“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?”