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.”