Years best sf 18, p.81
Year's Best SF 18, page 81
“All the rules of arithmetic that we learnt 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 not follow from those rules can then be seen anew, as a question about numbers. If this line follows from this 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, given any 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 there is 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 by doing 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 understanding anything is by doing arithmetic, and we’ve just found a statement that cannot 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? So we 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 I understand arithmetic, in a way that no electronic calculator ever will. What hope has a machine, then, of moving beyond its own most favourable 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 realised 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 that we 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 that we’ve all ended up with this mysterious ability to know that Godel’s statement is true? Where does this advantage, this great insight, come from? From our souls? From some immaterial entity that no machine could ever possess? Is that the only possible source, the only conceivable explanation? Or might it come from something a little less ethereal?
“As Professor Hamilton explained, we believe Godel’s statement is true because we trust the rules of arithmetic not to lead us into contradictions and falsehoods. But where does that trust come from? How does it arise?”
Robert turned the blackboard back to Hamilton’s side, and pointed to the cancellation rule. “If x plus z equals y plus z, then x equals y. Why is this so reasonable? We might not learn to put it quite like this until we’re in our teens, but if you showed a young child two boxes — without revealing their contents — added an equal number of shells, or stones, or pieces of fruit to both, and then let the child look inside to see that each box now contained the same number of items, it wouldn’t take any formal education for the child to understand that the two boxes must have held the same number of things to begin with.
“The child knows, we all know, how a certain kind of object behaves. Our lives are steeped in direct experience of whole numbers: whole numbers of coins, stamps, pebbles, birds, cats, sheep, buses. If I tried to persuade a six-year-old that I could put three stones in a box, remove one of them, and be left with four … he’d simply laugh at me. Why? It’s not merely that he’s sure to have taken one thing away from three to get two, on many prior occasions. Even a child understands that some things that appear reliable will eventually fail: a toy that works perfectly, day after day, for a month or a year, can still break. But not arithmetic, not taking one from three. He can’t even picture that failing. Once you’ve lived in the world, once you’ve seen how it works, the failure of arithmetic becomes unimaginable.
“Professor Hamilton suggests that this is down to our souls. But what would he say about a child reared in a world of water and mist, never in the company of more than one person at a time, never taught to count on his fingers and toes. I doubt that such a child would possess the same certainty that you and I have, as to the impossibility of arithmetic ever leading him astray. To banish whole numbers entirely from his world would require very strange surroundings, and a level of deprivation amounting to cruelty, but would that be enough to rob a child of his soul?
“A computer, programmed to pursue arithmetic as Professor Hamilton has described, is subject to far more deprivation than that child. If I’d been raised with my hands and feet tied, my head in a sack, and someone shouting orders at me, I doubt that I’d have much grasp of reality — and I’d still be better prepared for the task than such a computer. It’s a great mercy that a machine treated that way wouldn’t be able to think: if it could, the shackles we’d placed upon it would be criminally oppressive.
“But that’s hardly the fault of the computer, or a revelation of some irreparable flaw in its nature. If we want to judge the potential of our machines with any degree of honesty, we have to play fair with them, not saddle them with restrictions that we’d never dream of imposing on ourselves. There really is no point comparing an eagle with a spanner, or a gazelle with a washing machine: it’s our jets that fly and our cars that run, albeit in quite different ways than any animal.
”Thought is sure to be far harder to achieve than those other skills, and to do so we might need to mimic the natural world far more closely. But I believe that once a machine is endowed with facilities resembling the inborn tools for learning that we all have as our birthright, and is set free to learn the way a child learns, through experience, observation, trial and error, hunches and failures — instead of being handed a list of instructions that it has no choice but to obey — we will finally be in a position to compare like with like.
“When that happens, and we can meet and talk and argue with these machines — about arithmetic, or any other topic — there’ll be no need to take the word of Professor Godel, or Professor Hamilton, or myself, for anything. We’ll invite them down to the local pub, and interrogate them in person. And if we play fair with them, we’ll use the same experience and judgment we use with any friend, or guest, or stranger, to decide for ourselves whether or not they can think.”
The BBC put on a lavish assortment of wine and cheese in a small room off the studio. Robert ended up in a heated argument with Polanyi, who revealed himself to be firmly on the negative side, while Helen flirted shamelessly with Hamilton’s young friend, who turned out to have a PhD in algebraic geometry from Cambridge; he must have completed the degree just before Robert had come back from Manchester. After exchanging some polite formalities with Hamilton, Robert kept his distance, sensing that any further contact would not be welcome.
An hour later, though, after getting lost in the maze of corridors on his way back from the toilets, Robert came across Hamilton sitting alone in the studio, weeping.
He almost backed away in silence, but Hamilton looked up and saw him. With their eyes locked, it was impossible to retreat.
Robert said, “It’s your wife?” He’d heard that she’d been seriously ill, but the gossip had included a miraculous recovery. Some friend of the family had lain hands on her a year ago, and she’d gone into remission.
Hamilton said, “She’s dying.”
Robert approached and sat beside him. “From what?”
“Breast cancer. It’s spread throughout her body. Into her bones, into her lungs, into her liver.” He sobbed again, a helpless spasm, then caught himself angrily. “Suffering is the chisel God uses to shape us. What kind of idiot comes up with a line like that?”
Robert said, “I’ll talk to a friend of mine, an oncologist at Guy’s Hospital. He’s doing a trial of a new genetic treatment.”
Hamilton stared at him. “One of your miracle cures?”
“No, no. I mean, only very indirectly.”
Hamilton said angrily, “She won’t take your poison.”
Robert almost snapped back: She won’t? Or you won’t let her? But it was an unfair question. In some marriages, the lines blurred. It was not for him to judge the way the two of them faced this together.
“They go away in order to be with us in a new way, even closer than before.” Hamilton spoke the words like a defiant incantation, a declaration of faith that would ward off temptation, whether or not he entirely believed it.
Robert was silent for a while, then he said, “I lost someone close to me, when I was a boy. And I thought the same thing. I thought he was still with me, for a long time afterwards. Guiding me. Encouraging me.” It was hard to get the words out; he hadn’t spoken about this to anyone for almost thirty years. “I dreamed up a whole theory to explain it, in which ‘souls’ used quantum uncertainty to control the body during life, and communicate with the living after death, without breaking any laws of physics. The kind of thing every science-minded seventeen-year-old probably stumbles on, and takes seriously for a couple of weeks, before realising how nonsensical it is. But I had a good reason not to see the flaws, so I clung to it for almost two years. Because I missed him so much, it took me that long to understand what I was doing, how I was deceiving myself.”
Hamilton said pointedly, “If you’d not tried to explain it, you might never have lost him. He might still be with you now.”
Robert thought about this. “I’m glad he’s not, though. It wouldn’t be fair on either of us.”
Hamilton shuddered. “Then you can’t have loved him very much, can you?” He put his head in his arms. “Just fuck off, now, will you.”
Robert said, “What exactly would it take, to prove to you that I’m not in league with the devil?”
Hamilton turned red eyes on him and announced triumphantly, “Nothing will do that! I saw what happened to Quint’s gun!”
Robert sighed. “That was a conjuring trick. Stage magic, not black magic.”
“Oh yes? Show me how it’s done, then. Teach me how to do it, so I can impress my friends.”
“It’s rather technical. It would take all night.”
Hamilton laughed humourlessly. “You can’t deceive me. I saw through you from the start.”
“Do you think X-rays are Satanic? Penicillin?”
“Don’t treat me like a fool. There’s no comparison.”
”Why not? Everything I’ve helped develop is part of the same continuum. I’ve read some of your writing on mediaeval culture, and you’re always berating modern commentators for presenting it as unsophisticated. No one really thought the Earth was flat. No one really treated every novelty as witchcraft. So why view any of my work any differently than a fourteenth-century man would view twentieth-century medicine?”
Hamilton replied, “If a fourteenth-century man was suddenly faced with twentieth-century medicine, don’t you think he’d be entitled to wonder how it had been revealed to his contemporaries?”
Robert shifted uneasily on his chair. Helen hadn’t sworn him to secrecy, but he’d agreed with her view: it was better to wait, to spread the knowledge that would ground an understanding of what had happened, before revealing any details of the contact between branches.
But this man’s wife was dying, needlessly. And Robert was tired of keeping secrets. Some wars required it, but others were better won with honesty.
He said, “I know you hate H.G. Wells. But what if he was right, about one little thing?”
Robert told him everything, glossing over the technicalities but leaving out nothing substantial. Hamilton listened without interrupting, gripped by a kind of unwilling fascination. His expression shifted from hostile to incredulous, but there were also hints of begrudging amazement, as if he could at least appreciate some of the beauty and complexity of the picture Robert was painting.
But when Robert had finished, Hamilton said merely, “You’re a grand liar, Stoney. But what else should I expect, from the King of Lies?”
Robert was in a sombre mood on the drive back to Cambridge. The encounter with Hamilton had depressed him, and the question of who’d swayed the nation in the debate seemed remote and abstract in comparison.
Helen had taken a house in the suburbs, rather than inviting scandal by cohabiting with him, though her frequent visits to his rooms seemed to have had almost the same effect. Robert walked her to the door.
“I think it went well, don’t you?” she said.
“I suppose so.”
“I’m leaving tonight,” she added casually. “This is goodbye.”
“What?” Robert was staggered. “Everything’s still up in the air! I still need you!”
She shook her head. “You have all the tools you need, all the clues. And plenty of local allies. There’s nothing truly urgent I could tell you, now, that you couldn’t find out just as quickly on your own.”
Robert pleaded with her, but her mind was made up. The driver beeped the horn; Robert gestured to him impatiently.
“You know, my breath’s frosting visibly,” he said, “and you’re producing nothing. You really ought to be more careful.”
She laughed. “It’s a bit late to worry about that.”
“Where will you go? Back home? Or off to twist another branch?”
“Another branch. But there’s something I’m planning to do on the way.”
“What’s that?”
“Do you remember once, you wrote about an Oracle? A machine that could solve the halting problem?”
“Of course.” Given a device that could tell you in advance whether a given computer program would halt, or go on running forever, you’d be able to prove or disprove any theorem whatsoever about the integers: the Goldbach conjecture, Fermat’s Last Theorem, anything. You’d simply show this “Oracle” a program that would loop through all the integers, testing every possible set of values and only halting if it came to a set that violated the conjecture. You’d never need to run the program itself; the Oracle’s verdict on whether or not it halted would be enough.
Such a device might or might not be possible, but Robert had proved more than twenty years before that no ordinary computer, however ingeniously programmed, would suffice. If program H could always tell you in a finite time whether or not program X would halt, you could tack on a small addition to H to create program Z, which perversely and deliberately went into an infinite loop whenever it examined a program that halted. If Z examined itself, it would either halt eventually, or run forever. But either possibility contradicted the alleged powers of program H: if Z actually ran forever, it would be because H had claimed that it wouldn’t, and vice versa. Program H could not exist.
“Time travel,” Helen said, “gives me a chance to become an Oracle. There’s a way to exploit the inability to change your own past, a way to squeeze an infinite number of timelike paths — none of them closed, but some of them arbitrarily near to it — into a finite physical system. Once you do that, you can solve the halting problem.”
“How?” Robert’s mind was racing. “And once you’ve done that … what about higher cardinalities? An Oracle for Oracles, able to test conjectures about the real numbers?”
