Damon Knight - Orbit 16

“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.

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 had 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.