They do not brilliantly pull the induction axiom schema out of their asses or out of his mind, which they are not reading. They do mostly manage to follow along through all the rules of first-order arithmetic, and they seem to be having fun about it.

Once they've got a nearly full set of rules, Keltham remarks that the last puzzle piece for identifying the numbers, as well as they can ever be identified in a certain sense he's not going into right now, is one he really doesn't expect them to get unhinted.

And then he drops on them the infinite axiom schema for induction, trying as best he can to explain why you'd need it to pinpoint the numbers.

After clearing up any misapprehensions about that as best he can, Keltham is ready to move on to his next point.

"We're running through things a lot faster than I went through them as a kid, and I'm probably accidentally leaving out important ideas along the way - all of this would take more like a month, if you were eight years old and doing the exercises, even if you were doing nothing else.  But you may recall that some time earlier, I posed some puzzles about asking for examples of necessary truths, and why they were ever good for anything, and what it means to say that one equals one is a necessary truth - what you ought to expect to see happen as a result - especially given that a necessary truth should still end up being true no matter what happens to you, if it could happen inside any illusion depicted in full detail."

"We now have a language in which I can give some of my own answers there.  But before then, does anybody want to take a renewed shot at saying what it is we're talking about, and what we should expect to see happen, if we say that one equals one is a necessary truth?"

"...that if it's not true we just can't do any reasoning in a formal system at all?"

"Does reality need to care what you can't reason about?  Perhaps you can depict an illusion in full detail in which one does not equal one, and we will need to construct a new logic which does not take as primitive the assertion that every object equals itself, in order to describe that illusion."

" - you can't depict an illusion in full detail in which one does not equal one."

"What about clouds drifting across the sky and sometimes separating?  One cloud equals two clouds, it doesn't equal one cloud!  Divide both sides of the equation by cloud and there you have it."

They stare concernedly at him.

"More to the point, how would you depict one, the successor of zero, inside an illusion?  You can depict one cherry in a bowl of fruit, in an illusion.  How would you depict one, the successor of zero, as it appears in our collections of statements?"

"- you'd just have to put the symbols."

"Then we have merely depicted the symbols talking about one, in our illusion, not depicted one itself.  That's like making an illusion of a piece of paper with 'cherry' written on it, and saying you made an illusion of a cherry."

Some particularly daring girls, in the middle of the discussion of the induction axiom, sent around a crumbled piece of paper, as girls do; it flew between desks, as wizarding girls do; its text was in Infernal, as Chelish wizarding girls do. It read 'is Keltham a sadist? y/n'

The vote leaned yes.

Keltham's audience squirms anxiously at this question.

Keltham does not have the sharpness of unaided vision, let alone interpretation capacity, that would be required to perceive these nigh-imperceptible squirms.  He lives in a mental universe very far away from this reality, a mental universe where uncomfortable or unhappy students will of course speak up and tell you this fact as soon as they realize it themselves.  Though he has noticed his researchers' apparent lack of any visible emotions besides competitive enthusiasm, and is starting to wonder if they used a magical spell that's the equivalent of a mind-affecting drug that made them fixedly enthusiastic.

Well, at least this time he's about to say some words where enthusiasm seems appropriate.  Keltham tries to channel the demeanor of an appropriately specialized Watcher as best he can; the sort of Watcher who tries to make sure that kids get the full impact of things, and aren't cheated out of awesome stuff by older kids mistakenly trying to act like it's no big deal.

"In my language, we'd say that the subject matter of our discussion, when we talk about math, is which conclusions follow from which premises.  When we discuss numbers and say 2 + 3 = 5, it implies that if we observe cherries and come to believe that our number-axioms describe in reality the operations of combining bowls of cherries, we will expect to see in reality that pouring a bowl of two cherries into a bowl of three cherries yields a bowl of five cherries.  If we get six cherries instead, we might think we made a mistake in the math.  Or we might suspect that watching the bowl closely would let us see an extra cherry popping into existence.  In the latter case our beliefs about which conclusions validly follow from the addition-premises would have been right, but our guess that the addition-premises applied to combining bowls of cherries would have been wrong.  Being the fragile creatures that we are, and sometimes making mistakes in our reasoning, when we do math about a bridge and then the bridge falls down, we might be observing that the bridge disobeyed the premises about which we did math; or we might be observing that we made an error about what was a necessary connection, and our conclusion didn't follow even though all the premises were true."

"All of this is to say that we can observe the consequences, the shadows, of necessary truths, when we watch the empirical world; even to the point of our observations leading us to suspect errors in our own reasoning about what was necessary.  But observing the number 1 itself?  At best, maybe, somebody could make an illusion of an object representing zero, not the successor of any other object, connected by a successor relationship to the object that would therefore be one," and Keltham draws some green dots connected by red arrows to other dots.  "This would give us an illusion that would map very directly, in our external interpretation, onto a partial model that fits the number axioms.  But it doesn't make sense to say that the illusion is depicting the number one; there isn't a single thingy like that out there floating in the void, just a set of premises that actually existent things might obey, in which case we'd expect them to behave like the number one."

"The facts about which conclusions follow inevitably from which premises can't be said to be older than the temporal universe, because they're outside of time entirely; it's not that they existed before the universe began, or that they'll last after the universe perishes, but that they are somewhere above or below that.  Temporal and physical processes draw on necessities, mirror them, but cannot change them between one time and another.  There's a certain sense in which, in controlling our own decisions, we are controlling links between premises and conclusions - we are controlling, given the premise of a person like ourselves, the conclusion of which decision we come to - but this doesn't mean we are changing mathematical facts between one time and another.  An alternate plane of existence - or so I would suspect - can obey different premises in its physical behavior; it cannot alter which conclusions follow necessarily from which premises.  Those facts are, not eternal, but outside of time entirely.  This is one of the ideas invoked by a word in my language, which translates into your word 'Lawful': the concept of drawing on and mirroring the level of existence where certain facts may be viewed as absolute and unalterable."

"You have seen now how we can start with two apparently different concepts of validity - one that preserves truths about properties of objects connected by 'and' or 'implies', one that produces true numerical equations from true equations - and, at least for the case of whole numbers, reduced the equational subject matter to the predicate-logic subject matter.  Just like we were able to reduce 'and' to 'or', or 'or' to 'implies'.  I will tell you now a point that you will not be ready to prove yourselves for a while: the system of predicate logic I've introduced to you is one of several systems that are complete in the sense that all mathematics can be translated into them.  The topology you learned as wizards, unless it deals with some phenomenon absolutely foreign to dath ilan whose mere existence refutes this entire philosophy - which I am mostly not expecting, to be clear - is just another kind of math that could be translated into this system, or translated into several other systems I haven't shown you, all of which could also be translated into this system, and into which this system could also translate, moving between them as freely as we rewrite an 'implies' connector as an 'or' connector."

"This is one reason I could pop into another dimension and expect to have a reasonable conversation with Lrilatha, who belongs to a species that doesn't exist in my world.  If different formulations of validity can so freely translate between each other, it would seem more reasonable to hope that I, to the extent I had those concepts right, would use a version mappable to Lrilatha's, who is a Lawful being; so that her arguments would make sense to me, and my arguments make sense to her.  Either a conclusion follows with necessity from its premises or it does not.  Only mistakes about that subject matter could differ between people, between factions, between planes.  The right answer is the same everywhere.  And this is also part of the concept in my language, which the translation spell translates into your word 'Lawful'."

It is probably optimistic to, before she has even properly learned this herself, conclude that Cheliax could be teaching it so much better, but - Cheliax could be teaching this so much better that it suddenly hurts to realize how badly she learned it before. 

There is a right way to be. Devils are it; mortals aren't. Mortals were, for a while, at least controlled by gods, who are, but the control broke, and now mortals just wander around, missing the concepts that make up the right way to be, for the most part not smart enough to learn them. Cheliax emphasizes - that this is disappointing to Asmodeus, that it makes mortals less valuable to Asmodeus, and of course that's the angle from which Asmodeus cares about it, but the angle from which the humans ought to care about it is that they are worse. It is in their interests to be perfected, not because some god will get them when they die and the other gods waste even more of them, but because there is a commonality among all perfect beings, a shared language that the perfect can speak across planes, across time -

How is it that Chelish children learn they'll be perfected and are scared, instead of learning they'll be perfected and are joyful, impatient, full of the longing that filled Carissa when Contessa Lrilatha spoke - why aren't we teaching it like this - but the answer of course must be that Asmodeus couldn't divine the result of thousands of years of experiments across an entire other world, at least not more cheaply than He could grab someone from there - 

Keltham pauses, at least partially to catch his own mental breath.  He should probably call a break sometime soon, but he was working around to a point, some time earlier before the whole digression into Validity and first-order arithmetic, and he feels like it would be only polite to actually finish the digression and work around to what he meant to say, before then.  There's still some distance left to go before he can pop the stack - he wasn't actually expecting, on some wordless level, that it would take this long to teach grown teenagers first-order logic and how to axiomatize arithmetic.

"In saying all this, I'm jumping ahead in a dath ilani education and skipping over a number of exercises required to actually understand everything and a dozen dozen precautions we were given against common mistakes, some of which now seem pretty silly and obvious to me, but which might turn out to be a lot more necessary to otherwise unprepared minds, I don't know."

"For example, if somebody throws a ball and you need to catch it, trying to translate your thoughts into predicate logic about the ball having the property of flying and this implying an eventual fall given gravity, is going to utterly fail to help you.  You would be, first of all, better off thinking in your mind's native wordless language that tries to visualize the ball's fall and run to there, because your native language is faster and the ball will fall before you can think of anything logical.  You would, second of all, be engaging in a particularly naive kind of jumping ahead of your real capabilities, by trying to translate your thoughts about the ball directly into a logic with a falling predicate.  What you would actually need to predict the ball's fall using serious math is calculus, and not simple calculus either; it would include terms like how the resistance of the air to the ball's flight changed with the ball's speed.  If you try to summarize all that by saying that the 'falling' predicate on the ball was true, when that 'falling' statement would be equally true if gravity was pulling on the ball a different amount or if the air was more resistant, you're throwing away details that actually matter in order to try to squeeze down the issue into predicate logic - an amount of predicate logic that you find easy to handle, which is too little to actually solve the problem.  You also wouldn't be absolutely certain about the ball's position or trajectory, and managing uncertainty inside of mathematics is a whole separate topic I haven't broached to you."

"Actually solving the ball's flight using a full written-on-paper account of how conclusions about what's probably true, follow necessarily from some unnecessary premises you already believe about what's probably true, would require setting up a complicated problem in calculus and probability that would describe how to infer the ball's trajectory from what your eyes had seen of the ball.  And if you wanted that said in pure logic with all premises spelled out, you'd have to axiomatize that calculus problem into the more universal language of logic.  It is a whole lot easier to just run and catch the ball by thinking about it in your native brain's native style."

"I mean, maybe if you were a god you could solve the whole problem using explicitly valid reasoning - or not, I'm not sure how mentally powerful your gods are, exactly.  And if you were a god and you could do that, then you could get tossed into another plane of existence where gravity works differently, and rapidly recalculate how to catch flying balls from scratch and do that successfully on your first try, instead of having to relearn the rules your brain uses through a lot of trying and failing.  But I am not a god, yet, and you are not gods, also possibly yet, and I doubt that any intelligence headbands we can afford in the foreseeable future will let us do it either."

"So don't get ahead of yourselves just because logic is absolute, timeless, and universal.  If you can't think fast enough to solve a problem using explicit logic, or if it would require humanly unmanageable huge problem statements even in principle, or if you simply don't know how, then all that absolute and timeless stuff won't help you.  The fact there exists a better method that gods or super-gods could use to solve a problem doesn't mean that you can solve it that way, or that you can get closer to the super-god's solution by using bad, clownlike, and tiny imitations of the outer forms of the ideal methods they'd use."  Gods seem like a surprisingly useful metaphor for ideal cognitive processes, now that he's in a world that has gods; Keltham is a little surprised it wasn't a more common explanatory metaphor used in dath ilan.

"I don't know if any of you actually needed me to say that, to be clear.  But it's the sort of thing they tell you when you're 8 years old, and you get into your head the idea that if logic is so great, you should be able to use it to crush your opponents at sportsball by making only sharply logical muscle movements.  I mean, actually they just let me make the mistake and lose the sportsball game, but then afterwards, that was what they told me when I asked what I'd been doing wrong and how to do it correctly."

His audience nods seriously. 

"With that warning aside - I'll try to give you more of the dire warnings I got, as they come to mind and I remember what they were - there's a final thought that runs deep in the dath ilani conception of Law, which is why, when I heard that 'Lawful' was a godly concept rather than a human concept, I immediately thought, 'Heh, I bet I know what that's talking about, then,' and not 'Oh, it's probably something humans can't understand at all.'"

"If you start with a logical language that already has 'implies', you can add on the new connector, 'or', and then though you've made the statements a little easier for humans to understand, you haven't made the language any more expressive - your new innovation 'or' turns out to be reducible to the same old 'implies' combined with 'not'.  After trying out a number of innovations of this type, you might repeatedly find that you were unable to extend the real power of your language, and so venture a guess - a guess based on mere past experience, like seeing that every triangle tested was a red triangle - that you had reached some natural limit of logic's power."

"But when I asked my Watchers as a child, they did not tell me, 'We're guessing this logical system is as good as it gets.'  They told me, 'This logic you are learning is the most powerful form of logic that can exist.'  The Watchers where I come from are trained by Keepers and entrusted with the teaching of children; they are not there to set a poor example by just making stuff up, nor by taking great blatant invalid leaps, nor by saying with certainty what they have no right to be certain of.  It's not something that gods told us, either; there's no gods where I come from, remember.  So how did my Watchers know - what could they possibly have known - how could they possibly have obtained a piece of knowledge like that?  I do not expect you to guess this correctly, I couldn't have done that without being told; but I'd like somebody to say out loud a wrong answer; for it's easier to fill yourself with knowledge after you've explicitly noticed yourself not being already full."

"An...equivalence proof of some kind?" someone says after a moment. "That any kind of logic that does anything useful is the same as that one, specifies the same things?"

It continues to be disorienting to Keltham each time his audience of empirical-topologists throws around guesses built out of much more mathematically sophisticated language than you would associate with a dath ilani kid too young to know how 'p -> q' was defined.

"That's about the most plausible wrong guess I can imagine, so congratulations on that.  But no.  They didn't tell me right away why they knew, that time, the Watchers left it mysterious so I'd have some motivation to learn stuff myself.  And I apologize if I'm correspondingly wrecking your own education's most optimal ideal form under ideal circumstances, but we have a planet to industrialize, so I'm going to plow ahead anyways and just tell you, maybe someday your kids will learn it the right way.  What my Watchers secretly knew was a completeness or idealness proof, built from more powerful and sophisticated methods I wouldn't be ready to use myself until late in age 12.  They defined the most you could possibly reasonably ask for out of a logic, then proved they already had that."

"Consider again our worlds of blue circles, containing red triangles and green squares, and objects related by successorship and multiplication and all the rest.  We have said that the subject matter of logic is necessary connections from premises to conclusions.  Then the perfect or ideal logic would be one which, given some collection of premises, could derive through its permitted steps of inference every possible conclusion which actually followed from those premises."

"Well, with some fairly high-powered techniques, you can prove that first-order logic does in fact have this property - which means that if you created a new logic which is a single sentence more productive, in the sense that it says even one more thing follows from a premise set, which the logic I showed you cannot derive through its allowed steps, that new logic is making non-truth-preserving leaps; there will be some model, some world, where all the premises are true, but that extra derived conclusion is false."

"The key to that proof, incidentally - I sort of feel like I ought to say this, both to give you some hope that such a proof actually exists, and to make reconstructing all this reasoning easier, if it turns out that the food here is poisonous to me after all and it gets too expensive to keep resurrecting me - is a compactness proof.  Oh, nice, you have a word for that, so I'm guessing you used a similar concept in topology?  The compactness proof shows that if an infinite set of logical statements has no semantic model - if there is no depictable world or illusion in which all the premises are true - then some finite subset of those statements has no model.  We further prove that if a finite collection of statements has no semantic model, we can syntactically prove a contradiction from those statements in a finite number of steps.  Then if Q follows from a collection of premises in every possible model of those premises, we can adjoin ~Q as an additional premise to the collection, yielding a collection of premises which has no models; and obtain a contradiction in finitely many syntactic steps; and from this by double negation we can syntactically prove Q in finitely many steps.  So whenever Q follows from a collection of premises, we can prove it from those premises syntactically."

"That's the final reason I expected Lrilatha and myself to reason in ways that were not quite so different, even though she wasn't human and possibly hung out with gods.  Assuming the whole dath ilani philosophy was true across all planes - though I wasn't quite certain of that, and I'm still not - it wouldn't be surprising if Lrilatha could see some conclusions following from premises faster than I could.  But it would be surprising - considering the proof that logic is literally as good as it possibly gets and gives us everything we could possibly want - if Lrilatha could make premise-conclusion leaps of a qualitatively different kind that I could not follow even in principle, using new rules of deduction and permissible derivation that no dath ilani had ever encountered."

"That said, if you introduce the ability to directly quantify over functions or predicates, the proof I described no longer works, but most philosophers of mathematics in dath ilan claim that this can't really be improving the power of the logic, because anything you can actually derive in the syntax of a 'second-order' logic that quantifies over functions, can also be derived inside some corresponding 'first-order' system that doesn't, like this one doesn't.  I mention it because I'm now in some totally other plane and ought properly to be less sure of some things than I was yesterday, and if Asmodeus does show up using genuinely valid reasoning I can't follow even in principle, there'd be an obvious immediate guess that he was taking advantage of physical principles that don't exist in my universe but let him directly access the semantics of quantifications over predicates.  We were pretty sure that was physically impossible inside our own universe, but this plane might or might not be another story.  But, again, I am mostly not expecting that to be the case, and if Lrilatha could do that, she politely didn't do it around me."

"That's the final piece of the concept that 'Lawful' translates into my language - the ability of human beings, even if it's only a little, even if they have to struggle and work hard at it and often it's just faster to run and catch the ball instead of overthinking it - to sometimes know and make a more deliberate use of Laws that are timeless, universal, and even, sometimes, knowably optimal."

"And that's why I heard that Lawful was a god-concept and thought to myself, 'Heh, I bet I know where that's pointing to on at least some things.'  There are, in at least some parts of the Law, a single best way you can possibly do it, and then you can't do any better than that.  There may be more aspects to god-thought that I can't understand at all, for all that I presently know.  But if Validity is a part of the god-concept of Lawfulness at all, then I can take a pretty good guess at which version of Validity the gods are using, which rules they use to decide which arguments follow from which arguments.  Namely, any one that's inside the huge equivalence class of possible rules that allow deriving all the consequences of the premises you have, but not deriving any more than that."

"To be clear, I just popped into another dimension, I am guessing at some things rather quickly, I could be very wrong about all of this, and any more Lawful beings around are welcome to show up and tell me so before I mislead the lot of you any further.  I do think I have enough dignity not to take offense at being told I made some wrong guesses within my first two days of materializing in another world."

"But it is the obvious thing to suspect, when somebody tells you that 'Lawfulness' is a god-concept.  One at once suspects that the gods and smarter Lawful beings will be using forms of the Law that are optimal within certain dimensions - in some cases where I already know which kind of optimality to look for, and that it isn't a very impossible kind of optimality to have, if your brain isn't as completely messy as a human one."

It....seems likely, that Asmodeus is doing something that this isn't. Both because it seems heretical to say He isn't and because there's a -discontinuity, right, it's not that entities get more powerful and then some of them are debatably gods and then some are more unambiguously gods, either you're a god or you aren't, and it feels intuitively right, that that would be because gods have access to an entire form of valid reasoning mortals don't.

She is uncomfortably aware that none of those previous steps were valid reasoning. She thinks maybe she needs some practice at compartmentalizing so she can do well in logic class and not be aware of the validity of her reasoning all the time while she's trying to catch flying balls. Keltham specifically warned you shouldn't try using logic for that sort of thing.

"And remember again - it's not that humans contain nothing of Validity.  You have concepts like 'or', and 'and'.  You have, in fact, more concepts than you need in order to make first-order logic complete, and some of them are redundant.  The human problem is not so much what we can never manage to derive in infinite time, as that we are too slow, and, even more than that, we tend to derive an awful lot of stuff that doesn't follow.  In dath ilan the Very Serious People used to complain a lot about how we were all being terrible at this, and I used to think of myself as being willing to pursue even riskier and wilder lines of reasoning than that, but now that I've read a book in Cheliax it really puts a lot of that into perspective."

"But I digress.  Humans contain shards, pieces, of the higher mathematical structure we call Validity, the content of necessity, the rules governing premises and conclusions, whose optimal answer is pinpointed by the completeness theorem.  Without these shards of Law, humans couldn't function at all.  These shards of Law within us are not manifested in a centralized single engine whose voice we sometimes ignore; rather, there are bits and pieces and shadows and correlates of Validity, glommed onto us by mistakes retained in the tiny spirals specifying the starting biology of our brains.  It's not that there's a perfect engine of Validity inside us that's corrupted.  It's not even that the parts of the perfect engine are distributed here and there inside us.  The human versions of logical concepts like our version of 'or' - often implying exclusive-orness, which isn't the logical version I showed you, but sometimes not being exclusive either - are more like weird shadows or correlates of pieces of Law.  Same goes for the human native version of 'Z implies Q'.  In the human version it feels stranger to say that 'if I'm naked, that implies I'm wearing a shirt' is extremely true about me because I'm not naked and I am wearing a shirt.  You can make logic out of the human pieces."

"But for all that the human pieces were sloppily thrown together, it's no coincidence that you can make a valid logic out of them.  Generation after generation, for millions of years, there were slight advantages in reproduction to the ancestors of humanity, who we call hominids, when they could do a better job of deducing unseen truths from the truths they already knew or guessed.  The human versions of 'and' and 'or' and 'implies' were built into us in order to do jobs including that job.  And because there is only a single complete structure of Validity in the realm of math - because there is a Law and a best Law and it's not that hard a Law to find - all the bits and pieces of Validity that made their way into us, could have enough coherence and overlap in their messiness that a shadow of true Law could be born inside them."

"Validity is not the only principle with messy shards embedded into humans, in whose overlap and coherence the shadow of higher Law can be seen.  Another such principle is the one my people name Expected Utility, singled out as a unique answer by what we call coherence theorems.  If yesterday you trade two apples for twenty cherries, and then tomorrow you trade twenty cherries for one apple, you've gone around in a circle and ended up with fewer resources than you had when you started.  This is the bare start of a gesture in the direction of proofs that say, 'If you do a lot of deals with apples and cherries in ways that inconsistently value them relative to each other, you'll end up with strictly fewer of both apples and cherries than you could've had by doing different deals.'  If I'm rolling a die that must show either an even or an odd number, you'd be foolish to buy for eight silver pieces a gambling-ticket that pays seven silver pieces if the die shows an odd number, and six silver pieces if the die shows an even number.  This is the bare start of a gesture at the proof that, when you weight the probability of paths through time inside your mind, you should not weight a sub-possibility of a path more highly than you weight its whole."

"The principle of Expected Utility has indeed a sub-principle, which we call Probability, with rules and coherences of its own.  You may have noticed that a great number of conclusions that we need in everyday life do not follow with necessity from any facts that we are highly confident about; but there are also proofs about the best guesses you can extract from a state of uncertainty, and how you cannot do better than those without adding more data or more certainty into your premises."

"Validity, Probability, Utility.  Things being more or less likely, encountering new evidence and revising old beliefs, deriving the consequences of what we already know, wanting things, making plans.  It's not so much that humans have bits and pieces of the Laws glommed onto us, as that the shadow of those Laws within us explains, in a certain sense, why we function at all - why we can do even the little that we can do.  One of my pending questions about Golarion is whether Chaotic gods are still, like, mostly Lawful on a deep level and are just pursuing surface goals that are about humans behaving chaotically in social situations, or something like that, because otherwise I have a hard time imagining what it means to be a god, or intelligent, if your nature is contradictory to all Law.  The partial coherence that exists in the noisy bits of Law embedded in us, creating somewhat larger shadows of bigger pieces of Law, is what lets us form larger thoughts that make enough sense for us to ever figure out anything.  That all these bits and pieces of Law are bits and pieces of this larger coherent thing is part of the story behind how we can put together the human versions of 'or' and 'implies' and make larger useful thoughts out of them.  If Chaotic gods don't have that much Law embedded inside them, if they reject every bit and shard of Validity because it's Lawful, and therefore never think 'I guess that either Z or H will happen', I can't begin to imagine how a Chaotic god would work.  Which is one of the reasons why I wonder whether the concept 'Lawful' is translating correctly for me after all."

"But that's me being confused about this world, which is not our present priority."

"Our present priority is the industrialization of Golarion."

"And the reason I say all of this to you, is to make a certain point about our most important tool for doing that."

"Our path will be relatively simpler, easier, more direct - though still not easy - if a lot of the particular hidden orders I remember about dath ilani steel and dath ilani biology are also true here, albeit with some new hidden orders about magic that were not in dath ilan."

"But if that isn't true?  If snowflakes have six sides here for other reasons?  If my body was remade anew in Golarion so that I could eat the food?"

"Then the valuable knowledge I have to teach you will be the knowledge of how to discover hidden orders.  And this knowledge in dath ilan is said to be attained by using and operating shadows of Law that are purer, cleaner, more complete, than humans just throwing themselves at a problem with their own instincts.  The explicit math is mostly reified Probability, but the internal mental challenges are mostly those of being a little more Valid in which conclusions we jump to and which assumptions we mark as necessary."

"I am a lot more confident that Validity, Probability, and Utility are still singled-out mathematical structures whose fragmented shards and overlapping shadows hold power in Golarion, than I am confident that I already know why snowflakes here have sixfold symmetry.  And I wanted to make that clear before I said too much about the hidden orders of reality out of dath ilan - that even if the things I am saying are entirely wrong about Golarion, that kind of specific knowledge is not the most important knowledge I have to teach.  I have gone into this little digression about Validity and timelessness and optimality, in order to give you some specific reason to think that - even if the stranger proves to have no idea how Golarion is ordered - some of the knowledge he has to teach is sufficiently general that you have strong reason for strong hope that it will work in Golarion nonetheless.

"My memory is not perfect, and I was never a specialist in metal and fire.  To industrialize Golarion, what we must primarily use is not the recipes of dath ilan and its knowledge of hidden orders, for those I do not all have with me, even if they would work here.  What we need to do is operate the principles of thought and investigation, by which dath ilan found the hidden orders and recipes that worked in dath ilan; and, with those interplanar shards of Law, find the hidden orders and recipes that work in Golarion.  If I can accurately remember some of the recipes and hidden orders from dath ilan, and they prove to carry over here, I expect that will give a rather substantial boost compared to starting from scratch.  But I also expect that it cannot carry the day unless we do a little, or rather a lot, of saner thinking of our own."

"It is said also in dath ilan that there is a final great principle of Law, less beautiful in its mathematics than the first three, but also quite important in practice; it goes by the name Coordination, and deals with agents simultaneously acting in such fashion to all get more of what they wanted than if they acted separately.  Here, too, I considered myself relatively wild in this regard, compared to dath ilani standard, but that was before I came to Golarion and read about what all y'all were getting up to around here.  I expect I will have something to say on Coordination too, at some point or another."

"I'll pause here so we can all take a break for meals and washrooms, and resume in four third-hours... in one and a third hours.  Though I can also stick around here for another two dozen half-minutes... twelve minutes, if anyone wants to ask any immediate questions that you don't want to let fall out of your memory.  I mean, you can write them down, but I appreciate that there can be important messy thoughts that are hard to write down full notes to yourself about, and if so I can manage to stick around twelve minutes while you blurt them out before you forget them."

Meritxell wants to know what makes people able to become Keepers while they're still alive. Are they the smartest people? The most careful? Can you tell at age ten?

Not the question Keltham was expecting, but Okay Fine.

"I'd guess the smartest, the most careful - we have a specialized term for that as a usually-mostly-stable quality of a person, maybe 'conscientiousness' would be the best translation here?  I expect they'd look for things you can test in childhood that somebody has shown to correlate with keeping oaths in adulthood later and being very unlikely to go unstable under stress.  I mean, the real answer to your question is that we have prediction markets, people betting on outcomes, with which a lot of people betting, operates as a kind of summary of everyone's best guess at the probabilities of definite observations being made later.  And, I would strongly expect, the Keepers have secret prediction markets that only Keeper institutions can bet on, because it's a secret what exactly they're betting on, because they don't want parents pushing their kids into faking their way into joining the Keepers.  But I imagine the secret prediction market topics say, is this person going to end up passing the following competence tests, will they end up measurably mastering the Way that Keepers keep, will it be recorded that any spilled secrets get traced back to them, will they ever be observed to have broken an oath they took.  Are they going to get along with other Keepers the right amount, neither too conforming nor too iconoclastic.  Will they end up being promoted, are they going to report enjoying their work and be happy at it in observable ways... now that I say it out loud, I feel like there's probably more in the secret markets than that.  That's the kind of market you run to find out if a kid is going to be a good matchmaker or doctor, not to find out who ought to be a Keeper.  The thing is, prediction markets are ultimately betting markets and they have to resolve in definite observations at some point.  So there's some sort of observable thing that would happen to you over the course of your career as a Keeper that a bet would have to be about, in order for it to ever pay out.  In terms of your local system - I don't quite know if they'd qualify as 'Good', they do get paid for what they do, in both money and reputation, but they definitely lean further Good than average - they are, in the end, spending their lives taking care of other people."

"I've always felt weird about the aspect where Keepers are significantly more Good than I am, to be frank.  Even if you nod respectfully at them and pay a tiny fraction of their salaries, they're still doing you this huge favor, that you didn't ask for - some of which probably has to be done in order to make society livable for you at all - but they're doing more of it than I'd ask for, if it was up to me - supposedly on my behalf.  And they aren't doing it wrong, that I know about, or hurting me in any way, that I know about.  But they're still doing more of what they do, than I'd have really asked for... though I'm not a typical dath ilani, the typical dath ilani probably feels more on median-average like there's the right amount of Keepering going on.  Though actually, by the nature of their jobs, there's got to be more of it going on than we really know a specific reason for?  So some reasons for the Keepers' existences are hidden, and maybe my own first-impression feelings are closer to average and I'm just failing to adjust for predictable updates on the hidden info if I could see it... the whole Keeper thing is probably one of the objectively weirder institutions in dath ilan from an outside viewpoint, along with the Surreptitious Head Removers, the Official Government Con Artists, and the Planetary Emergency Rehearsal Festivals.  All of which have completely logical and reasonable reasons behind them, and are still understood and acknowledged even by Civilization generally to be some of the weirder things they have talked themselves into doing."

"Though, I mean, I don't disagree with the reasons, I can see why something like the Keepers need to exist.  Very stable geniuses can extensively develop thoughts that will wreck less stable people's minds, often without them even meaning to do that.  Even pursuing Lawfulness too far can sometimes end up that way.  Human beings are not designed to work great when we push ourselves harder and harder in the direction of Lawfulness - I mean, we're not designed at all, but I doubt it's something our distant ancestors bred themselves to be able to do safely.  I imagine that Keepers are people who by nature are smart and resilient and exceptionally stable in the face of internal insult, able to tolerate weird stuff going on inside or outside their own heads, and what they spend that internal resilience on is going way further in the direction of Law than their ancestors a million years earlier were pseudo-designed to do."

"And, I mean, I'm sure the Keepers have got a pretty good idea about who can do that by age ten.  But that's not because I could take one glance at a ten-year-old and figure out who'd be a good Keeper.  It's because, I confidently predict, the Keepers observe a lot of facts about ten-year-olds, and they keep excellent records of long-term outcomes, and they train people with very high measured intelligence to make good predictions about it.  Come to think, I wouldn't be surprised if the Keepers had a secret prediction market about me somewhere in their systems, saying exactly what my chances were of succeeding at my life goals, and people like me aren't told those predictions because that's exactly the kind of information that - can be a bit - more Lawfulness than we're really happy having in our lives.  And if you can predict that actually a kid is going to be totally okay with knowing that information, then maybe you try to make them a Keeper.  Or maybe what they predict isn't so much kids starting out imperturbable, as that you'll end up driven to face down whatever kind of internal bumps you face, in order to be able to face any kind of disturbing truth and not allow your potential to be limited by the disorder of your own mind... I don't know.  I didn't want to be a Keeper.  They weren't the kind of weird I wanted to be."

She nods. "And - if smartness is part of it - then probably our world just doesn't have people smart enough to be Keepers, yet?"