Damon Knight - Orbit 16

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.