Galileo's Dream

Chapter eleven The Structure of Time

He stood by the reclining chair he had taken his tutorial in, high in Rhadamanthys Linea, the Venice of Europa. Aurora was indeed there to greet him. “You look unwell,” she said, staring at him curiously.

Galileo said, “I am fine, lady, thank you. Please, may we continue your tutorial where we left off? I need to understand better how things work, in order to alter my life away from a bad result. You said when we parted that I was only at the beginning of your science. That there was some kind of reconciliation that would solve the paradoxes we were mired in. That I am mired in.”

Aurora smiled. She had in her gaze the glow that her name led Galileo to expect, even though she was obviously aged. “There is a reconciliation,” she said. “But it will require you to go much further than we did before. That session took you through four centuries, as I said. To get to the theory of the manifold of manifolds, you must keep going for a thousand more years. And mathematical progress has often accelerated in that time. Indeed there is one century called the Accelerando.”

“I like those in music,” Galileo said, climbing into the tutorial chair. “Was it then followed by a ritard?”

“Yes, it was.” She smiled as the Aurora of myth would have at old Tithonius. “Maybe that’s part of the definition of an accelerando.”

Warmed by her glance, anticipating with pleasure another flight with her into the future of mathematics, Galileo said, surprising them both, “I never knew a woman mathematician.”

“No, I suppose you didn’t. The power structure in your time was not good for women.”

“Power structure?”

“Patriarchy. A dominance system. A structure of feeling. We are cultural creatures, and what we think of as spontaneous and natural emotions are actually shaped in a culture-made system that changes over time, as with arranged marriages to romantic love, or vengeance to justice. There are of course enduring hormonal differences in brains, but they are minor. Any hormonal mix can result in someone good at math. And everyone is a mathematician.”

“Maybe in your world,” Galileo said, remembering some of his more hopeless students with a little snort. “But please, give me the preparation, and let’s be on our way. And I think it might go better for me this time, if you were to help the machine more often than you did before.”

Aurora looked amused that he would presume she was at his service. But he was too hungry for knowledge to be concerned about courtesy, and perhaps she could see that too. “I’ll listen in,” she said. “If I feel I can help, I’ll speak up.”

Her assistants brought the wiry helmet to him, and the alchemical preparation.

Humans sensed only a small part of reality. They were as worms in the earth, comfortable and warm. If God had not given them reason, they would not by their senses know even a minim of the whole.

As it was, however, by the cumulative work of thousands of people, humanity had slowly and painfully built a picture of the cosmos beyond what they could see. And then had found ways to use that knowledge, and move around in the cosmos.

Galileo flew again in the space of ideas, as if through patchy white clouds, following the construction of mathematics’ monumental edifice step by step through the centuries. He was thankful for the velocinestic, because he needed to be quick to apprehend what the machine was saying, and what Aurora added to its speech. This heightened apprehension now took him speedily beyond thought as he was used to it, into some larger realm of understanding, full of feeling and movement, something like a bodily music. He did not just see or sing the music, but became it. Math was his body. Words, symbols, and images all formed in the vague enormous clouds inside him, all moving in a continuous dance of equations and formulas, operations and algorithms, together melding into an ongoing polyphonic chorus. He was singing along and being sung. This meant taking certain things on faith, hoping that his performance of them indicated a subsequent firmer understanding that would grow and hold.

Here Aurora helped him to hew to the main line, reassuring him that he was proceeding just as all the rest of them had at one point or another, enduring confusions to follow a line through them. “No one can know everything,” she said. Galileo found this hard to accept. But in order to keep flying he ignored the bitter taste of his ignorance, of his faith in things he had not mastered. There were more important matters at hand than his sense of complete understanding. Apparently no one got to have that but God.

And so he flew on, diving into the new fields and methods, gauge theory, chromoelectrodynamics, symmetry and supersymmetry, multidimensional topology, manifolds, on and on it went, smaller and bigger, more complex and simpler—and after an extended protraction of his mind he found the long looked-for reconciliation of quantum mechanics and gravity physics. It came only very late in the story, when they got down into the very finest grain of things, regarding sizes that were so small that Galileo marveled there could be any knowledge of them whatsoever. But apparently it had been done.

As the generations of scientists had succeeded each other, each step of comprehension had served as scaffolding on which to stand and erect the next level. At every step of the way, quantum mechanics had proved itself accurate and useful. And so one aspect of it, Pauli’s exclusion principle, could be combined with the speed of light to establish minimum lengths and times: these were true minimums, because further division would break either the speed of light or the exclusion principle. The minimum width established by this principle turned out to be of a meter, and traveling at the speed of light a photon would cross this distance in of a second—a second being about the equivalent of a pulse, which Galileo measured as the speed of his calm heart beating. The ultimate minim of time, in other words, was a billionth of a billionth of a billionth of a billionth of a heartbeat, more or less. That was brief! The universe was very fine-grained indeed. Just thinking about it gave Galileo a shiver. It was stunning to feel in himself that fine grain, the dense texture of the glossy plenum—to sense in that density also God’s sense of artistry, His meticulousness or pulitezza. His love of math.

He flew on, doing his best to catch up to Aurora, who was continuing as if the minimum units were not stupendously, unimaginably small. She was used to the idea of them, and moving on to the question of how physicists had dealt with the idea that all space and time might be created out of the vibration of objects of the absolute minimum size and duration. Their most powerful experimental machines would have to be 10²⁰ times more powerful than they were to be able to investigate these minimal particles or events; in other words, an accelerator ring large enough to create the energies needed would have to be as big around as the galaxy. The particles they sought were so small that if one of them were expanded to the size of the Earth, the nucleus of an atom, to stay proportional, would have to be expanded to ten times the size of the universe.

Galileo laughed at this. He said, “It’s the end of physics then.”

For it meant that a stupendous abyss lay between humanity and the fundamental reality that would explain things at all the larger scales. They couldn’t cross that abyss. Physics was therefore stumped.

And indeed, for a long time mathematical physics and cosmology skittered around and appeared to stall, as physicists struggled to concoct scaffolding that they could cast all the way across the abyss in a single throw—that would give them even questions to ask.

“To an extent we are still there,” Aurora said. “But a mathematician named Bao made a bridge that seems to have held, and allowed us to build from it. Let’s go there now.”

Before Bao’s time, Galileo saw, which was precisely the beginning of the period known later as the Accelerando, the goal of physicists was to explain everything. He recognized that; it was the reductio ad absurdum of science: to know everything. The unspoken desire in that urge was the hope that, knowing everything, humanity would also know what to do. The blank that was their sense of purpose would perhaps also be filled.

But to know everything was asking too much. “They want to be like God!” he said.

“Maybe God is only a prolepsis,” Aurora said. “Our image of what we could be, imagined by contemplating our future.”

“Which would make it an analepsis, no?”

She laughed as they flew. “You like paradoxes, but of course this one is just entanglement all over again. We are extensive in time. Fly on and you will see.”

So they flew. Progress in physics struggled on. Theories of what occurred at the minimum and in postulated extra dimensions were elaborated, considered, challenged, refined. Predictions were made that could sometimes be checked against observation, or involved findings just beyond the realm of current observation. Ideas thus drove technologies. Slowly progress was made. But the unbridgeable abyss made every theory speculative. The wind from Galileo’s flight could have knocked down some of these houses of cards, and the collapsed theories he flew through had perhaps been knocked down in just that way, by the offhand remark of some observer like Bao, surveying the whole landscape and taking a completely new line over it.

It wasn’t until the twenty-eighth century that a theoretical structure accomplished a substantial part of what had been begun so long before. It was a physics based on Bao’s bridge to the minim, and on experiments spanning the solar system—controversial experiments that had entrained significant portions of the system’s total potential energy. Bao’s work had clarified the ten-dimensional manifold of manifolds theory that had been proposed since the time of Kaluza and Klein. Bao’s version had created many cosmological and subatomic questions and predictions that had given them experiments to try, observations to make, the results of which then gave them corrections and surprises, but mainly confirmations, a sense that they were on the right track at last—and in some ways had been all along, if one made allowances for the usual eddies and dead ends. Each generation had served as scaffolding for the generations that followed, and the work continued through collapses and reversals, almost one might say mindlessly. “It’s like watching ants building a mound,” Galileo observed as he flew through the elaborations. “The mass just keeps grinding.”

“Yes, although it’s a strange thing to say about a process that has taken so much brain power.”

“Tell me more about the ten dimensions,” Galileo requested. “Something more than their math. What do they mean? What can they mean?”

Aurora flew next to him, so closely that he felt they were intertwined. He dipped and turned, dropped or soared, stooped or gyred, always trying to stay next to her, and he found that she could make writing appear as clouds, or red ingots in the air before him. His body was a flock of bannering thoughts, flying around her in a dance. The landscape under them was a mountain range made of symbols and numbers piled one on the next, gnarled tectonically.

“Recall the Euclidean space that you know and sense,” she said, “having the three dimensions of length, breadth, and height. With Newton we added a different kind of dimension, which is time—”

“But I did that!” Galileo objected again. “Falling things accelerate as a square of the time passed! This I found out, and it meant time and space were bound together somehow.” Although, he recalled uneasily, the finding lay still unpublished, buried in his folios out in the workshop.

“All right, call it Galilean space,” Aurora said easily. “Whatever you call it, these four dimensions were understood as if they were an absolute, an underlying invisible gridwork through which physical phenomena moved. That’s when you have Laplace declaring that with a sufficient physics and database you could predict the entire past and future of the universe just by entering the numbers for the current moment, and running them through the equations either forward or back, as in an astrolabe. It was a thought experiment only, because no one would ever have the data set to do it. But the implication was that God, or something like it, could do it.”

“Yes. I can see that.”

“It implied a predetermined, clockwork universe that many found depressing to contemplate. We weren’t really choosing to do anything.”

“Yes. But your quantum mechanics destroyed all that.”

“Precisely.”

“Or imprecisely.”

“Ha, yes. With relativity and quantum mechanics we began to understand that the four dimensions we sense are artifacts of our perception of dimensions far more numerous than we knew. We began to see things that made it clear four dimensions were not adequate to explain what was happening. Baryons rotated 720 degrees before returning to their starting positions. Particles and waves both were confirmed even though they contradicted each other as explanations, as far as our senses and reason were concerned. In some cases our observations seemed necessary to make things exist at all. And something otherwise undetectable was exerting very marked gravitational effects, that if caused by a mass would outmass the visible matter of the universe ten to one. Then there appeared to be a kind of reverse gravity effect as well, an inexplicable accelerating expansion of space. People spoke of dark matter and dark energy, but these were names only—names that left the mysteries untouched. What they were was better explained by the existence of extra dimensions, first suggested by Kaluza and Klein, and then put to use by Bao.”

Galileo said, “Explain them to me.”

He felt himself become equations in the clouds inside him. Formulas described the motions of the minims, vibrating at the Planck distance and duration, thus small and brief beyond telling, and vibrating in ten different dimensions, which combined into what Bao called manifolds, each with its own qualities and characteristic actions.

“Investigations have by now found evidence for all ten dimensions,” Aurora said. “Even confirmation. The best way to conceptualize some of the extra ones is to imagine them enfolded or implicate in the dimensions we sense.” A long, flat red sheet appeared before him; it rolled lengthwise into a long thin tube. “Seen in two dimensions this looks like a ribbon, but in three dimensions it’s obviously a tube. It’s like that all through the manifolds. Dark matter has to be very weakly interacting but at the same time registering gravitationally at ten times the mass of all visible matter. That is an odd combination, but Bao considered it as a dimension we only were seeing part of, a hyperdimension or manifold that enfolds our dimensions. That manifold happens to be contracting, you could say, which gives the effect in our sensible universe of the extra gravity we detect. So that’s dimension four.”

“I thought you said time was the fourth dimension,” Galileo said.

“No. For one thing, what we call time turns out to be not a dimension but a manifold, a compound vector of three different dimensions. But put that aside for a second, and let’s finish with the spatial manifold. Dimension four we still call dark matter, as a gesture to our first awareness of it.”

“Four,” Galileo repeated.

“Yes, and dimension five in some ways counterbalances the action of four, as it is the perceived accelerating expansion of space-time. Aspects of this dimension are called dark energy.”

“Do these dimensions pass through each other, then?”

“Do length and breadth and height pass through each other?”

“I don’t know. Maybe they do.”

“Maybe the question as formulated does not have an answer, or maybe the answer is simply yes. Reality is composed of all the dimensions or manifolds, compounded or coexisting in the same universe.”

“All right.”

“Now let us come to time. Mysterious from the start, it seems mostly absent from our perception, but crucial as well. Past, present, and future are the aspects of time commonly spoken of as perceived by us, but they and other phenomena are the result of sense impressions compiled by living in three different temporal dimensions, which together make the manifold, in the same way our impression of space is a manifold. All three temporal dimensions impact on us even though we mostly have a very strong sense of moving forward in a manifold, so that we can only remember the past, and only anticipate the future, both of which remain inaccessible to us in any sensory way. Our senses are stuck in the present, which appears to move in only one direction—into the future, which does not yet exist, leaving behind the past, which exists only in memory but not in reality.

“But that present moment: how long is it, of what does it consist? How can it be as short as a single Planck interval, 10⁴³ of a second, while even the briefest of phenomena that we are aware of takes much longer to happen than that theoretical minim? What can the present be? Is it a succession of Planck intervals, a clutch of them? Is it even real?”

“God knows,” Galileo said. “I count it in heartbeats. The beat of the moment is my present, I pray.”

“That’s a long durée, in effect. Well, look at Bao’s temporal equations, and see how neatly every present that we sense, like your long durée of a heartbeat, gets explained.”

They flew into something like a cathedral, or an immense snowflake, made of intersecting numbers and figures: a lacing of equations, the details of which now completely escaped Galileo. He tried to hold to the architectural shapes they made, but he was no longer following the math.

“Her equations postulate a temporal manifold made of three dimensions, so that what we sense as time passing, what we call time, is a compound with a vector made up of the three temporalities. We can see it here, in something like a Feynman drawing for elementary particles. Indeed we can fly in the drawing, see? The first temporality moves very fast—at the speed of light, in fact. This explains the speed of light, which is simply the rate of movement in this dimension if you consider it as a space. We call that time therefore speed of light time, or c time, from the old notation for the speed of light.”

“How fast is that again?”

“Two hundred thousand miles a second.”

“That’s fast.”

“Yes. That component of time is fast. Time flies! But the second temporal dimension is very slow, by comparison. It’s so slow that most phenomena seem suspended within it, almost as if it were that absolute grid of Newtonian—I mean Galilean—space. We call this one lateral or eternal time, thus e time, and we have found it vibrates slowly back and forth, as if the universe itself were a single string or bubble, vibrating or breathing. There is a systolic/diastolic change as it vibrates, but the vibration is weakly interacting with us, and its amplitude appears to be small.”

“All things remain in God,” Galileo said, remembering a prayer he had once learned, when as a boy he had briefly attended the monastery school.

“Yes. Although it is still a temporality, a kind of time we are moving in. We vibrate back and forth in this time.”

“I think I see.”

“Then lastly,” she went on, “the third temporal dimension we call antichronos, because it moves in the reverse direction of c time, while it also interacts with e time. The three temporalities flow through and resonate with each other, and they all pulse with vibrations of their own. We then experience the three as one, as a kind of fluctuating vector, with resonance effects when pulses from the three overlap in various ways. All those actions together create the perceived time of human consciousness. The present is a three-way interference pattern.”

“Like chips of sunlight on water. Lots of them at once, or almost at once.”

“Yes, potential moments, that wink into being when the three waves peak. The vector nature of the manifold also accounts for many of the temporal effects we experience, like entropy, action at a distance, temporal waves and their resonance and interference effects, and of course quantum entanglement and bilocation, which you yourself are experiencing because of the technology that was developed to move epileptically. In terms of what we sense, fluctuations in this manifold also account for most of our dreams, as well as less common sensations like involuntary memory, foresight, déjà vu, presque vu, jamais vu, nostalgia, precognition, Rückgriffe, Schwanung, paralipomena, mystical union with the eternal or the One, and so on.”

“I’ve felt so many of those,” Galileo said, buffeted as he flew by memories of his lost times, his secret times. “In the sleepless hours of the night, lying in bed, I feel these phenomena often.”

“Yes, and sometimes in the broad light of day too! The compound nature of the manifold creates our perception of both transience and permanence, of being and becoming. They account for that paradoxical feeling I often notice, that any moment in my past happened just a short time ago and yet is separated from me by an immense gulf of time. Both are true; these are subconscious perceptions of a delaminated e time and c time.”

“And the sense of eternity that occasionally strikes? When you ring like a bell?”

“That would be a powerful isolated sense of e time, which does in fact vibrate in a bell-like way. Then in a different way, the sense of inexorable dissolution or breakdown we sometimes call entropy, also the feeling called nostalgia, these are the perceptions of antichronos passing backward through c and e time. Indeed Bao’s work leads to a mathematical description of entropy as a kind of friction between antichronos and c time running against the grain of each other, so to speak. By their interaction.”

“Things get ground up,” Galileo agreed. “Our bodies. Our lives.”

“That’s the effect of being in a manifold made of three different motions.”

“It’s hard to see.”

“Of course. We mainly experience time as a unified vector, much as we experience space as a plenum made up of the three spatial macro-dimensions. You don’t usually see the plenum as length, breadth, and height, you simply experience space. Time is similarly triune but whole.”

“Like tides in a river mouth,” Galileo ventured. One time as a boy he had watched the seaweed flow first one way, then the other. And at the moment the tide changed: “Sometimes there is flow both ways, and the interference chop can be either obvious or subtle. And the water is always there.”

“There are interference patterns, yes. Other people talk about Penelope’s Loom, and how we are all in our place of the tapestry busily embroidering it, and now the analepts are hopping back and re-embroidering certain parts. Anyway, time is not laminar. It shifts and flows, breaks up and eddies, percolates and resonates.”

“And you have learned to travel on these currents.”

“Yes, a little bit. We learned to shape a charge to create an eddy of antichronos, and push something along in it, and when that eddy touches c time again a complementary potentiality is created. That was enough to do a limited sort of time travel. We could perform analepses at certain resonant entanglements in the manifold. But it required very large applications of energy to make the first shift of the transference devices back in time. The required energies were so large that we were only able to move a few entanglers to bilocated past potentialities. Black holes sucked down large fractions of the gas of the outer gas giants for each entangler sent back. After that they were in place to be used as portals for entanglements of consciousness. These entanglements require much smaller energies, being a sort of field of induced or potential dreaming. The entanglements create a complementary potential time with every analepsis and prolepsis, and for this reason and others, the entire process remained controversial throughout the time it was being actively pursued. Shifting ten or a dozen entanglers required the complete sacrifice of the two outermost gas giants. That was felt to be enough, or too much. So really, this was mostly a technology of about a century ago, when analepts were going back frequently, and sometimes fighting over their changes, as Ganymede did more than anyone. It has all since been reconsidered. By no means does everyone agree it was a good idea.”

“I should think not,” Galileo said. “Why did they do it at all?”

“Some wanted to retroject science analeptically into a time earlier than it had naturally appeared, in the hope of making human history a bit less dire.”

“Why bother, now that you are here?”

“The intervening years were more dire than you know. And we are not just here; we are there too. You are not really comprehending what I have told you. We are all connected and alive in the manifold of manifolds.”

Galileo shrugged. “Things still seem to happen one after the next.”

She shook her head. “In any case, what you see here is a damaged and traumatized humanity. It was felt for a while that work on the past could make that better. A kind of redemption.”

“I see … I think. But, speaking of what you have taught me—that’s only eight dimensions, if I have not lost track. Five of space, and three of time.”

“Yes.”

“And the other two?”

“One is a truly implicate microdimension, inside all the rest. Each minim holds a universe in that dimension. Then all these and ours too exist inside a macromanifold, you might call it. This enfolds a multiplicity of universes—a kind of hyperspace of potentialities, well beyond human perception, although discoverable by observations of cosmically high energies, and of the background radiation. It’s said that in this manifold there are as many existing or potential universes as there are atoms in this universe, and some say even many orders of magnitude more than that, like 10³⁰⁰⁰.”

“That’s a lot,” Galileo said.

“Yes, but it is still not infinity.”

Galileo sighed. He saw they were no longer flying, but in a room the size of a lecture hall in Padua. Aurora could point at one wall and mime writing, and equations appeared on the wall before them. She walked him through the mathematics of the tenth dimension, the manifold of manifolds, and Galileo, as he struggled to follow her, was comforted by the idea that even here her work was a kind of spatial geometry, things laid out in relationships, with proportions, just as always. Maybe that was for his sake, but it all fell into place. Everything could be explained: the bizarre paradoxes of quantum mechanics, the strange billowing of the universe out from a single point that had never been anywhere. All the laws of nature, all the forces and particles, all the constants, and all the various manifestations of time, of being and becoming, their suprachronological travel in time, the bizarre giant reality of universal entanglement, were explained. It was a whole, a quivering organism, and God was indeed a mathematician—a mathematician of such stupendous complexity, subtlety, and elegance, that the experience of contemplating Him was inhuman, beyond what any human feeling could encompass.

“My head hurts,” Galileo admitted.

“Then let’s go back,” Aurora said.

As she was flying him back into the world, Galileo experienced a moment of selfish curiosity. In his first tutorial, he had gotten a glimpse of his hero Archimedes, as clearly as if he had been through the tele-trasporta and seen the Greek face-to-face, or even lived his life. Someone had mentioned something about Ganymede visiting Archimedes before he visited Galileo; perhaps that explained it. Now, with Aurora absorbed in a private conversation with her assistants, Galileo murmured a request to the teaching machine to show him the historical background of the astronomer Galileo Galilei.

Immediately he was cast into a space like that which had surrounded Archimedes; not a moment but a life—his life. Instantly he was filled with his own life, in Florence, Pisa, Padua, then Bellos-guardo, then a smaller house he didn’t recognize, in a village. All of it filled him at once, fine-grained to the minute, and fearfully he cried out, “Stop! Take it away!”

Aurora now stood before him, looking surprised. “Why did you do that?”

“I wanted to know.”

“You thought you did. Now you will have to forget.”

“I hope I can! But I suppose you can give me an amnestic that will help me to deal with it?”

“No,” she said, looking at him curiously. “I can’t. That’s Hera’s kind of thing. You will have to cope with whatever you learned yourself.”

Galileo groaned. He struggled up from his big reclining chair, Aurora’s helmet on his head. He felt drained, frightened. The sensation of immediate powerful apprehension was still with him, but it all had to do with his life now. His past—the present moment—

People were talking. Aurora and her assistants. For a time he lost the sense of it. Thoughts in language, like the voice speaking in him; they were such simple things, like the twittering of birds. Pretty, even sometimes beautiful, but nowhere near as expressive as mathematics. Now he tried hard to remember, he tried hard to forget; some of it was there and some of it was gone, but not in the ways he would have hoped. Nothing to be done about it. The tutorial had happened in him, it had left marks; it would remain somewhere in him, in what they called e time, or in that evanescent present that always bloomed at the edge of c. Or headed back through antichronos, all the way back to the curious boy looking at the lamp swinging in the cathedral. Memory as a kind of precognition.

He regarded Aurora freshly. An ancient woman, who had, he now knew, a knowledge of mathematics, and of the physical universe, that far, far, far transcended his. That was rather amazing. He had never thought that any such person could exist.

“Do you believe in God?” he asked her.

“I don’t think so. I’m not sure I grasp the concept.” She hesitated. “Can we get something to eat? Are you hungry? Because I am.”

They sat beside a low table next to the far railing. It was an altana, it seemed to him; just as in Venice, they made their ground on their rooftops. He sat by the railing and looked at this Venice under its pulsing green-blue sky. On the table between them were plates of small cubes and slices of a vegetable substance unknown to Galileo, the bits flavored with ginger or garlic or various peppery spices he was not familiar with, which made his tongue buzz and his nose run. The water was berry-flavored; he drank deeply, feeling suddenly very thirsty. He surveyed the dim turquoise and cobalt buildings beneath them. Europa was a world of ice, Io was a world of fire. Were Ganymede and Callisto then earth and air?

“Have you had more conversation with the thing under us?” he asked Aurora. “You were telling me about it before. It seems to know gravity well, you said?”

“Yes.”

“What about the compound temporality, the vector of three times?”

“That’s been hard to determine.”

“Show me the exchanges with it.”

Aurora smiled. “It’s been eleven years since the ice was broached and the sentience confirmed. Most of the interactions have come to dead ends. But an abstract of it can be found here.”

She indicated their table, and Galileo looked at it and saw long strings of mathematical symbols and graphically organized information. The tutorial pulsed in his head like a kind of headache. He tried to pilot that knowledge into this new problem.

“Interesting,” he said at last. “What physically constitutes the sentience, do you know? Have you located the bodily source of its mind?”

“It fills the ocean below us, but is not the ocean. The things like fish that you yourself saw, I believe—”

“I saw spirals of blue light, more like eels than fish.”

“Yes, well, these came from parts of a larger whole. Like brain cells of a sentience distributed across the group. But still it does not appear to be consciousness as we would recognize it. There is a kind of absence in its cognition, having to do with self-awareness and other-awareness. An absence that makes some suspect that what we are conversing with is part of a larger whole.”

“But what?”

“We don’t know. But there are people who want to find out.”

“Not all of you?”

“Oh no, not at all. There is a … disagreement. A very basic philosophical or religious disagreement. One might call it a dangerous disagreement.”

“Dangerous?” Galileo was apprehensive: “I was hoping you were all past that kind of thing by now.”

She shook her head. “We are human, and so we argue. And this is an argument that could lead to violence.”

Dissent among the Galileans. Well, he already knew that. Hera had kidnapped him, and Ganymede had rammed his ship into the Europans; he should not be surprised. It was people changing their nature that would have been surprising. “Actual violence?”

“People are much more likely to kill each other over ideas than over food,” she said. “It’s very clear in the historical record, a statistical fact.”

“Maybe,” Galileo ventured, “when food is secure, the grasp for certainty moves elsewhere.”

“Certainty,” she scoffed. “In the manifold of manifolds!” And she laughed.

As if to illustrate her point, out of the glass antechamber appeared Hera herself, ivory-armed and magnificent. She was trailed by her Swiss guard equivalent, a dozen bruisers even bigger than she.

Now she approached Galileo, shaking her head as if at a child who did not comprehend his transgression.

“You again!” he said sharply, angered by this look. “What is it this time?”

Then a loud group of locals spilled out of the next antechamber over. Hera saw them and said, “This rabble is trying to keep us from joining you, here in a public space. One moment—”

She and her gang ran at the Europans, and a brawl began. In Venice such a thing would have been dangerous, with knives pulled from sleeves. Here it was just shoving and shouting, and the occasional flailing roundhouse. Hera shouted, “You’ll be charged with assault! I hope you’ll get exile!”

“You’re the one who made the assault,” one of them shouted, and appealed to Aurora: “We did what we could. She stops at nothing.”

The mathematician regarded them without expression. “Then let her speak.”

Hera returned to Galileo’s side. “Take the entangler,” she said to her people, gesturing at the pewter box. She said to Aurora, “I’m the one who should have it, and you know it.” One of Hera’s guards went to the box and picked it up. Then without warning Hera grabbed Galileo by the arm, lifted him off his feet, and walked with him toward the glass closets, leaving a rear guard behind to protect her retreat.

“Kidnapping again?” Galileo inquired caustically, struggling to free himself from her grasp. It was galling that he could not even slow her down.

She rolled her eyes as she set him down and dragged him with her, forcing him to step fast. “Those drugs Aurora gives you,” she said emphatically, “and the lessons, they do more than teach you our math. They change you. By the time you’re done, you won’t remember what I showed you before! Do you remember that? Do you remember how they burned you?”

“Of course I do! I’m not going to forget that. How could learning more mathematics cause anyone to forget that?”

“By changing you so that even if you do recall it, you lose your understanding of why it happened.”

“I never knew why it happened!” Galileo shouted, suddenly furious. He took a big swing at her, which she easily avoided. “I’m still trying to figure that out!” He swung again and caught her on the arm, but it was like hitting a tree. “Everything I’ve done since you showed it to me only seems to bring it closer! I’ve been destroyed. And it can get worse. That’s precisely one of the reasons I want to learn more!” And he yanked his arm free of her grasp.

She took it again, with a grip like an eagle’s. “You don’t understand. Your fate doesn’t have to do with the math and the physical theories. It has to do with your situation at home, and with you yourself—your nature or your characteristic responses. The kind of conclusions you draw, and how you react in a crisis. You are your own problem.“

She pulled him into the glass closet and let him go. Glowering, she poked buttons on the panel next to the door. “I guess I have to teach you that part, just as Aurora taught you physics.”

“But we were working here. They’re making an attempt to contact the thing inside Europa, and I was helping them.”

“That’s none of your business. And there are people who think they understand the thing already. Including Ganymede, in fact. He and his followers are the ones causing the problems.”

“How so?”

“They still consider the Europan thing a danger to us, a mortal danger.”

“But why? How could that be?”

“It doesn’t matter.”

“Of course it matters!”

“Not to you it doesn’t. What matters for you is doing what you did in your time without getting burned for it. Do you want to be burned?”

“No! I just don’t see how me knowing more can change that.”

She shook her head, red-cheeked and still breathing hard, looking down at him with a grim expression. As they left the moving closet, now on the ground, she said, “You understand nothing. Especially your self. All that celebrated ceaseless activity of yours, performed in ignorance.”

“I know as much as any man! Indeed more than most. You know less than I about how the world works, even with fourteen centuries’ advantage. You have nothing to teach me.”

“There is no hatred like that of ignorance for knowledge,” she quoted him sardonically. “Especially self-knowledge. Do you want to be burned or not?”

“Not.”

“Come along, then.” She made a brief gesture at a new group of her retainers, waiting beside a long low boat, like a gondola. From behind them, the guard with the teletrasporta ran up and put it beside Hera.

“I need to join the grand council on Callisto,” Hera told Galileo as she gestured at the gondola. “The transit will take several hours. You will come along, and we can talk. There are some things you need to see in your life.”

“Spare me,” he said.

She wheeled and glared at him, face inches from his. “I will not spare you! I’ll put you through your life as many times as it takes.”

“As it takes for what?”

“For you to get it right.”

This was sounding bad.