As I mentioned in “Against Recurrence #1” contemporary cosmologists are inclined to say that the space of our universe is infinite. The whole infinite space was effectively filled with a giant flash about four billion years ago.

In 2008, I was interested in a fairly simple model for this called the cyclic universe. Invented by Paul Steinhardt and Neil Turok, it basically postulates a pair of “branes” that oscillate back and forth, periodically passing through each other and thereby filling all of infinite space with a revivifying and reseeding flash. I posted about it several times, and I even discussed it on the phone with Steinhardt.

Eventually, Bruce Sterling and I even wrote a story based on this idea, it was called “Colliding Branes,” and you can read it online. I sent the story to Steinhardt, but he never wrote me back. I guess he was like, “That is not what I meant at all. That is not it at all.”

Click for a larger version.

Now, in 2015, the cyclic universe model seems to have lost ground to the eternal inflation model that I mentioned the other day in “Against Recurrence #1“. This is the idea that one speck of supermatter expands forever, and that successive regions of space “boil off” from the expanding speck, creating a diagram like the one I found in Max Tegmark’s book—I’m showing it here again, and I’ve added a drawing below.

We’re looking at a spacetime diagram, with time running up forever. The idea seems to be that the one little scrap of “starter dough” has two regions that last forever, shown on the left and the right. The “steaming off” galaxies from the starter dough keep endlessly appearing, each with its own timeline that in turn bends up and runs toward eternity. The poor saps in these galaxies (us) imagine that all of them arose at the same time. The other tricky angle is that that finite-looking space inside the “U” between the two borderlines—that space is unbounded, and in fact infinite.

Back in the ‘50s and ‘60s some cosmologists used the word ylem to refer to the starter dough, so I might use that word here too.

Really this drawing should be shaped more like a V than like a rectangle, the idea being that the horizontal dimension is tiny and finite at the bottom, and it opens up endlessly as we run up the pseudotimeline of the ylem blobby. But, as I say, we see that whole U-line as being the one moment of universal creation of an endless world.

And it’s the quantum fluctuations and wriggles in the starter dough that are seeding the infinitely many galaxy births. In other words, the starter dough contains an infinite amount of information.

What gets me attention is that the eternal inflation model is in effect ascribing infinite complexity to that finitely large initial glob of ylem. This tends to support a view of mine which is, these days, complete out of favor with scientists. That is, I advocate the doctrine of “infinities in the small.”

The current philosophy of science is dominated by the pernicious and unreasonable doctrine that space and time are quantized into pixels. But why should we believe this? It’s like the situation where, to a hammer, everything looks like a nail. We use computers, so to us everything looks like a computer screen.

Quantum mechanics is only one stage of our ongoing exploration of the world. Inevitably we’ll move beyond it, onwards into the subaether, the subdimensions, the ultra particles, the rootless tree of matter, the absolute continuum, whatever. If the universe is infinitely large, why can’t we have infinities in the small? If the primordial ylem contains infinite information, then why shouldn’t the tip of my finger?

Regarding physical infinities, here are two relevant quotes from Georg Cantor, the founder of set theory, which is the mathematical study of infinity.

The fear of infinity is a form of myopia that destroys the possibility of seeing the actual infinite, even though it in its highest form has created and sustains us, and in its secondary transfinite forms occurs all around us and even inhabits our minds.

The actual infinite arises in three contexts: first when it is realized in the most complete form, in a fully independent otherworldly being, in Deo, where I call it the Absolute Infinite or simply Absolute; second when it occurs in the contingent, created world; third when the mind grasps it in abstracto as a mathematical magnitude, number or order type.

—-Georg Cantor, writing in the 1880s, passages translated by me for my nonfiction book, Infinity and the Mind.

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Okay, fine. At this point you might wonder why I’m so obsessed with arguing for infinities in the small. Well, my issue, once again, is that, if we are going to have an infinite space with infinitely many planets, I don’t want to frikkin’ squander this endless grandeur on lame-butt wallpaper-like repetitions of the same things over and over and over. I don’t want the cosmos to be sit-com reruns. I want our infinite space to be filled with infinitely many worlds with no two alike! And if each of us is in fact an infinite being, then it’s clear that there’s no need to repeat. QED.

And now we get to my real underlying purpose in these posts. I’m working on a novel called Million Mile Roadtrip. My teen characters are in a car on an alternate universe that has the form of an infinite more-or-less flat plain (albeit with mountains and seas). The plain is divided into basins about ten thousand miles across. Each basin is, in effect, like the surface of a new and different populated planet, no two the same.

My question is this: what is the process that seeds a universe with an endless number of different worlds? Where does all that tasty info come from?

I’m not at all into the defeatist shoulder-shrugging non-explanation along the lines of, “Well, there’s infinitely many planets, and each one is completely random, the product of quantum coin flips so pretty much everything possible has to crop up.” This multiversal line of thought is currently much in favor, but I don’t like it. I want some frikkin’ answers, man!

[Rings from Isabel Jewelry.]

Although it’s not quite relevant, I’ll quote from an 1880s essay by Hermann Schubert, attacking the then popular spiritualist notion that there are ghosts which live in the fourth dimension. Schubert winds up his essay with these stirring words:

The high eminence on which the knowledge and civilization of humanity now stands was not reached by the thoughtless employment of fanciful ideas, nor by recourse to a four-dimensional world, but by hard, serious labor, and slow, unceasing research. Let all men of science, therefore, band themselves together and oppose a solid front to methods that explain everything that is now mysterious to us by the interference of independent spirits.

Great rhetoric. To make it fit what I’m talking about in this post, I’d need to change “independent spirits” to “unseen, endlessly spinning roulette wheels.”

What if we say that there is only one universe, and its infinite, and it’s not random at all, no, it’s filled with lovely, beautiful forms such as might be crafted by an infinite, omnipotent, eternal Mind? What if our universe is a supreme work of art? [Religion alert! Angry buzzers! Flashing red light!]

But wait. What if the supreme mind isn’t some distant, bossy God. What if the Mind is omnipresent, an ocean in which we swim, a great dance in which, by thinking our own thoughts, we participate. What if you’re infinite—and you made the world? Look within yourself. Could it really be otherwise?

Okay, fine, I enjoy writing mystical effusions, they get me high. But then I like to try and tighten it down a little more. What would be a reasonable process by which some great Designer might come up with endlessly many cool planets without having to comb through a superexponential amount of junk?

My Twitter follower R. R. Mutt reminded me of a recent article in the New Yorker about a guy, Sean Murray, who’s using what they call a “procedural algorithm” to generate endless numbers of planets for the artificial world of a forthcoming videogame, No Man’s Sky.

The idea is to come up with an algorithm that accepts some random parameters, and use that for creating your endlessly many populated planets. The desiderata are that (a) the algorithm produces something fairly reasonable-looking no matter what the parameters are, and (b) the algorithm’s parameter space contains many bifurcations—meaning that different parameters can produce totally different-looking worlds, and (c) the parameter space’s basins of attraction have a fractal quality, meaning that, by digging down into more and more precisely specified parameter sets, you can obtain infinitely many interesting and distinct outputs.

The DNA in our cells is, in a way, a parameter set for the procedural algorithm of life on Earth. And the algorithm is very richly sensitive to the parameters.

I once heard the biologist Richard Dawkins talk about the pre-Cambrian explosion of new species being sparked by something that somehow enhanced the “evolvability” of the organisms.

So as universal gods, we’d need something like DNA, but better—for spinning out an endless range of cool worlds.

And that’s all I’ve got today!

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!