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RULES WITHIN RULES

Patrick and Michael Flynn were twin brothers, the sons of a family doctor who practiced in a sleepy Irish village called Ballaghkelly. The village was little more than a crossroad buried in a huddle of houses and outlying farms, and boasted one church, a school, a few tiny stores, a newsagent's that was also the post office, and three pubs. Little ever happened to change the routine of a typical day in Ballaghkelly. At six o'clock every morning, Willie Maherty's one-horse dray would clatter off on its round of the farms to collect the churns from the previous day's milkings before the day warmed. Twice in the day and once in the evening—exactly when was always a standby topic of conversation if the weather looked settled—the bus from Kilkenny would arrive and depart again. And after nightfall, the farmers would begin gathering in O'Toole's, Mulligan's, and O'Shaughnessy's to nod and murmur over their pints of porter while they reiterated worldly wisdoms that had been handed down from father and grandfather for generations. Life had always been that way, and nobody—in the unlikely event of such a thought entering his head—could have conceived of its being any other.

Patrick was a voracious reader and did brilliantly in school. Kevin Halloran, the schoolmaster, became convinced that the boy was a prodigy and persuaded him to try for a scholarship to study mathematics at the university in Dublin. Patrick was accepted, gained a degree with honors after some years, and then departed to pursue postdoctoral studies in America, which was somewhere across the sea, in the opposite direction from England—whichever way that was. Michael, too, was studious, but his vocation lay in a different direction. He left home to train at the Catholic seminary in Waterford, and shortly after Patrick left for America, Michael was installed as the new parish priest in a town not far from Limerick.

Although the brothers corresponded, they did not meet again for several years. Then Patrick wrote from America to say that he had been offered a research fellowship at Cambridge, England, and that before taking up his new position he would come back to Ballaghkelly on a short holiday, which he referred to in his letter as a "vacation." Michael, too, obtained leave of absence and came home for the big family reunion. There was a fine party at the Flynns', as good as any wedding or funeral wake that even the oldest in Ballaghkelly could remember, and in the course of the evening it seemed that every pair of feet from twenty miles around had passed over the threshold, bearing some well-wisher to join the family in a glass of stout and perhaps sample one—or two, or maybe a few—of the choice blends of whiskey brought in for the occasion. At last, after it was all over, and their father had tottered unsteadily but contentedly away to his bed, with their mother closing the door after him and warning them sternly not to touch any of the mess until morning, Patrick and Michael found themselves alone with their glasses, staring at the logs crackling in the large brick fireplace in which, as boys, they had watched faces and dragons together, many years ago.

"Ah, 'tis a wonderful place ye've been tellin' me about, Patrick," Michael said distantly. "With them Americans drivin' their motorcars along roads as wide as ten boreens, and flyin' around in the sky as naturally as the likes of us would take the bus into Kilkenny . . . and walkin' around on the moon itself, if you're not after pullin' me leg."

Patrick took a sip from his glass. "And that isn't the half of it, Michael. There's the size of the place. . . . Do you know, the nearest part of the U.S.A. is two hundred times farther away from here than Dublin, yet it's still nearer to us than it is to the other side. Can you imagine that?"

Michael tried to visualize what it meant, but in the end shook his head with a sigh and sank back in his chair. "Anyhow, about yourself, Patrick," he said. "Ye told us enough about America in the letters ye wrote. But there was never a time when ye said anythin' about what ye were doin' in that university." His brow creased uneasily. "Ye wouldn't have gone and got yourself mixed up with the makin's o' these bombs and things, now, would ye, Patrick? 'Tis the devil himself's work ye'd be doin'."

Patrick gave a short laugh. "You don't have to worry yourself about anything like that, Michael. There are some areas of research that aren't connected with the weapons program, you know." Michael looked relieved. Patrick went on, "In fact I'm not involved with any aspect of applied research at all. My work is all to do with pure mathematics—in fact, an area called number theory, if you'd really like to know." Modesty had prevented him from saying much about this in his letters. Now, however, with the euphoria of being home again—and perhaps also from the effects of the party and the drink—he was unable to keep just a hint of a swagger out of his voice.

Michael seemed not to notice. "Ah, so that's what it's called, is it?" he replied, nodding slowly. "And what would ye be doin' with the numbers? Is it some kind o' computin' with them electronic machines?"

"We use computers a lot, but that isn't really what it's all about," Patrick said. "Number theory is simply the study of the properties of the whole numbers themselves, and of the rules for manipulating them."

"And that could keep an honest man busy for a lifetime?" Michael sounded dubious.

Patrick laughed again. "A lot more than that, Michael, believe me. People have been developing it for centuries, and they've still only scratched the surface. My work only touches upon one little piece of it."

"Is that a fact, now?"

"It has to do with the implications of something known as Gödel's Incompleteness Theorem. It, er . . . it deals with the inherent limitations of any formal system of rules, no matter how complex."

"Ah." Michael gazed silently into the fire for a few seconds. "Gödel, you say, eh?"

"Kurt Gödel . . . an Austrian mathematician. He formulated the theory in 1931. What it says is that all consistent axiomatic systems of number theory include undecidable propositions." Patrick paused for a second. "Well, actually the original was expressed in more technical terms, and it was in German, of course . . . but that's about what it boils down to."

Michael squinted and rubbed his nose with the crook of a finger. "Well, it might as well be in German still, for all the sense I can make of it," he confessed. "It's havin' a bit o' fun at your brother's expense, ye are, if I'm not mistaken. Now could ye imagine me in the pulpit on a Sunday, railin' me congregation with that kind o' talk? Why, wouldn't Mother McCreavy from the village be down to the post office at the crack o' dawn the next mornin', writin' letters to the Holy Father himself? Away with ye now, Patrick. If what ye just told me can't be said in God's own English, then it's likely as not that it's without any meanin' at all. I'm thinkin'."

Patrick grinned apologetically. "I guess I just couldn't resist it." He refilled their glasses from the bottle standing between them. "It concerns the systems of rules that govern mathematical proofs. What it says is that no set of rules can ever be complete enough in itself to enable every true statement to be proved. There will always be at least one statement that can only be proved by bringing in another rule from outside the system. And if you add that rule to make a new, bigger system, then the new system will contain at least one statement that would require yet another rule to prove it, and so on. There's no end to the process. However big you make the system, it can never be complete enough to prove all true statements. That's why it's called the Incompleteness Theorem."

A silence fell while Michael tasted his drink and sat back to reflect on this. Eventually he said, "You'd think, now, wouldn't you, that rules and such would be somethin' a priest would know all about, for isn't every day of his life just a matter of passing on a few simple rules of livin'? But I'm blessed if I can tell the head from the tail o' what you're tellin' me now, Patrick—blessed if I can at all."

"Actually, that's not a bad analogy," Patrick said. Michael looked puzzled. Patrick sat forward and spread his hands to explain. "People, nations, society in general . . . they all have systems of rules—laws—that govern the ways they behave. Now this comparison is only a loose one, you understand, but it gives an idea—no system of social rules is ever complete, is it? For who writes the laws that govern the lawmakers? You see my point—such laws would have to be written from outside the system. But then the same questions would still apply: Who would write the laws to govern whoever wrote those laws? You could go on as long as you like, but you could never completely solve the problem."

Michael considered the proposition for a while. " 'Tis a sad picture of the human race that you're paintin'," he commented at last. "Ye make it sound as if everybody in the world is unable to live a decent life without rules to stop them from robbin' each other and cuttin' each other's throats."

Patrick sighed. "True, but what can you do? That's the way the world is out there. They're all in a rat race, scrambling and trampling over each other to get a bigger piece of the cake. And when a bunch of them get into a position where they can write their own rules, it brings out the worst. Sooner or later they have to be regulated somehow, but that never solves the problem. All it does is shift it another level higher up."

"How would ye be meanin'?" Michael asked.

"Oh, the legal system over there in America is a good example," Patrick replied. "The lawyers make money by complicating the problems that they're supposed to be solving, which suits them fine, but doesn't suit the clients. They can get away with it because the rules let them, and they write the rules. It's the same with price-fixing cartels in business, or the lobbying that corporations can do to get tariff laws and other restrictions passed to get an edge over their competitors. And it's the wealthy who get to influence how loopholes are written into the tax code. You see—whenever a group can write its own rules, it writes them in its own favor. Everyone else loses."

Michael shook his head sadly. "That's a terrible thing . . . that with all these machines and all, and them fellas walkin' around on the moon, they'd still be havin' this kind o' trouble with each other. Ye'd think, now, that with all their talk about puttin' the Russians in their place, the government would have somethin' to say about these carryin's on. Have they no care at all for the people that's payin' the money for the motorcars for them to go paradin' themselves around in with all their grand speeches and smilin' faces?"

"That's my whole point, Michael," Patrick agreed, nodding. "You're right—in theory the government ought to be able to prevent things like that through the power of law. But in practice it doesn't work out, because the government in turn isn't subject to any law but its own, and the same thing happens. The federal bureaucracy is out of control. A bureaucrat's status and income are geared to the size of the department he runs and the number of people in it. Therefore they all want their departments and budgets to grow bigger. To justify that, they have to have problems that are getting bigger instead of being solved, and the legislation that emerges makes everything worse, not better. Take the welfare system as an example—it costs more than anything else in the government budget. If everybody in the country became self-sufficient, the agencies would be out of business. So laws get passed which guarantee that there'll always be a bottom layer of dependents who'll never able to climb out of the trap—such as minimum wages that make them unemployable, or housing standards that drive prices above anything they can afford. And then, of course, we've got the defense industry and the military—they don't want any relaxation of tensions with the Soviets. And the banking system that finances the government also owns the contractors that the government spends the loans with—so they're making money with both hands by billing the taxpayer for lending their own money to themselves. It's a good deal—you sell a few billion dollars worth of sophisticated weaponry, put it down holes in the ground for ten years and wait for it to become obsolete, then scrap it all and start over again. And while the banks are owed money, they write the rules."

Michael stroked his chin thoughtfully as he stared into the fireplace. "And there's honest people toilin' away the lives that God meant them to enjoy, just to pay for it all, eh? And to think, with all them machines and the clever fellas in the universities, couldn't they all be livin' like kings?"

"I guess so." Patrick sighed. "But they won't. Too many people would have to come down a peg or two, and they're the ones who write the rules. Oh, I suppose you could think in terms of a world government or something one day, but the same would happen eventually because there wouldn't be any rules to control them."

"Maybe they should try lookin' in that number theory that ye were tellin' me about," Michael suggested. "Wasn't it all to do with writin' rules within rules, and isn't that what we're talkin' about now?"

Patrick frowned. "Number theory doesn't apply here," he said. He kept his voice even to avoid sounding as if he were talking down to his brother. "The analogy was only a loose one. Gödel's theorem merely deals with sets of formal symbolic axioms. It doesn't have anything to say about social rules and how people should behave toward one another. You're reading too much into it."

"I have to disagree with ye, Patrick," Michael said. "It has everything to say—about some very good social rules that were written down a long time ago now, and very concisely."

Patrick stared across with a puzzled frown. "I'm not with you, Michael," he said. "What do you imagine Gödel's theorem says that has anything to do with people?"

"Tch, tch." Michael shook his head reproachfully. There was a faint smile on his face, and his eyes twinkled in the glow from the fire. " 'Tis a shame ye've never seen it, Patrick," he replied. "It's been all these years now, and ye mean to tell me that ye've never noticed what the first three letters of Gödel are?"

 

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