"You could look at ...how good they were at making those guesses? How often when they guessed they were right, how often they missed a pattern..."

"Measuring how good people are at guessing final conclusions in reality - whether, when they say 'I assign 90% probability this triangle is large', the triangle is actually large 9 times out of 10 - sure is a metric of how much Law people contain and are using correctly!  But there's more than one kind of Law you need to build an agent, and the piece of Law we're trying to isolate is the one that's about using necessary truths correctly.  One way of looking at that part is that it's about which conclusions follow from which premises.  To demonstrate -"

Keltham has seen one or two fragments of algebra in his reading, enough that he has some idea of what Chelish algebra conventions look like.  Though it's a bit weird that they teach algebra without, like, teaching people what algebra means.  Hopefully it's not a piece of knowledge that's infohazardous here but not in dath ilan.

He sketches a series of equations:

Otolmens is now in EMERGENCY PANIC OVERDRIVE, which you would be able to distinguish from her usual state of being if you looked carefully.  This particular proof of an inconsistency in first-order arithmetic is safely flawed, but if the foreign mortal is plotting to produce a valid proof of inconsistency - why won't they move the mortal somewhere prophecy still works?

She can't trust Abadar anymore, fellow Lawful Neutral god or not.  Abadar might not be useful in this emergency even if she could trust Him; He's scarcely better at decoding mortal minds than Herself. 

Otolmens sends a message reading simply HELP, tagged with a location.

"Now I'm not so much asking 'What is the flaw in this proof?'," Keltham is saying, now that he's given the classroom the few required seconds to look over his derivations, "as asking, 'How would you go about finding the flaw, if you couldn't spot it at a glance or on your first try at looking?'"

Irori has never once received an emergency summons from Otolmens that was actually important.

He nonetheless maintains a habit of responding with alacrity, just in case.  The concept of 'anthropic selection' is not lost on him, and zero urgent summonses from Otolmens is not quite as reassuring as a mortal might think.

Yes?

You USED to be a MORTAL.  I request you to read this mortal's mind and inform me whether it is plotting to write down a series of VALID proof steps proving an inconsistency in first-order arithmetic.

Otolmens isn't sure, for obvious reasons of resulting inconsistency, but She suspects that She internally uses ordinal induction up to epsilon-zero.  They'd have to boot up Metatolmens to fix Her!

Ex-mortal or not, from where Irori truly stands far above Golarion and other places, it isn't easy for Him to look inside the mind of a mortal not pledged to Himself and praying.  Otolmens only needs to pay attention to relatively few things going on, inside the multiverse, and then She is a relatively materially-focused entity on top of that, designed to be able to check all the electrons in a room to make sure none of them have the wrong mass.  Irori, if He hasn't formed an avatar and sent it into the room, cannot read the writing on the whiteboard the way Otolmens can; He can barely tell that these souls are in a library surrounded by books.  He definitely can't hear the sounds, the pressure patterns transmitted through the air as vibrations.

Still, it is Otolmens who calls, and the mortal is more Lawful Neutral than usual even for those that register Lawful Neutral.

From the mortal's general spiritual posture, Irori can already guess what He'll see.  But just in case, Irori expends the energy to take a very brief look at the surface of the mortal's mind.  It's not as difficult as it would be at other times of this mortal's life, given his current endeavors.

...he's not planning to destroy mathematics.  He only intends to teach of his Way to others.

Irori shifts most of His delegated attention back to other aspects of His businesses, leaving only a tiny fragment to look at the Chelish place a bit longer.

...Irori shifts somewhat more of His attention back to that location.

Carissa feels that she could grasp what Keltham is pointing at a lot faster if she were reading his mind but that's disallowed, now, he's a fourth-circle caster and reasonably likely to notice. She can't even ask him whether it'd be all right if she read his mind because they haven't acknowledged mindreading to be a thing that magic can do.

It remains bizarre, to think that Law has anything to do with formal mathematical logic. You don't need to understand the gods to be Lawful, you just need to obey them. But - but Keltham's world is more Lawful than hers, and -

- so there's nothing heretical about the claim that humans are using a mediocre approximation of Law, which is a god-concept that doesn't mean quite what humans understand it to mean. And there's nothing heretical about the idea that humans ought to use the real thing, except that they're too stupid and limited to understand it, so they have to settle for their wrong approximations. And there's...nothing very heretical about the claim that, actually, there's a way to teach humans the real thing, despite their stupidity and limitations -- at least, to teach smart humans, to teach humans in Keltham's world with a median INT of 16 or 17, and the people in this room have a median INT of 16 or 17, so the people in this room can learn it. 

And the true structure of Law would be mathematical, because it's about - regularities, consistencies, treaties among the gods aren't promises so much as fundamental changes, becoming the kind of structure of which the promise is true, and there is, actually, an obvious parallel to math there, even if she can't properly articulate it. The way the gods are is inevitable; in many ways they vary much less than humans, because there is only one way to be right and many many ways to be wrong. 

And the gods wouldn't be very suited to figure out what math, specifically, to teach to humans, especially if it requires obnoxiously counterintuitive tactics like making everyone limp their way through the lesson guessing - and perhaps, too, this wouldn't even have been worth trying anywhere in the world until quite recently, you need a bunch of smart people in a room and Cheliax is the first society in recorded history to look for all their smart children and teach them math -

"Well, you'd know there has to be an error somewhere, since you got it wrong."

"You could - check each line and see where the error showed up first -"

"Check each line to see where the error showed up first?  How would you check a line for error?"

" - well, there's obviously a problem in the eighth line, where if you substitute in '1' for X and Y you've got the error already. And there's...not a problem in the seventh line, because that one comes out to 2*0 = 1*0. Which is true."

Keltham takes a quick look at the nametag of whoever that was.  Why the Chelians collectively aced this problem but not the predicate-logic one... presumably it's just down to more actual practice with algebra?

"Precisely.  If we substitute in 1 for x and y, and evaluate the left-hand sides and right-hand sides of each equation, we get the following assertions:"

"The tactics of algebra - like being allowed to add 3 to both sides of an equation - are meant to preserve truth, not create it from scratch.  If an equation starts out true, a tactic in algebra should not produce a false equation from that true equation."

"This way of thinking holds even if the elements of the equation refer to things in the outside world.  Let x be the number of people sitting in the brown chair, 2 as it happens, and let y be the number of people sitting in the red chair, currently 3.  It is then an unnecessary truth, not a necessary truth, that x + 1 = y, as I have defined those terms to refer to the outside world.  In our world, x + 1 = y evaluates to 3=3, which happens to be true; but if you cast an illusion showing two people sitting in the brown chair and two people sitting in the red chair, the equation in that world would evaluate to 3 = 2, which is false.  And if I said x + 10 = y, that would be an unnecessary falsehood; in our world it evaluates to the false statement 12 = 3."

"Now apply the rules of algebra, add 2 to both sides, and transform the first equation x + 1 = y to the new equation x + 3 = y + 2.  In our world, this evaluates to 5 = 5, which is again true.  If we apply the same tactic to x + 10 = y, it yields x + 12 = y + 2, which evaluates to 14=5, again false."

"We term a step of inference valid when it is truth-preserving; when it transforms true statements into only other true statements.  It doesn't have to preserve falsehood; multiplying both sides of an equation by zero will produce truth even where it didn't previously exist."

"What makes the tactic of adding 2 to both sides of an equation, allowed in math, is not that some Watcher or representative from Governance told you it was allowed."  This part got hammered into Keltham and his agemates a lot as a kid, so it was probably determined to be important in practice to emphasize??  "What makes it an allowed step is that, if you have two weights balanced on either side of a scales, and you add two identical rocks to both the left side and the right side, the scales will still balance after that."

"If you look back at the original flawed proof that 2=1, it goes from a true statement in step [7], to a false statement in step [8].  Then between [7] and [8] we must have applied some operation of inference which is not 'valid', which has the ability to take in a true statement and spit out a false statement.  This tactic was canceling the multiplication by (x - y) from both sides, which is to say, dividing both sides by x - y.  Dividing both sides of an equation by 2 is valid; if you have a scales in balance, and remove half the weight from each sides of a scale, it will still be in balance.  Here, we see that division by 0 is not valid, because it can produce falsehood from truth.  What makes division by 0 unlawful is not that your Watcher told you not to do it while doing algebra; it is that division by 0 is not generally truth-preserving.  We can find some equations that will still be true after dividing both sides by a term equal to 0, but it is not a safe step in general."

"Sorry if that part about Watchers seems overly obvious, by the way.  It's just that apparently human brains by default try to reuse the part of ourselves that learns from adults not to steal cookies outside of mealtimes or we'll get slapped on the wrist, in order to relate to the rules for manipulating necessary truths that existed outside the start of Time.  And these are actually quite different topics; like, rules change sometimes, when Legislators vote on them, but algebra doesn't.  So you want to be explicitly aware of the difference, and not go bugging adults to let you divide by zero just this once."

"So the argument is that part of Law is - the habits of mind so you only reason in truth-preserving ways?" Meritxell, who was also fastest on the algebra, says. 

"I am still not entirely sure what the word 'Lawful' means to y'all.  Multiple different words in my native language all come out as 'Lawful' in Taldane and I'm mostly running with those.  Cheliax is supposedly a 'Lawful' country, but the books are written with what look to me like appalling jumps of reasoning, and somebody seems to have taught y'all algebra without teaching you what math is or why it works.  But Lrilatha-whose-job-title-I-already-forgot is supposed to be more innately Lawful, and she did not talk with those appalling jumps in her reasoning.  Which suggests to me that the word 'Lawful' is translating to me mostly correctly, or that the concept I hear is at least a real part of what 'Lawfulness' is; and the humans here simply are not being taught about that part of Lawfulness, or how to flow along with it on purpose instead of by accident."

"That said, not being taught something is not the same as having none of it inside you.  Your eyes can see without you being taught how the - part of the mind that handles vision - is doing the work it does.  And if you could never see the implications of other guesses you'd already made, you wouldn't get far enough in life to reproduce.  Everyone here has bits and pieces of them that imperfectly echo the shard of Law about which conclusions follow from which premises.  I also happen to have studied that Law explicitly and went through standard training for not being quite as messy about it.  That's part of the process that dath ilan went through to put together aeroplanes that could fly across oceans.  We aren't perfect at it, to be clear, just better than whoever wrote the so-called books in this library.  I really want to see what happens if we match up Lrilatha against a Keeper - one of the people from my world who are actually specialized in being more perfect reflections of Law - but I doubt we'll ever get a chance to try."

"....you think that in a Lawful country all the books should only use truth-preserving arguments?" someone says, somewhat dumbfounded.

It makes sense, though. Mortals didn't have free will. Now they do, and it displeases Asmodeus, but no one has a complete account of what free will is, because they're not gods, and don't understand what exactly displeases Asmodeus. But that might just be it. Gods, innately, reason in truth-preserving ways. Of course they would. Lying to yourself for self-preservation is a thing you only have to do if you have wrong beliefs and can't argue yourself out of them because you don't know the counterarguments, and so you have to stop thinking about them. That is not a problem gods have. Gods just reason correctly. And in Keltham's world - there's still the concept of infohazards, things you're not supposed to learn, presumably because you're only human and can't properly have the kind of mind that entertains that fact in a way that allows for continued useful functioning -

- something about that frame isn't quite right but despite that she feels like everything is coming together.

Minds should reason in truth-preserving ways. Someone, a long time ago, robbed humans of that, and Asmodeus is angry. Carissa is angry! That was her birthright, and she wants it back. And Asmodeus thought, until Keltham arrived, that the scars they'd wrought on human souls could only be corrected in Hell - or at least could most cheaply for Asmodeus be corrected in Hell - but in Keltham's world, where humans do not magically reason in truth-preserving ways, they figured out, possibly over many thousands of years of careful experiments, how to teach it. And Asmodeus saw that and immediately told them not to hurt Keltham, because -

- okay, that line of thought she's going to tuck away for later, it seems maybe ill-advised. Sufficient that Keltham got Asmodeus's endorsement immediately.

Minds should reason in truth-preserving ways. The books ought to have good arguments. Devils are masters of propaganda, but aren't convinced by it. Carissa - doesn't think of herself as convinced by it, the books are really presenting their conclusions not their arguments, but - but that's because the books think humans aren't doing reasoning well enough to be persuaded by argument, and humans can learn that. At least smart ones. And if they knew it, then you could just argue everyone out of all the heresies, their minds wouldn't possess the weaknesses that make that strategy doomed, that make it necessary to present them with conclusions they won't be able to understand. Or at least - less of it. Keltham did have the concept of things he was not meant to learn. 

(More things that suddenly make sense: what the Starstone does to you, why it changes some people more than others. Godhood, even more than devilhood, would preserve you to the extent that you are worth preserving - to the extent that you have learned the processes of reasoning - Irori ascended just by becoming perfect, and everyone writes that off as a strange one-off that only Irori could do but in dath ilan they teach it -)

It has to be done all at once, she realizes. There's a terrible middle ground where you are trying to reason things out, but you are incompetent to do it, and so you run right into all the heresies that you could have been protected from by not trying to reason. You would absolutely fail a loyalty check, in the middle of trying to learn how to think. But at the end of it - Asmodeus arrived at His beliefs through reason. And He hates it, that humans were changed, so they can't, and He wants them changed back.

She rereads everything on the board, though there's not much written on the board. The new thing she's learned here isn't that there are necessary truths and empirical truths, or that you shouldn't divide by zero, it's that it is possible for humans to learn how to reason well enough they're better off trying it.

"If you found yourself in an unfamiliar country and you opened up a book and it was like, 'The sky is green.  How do we know this?  Because teddy bears are cute!  My dad once bought me a cookie!' would you suspect you were in a Chaotic country or a Lawful one?  Now, I admit this example is unrealistic; generalizing from my reading experiences, a Chelish author would never explicitly ask 'How do we know this?'  And yes, I'm sure places outside of Cheliax are even sillier but your book authors are still all very silly and if Lrilatha had infinite free time I would lock all of them in a room with her until they learned better."

"That's kind of what Hell is," someone offers. The other people who were totally thinking that but not sure if they were allowed to say it giggle. 

"The Worldwound isn't in Hell, it's here.  And I don't know why you can't have people train in Lawfulness in the whole post-life thing for a few years, and then resurrect them here, if that's a thing in the first place; or why Lrilatha hasn't been able to train teachers who could train teachers who could train you.  But the Worldwound isn't in Hell, it's here, and it's this world that needs to become saner and wealthier and better at repelling demons, or die." 

Those questions don't...sound like they're meant to answer them? Instead, they nod vigorously. 

- no, actually, she thinks they're meant to answer that. Or she thinks they ought to, regardless of whether they're meant to. "Becoming a devil in Hell takes centuries," she says. "You can't be resurrected after that long. It's been widely assumed there just wasn't any way to make a useful amount of progress on - being Lawful the way devils are - in a human lifetime. Or in time to close the Worldwound. But it seems to me that the reason Asmodeus intervened directly to tell us to make this a priority is that - the way you know is a lot faster."

"Asmodeus would also bet significant resources on that even if he only estimated a small probability of it working, so let's not get overconfident.  But yeah.  I don't know how long dath ilan took to get where we did, starting from scratch and baseline - we had to screen off our history, for reasons that are apparently also infohazardous to know about.  But the pieces all fit together, and you should be able to complete the whole thing once you have enough hints from me.  Even if there's no spell to give me perfect recollection of all the training I went through, I'm hoping it should be possible to get, like, 80% of the benefit from going off my memory of, hopefully, the most critical parts.  Not to mention, you're not all 8 years old and that should count for something when it comes to learning this part a little faster."

Keltham turns back toward the whiteboard, completely unconscious of any effect the declaration about 8-year-olds might've had on the rest of his audience, who are all concealing their reactions anyways.