I’m nowhere near finished reading Max Tegmark’s new book, *Our Mathematical Universe*, and my thoughts are still evolving. Looking ahead, I have a feeling the last few chapters of the book are weak, but his explanation of inflation is great. The stuff I’ve been discussing is in Chapters 5 & 6. Today’s post follows up on my post “Against Recurrence”: #1.” And I’m prompted to say more by the good comments on that post.

What I’m talking about has to do with how we might emotionally “feel” about the idea of an infinite universe that may contain identical copies of ourselves. I don’t like the idea of identical copies of me, it seems wasteful, and in some sense it makes my life seem pointless.

Rather than wholly giving way to emotion, however, I want to reason about the recurrence proposal. So, once again, the idea is to go with the idea that we have an infinite number of stars and planets and see where that takes us.

It’s hard for us to grasp how really big an infinite set is, and how strongly it differs from a finite set. If you get infinitely many tries, you really can expect to flip a googol heads in a row.

If we say the universe is akin to a 3D chessboard made up of minimal spacetime cells, and if we say that each cell can only be in some limited number of states, then our local visible universe volume is akin to an array of numbers, and therefore in the endless number of “hands of cards” that infinity deals out, it’s quite possible that the same pattern could re-emerge, and not just once, but many times or infinitely many times over.

As I say, my (emotional) issue is that it seems like waste to have an infinite universe and then to be cluttering it with repetitions of things. So I’m alternating between (a) looking for a way out and (b) coming to accept this.

One way out, as I already hinted, might be to argue against the cell-grid image of spacetime. I think such a worldview has more to do with our current cultural obsessions than with ultimate reality. A lot of scientists are, after all, geeks. People lacking in empathy, or in a poet’s “negative capability” for tolerating ambiguity, or in a relish for old-fashioned sloppiness. People who want things to be orderly, straight, square, lined up. (And, admittedly, as a mathematician, I myself have tendencies in that direction—counteracted, of course, by being a beatnik SF novelist.)

Even if we suppose there’s a smallest possible space size and a smallest time length — that is, quanta of space and of time — would these quanta really be arranged in a dry, precise grid? Not likely. Nothing in nature is dry and precise. Look at moss on a stone, look at the leaves on a tree, look at the sand on a beach. Nature duplicates herself—but only approximately.

Natural systems are chaotic. A waving leaf, a fluttering flag, a human heartbeat, or one’s flow of thoughts—in the analog idealization, these processes never ever repeat themselves. Even though, in a rough qualitative sense, they’re always doing the same thing. Chaos theory is a great teacher. *Always different, yet always the same.* That’s a viable option. You’re surrounded by seeming sameness in daily life but yet—ah, *but yet*—nothing is ever really same. You never step twice into the same river. The world is continually dancing, unfolding, jiving, and coming up with fresh variations.

Temporal sequence is a source of variation as well. It may be that, once in a blue moon, a fluttering flag takes on what seems to be the same configuration. And it may be that, at certain *deja-vu*-type times you feel like your head is right back where it once was. But then the next tick of time feeds in and, ah yes, the progression is, after all, different than before.

So what about the boring, jive-ass, Lego-like grid of spacetime quanta postulated by our contemporary cult of the mighty Computer? I’m saying that, far from being like 3D or 4D graph paper, we’d be looking at something more like the slightly wonky and irregular units of a honeycomb or knit scarf, with the spatial locations of cells jiggling or moving around as time went on.

How might we express such an irregularity in the spacetime cells? Well, under the current scientific dispensation it wouldn’t be kosher to talk about analog real-number distances between he cells—although I’d like to. We might to turn to something more digital like, say, “adjacency.” This brings us to the mathematical disciplines of “graph theory” or, better, “network theory,” which look at structures made of vertices (spacetime cells) with lines (adjacencies) connecting some of them.

But if we take the network theory route, we’re still stuck with the visible universe being a finite digital pattern. Rather than a 3D graph-paper-like grid, it’s now a heap of dots with lines connecting them. So we’re still stuck in Squaresville. Recurrence Land.

But wait. Let’s go back to the idea that the network is dancing, with connection lines popping in and out of existence. A Big Aha here, a Small Aha there. “Have you met my friend?” “I’m never speaking to you again.” “Let’s do-si-do.”

You might find a finite region within our infinite space that momentarily looks just like our current home region. But then, ah yes, with the next tick of time, our visible universe and the twin visible universe would progress on to different states.

Cornered now, the world-numbing advocates of Recurrence might protest, “No, that can’t happen, physics is deterministic, and any two momentarily identical systems have to stay identical forever after.” Not true. For two reasons. (a) Any region of space is going to be receiving inputs from the rest of space. Noodges and jiggles that upset and scramble whatever teetering Cat-in-the-Hat pile of plates you’ve got. (b) Even if there were nothing “outside” of these two seemingly identical regions of space—even if we were talking about two identical pocket universes—we still have the saving fact that *Physics is not deterministic*.

What? Not deterministic? Live by the sword, die by the sword, quantum mechanics!

Yes, stodgy, boring, spoil-everything, quantum mechanics insists that we can’t have wonderfully smooth and infinitely variously marbled matter with patterns all the way down into gloriously infinites subsubsubsublevel after subtillionlevel. But QM also delights in saying that physics is fundamentally random—in the sense that any observation that chooses between two options produces completely unpredictable outputs. The refer to the built-in randomness as the Measurement Problem. They love to get all ecstatic and mystagogic and woo-woo about it. “The QM universe is stranger than anything we can possibly imagine.”

“Measure *this*, mofo.” The fickle finger of Fate appears in zombie universe B, stirs the porridge, and, having writ, rocks on.

But…might it not be that, among all the seeming identical copies of our visible region’s Now Moment, there would be some other, particularly dogged, zombie regions that are tracking our moves? And here, ah yes, we’re saved by Mamma Mathematics.

If we accept the digitization of space, the number of possible visible regions in an infinite space is going to alef-null. The lowest level of infinity. But—even accepting the digitization of time—the number of possible future universe-histories is then going to be 2^{alef-null}, which is known to be a transfinite number larger than alef-null.

Georg Cantor, father of our strange transfinite country, thought this higher-order infinite number would be alef-one, but these days, the mathematician hep-cats think it’s more likely to be alef-two. Bigger than alef-null in any case.

And this means that, in fact, the probability of finding another volume of space that behaves just like ours forever is 0. Not absolutely impossible, you understand, but an event so unlikely that the probability is formally 0.000000000… With those no-effin’-way 0’s running out forever. We’re risen from the tomb.

These posts are rhetorical as well as scientific!

By the way, *ahem*, there *can *be regions that behave just like us for arbitrarily long finite times—a different region for each specified length of time—wihout there being any region that matches us forever. But any given region eventually divergees from us. I might say more about this somewhat subtle point later.

Math is strange and wonderful. I look forward to reading the rest of Max T.’s book which, after all, has *Mathematics *right in the title!

May 28th, 2015 at 11:09 am

Julian Barbour got to me before you did and so I tend to have trouble dragging my mind back from Hilbert space, or configuration space as he prefers to call it, into the 3+1D realm you are describing. These problems may be all beside the point if the underlying reality is actually elsewhere.

But even here Barbour has a relevant insight. He has reasoned (or so it seems to me) that to the extent there is a spacetime metric at all, the notion of distance is entirely derived from the angles between events. In other words, the position of everything is only relative to everything else, and if thats true then its those relationships that get the quantum jiggles.

I wonder how may times more chances we get to replicate our universe exactly if we are allowed to count copies that are identical if we only apply a rotational transformation? And ditto mirror symmetry. And transformations of scale, which Barbour might even dismiss. Would these tilt the likelihood of finding duplicates in our favour?

May 28th, 2015 at 11:27 am

It seems to me now upon revisiting your previous blog that configuration space might not have been very far from your mind when you wrote it, and maybe you were trying very hard not to get too technical too quickly.

Barbour wasnt the first writer I encountered using those ideas now that I think about it. It might have been Roger Penrose. He lost me though somewhere between Spinors and Twistors.

Penrose has some other fascinating ideas about a Conformal Cyclic Cosmology that also make the spacetime metric go away. Your illustration from Tegmark reminded me of it, with its spatially stretched out c-boundary. Was it actually CCC he was talking about?

May 30th, 2015 at 1:08 pm

For some reason, I actually like the idea that there are an infinite number of iterations of each of us. Think it’s a cool idea that there are an infinite number of me, who have lived all possible variations of my life. And infinite versions of Earth…

Maybe the observable universe is a tiny part of a larger structure, and those structures that repeat on huge scales are arranges in even larger forms that repeat on yet larger scales…with possible iterations, of course.

If it was proven that this infinity of iterations existed, I would find it…comforting.

May 31st, 2015 at 5:07 pm

Rudy,

Is there a flavor of infinity that excludes recurrence? All possibilities, but only once? I guess that would be hard to “enforce” since it would prohibit five heads in a row for penny flips from happening more than once. And I think I heard of one person who did find two snowflakes the same. Seems like recurrence does happen at the lower levels, but you are looking for a mechanism that prevents it at higher orders of complexity. Does it come down to

deterministic = recurrence

non-deterministic = non-recurrence

?

June 4th, 2015 at 6:04 pm

Kevin, certainly if we simply look at the set of natural numbers, each possible number is in there, but only once. Or take all the prime numbers as a sparser set. Or the Godel numbers of provable mathematical theorems.

But the hassle that we keep running into is that you only look at, say, Earth sized planets, it’ shard not to repeat unless (a) we suppose that the planets are made of smooth continuous matter so you can have deep marbling or (b) we compare planets by their entire past and future history.

Ed, I have at times liked the idea of having endless copy of me out there, but nowadays I decided I don’t like it, as it makes my life seem (more) pointless. I’m not sure if the U-shaped infinite universe matches an idea of Penrose’s. I do say a bit more about it in the next post.

Ralph, I haven’t checked out Julian Barbour, will take a look.