Bron nodded. “Fine. I’ll just get—” getting up.
“May I sit in this—?”
“Sure. Make yourself comfortable. How do you take your—?”
“Black,” she said from the sling chair, “as my old lady,” and laughed again (while he reached into the drawer at his knee and dialed. One plastic bulb, sliding out, hit his knuckles and burned). “That’s what my father always used to say.” She put her hands on her knees. “My mother was from Earth—Kenya, actually; and I’ve been trying to live it down ever since.”
Bron smiled back, put one coffee bulb on the desk, reached down for the other and thought: Typical u-1 ... always talking about where they come from, where their families started. His own parents had been large, blond, diligent, and (after years of working as computer operators on Mars, when their training on Earth, outmoded almost before their Martian emigration, had promised them glorious careers in design) fairly sullen. They were in their midforties when he’d come along, a final child of five. (He was pretty sure he was a final child.) Was that, he wondered again, why he liked sullen-looking women? His parents had been, like so many others it was embarrassing, laborers in a new world that needed such labor less and less. He had not lived with them since he was fifteen, had not seen them since he was twenty, thought about them (usually when someone was talking about theirs) seldom, talked about them (in concession to a code of politeness almost universal outside the u-1 that, once he had realized it existed, he’d found immensely reassuring) never.
Bron handed Miriamne the second bulb. “All right. Metalogics ...” Back behind the desk? No, better prop himself on the front again, for effect. “People—” He settled back on crumpling flimsies—“when they go about solving any real problem, don’t use strict, formal logic, but some form of metalogic, for which the rules of formal logic can be considered—on off Thursdays—the generating parameters. You know the old one: If a hen-and-a-half lays an egg-and-a-half in a day-and-a-half ... ?” He raised an eyebrow (the real one) and waited for her to sip:
Her plastic bulb wrinkled in miniscule collapse. She looked up.
“The question is: Then how many eggs does one hen lay in one day?”
“One?” she suggested.
“—is the quote logical unquote answer people have been giving off the top of their heads for over a hundred years. A little thought, however, will show you it’s really two-thirds of an egg—”
Miriamne frowned. “Cybralogs are speech/thought representation components—I’m a hardware engineer: I don’t know too much about logic, meta or otherwise. So go slow.”
“If a hen-and-a-half lays an egg-and-a-half in a day-and-a-half, then three hens would lay three eggs in the same day-and-a-half, right? Therefore one hen would lay one egg in that day and a half. Therefore one hen lays—”
“Two thirds of an tgg—” She nodded, sipped. The bubble collapsed more—“in one day.”
“We got into metalogic,” Bron explained (thinking: With the sullen, intelligent ones, that look of attention means we’re getting further than we would if they were smiling), “when we ask why we called ‘one’ a ‘logical’ answer in the first place. You know the beginning tenet of practically every formal logic text ever written, ‘To deny P is true is to affirm P is false’?”
“I vaguely remember something about denying the Taj Mahal is white—” Miriamne’s bubble was all wrinkled plastic between bright nails—“is to affirm that it’s not white ... an idea that, just intuitively, I’ve never felt very comfortable with.”
“You have good reason.” Bron sipped his own and heard the plastic crackle. “The significance of ‘white,’
like the significance of any other word, is a range of possibilities. Like the color itself, the significance fades quite imperceptibly on one side through gray toward black, and on another through pink toward red, and so on, all the way around, toward every other color; and even toward some things that aren’t colors at all. What the logician who says ‘To deny the Taj Mahal is white is to affirm that it is not white’ is really saying is: 7/ I put a boundary around part of the range of significance space whose center we all agree to call white, and // we then proceed to call everything within this artificial boundary “white” and everything outside this boundary “not white” (in the sense of “nonwhite”—now notice we’ve already introduced a distortion of what we said was really there), then any point in the total range of significance space must either be inside or outside this boundary—already a risky idea; because if this boundary is anything in the real universe, from a stone wall to a single wave pulse, there has to be something underneath it, so to speak, that’s neither on one side nor the other. And it’s also risky because if the Taj Mahal happens to be made of white tiles held to brown granite by tan grotte, there is nothing to prevent you from affirming that the Taj Mahal is white and the Taj Mahal is brown and the Taj Mahal is tan, and claiming both tan and brown to lie in the area of significance space we’ve marked as ‘nonwhite’—”
“Wait a second: Part of the Taj Mahal is white, and part of the Taj Mahal is brown, and part of the Taj Mahal is—”
“The solution’s even simpler than that. You see, just like ‘white,’ the words ‘Taj Mahal’ have a range of significance that extends, on one side, at least as far as the gates around the grounds, so that once you enter them you can say, truthfully, ‘I am at the Taj Mahal,’ and extend, on another side, at least down to the individual tiles on the wall, and even further to the grotte between them, so that, as you go through the Taj’s door and touch only your fingernail to the strip of no-colored plaster between two tiles, you can say, equally truthfully, T have touched the Taj Mahal.’ But notice also that the grounds of the Taj Mahal have faded
(until they are one with) the area of significance of the ‘surface of India,’ much of which is not the grounds of the Taj at all. And the grotte between the tiles has faded into (until it is one with) Vriamin Grotte—grotte mined from the Vriamin Clay-pits thirty miles to the south, some of which went into the Taj and some of which went into other buildings entirely. Language is parametal, not perimetal. Areas of significance space intermesh and fade into one another like color-clouds in a three-dimensional spectrum. They don’t fit together like hard-edged bricks in a box. What makes ‘logical’ bounding so risky is that the assertion of the formal logician that a boundary can be placed around an area of significance space gives you, in such a cloudy situation, no way to say where to set the boundary, how to set it, or if, once set, it will turn out in the least useful. Nor does it allow any way for two people to be sure they have set their boundaries around the same area. Treating soft-edged interpenetrating clouds as though they were hard-edged bricks does not offer much help if you want to build a real discussion of how to build a real house. Ordinary, informal, nonrigorous language overcomes all these problems, however, with a bravura, panache and elegance that leave the formal logician panting and applauding.” Bron rocked on his desk once. “Visualize an area of significance space—which is hard to do at the best of times, because the simplest model we’ve come up with has to represent it in seven coordinates (one for each sense organ): and the one we use currently employs twenty-one, some of which are fractional—which isn’t any harder than working with fractional exponents, really—and several of which are polar, because the resultant, nondefinable lines between bipolar coordinates nicely model some significance discontinuities we haven’t yet been able to bridge coherently: things like the slippage between the denotative and the connotative, or the metonymic and the metaphoric. They take a smidge of catastrophe theory to get through—incidentally, did you know catastrophe theory was invented back in the twentieth century by the same twentieth-century topologist, Rene Thorn, the neo-Thomists are always going on about?—but not as much as you might think ...”
To outline the parametal model of language, he used the fanciful analogy of “meanings” like colored clouds filling up significance space, and words as homing balloon
s which, when strung together in a sentence, were tugged to various specific areas in their meaning clouds by the resultant syntax vectors but, when released, would drift back more or less to where, in their cloudy ranges, they’d started out. He wasn’t sure whether he’d gotten the analogy from something he’d browsed over in the office library—perhaps one of the feyer passages in the second volume of Slade’s Summa?—or if it were something from back in his training course. Possibly it had been both.
“I think I’m actually with you,” Miriamne said at one point. “Only, a little while ago, you mentioned things-in-the-Universe. Okay: where in real space is this significance space? And what in real space are these seven-to-twenty-one dimensional fuzzy-edged meaning clouds, or ranges, made of?”
Bron smiled. “You’ve got to remember that all these visualizations, even n-dimensional ones, are in themselves just abstract models to—well, explain how what-there-is manages to accomplish what-it-does. What there is, in this case, is the highly complex organic matrix of the micro-circuitry of the human brain in interface with a lot of wave fronts distorted by objects and energies scattered about the cosmos. And what it does, in this case, is to help the brain to learn languages, produce arguments in those languages, and analyse those arguments in formal logical terms as well as metalogical ones. If you’ll allow me to run some syntax vectors among some balloons that will shunt them all off over wildly metaphoric slippages: The space is in the brain circuitry, and the clouds are composed of the same thing the words are once they pass through the eardrum, the same thing the image of this coffee bulb is composed of once it passes the retina, the same thing the taste of a bar of Protyyn is composed of once it passes the taste buds, the same thing as the vectors that bind the balloons together, or the homing forces that anchor them within their clouds more or less where learning first set them: a series of routed, electrochemical wave fronts.” Miriamne smiled. “And I still think I’m with you/’”Good. Then throw out all the visualizations you’ve been forming up till now, first because there just isn’t any way to visualize a directed wave-front mapping of seven-to-twenty-one dimensional space full of spectrally related meaning ranges that is less than—to put it mildly—oversimplified. And because, second, we’re going to start all over with ‘To affirm P is to deny not-P’ and go running up a completely new set of stairs. Are you ready?”
“Off you go,” Miriamne said. “Right behind you.”
“All right: the two goals of metalogics are, one) the delimitation of the problem and, two) an exploration of the interpenetration among the problem elements in significance space. In old Boolean terms (Did you see the public-channel coverage about that stuff a few weeks back?), you might call it a rigorous mapping of the Universe of Discourse. Suppose we’re constructing some argument or discussion about Farmer Jones’s ice-fields, and we know the resolution will be in terms of some area on Farmer Jones’s land; and most of the problem elements will be things already on the land or things that might be brought to the land. If we decide to call all of his land south of the Old Crevasse ‘P,’ then, depending on how and what things on the land affect the problem—that is, interpenetrate with it—, there may be no reason not to call all the land north of the Old Crevasse ‘not-P.’ Or, we might want to call all the land to the north and the things that are on all the land north and south, ‘not-P.’ Or, with a different type of problem, we may want to call all the land to the north and all the things on all the land north and south and all the things that might be brought to the land ‘not-P.’ Or, indeed, we might, depending on the problem, make some other division. Now remember, in formal logic, ‘not-P’ had to be taken in terms of ‘non-P,’ which (if P is Farmer Jones’s south acres) includes not only the north acres but also the problem of the squared circle, the inner ring of Saturn, and grief—not to mention the Taj Mahal. But given what we know of the problem, it would be a little silly to expect any of these things to come into a real solution. Dismissing them from our consideration is a metalogical delimitation, resulting from an examination of the significance space around various syntax vectors connecting various words of the problem. This means our delimited area for P and not-P can be called metalogically, if not logically, valid. In other words: To deny meaningfully that the Taj Mahal is white, while it is certainly not to affirm, it is most certainly to suggest, that the Taj Mahal is some color, or a combination of colors; and it suggests it a lot more strongly than it suggests that the Taj Mahal is Brian Sanders, freedom, death, large, small, pi, a repeating decimal, or Halley’s comet. Such suggestions hold significance space together and keep it in order. Such suggestions are what solve real problems. Getting technical again for a moment and returning to our n-dimensional nightmare ...” From here he skirted into the various topological representations of metalogical interpenetrations of ‘P’ and ‘not-P,’ in whatever n-space volume the two were represented: “... not-P can cut off a small piece of P, or it can be a shape that pierces P like a finger through a ball of dough, sticking out both sides. It can be a shape that cuts through P and cleaves it in two—actually three, considering the result as two sides and a slice out of the middle. We have a very useful P/not-P relation where we say that, for whatever the space, not-P is completely contained by P, is tangent to it at an infinite number of points, and cleaves it into an infinite number of pieces—that’s such a common one we have a special name for it: we say that not-P shatters P. That’s the metalogical relation the hen-and-a-half wrongly suggests you use to get a quick answer of ‘one.’” Bron took a large breath and found his own eyes wandering around the cubicle, pausing at the bulletin board, the wall and desk consoles, filing drawers, shelves, and readers. “And what we do here, in this department, is to take the programs for some very complicated problems, their verbal synopses and the specifications for answers—often the problems themselves contain millions of elements and millions of operations—and do a quick survey from which we try to map which one-space, two-space, or seventeen-space the problem/answer belongs to; and then suggest a proper topological interpenetration for the constituent P’s, Q’s, R’s and S’s that make up the thing, thus yielding a custom-tailored metalogic, that, when it goes into the computer, reduces the whole thing to manageable size and shape; however we do not send our results direct to the computer room, but shuttle them off to another department called, simply 70-E, which completely reworks our results into still another form and sends them on again to yet another department known as Howie’s Studio (though I believe Howie has not been with us for over seven years) where still further arcane and mysterious things are done to them that need not concern us here. Getting back to what does ...” and as his monologue slowly became a discussion once more, he discovered that many of the more technical aspects (“... if we fail to generate a coherent problem-mapping onto a space of n coordinates, a cross-indexing of the mapping onto 3-space through a set of crossproduct matrices represented by #1, #2, *3... #n can often suggest whether an approximation of coherence can be obtained on a space of if + 1, n + 2, n + 3 ... n + r coordinates. Which is very nice, because all you have to do, for a given map, is take the volume which only leaves you to figure out certain metalogical aspects of its translation which we can model as regressive acceleration with respect to the specific products of the noncommutative matrix i, j, and k ...”) she seemed familiar with from other applications. The easy analogies, ultimately full of holes (the incoheren-cies the technical tried to fill), she poked through in the proper places immediately. He was beginning to suspect that, in anticipation of the job, she had pulled a Sam. Now and then she listened extremely attentively when he found himself veering toward a muzzy eloquence. In the midst of one such veering, the thought struck: Somewhere in real space was the real Taj Mahal. He had never seen it: he had never been to Earth. It and rain and unshielded daylight ... what with the current political situation, he probably never would. Then, rising to replace the whole discomforting notion (though why it discomforted he did not know) was: If I’m real
ly out to start an affair with this woman, perhaps I’ve been going on a bit ... ? His eyes came back to Miriamne’s. He waited for her to say, to whatever last thing he’d said, that she understood, or that she didn’t understand, or that the view from the top of these steps was a little heady (he always found it so), or ask a question about some part of it, or finally admit that her attention had wandered and that she’d missed some; and would he please repeat.
What she did say, after she had crumpled her coffee bulb in her fist, looked around for a disposal, not found one, so finally tossed it into a corner with a lot of other crumpled bulbs he himself had inelegantly dropped there over the past month, was: “You know, I think you met a friend of mine yesterday over in the u-1. She runs a theater commune ... the Spike?”
What happened next was that his heart began to pound. (The Taj crumbled in a welter of granite, grotte, tile ...) He kept his smile in place, and managed to say, hoarsely, “Oh, you mean you know ... ? Now that’s a coincidence!” The pounding rose so high it hurt his ears—”... the Spike?”—then ebbed.
Over the next six hours, by some process logical, metalogical, or random, he learned that Miriamne lived at the u-1 co-op (Three Fires) which had offered the Spike’s company the empty set of rooms on the basement level; that Miriamne had struck up a friendship with the Spike about a week ago; that the Spike had mentioned to her, last night, that they’d done a performance for someone who probably worked in the big computer hegemony off the Plaza of Light—no, the Spike didn’t know his name, but he was into metalogics and wore one metal eyebrow. During all this, Bron took erasable writing slates out of his drawer, erased some, put them in other drawers, realized he had put them in the wrong drawers, kept smihng, briefed her on the Day Star project (with an explanation that, by the time he was halfway through it he realized, she couldn’t possibly follow because it was simply incoherent, finished it anyway, and discovered she’d followed a good deal more of it than he’d thought), learned that when she’d been hired, she’d been told pretty certainly that she would not end up in her own field but, with things in the economic state they were in, you had to make do with what you could get. When they’d told her they’d try her in Metalogics, why, she’d wondered if she’d run into the tall blonde with the gold eyebrow. Yes, she had been surprised when she realized that he was the person sitting behind the desk, whom she had been assigned to as an assistant. Yes, Tethys was a small city. In the middle of all this, lunch-time came and he told her where the cafeteria was in the building, sent her off up there, having decided to eat something wrapped in plastic by himself in the office. Five minutes after she left, he remembered he was trying to start an affair with the woman. Sending her to lunch alone wasn’t very smart if that was his goal, so he hurried up after her.
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