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 balloons 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 really 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.