But how do you make 4-dim molecules? Goldy thought he had a way, and Ted Anderson had checked over the math and pronounced it workable. The notion rested in the middle of the lab: a queer, half-understood machine of mind and matter called a Grahm-Schmidt generator.
“Jeez, Ed! This lab looks like your old room back in Diego Borough.”
“Yeah,” said Goldwasser. “Johnny said it would be a good idea. Orientation against that.”
That was the outside of the lab, raw topological space, without energy or matter or time. It was the shape and color of what you see in the back of your head.
I looked away.
Goldwasser’s room was a duplicate of his old home—the metal desk, the electronics rigs, the immense bookshelves, half-filled with physics and half with religious works. I picked up a copy of Stace’s Time and Eternity and thumbed through it, then put it down, embarrassed.
“Good reading for a place like this.” Goldwasser smiled.
He sat down at the desk and began to check out his “instruments” from the locked drawer where he’d kept them. Once he reached across the desk and turned on a tape of Gene Gerard’s Excelsior! The flat midwestern voice murmured in the background.
“First, I need some hands,” said Ed.
Out in the nothingness two pairs of lines met at right angles. For an instant, all space was filled with them, jammed together every which way. Then it just settled down to two.
The lab was in darkness. Goldwasser’s big form crouched over the controls. He wore his engineer’s boots and his hair long, and a beard as well. He might have been some medieval monk, or primitive witchdoctor. He touched a knob and set a widget, and checked in his copy of Birkhoff and MacLane.
“Now,” he said, and played with his instruments. Two new vectors rose out of the intersections. “Cross-products. Now I’ve a right- and a left-handed system.”
All the while Gene Gerard was mumbling in the background: “ ‘Ah, now, my pretty,’ snarled the Count. ‘Come to my bedchamber, or I’ll leave you to Igor’s mercies.’ The misshapen dwarf cackled and rubbed his paws. ‘Decide, decide!’ cried the Count. His voice was a scream. ‘Decide, my dear. SEX—ELSE, IGOR!’”
“Augh,” said Goldwasser, and shut it off. “Now,” he said, “I’ve got some plasma in the next compartment.”
“Holy Halmos,” I whispered.
Ted Anderson stood beside the generator. He smiled, and went into topological convulsions. I looked away, and presently he came back in to shape. “Hard getting used to real space again,” he whispered. He looked thinner and paler than ever.
“I haven’t got long,” he said, “so here it is. You know I was working on Ephraim’s theories, looking for a flaw. There isn’t any flaw.”
“Ted, you’re rotating,” I cautioned.
He steadied, and continued. “There’s no flaw. But the theory is wrong. It’s backwards. This is the real universe,” he said, and gestured. Beyond the lab topological space remained as always, a blank, the color of the back of your head through your own eyes.
“Now listen to me, Goldy and Johnny and Kidder.” I saw that Pearl was standing in the iris of the tube. “What is the nature of intelligence? I guess it’s the power to abstract, to conceptualize. I don’t know what to say beyond that—I don’t know what it is. But I know where it came from! Here! In the math spaces—they’re alive with thought, flashing with mind!
“When the twiddles circuits failed, I cracked. I fell apart, lost faith in it all. For I had just found what I thought was a basic error in theory. I died, I vanished . . .
“But I didn’t. I’m a metamathematician. An operational philosopher, you might say. I may have gone mad—but I think I passed a threshold of knowledge. I understand . . .
“They’re out there. The things we thought we’d invented ourselves. The concepts and the notions and the pure structures—if you could see them . . .”
He looked around the room, desperately. Pearl was rigid against the iris of the tube. Goldy looked at Ted for a moment, then his head darted from side to side. His hands whitened on the controls.
“Jimmy,” Ted said.
I didn’t know. I moved toward him, across the lab to the edge of topological space, and beyond the psychic ecology. No time, no space, no matter. But how can I say it? How many people can stay awake over a book of modern algebra, and how many of those can understand?
—I saw a set bubbling and whirling, then take purpose and structure to itself and become a group, generate a second-unity element, mount itself and become a group, generate a second unity element, mount itself and become a field, ringed by rings. Near it, a mature field, shot through with ideals, threw off a splitting field in a passion of growth, and became complex.
—I saw the life of the matrices; the young ones sporting, adding and multiplying by a constant, the mature ones mating by composition: male and female make male, female and male make female—sex through anticom-mutivity! I saw them grow old, meeting false identities and loosing rows and columns into nullity.
—I saw a race of vectors, losing their universe to a newer race of tensors that conquered and humbled them.
—I watched the tyranny of the Well Ordering Principle, as a free set was lashed and whipped into structure. I saw a partially ordered set, free and happy broken before the Axiom of Zemelo.
—I saw the point sets, with their cliques and clubs, infinite numbers of sycophants clustering round a Bolzano-Weirstrauss aristocrat—the great compact medieval coverings of infinity with denumerable shires—the conflicts as closed sets created open ones, and the other way round.
—I saw the rigid castes of a society of transformations, orthogonal royalty, inner product gentry, degenerates— where intercomposition set the caste of the lower on the product.
—I saw the proud old cyclic groups, father and son and grandson, generating the generations, rebel and blacksheep and hero, following each other endlessly. Close by were the permutation groups, frolicking in a way that seemed like the way you sometimes repeat a sentence endlessly, stressing a different word each time.
There was much I saw that I did not understand, for mathematics is a deep, and even a mathenaut must choose his wedge of specialty. But that world of abstractions flamed with a beauty and meaning that chilled the works and worlds of men, so I wept in futility.
Presently we found ourselves back in the lab. I sat beside Ted Anderson and leaned on him, and I did not speak for fear my voice would break.
Anderson talked to Johnny and Ed.
“There was a—a race, here, that grew prideful. It knew the Riemann space, and the vector space, the algebras and the topologies, and yet it was unfulfilled. In some way— oddly like this craft,” he murmured, gesturing—”they wove the worlds together, creating the real universe you knew in your youth.
“Yet still it was unsatisfied. Somehow the race yearned so for newness that it surpassed itself, conceiving matter and energy and entropy and creating them.
“And there were laws and properties for these: inertia, speed, potential, quantumization. Perhaps life was an accident. It was not noticed for a long time, and proceeded apace. For the proud race had come to know itself, and saw that the new concepts were . . . flawed.” Anderson smiled faintly, and turned to Ed.
“Goldy, remember when we had Berkowitz for algebra,” he asked. “Remember what he said the first day?”
Goldwasser smiled. “Any math majors?
“Hmm, that’s good.
“Any physics majors?
“Physics majors! You guys are just super engineers!