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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
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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
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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
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