Finity

by John Barnes

  “I’ll do that. Really. And if you’d like a good reference for your file, let me dictate you one.”

  “Thanks a bunch, Mac! Sure.”

  I spent a few minutes blabbing on about what a splendid limo this limo was, and how pleased I was with it, which it recorded into a file of references for later; I figured it would be no bad thing for this one to get commended.

  The coast of Java had just disappeared over our horizon in the rear window as the Big Sapphire seemed to rise from the sea in the windshield. It was called that because it was a gigantic blue regular dodecahedron, balanced on a single slim column that held it about a hundred feet above the water, and it was a bright blue that seemed to glow with an inner light, a neat effect achieved by fiber optics that took light on any one surface and relayed it through the half-kilometer of building to the corresponding point on the opposite surface. Probably it was the most famous building in Asia these days, since it appeared in so many ConTech ads.

  The slim column, in fact, was only proportionately slim—it was thick enough to be a creditable skyscraper, and as we approached I saw the doors slide open in the base for us, and a ramp extend down onto the water. “Got you here now, Mac,” the limo said, and we glided up the ramp, into the column, and to a stop inside a freight lift that whisked us upwards for what seemed the better part of a minute. Then the lift doors opened onto a big lobby area, and the limo said, “I enjoyed driving you today, Mac, take care and good luck with the interview.”

  The door beside me opened, and I got out and removed my bags from the boot as soon as that opened. Then I walked forward into the lobby; behind me, the lift took the car away. I looked around and wondered what I should do next.

  “This way, Dr. Peripart,” a pleasant voice said, and I walked toward it. Beyond a set of dividers and a row of potted plants, there was a large desk and a set of worktables, and seated at the desk was Geoffrey Iphwin. He got up and came around the desk, and we shook hands.

  “A pleasure to meet you, Dr. Peripart. May I call you Lyle?” I nodded. “Thank you, Lyle. And although some people try to call me Geoff or Geoffrey, I really am more used to Iphwin. Not Mr. Iphwin, for the love of god. We’re both American expats and we are renowned for our informality, aren’t we?”

  “I suppose so,” I said.

  “Exactly! Excellent! Have a chair—coffee?”

  He was absolutely the most energetic person I had ever seen in my life, seeming to zip from place to place in his spacious working quarters as if he were on high-speed rails. There were times when I could have sworn that he wasn’t visible between where he started and where he ended up; he had offered me a chair, one of eleven in his office space, and—I know because I was fascinated enough to watch for it—during the first part of the interview, he sat in all ten of the other chairs at least twice, besides also perching on his desk and on a worktable.

  Physically he was a slight man, with a crooked, vaguely beaky nose, prominent teeth that didn’t quite form an overbite, and large, close-set, washed-out blue eyes. His mouth small, his thick protruding lips a deep red, chin narrow, and the overall effect was of a man put together from spare parts. When he smiled or laughed, which was often, he seemed to be one of those people who does it with his whole body and soul.

  “To begin with and to put your mind at ease,” he said, “I read your dossier and you are the guy for the job, so the only problem of this interview is to persuade you to take it and to become acquainted with you.”

  “That’s good to hear,” I said. “But I think before we start I should show you something.” I pulled out the blue note that had been left for me that morning, and watched as he read it, frowning in concentration. It seemed very strange that the wealthiest private citizen on Earth should move his lips when he read, but Iphwin did.

  “Lyle, if I may have this,” he finally said, “I would very much like to see what my security people can find out about it. They might or might not have something to tell us even before the interview is over—I’ve known them to be that good.” He spoke into the air. “Security here for a piece of evidence for analysis.”

  A steel door opened in the wall behind him; it had been all but invisible between the two windows before then. A uniformed guard came, took the note, asked me the obvious questions about whom I had told and whether I had any personal enemies, and departed by the same concealed elevator.

  “No way of knowing what they’ll find and I detest theorizing in advance of data,” Iphwin said. “Well, back to the problem at hand, then. I assume that if we can adequately clear up this threatening note—that is, we can establish that the person who wrote it cannot harm you, and that I am your real friend and that person is not—you are still interested in the job? And since you don’t know us here at ConTech at all well, I also assume that you will want some proof and evidence of good faith? I know I would in your position.”

  In my position, I thought, I still don’t believe that this is happening. I thought at best I might be hired to do statistics for some research project using a mathematical method similar to the abductive statistics I’ve used in my work. I am not accustomed to having a car—even a very friendly and pleasant car—tell me that I am about to meet an international celebrity, less than fifteen minutes before I do. Out loud, I said, “That’s extremely reasonable and it already increases my trust. I’m sure we’ll be able to work something out soon enough. But, sir—”

  “Iphwin.”

  “Er, yes.” I swallowed hard. “Iphwin, this whole situation makes no sense to me. I’m not a particularly distinguished astronomer. It makes some sort of sense that you want me as a statistician, because that is the one area where I’ve done considerable original work, but all the same there are mathematicians out there who could do rings around me—rings and groups and matrices and tensors, to tell the truth.”

  He didn’t laugh; inwardly I cursed whatever it was that had prompted me to make a feeble mathematical pun. Then abruptly he did laugh, and said, “But if they don’t do Abelian groups, they’ll have to live here in the building, since they can’t commute.”

  Startled, I laughed.

  “You see,” he said, “we have similar senses of humor.”

  But yours seems to run on a schedule different from mine, I thought. “Anyway, sir—I mean Iphwin—it just seems to me that you could easily get someone better for whatever job you could possibly have in mind.”

  Iphwin hopped up on his desk and crossed his legs, peering at me over his knee, like a small boy about to spring a transparent practical joke. “Who else has even tried to develop a statistics of abduction?”

  “Er—eight or nine people. And only four of us are alive. But that’s a pure hobbyhorse of mine. If Utterword weren’t the editor of that little journal, I wouldn’t even be getting published.”

  “But you get results.”

  “I think I do.”

  His smile grew more intense, his eyes twinkled, and he said, “Tell me everything about abduction.”

  “That’s a tall order,” I said. “At least let me try to summarize. About 170 years ago, the great American polymath, Charles Sanders Peirce—”

  “I thought it was Pierce,” he said, pronouncing it with a long e.

  “He pronounced it like ‘purse,’ “ I said. “Anyway, Peirce did an enormous amount of work on logic, developed a very eccentric theory of semiotic, and made contributions to half a dozen sciences and to philosophy, but this is one of his strangest ideas—and he had some very strange ones.”

  “Strange but not bad?” Iphwin asked.

  “Not bad, or at least not all bad. Peirce said that there were two common kinds of logic—deduction and induction. Deduction is deriving the behavior of the particular case from the general case, like the famous syllogism where you figure out that Socrates is mortal. Induction is the other way round, figuring out general laws from some number of particular cases, like noticing that some energy is always lost irrecoverably as heat in every physical ex
periment you can run, and coming up with the theory that entropy always increases. Induction gives us general laws, and deduction lets us use them in our particular cases; one gets us ready to cope with a situation and the other is the process of coping. They’ve served humanity pretty well.

  “But, Peirce said, that set is incomplete. There’s one more kind of logic not covered there.

  “Now one reason he might have thought that is that in Peirce’s thought everything is always organized into threes, so anytime there’s a pair, it’s incomplete, and a third member must be found. It might be no more than that. But Peirce proposed a problem that turns out to be surprisingly difficult to resolve in a satisfactory way, which seems to indicate that there really ought to be one more kind of logic.”

  Iphwin jumped up and paced; it was just as if this was all news to him, and yet if he had really been interested in Peirce and in Peircean thought, he could probably have found a Peirce scholar cheap—studies of obscure philosophers do not make for lucrative careers—and gotten a much better exposition than I was giving him. The pacing and gesturing seemed as if he were playing the part of a man consulting an expert, and he expected me to play the role of the expert. “So,” he said, “Peirce proposed a problem?”

  With a small tremor of guilt, I realized I had gotten fascinated with watching him, and had not talked for several seconds after his question. “What Peirce proposed was a problem which ought to have a logical solution—that is, one you could arrive at by stepwise objective reasoning that anyone with adequate training could copy or evaluate—for which he could show that both induction and deduction could not lead to the solution. If it was soluble, then it had to be soluble by some other means.” I was warming to the subject, now, I confess, and at the same time I was very worried that I might bore him or begin to lecture and thus lose the friendly warmth he had been beaming at me since I arrived. “What he said was that all logic is basically made up of terms, propositions, and arguments—names of things, statements about names, and groups of statements from which you can generate more statements. ‘Socrates’ is a term, ‘Socrates is a man’ is a proposition, and the syllogism is an argument. Now, Peirce says, it doesn’t matter where we get terms because they’re not subject to logic and are purely arbitrary—all we have to do is remember that we called it a ‘glump’ last time, and we can just go on calling it a ‘glump’ forever. And obviously arguments are deductive or inductive logic, so we know how we get arguments—we take propositions and apply the rules of induction or deduction to connect them with each other.”

  “But!” Iphwin shouted. “But!” He leaped up and spun around.

  By then I was about half ready to join him; his enthusiasm was so contagious and it would have made as much sense as anything else. I couldn’t help smiling but I otherwise restrained myself and went on. “We know where we get some propositions—we make them out of other propositions, using arguments. But where do the starter propositions come from? How do we link terms to form propositions without going through the stage of argument—since we can’t make arguments if we have no propositions? And Peirce’s answer was that we must have a way of choosing propositions out of the whole vast welter of possible ideas, and of knowing that some propositions are more likely to yield worthwhile results than others. And that way of choosing is his third kind of logic—which he calls abduction. ‘Deduction’ is Latin for leading an idea down—that is, down from general to particular. ‘Induction’ is Latin for leading an idea to or into something—that is, to or into the general from the particular. But abduction is leading away—taking some combination of words, symbols, thoughts, or whatever out of the vast swamp of what it’s possible to think of, and picking one that has a chance of being true, so that when we perform induction and deduction on it, we stand a good chance of gaining either a general law or an understanding of a particular situation.”

  “Where do statistics get into it?” Iphwin asked.

  “There’s a trivial argument that if you could look at all the possible propositions—a list that would include things like ‘Ice cream comprehends lions and dislikes beauty,’ ‘It always rains on vacuum-flavored machine tools,’ and ‘The king is polynomial’—most of them would be inapplicable to the real world, not testable by any means whatever, which is another way of saying we wouldn’t be able to know if they were true. Another large group is testable but not useful or interesting—’Monkeys wear red dresses to seduce geraniums.’ The number that would be interesting if true is a fairly small proportion—and of course the true ones are a small subset of that. It turns out that the important question is, what’s the shape of the population of possible ideas? And how many of those ideas are useful, which is shorthand for ‘capable of being true in some circumstance where it would matter to someone?’ And how is it possible for any finite mind to sample effectively from that population?

  “As soon as you realize that the number of useless statements must be much, much greater than the number of useful ones, you have to see that people can’t possibly be generating propositions on a purely random basis, testing all of them, and keeping the ones that work. They must have a way to find a good-enough place to start, some way to come up with the subset of propositions worth examining, a process of some kind, because we don’t see people paralyzed about what to buy Uncle Ned for his birthday because first they have to think of all the possible statements involving buying, then all those involving Uncle Ned, and then all those involving birthdays.

  “Well, I thought, the world has so few astronomers this century, they can’t possibly look for all the interesting things that might be happening in the sky—so how do they choose a proposition to test? With so few of us, could we really just rely on intuition? Or luck? But if you admit that there is some use in intuition, that it does something better than a random statement generator could, it must be a human capability of some kind, rooted in the real world, which means that very likely it can be developed and trained to make someone better at it—which might even be the same thing as making him lucky. And if you can invent a method for training intuition, you have to be able to describe what it does-—and in math a description is always at least halfway to a solution. So I started to think that maybe I could invent a way to imitate, computationally, what intuition does.

  “From that initial idea, I developed some theories about sampling and about how to find the answer next to the answer next to the answer that’s the right answer, and so forth, and I’ve been publishing ever since. With, I might add, hardly any reaction worth talking about from any of my fellow astronomers, who are mostly just guys that like to photograph stars.”

  Iphwin nodded. “The lack of reaction is profound, and more profoundly it is to be expected.” I wanted to ask him what he meant, but he went on before I could. “And yet, however large, the number of possible propositions must be finite, since it’s generated from a finite list of terms, and we know the list is finite because there’s only so many things in the universe, or at least only so many things that we can encounter between the beginning and the end of our species. Am I right?”

  “I guess as far as it goes. But you know, there may be as many as half a million words in English, so that just the number of possible statements of some short length—maybe one hundred bytes and shorter—would have to be more propositions than could be thought of between the Big Bang and the end of time, even if the universe were made up of nothing but proposition-writing computers. No reason to be concerned about a number being finite when it’s infinite for every practical purpose— abduction from an infinite set is not materially different from abduction from an extremely large one.”

  He sat back in one of the chairs, stretched, and put his hands behind his head. “So you have worked out the rudiments of a method for doing abduction mathematically, instead of just trusting whatever it is that human beings have and robots don’t.”

  “Rudiments is the word,” I said. “I have little bits and pieces of a method and not th
e slightest idea whether the pieces could ever come together to form a coherent theory.”

  “Have you solved the problems I asked you to solve?”

  “I think so,” I said. “Let me pull out my computer and I’ll show you what I have.”

  The problems had all been very peculiar—the first question was “Which English language poetic forms would be the best ones to study in order to understand the concept of triteness?” Another one was to explain why “meaningful” and “nonmeaningful” were or were not meaningful as categories applied to integers. Yet another problem was “How many published physical experiments would be required within a period of twenty years to cause all physicists worldwide to believe that there is a fifth fundamental force, and what is the likelihood that they would believe so correctly?”

  Originally I had developed the abductive statistical methods because the number of possible hypotheses in astronomy, about things big and little, general and particular, and all, was so large relative to the number of astronomers actually working that it seemed unlikely to me that any astronomers at all, out of the whole population, were working on anything particularly important. The world only had one-tenth as many trained professional astronomers in 2050 as there had been in 1920, and yet the thousands of amateurs had flooded the databases with innumerable observations. I was looking for a way to choose the most productive paths of research—but since a path of research is a set of propositions about what hypothetical propositions should be tested by argumentation against a set of propositions about what did happen, that’s just another way of saying I needed a method of abduction, and the abductive problem in front of me was much bigger than the abductive abilities of the naive human brain.