Neal Stephenson - Anathem

“Please do,” I said.

“Be my guest,” Lio said.

“All right. The first assertion is: entities that are the subject matter of theorics exist independently of human perceptions, definitions, and constructions. Theors don’t create them; theors merely discover them. And the second premise is that the human mind is capable of perceiving such entities; which is exactly what theors are doing, when they discover them.”

“We’re with you so far,” I said.

“Very well,” Criscan said, “now, if you want to proceed beyond merely rattling off those two premises, you need to supply an account of how it is that the human mind is capable of obtaining knowledge about theorical entities, which, according to the first premise, are non-spatiotemporal and do not stand in a normal causal relationship to the entities that make up the cosmos as we know it. And various arguments have been put forward over the millennia as metatheoricians have tried to supply that account. For example, Halikaarn took a lot of heat from the Procians because he thought that our brains contained an organ that was responsible for this.”

“An organ? Like a gland, or something?” Lio asked.

“Some interpreted it that way, which helps explain why he took so much heat for it. But this was probably a translation error. Halikaarn was pre-Reconstitution, of course, so he was not writing in Orth but in one of the minor languages of his day. The person who translated his works into Fluccish did him a disservice by choosing the wrong word. Halikaarn wasn’t thinking of something like a gland. He was thinking of a faculty, an inherent ability of the brain, not localized in any one specific lump of tissue.”

“That’s a little easier to take seriously,” I said. “Fine.” Because I had the sense that Criscan was getting ready to veer off into a long tedious defense of Halikaarn. “So how does this faculty figure into his account of what’s happening in this diagram?”

“There is some other type of given, other than what we can detect with our eyes, ears, and so on, that somehow reaches the Arbran Causal Domain and that is perceived by Halikaarn’s Organ,” Criscan said.

“That almost raises more questions than it answers,” Lio pointed out.

“It doesn’t answer any questions at all,” Criscan returned, “this is not really an attempt to answer questions but a way of setting one’s pieces out on the board, agreeing on terminology, and so on. So. The theorical entities in the HTW—triangles, theorems, and other pure concepts—are called cnoöns.”

“Cnoöns, check!” Lio said.

“Between us and the HTW is a relationship, the details of which are subject to further debate, which Halikaarn didn’t name, but it’s symbolized by this arrow, and so people have ended up calling it Halikaarn’s Arrow.”

“Halikaarn’s Arrow, check!”

“A Halikaarn’s Arrow is a one-way conduit for givens about the cnoöns. These givens enter the Arbran Causal Domain through a poorly understood process called the Hylaean Flow and there impinge on Halikaarn’s Organ, which is how we become aware of them.”

“Hylaean Flow, check!”

Criscan had decided that he didn’t like Lio very much, but was making a visible effort to tolerate him. I stepped into the position of interlocutor, shouldering Lio aside. Lio reacted melodramatically, sprawling off to the shoulder of the road as if he had been struck by a speeding fetch. I ignored him. “So,” I said to Criscan, “now that we have the terminology bolted down, where are we going with it?”

“Now we’re going to skip ahead a millennium and a half,” Criscan said, “and talk about the move that Erasmas and Uthentine made, when they decided to see what happened if they construed this diagram as just one, particularly simple example of a Directed Acyclic Graph or DAG. Here ‘directed’ just means ‘arrows are unidirectional.’ The modifier ‘acyclic’ means that the arrows can’t go around in a circle, i.e., if we have an arrow from A to B, we can’t also have an arrow from B to A.”

“Why bother stipulating that, I wonder?”

“The property of being acyclic is required in order to preserve the fundamental doctrine of Protism: that the cnoöns are changeless. If it were possible for the arrows to go around in a circle, it would mean that events in our universe could alter things in the Hylaean Theoric World.”

“Of course,” I said, “pardon me, that’s obvious now that you mention it.”

“This diagram,” said Criscan, drawing my attention back to his two-box sketch, “just seems wrong, to a metatheorician.”

“What do you mean, just seems wrong? How can you get away with statements like that?”

“It is a legitimate move in metatheorics. You have to be continually asking yourself, ‘why are things thus, and not some other way?’ And if you apply that test to this diagram, you immediately run into a problem: there are exactly two worlds. Not one, not many, but two. One might draw such a diagram having only one world—the Arbran Causal Domain—and zero arrows. That would draw very few objections from metatheoricians (at least, those who are not Protists). One might, on the other hand, assert ‘there are lots of worlds’ and then set out to make a case for why that is plausible. But to say ‘there are two worlds—and only two!’ seems no more supportable than to say ‘there are exactly 173 worlds, and all those people who claim that there are only 172 of them are lunatics.’”

“Okay, if you put it that way, I agree that there is a certain odor of crankiness about it. Like when Deolaters claim that there are thirty-seven books making up their scripture but that anyone who proposes a different number must die.”

“Yes, and this accounts, at least in part, for the way Protism raises hackles in some quarters. So the Erasmas/Uthentine move is simply to say ‘what’s true of one DAG ought to be true of another’ and to consider other DAGs having other numbers of worlds.”

Criscan took up his stick again, and scratched out a diagram like this one:

“They called this one the Freight Train,” Criscan announced. “In the Freight Train topology, there is a (possibly infinite) plurality of Hylaean Theoric Worlds that stand in a hierarchical relationship, each ‘more Protan’ than the last and ‘less Protan’ than the next. This introduces the notion of Analog Protism. In Simple Protism, being Protan is a binary, digital property.”

“A world is either Protan, or it isn’t,” I translated.

“Yes. Here, on the other hand, gradations of Protanness are possible.”

“Not just possible,” I pointed out, “they are required.”

“Yes,” Criscan said, a little distractedly, for he was already at work making another diagram.

“This is the Firing Squad,” he said. “In the Firing Squad topology, some number of Hylaean Theoric Worlds are connected by direct linkages to the Arbran Causal Domain. This introduces the notion of separate Protan domains that have nothing to do with one another. In Simple Protism, all possible theoric entities are lumped together in one box labeled ‘Hylaean Theoric World,’ which seems to imply that, within that box, they can stand in cause-and-effect relationships to one another. But perhaps this is not the case, and each mathematical entity should be isolated in a separate World as above.”

He now spent a while drawing a much more complex diagram: