David’s proof had checked out perfectly in all situations, which did not surprise him at all. The program—which turned out to supply much more detailed output than he remembered from CONPROOF 1—even made some suggestions on simplification of certain steps in the proof. It was indeed valid. There could be no doubt of that. He began writing up a short report on the program, which he would eventually include in his report to the Executive Board.
All day Wednesday was spent in conference with the head of the computer division, explaining in detail the results of David’s proof and its connection with the rest of mathematics. They finally decided that the social projection could be done fairly easily with existing programs and data tapes, and it wouldn’t be necessary to confer with Sociology—at least not until after the initial run. Lucus found himself working especially well with the man, developing an instant rapport and communicating the details of the problem much better than he had with the Director. In fact, the entire day he felt especially energetic and happy, almost euphoric, and he finally went home after seven with a genuine sense of accomplishment, disturbed only by the itching occasional thought that there was something he had meant to do but forgotten—nothing very important, but some detail that was left out, that destroyed the symmetry of the day. But this thought was eventually buried by the mass of other details, important details, enjoyable details, that competed for his time until late Wednesday evening.
Thursday and Friday were spent shuffling between two projects: the social projection and his report to the Executive Board. The Executive Board meeting, which had been scheduled to begin at two o’clock on Tuesday, was rescheduled to the morning, and all section heads were advised that some very important business might well cause it to run into most of the afternoon. The mere fact that the nature of this business was not mentioned, of course, tipped them off that it was “Limited Interest” and probably involved the sort of executive action that the Institute was theoretically not empowered to take. Actually, the Institute did stick literally to the guidelines in the Congressional bill which had authorized its founding. It served in an advisory capacity to the agencies which carried out the occasional difficult decisions which the board was sometimes forced to reach despite the seeming incompatibility of these decisions with the supposed concerns of the Institute. And if employees of the Institute were called upon to aid these other government agencies in the regrettable but necessary enforcement of decisions made in the interest of the general good, they clearly cooperated with their government as private individuals, usually in “special consultant” positions, and not as employees of the FBRI itself.
And so by adhering to the letter of its charter, the Institute managed to stretch the spirit of the charter when that spirit became a threat to more important considerations. No one on the Executive Board took this responsibility lightly. Indeed, it was the gravity, the solemnity with which they were bound to weigh questions of ethics and then exercise their own benign power in the interests of the whole of society—it was this gravity, the awesome weight of obligation and the crucial necessity of judicious application of their superior skills which secretly thrilled many of the board members and added an unequaled zest to these meetings. There had been one department head who opposed all such actions, but he had left the Institute to return to teaching some years back. Now there was usually a broad and healthy range of opinion and discussion on questions of “interference,” as it was called, Genetics holding out against exercise of such power except in the most extreme cases, Biophysics being perhaps a bit overzealous in his enthusiasm for the Institute’s potential control of future events, and the rest of the departments arranging themselves variously between these two poles as befitted their individual politics, esthetics, professional ethics, temperaments, and digestive difficulties.
Mathematics, that is to say, Dr. Donald Lucus, was never entirely sure where he belonged in the spectrum, being, he knew, too easily swayed by each side of the debate in its turn, and most often casting his ballot with the majority. The issues were always too vague, uncontrollable, and, as he put it, “political,” and they seemed very far from his real concerns in his work. This time, however, he had no doubt which side of the issue he would take. He would have to hold the floor himself, and he knew for certain that the Institute must take appropriate measures to head off the catastrophic events that could be instigated by David’s proof.
The initial surprise of the board when they realized it was he who was going to read the report was the customary reaction he got from everybody whenever they learned that Mathematics might be involved in something important. He was the out man in the building, and he now felt a bit of pride in presenting his case, in being allowed to overshadow their scientific concerns with the problems of his field for an entire morning, perhaps for a whole workday.
He began by giving a precise and ordered account of David’s proof, its connection with the previous history of mathematics, and the interrelation of geometry with the foundations of all modern mathematical theory.
After this, and before the presentation by Computers and Sociology of the social projection, there was a period of questions directed to Lucus, as was customary, to ensure that everyone had a clear understanding of the issue. As it was, a number of them did, but the others were often reluctant to question a speaker in another field, less for fear of exposing their own ignorance than out of professional courtesy and a desire to avoid any question which might appear as a challenge to the speaker’s competence. It was usually understood that each head considered his colleagues as the final experts in their own fields, and their private terrain must be respected, as they respected his. An attempt to gain too complete an understanding of his territory was dangerously close to a takeover of his sovereign province. So the questions were only halfhearted requests for clarification about Hilbert’s axioms and the independence of the parallel postulate. Until Genetics raised a fluttering hand and asked with a Socratic smirk, “You say that the principles of Euclidean geometry can be derived from set theory, and, of course, this can be verified on the 666?”
“That’s right.”
“And so it all falls back on the principles of symbolic logic, and theoretically you could put the whole thing in terms of a proof in logic?”
“Yes, in fact, there is a program which can do just that with most mathematical proofs, and once I get around to it I intend—”
“Yes, yes. Well, very good. But isn’t it true that your whole method of proof is based on symbolic logic?”
“Yes.”
“And since you use this same method of proof in David’s Theorem—”
Lucus smiled. He knew what was being suggested.
“Since you use this same method of proof in David’s Theorem, and since you have shown that this method of proof is not valid —I mean, that’s what you’ve shown by proving the inconsistency of symbolic logic itself—then you really haven’t proved David’s Theorem at all, have you? I mean, have you?”
Naturally, most of the board members were made acutely uncomfortable by this want of tact, and especially by Genetics’ toothy grin as she spread her hands and waited for an answer. She was most unpopular, and it was rumored that she was not likely to remain much longer at the Institute. These rumors, however, had been circulated for a number of years with no noticeable effect on her position or her unwillingness to initiate a change of career herself. She was fifty-five now, and if she remained at the Institute many more years she was quite likely to become the Grand Old Lady, in which case her position and power would be unchallengeable until her own gracious retirement.
But now she was a minority of one, smiling at Lucus that same wide friendly smile that her secretary had conscientiously striven to imitate, smiling and waiting. He was not prepared for this particular question, but he had taught basic math courses long enough to be familiar with its general tactics. It was, put on the grossest level, to dismiss mathematical jargon as a lot of nonsense. But, applied more subtly,
it consisted of pointing out illogicality in the nature of the mathematical approach, in the detection of ubiquitous paradoxes, all of which eventually boiled down to some variation on the Russell paradox: the serpent of mathematics was forever swallowing its own tail. But Russell had long ago found a solution in the simple expedient of multiplying the number of serpents and lining them up to swallow each other’s tails, a much more plausible situation, and happily one which introduced no further problems until the level of transfinite numbers was reached, and here the mathematician was again swimming in his own medium.
“But your argument can only serve to confirm David’s proof. You are arguing from a paradox, from the absurd.” Lucus returned her smile, feeling around his shoulders the temporary, illusory mantle of the Grand Old Man.
“How so?” she asked.
“Well,” he continued, “simply because the method of proof of David’s Theorem is invalid, that does not insubstantiate its result —no, wait, let me finish—at best you would have to conclude that it is proven neither true nor false. Now suppose you assume that the basis of logic is in fact valid. Then you are forced to accept David’s Theorem and the proof of the invalidity of your logic; you are led to a contradiction. Therefore the assumption of the validity of logic is untenable.”
“But, on the other hand,” she objected, “if you assume that logic is invalid, then David’s proof is invalid.”
“Precisely. And that is a perfectly consistent position. David’s Theorem does not say ‘This sentence is false.’ It says ‘This sentence is unprovable,’ and therefore it must be true.”
“I think you’re talking in circles,” said Genetics.
“It’s all very well for you to think that, but the fact remains that this position is sound—it can easily be verified on the 666.”
The remaining questions were dutiful inquiries into the nature of the Russell paradox, and each answer was followed by a polite “Oh, yes, I see.” Lucus was calm and confident by the time Computers and Sociology began their description of the social projection.
The problem had been of quite a different nature from that of most of the projections that had been introduced to the board. Usually, a discontinuity was introduced into a percentage prediction pattern and the other initial conditions were varied within certain ranges, so that the effects of the invention of some new device or some new discovery in physical law could be ascertained, both short- and long-range effects. The discontinuity had some direct and immediate effect on material or political conditions, on arms capabilities or projected population figures or the economy. A good deal of such research had been done, it was true, with discontinuities of a religious or philosophical nature, and the sort of results obtained was quite well understood by those in the field. The Institute as a whole was seldom concerned with such results. It was the responsibility of other branches of the government to deal with the possible detrimental long-range effects of new religious or philosophical movements.
The projection for David’s Theorem showed remarkably unperturbed figures for a long period. Even in the mathematical world, it was predicted, little notice would be taken of it for fifteen to twenty years, notwithstanding its immediate effect on Dr. Lucus. It would be dismissed and ignored—at first. But within thirty years the disruption of the mathematical world would become violent and begin spreading into other fields. Still it would remain an academic debate. There would be much name-calling and side-taking, the introduction of heated emotions into decisions of hiring, tenure, structuring of mathematics and science departments. But still the public at large would remain entirely unaware of the issue. New schools of philosophy would arise to address themselves to the problem. Within forty years the issue would be taken up in the public press, the result being an increase in the polarization of scientists in all fields, and within fifty years, sixty at the outside, a violent antiscience reaction at all levels of government, huge cutbacks in funding, reduction of departments in universities all over the country, massive shutdowns of laboratories, and even elimination of many industrial research programs. The original issue would be mostly forgotten by this time; it would be the widespread fear of being dominated by scientists, scientists pictured as caricatures from Vincent Price movies, that would be the main concern of the public. But the effects would be disastrous for the scientific community.
There was heated discussion on this projection until one o’clock, when the board took an hour break, and then reconvened for the afternoon session. There was particular objection to the long-range nature of the projection. Many members of the board felt it was not their duty to be concerned with developments half a century in the future, and some of them were highly skeptical of the accuracy of the 666’s figures for such a period, although error estimates were given for all figures, and they usually didn’t get beyond five or ten percent at the sixty-year level.
To many it still seemed incredible that a mathematical theorem could have such an effect, but the evidence of the 666 was hard to dispute. Only Genetics objected to the input and assumptions of the projection program itself.
“What you don’t assume,” she said to Lucus, again smiling with all her teeth, “is the ingenuity of the world’s mathematicians. The program projects the proof of one theorem into the future, without considering what else will be proved in the future. If you scrap geometry, why shouldn’t a new geometry arise? If you scrap Aristotelean logic, why shouldn’t a new logic arise?”
“I assure you, madam,” answered Lucus, “the proof is valid for all logics, for intuitionism as well as for the many-valued systems.”
“And there are no other roads?” she asked cynically.
“And there are no other roads.”
“Well, then, I would suppose your field is likely to come to a stagnant standstill in any case, with or without any help from Professor David.”
There were shocked murmurs of censure up and down the table. Cryogenics, who had intended to question the methods of formulation of the theorem for the 666, thought better of it and folded his hands in silence.
The debate went on, but there was a marked increase of support for Lucus. By seven, when the final ballot was taken, he sat content, watching the other men and women folding their slips of paper. There was a sense of camaraderie, of common purpose, that he had never felt before with the Institute personnel. It was much like serving on the Honor Code Committee in college: the shared duty, the secret debates, and the final pride in the satisfying justice of the verdict, a day very well spent creeping, creeping to a close.
It was the consensus of the Executive Board that Something Must Be Done, and with that most of them could entirely forget the problem in good conscience. For Lucus, however, there were more meetings with the Director and Sociology, conferences in Washington, and an eventual temporary advisory post in an agency of the executive branch.
He returned to his work at the Institute, allotting a few hours a week to his new advisory position, the few hours he used to devote to research, research which seemed to lose some of its insistent appeal now that he had other important duties. The fall Quarterly came out only two weeks late, minus David’s article, with a brief editorial apology for the delay. Letters had to be written, editors conferred with to assure the difficulty of David’s publishing his article elsewhere in the near future. Meanwhile, the foundation was laid for the discrediting of any publication he eventually managed to achieve. The strategy was all laid out by experts in Washington who had handled similar cases, and Lucus had only to implement a few of the moves which required the prestige of his position in his field. Fortunately, David was up for tenure at his university that year. When it was not granted, he had immense difficulty obtaining a position elsewhere. He was quite a meek man, and certainly not paranoid enough to accuse anyone of being involved in a conspiracy against him. He ended up turning to high school teaching, which allowed him less and less time for any serious research.
The whole problem was neatly and efficiently
disposed of, and Lucus could not help admiring the simplicity of the plan of action. He had carried out his own part of the program carefully and professionally. No one could reproach him. He had represented his profession admirably.
* * * *
5
“You’re a creep!” said Hans, coming up behind him on his way back to the dorm after class.
“What do you mean by that?” he asked without turning, gripping his books tighter, feeling the strength in his fingers against them.
“You’re a creep, Lucus, that’s what!” Hans held up the morning’s edition of the school paper. “Look at this, you creepy bastard! That was really a rotten thing to do.”
“Look, the whole committee voted on Jonathan!” Don left the brick walk and cut across the lawn, anxious to be rid of Kaefig.
“And I know how you voted too, you creep!” Hans insisted. “And now the poor kid’ll be expelled, just because he got caught cribbing on one little exam . . . and you can be smug about it. Can’t you find a better way to save the honor of your precious code?”
“It wasn’t one little exam, Hans, and besides—let go of me!— and besides, it wasn’t just Jonathan we had to think of; it was the integrity of the whole school and the Honor Code. How long do you—I said let go of me!—how long do you think the faculty would let us keep the Honor Code if we let everybody off who broke it? Hey, get away! I’ve got to go to lunch!”
Orbit 16 - [Anthology] Page 28
