Today I’m returning to a question that’s nagged at me for about fifty years. You might call it the question of recurrence.

Would an infinite universe necessarily contain identical copies of me?

My post today uses some of the material that I wrote for the preface of the 2004 edition of my nonfiction book, Infinity and the Mind.

As is usual for my blog posts, most of images I include will have nothing to do with the text. At least not at the conscious or deliberate level. At the intuitive and synchronistic level, of course, each of them is a sly commentary and a perfect fit.

I’m thinking about the issue of recurrence again because I’m reading Max Tegmark’s stimulating 2014 book, Our Mathematical Universe. It’s has the best (only?) explanation I’ve ever seen of how cosmic inflation can be viewed as impling that the space of our universe is infinite—this is a connection I’ve always wondered about. The latter part of the book gets into the more familiar ideas that quantum mechanics might lead to an Evertt style multiverse, and to the very shaky idea that the world “is” a mathematical object, and that every mathematical object “is” an existing physical universe—which is, in the humble opiion of this mathematician, bullshit. You can find an early survey of TEgmark’s ideas in his 2003 article by Tegmark, “Parallel Universes,” in Scientific American.

Anyway, as I say, Tegmark is great on infinite space. Right off the bat, let me point out that there’s no mathematical certainty that a supertraveler across endless space with endless galaxies would necessarily find another Earth just like the one right here. Consider for instance an infinite set of integers which has only one odd member, the number 3. Someone who starts at 3 and looks for another odd number is going to be disappointed.

{2, 3, 4, 6, 8, 10, 12, … , 2n, …}

So, as a mathematician, I’ve always bridled at the physicists’ claim that an endless world has to repeat. An infinite set doesn’t have to exhaustive.

As a higher-order rebuttal to the tedious insistence that an infinite world must repeat, I’d also point out that it might be that the objects in our world are in fact infinitely complex—and are not like the Lego-style pixel-by-pixel structures that physicists like to image them to be. If matter is endlessly smooth, then there’s room for endless, non-repeating variety.

But let’s look at Tegmark’s reasoning before I argue about it any more. The thing to keep in mind is that nowadays many cosmologists believe that our space really is truly infinite, with an endless number of stars and planets. The initial Big Bang singularity is to have happened not at any single point, but across an infinite space.

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If the old image of the Big Bang was like a white dot appearing in a plane, the new image is like an entire endless plane becoming suddenly illuminated in every part. You might visualize a sheet of light settling down upon the plane.

More precisely, Tegmark describes a sequence of events before the big bang: a tiny bit of darkly energetic supermatter undergoes an insanely rapid inflation which will last forever. The inflation fills out a finitely wide region of space and an infinite amount of future time. And what we perceive to be the big bang is a long, skinny, infinitely extended U-shaped hypersurface within the inflationary zone.

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Very hard to wrap your mind around this, but Tegmark gives a good presentation of the idea in Our Mathematical Universe, using the figure shown above — which is © Max Tegmark, 2014.

So what we call the Big Bang is that bumpy U-shaped line between the light (inflationary) zone and the dark (normal space) zone. And from our perspective that line is one simultaneous moment that happened fourteen billion years ago. And it’s infinitely large.

This idea is so novel, startling, and hard to understand that I’ll go over it again in my folow up post, “Against Recurrence #3“, that is in the post after next.

I’m going to have to talk about some very large finite numbers now. And the way we do that is with exponents. But it’s hard to show an exponent in HTML. So instead we’ll use a caret symbol, like ^, to mean “to the.” That is, we can think of 100 as 10^2, and 1,000,000 as being 10^6. Fine. And now we’re going to go crazy with this.

The region of our endless space that’s presently visible to us is a sphere with a diameter of some 10^27 or one octillion meters. (I should mention that the standard name for a number of the form 10^((k+1)∙3) has the general form k-illion.) This octillion-meter-wide sphere, which we can call our home volume, contains those objects that are close enough so that light from them has had time since the Big Bang moment to travel to us.

Suppose we make the unexceptionable assumption that our home volume has an average temperature of less than a hundred million degrees centigrade (the sun’s surface is a mere five thousand degrees centigrade). In this case, according to Tegmark, the home volume has room for some 10^118 protons. To get a handle on this number it’s useful to use the number googol, which is written as a one followed by a hundred zeroes, that is, a googol is 10^100.

10^118 = 10^(18 + 100) = 10^18 x 10^100 = 10^((5+1)∙3) x 10^100 = quintillion googol.

Now we can wonder how many distinct possible home-volume-sized regions there could be. Okay, this is where Tegmark and his camp “beg the question,” that is, this is where they slip in the conclusion that the want to reach, importing their eventual conclusion as a plausible observation about the state of affairs.

Here it is: “We can think of the home volume as a jungle-gym grid with a quintillion googol slots. One can specify an arbitrary random visible universe by deciding what to put in each slot — one might leave a slot empty, put a proton or neutron in there, or perhaps stick in an electron or some other kind of particle.”

So, okay, I’m going to argue with that. But for now, let’s follow the flow of the Tegmark argument.

To keep things reasonably simple, let’s suppose we have ten alternate ways to fill each of the quintillion googol proton-sized slots. In that case, the number of possible ways to populate a home volume with matter consists of choosing among ten options a quintillion googol times in a row, which is ten to the quintillion googol power. In describing this number, it will be useful to use googol’s big brother, the googolplex, which is a number written as a one followed by a hundred zeroes. If googol is 10^100, then googolplex is 10^googol, or 10^(10^100).

10^(quintillion googol) = 10^(10^118) = 10^((10^100)x(10^18)) = (10^(10^100))^(10^18) = googolplex^quintillion

So now we know that there are at most googolplex-to-the-quintillion possible versions of how our visible universe could appear. A large number, yes, but if our universe is truly infinite, there will be an infinite number of possible home volumes besides ours, and it seems likely that one of them could be an exact match for your own. (Although, again, this is not a certainty.)

How far off might the first copy of our visible universe be? One idea might be to set out in a straight line and whip through the first googolplex-to-the-quintillion home volumes. Just for fun, let’s give this distance a made-up name: one striiide. Given that a home volume has a diameter of a octillion meters, a striiide is an octillion googolplex-to-the-quintillion meters. Would traveling this far guarantee a hit? Not quite.

A little calculating of probabilities indicates that if I travel one striiide, I have a 63% chance of encountering a home volume exactly like the one I started from (the precise probability is very close to 1 – 1/e, where e is the base of the natural logarithm). But as I travel more striiides, the odds go up, and after ten striiides, my chances of having found a visible universe exactly like ours is better than 99.99%.

So, the argument concludes, if the universe really is infinite, then there probably are other people exactly like us somewhere out there. I find this reductive and dispiriting—but I suppose someone might view it as liberating. Like, even if you goof up your life, some other you will get it right?

Nah, to hell with that. It’s like saying nothing I do in my life matters because one of these days I’ll be safe in heaven. Don’t bet on it!

In my next post, maybe next week, I’ll say more about how we might evade the dull Lego-block view of our world as being finite in the small. If it’s infinite in the large, why shouldn’t we be enjoying infinity right here and now. Right in our bodies, and in our minds!

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Your mind is an infinite Jackson Pollock. Stay tuned.

I have a lot of old photos that I never got around to putting into a post. So I thought I’d reduce my inventory in a few long Photo Bin posts.

But first a word from your sponsor.

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Nice blurbs for my Journals from two of my friends.

“The publication of Rudy Rucker’s Journals beautifully supplements his astonishing autobiography, Nested Scrolls. For anyone who has marveled at Rucker’s gonzo, idea-rich fiction, this behind-the-scene account of his daily inspirations, brainstorms, detours and dead ends will be essential reading. But far more than that, it shows us the essential man behind the keyboard, so to speak: father, husband, citizen of the galaxy.”
—Paul Di Filippo, author of The Steampunk Trilogy.

“Rucker’s Journals are great. I fear he will be famous, long after he’s gone, for providing the best picture of late American society. Out peeps Pepys.”
—Terry Bisson, author of Any Day Now .

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Two days ago at midnight I learned how to use the pan function of my Fuji X100S. I like seeing my foot at ease there.

The next five or six photos are lo-fi iPhone 5 photos from yesterday. As they say, the best camera is the one that you have with you that day. But I do feel regret when I’m somewhere interesting with only the iPHone.

A passel of punk stickers on…some technical object. Along Bear Creek Road between Boulder Creek and Los Gatos, where there’s this one pull-out and you can look out across the big basin and see all the way to the open waters of the sea.

A skull in the Boulder Creek sage/hermit cabin of Nick Herbert, a.k.a. Frank Shook.

Saint Nick himself. Nick wrote a brilliant popularization of quantum mechanics: Quantum Reality. And his profound paper, “Holistic Physics: An Introduction to Quantum Tantra” is so important a key to my work that I keep it handily available online on my site.

My novels Frek and the Elixir, Postsingular, Hylozoic and The Big Aha were all profoundly influenced by this epochal paper.

Why isn’t Nick better known? Jorge Luis Borges put it in his essay on Herman Melville: “’Vast populations, towering cities, erroneous and clamorous publicity have conspired to make unknown great men one of America’s traditions.”

A superheterodyne laser collimation unit on Nick’s ceiling.

Two nuts.

And now back to the Fuji x100T.

Me lying on the grass staring lovingly into the lens of the new toy. Like a high-school swain on his first picnic date.

A bottle of Tide with someone’s fingers.

Mandatory calibration shot of Zhabotinsky geranium leaves.

Fuji-seen light through a towering oak.

The rest of the shots in today’s post are from my old Sony RX100, presently laid low by zoom-barrel jam.

“Hell Courtesan” scroll, ~1850, by Utagawa Kuniyoshi ( Click for a larger version of the painting.

Saw this at the Asian Art museum show “Seduction” in SF. Utagawa Kuniyoshi is awesome. He did about six paintings of the “Hell Courtesan,” a legendary geisha who was converted to a higher way by a Zen monk. Here’s a diffuse but interesting post about zen monks and prostitutes. I saw a post somewhere with a giant tattoo of the monk and the courtesan, but I can’t find it now.

Manhole mural inside the Coit Tower.

SF is full of nice ironwork.

Classic EAT sign at Gott diner at the Ferry Building in SF.

Children’s birthday-party hats. Wee wizardry. I always love the reflections of curved surfaces.

Ah, the heart’s nostalgic clutch at the sight of long grass and a motorcycle-tire swing.

This was in the nice AirBnB cottage where Sylvia and I stayed near Occidental, CA. The lady of the estate, Marylu Downing free-hand painted this great star on the wall. Her husband, Roger House, did great proofing work on my Journals.

Heaven for cows around here. I always tell the kids that after I die, I’d like to be reincarnated as a cow on a slope above the Pacific. In Big Sur maybe, or maybe above Bolinas here. That cow is me in twenty years. Another good reincarnation option would be as a pelican. Pelicans as the Hells Angels of the sky. Trundling past in a flying V. Vooden vooden.

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Panorama, seen from the road to Bolinas from Occidental, Coleman Valley Road. Cosmic, uplifting, beyond the beyond. And really only a couple of hours from our house.

Big stump at the mouth of the Russian River, where it hits a sandbar island and trickles into the sea.

Lone windblown Monterey pine near Bolinas Head.

Photogenic sail off the cliff of the Bolinas Head. So heart lifting to stand there in the wind. We have lots of room, lots of room.

My art show at Borderlands opened this week. It’ll run through July 6. We’ll have a reception part on Saturday, June 13, at 3 in the afternoon. We’ll hang out, I’ll do a reading from Journals 1990-2014, and give a little tour of the paintings.

Click for a larger version of the poster.

And here’s a panorama shot of one of the walls in the Borderlands cafe.

Click for a larger version of the pan shot.

Many thanks to Rudy, Jr. and fellow Monkeybrainer Devin for helping me set up. No way could I hang all these paintings alone. You can find prices on my Paintings page.

“The Sage and the Messenger” oil on canvas, May, 2015, 28” x 22”. Click for a larger version of the painting.

Over the last couple of weeks, I finished two new paintings for the show. The Sage and the Messenger relates to a short story I’m working on with Bruce Sterling. One of the characters is sage or hermit who lives inside in a hollowed out spot high up in the trunk of a sequoia tree. And a artificial biotweaked organism comes to bring a message to him. Wanting to lure him into a wild and hare-brained adventure. The messenger iss a thing like biological drone, or like a flying jellyfish. I like the interplay of the expressions among the sage, the jellyfish and the squirrel.

“Cells” oil on canvas, May, 2015, 24” x 24”. Click for a larger version of the painting.

About four days before the show I dove into Cells. I had no real idea what I’d paint when I started. First I did an underpainting in acrylic with a heavy gel medium to get some texture, and to have some extra color glowing through. But I don’t like how flat acrylic looks, so I layered an oil painting on top of that. I outlined some blobs in my original painting, and then filled them in to look like living cells. I used a fan brush for the halo effect, and I flicked the bristles of the fan brush to add some life with splattered dots.

A big push.

My old Sony RX100 camera died this month. The telescoping zoom lens seized up and won’t properly go in and out. I’ve had thee or four pocket digital cameras die in this same way. The lens is a definite weak point. When it breaks, it costs almost $200 to fix, so it’s questionable if that’s worth doing. So I decided to get a small “prime lens” camera, that is, a camera with a non-zooming lens. So there’s not the telescoping crap to break.

I sold a couple of paintings this month, so I splurged and went for a Fujifilm X100T. It’s a compact digital camera (despite being called Fujifilm), solid, great lens, solid metal frame, and with a nice old-school look, kind of like a vintage Leica M-Series camera, but at relatively reasonable price. Not a pocket camera, but it’s small and light.

So I’m going around taking lots of pictures. Some reviewers like to gush that the camera is so good that they use the X100T images just as they came out of the camera. Me, I pretty much run every single shot though Lightroom and/or Photoshop. That’s my work flow. That way I can crop, possibly lighten the shadows, maybe sharpen the image or warm the tone. But this particular shot is right out of the camera. A sweet shop.

I call this terrifying “hand puppet” Cousin Millie. I’ve been showing her to my kids and the grandkids for years.

And the camera is automatic enough that you can hand it to someone that the shot comes out good. I’m till learning the ins and outs of fine points of the controls. The (online PDF) manual is well over two hundred pages long.

The thing about walking around looking for shots is that I dig below the smooth familiar reality to find little bits of oddness. A dial with numbers. What might this mean?

You can never go wrong photographing street-workers’ markings on the asphalt.

Midnight in the china closet. The X100T really fills out the three-dimensional space.

Exultant in son Rudy’s car, riding through the Mission after my paintings…with my new shooter in hand.

I’ve photographed these phone/electric/cable lines a dozen or more times over the years. I think this one is better than usual. Thank you Fujifilm!

My grad-school friend Jim Carrig’s son Eamon showed up the other day. I took him to San Jose’s finest fast food stand, named simply Falafel. It’s on Stevens Creek Blvd near Bascom Ave. They’ve been there since 1966. Wonderful, wonderful falafel. Green inside, freshly cooked and mashed beans, crisp on the outside, hot.

Eamon Carrig himself. He’s started a company that’s designing small robotic sail boats. Sailing drones for the high seas.

Weirdly enough a woman reporter named Leona was at Falafel. She’s from LA, but is writing an article on falafel restaurants in California for Brownbook magazine, published in Dubai! Once the sheiks hear about Falafal of San Jose they’ll be jetting in no doubt.