I’m nowhere near finished reading Max Tegmark’s fantastic new book, *Our Mathematical Universe*, and my thoughts are still evolving. The stuff I’ve been discussing is in Chapters 5 & 6. of Tegmark’s book. 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!