Galileo's Dream (29 page)

The Structure of Time

H
E 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 10
²⁰
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.”