Orbit 16 - [Anthology]

by Ed By Damon Knight

  Lucus thanked him and hung up. His palms were covered with sweat. The use of the term “Limited Interest” had frightened him and impressed on him the seriousness of the thing he was doing. There was no classified research at the FBRI; its fundamental philosophy was one of “basic research in a free and open environment.” All the work done in the building or under FBRI grants was published and widely disseminated. However, it occasionally became clear to the Institute officials that certain results could prove dangerous in one way or another if prematurely released to the public or to the scientific community at large. Therefore the code “Limited Interest” had been developed to refer to such work: unclassified, but kept strictly under wraps.

  It was nine thirty, and he had done all he could until he talked to the Director. The tower stood flat against the blackboard, a dead, crystalline, cutting blade of red and blue and orange. Outside the window, a squirrel darted along a branch and vanished down one of its countless customary routes in the maze of almost leafless branches. The Institute was built into the side of a hill, so that Dr. Lucus’ office, which was on the sixth floor if seen from the front, actually appeared to be no higher than the third floor. He had a peaceful view of grass and sky, held fast by the swift, layered lattice of branches. Often he felt that he did his best thinking while he was standing here, running his eyes peacefully along the branching lines, like one of Kaufmann’s illustrations in Graphs, Dynamic Programming, and Finite Games. But today he could not think in leisure. The soap-white walls that rose to enfold his world were too close now, and he was trapped, trapped and falling.

  He did have one thing left to do, although he did not feel like seeing Ruth. It must be done, and it would give his mind something to grasp until he could talk to the Director. He had her come in, and he dictated a polite letter, a bit too long, to the editor of the AMM, explaining that an embarrassing error had crept into the fall Quarterly and asking that it be returned, so that it could be replaced by a corrected copy.

  As Ruth got up to go, her steno pad pushed the yellow paper off his desk to the floor. He bent to pick it up and smoothed it on his desk, staring blankly at the calculations on the back. His watch said 10:05.

  At ten fifteen he still sat, frozen, his eyes open and filled with the erected sword shape plastered flat against the blackboard. It was cutting deep into his retinas, but his mind was suspended in dreamless waking sleep that numbed the wound and held him in inanimate rigid repose. Solidity of metal and wood near him and touching him melted, and only the neat impersonal sword hung above, simple, clear, no longer threatening, neutral now as all things were within the asbestos web which held him.

  He had to force his head down, to force his eyes to see the desk, the wrinkled yellow on white metal. To force his arm up, pull back the sleeve, and decide to act. Only then did his focus return, caught by the jittering watch, and then the office took clear familiar shape again around him. And fear returned.

  “Ruth? Would you call the Director’s office again and see if he’s in yet?”

  When the Director finally came on the line, he was curt and impatient. He didn’t like to be bothered by petty problems; Lucus knew that.

  “It’s really quite urgent. I don’t like to upset your schedule, but I’m afraid it can’t wait until tomorrow, sir.”

  “Well, what is it in Math that you can’t handle yourself, Lucus?” His voice was overamplified by the receiver, and there was no comfortable position for it. “If there are problems with funds or payroll, that shouldn’t be handled through my office. I should think you would be able to take care of your own distribution of grants.”

  “Well, no, sir, it’s a more important problem than that. It’s research that I feel needs ... uh ... special attention. I mean there seem to be possible . . . possible dangers in publication of certain discoveries.”

  “In math, Lucus? You’re exaggerating. What sort of research in math could produce . . . uh, dangers? I mean, you’re surely getting carried away with your formulas, aren’t you?”

  “I’d rather not discuss it over the phone, sir. It is of rather . . . of rather Limited Interest.”

  “Oh?” Lucus could feel the younger man’s eyebrows rising. “What sort of ‘Limited Interest,’ Dr. Lucus?”

  “If I could make an appointment, sir . . .”

  “My schedule is terribly busy, Dr. Lucus, and I don’t see how I can fit in another appointment—unless you will tell me the nature of the problem.”

  Lucus was not ready for this, not ready to reveal to anyone else the secret that, as far as he knew, he alone shared with Professor Paul David. He had not thought this far. He would have to tell another man the horrible thing that had been discovered, the horrible thing that had lain in wait for discovery all these centuries. His face covered with sweat, his hand sticky against the plastic receiver, he controlled his voice as much as he could and said, “A disproof of Euclid, sir. One of our fundees has produced a proof of the inconsistency of Euclid . . . that Euclid is not true, cannot be true ...”

  There was no reply, no sound. He didn’t know if the Director was as shocked as he, or if he was incredulous, unable to believe such a thing. Did he perhaps share Hans’s sense of the cosmic joke of the whole thing? Was he smiling with the chemist’s triumphant smile at the defeat of the abstract theoretician? How would any man react to such knowledge? Lucus decided that the Director did not believe him, that no man could accept such a horrifying conclusion without rigid proof. Surely he himself had spent two days and sleepless nights in the attempt to shake the unshakable conclusion. Finally the voice answered. It was a short answer and made its point perfectly clear.

  “Is that all? I think you should be able to clear that up by yourself, Lucus. After all, I don’t know much about math, and I don’t see why you have to bring it to my attention.”

  And that was it. He was unimpressed. It meant nothing to him.

  “I think, sir, that it is very important that I explain the problem to you in more detail.”

  “All right, Lucus, all right. Come by at three, will you? I have an important call on the other line. Sorry, I have to hang up. At three.”

  The sigh of resignation in his voice had been almost theatrical. He hated to be bothered with petty departmental problems.

  Dr. Lucus cradled his head in his arms on the desk, shaking uncontrollably with the release of tension, still alone in his fear and in his knowledge.

  * * * *

  2

  “Don, you’re using strategy again. It’s the same strategy; it’s a textbook play.” Hans Kaefig blew a thick puff of smoke at the stilled Go board. “Blurd your vision a little, Donnie. Come on, find a play that doesn’t have a proverb to go with it.”

  Hans leaned back in his chair, hands behind his head, cigarette holder rising out of his bushy grey beard like a radius vector tracing minute burning circles in the air. He closed his eyes tight in a pantomime of cogitation. “What you need to do, Donnie, what you need to do is . . .”

  Don smiled and fumbled with his pipe. He knew Hans could go on like this all night, fighting his own eternal battle with rational thought out loud, using Don’s career as his battlefield, giving him advice, often self-contradictory, on how to break from the confines of Aristotelean logic and soar like a bird on the soul of his intellect. Or the intellect of his soul. The words varied proportionally to the amount of brandy consumed every Tuesday.

  “. . . what you need, Donnie, is to state a theorem without a proof—with no hope of a proof. Write a paper, Don, with ten or twenty wild, impossible theorems and lemmas and corollaries— no proofs . . . absurd theorems. I’ll help you. I’ll give you some ideas, you can rewrite them to sound mathematical. We’ll publish them—inside a year someone will have proved half of them, done all the work, but they’ll all be called Lucus’ Theorem or Kaefig’s Conjecture—and we’ll have it made.”

  “Wouldn’t work, Hans. No one would publish them without proof.”

  “Well, then—we’
ll publish it as a novel. That’s it, a novel. You write the theorems, I’ll write the sex. We’ll call it Propositional Calculus, and start a rumor that it was written under drugs. We can cut the verbs out of all the sentences and make it look all Burroughs-y. That’s the way to do mathematics, Don. Get out of the mainstream . . . underground math . . . subversive topology, that should be your field, luv.”

  “Hans, I appreciate your help with my career—”

  “It’s only that I pity you—I’m determined to make an artist of you, if you don’t make me into a scientist first.”

  “Now have I tried to do that?”

  “Oh, you’re subtle, Donnie. You’re subtle. And that’s what I’m not. You can see my plan of action right away. But you—well, you just leave those books lying around open so I’ll sneak a peek. You try to draw me to those dirty pictures: a truncated cube, a stellated dodecawhatsit, two pyramids stuck through each other . . . it’s warping my brain. I go to my studio and find my mind all hung up in your simply connected sets and those tragic asymptotic curves—Tantalus damned to approach without reaching forever. What can I do? You have told me a doughnut is a coffee cup, and I have believed you. I used to paint the city and garbage and reality. Now all I know are points in space. I dream each night of being trapped in Konigsberg, forever recrossing those bridges, while Euler stands by the river and laughs. Oh, don’t deny it, Don. You are slowly turning me into what you are, enveloping me in symbolic logic and set theory. And I keep coming back and asking for more.”

  “And why do you come back?” asked Don, finishing off his brandy.

  “Ah, you force me to say it! You are my muse, Professor Lucus. Without you my art would die. Without you, my dear friend, I would paint only the city and garbage and reality.”

  “Oh ... I thought Mary was your muse.”

  “Mary? Of course not. A muse must be unobtainable, mysterious, the artist’s opposite, the soul of what he can never be. You might as well say I am Mary’s muse. After all, she did dedicate a quartet to me. Am I flattered? No. We’re getting divorced this year, or next year—whenever there’s time.”

  “You’re not serious, Hans!” Don hated the terrible uncertainty. In fact, he never did know when Hans was serious about any subject.

  “Of course I am. It was her idea. Or maybe it was mine. Anyway, we talked it over and thought it would be fun. But now you’ve sidetracked me. I was explaining your role in my art. Ever since I started playing Go over here, look how I’ve improved. Look at that sketch I did of your continuous function theorem. The critics love it. They think I’m a genius.”

  “It’s ugly, you know. It’s really ugly.” As much as he liked Hans, Don had never been able really to appreciate any of his work. He found it childish, simple, and sometimes repulsive. Since Hans had begun basing his things on diagrams in Don’s math books, he liked them even less. They seemed to make art lifeless and mathematics unprincipled.

  “Yes, it is,” agreed Hans. “Very ugly. And you see, before you inspired me I’d never been able to paint anything quite that ugly. I’d tried . . . Lord, I have as good a sense of what is offensive to the eye as any other artist, but I’d never been able to put it down on canvas. I would walk around in the slums and look at the dreck in the alleys and think it was ugly. I would eat starch sandwiches at the Automat and wipe my beard with a used napkin and think that was ugly. But then when I looked into your Hocking and Young and saw that wild sphere—I knew I had found it! I knew other men had seen the true vision, and I could learn from them.”

  “You’re crazy, Kaefig,” Don intoned, shaking his head.

  Hans clicked his teeth against his empty cigarette holder and drew a pack of Camels from the pocket of his Levi jacket. “You’ve said that ever since college, my dear Professor, and it hasn’t made it any less true, you smug sane bastard. Let’s put away the game and get drunk. I think I’ve done quite enough to try to save your soul for one night.”

  And so it went for over seven years, from Don’s thirty-eighth birthday deep into his forties, until Hans and Mary finally split up and he moved to the Midwest.

  And Don never tired of his friend’s harangues, because he knew there was something important there, something he should hear. And so he listened to all the nonsense and rambling, trying to sift out the bit of informational content, the little he could really learn from Hans.

  Mary had been hired as conductor of the Denver Symphony, and Don had heard little from either of them since, except for sporadic Christmas cards. He had left teaching to come to the Institute at fifty-five, and there he had remained, sitting—how had Hans put it?—sitting on top of that pile of elephant tusks, lord of what little he surveyed.

  A few years ago, a journalist had interviewed Dr. Lucus, because Hans had said he was the only man who could explain his sculpture Ragtime Band, that sprawling monstrosity that was the culmination of his fascination with the wild sphere in Hocking and Young and, according to many critics, the culmination of his career.

  And now all that remained of those endless games of Go were a couple of Hans’s paintings in Donald Lucus’ house, the wire sculpture on his desk, and a telegram with few hasty calculations on the back.

  He worked on the program until past his usual lunchtime, carefully cross-referencing the manuals spread out on his desk. The first step had been to write out the entire proof, as well as he could remember it, but with the diagram on the blackboard to help him. This was written in his own private notation, a hybrid of Fortran, mathematical symbology, and abbreviated English. The next step was to translate this into symbolic logic, using the special terms and syntax laid out by CONPROOF 2 and EUBERT. Not only was it necessary to translate from one code to another; in order to avoid an impossible mass of detail, Lucus also had to augment the Hilbert axioms for Euclidean geometry with statements of all the Euclidean propositions called upon in David’s proof. There was a list of these in the supplementary notes on EUBERT, and so he didn’t have to worry about coding them, only that he had inserted all the necessary ones and correctly labeled them.

  This was only the Euclidean part. For a while, he was afraid it would be necessary to duplicate all this work to program his Lobachevskian and Riemannian checks. But then he discovered a special tie-in in LOBACHEVMANN which would allow him to use the exact same input as was used for EUBERT and have validity checked in both non-Euclidean geometries at the same time. He was familiar enough with the use of the old CONPROOF 1, but only in conjunction with such systems as TOPOSPACE and ENSN, which he used constantly in verifying topological proofs. The axiom systems for synthetic geometries had been a complete mystery to him for over thirty years, and what he had relearned in the last three days was hasty and incomplete.

  So his office, normally neat to the point of sterility, took on the aspect which it had only a few days a month, those few days of feverish inspiration when he had all the business details of his position out of the way and could allow himself the luxury of creation. Directly in front of him were a programming pad, on which he was writing his final version, and a pile of scratch work. Across the upper part of the desk the three program manuals lay open; to his left was his recent reproduction of David’s proof, continuous on the back of last month’s budgetary output, and a stack of used scratch paper which contained calculations important enough to be saved; on his right, a well-worn Fortran manual on top of the three books which were almost the only customary adornment of the desk: Webster’s New Collegiate Dictionary, Whittier’s Trilingual Mathematical Encyclopedia, and a book of Go proverbs. The wire sculpture and desk calendar had been moved to the file cabinet to make room for all the necessary reference material.

  By twelve thirty Ruth had gone to lunch, but the programming was not nearly completed. He had to pull himself away from the pad and pencil almost violently. His hand, his whole body, and a portion of his mind were unwilling or unable to stop writing. Once the trance was broken and he was putting on his coat, it began to frighten him. Surely he wo
rked efficiently in such a hyperactive state, but it was dangerous. He could easily push himself too far, almost unknowingly, uncaringly, if he were allowed to give himself up to the immersing impulse too often. Even as he walked through the outer office, he noticed a stiffness in his legs and neck, an ache in his back and hand, that he had been oblivious to minutes before. Returning to awareness of his body’s torture, he found the temporary divorce from objectivity even more frightening, as though it had been imposed not by himself but from the outside, as though he had been driven too hard by some other being, with little or no concern for his complaints or his safety. He had been abused. And he was tired, very, very tired.

  * * * *

  His stomach was feeling upset—from the two cups of coffee he had had with lunch, he supposed—when he returned with his briefcase and David’s paper at one fifteen. He found that once he lowered himself into the swivel chair it was necessary to sit still for several minutes to catch his breath. He knew he needed rest, but there was much more to do before he saw the Director. He tried to weigh the priorities in his mind, to reach a reasonable plan of action, but it was difficult to pin down ideas, and his thoughts were constantly intruded upon by images of congruent triangles and hyperbolic planes. Each attempt to list the tasks of the afternoon and assign time estimates to them was met with frustration, and his ears rang with the faint sound of laughing voices chattering in FORTRAN.