“If I have seen less far than others,” Galileo complained in irritation to Aurora, “it is because I was standing on the shoulders of dwarfs.”
She laughed out loud. “Don’t say that to anyone else.”
They flew over and through number theory, theory of equations, probability theory—which was ever so useful, and instantaneously true to experience as well. It was the way of the world, no doubt about it, the way of the world mathematicized; oh how he could have used that! And how broadly it could be applied!
Quickly with these tools they flew into differential equations, and then to advances in number theory, and what he learned to call differential geometry. Indeed at times it seemed to him that geometry continued to underlie everything, no matter how elaborated and abstracted it became. Geometry converted to numbers, the numbers then mapped by further more complex geometries; thus trigonometry, topology—and all along he could still draw lines and figures to map what he was learning, though sometimes they looked like snarls of wool.
When Aurora led him further on, and they flew into the non-Euclidean geometries, he laughed out loud. It was like pretending that the laws for perspectival drawing were a real world, so that parallel lines met at a hypothesized horizon, which was infinitely far away and yet susceptible to ordinary calculations. A very funny idea, and he laughed again at the pleasure of it.
When Aurora then told him that these impossible geometries often made a better match for the real world of invisible forces and fundamental particles than did Euclidean geometry and Newtonian (which was really to say Galilean) physics, he was amazed. “What?” he cried, laughing again, but this time in astonishment. “No parallel lines anywhere?”
“No. Only locally.”
It struck him funny. That Euclidean geometry was a formal artifice only—it was profound, it overthrew everything. There was no underlying Euclidean grid to reality. And it was true that he himself had once said that no one could build a true plane of any great size, because of the curvature of the Earth. So he had had an intuition of this non-Euclidean world, he had almost seen it all on his own—as with everything else he had learned so far! Oh yes, he had been right; the universe was a wild place, but mathematical. And God was not just a mathematician, but a superhumanly complex mathematician—almost, one might say, perversely inventive, such that He was often contrary to human sense and reason. Although still rigorously logical! And so: integration theory, complex variables, topology, set theory, complex analysis, theory of infinite sets (in which there was a paradox called Galileo’s Paradox that he didn’t recall ever having proposed, so that he was distracted momentarily as he focused on it and tried quickly to learn what he would otherwise have to discover). Then came the mathematization of logic itself, finally and at last—though when he flew through it, he was surprised how limited its usefulness seemed to be. Indeed it mostly seemed to prove the impossibility of logical closure in any mathematics or logics, thus destroying both its parents at one blow, so to speak—a double parricide!
That was confusing enough, but then they flew on. And just as non-Euclidean geometry had made him laugh, quantum mechanics made him cry. He tumbled and fell rather than flew. The live hum of intelligence, even wisdom, that the velocinestic had filled him with, also had in it a huge emotional component, he suddenly understood; and these two aspects of understanding were all entangled with each other. Learning so much so fast, he had been filled with joy; now that ended so abruptly it was like smashing into a glass wall that one had not seen. It hurt. He cried out in startled pain, tumbled downward, shocked and dismayed.
He became light. He was a single minim of light and he flew through two parallel slits in a wall, and the interference pattern of his collision with the wall beyond showed without doubt that he was a wave. Then he bounced through a half-mirrored glass and it was obvious he was an incredibly tiny particle, one of a stream of minims moving one by one. Depending on what flight he was made to fly, he was either particle or wave, so that it seemed he had to be both at once, despite the contradictions involved in that, the impossibilities. Maybe thoughts were minims and emotions were waves, for he was stuffed to exploding with both at once—the emotions in their waves also a myriad of pricking jolts, little affectinos that flew in clouds of probabilities and struck like icy snow. It was true but impossible.
Before he could even try to puzzle this out, he found himself looking at one of these minims, like a chip of sunlight on water. But to see it meant that a minim of light had hit that chip and bounced to his eye, and this minimal hit had knocked the observed minim off course, so he could not make a measurement of its speed by taking two looks at it, because each look cast it on a new course that wrecked the calculation. There was no way to determine both position and velocity of these minims, and it wasn’t just a measurement problem either, a matter of knocking off course. The two aspects existed at cross purposes and canceled each other out at the smallest level. The probability of a course was all there was, a wave function, and measurement itself set one possible version in place. These blurs were the minims themselves, and everything in the world was made of them! Some kind of smears of probability, with mathematical functions describing them that often involved the square root of negative one, and other flagrant irrealities. The wind on a lake, the sun beating down on it, a flutter of light on the water, points piercing the eye.
Galileo flew into another tilted mirror, and both shot through it and bounced off it at the same time, either reintegrating or not on the far side, breaking up as he became whole—
“Wait!” he shouted in panic to Aurora. “Help. Help me! This can’t be right, it makes no sense! Help!”
Aurora’s voice croaked in his ear, full of amusement. “No one understands it in the way you mean. Please, relax. Fly on. Be not afraid. Bohr once said if you are not shocked by quantum mechanics, you have not seen it properly. We have come to an aspect of the manifold of manifolds that cannot be understood by recourse to any images from the sensorium, nor by your beloved geometries. It is contradictory, counter to the senses. It has to remain at the level of the mathematical abstractions that we are moving among. But remember, it has been shown that you can use these quantum equations and get physical experimental results of extraordinary accuracy—in some cases as much as one in a trillion. In that sense the equations are very demonstrably true.”
“But what does that mean? I can’t understand what I can’t see.”
“Not so. You have been doing that quite frequently now. Rest easy. Later the whole of quantum mechanics will be placed in the context of the ten-dimensional manifold of manifolds, and there reconciled to gravity and to general relativity. Then, if you go that far, you will feel better about how it is that these equations can work, or be descriptive of a real world.”
“But the results are impossible!”
“Not at all. There are other dimensions folded into the ones our senses perceive, as I told you.”
“How can you be sure, if we can never perceive them?”
“It’s a matter of tests pursued, just as you do it in your work. We have found ways to interrogate the qualities of these dimensions as they influence our sensorium. We see then that there must be other kinds of dimensions. For instance, when very small particles decay into two photons, these photons have a quantum property we call spin. The clockwise spin of one is matched by a counterclockwise spin of the same magnitude in the other one, so that when the spin values are added, they equal zero. Spin is a conserved quantity in this universe, like energy and momentum. Experiments show that before a spin is measured, there is an equal potential for it to be clockwise or counterclockwise, but as soon as the spin is measured it becomes one or the other. At that moment of measurement, the complementary photon, no matter how far away, must have the opposite spin. The act of measurement of one thus determines the spin of both, even if the other photon is many light-years away. It changes faster than news of the measurement could have reached it moving at the speed of
light, which is as fast as information moves in the dimensions we see. So how does the far photon know what to become? It only happens, and faster than light. This phenomenon was demonstrated in experiments on Earth, long ago. And yet nothing moves faster than the speed of light. Einstein was the one who called this seemingly faster-than-light effect ‘spooky action at a distance,’ but it is not that; rather, the distance we perceive is irrelevant to this quality we call spin, which is a feature of the universe that is nonlocal. Nonlocality means things happening together across distance as if the distance were not there, and we have found nonlocality to be fundamental and ubiquitous. In some dimensions, nonlocal entanglement is simply everywhere and everything, the main feature of that fabric of reality. The way space has distance and time has duration, other manifolds have entanglement.”
“My head hurts,” Galileo said. He flew after her toward a beam of violet light. “Spin is something I understand,” he said. “Go back to that.”
“This spin is not like your spin. There can be two axes of spin going at once in the same particle. In the particle called the baryon, there is a spin such that it has to rotate 720 degrees before it returns to its original position.”
“My head really hurts,” Galileo confessed. “Could it be the preparation?”
“No. It’s the same for everyone who comes to this point. Reality is not a matter of our senses. It can’t be visualized.”
“And so time?” Galileo said, thinking of his travels.
“Time in particular is impossible to properly perceive or conceptualize, and very much more complex than what we sense or measure as time. We keep mistaking our sense of time for time itself, but it isn’t so. It isn’t laminar. It bubbles and eddies, percolates and disappears, is whole but fractionated, exhibits both the wave-particle duality and nonlocal entanglement, and is always changing. The mathematical descriptions we have of it now test out in experiments, even to the point of us being able to manipulate entanglement interference, as you know very well because of your presence here. So we know the equations must be right even when we can’t believe them, just as with quantum mechanics.”
“I don’t know,” Galileo objected, growing more and more afraid. “I don’t think I can come to terms with this. I can’t see it!”
“Perhaps not now. It’s been enough for one lesson, or too much. And some people have arrived here who want to talk to you.”
He came out of the visionary flight as if out of a dream that did not slip away upon waking. He found himself back on the roof terrace of the tower, dazed and raw in his feelings. Clarity and confusion, a beautiful impossibility … He helped Aurora’s assistants remove the helmet from his head, then looked down at a glowing mirror in his hand, which was covered with his notes, his crabbed handwriting made big and crude by using his fingertip as the pen. A large diagram of the two-slit experiment filled the top of the pad like a sigil, reminding him that the world made no sense. He inspected the back of the mirror, which appeared to be made of something like horn or ebony.
He said, as if reaching for something to hold on to in a fall, “So it is true, then, that God speaks in mathematics.”
“There is a relationship between observed phenomena and mathematical formulations, sometimes simple, sometimes complex,” Aurora replied. “Philosophers are still arguing about what that means, but most scientists accept that the manifold of manifolds is some kind of mathematical efflorescence.”
“I knew it.” Though mentally exhausted, and confused, there was a glow in Galileo that he recognized, a kind of humming in him, as if he were a bell that had been rung some time before. Then maybe the bell had cracked. “That was quite a lesson.”
“Yes. About four centuries traversed. That’s a lot. But you have to remember that we covered only a small portion of the whole story, and much of what you learned today would in later lessons be overthrown, or superseded, or integrated into a larger understanding.”
“But that’s bad!” Galileo exclaimed. “Why then did you stop?”
“Because to go on would be too much. I trust we will continue later.”
“I hope so!”
“I don’t see why not.”
“Can I call on you?”
“Yes.”
“And will you come when I call?”
She smiled. “Yes.”
Galileo thought over what he had learned. It was impossible to grasp. In a different way than the experiences of his previous trips to Jupiter, it lay just a bit beyond his reach. He remembered it clearly, but he couldn’t comprehend or apply it.
Aurora was looking down at the canal running up to their tower. Galileo, following her gaze, said, “What about the thing that lives in the ocean below you?” he inquired. “Have you tried giving these lessons to it? Have you learned its language, or even hailed it and gotten an answer?”
“We have communicated with it, yes. And the communication has been entirely mathematical, as you have guessed.”
“What other way would there be?”
“Exactly. So, first we tried to find out if the sentience perceives some of the same mathematical operations in its natural phenomena as we do.”
“Yes, of course. And what have you found?”
“It is in agreement with us on the existence and value of pi. That was a first success, established with simple diagrams and a binary number code. Also, it appears to pick out the first twenty or fifty prime numbers, and the usual sequences like the Fibonacci sequence and so on. In short, you may say that when it involves real numbers, or the simplest Euclidean geometry, we appear to be in substantial agreement with it.”
“But?”
“Well …” She hesitated. “When it comes to various higher mathematics, when we have been able to formulate clear questions, the sentience does not seem to recognize what we are saying. Quantum mechanics, for instance, appears not to register.”
Galileo laughed. “So it’s like me!”
She regarded him without joining his laughter. He reconsidered.
He said, “Is this why you agreed to teach me? Because you think I am as sequestered as this thing in its ocean, so that you can use me to get ideas to communicate with it better?”
“Well,” the old woman said, “it’s true that a different perspective on the problem might bring new insights. You are well remembered here on the Galilean moons, as you might well imagine. I believe Ganymede entangled you into this time for other reasons of his own, but some here think you might bring a certain freshness to our local problem too. Others feel your context is just a handicap, and that you can be of no help. In any case, while it’s possible the Europan sentience exists in a mathematical moment roughly corresponding to your own, I think it is more likely that it senses principally in different manifolds than we do. That may form the basis of the problem. Mathematicians with a philosophical bent are having a heated discussion of the ontological and epistemological questions brought up by the situation, as you can imagine.”
“It may think it is dealing with a simpler mentality than it,” Galileo suggested ironically. “Like you think you are doing with me.”
“It is capable of generating very complicated geometrical patterns,” she said, “conveyed to us by sound arranged in a binary code. But there are gaps that suggest it lives in some of the other manifolds.”
Galileo didn’t know what this meant. “The creature must be blind, no? It was really dark in there.”
“It may sense parts of the spectrum not visible to us that would serve as equivalents to our sight. We continue to work out codes of communication in which it sings to us information that we can display as visual patterns for our own comprehension. So in that sense you could say that it sees, I think. Indeed, when we sent to it a schematic of the gravitational patterns created by all the bodies in the Jovian system, it sent us corrections that make us think it knows very subtle aspects of gravitation, aspects like gravitons and gravitinos, which are apparent only when seen in the context of the full ma
nifold of manifolds theory. For us, working with that model is only a recent development. So this is rather thought provoking.”
Then there was an eruption of shouting at the vertical antechamber. It proved to be Hera and a retinue of followers, bulling their way through Ganymede’s crowd. Hera was in the lead, angry and unstoppable.
“Oh dear,” Aurora said. “She appears unhappy.”
Galileo snorted. “Is she ever otherwise?”
Aurora laughed. Hera approached and loomed over them, her white arms thick, bare, muscular, and tensed, as if she were only just restraining herself from thrashing them both, and Aurora’s assistants as well.
“I hope you have not been disturbed by this wandering ghost?” she inquired of Aurora.
“Not at all,” Aurora replied, looking amused. “It was our pleasure to converse with such a famous person.”
“Do you know that such conversations can be dangerous? That you may alter the manifold analeptically enough to change us all, perhaps right out of existence?”
“I don’t think anything that happens to Galileo here could have that kind of impact,” Aurora said.
“You have no way of judging.”
“Measured inertias of temporal isotopies give me a grasp of the chances involved,” Aurora said, in a tone that suggested Hera could form no such grasp.
“Ganymede is trying to use Galileo to change things,” Hera replied. “So he must think it works.”
“Perhaps so. But I don’t think what happens to Galileo here is properly located to make any such change. Besides, Galileo has always had a remarkably strong sense of proleptic intuition. Indeed, when judged by that rubric, of anticipating future developments, I’ve read commentaries that rate him as the third smartest physicist of all time.”
“Third,” Galileo scoffed. “Who are these supposed other two?”
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