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 the set of finite natural numbers {0,1,2,3,…}. This “power set of the natural numbers” is the same size as the set of all real numbers.

[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. In 2008, the ezine Tor.com published my SF story along these lines called “Jack and the Aktuals, or, Physical Application of Transfinite Set Theory.”

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.

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.