by Rudy Rucker
The three-dimensional space of our universe consists of a continuum of c idealized mathematical points. There are two types of substance moving about in this space: mass and aether.
We all have a pretty good idea of what mass is, but aether? Aether is a very tenuous sort of substance associated with the transfer of energy. We do not necessarily assume that aether fills up all of the space between the bits of mass. All we know is that some regions of space contain mass, some contain aether, and some are empty.
Now any massive object—such as a rock—can be endlessly cut into smaller and smaller pieces. In the limit one ends up with alef-null infinitely small bits of mass. These indivisible bits are called mass-monads. In general, then, any massive object is an arrangement of alef-null point-sized mass-monads.
Aether is also infinitely divisible—but even more so! Any aethereal object is to be thought of as an arrangement of alef-one point-sized aether-monads. Since we have c points in space, and since c is at least as great as alef-one, there is certainly room for all these monads.
At any instant, then, the state of affairs in our universe can be specified by stating which of the c possible locations in our space is occupied by a mass-monad, and which locations are occupied by aether-monads. To put it another way, space contains a set M of alef-null points occupied by aether. The state of the universe at any instant depends only on the properties of the two sets of points M and A.
Cantor spends most of the 1885 paper describing a special way of splitting M and A into five significant subsets. He closes with these words: “The next step will be to see if the relations between these distinct sets can account for the various modes of existence and action exhibited by matter—such as physical state, chemical differences, light and heat, electricity and magnetism. I prefer not to explicitly state my further speculations along these lines until I have subjected them to a more careful consideration.” Cantor loved italics.
When I finished reading, I sat there looking out the library window for a while. The clouds had blown away during the night, and it looked like we were going to get one last taste of Indian summer. The blue sky was like a taut stretched film of color. Dry leaves scuttered up and down the asphalt campus paths.
My mind was exceptionally clear, and I could remember every word of the remark Cantor had made to me in the tunnel to alef-one. “If there were a third basic substance in addition to mass and aether, then we would know that c has power at least alef-two.”
I thought this over. Say there were a third substance—call it essence. Mass, aether, essence. What would essence-objects be like?
If mass is like a pile of sand, then aether is like water. Essence would be even subtler, even more continuous. Perhaps the white lights were made of essence. Higher levels.
One thing was clear. To differ from mass and aether, essence-objects would have to be made of alef-two monads each. But if there were essence objects in our space, then space would have at least alef-two points—and the Continuum Hypothesis that space has alef-one points would be disproved.
Well and good. But how… Suddenly I heard the clock tower strike two. It was time for my Foundations of Geometry class. Hurriedly I gathered together the sheets of paper I’d been writing on. I left the Cantor book on the table by the window and rushed out of the library.
A fitful little breeze was herding things around. Pale orange leaves nipped my ankles. I tried to imagine that there were subtler forms around me as well. Aethereal forms, astral bodies, spirits, bloogs.
It seemed reasonable to assume that most of the things I’d seen in Cimön were made of aether. Alef-one aether-monads each. For now, I didn’t want to try to think any more about the possibility of essence-objects. Today it would be enough just to grasp the implications of Cantor’s original idea about mass and aether.
It was a long downhill walk from the library to my Foundations of Geometry class. The class was held, due to some quirk of the scheduling office, in one of the phys-ed classrooms connected to the gym. The gym was all the way down at the bottom of the campus.
Lake Bernco floods often, and is surrounded with rich flat soil. Fields jigsaw around the lake and its tributary rills. The largest of the feeder streams is edged with trees and there is a dirt road running along next to the trees. As I ambled along, I could make out puffs of dust from some farmer’s car, hurrying along the stream’s great curve. There were flies in the air, and their buzzing seemed the very sound of sunshine. I kept asking myself what it would mean for ghosts to be made of alef-one aether atoms each. Two ideas came.
First point. Given a shoe-box you can either fill it with alef-null mass-monads or alef-one aether-monads. Even though both types of monads are vanishingly small, it would have to be that the mass monads somehow behave as if they are coarser, rougher, less densely packed. Presumably an aethereal body can trickle through the interstices in a solid mass-object. Therefore ghosts can walk through walls. Good.
Second point. A beast with four feet has no difficulty in counting up to three. A physical body has alef-null mass-monads, and is happy with smaller numbers like ten or ten thousand. If an astral body has alef-one aether-monads, then it stands to reason that it can handle alef-null. Therefore astral bodies should be able to carry out infinite speed-ups, but would have trouble with alef-one. Good again.
My students were waiting for me outside the gym annex. some of them started laughing when they saw me coming. I put on a friendly professional face as I walked up to them.
The tall kid with the mustache—Percino—spoke up, “How are you feeling, Dr. Rayman?”
“Fine.” I said it as blandly as possible. I recalled that Percino was the boyfriend of that barmaid at the Drop Inn. Mary. I still hadn’t had a chance to ask her what my body had done there Wednesday night.
I unlocked the door, and the class followed me into the quiet hall. Sunlight was slanting in through a window at the end, lighting up a shifting multitude of dust-specks. Something I had read about the Pythagoreans popped into my mind. They believed that there are as many spirits about us as there are motes in a shaft of sunlight. For an instant I could feel an endless hierarchy of spirits teeming around me.
“Mary says you were pretty twisted the other night,” came the confidential mutter. Percino was walking along next to me. He waited avidly for my response.
“I had a couple of beers,” I said, stonewalling.
Fortunately we reached the room then, and I was spared his follow-up. I could see some half formed scheme of blackmailing me for a good grade percolating behind his murky eyes. He was the one doing his term-paper on UFO’s. I hoped he would get me off the hook by doing a good paper.
The students filed in and sat down. I began to talk, pacing slowly back and forth in front of the blackboard.
24: Teaching
“Last time, as you may recall, I talked about the writing of C.H. Hinton. His great concern in life was to make the fourth dimension into something real. I find myself in a similar position today. I want to convince you that infinity is real.”
Some of the students looked uncomfortable at this. There was one girl in particular, a tough cookie with blond hair cut shorter than mine. She regularly asked me what my lectures had to do with the geometry she intended some day to hammer into high-school students.
Glancing at her I lied, “The concept of infinity is crucial for a proper understanding of the Foundations of Geometry.” Some of the students sensed a cover-up and chuckled a little. I had already used the same excuse for lecturing on the fourth dimension for three weeks. “Just give me today,” I said with a pacifying gesture. “I’ve got to talk about infinity today.” There were some smiles and some sighs, but everyone looked ready. I began.
“The idea I want to develop today is that the human mind is infinite. I mean this quite literally. If this class is a success, you will all leave this room with the ability to think of infinit
e things.
“Now, people often assert that it is impossible for us to fully conceive of infinity because our brains are finite. There are two rebuttals to this. First of all, how do you know your brain is finite? It is, after all, quite possible that any bit of matter is made up of smaller bits—so that any material objects actually has infinitely many bits of matter in it. Just before I came here I was in the library reading an article by Georg Cantor. He claims that each piece of matter contains alef-null indivisible bits—what he calls mass-monads.”
The students looked blank, and I back-tracked. “The point is that maybe the brain isn’t finite. Maybe it has infinitely many tiny bits in it, so that you really can have infinitely complex patterns in your head. Can you feel them?”
My head was beginning to tingle a little. A fat girl in the back row nodded encouragingly and I continued. “That’s the first line of defense. Now for the second. Suppose the brain were completely finite after all—just a sort of finite network with only finitely many possible configurations. I want to claim that even then it would be possible to experience infinite thoughts.
“The reason is that we are not just made of mass, of flesh and blood. We have souls, ghosts, astral bodies—there is another order of existence. And on that level we are surely infinite.”
Some of the students glanced at each other with smiles. One of them spoke up, a physics major named Hawkins. He talked slowly, with a heavy Long Island accent. “That’s just your opinion, Doctah Rayman. You think you have a soul. I think you’re just a complicated machine. We could argue all night about it—but why waste the time? There’s no way to win.”
I was beginning to see green and pink flashes. I tried to collect my thoughts. There had to be a way to bring infinity to Earth. “That certainly seems like a reasonable point,” I said smiling. I liked Hawkins for always disagreeing with me. “I guess it’s a matter of put up or shut up. Either I show infinity to you right now, or I admit that it’s just a convenient mathematical fiction. Now let’s see…”
I looked out the window for a second, lost in thought. The soccer team was practicing. Looking at one of the distant players I had a momentary shift of consciousness. I could see through his eyes, feel the ball against my toe. I shifted back and forth between single and double consciousness, between One and Many. I began to feel something. Suddenly I saw bloogs outside. I turned back to the class.
Kathy was sitting in the first row, smiling at me uncertainly. She was made of greenish aether and there were pink bloogs all over the ceiling. Kathy’s lips moved. I could only hear the pounding of my heart. She really had followed me back to Earth. And Satan didn’t have her after all. I walked over and tried to touch her, but my hand went through her head.
I realized then that the students were watching me curiously. I began again to lecture, talking almost at random.
“Take self-consciousness. You know that you exist. You have a mental image of yourself. In particular, you have a mental image of your state of mind.” I drew a thought balloon on the board with a variety of shapes inside it.
“Say that this is your mind. Now suppose that you decide to think about your mind as well as about the other things.” I squeezed a small thought balloon into the bigger one, filling it with the same shapes as before. Then I drew an even smaller thought balloon inside the small thought balloon. Some of the students began to laugh.
“You see the problem,” I said, turning to face them. Kathy had a bag of something in her lap. I couldn’t let myself look at her for too long, for fear of falling into her eyes. I picked up the thread of my argument.
“The idea is that if you form an image of your mind, then this image has an image of your mind, which has an image, which has an image… And so on. We are capable of thinking infinite regresses.”
Hawkins spoke up. “You can’t draw that picture more than about five levels deep.”
“But I can think it all the way through. That’s what real higher consciousness is all about. That’s the first step to merging with the Absolute.”
The girl with short blond hair lost her patience at that. “Isn’t this supposed to be a Geometry course?” Everyone roared, and I fumbled for an answer.
“Just give me a little slack,” I said for the second time that day. Meanwhile Kathy had stood up and dumped the contents of her bag out on my desk. Fuzzweed. A mound of fuzzweed. I nodded vigorously to her. She’d come to help me.
“I’ve been thinking about infinity a lot recently,” my voice was saying, “And you should remember that space is made of infinitely many points—though no one knows what the exact level of that infinity is.” Kathy took out a cigarette and lit the mound of fuzzweed. It began to smolder like a pile of autumn leaves. Light blue tendrils stretched towards the students.
“I really think there’s a chance that some of you could grasp the notion of infinity right now.” I walked over and closed the window. “Just relax and try to form an image of your mind.” There were titters, but I raised my voice. “I mean it. Let’s just meditate together for a couple of minutes, and then we can all go home. I promise to have a more together lecture on Monday.”
I sat down and propped my head in my hands. Some of the students followed suit, some leafed back through their notes, and some stared out the window. It didn’t matter, though. A blue haze of fuzzweed smoke had filled the room.
I pushed my face forward into the plume of smoke and inhaled. I felt a loosened sensation all over my body. Kathy breathed in some smoke, whited out, snapped back. We smiled at each other.
The students were beginning to look a little dazed. Percino yawned, then stretched his arms out. Only they weren’t real arms. He realized this, and jumped to his feet in surprise. His body stayed in his chair. Hurriedly he got back in it.
It started happening to all of them then. We weren’t whiting out, just coming loose from our bodies, getting into aethereal consciousness.
There was no time to waste. It’s possible to grasp alef-null-sized collections once you’re in your aethereal body—but you need some to look at. My job right now was to generate infinities.
“La,” I said, “La, La, La,…” I tried going into a speed-up, but my physical tongue tangled with my astral tongue and I stuttered to a halt. I would have to try something else.
I slipped out of my physical body and began running around and around the room. I did alef-null laps, took Kathy’s hand and did alef-null more. Percino jumped out of his body and joined me for the next set, and then the whole rest of the class joined in—even the hard-faced blonde.
As the fuzzweed spread its smoke between the two sheets of reality, we slid faster and faster around the room. We started running on the walls. My body had started out as the usual pale green nude copy of myself, but as I ran I grew more and more streamlined.
There were people all over the walls—no one could have said who was first or who was last. Some of the students streamlined themselves as I had, but others added new complexities to their forms. I saw a lobster whizz by, and then gryphons and dodo birds.
All the while our physical bodies sat slackjawed in their chairs. We whipped through alef-null more laps and fell into a laughing heap in a corner of the classroom, too excited to talk.
I looked for Kathy, but she had disappeared. The fuzzweed on the desk had burned out, leaving no ashes. My consciousness was jittering back and forth between my astral and physical bodies. The gap was upsetting.
I walked back to my body and slipped in, waiting for the tight feeling that would signal I was bonded again. The students—lobsters, turtles, nudes—crept into their bodies too. I wondered what would happen if two of them were switched.
Suddenly things lost their aethereal shimmer, the bloogs disappeared, and I was locked back into my meat. I wished I knew how to enter and leave at will. For a minute I tried to bring back that loosening sensation the fuzzweed had given m
e, but I couldn’t quite get it. There was something so simple, yet so…so elusive about the transition.
“What happened?” a kid with glasses and dandruff asked. “Did you hypnotize us?”
“It was drugs,” the blonde girl said, looking upset. “He was burning something on the desk.” She stood up to go, probably to the dean.
“There’s nothing on the desk,” Percino pointed out. “It was a close encounter of the third kind. Didn’t you see that green, glowing being?”
“The main thing is that you saw infinity,” I said, standing. “I hope. I’ll see you Monday.” I had to go find Nick DeLong.
A few of the students left, a few just sat in their chairs, and one or two came up to talk to me. Each had his own interpretation of what had happened. This was, in a way, disturbing—for it made me wonder if there was any reason to believe that my version was the correct one.
The nicest description of it all came from the fat girl who always sat in back. She understood everything and wrote perfect exams, but rarely spoke up. “It was a caucus race,” she said to me in a low voice. “Just like in Alice in Wonderland.”
I walked up through the campus alone, struggling to capture the exact feeling I’d had just before I’d seen Kathy. Percino was walking up the hill fifty yards ahead of me, and as I gazed absently at his back I again had a feeling of shifting consciousness.
I could feel his tight shoes, see through his inexperienced eyes. I was equally present in both our bodies. There were other figures here and there on the campus, and I reached out to them too. I was a jelly-creature with dozens of eyes—all equally important. Suddenly I knew how to leave my body. Many to One.
I pulled myself back from the bodies with a sudden twitch. I congealed into an astral body twenty yards away from my physical body—about halfway between it and Percino. Suddenly there were two of me on the hillside—one made of aether and one made of mass.