But if the universe is a 3-torus, all three axes have a preferred orientation, — noAxioms

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.

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. — noAxioms

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.

MWI is not an ontological stance. No creation of new universes ('somewhere' as fishfry puts it) occurs — noAxioms

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?

A bit off topic, but I've always noted that the orientation of the three spatial axes (X, Y, and Z) is arbitrary. If there is an actual x axis, which way is it? But if the universe is a 3-torus, all three axes have a preferred orientation, and this defines a preferred frame as well, even if not an inertial one. If the spatial axes are fixed, the temporal one, orthogonal to the others, is fixed as well.
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. — noAxioms

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.

In math, the idea of a manifold is more abstract than that. When we imaging 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. — fishfry

Is this true? I play asteroids in a flat 2-torus space, not on the surface of a donut embedded in three-space. If I fly along either axis, I return to my starting point, but if I angle it a little, I don't. The two axes are obvious even if I'm presented only with a circular screen displaying the local area.
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.

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? — fishfry

Not always there. No new thing at all, so nothing to always have been there.
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.

Is this true? — noAxioms

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.

I play asteroids in a flat 2-torus space, not on the surface of a donut embedded in three-space. — noAxioms

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.

Is this true?
— noAxioms

That a manifold is a locally Euclidean object that exists on its own and does not live in an ambient Euclidean space? Most definitely. — fishfry

I should have quoted more.
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.

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. — apokrisis

Well, only if the space was smaller than the hubble-sphere (which it very much is in asteroids). You could see the repetitive things line up in certain directions. With larger space, that can't be done. The grid is objectively there but the event horizon is too far away to detect the grid.

I should have quoted more.
I meant to ask if there was no grid in the sky if the space was torrid manifold. — noAxioms

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.

I then gave the example of the asteroids game which is played on such a manifold, — noAxioms

Asteroids is a torus represented on a 2-D grid. Not an abstract 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. — 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.

Well, only if the space was smaller than the hubble-sphere — noAxioms

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. :)

Oh I see your point. Yes, the torus is orientable. — fishfry

OK, My terminology is wrong. Orientable, yes. There is no obvious origin, hence no actual grid. It is the orienting that suggests a preferred reference frame.

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.

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. :) — apokrisis

I comment on something like this earlier. Such a space is not flat, so it would need to be big enough to account for whatever measure of flatness they've made so far. It's enough that there are places that cannot be reached by light from here, ever. So steering things back on topic, that means it has never been a really alien concept that there are undetectable portions of our universe. Are those places other universes? Not like they're discreet with boundaries where one stops and the next starts.

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.

"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". — Michael Ossipoff


NoAxioms and I just had a lengthy conversation disproving this very point. — 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.

Could you please review those posts?

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.

What you say is simply not true. At best you have a probabilistic argument that falls short of certainty.

I refer you to NoAxoms' argument. Read it carefully this time.

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.

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

it has never been a really alien concept that there are undetectable portions of our universe. — noAxioms

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.

Are those places other universes? Not like they're discreet with boundaries where one stops and the next starts. — 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. :)

So spatial infinity would seem to guarantee that there should be an infinity of Earths — apokrisis

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.

If there are two states and infinitely many universes they could be 0, 1, 1, 1, 1, 1, ... — fishfry

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.

If 0 is the "earth" state, there is no other earth. 1 is maybe Mars. So Mars exists infinitely many times but not earth. — 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?

Maybe there's only one earth even though there are infinitely many copies of Mars. It's perfectly possible. — 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.

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. — apokrisis

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.

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. — 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.

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. — apokrisis

If there are zillions of states, it's the same argument. The two state example is only a simplified illustration.

A bit off topic, but I've always noted that the orientation of the three spatial axes (X, Y, and Z) is arbitrary. If there is an actual x axis, which way is it? But if the universe is a 3-torus, all three axes have a preferred orientation, and this defines a preferred frame as well, even if not an inertial one. If the spatial axes are fixed, the temporal one, orthogonal to the others, is fixed as well.
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. — noAxioms

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.)

I refer you to NoAxoms' argument. Read it carefully this time. — Michael Ossipoff

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."

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. — fishfry

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.

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. — 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.

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. — fishfry

If the universe is infinite and ergodic, it is impossible that state 0 will not repeat.

You obviously can't fit in an infinite number of copies of each, which is what you need for a multiverse. — tom

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.

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. — fishfry

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.

It's called ergodicity. I have mentioned that ergodicity is a requirement for Type 1 multiverse several times. — tom

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

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. — fishfry

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.

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. — apokrisis

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.

Each of your regions would contain about 10^120 degrees of freedom. That would be the entropy content of a Hubble radius. So, naively, we would be talking about the chances of one configuration of that magnitude being exactly repeated.

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.

It might not. It might land on one face an infinite number of times. It might have been Mars all the way down. — Michael

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.