Galileo's Dream

Chapter nine Aurora

On his uneasy journey back to Florence, Galileo wrote letters to all his correspondents, explaining to them why his visit had been such a success, even more so than in 1611. All of them had already heard the story from faster sources, and so did not believe his account, but many wrote back to him reassuringly. A success, no doubt about it.

Every night he complained about the inn food, the flea-bitten beds, the creaky floors, and the endless snoring of the other wayfarers (he himself was a prodigious snorer), so that rather than retire he went out to sleep on the cushioned seat of his litter, or on his telescope stool under a blanket.

One night, at the inn on the road below Montepulciano, he could not sleep at all, and so he sat wrapped in his blanket by his telescope. He crouched to look through it at Jupiter, his own little emblem and clock, and in so many ways the home of his troubles. At this moment it was near the zenith. He marked down the positions of its moons in the chart in his workbook.

After staring at the little orrery of white points for a long time, he got up and went into the stables, where he knew Cartophilus preferred to sleep. He thumped him ungently on the back.

“What?” the ancient one croaked.

“Bring me your master,” Galileo demanded fiercely.

“What, now?”

“Now.”

“Why now?”

Galileo seized the man by his scrawny throat. “I want to talk to him. I have questions for him. Now.”

“Gah,” Cartophilus croaked. Galileo let go of him and he rubbed his neck, frowning resentfully. “Whatever you say, maestro, your wish is my demand, as always, but I cannot produce him immediately.” He reached for a jug of water he kept by his bed at night, took a pull and offered it to Galileo, who waved it off. “I will as soon as I can. It may take a day or two. It would be easiest back in Florence.”

“Quickly,” Galileo ordered. “I’m sick of this. I have some questions.”

The old one gave him a brief glance and looked into his jug. “This trip to Rome was perhaps in reference to him?”

“In a manner of speaking.” Galileo put his big right fist under the man’s nose. “You know more about it than I, I’m sure.”

Cartophilus shook his head unconvincingly.

Galileo humphed. “Of course not. Are you really the Wandering Jew?”

The old one waggled his head equivocally. “The story isn’t really right. Although I do feel cursed. And I’m old. And I have wandered.”

“And are you a Jew?”

“No.”

“Did you mock Christ as he carried the cross to Golgotha?”

“Definitely not. Huh! That’s a story the Gypsies used to tell. A band of them would come into a town, a couple of centuries ago, and explain that they had been made immortal penitents, because they had accidentally insulted Jesus. Practically every town we told the story to opened their gates and treated us like royalty. After that it was a case of transference.”

“So the Wandering Jew came from Jupiter.”

The old man’s eyebrows arced high on his forehead. He took another pull before replying. “You remember something from your last syncope, I take it.”

Galileo growled. “You know better than I.”

“I don’t. But I could see that you wanted to get to Rome to defend yourself.”

“Yes.”

“But it didn’t work as you had hoped.”

“No.”

Cartophilus hesitated for a long time. Just as Galileo thought he had fallen back asleep, he ventured, “Often it seems to me that when one tries to do something based on … knowledge—or even let us say foreknowledge, or a premonition, what the Germans call Schwanung—that whatever you do, it … rebounds. Instead of forestalling it, or fulfilling it, your action has the effect of bringing about exactly the opposite of whatever you might have been trying for. A complementary action, so to speak.”

“You would know better than I, I’m sure.”

“I don’t.”

Galileo lifted his fist again. “Just get your master to me.”

“As soon as I can. In Florence. I promise you.”

Back in Florence, Galileo moved into his newly rented house in Bel-losguardo, the Villa del Segui, a fine establishment overlooking Florence from a hill to the south of the river. He had a real home again, for the first time since Hostel Galilei in Padua. Here he was, back in his gardens, back in La Piera’s care, back in the arms of his girls (or Virginia’s anyway).

He was barely settled in, and had gone out into the garden one night to complete his ablutions, when a movement against the stable wall caused him to flinch.

A black figure emerged from the murk, and he was about to cry out when he saw that it was the stranger. At the sight of that narrow face, the unganymedean face of Ganymede, he experienced a big if vague abreaction; all of the blurred uncertain memories of what had happened to him on the Jovian moons came back to him in force. The memories of his earlier night voyages were like dream memories, with certain moments sticking out more distinctly even than events of the present moment—in particular, in this case, the fire—but the rest fuzzy beyond what was usual for his memory, perhaps because of the dreamy content. They had done things to his mind, he knew that; the woman Hera had helped him to counteract one preparation with another, he recalled. So odd effects were not surprising. In any case, now the earlier voyages had bloomed in him, and all from the sight of the stranger’s hatchet face. Galileo’s heart beat in his chest at the vivid memory of the fire, which had never really left him. “I want to go back,” he demanded. “I have questions to ask.”

“I know,” Ganymede said. “There are questions for you there as well. I have taken steps to secure the device at the other end.”

Galileo snorted. “You hope you have. But I want to see Hera in any case.”

Ganymede frowned. “I don’t think that’s wise.”

“Wisdom has nothing to do with it.”

This time Ganymede merely twisted a knob on a pewter box he was carrying crooked in his elbow, and there they stood, inside one of the green-blue ice caves of Europa.

“Hey,” Galileo said, shocked. “What happened to your teletrasporta?”

Ganymede tilted his head. “All that was done to give you a way to comprehend what was happening. It was felt that if you were bilocated without some way to explain the prolepsis to yourself from within your own frame of reference, you might be excessively disoriented. Some feared you would experience a mental breakdown, or otherwise fail to accept the reality of the prolepsis. Perhaps assume you were dreaming a dream. So we constructed a simulacrum of a translation that would make sense in local terms—in your case, a flight through space. We made the entangler look like something that could cast your vision to us. Then the experience of flight was given to you after you had already been bilocated.”

“You can do that?”

The stranger gave Galileo a pitying look. “Simulated experiences can sometimes be distinguished from real ones, but in data-poor environments, like space, it’s hard to do.”

Galileo gestured at the great ice cavern extending away from them in every direction, its aquamarine roof starred by cracks. “If this cave were not real, how would I tell?”

Ganymede shrugged. “Maybe you couldn’t.”

“I thought not,” Galileo muttered. “These are all dreamscapes.” He thought again of his immolation at the stake. More loudly: “What keeps us warm?”

“Heat.”

“Bah. Where comes the heat? Where comes the air?”

“There are engines creating them.”

“Engines?”

“Machines. Devices.”

“So illuminating!”

“Sorry. The details would mean nothing to you. Very few people here understand them. The heat and air are simple, in any case. It’s protection from Jupiter’s radiation that is difficult. That’s why we stay below the surface most of the time when on Europa. One of the reasons they’ve gone mad, if you ask me. On Ganymede we were out under the sky. On Io, we take advantage of the new bubble fields. But here they have older structures for dealing with the problem.”

“Radiation? Isn’t that another name for heat?”

“Well, but there are vibrations along a spectrum of sizes. What our eyes see are wavelengths of a certain size, but that band of the visible is just part of a range that extends far to either side. Shortest are gamma waves, then longer wavelengths range from braccia to the width of the universe, more or less.”

Galileo stared at him. “And these other waves manifest as?”

“Heat, sometimes. Damage to flesh that can’t be felt. I don’t know exactly how to explain it to you.”

Galileo rolled his eyes. “Then take me to someone who can.”

“We don’t actually have time for that, sorry—”

“Take me to someone who knows! Because you are an idiot.”

Ganymede rolled his eyes. “I forbear—”

“Take me!” Galileo shouted, and shoved the man hard in the chest. At home he would have beat him, so why not here? He wasn’t convinced any of it was real. He kicked Ganymede in the shins, yelling fit to turn all the blues of the place red. “Come on! Someone who knows something. Surely there must be someone who knows something!” He raised his big fist.

“Stop it,” Ganymede complained. He was wispy despite his height, and looked confused to be assaulted. “Quit trying to bully me. We aren’t in one of your downriver alleys here. People will notice what you do, and conclude you aren’t really civilized.”

“Me? It’s you who are uncivilized. You don’t know even the basics of how your machines work.”

“Spare me. No one knows all these things. Could you tell me how every machine of your time worked?”

“Yes, of course. Why not?”

Ganymede pursed his lips. “Well, it is no longer possible.”

“I don’t accept that. The principles at least must be clear, if you make the attempt to understand.”

“You’ll see.” And he muttered to the side, as if to an invisible angel.

“Take me.”

“I’ll take you.”

The gallery they were in was a kind of giant open antechamber to another under-ice city. Broad spaces extended so many miles away from them that in the distance the blue ceiling curved down and met the floor, cutting off any farther sight. Picking out one particular bright silver building ahead of them, just where the ceiling appeared to meet the floor, Galileo found it took only about fifteen or twenty minutes to walk to it. A close horizon. The alleys and strada of this cold town were sometimes crowded with tall graceful people, moving as if in water; at other times the streets were nearly empty. The people wore clothing like Ganymede’s, simple but fine, warm pastel tones making them appear illuminated in the green light.

They continued beyond the silver building for about an hour, he reckoned, passing crowded plazas extending to left and right, some of them open to the black sky, most roofed by ice. As the hour passed, he learned better how to walk in the light downward pull. This strange lightness was suggestive of all kinds of things, including the idea that weight was perhaps proportional to the size of the planet one stood on. Another sign that Europa must be fairly small.

“Where are you taking me?” he said.

“To a person who may be able to answer your questions. Or maybe you would call it a machine.”

“A machine? So none of you know?”

“No no, this person is a kind of … composite. A person quite like you, in fact—a physicist and mathematician, quite famous.”

“Good,” Galileo said. “I want some explanations.”

They came to a lake, and stepped down into a long low boat, like a gondola. When they were settled in its bow, a boatman cast off and they hummed slowly over clear blue water, leaving a wake that ran in a clean curl that was slower than it would have been on the lagoon. Greenish blues pulsed overhead and in waves around them, and Galileo could not tell how deep the lake might be, as the many subtle shades of creamy blue bobbed darker and lighter, but always opaque. Royal blue, sky blue, azure, turquoise, aquamarine—all these bounced against each other in long bands, and it also seemed that waves of cobalt were passing through the other blues, staining them as it pulsed by, as if they boated through the veins of a beating blue heart. The buildings behind the broad fondamenta to their left looked like clean blocks of ice, painted in pastels that held their color manfully even in the omnipresent green-blue glow, contradicting what Galileo thought he knew of color theory. The sight of one curving row of waterfront buildings reminded Galileo strongly of the Grand Canal, and he saw the city was a kind of Venice carved in ice. “Why doesn’t it melt?”

“It’s all cladded. Sheathed in diamond, in fact.”

People promenaded on the fondamenta just as they would have at home. Some of them looked out on the water, but not at Ganymede and Galileo; theirs was only one watercraft among many. All the wakes on the water created a fine curvilinear slow-motion cross-chop. The ice ceiling overhead was thicker in some places than in others, judging by the differences in the green-blue. Pulses most definitely were running through it.

“What are those waves of color running through the roof?” he asked.

“The other moons exert tidal forces against the tug of Jupiter proper. We shine a type of light through the ice to reveal the stresses in it, so we can see these tides’ interactions.”

“How do you keep these canals and lakes liquid?”

“We heat them,” Ganymede said patiently. “In places you will see steam. In other places we will break through a skim of ice as we progress along certain canals.”

“But you don’t know how the water is heated, do you?”

“That is not one of the more difficult accomplishments of our technology believe me.”

Their boat hummed up to a fondamenta made of something like black stone. As they climbed out of the boat, Galileo asked, “Where do you get rock?”

“From meteorites, called here dropstones. One or two big ones will supply enough material for an entire city, as it just supplements the local ice.”

“How many people live in this Venice of yours?”

“This is Rhadamanthys Linea. About a million people.”

“That many! And how many cities like this are there on Europa?”

“Maybe a hundred.”

“A hundred millions!”

“It’s a big moon, as you know.”

Overhead the broad crossing arcs of cobalt and violet pulsed from before them to behind them. Galileo said, “The patterns of light are so complicated, it seems there must be more than four influences.”

“All the Jovian moons pull a bit on the rest.”

“But are there more than four moons?”

“There are about ninety.”

“Ninety?”

“Most are very small. Some are out of the plane of the rest. In any case they all have a pull, no matter how slight, and with the ice overhead charged as the locals have charged it, every change in tug registers piezoelectrically.”

“Why do they charge it that way?”

Ganymede shrugged. “They like the way it looks.”

They were now walking down a broad crowded street flanked by long low buildings. Low carts moved at a running pace, without anything pulling them. Before them a cluster of very tall angular buildings reached right up to the ice ceiling.

“It must be the Tower of Babel,” Galileo said.

“Well, there is a great deal of confusion inside it, to be sure. And people who want it to fall.”

Soon they reached these tall buildings, and outside one they entered a glass antechamber, which then rose on the outside wall so fast that Galileo’s ears popped, surprising him. He always had a small earache in his right ear, and now it throbbed unhappily. So it seemed that in some sense his body was here too. “If I am here, how am I also back in Italy, lost in one of my syncopes?”

“You are here in a complementary potentiality.”

The glass antechamber stopped and a door opened on its inner side. They stepped out on a smooth broad roof terrace the color of malachite, just under the ice ceiling. Ganymede led Galileo to a small group of people congregated against a railing that overlooked the city. From here Galileo could see far down the canal; it developed a mirror surface in just the place a waterblink would have appeared on Earth, about halfway to the horizon. From there on it looked like a silver road through undulant blue buildings. Venice had looked just so on certain moony nights, and again Galileo wondered if he were dreaming.

Ganymede said, “This is Galileo Galilei, the first scientist, here in a proleptic entanglement.”

“Ah yes,” said a tall old woman at the center of the group. “We heard you were coming. Welcome to Rhadamanthys.”

Though old she was still straight, and stood a head taller than Galileo. Pendulant silver earrings emerged directly from her ear holes and then curved and seemed to dive into her neck. He bowed to her briefly, looked to his guide, muttered, “And where is the mathematician?”

Ganymede indicated the old woman. “This is she. Aurora.”

Galileo tried to conceal his surprise. “I thought you said it was a machine,” he said to cover himself.

“That’s partly true,” the willowy crone said. “I am interfaced to various artifactual entities.”

Galileo kept a straight face, although the idea struck him as monstrous, like jamming one of his military compasses through an ear into one’s brain. And in fact there were those earrings.

“Come with me,” Aurora said, taking him by the arm and moving him down the altana railing a short distance. Low creaks and hums that seemed to come from the ceiling kept them from being able to hear the other conversations on the terrace.

“It’s a pleasure to meet you,” the ancient woman said politely. She had a voice like Ganymede’s, hoarse and croaky, and her Latin had the same odd accent. “You are often called the first scientist.”

“That would be an honor, but I was not the first.”

“I agree with you. But you were the first mathematical experimentalist.”

“Was I?”

“So it seems from what we read in history, and see in the entanglements. One must always make assumptions, of course. And the past is always changing. But as far as we can tell, you tried only to assert what you could demonstrate and describe mathematically. This is science. Wasn’t it you who said that? That the world is written in mathematics?”

“I like that,” Galileo admitted. “If it’s true.”

“It’s partly true.” Although she looked troubled. “Reality is mathematical, as long as you understand that uncertainty and contingency can be mathematically described, without them becoming any more certain.”

“Teach me,” Galileo said. “Teach me how you breathe here, and what these tides of color are, and—teach me everything. I want to know everything! Teach me everything you have learned since my time.”

She smiled, pleased by his effrontery. “That would take a while.”

“I don’t care!”

She glanced at him curiously. “It would take years, even for one of your intelligence.”

“Can’t you do it quickly? Give me the short version?”

“The short version doesn’t give you real understanding. It’s only a matter of metaphors, images that don’t really convey the situation. The mathematics is what you want, and that took a great number of people many centuries to develop. Now no one learns more than a small percentage of what there is, and even that takes many years.”

“Maybe not for me!”

“Even for you.”

Galileo shook his head. “I don’t want to take years. I don’t have years.”

Aurora seemed to consult the patterns of intersecting waves in their low ice sky. She said, “There is a drug complex we can give you that would enable you to learn faster. A synaptic velocinestic, it is called, made of a particular mixture of brain chemicals. With the help of it, one can accomplish a certain forcing. Networks bloom in the brain extremely rapidly. It’s useful in certain situations.”

“An alchemical preparation?”

“Yes, if you like.”

“Is it safe?” Thinking of the half-crazed alchemists he had met, pursuing something like witchcraft in their foul workshops, poisoned by their own hand.

“Yes, we think so. It is mildly carcinogenic, but it won’t kill you. Although some people have felt distressed afterward, I’ve heard. But I have taken it, and felt no such thing.”

This, from a machine mind. Galileo could not stop a snort from emerging, though he curbed his tongue. After considering it briefly, he said, “Give me this preparation of yours. And then who will teach me the mathematics? You?”

She gave him an amused look. “One of our machines.”

“Another machine?”

“It’s a standard curriculum, designed for use with the velocinestic. It will be faster than I could be, and clearer too. I will oversee the process.”

“Do it then. I want to know!”

Her people gave him a tight-fitting helmet, made of a mesh of metals in a dense weave. They insisted he sit down, and got him settled into what looked like a small throne tilted onto its back.

Recumbent in it, he stared at the ice ceiling. It was pulsing rapidly in dense interference patterns, waves from three directions tossing off brief glints of sapphire iridescence. These triple peaks formed their own moving pattern, like sunlight on windblown water. Even if there had been just the four big moons, the Galilean moons (such a good name), their tugging would of course create a very complex pattern. He had been so sure that the tides on Earth were the result of the ocean sloshing around in its basins of stone, shifting as the Earth both rotated and flew around the sun, creating differential speeds. But here they said it was not true. In that case, what caused tides? The tug of celestial bodies—but that was astrology all over again. And yet they seemed to be saying it was so. Was astrology right, then, with its celestial influences and its action at a distance, action without any mechanical forces applied? He hated such nonexplaining explanations!

And yet here they were. He looked at Aurora’s assistants, hovering over the bank of machines against the wall. He hoped the treatment would work, that it would not kill or derange him.

They slipped their preparation into his blood using a hollow needle that they inserted painlessly into his skin—an ugly little experience. He held his breath as they did this, and when he finally exhaled and inhaled, the world ballooned. He saw immediately that he was thinking several trains of thought at once, and they all meshed in a contrapuntal fugue that his father would have very much enjoyed hearing, if it were music, which in a sense it seemed to be: a polyphonic singing of his ideas, each strand taking its part in the larger music. To a certain extent his thinking had always felt that way, with any number of accompaniments running under the aria of the voice of thought. Now these descants were choral, and loud, while at the same time architectonically fitted to the melody. He could think six or ten thoughts at once, and at the same time think about his thinking, and contemplate the whole score.

There remained a main melody, or a path through a maze—a maze that was like the delta of the Po. He seemed to look down on it as he sang it. A great number of channels were weaving down a slightly tilted plain. Each channel was a mathematical specialty—some of them shallow and disappearing into the sand, but most making their loop and reconnecting to other flows. A few were the kind of deep channels that ships would use. Upstream they coalesced until there were fewer, scattered streams. Fewer tributaries rather than more, leading up in different directions to sources, often at springs. Water out of the rock.

This was, he saw, an image of mathematics in time. Or maybe it was all time, or humanity in time; but it was the mathematics that sprang out at him.

The fewer channels upstream, in the distant past, well before his time, were where Aurora’s tutorial now led him. Then he was flying over the time stream, or in it, sometimes returning upstream to view a contemporaneous discipline. Mainly he had a general sense of flying downstream, over or occasionally inside some eternal landscape, the nature of which could not be discerned. He inhabited an image he had heard some time before, of history as a river, in which people were water, eroding the banks and depositing soil elsewhere downstream, so that the banks slowly changed and the river ran otherwise than it had, without the water ever noticing the changed courses of the braiding stream.

He tried to turn all the mathematics into geometry, so that he could see it and thereby grasp it. It often worked. It was definitely true what Aurora had said about the preparation. He grasped things he saw the moment he saw them; aspects even leaped out to him in advance as implications, shooting out before him like arrows. He was both in and out at once, back and forth, up and down, ranging widely, flying in stoops and gyres, and always looking forward with an eagle’s eye. The voice of the machine tutor was Aurora’s own hoarse voice, and Aurora herself flew beside him or in him, and sometimes she spoke too in her odd Latin, so that it seemed there were two of her talking. Sometimes Galileo asked questions and all three of them spoke at once and yet he could follow all three lines of thought, which merged in his mind into music, into a trio for lute and two squawky fagatto.

He was shown glimpses of people and places, but always the main thrust of the tutorial was mathematical. He recognized Euclid and Pythagoras, and for a short but incredibly packed moment he was actually with his hero Archimedes, still crucial to the story, hurrah! The Greek’s entire life bloomed in him at once, an island or bubble in the flow of the stream, and for a moment he knew it completely—and thought he saw Ganymede too standing there, and the burning mirror—also the Roman soldier at the terrible end—

Startled, for this was not like the rest of the lesson, he jerked up in his flight, feeling like a crow frightened out of a tree. Then he recognized Regiomontanus, and all that that brilliant man had rescued from the Greeks by way of the Arab texts, and was distracted that way. Then on to Harriot with his algebraic symbols, which Galileo had known would be useful the very first time Castelli showed them to him. Then Copernicus and his system, and Kepler and his polyhedraic formula for planetary distance, which Galileo had not thought was correct, and indeed it was not.

His own sense that all things moved naturally in circles was also shattered, however, as he was introduced to inertia—but that idea had always been on the tip of his tongue, indeed he had said it in slightly different words, as he cried out when he saw it. And then to the law of gravity—Newton’s equation for it caused him to soar up, startled; such a simple deep thing! He had seen the evidence for the laws of both inertia and gravity, he had used them in his parabolic description of falling bodies, but he had not understood what he had used, and now he floated above them, abashed, glowing before their utter simplicity. The force of gravity was simply an inverse power law, easy as kiss your hand, and resulting in obvious solutions to things like Kepler’s orbits, which Kepler had only groped his way to after years of observation and analysis.

So planetary orbits were naturally ellipses, with the sun occupying the major focus, and the other gravitational pulls together locating the minor focus. Of course! Too bad he had never read far enough in Kepler’s crazy tomes to get to these observations; it might have alerted him to the absence of circularity in the heavens—though he might have concluded they were just circles distorted by something he didn’t see. Certainly any idea one had in mind altered what one could see. And yet still, despite his ideas against it, here was attraction and influence at a distance again, without a mechanical force or cause! It was a mystery. It could not be the whole story, could it?

He was not aware he had asked this aloud, but heard Aurora reply: “This is the question that keeps coming up, as you will see. You are by no means the first or the last to dislike what one of us called spooky action at a distance.”

“Well, of course. Who could like that?”

“And yet as you will also come to see, such action is simply everywhere. You will find that there are serious problems with any simple concept of distance. Eventually distance becomes as problematic as time.”

“I don’t understand.”

But already she and her machine voice had flown off to analytic geometry, and then to a form of analyzing motion called the calculus, which was just what he had always needed and never had. And it seemed to have appeared just after his time, worked out by people young when he was old: an irritating Frenchman called Descartes, a German named Leibniz, and the English maniac Newton again, who to Galileo’s chagrin had distilled Galileo’s dynamics in just the way Galileo had struggled to do all his life. So simple when you saw it!

“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 manifold 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?”

“The second was a man named Einstein, the first a woman named Bao.”

“A woman?” Galileo said.

Hera shot him a look so full of contempt and pity, disgust and embarrassment that Galileo cringed, unfortunately shifting his balance on the slick floor such that his feet shot out sideways and he crashed down. By chance his bounce off the floor returned him right to his feet, where he could only blush and smooth down his jacket sleeves as if nothing had happened.

“Come with me,” she said to him peremptorily.

He followed her, greatly apprehensive, but aware that if he didn’t cooperate she would drag him away. “What is it?” he complained.

She glared at him. “Leave us,” she ordered her retainers, “and keep anyone else from following.”

She took him by the arm and pulled him with her as one would drag along a reluctant five-year-old. Under her fingers a shock tingled up his arm and all along that side of his body, from his ear to his foot.

Ganymede then emerged from a knot of his retainers on the other side of the terrace, and hurried over to them. Hera cursed under her breath and said to Galileo, “Stay put.”

She went to Ganymede and confronted him, and they argued in undertones that Galileo could not hear. When Hera returned to his side, she wore a look of grim satisfaction. “Come,” she said again, and pulled him across the terrace. “He’s not supposed to be on Europa at all anymore, so he can’t stop us.”

From the railing on this side of the terrace they looked down on a veritable maze of white rooftops cut by canals.

“Do you not remember what I showed you last time you were here?” she demanded of him.

“Yes, I remember!”

“Why did you come here then?”

“I wanted some answers, “Galileo said mulishly. “I told Ganymede to take me to someone who could give me answers, answers that you had not given me.”

She was not moved by this. “You can tell him to give you anything you want, but that doesn’t mean you’ll get it. Understand me: he wants you to end in just the way I showed you ending. In the fire.”

“Yes yes, but look. I took the preparation you gave me last time, but they made me breathe the mist you warned me against. I remembered part of what you showed me—certainly the, the essentials. So I went back and did everything I could to make sure that that event could never come to pass. But it didn’t work. In fact it only made things worse! Now I have been forbidden even to mention the Copernican theory. And yet there it rests, at the base of everything else. It’s God’s truth, and a rather elementary truth at that—and we have finally perceived it, yet I can’t say a word about it! If I say anything at all, that could be it. And I have enemies watching my every breath. I might as well cut my tongue out of my head!”

She shook her head. “You can find ways to say what you want to say. Meanwhile, you have to consider what will happen if your understanding is brought up to our time, and then you return to your own time. If you try to counteract that, and take the strong amnestics and forget everything, you will forget the fate you are trying to avoid. You may walk into your fiery alternative unaware. If, on the other hand, you take anamnestics like those I gave you before, and preserve your memory of this visit, you will know too much. Your work will be skewed, and you may change things in ways that would be disastrous to your future, and ours too. You will put yourself on the horns of a dilemma, or in the clutches of a double bind.”

“Can’t you give me a preparation that would keep some memories and suppress others?”

“It doesn’t work that way.”

“It seemed that it did. In my last few years, I remembered this, but it was only a very partial memory, like a dream. I remembered the fire, and you warning me, but it was all confused.”

“Possibly so, but there is no way of controlling it so finely as to be sure. Memory is very diffuse in the brain, it relies on multiple systems in concert. It’s quite a feat to manipulate it as much as we do. You can’t take the chance of knocking out too much.”

Galileo threw up his hands. “But I want to know things; I’m made to know things! And I don’t see how knowing more can possibly harm me! If you are trying to help, as you say you are, then help! But don’t help me by telling me to stay more ignorant, because I won’t accept that. I’m sick of being told not to know things!”

She heaved a sigh, looking grim. “Prolepses are awkward,” she said. “I wish Ganymede had not done this to you. Now we need to make a plan. In your own time, you should certainly stop talking about the Copernican theory for a while. Bide your time and work on other things. It isn’t as if you understand very much of basic physics, after all, as you now know all too well. You could focus on that. I tell you what—I’ll give you an amnestic that will obscure short-term memory. It will allow you to retain what you remembered before this little tutorial, but make this trip’s contents hard to recall. Hopefully that will serve to keep your part in the flow of events consistent.”

“I want to know,” Galileo said. “I don’t see how it can hurt.”

“You don’t understand—not us, not time, not yourself.”

Now Ganymede and his gang on the other side of the terrace were pushing Hera’s people to the side, approaching Hera and Galileo in a swirl of tussling and curses. Hera put a forefinger under Galileo’s nose.

“I’m the one helping you to avoid your fate,” she reminded him as she took a pewter box from one of her retainers. “So listen to me. You can’t be one thing here and another there. You need to knit your selves together. You either make yourself whole, or else die in the fire.”