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John Conway and the Absolute Continuum

August 31st, 2015

I just read this wonderful biography of the mathematician and recreational-math maven par excellence, John Horton Conway, best known (to his mild annoyance) as the inventor of the cellular automaton rule known as the Game of Life. The book, Genius at Play , is by Siobhan Roberts.

Some great quotes. John Conway’s daughter: “There goes somebody looking strange. Ergo, it must be a friend of Dad’s!”


[This photo has no literal connection to what I’m talking about, although the musician is, in a way, like Conway. It was shot inside the San Jose California Theater during the Jazz Festival, it’s a group called Bombay Jazz.]

A lot of my favorite mathy people are in the book, many of whom I’ve managed to meet over the years: Kurt Gödel, Stephen Wolfram, Bill Gosper, Martin Gardner, and more. Bill Gosper (discoverer of the Life glider gun): “Conway is approximately the smartest man in the world.”

One of the things Conway is most proud of is his expansion of the familiar real number line so as to become the so-called surreal numbers. The surreal numbers are very densely packed—technically speaking there’s so many of them that they comprise what set-theorists call a “proper class,” which is a collection so vast that it is in some sense impossible to think of it as a finished thing. The less daunting collections are called sets: an example is the collection of all subsets of natural numbers 0,1,2,3,…


[Original Tenniel drawing of Carroll’s Alice holding a baby who turns into a pig.]

So how do the surreal numbers arise? There’s a good popular book about them: Donald Knuth’s Surreal Numbers, available in paperback and also for free online. The basic idea is fairly simple. Whenever you have a set L of numbers that are smaller than the numbers in a different set R, then there is going to be a surreal number in between the two sets…the number can be called {L | R}.

In the regular real number system we use a principle like this to say that there’s a number on the line just beyond 3, 3.1, 3.14, 3.141, 3.1415, 3.1459, etc. This number is our friend pi.

In Conway’s system, we go further, and squeeze more and more stuff in. Like there’s a number that’s bigger than 0, but smaller than 1/2, 1/3, 1/4, … , 1/n, … for all finite n. This infinitesimal number can be called 1/omega, where omega is a name often used for the simplest infinite number that lies beyond 1, 2, 3, …, n, …

In Conway’s rich system, however, omega isn’t the very first infinite number, you’ve got omega-1, omega / 2, and even the square root of omega.

Getting technical, the surreal numbers are what the early 20th C mathematician Felix Hausdorff would call an eta-On set, where On is the class of all finite and transfinite ordinal numbers—an ordinal being a generalized counting number. The class On is a subset of the surreal numbers…like the spine of integers inside the real number line. On goes on and on, including infinite numbers like alef-null, alef-one, and so on. When we fill in the surreal numbers, we can have cool things like (((alef-one divided by (square root of alef-one)) minus (pi divided by alef-seventeen) divided by two) minus 48.) It’s all there!

The surreal numbers make up what we can call an “Absolute Continuum” where there’s always yet another number lurking between any successive pairs of sets of numbers. Physics would make more sense, suggests Knuth, if it was based on surreal numbers instead of the so-called real numbers, and I think he’s right. Assume we have endlessly transfinite infinitesimals in the small, and space gets really smooth.

I’m fascinated by the notion that our physical space is in fact an absolute continuum. This is what we might call “infinities in the small” instead of “infinities in the large.” I’ve always felt that it’s a mistake to simply stare out at the stars and yearn for the vastness out there. In my opinion, we have huge infinities right here. Underfoot.

In a slightly different vein, the founder of transfinite number theory, Georg Cantor wrote a couple of great passages.

“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.”

I wrote about infinity in two of my pop math books, Infinity and the Mind and Mind Tools. But getting back to the idea of the absolute continuum, let’s just marinate a little more in the idea that physical space is not *feh* quantized, and it’s not just some mere real number line—it might have levels, sublevels, subsub………subsublevels. Inconceivably rich smoothness.

That quantum bump stuff is just a glitch, like a rumble strip by a toll booth, we trundle past that in our Shrink-O-Tron device, going ever downward in our hypertransfinite subdimensional bathyscaphe.

If you have having absolute continua (plural of continuum) for your space and, indeed, the very substance of your body, then there’s no reason to suppose that you will ever be repeated, as a pattern, in the transfinite universe. Those pawky bean-counting quantum Lego-block arguments don’t work if we’re transfinitely smooth. I was talking about that issue in a series of 3 posts called “Against Recurrence” in May, 2015—disputing the common (and false) claim that, “in an infinite universe everything repeats.” No repetition if your gnarly down to the alef-seventh level and beyond, my friend.

I’ll work the details out for you with my occult analog continuum computer, which merely appears to be an electric-cord entangled with a semi-reflective mirror…

I’ll leave you with this image of Davina and the Vagabonds at the San Jose Jazz Festival. Davina was great, she never stopped mugging and making faces. Funny rootsy old songs. Davina looks like a WWII poster of a working woman here. She is, very clearly, embedded in an Absolute Continuum.

Having read to Siobhan Roberts’s charming bio, I get the impression J. H. Conway would enjoy meeting Davina and she, in turn, might well find him fascinating. A mathemagician among us.

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Podcast #87. Interviewed by Doug Lain

August 11th, 2015

August 11, 2015. Interviewed by author Doug Lain for his podcast Zero Squared. We mostly talked about what it’s like to have a career as a writer, with some mentions of my recent Journals 1990-2014. An edited version of the interview appeared as part of an episode on Lain’s podcast Zero Squared, with better sound for Doug’s voice. The version I’m posting here, six weeks later, is the unedited full conversation, recorded at my end at the time of the interview, which was June 26, 2015.

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New York #2. WTC, Seaport, Eddie, Midtown. Plus Backlog.

August 9th, 2015

Here’s a few more photos from Manhattan, followed by some pictures from my backlog.

One of the best deals in the city is the East River Ferry, which runs from 34th St. to Wall St. stopping in four or fives Brooklyn docks on the way. Takes about half an hour, great views and fresh air, uncrowded, $4.

Near Union Square. I love the ever-changing kaleidoscope of people, signs, and machines in the big city. The sign vaguely sinister here, like Big Sister is Watching You. Fashion police?

Cool wall painting near Wall St. This part of town is the oldest, going back to colonial days. I like poking around here. And you see genuine financiers.

We were resting on a bench and two ambulances pulled up—a regular one and a Jewish one. Some bond-trader type guy had had a heart attack or maybe just a panic attack, and after fifteen minutes he came walking out, a little shaky. He went for the Jewish ambulance.

The big new world trade center building is done, they call it One World Trade Center or Freedom Tower. Very cool to see it suddenly there amid the streets down around Wall St.

We made our way there, and it’s really awesome. I felt this surge of patriotism and gladness. We’re back! In your face! Bigger and better. That thing made of fins is a glorified subway stop, like for suburban trains as well.

There’s this little old colonial church by the World Trade Center, it’s St. Pauls Episcopal Church, and it has an ancient graveyard, you can go in there and sit on a shady bench. Kind of dizzying to sit there and imagine a fast-forward movie of the neighborhood as seen from the once-peaceful graveyard—the growth of the city, the rise and collapse of the first towers, the construction of the new ones…and then what?

Such gorgeous ornamentation and elaborations on those 40s buildings. Fractal. When will the long desert of flat glass end?

We made our way over to the South Street Seaport, kind of an urban mall, with cobblestones. Not plastic, exactly, but a bit chainy. Had some great crab cakes amid numerous colorful locals on lunch break. The good part there is a pier with old ships. It was a trip to look at the old ships’ rigging against the Wall St. towers.

I always dig taking high-speed photos of water in motion. It’s all a matter of time-scale. A lot of things in ordinary life can be thought of as slow-motion fluid flow. Our bodies, our cities, out landscapes.

Gotta have a least one photo of a big-ass NY truck. This one’s near Union Square, which is always a good spot to visit. Lots of benches, good people-watching. I bought some new walking shoes near here, New Balance 990s—I got the idea from seeing an article in the Sunday Times, some guy swearing by them. The salesman was great, urban gay, with that tired kind of voice, he observed that he’d sold a lot of this model that day, and I told him about the Times article and he looked it up. On a trip, you’re happy if you manage to have a conversation with anyone.

I always love how the sidewalk scenes are mirrored in the windows.

We got together with our old friend Eddie Marritz for dinner at cool West Village place he recommended, Mary’s Fish Camp. We’ve known Eddie since 1968, when he showed up at Rutgers U. in New Brunswick, New Jersey as a freshman. Sylvia and I were in grad school there—we’d gone to college with Eddie’s older brother Don. Just about fifty years ago. Half a frikkin’ century. Eddie’s a very successful cinematographer now, he gets work shooting a lot of documentary films. It’s so relaxing to talk to a very old friend, the complete trust, the sensitivity to subtle allusions. Sylvia took this photo with her iPhone.

I love the rich architectural details in Manhattan. Like this “awning” over the door in a mid-down.

I’ve got some more NY photos, but I’ll post them another day.

For now, I’m noticing that I have quite of backlog of older photos, so I’ll go ahead a post some of them today. On the shell theme, here’s some of those nice, crisp black scallop (?) shells that Sylvia collected at the Outer Banks. Sitting on a glass table, hovering in space.

What would American photography be without neon signs amid window reflections?

A “Rudy” photo of the pool skimmer at our OBX McMansion-cottage. I dig these kinds of collage / angles / shades photos.

An iPhone photo of my car’s windshield below a palm in Los Gatos a couple of months ago. The thing about iPhones, they basically suck in terms of photo quality, but sometimes they’re all you’ve got. That old saying, “The best camera is the one you have with you.”

I kind of wonder about that recent Apple ad campaign, with high-art black-and-white full-page magazine photos labeled, “Shot with an iPhone 6.” It’s certainly safer to go black-and-white, as then there’s less chance of digital crud. And of course they’d be running the photos through something like Lightroom—dialing up Clarity and Sharpness a bit, and cleaning up pixel stutter with the Noise sliders. And if it’s a wonderfully envisioned and framed shot, it’s gonna look good. If you have strong lighting that helps too. And you’d want to shoot it in such a way that you don’t have to crop, so that the meager iPhone pixel count isn’t being stretched too much for the blow-up.

What’s this? Bolts of cloth and buttons in Hart’s fabric store in Santa Cruz. I end up accompanying Sylvia here quite often, and to pass the time, I tend to take a photo or two. Sf concept: a “fabric store” where supernal god-like aliens buy materials for making quilt-like universes.

Going back another month or two, here’s a seagull at Fort Mason in SF. Haughty aplomb.

A red flag at Ft. Mason! Rothko style. I’m crazy about weathered old walls. Easy to shoot, as they’re flat, like a photo. All you need to do is to see the shot, do a minimally competent job with the camera (never a gimmie!), and fix the light in Lightroom. Got this one the day I went there for the beatnik con with V. Vale and Marian, and I lost my glasses (now replaced).

And that’s about enough for today.

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Outer Banks & New York #1.

August 2nd, 2015

My wife and I went to reunion with all our children and grandchildren at a rented house in Corolla on the Outer Banks of North Carolina for a week, and then we we two went alone to New York city for a week. So I’ll be posting some photos of all that.


[Tail of a Lego-built dragon in New York Lego store.]

Before I get going on the trip pix, I want to mention that on the plane, on our way home, I watched the movie Chappie, which I didn’t manage to see when it came out earlier this year. For some reason the reviews were fairly lukewarm, but I thought it was great.

Chappie is a real cyberpunk robot—he’s got graffiti on his body, he wears chains, he bops when he walks, he curses, he robs an armored car, he beats a militaristic paramilitary guy to death, and he saves the life of his maker. How? He saves his maker’s life by uploading the man’s consciousness into a robot body.

Just like my robot character Ralph Numbers did for his maker Cobb Anderson in my 1980 novel Software.

I know I’ve said this before, but the uploading-human-mind-to-robot-body is something that I frikkin’ invented—in Software, and I elaborated it in all four of my Ware novels, which you can still buy in paperback or ebook (and you can also read it for free in a CC edition.)

I don’t know why I never seem to get much credit for inventing this move, which has been in, like, two hundred movies by now. It’s not like it was an obvious idea when I wrote about it, anymore than a time machine was an obvious idea when H.G.Wells wrote The Time Machine. It took me nearly a year to really figure out the idea, simple as it now seems. I was studying the philosophy of computation at the University of Heidelberg, reading and pondering the essays of Alan Turing and Kurt Gödel. It’s some serious shit. But I chose to present it in cyberpunk format. So no po-faced serious analytic type high literary mandarins are ever gonna take my work seriously. At least not till I die. Or maybe not even then—posthumous recognition is a classic writer’s pipedream, and WTF diff would it make anyway. Rant, rant, rant, rave, rave, rave. Am I eighty years old yet? Move over, Harlan.

The Outer Banks is a long thin island, or group of islands, running long the coast of North Carolina. Back around 1975, Sylvia and I used to drive down there from Geneseo, NY, where I had my first teaching job. Us and the three kids. The Outer Banks were sparsely built then, and we stayed a couple of times in 1940s vintage motel, plywood shacks with linoleum floor, right on the all-but-empty beach, the Lamplighter Inn. Paradise. Those wheat-like plants atop the sunset dune in this photo are called sea oats.

Dig this treasure hunter, just a guy on the beach, not somebody I know. Our son Rudy was very keen to get one of these devices when he was about ten. Mostly he found cans.

Kites are still big here. Lovely to see them at sunset, and sometimes with a crescent moon.

Anyway, the OBX (as they now term the Outer Banks) are fully built-up now, the coast lined with developments of McMansion style beach cottages. We were there as a party of 14, and we got a three story house with a pool and an elevator and a movie room, and the rent was about fifty times as high as that of the long-gone Lamplighter.

But it was great to be with the children and grandchildren, and the ocean was very swimmable—not too cold, and the waves not too big—and there were some good shells.

As the jesting fates would have it, there was a huge OBX shark scare in progress when we arrived. Initially we were nervous, but when I didn’t get bitten in the first ten minutes, I pretty much stopped worrying. The human mind’s risk assessment. Anyway the closest shark attacks had been about a hundred miles away, down in Ocracoke. We were all the way up north in Corolla.

The crabs didn’t eat us…we ate the crabs. A bushel of crabs. What a concept. What if someone snuck in while you were sleeping, and poured this many live crabs onto you in bed? There’s a Grimm Brothers fairy tale along these lines, about a boy who couldn’t feel fear, and he learns when someone dumps a bunch of fish on him in bed like that.

One of the fun attractions on the way down was the home of the monster truck known as Grave Digger. Lots of earlier versions of Grave Digger on display here in Grave Digger Garage. In yo’ face, mofo! My grandchildren are endlessly fascinated by YouTube videos of Grave Digger in action, accompanied by the Grave Digger theme song, George Thorogood and the Destroyers playing “Bad to the Bone.”

Plus the original ur-Grave-Digger prototype vehicle, an awesome sight, like seeing the first fish with legs. Or like seeing the Wright Brothers’ original plane. One reason I’m so interested in these vehicles is that my characters in my novel-in-progress Million Mile Road Trip are driving a station wagon that’s been tweaked into something like a monster truck.

At the beach we saw a lot of awesome clouds. Imagine if there were only a few places on Earth where you can see clouds. How you’d value them. And yet we tend to ignore them, take them for granted, or even gripe about them.

We saw a thunderstorm or two as well. I loved this bright white puff beneath a vast dark anvil. So invigorating to see rain, if you’re a Californian.

Speaking of rain, I cranked my awesome Fujifilm X100T digital camera down to 1/2000 sec exposure time, turned on the flash, and got some shots of raindrops in a storm outside our 10th floor room in Manhattan the next week. That’s not rain on the window pane, you understand, that’s raindrops falling in midair, frozen (more or less) in flight by the magic of postmodern photography.

Here’s another shot of the raindrops, I took this picture about ten times in a row, trying to get it right. Surprise: raindrops do not look like cartoon teardrops. They’re wobbly globs, although, yes, it seems the larger ones are indeed fatter on one side.

After I finished shooting the raindrops in New York, I took a shot of the building facing us across 41st Street, and later, when I examined the photo, I had this Antonioni Blow-Up type discovery that a man in an office across the street was staring at me, probably wondering what I was doing taking flash pictures out my window.

My camera has a fixed wide-angle lens, and really high resolution, and I was able to zoom in on previously unnoticed details in a lot of my Ney York shots. Like here I’m in the lobby of the Chrysler Building, taking a picture of a weird pseudo-digital Deco clock labeled “TIME” in case you don’t know what a digital clock is (and who did, back in the 1930s). And a guard is looking out at me from a door in the wall, smiling, like the friendly bird inside the cuckoo clock, and I didn’t even know he was there.

Another cropped-down zoom photo from NY: a chic woman among the marching ants in a crosswalk at 41st and Madison, which is where our hotel was.

Such awesome people-watching in the big city. We saw this woman at the new Whitney Museum, down on the old meat-packing district. Awesome building, same old collection, but with more of it on display than before.

I’ve never been sure if I liked Willem de Kooning, but I saw a kind of landscape by him called Door to the River, and it really knocked me out. There’s something about it, maybe hard to see in a reproduction or a tiny computer screen image, it’s like the painting captures the glancing quality of light, the way that when you look at something you see patches of brightness and glare even before you overlay your notion of what it “is.” And the title “Door to the River” is kind of uplifting, I mean that’s what we’re always looking for, right, a magic DOOR to the river of LIFE. And, while we’re at it, a frikkin plot for our novels.

As chance would have it, right while I was standing there admiring this painting, my very favorite of that day’s visit to the Whitney, a woman my age walks over to me and says, “Looking at this painting, I’ve finally decided for once and for all that de Kooning was a fake.” I tried to disabuse her of her errorneous opinion, with little effect. Oh well!

Yet another street-photograph of a New Yorker. Note the big fan. It was about 95, incredibly humid, with the sun like a sledgehammer. You had to walk on the shady side of the street.

The beach on OBX was really hot, too, but there you had the option of jumping into the water. And then a half hour in the waves I’d even be cold.

That’s it for today. Naptime. I’ll post more photos of New York later this week.

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