
I think you need to expand on what you mean by these terms since we seem to be talking past each other.
If you perform a quantum measurement - e.g. a measurement of z-spin of a particle prepared in x-spin-up configuration, and choose your spouse based on the result, in half your futures you are married to Mary, in the other half it's Jane. Same past different futures.
Determinism is dead. Long live Unitarity!
I just now see Michael's edit where he notes the same view shift.
In half your futures you are married to Jane. In the other half, you are married to Mary. Are you suggesting I have ever promoted polygamy?
Of course I am a determinist, it's just that the picture of how determinism applies in Reality has to take into account quantum mechanics. This is different from classical determinism.
I think all you need is a specification of an event, frame, and state. The frame is there for future state only, not present state. So:
Earth CoG at stroke of midnight Y2K, in the arbitrary frame of Earth CoG at that moment, in the state that is the direct causal history of me making this post 17 years later. That defines a temporal line of that point in space that goes out forever. A different Hubble Volume is identical to that one if the wave function of any point along that line is the same in both volumes.
Does that work? It totally leaves out the definition what is the 'current state' of our volume, and I think it might even be independent of QM interpretation.
I have recently concluded that whether or not multiverses are real is not a very interesting question since currently we have no way of observing the truth of it.
However, I do find that considering reality in 10 dimensions, the 7th and beyond relating to multiverses, is a useful tool when determining the characteristics of the universe we can observer by comparing it to the arbitrary characteristics of other "possible" universes.
Well, I cannot answer for Vilenkin or Tegmark, but I think they were speaking informally.
How we interpret these results depends on how we think about probability. If we interpret probability as a quantitative measure of credence, or degree of belief, then there isn't really a difference between "almost surely" and "surely": in either case, the credence is exactly zero. This failure to make a distinction between possibility and impossibility may be a deficiency of the epistemic interpretation of probability (not to mention the problems of formal probabilistic modeling that have been raised here).
But if we further think about our concepts of probability and possibility, this might be argued to be a distinction without a difference. We can hardly tell the difference in credence between an event that has a probability of 10[sup]-10[/sup] in a single trial and one with a probability 10[sup]-100[/sup]. We stop making a difference long before "almost surely".
There is still a possible/impossible distinction though. But is there, really? If "an event A is impossible" means for you that you should live your life as though A will never happen, then events with an extremely low probability are as good as impossible. You live your life assuming that the air will not suddenly evacuate the room through the window, leaving you choking on the floor, even though science says that such an event is possible (and even has a well-defined, finite probability!)
Right, but if we're thinking about the universe at large, then all these low probability events could be happening elsewhere, assuming a large enough universe. There would even be worlds where the low probability events are common, and assuming anything intelligent can manage to survive long enough there, it might come to a different conclusion about how low those probabilities are.
In an infinite universe, aren't we almost surely guaranteed a world where our doppelgangers walk through walls (the molecules align just right) after saying an incantation? Maybe doppleganger Jesus really did take a stroll on the water.
The general point that I wanted to make is that if there are separate systems with a finite number of possible states between them, then for them to be found in the same state at some moment, they do not have to have identical histories up to that moment. Even in a purely deterministic universe, as these systems transition from one state to another, they may end up in the same state at some point simply by chance. What that chance is - high, low, "almost surely" - will depend on a more detailed analysis.
I've quoted from Vilenkin's book. Nerither he not Tegmark were speaking informally.
Quoting SophistiCat
It's nothing to do with probability, or our interpretation of probability. All initial conditions are realised by inflation, deterministically.
Quoting SophistiCat
It's nothing to do with credences either.
Quoting SophistiCat
Impossible events are those forbidden by the laws of physics.
And, this is also true for Hubble Volumes in the Level 2 Multiverse. Indistinguishable Hubble Volumes will have different histories due to different laws of physics operating.
We're talking a hubble-volume in this case, which has a finite but large degree of freedom. My wave function was based on that. Interestingly, I think it was a mistake to specify an inertial frame in my description. The full wave function of the one event is enough. If another event somewhere has the same wave function, it defines a clone Hubble sphere to ours.
It is also a definition free from tom's concern about the two universes staying identical. The definition is of an event which doesn't become something else.
My definition breaks down with Bill Clinton's oddly applicable statement: "It depends on what your definition of 'is' is". How can anybody assert that the state of some event outside our sphere 'is' in any particular state? Our definition sort of assumes a measurement taken from 'here', and by that definition, those distant events have no measurement and are in complete superposition. The nearest Earth clone is massively closer than the figure Tegmark quotes where the number of finite states is computed and divided by distance, something that seems invalid without a measurement being taken, from here no less.
The distance then becomes a function of the furthest historic matter that made a difference to our state now. That's further out that the Hubble-Volume, which is defined as the stuff that can make a future causal difference to here, not that which has made a past causal difference.
Agree with this. Yes, I think I alluded to the opposite at first, but you're right. This was pointed out to me in a prior post.
They do not say that they are speaking precisely and formally in their books. It is only you that says that. The evidence points to the opposite being the case. The absence of equations is a big clue.
In any case, the books are not holy scripture and we are not in the helpless position of those trying to interpret holy scripture and work out what the Author intended. Either mathematical analysis supports a conclusion that there does not exist a single level 1 spacetime lacking a duplicate Earth, rather than the set of such spacetimes merely having measure zero, or it doesn't. If it does, you should be able to point to a rigorous proof of the former. So far you have not done so.
I went to the Wiki article on the multiverse. I searched for "ergodic," and found this:
A prediction of chaotic inflation is the existence of an infinite ergodic universe, which, being infinite, must contain Hubble volumes realizing all initial conditions.
Now we see a couple of things. First, here is the popularized explanation of the ergodic business. But note the imprecision and inaccuracy:
... which, being infinite, must contain Hubble volumes realizing all initial conditions.
The language here clearly shows that the author thinks that the mere fact that there are infinitely many Hubble volumes implies that all initial states must be instantiated. But of course we have seen that this is not true.
I do realize that the fault here is not with Tegmark or with any other serious multiverse theorists, but rather with the Wiki author. But this is a good example of the type of loose thinking and careless use of infinity that seems to be a big part of this subject.
Now two paragraphs later, we find this beautiful resolution to our entire problem.
Given infinite space, there would, in fact, be an infinite number of Hubble volumes identical to ours in the universe. This follows directly from the cosmological principle, wherein it is assumed that our Hubble volume is not special or unique.
AHA!!!! Mere infinity is not sufficient, we all agree on that. And as I've pointed out, ergodicity is not sufficient, because a distribution could be ergodic yet still behave badly on a set of measure zero.
We need another assumption. the cosmological principle, which says in effect that there are no measure zero misbehaviors!
So ergodicity says that the large scale behavior is statistically well-behaved; and the cosmological principle says that there are no measure zero exceptions. Ergodicity allows that for all we know, our own earth is a statistical exception. The cosmological principle says that there are no statistical exceptions.
None of us know (though some claim to know) how the world got started, if was really a big bang multiverse or whether it's just turtles all the way down.
But as philosophers we can at least point out which arguments correctly follow from which assumptions.
As far as I can determine, the fact that there must be a duplicate earth does not follow from ergodicity; but rather from the additional assumption of the cosmological principle. It's the cosmological principle that rules out measure zero exceptions. And of course the cosmological principle is an assumption not directly supported by evidence. It's simple an assumption whose purpose is to make the duplicate earth theory true. Without that extra assumption you haven't got certainty that there's a duplicate earth.
This is something that troubles me deeply. You are claiming that physicists assume that Zermelo-Fraenkel set theory applies to the physical universe.
If that were true, wouldn't their be clever physics postdocs applying for grants to see if the axiom of choice or the continuum hypothesis or any of the large cardinal axioms are true?
If you claim that the collection of clones may be placed into bijection with the set of positive integers, then how large does physics say is the collection of the set of subsets of the clones? Is it Aleph-1, as the continuum hypothesis would imply? Or do you think it might be Aleph-2, as ?Gödel suspected? Or perhaps far larger than that, as Paul Cohen believed?
It seems to me that these questions are utter nonsense. Nobody knows whether ZF applies to the real world. Perhaps category theory or homotopy type theory, two modern alternatives to set theory, better describe the mathematical foundations of the world.
You have a reference for the claim that physicists believe there are countably many clones? Or that Zermelo-Fraenkel set theory has relevance to physics? This I would really like to see.
Along the same lines, if you claim there are uncountably many indistinguishable Hubble volumes, what is the cardinality of this uncountable quantity? Is it one of the Alephs? Or perhaps the axiom of choice false and the uncountable cardinality is not an Aleph at all. Could that be the case?
I don't think physics has given any credence to these questions at all. In which case any claim about the cardinality of clones is nonsense.
As such, it is, as you say, an assumption, not some mathematical certainty that we're not unique.
Well the probability is zero. But that is not the same as guaranteeing with absolute certainty that there is another earth. That's the claim on the table. The difference between absolutely certain or almost certain. If someone wants to claim that the multiverse theory says that it's almost certain [in the technical sense] that there's a duplicate earth, I'll accept that this conclusion follows from the premises. But @Tom is claiming absolute certainty, and I don't see it.
Well actually the author has misused the cosmological principle, which implies nothing of the sort. The cosmological principle states that each constant-time hypersurface of the universe ('this spacetime') is homogeneous and isotropic at the large scale. When formalised (which is quite tricky to do - see this discussion), this is a statement about observed average quantities as the size of the hypersurface subsets we average over approaches infinity.
So the cosmological principle says absolutely nothing about microphenomena such as whether a particular teensy-weensy arrangement of molecules like the Earth recurs - even though the author of that article appears to think it does. I am confident that neither ergodicity nor the cosmological principle, either alone or together, can imply the conclusion that there is certainly a duplicate Earth in this spacetime.
I suspected that, but didn't want to complicate my point.The Wiki article on the cosmo principle does note that the sun is different from the earth, so that the cosmo principle doesn't apply at such small scales. But if there are two Hubble volumes that are identical, that must (might?) mean that those volumes are particle-by-particle identical, which would imply a duplicate earth. Is that true? If there are two identical Hubble volumes, what does identical mean? Are they quark-by-quark the same? Ih which case they have duplicate earths.
I must say that I don't really believe there are duplicate earths, duplicate people, or for that matter an actual infinity of anything. I'm with Lee Smolin, who coined the phrase, "the trouble with physics." He was talking about string theory, but multiverse theory strikes me as suffering from the same type of problem. We are at the limit of our ability to do experiments, so the theorists are running amok and no longer doing science.
https://www.amazon.com/Trouble-Physics-String-Theory-Science/dp/061891868X
Aren't you neglecting that the matter density must be uniform? You have to count the contents too. Spacetime won't be flat unless the matter is presumed to be evenly spread.
Quoting fishfry
And remember that Linde's eternal inflation would presume that each bubble universe would start off at a planckscale energy density and so the initial state would be a relativistic gas, a quark-gluon hot soup. So the material content would be at thermal equilibrium. The only fluctuations - the seed forming inhomogeneities that result in the later gravitational/material structure - would be thermal quantum ones.
So both protons and electrons, stars and galaxies, are local inhomogeneities that pop out way after any such structure has been washed clean by an initial thermal equilbration.
Of course that then is a constraint on the odds of the history of a universe actually repeating "particle for particle". However no one wants to talk about the real combinatorial issues here. :)
It's possible to dislike multiverse hypotheses but not blame it on physics, because it's all unfalsifiable and hence doesn't count as science. I regard it as metaphysics.
Smolin's complaint is that the same applies to string theory. If he's correct (I don't know enough about string theory to comment) then string theory also is not science and so should not be getting large parts of physics funding. Also, it should be called 'String Hypothesis', as a requirement of any 'theory' is that it be falsifiable.
Heat is a measure of average energy over a region. Thermal equilibration is a statistical process. My remarks stand.
Quoting apokrisis
I'm perfectly happy to talk about combinatorics. Although I'm ignorant of the specific physics, I gather that the argument takes the form of modeling the number of particles in a region, and all the states they can be in, and figuring out how many possible configurations of all the particles there are.
But it doesn't matter. The number, whatever it is, is finite. As far as the mathematics, the situation is best modeled by assuming there are exactly two states, heads and tails. Many here are making detailed technical points of physics. But if you had a "zillion" possible states -- Tom objected to that earlier, a zillion is just whatever big finite number of states you like -- then you could code each state in binary and combine them by some rule and you'd map your entire state space into the set of all possible binary sequences.
Coin flips are exactly the same as fancy physics-y states for purposes of this discussion. Because you could express all the states in binary and have the same conversation.
Now Tom has claimed in one post that there are infinitely many states. THAT I do NOT believe. There must be finitely many states in order for this particular argument to go through at all. The argument is that there are infinitely many Hubble volumes, and each one can take on one of a bounded set of finitely many states. In other words there's some number S and that's the most states you can have. [In other words you can't have 1 state in Hubble 1, 2 states in Hubble 2, and so forth. There is a max number of states ANY region can have].
Otherwise this entire argument does not work. There are finitely many states. And when you're making statistical arguments, coin flips work just as well as huge numbers because you can always code any huge number in binary. The specific details of the physics and the calculation of the possible number of states is a huge distraction that's causing people to miss the fact that they are making statistical arguments. Statistics only tell you about populations, and never individuals.
We’re waiting for the quantum computer to come along.
Yes, I think Vilenkin entertains similar fun scenarios, but frankly, not having followed the derivations, I am a little hesitant to commit to such specific predictions.
Superposition states are states too (they are also called "mixed" states, as opposed to "pure" states). But I think I get your point: if we haven't been in contact with some remote region of the universe, then within that interval of time its wavefunction has been evolving independently from us, and there is no coherence between us and any one of its branches.
This is simply false. In Inflationary cosmology, no assumptions about randomness, or all initial states being instantiated, or probability distributions, or typicality, or mediocrity is required. Inflation guarantees that this type of "exhaustive randomness" is in place.
Which is why I have been describing the initial conditions as ERGODIC from the beginning.
Neither are bounded by the Hubble-Sphere. The type-1 universe is bounded by the event horizon (which IS frame dependent, despite my expressed hesitancy in the prior post), but the Earth copy requires that pure quantum state which is bounded by the particle horizon.
The former is a ball about 31 Glyr in diameter (units in proper distance), but the latter is a frame-independent ball about 92 Glyr in diameter, beyond which all quantum states are pure from our standpoint. That means the nearest copy of us is only 92 Glyr away There are closer ones, but there is no coherence between us and them, so they don't really exist in a type-1 sense.
Comment on my QM thingy instead. I just stated that there is a copy of us quite nearby, to the point of giving a fairly specific figure for it.
Here's Tegmark being as explicit in one of his papers as in his book:
"In particular there are infinitely many other inhabited planets, including not just one, but infinitely many with people with the same appearance, name and memories as you. Indeed there are infinitely many other regions the size of our observable universe, where every possible cosmic history is played out. This is the Level I multiverse."
All relevant papers are available on arXiv.
EDIT:
For example the abstract of the paper you claimed to have read:
"A generic prediction of inflation is that the thermalized region we inhabit is spatially infinite. Thus, it contains an infinite number of regions of the same size as our observable universe, which we shall denote as O-regions. We argue that the number of possible histories which may take place inside of an O-region, from the time of recombination up to the present time, is finite. Hence, there are an infinite number of O-regions with identical histories up to the present, but which need not be identical in the future. Moreover, all histories which are not forbidden by conservation laws will occur in a finite fraction of all O-regions. The ensemble of O-regions is reminiscent of the ensemble of universes in the many-world picture of quantum mechanics. An important difference, however, is that other O-regions are unquestionably real.
Actually it's the other way round. Superpositions are pure states, mixed states are statistical mixtures.
I asked you to link to a proof for your claim. Those papers contain no proofs.
The closest the Vilenkin paper comes is on p7 (2nd para of section IV) where it says:
Right there, in that sentence, Vilenkin asserts that E having a nonzero probability in a single trial entails that it is impossible for there to be an infinite sequence of trials in which E does not happen. That is, he simply assumes the conclusion that you assert. He does not prove it.
That's because he's writing informally. That becomes blindingly obvious in the paragraphs that follow, where he whimsically contemplates things like O-regions in which Elvis is still alive.
He explicitly says that the infinities that come up in multiverse theory are breakdowns in our theory, and not actual infinities of universes.
http://blogs.discovermagazine.com/crux/2015/02/20/infinity-ruining-physics/
Also see my extended discussion here.
https://thephilosophyforum.com/discussion/comment/142382