lying entirely in unthinking matter.
“There might.” Hamilton smiled wistfully. “I had to concede that possibility, because I only had my instinct, my gut feeling, to tell me otherwise.
“But the reason I only had my instinct to guide me was because I’d failed to learn of an event that had taken place many years before. A discovery made in 1930, by an Austrian mathematician named Kurt Godel.”
Robert felt a shiver of excitement run down his spine. He’d been afraid that the whole contest would degenerate into theology, with Hamilton invoking Aquinas all night-or Aristotle, at best. But it looked as if his mysterious adviser had dragged him into the twentieth century, and they were going to have a chance to debate the real issues after all.
“What is it that weknow Professor Stoney’s computers can do, and do well?” Hamilton continued.
“Arithmetic! In a fraction of a second, they can add up a million numbers. Once we’ve told them, very precisely, what calculations to perform, they’ll complete them in the blink of an eye-even if those calculations would take you or me a lifetime.
“But do these machinesunderstand what it is they’re doing? Professor Stoney says, ‘Not yet. Not right now. Give them time. Rome wasn’t built in a day.’” Hamilton nodded thoughtfully. “Perhaps that’s fair.
His computers are only a few years old. They’re just babies. Why should they understand anything, so soon? “But let’s stop and think about this a bit more carefully. A computer, as it stands today, is simply a machine that does arithmetic, and Professor Stoney isn’t proposing that they’re going to sprout new kinds of brains all on their own. Nor is he proposinggiving them anything really new. He can already let them look at the world with television cameras, turning the pictures into a stream of numbers describing the brightness of different points on the screen…on which the computer can then performarithmetic. He can already let them speak to us with a special kind of loudspeaker, to which the computer feeds a stream of numbers to describe how loud the sound should be…a stream of numbers produced by morearithmetic.
“So the world can come into the computer, as numbers, and words can emerge, as numbers too. All Professor Stoney hopes to add to his computers is a ‘cleverer’ way to do the arithmetic that takes the first set of numbers and churns out the second. It’s that ‘clever arithmetic,’ he tells us, that will make these machines think.”
Hamilton folded his arms and paused for a moment. “What are we to make of this? Candoing arithmetic, and nothing more, be enough to let a machineunderstand anything? My instinct certainly tells me no, but who am I that you should trust my instinct? “So, let’s narrow down the question of understanding, and to be scrupulously fair, let’s put it in the most favorable light possible for Professor Stoney. If there’s one thing a computerought to be able to understand-as well as us, if not better-it’s arithmetic itself. If a computer could think at all, it would surely be able to grasp the nature of its own best talent.
“The question, then, comes down to this: can youdescribe all of arithmetic,using nothing but arithmetic?
Thirty years ago-long before Professor Stoney and his computers came along-Professor Godel asked himself exactly that question.
“Now, you might be wondering how anyone could evenbegin to describe the rules of arithmetic, using nothing but arithmetic itself.” Hamilton turned to the blackboard, picked up the chalk, and wrote two lines:
If x + z = y + z then x = y “This is an important rule, but it’s written in symbols, not numbers, because it has to be true forevery number, every x, y, and z. But Professor Godel had a clever idea: why not use a code, like spies use, where every symbol is assigned a number?” Hamilton wrote:
The code for “a” is 1.
The code for “b” is 2.
“And so on. You can have a code for every letter of the alphabet, and for all the other symbols needed for arithmetic: plus signs, equals signs, that kind of thing. Telegrams are sent this way every day, with a code called the Baudot code, so there’s really nothing strange or sinister about it.
“All the rules of arithmetic that we learned at school can be written with a carefully chosen set of symbols, which can then be translated into numbers. Every question as to what does or does notfollow from those rules can then be seen anew, as a question about numbers. Ifthis line follows fromthis one,”
Hamilton indicated the two lines of the cancellation rule, “we can see it in the relationship between their code numbers. We can judge each inference, and declare it valid or not, purely by doing arithmetic.
“So, givenany proposition at all about arithmetic-such as the claim that ‘there are infinitely many prime numbers’-we can restate the notion that we have a proof for that claim in terms of code numbers. If the code number for our claim is x, we can say ‘There is a number p, ending with the code number x, that passes our test for being the code number of a valid proof.’”
Hamilton took a visible breath.
“In 1930, Professor Godel used this scheme to do something rather ingenious.” He wrote on the blackboard:
There DOES NOT EXIST a number p meeting the following condition: p is the code number of a valid proof of this claim.
“Here is a claim about arithmetic, about numbers. It has to be either true or false. So let’s start by supposing that it happens to be true. Then thereis no number p that is the code number for a proof of this claim. So this is a true statement about arithmetic, but it can’t be proved merely bydoing arithmetic!”
Hamilton smiled. “If you don’t catch on immediately, don’t worry; when I first heard this argument from a young friend of mine, it took a while for the meaning to sink in. But remember: the only hope a computer has for understandinganything is by doing arithmetic, and we’ve just found a statement thatcannot be proved with mere arithmetic.
“Is this statement really true, though? We mustn’t jump to conclusions, we mustn’t damn the machines too hastily. Suppose this claim is false! Since it claims there is no number p that is the code number of its own proof, to be false there would have to be such a number, after all. And that number would encode the ‘proof’ of an acknowledged falsehood!”
Hamilton spread his arms triumphantly. “You and I, like every schoolboy, know that you can’t prove a falsehood from sound premises-and if the premises of arithmetic aren’t sound, what is? Sowe know, as a matter of certainty, that this statement is true.
“Professor Godel was the first to see this, but with a little help and perseverance, any educated person can follow in his footsteps.A machine could never do that. We might divulge to a machine our own knowledge of this fact, offering it as something to be taken on trust, but the machine could neither stumble on this truth for itself, nor truly comprehend it when we offered it as a gift.
“You and Iunderstand arithmetic, in a way that no electronic calculator ever will. What hope has a machine, then, of moving beyond its own most favorable milieu and comprehending any wider truth? “None at all, ladies and gentlemen. Though this detour into mathematics might have seemed arcane to you, it has served a very down-to- Earth purpose. It has proved-beyond refutation by even the most ardent materialist or the most pedantic philosopher-what we common folk knew all along: no machine will ever think.”
Hamilton took his seat. For a moment, Robert was simply exhilarated; coached or not, Hamilton had grasped the essential features of the incompleteness proof, and presented them to a lay audience. What might have been a night of shadow-boxing-with no blows connecting, and nothing for the audience to judge but two solo performances in separate arenas-had turned into a genuine clash of ideas.
As Polanyi introduced him and he walked to the podium, Robert realized that his usual shyness and self- consciousness had evaporated. He was filled with an altogether different kind of tension: he sensed more acutely than ever what was at stake.
When he reached the podium, he adopted the posture of someone about to begin a prepared speech, but then he caught himself, as if he’d forgotten something. “Bear with me for a moment.” He walked around to the far side of the blackboard and quickly wrote a few words on it, upside-down. Then he resumed his place.
“Can a machine think? Professor Hamilton would like us to believe that he’s settled the issue once and for all, by coming up with a statement thatwe know is true, but a particular machine-programmed to explore the theorems of arithmetic in a certain rigid way-would never be able to produce. Well…we all have our limitations.” He flipped the blackboard over to reveal what he’d written on the opposite side:
If Robert Stoney speaks these words, he will NOT be telling the truth.
He waited a few beats, then continued.
“What I’d like to explore, though, is not so much a question of limitations, as of opportunities. How exactly is it
