Finally he decided that the only really necessary task was to finish the programming foe CONPROOF 2 and have it sent to the computer division.
This look about half an hour. The work went much slower than it had in the morning. He constantly found himself looking blankly at his own notes, confusing output statements with axioms, losing his tenuous grasp on the details of David’s proof.
When the whole thing was finally sent out to be punched up and compiler-checked on the A50 unit, it was ten minutes to two. He told Ruth to call him at five to three and gratefully laid his head on his folded arms on the desk, not even bothering to darken the room.
At first sleep would not come, only sharp-edged pictures, alternately threatening and soothing: his small house, his books, Mary as she was thirty years ago, when he had thought he might marry. Unrelated images swam about in his mind: a snatch of old rock music, a lemma from the Side Approximation Theorem, the smell of lilacs, and slowly one figure emerged from the mass and began to grow and dominate it all. At first it appeared to be only a smooth, featureless, somewhat metallic topological sphere. But this was only the bottom portion. Above, the figure split into two parts, not so much like a branching tree as like a squid with two plump arms, spread in a gentle flattened circle and coming together again—but not quite. Before the arms met, each of them split into two parts again, a thumb and an index finger, which linked together like a chain—but not quite. Before each finger reached its thumb, it was bifurcated, as was the thumb, and again the two arcs linked to form a chain—but not quite. This process continued infinitely, each step increasing the number of parts geometrically. The result was a figure simple in its construction, frighteningly complex in its final appearance.
“My God in heaven!” exclaimed Hans. “What in merciful hell is that? I never expected to find a book on demonology in your home, Donald.”
Don had been at the bar pouring brandy and didn’t know what Hans was referring to. By the time he got back to the table, Hans was standing silently, biting the side of his thumb and staring fixedly at the open copy of Hocking and Young’s Topology.
“That’s Alexander’s horned sphere,” explained Don evenly. For a moment he too felt an uncanny horror at the picture. But it passed, and what was to Hans the image of Satan became only a wild embedding of the two-sphere in Euclidean three-space.
“You see,” he went on, setting down the drinks, “it’s topologically a sphere, but its complement is not the same as the complement of an ordinary sphere. For instance, you could link a circle around it, just like you would around a torus, and it won’t come off. In fact, there are an infinite number of ways you can do it.”
Hans seemed not to be listening. He did not respond when Don sat down, but continued to bite his thumb rhythmically.
“The one thing that amazes me,” he said at length, “is that I have seen it—” He paused, running his fingers through his beard. “—that I have seen it . . . and I still live. What is it, Donald, and how has it found its way into your neat religious parlor?”
“I told you: it’s a wild sphere. It’s called Alexander’s horned sphere. It’s really not so extraordinary, Hans. There’s a wild arc on the next page. Here’s your brandy; now let’s get started with the game.” Don was impatient to forget the thing, for Hans to close the book and change the subject. He didn’t like this reaction to a simple mathematical object, as though it were something more than it actually was.
“One could imagine Dali painting it,” Hans continued, still fixing his gaze on the picture, “all ugly, bleeding lumps of flesh. That would be the obvious way. But I think I see it as a sculpture. It would have to be tremendous—say, thirty feet high—so that the branches start out fat as sequoias and end up—and end up microscopic—and never end. They should go on to the atomic level and beyond. Alexander’s horned sphere. Can you see it squatting in the sun in the middle of Chicago, like some horrid, slimy crab? Yeah! Come on along, come on along. A sculpture, yes, a sculpture, that’s the way I’d do it.”
He slammed the book abruptly and laid it on the floor.
“Donald,” he said, more in his natural voice. “Donald, if you have looked at that picture before and not felt the fear of the darkness in your veins, then all I can say is—you are hopelessly lost in your salvation. Have a Camel?”
Don shook his head, smiling, and Hans fitted a cigarette into his holder, continuing to talk between his teeth as he puffed life into it.
“Donald, the only man who is on such good terms with the devil that he can look him in the eye so casually is the satisfied theologian. You are a priest, not a prophet, and you must learn that even the bestest church what am will not protect you, your doctrinal orthodoxy will not save you, when the prophets begin to quake and wail outside the temple.”
“And you are the prophet?” asked Don, egging him on.
“Hah! No, not your prophet. No, honey lamb, you’ve missed my point. Or else my analogy doesn’t work out right. The prophets. Donald, I’m talking about mathematics, not art.”
“Well, then I wish you wouldn’t use a religious analogy. There is a fundamental difference in approach between religion and mathematics, and— No, let me finish. I know what you’re going to say: that mathematics is predicated on the worship of reason. Well, that’s wrong. Reason is only a tool to certain ends.”
“Well, I agree, of course, Don. Reason is a tool to certain ends, and in your case those ends are basically theological. It’s clear, you know, in this baroque fascination you have with the intricacies of your own proofs. You’re only interested in plastering over the cracks in the temple. You’ve grown too dependent on it; you’re afraid to worship in sunlight. Don’t hide behind reason, Don. Your enemy will use the same tool. It’s not reason that’s against you, sweetheart; it’s history.”
“Anyway,” Don interjected, annoyed at this turn in the conversation, “let’s start playing or we’ll never be done by ten.”
“All right. But, Donald, I am going to do that sculpture, that Alexander’s whatsit, someday. If ever I get a big commission. And you will be the only man who will understand all of its . . . all of its deeper meanings, my friend.”
It was to be another fifteen years before he would get that commission and carry out his threat. It would be only twenty feet tall, not thirty, and in Cleveland, not Chicago. But it would shock, amaze, and frighten thousands of art lovers/haters, just as the original conception had shocked, amazed, and frightened Hans Kaefig.
Hans Kaefig, whose thoughts enveloped, surrounded, like rows of black disks, moving, shifting, unpredictably, while Donald Lucus’ white disks coiled and struck, each move a step in a plan, each play a proverb. The patterns of black and white tesselations became too intricate to follow, and then there was no pattern at all, only the flashing black and white and a buzzing behind them, an insistent buzzing; as seconds stretched and expanded, he groped for his thoughts, sorting out the buzzing, reached for his glasses and the button on the intercom.
“Yes, Ruth?”
“It’s two fifty-five, sir.”
“Thank you, Ruth. Thank you.”
He sat another minute, not really awake. Then, both hands on the chair, he lifted himself to his feet and did his best to tidy his suit. The Director was twenty years his junior, and yet he felt like a truant student being sent to the principal’s office to explain himself, and knowing that the principal is never disposed to hear explanations.
* * * *
3
He hadn’t prepared a lecture in years. He had spoken of math only with other mathematicians. He had lost the knack of translation. There were English words, phrases, similes, that could say the same thing as a few swift logical statements, but he had forgotten them. And so he did not prepare a lecture. He had no idea what he was going to say to the Director, whether to present him with a neat, clear proof or simply to shout “Goit ist tot, “ and make his point loudly and emotionally. He knew the Director was not a believer in any of the fun
damental truths that were at stake. He would view a breakdown in the fabric of logic the same as a breakdown of the subway system. It was a nuisance to him, but it was certainly not his job to address himself to the problem; that was what metro engineers were paid for. That was what mathematicians were paid for. Lucus could not approach him on that level. What level he should direct his strategy toward he was not certain. He had, however, foreseen his opponent’s moves well enough to expect the reception he received from the Director’s private secretary.
“I’m sorry, Professor Lucus,” the young man clipped, his glasses sliding down his nose in what seemed a studied parody of Ruth. “I’m sorry, but the Director is extremely busy this afternoon. I can make an appointment for Wednesday, I think . . . Of course, the Executive Board meeting is coming up next week . . .”
“That’s all right, young man, I arranged to see him for a few minutes at three. I’ll just slip in, and you can go back to your datebook.” He would not be stalled any longer by the technical shunting about of the organization.
The Director looked blankly up from his desk as Lucus shut the door. He mumbled something feeble about thinking it was tomorrow that they were to meet, hoping to be rid of Lucus. As it became apparent that the math head had no intention of being put off further, he graciously conceded the skirmish and turned in an overly friendly manner to the problem itself.
As they talked, he leaned back in his chair, making full use of the physical advantage of his position. He sat comfortably in his shirtsleeves, collar open, bulky arms raised with his hands behind his head. Lucus alternately stood and sat—neither position was comfortable—in coat and tie, sweating through his shirt in the overheated office.
The Director was in his early forties, had held his position for three years. He divided his time unequally between his office, wife and children, and a girlfriend in San Jose. He drank more than Lucus had at his age, but seldom drank brandy. He was a mediocre chemist and an excellent administrator. He played golf one weekend out of two and worried that he was growing too fat. Lucus knew all this and very little else about the man. It was probable that he had studied calculus in college and forgotten a good deal of it by now, that he would be surprised to learn that an excellent mathematician might be very bad at arithmetic.
“Wait a minute,” he protested before Lucus was very deep into his subject. “I thought Lobachevsky did that. I don’t know much about math, but isn’t that what non-Euclidean geometry is? Didn’t they prove that Euclid was wrong? If it wasn’t a big catastrophe then, why should it be now?”
And so he had to backtrack and try to give a ten-minute summary of the history of axiomatics. That Euclid’s main contribution was not in his specific theorems, but in his method of assuming a very small number of “self-evident truths” and deriving all his results from them alone. That the question in the nineteenth century had only been over the notion of “self-evident,” and then only over the fifth postulate, the so-called parallel postulate, and the exterior angle theorem. That non-Euclidean geometries had never denied the consistency of Euclid, but had only proposed alternative, equally consistent systems.
The Director balked at the word “consistency.”
“But what’s the difference between consistent and true?” he asked innocently.
“Truth has no meaning in mathematics,” Lucus began. At the Director’s scowl he corrected himself, for he was no logician, and these distinctions did not come quite naturally to him. “Or rather, truth is defined only relative to a given system of assumptions, you see. A statement is true in this system if it can be proved . . . I’m not sure if that’s quite right . . . Well, anyway, if it necessarily follows from the assumptions. But a system of assumptions is consistent if you can’t prove a contradiction from them, you see? If they could be a description of something that really exists.”
“Okay, let me get this straight,” said the Director, fishing a pack of Marlboros out of his pocket. “Something is inconsistent if you can prove a contradiction from it, right? And what your Professor David seems to have done is prove that Euclid’s postulates—is that the right word?—that his axioms or postulates or whatever are inconsistent. Am I right? So that means the whole notion of Euclidean geometry is nonsense. Well, I’m no mathematician, but I don’t see the problem. Luckily this Russian has given you an alternative. So if, as you say, Euclid is scrapped, you still have this hyperbolic geometry and this other one, the one like the sphere, to choose from. It’s very interesting, but hardly the kind of thing that requires any sort of executive decision.”
Lucus bit his lip in frustration. He had always been a bad teacher, and Hans had said that . . .
“The non-Euclidean geometries were proven consistent by Riemann and Lobachevsky—” he began.
“Yes, well, that takes care of it, doesn’t it?” the younger man interrupted.
“No, it doesn’t!” said Lucus, too loudly. He sat down and tried to control his voice. “Non-Euclidean geometries were proven consistent by constructing models of them within Euclidean space. They are conditionally consistent. They are consistent only if Euclid is consistent. And, in the same way, Euclid depends on them. David’s proof is valid for all three.”
“You mean to say that every system of geometry is ... is inconsistent ... is meaningless?”
“Yes, sir. Not just geometry. Euclid can be derived from the real numbers. The real numbers can be derived from set theory. If Euclid is inconsistent, then the whole basis of mathematics is demolished. David’s proof comprises the futility—” Donald Lucus’ vision began to blur. His heart pumped blood deafeningly into his temples. There was a sharp pain in his chest. He spread his soaked and empty palms and spoke hoarsely. “—the futility of everything.”
The Director was not unmoved by this display. He expected such an emotional plea on the part of a suppliant for a research grant on occasion. He was used to tearful outbursts from his girlfriend in San Jose, and he could react gently but unfeelingly in most emotional situations. But old men made him acutely uncomfortable. Emotional involvement in one’s professional work puzzled and frightened him. He did not even yet understand the importance of the revelation which had been disclosed to him, but he did understand that it must be of some importance to bring this staid and dry old man to tears.
“You’ve checked it on the 666?” he asked.
Lucus looked away from him, embarrassed, fighting for breath, but trying not to breathe too deeply. “Not yet,” he answered. “I have computer time tonight. I’ve made arrangements to have a social projection done this Friday, dependent on your approval.”
“My approval?”
“For Limited Interest status.”
“Oh.” The Director rounded his lips meditatively and put his hands behind his head again. His cigarette lay in the ashtray, a long grey ash extending from the filter.
“Oh,” he repeated. “Well, yes, of course. I suppose if it checks out, that you feel Something Must Be Done?”
“Yes, sir. I think there may be indications that Something Must Be Done about the problem.”
And so Lucus knew that he had won the minimal confidence that he needed from the Director. The matter was to be given priority at the Executive Board meeting next week. He would have to go through the whole explanation again, many times. But it would be easier, the responsibility would no longer be entirely his. His white pieces coiled and struck across the board like a snake, squeezing the black ones out of strategic positions, reducing Hans’s forces to a few holdouts near the edge. The brandy was sharp and exhilarating this evening.
* * * *
4
With that trying interview over, Lucus felt a change in his mind and body. The oppressive burden was gone, and he could look forward to a great deal of time- and energy-consuming work. Responsibility was his, but it was the sort that he could be comfortable with, responsibility to get things done, to keep things moving. He spent the rest of Monday afternoon debugging the CONPROOF 2 input, which ha
d arrived from the A50 during his absence. The work was routine, undemanding, and gently satisfying. By five thirty it was in shape to run, and Lucus went home to dinner and eleven hours of cool and dreamless sleep.
Tuesday he worked continuously, stopping only for half an hour for lunch, fortified during the day by three cups of Ruth’s dark but tasteless coffee. He had called Bibliography as soon as he got in and had his checklist headings augmented greatly. Every month articles containing certain key words or phrases in their titles or abstracts were sent to all the department heads. The controlling program was sophisticated enough to produce some very worthwhile information and very little that did not hold at least some interest for him. Now, in addition to his standard topology codes, he added a few checks in various kinds of geometry—it was, after all, likely to be his field of specialization into the foreseeable future.
Orbit 16 - [Anthology] Page 27
