Musings on the infinite

[steuard]

A few weeks ago, my friend and former colleague Sean Carroll was a guest on the Colbert Report to promote his book about the nature of time. Toward the end of the interview, they discussed the idea of the "multiverse", which Sean uses to refer to the (possibly) infinite number of "universe-sized" regions within the vast web of space and time where we live. The notion is that if we could somehow travel far enough (faster than light) to regions many times more distant than our telescopes can see, we could find countless independent "universes" that can never talk to each other at all. Some of them would be much like our own but others could be very different, maybe even with different laws of physics. Steven Colbert seemed quite interested:

Colbert: Am I in these other universes?

Carroll: There will be people very much like you.

Colbert: In these other universes, is it possible that my show's on at 11 and John Stewart is at 11:30?

Carroll: Maybe more often!

It's a cute exchange, and it's a variant on the old idea that "in an infinitely big universe, everything that could possibly happen must happen somewhere."

Trouble is, I don't know that I buy that argument, for rather subtle reasons. However we define them, the number of "independent universe-sized regions" of space and time is countably infinite: we could in principle come up with some way of labeling each one by an integer. But many sets (like the real numbers) are uncountably infinite: no matter how you try to label each real number by an integer, you'll miss the vast majority of them. The real numbers are just a much bigger infinity than the integers are. Going on, the set of all possible curves in space is a yet larger infinity. (Assuming space and time are continuous! If they turn out to be discrete, then the set of curves has the same infinite size as the real numbers.)

The thing is, the set of "everything that could possibly happen" is a lot more like the set of all curves than like the set of integers: if anything, it's a still larger infinity. So no matter how large our multiverse may be, it's mathematically impossible for every possible history to occur somewhere. Does that mean that our Steven Colbert (on at 11:30) is the only one? Quite possibly so. I'm not convinced that the multiverse idea opens up as many possibilities as people sometimes think.

Comments

[steuard.livejournal.com]

Oh, blast. Maybe there is a way out, at least sort of, for a reason comparable to the status of the rational numbers in this whole story. You can prove that the rational numbers are countably infinite (by assigning labels based on their integer numerators and denominators, which are countable). But the rationals are also "dense" within the real numbers: if you pick any real number and any allowed deviation (no matter how small, as long as it's not zero), I can find a rational number that's "close enough" to your real.

So even though it's impossible for every single possibility to occur somewhere in the multiverse, perhaps there will always be some region that's "close enough" to any given outcome that you might desire. I don't know if there's a proof of that (I suppose it's possible that the sizes of the infinities involved are too different), but it's at least plausible. Rats.

[beth_leonard]

Maybe 11 is "close enough" to 11:30? ;-)

I've always wondered about the infinite monkeys producing the works of Shakesphere on a typewriter. If you could prove that a monkey's fingers were too large to independently push a single button on the typewriter without pushing another one, or a multitude of other sub-premises about how monkeys type... anyway, I just don't think it's possible even with an infinite number of monkeys and infinite time.

--Beth

[jon-leonard.livejournal.com]

The distinction between countable and uncountable infinities seems like a distraction: Is there anywhere in physics that you actually need the real numbers? (As opposed to, for example, the expressable subset of the real numbers?)

It may be better to approach this sort of problem in terms of limits. Consider, for example, the problem where you pick one individual from each of an infinite number of universes, and stick them (one each) at random back into the universes. What's the probability that no one winds up home? This has a well defined answer, and it's even between 0 and 1: It's 1/e.

I'm reluctant to apply much in the way of intuitive reasoning to this sort of problem, though: Human intuition is pretty bad at dealing with rare events (thus the popularity of things like lottery tickets), and this sort of problem really isn't any easier.

As an aside, the set of continuous curves in space is only the same size as the real numbers.

[steuard.livejournal.com]

The distinction between countable and uncountable infinities seems like a distraction: Is there anywhere in physics that you actually need the real numbers? (As opposed to, for example, the expressable subset of the real numbers?)

Oh, constructivism. :P I wouldn't be entirely surprised if physics (or most of it) could get by with just the rationals. For that matter, if at some point space and time turned out to be quantized in a fundamental sense, there might well exist a manifestly countable basis for the space of wavefunctions. But reality certainly feels like R^{3,1} to me... even if that is just years of indoctrination talking. (I like my limits to exist...)

I'm not sure I entirely see the connection between your comment "in terms of limits" and the issues that I'm considering here, but it's possible that we're just paying attention to different aspects of the question.

I'm reluctant to apply much in the way of intuitive reasoning to this sort of problem, though: Human intuition is pretty bad at dealing with rare events (thus the popularity of things like lottery tickets), and this sort of problem really isn't any easier.

I think this one is even less intuitive: at least with lottery tickets we've got some sort of experience dealing with large numbers. Dealing with infinity (and worse, different cardinalities of infinity) is entirely divorced from our intuitive sense. But it really was the nagging suspicion that "the set of all possible histories" might be of much higher cardinality than the integers that got me thinking about this. I'm honestly not sure what that cardinality might be: anything between the reals and goodness knows what. (And as noted, it's even possible that a correct quantum treatment could make the whole thing countable after all.)

As an aside, the set of continuous curves in space is only the same size as the real numbers.

That's a handy fact. Thanks! (I hadn't heard it before, but it feels quite plausible now that I have.)

[marphod.livejournal.com]

There are many places in physics where the rationals are insufficient, and the reals are required. Anything that has a square effect (magnetic field strength, distance covered in an inertial reference frame under acceleration, etc.)

[jon-leonard.livejournal.com]

There are lots of kinds of numbers between the rationals and the reals: Quadratic surds (stuff you get by adding the square root operator), algebraic numbers, etc. Admittedly neither of those are sufficient for physics, since pi keeps showing up, but it's still not clear to me that you need all the reals. Well, all the complex numbers. You can't really do quantum physics with only real numbers.

[steuard.livejournal.com]

Yeah, pi and e are two of the big sticking points that I'd worry about when trying to develop a rationals-only physics. Oh, and by "rationals" I should be clear that I was thinking "complex rationals". (Now I'm starting to worry about whether that's well-defined in the ways I'd need, though.)

Though to be honest, my original thought was actually just that the space-time manifold would be Q^{3,1} rather than R^{3,1}; I hadn't followed that up in my head to conclude that the whole formalism would have to be based only on rationals. I'm still not sure.

[steuard.livejournal.com]

I guess what I'm thinking of here isn't "can you find a closed-form solution to an arbitrary equation of physics with just rationals" but rather "would it be possible to write a fully predictive theory with just the rationals". So I'm picturing less "find the speed of a block sliding down an inclined plane" and more "compute the scattering amplitude for this process in particle physics". And for that, all I really need to have in hand is the Lagrangian of the system (which can usually be written as a polynomial function of position and momentum) and some fancy integration (which I suspect could be sensibly defined over the rationals if one put in some effort).

[jon-leonard.livejournal.com]

Restricting yourself to a countable subset of the reals doesn't mean you have to give up on limits. I'm just making the (pointless?) observation that if you're talking about a specific real number, than your description of it is a finite length string of a finite number of symbols, and there are only countably many such descriptions. Unless you're actually invoking the Axiom of Choice somewhere, the countable/uncountable thing (and the rest of the reals) really shouldn't matter.

However, that's far from the only mathematical tool for dealing with inifinities: Ordinals are often more useful than cardinals, for example. I can make a meaningful claim that a third of the positive integers are divisible by 3, but the cardinal infitities (countable, uncountable: reals, uncountable: real functions, ???) aren't useful in discussing that. It's more a formal claim that in the limit as n goes to infinity, (whole numbers <= n divisible by 3)/(whole numbers <= n) is arbitrarily close to 1/3. I think Sean Carroll's "Maybe more often!" has to be taken in a sense like that.

[steuard.livejournal.com]

I guess to me, the difference between "the expressible subset of the reals" and "the reals" feels quite artificial, perhaps quite anthropocentric. "Expressibility" feels much more like a philosophical criterion than a mathematical one.

I see what you mean about limits now. I guess my original point was at a lower level than such arguments: I was objecting to the suggestion that in an infinite multiverse there would likely be other near-copies of Steven Colbert in the first place (not to the likelihood of their time slots). As noted, the complexity of the history of the universe that produced our Colbert feels to me like it's likely to be much "bigger" than the number of independent chunks in an infinite universe.

So is the set of all quantum histories a separable space, or is it too big for that? If it's not separable, then there's no point arguing what fraction of the Colberts are on at 11, because the odds of having more than one of him are essentially zero anyway.

[jon-leonard.livejournal.com]

We may need to establish what it takes to be a "Colbert", for that matter. If you divide up a volume of space into roughly atom-size pieces, then specifying what atoms (or gaps) go into each one and how they're bonded together gives you only a finite number of possibilities. So while the question of how many quantum histories there are is interesting, subtracting out stuff like ordinary thermal variation really shrinks the number of possibilities to consider.

So, if you really think there are an infinite number of universe-like regions, and some object of interest is possible, I don't see how to avoid the conclusion that it shows up an infinite number of times. (Infinity times a positive real number is still infinite.)

I'll happily accept that that's not useful, but still...

(Oh, and I do think quantum histories are separable, but maybe I need to learn more quantum to be sure, or even be convinced that it matters.)

[jon-leonard.livejournal.com]

On the continuous curves thing: A continuous curve can be defined by its values on any dense subset (otherwise it'd be discontinuous!), and a countable product of reals still has the same cardinality as the reals. (And there are countable dense subsets, etc.)

[steuard.livejournal.com]

I like it.

[coldtortuga.livejournal.com]

Is there anywhere in physics that you actually need the real numbers?
We've been arguing about this at work. (Not that it's work-related, just that there are a lot of physicists and mathematicians looching about to have such arguments with :-)

Some branches of physics regularly demand the Axiom of Choice in its full generality. So, at least according to "that sort" of physicist, the answer is "Just the reals are not enough!" Although I'm not sure about the semantic content of any physical theory that requires volume to be meaningless.

[steuard.livejournal.com]

I'm glad to have provided a fun topic of conversation! (My work here is done.)

[madmanatw.livejournal.com]

This kind of stuff was exactly why I almost went for my bachelor's in physics before I realized that my favorite things about physics were the things that were actually math. ;) This has been a fun post/thread.

[coldtortuga.livejournal.com]

Um, sorry --- we've been arguing about it for *months* :-)

[spoonless.livejournal.com]

I think must be a lot of assumptions you're making here that I'm missing.

However we define them, the number of "independent universe-sized regions" of space and time is countably infinite: we could in principle come up with some way of labeling each one by an integer.

Can you elaborate on that? There are a lot of different kinds of multiverses in physics, with all sorts of rules... is this in Sean's particular set of ideas? or some theorem for eternal inflation in general? Or something based on your own reasoning?


The thing is, the set of "everything that could possibly happen" is a lot more like the set of all curves than like the set of integers: if anything, it's a still larger infinity.

I think I disagree with this. As long as you believe the holographic principle (and for the most part, I do), there are only a finite number of quantum bits in our Hubble volume anyway. So I think(*) that means the number of all possible configurations for a Hubble volume of our size is finite, not infinite. If you allow for any possible size, then it's infinite, but still only countably infinite since you can pick the size by picking the number of pixels that make up the holographic screen corresponding to our Hubble horizon.

(*) I guess the part I'm glossing over here is what about the actual "parameters" of the theory, eg things like the cosmological constant? Well, I'm not sure I understand that part very well but if pure string theory is really parameter-free and each parameter is really determined by degrees of freedom that are frozen within a particular region of the multiverse, then maybe it doesn't change what I'm saying. Or, maybe I'm just looking at this in totally the wrong way... could be.

[marphod.livejournal.com]

I've got to agree with this. How do you come to the assumption "that the number of the number of "independent universe-sized regions" of space and time is countably infinite? That's a HUGE assumption.

I've never seen such posited as an axiom for multiverse physics. In fact, most theories of multiverse physics I have seen make an assumption that, in the end, is very much the opposite.

[madmanatw.livejournal.com]

I'm only into this as a hobby, but it is my suspicion that "multiverses" as used by the Many Worlds interpretation of QM and "multiverses" as used by M-theory, branes colliding, and theories involving gravity "leaking" are pretty completely different. The MW version is an interesting, self consistent way of dealing with QM weirdness, but I don't think that the theories try to, well, _put_ those "alternate universes" anywhere. Whereas with branes, etc you have "universe sized regions" (well, universes) all embedded in a higher dimensional space (or something). The number of universes in a many worlds branching system may be a different cardinality of infinity than the number of independent universe sized regions... though I admit I don't know why number for the latter would need to be countable. Seems plausible, but, ya know, I'm not the one with the PhD in this stuff. :)

[steuard.livejournal.com]

I've alluded to this elsewhere in the thread, but I'm a bit annoyed by the sheer number of wildly different ways that people use the term "multiverse". On the one end you have Sean's usage here, which to me feels too "small" for the word: causally disconnected regions of our own space-time manifold. On the other end you have people who use "multiverse" to refer to the whole tree of branching quantum histories, which feels entirely too large to me (or, well, like a misuse of the word somehow).

If I had my druthers, I think I'd at least want the different "universes" in the "multiverse" to be geometrically disconnected in some way. I'm really not sure which of the proposals out there are consistent with that: brane-world scenarios probably are, for example. But as far as I can tell, Sean was discussing one very specific and limited case, whatever we call it, and I'm pretty sure that case must be countable.

[madmanatw.livejournal.com]

The one I've seen used for that is that two things/points/whatever are in the same "universe" if their source is the same local big bang (or functional equivalent). We pulled it out in a recent conversation I was in elsewhere where confusion between regions of our universe outside our own light cone (spacially separated universe sized regions, perhaps even) and separate "nearby" universes needed to be cleared up. (We were discussing the Dark Flow and possibility of gravity "leaking".)

It unfortunately doesn't work as well for the Many Worlds 'multiverse' usage, since you could argue that's all sourced in the same Big Bang, so for now I'm just going to hand wave them.

[steuard.livejournal.com]

two things/points/whatever are in the same "universe" if their source is the same local big bang

Well, by that definition I think much of what Sean is talking about really would be a multiverse. Perhaps that is a fair definition, or at least as fair as my "disconnected spacetime manifold" suggestion. (Heck, my suggestion may be too strong to be interesting, depending on how you interpret it.)

There was a paper I saw a while back (I don't recall enough to track it down easily) that attempted to define all the ways in which "multiverse" is used. My memory is that it identified a good half dozen distinct senses of the term.

[steuard.livejournal.com]

the number of "independent universe-sized regions" of space and time is countably infinite

Can you elaborate on that? There are a lot of different kinds of multiverses in physics

As I understand the context of Sean's conversation with Colbert, he was using "multiverse" to refer more or less to "causally disconnected regions of the full spacetime manifold". So he's essentially discussing different regions of space, not any of the more complicated (and likely less countable) versions of what "multiverse" might mean. (I feel like that term gets overused.)

the set of "everything that could possibly happen" is a lot more like the set of all curves than like the set of integers

As long as you believe the holographic principle (and for the most part, I do), there are only a finite number of quantum bits in our Hubble volume anyway.

This is a very fair point. I tried to nod toward related concepts a bit with my comment "assuming space and time are continuous": I think you may get the same reduction of degrees of freedom from uncountable to countable whether you take a holographic approach or just discretize space into Planck-length-sized cells (as I understand it, the big difference is whether the number of qbits scales like area or volume). I'm not even sure that you need all of holography to make the area-law argument: I seem to recall that there's a proof based just on quantum fields in GR that black holes have the highest possible entropy density.

I think I'd need to know more about quantum information theory to be sure whether this really does break my argument, though. (Hmm, the more I think of it, the more I suspect it does.) Perhaps the real lesson from all this is that our (my?) intuitive notion of smooth paths and histories is just wildly inaccurate as a guide to true underlying physics. (Amazingly inaccurate.) I've known that for a while, but I clearly still haven't gotten my brain fully around it.

[akjdg.livejournal.com]

Sometimes I'm real glad I became an engineer. Real glad, see.

:)