"Now there's something that might be true everywhere, which, you might think, would make it an important fact; and if it's important, then it's important to know exactly what it is, that's true everywhere.  So what do you mean, when you say that one equals one?"

"I mean, I'm not at all sure it's an important fact, it's mostly just saying that we defined equals, and the way we defined equals, the things on both sides of it are the same, and things are the same as themselves. But it does seem like it'd be true everywhere."

"It's something of a mischievous question, but mischief is also important in learning, so I'll ask.  One common way to ask what something means, is to ask what you experience when that proposition is true.  If you say 'water is liquid', for example, and I ask you what that means, you might tell me that 'water' describes the clear stuff inside a glass you hold up, and that 'liquid' means that a substance tries to cling to itself but has no set shape, and so conforms itself to the shape of its container; and when I see you pour the water from the glass, onto the floor, I should expect to see it spread out across the floor, while still locally clinging to itself and staying in contiguous puddles.  Now, what do you see when one equals one?"

This is SO STRESSFUL. 

"If you use a spell to duplicate something it'll have all the same properties as the original."

"You don't see anything, it's just a definition."

"Things ...exist at all? ...that'd imply it's not true in the Maelstrom, though -"

"If you try to do math and you assume it, your math will keep making sense."

"Positive reinforcement for continuing to be wrong instead of quiet!  Now, really I only told you half of a proverb, just then.  The real proverb says that to ask what a proposition means, we ask what you should see that's different, depending on whether the proposition is true or false.  Yesterday, water was liquid; tomorrow, water won't be liquid.  How are yesterday and tomorrow different?  Well, yesterday, when I poured water from the cup, it spread out over the floor, in puddles where it clung to itself.  So if tomorrow, I pour out water, and it stays in the same shape as when it left the cup, then tomorrow, 'water is liquid' is false.  Yesterday, you used a spell to duplicate something - let's say a small flower, a dandelion - and the duplicate dandelion seemed just the same as the original.  Tomorrow, you use a spell to duplicate a dandelion, and the resulting flower is blue instead of yellow.  Is one no longer equal to one, tomorrow?  Yesterday, one equaled one; tomorrow, it won't.  What will you see tomorrow that's different from yesterday?"

AAAAAAAHHHHHHHH

"I...don't think tomorrow sustains conscious life that's observing things."

"That's a cop-out, whatever, you're scrying the place where this is true."

"I still think - you try to do math, and your math doesn't work anymore."

"'Tomorrow, it won't' can't be true."

"Can't be true?  Well, if it can't be true that something is false, that would make it a necessary truth, I suppose.  Dath ilan might imagine that it'd managed to deduce what was true in all planes, if it couldn't be false.  But if for that reason you can't tell me what you expect to see, what will happen to you, as a consequence, does your necessary truth really mean anything?  After all, if it meant only some things could happen to you, but not others, it would cease to be true if you traveled to a plane where other things happened to you instead.  So whatever is true no matter what happens to you, never helps you figure out what will happen to you; and, therefore, is absolutely useless.  Now I have just proven to you that all necessary truths are absolutely useless.  And some of you have suggested that math is made of necessary truths.  So have you just proved that math is absolutely useless, since, whatever could happen to you, that wouldn't make math false, and therefore math can never say anything about what will happen?"

Otolmens is watching this classroom SO HARD right now.  The mortal had BETTER not be going anywhere weird with this.  Physics disasters are BAD but math disasters are SO MUCH WORSE.

 "You can use math to derive how to move a spell, and then the spell works or it doesn't."

"And target a catapult."

"And build a bridge."

"If I have one hat and one head, one equalling one means that after I have put the hat on the head there won't be any spare hats or any spare heads. It seems - possible to imagine observing instead that if you have one of something and one of another thing it doesn't mean they match up to each other with none going spare."

The group is divided on whether this is in fact possible to imagine. 

"Just to check, Carissa-Sevar, can you describe to me in additional detail what you'd imagine it to be like to observe that?"

Keltham has had a pretty strange couple of days and is, in fact, less sure of some things than he used to be.

"I mean if it happened I'd assume someone was messing with my head, or I was dreaming, but - well, imagine instead we have five weapons and five spots on a weapons rack, it's not hard to imagine that you put a weapon in each slot but then there's still one slot left over, and you go back and count and there are five slots, one of them empty, and you count the weapons and there are five, all in a rack. It's harder to imagine with one because in dreams sometimes counting to five doesn't quite work but counting to one still does."

"Saying those words out loud is one thing; could you create a detailed illusion of it happening?"

"Not a motionless one. I bet I could - do one that took advantage of how people can't look at a whole landscape at the same time and changed where they weren't looking at it. You'd just be tricking them, though, even if you did it perfectly, you wouldn't have changed what one equalled."

"If it's not possible to create an illusion of something being false, you might not need to travel to other planes to guess it would be true there.  But I offer the same mischievous objection as before:  To say that you can't make an illusion of something, doesn't narrow down what kind of future follows from the past - we can make an illusion of a plane where jumping up puts you at the bottom of an ocean, instead of off the ground.  Even if in all previous history, jumping just lifted you off the ground a bit, we can make a detailed illusion of a world where that happens the first trillion times, and on the trillion-and-first time, jumping teleports you under the ocean instead.  So if math is about truths we can't make even an illusion deny - then why is math any good for building bridges?  We can make an illusion of a bridge falling down."

They are so confused and varying degrees of distressed about it.

"Actual bridges fall down more if you did the math wrong."

"Making an illusion of casting a spell isn't - the same thing as actually casting the spell - sometimes the way to pass the test is to be able to actually do it, not just to make it look like you can -"

(Keltham does not have the faintest chance of noticing that somebody who did well in a Chelish academy is leaking tiny signs of distress past their routine concealment thereof.)

"Well, I think I've created enough explicit confusion that you'll notice learning something that makes you feel less confused," Keltham says, and then makes a brief sad face about how this snappy statement sounds so ridiculously long in Taldane.  What kind of language makes confusion a three-syllable word, anyways?  One that has no idea what its nearly neural-level cognitive primitives are, presumably.

Keltham goes to the improvised whiteboard, and starts drawing squares and triangles, red and green, large and small, inside some bigger blue circles.

"Consider each of these blue circles and their contents as depicting - we would say in Baseline - possible worlds.  By possible, I don't mean it's especially likely that you'll find yourselves in them; these possible worlds I'm depicting are much too tiny to support intelligent life.  They've only got a few squares and triangles inside.  By 'possible' I do mean that one could make a fully detailed illusion of the world, given the ability to cast arbitrarily large illusions; my using markers to draw a world in complete detail similarly shows that world to be 'possible'.  Now, consider these propositions -"

Keltham writes, in black marker:

Z.  All triangular things are red.
H.  All red things are large.
Q.  All triangular things are large.

(Dath ilan has some different conventions for symbols to use in equations, for example, all the symbols should be as topologically and typographically distinct as possible.)

"As you can see, I have shown worlds where Z is true, and worlds where Z is false.  I have shown worlds where H is true, and worlds where H is false.  I have shown worlds where Q is true, and worlds where Q is false.  None of Z, H, and Q, then, are necessary truths, nor necessary falsehoods; for they are all true in some illusionable worlds, and false in others.  Then is there anything useful here for math, logic, and necessity to say?"

It takes a couple of minutes of muttering and frowning and guessing "no?" and "there are triangular things in all the world- oh, no, not that one -" before - "well, if Z and H are true, then Q is, you can't have any with Z and H but not Q."

That took them longer than Keltham expected.  He frankly would not have expected that all the exercises he had to do as a kid were, like, required for getting that point instantaneously as an adult.  Not to mention, they know topology but not predicate logic?  Right, because you need topology for spells, but not, apparently, predicate logic.  If he'd realized he sure would've told them to learn that in yesterday evening's afterhours instead of calculus.  Oh, well, he'll plunge on and see how far he gets.

Keltham goes to the whiteboard and draws some conscious observers inside his blue circle-worlds.  Much as some other world might indicate observers with smiley-faces, dath ilani convention calls for Keltham to draw a number of glaring eyes inside his worlds, creating a tableau that somebody from a differently-troped world might regard as eldritch.

"Well, now I've put some conscious observers inside these worlds!  Not that my tiny drawings embody real experiences, of course, they're not detailed enough drawings for that; so now these pictures are no longer being drawn in full detail, which is something we might need to watch out for if this was a more complicated debate about conscious experiences."

"Some of these observers, in the worlds where Z is actually true, might see twenty triangles being red, and zero triangles being green, and hypothesize a general law: all triangles are red.  They might be able to deduce, without having to actually scry into other planes, that Z was not a necessary truth; they might be able to cast illusions, draw on walls, or just use their imaginations to see that.  So they would not be certain that all triangles are red.  For all they know, the world might up and present them one day with a green triangle.  But the next time they saw a triangle, even if their world made them slower to see colors than shapes, they could guess even in advance of observing; they would guess the triangle was red."

"Let's also suppose that you can tell whether an object is small or large, but it's an expensive measurement; an observer has to actually wander over close to the object, to determine its size; because if they're looking at the object from a distance, they're not sure if it's nearby and small, or large and far away.  These observers have only one eye, as you can see; no binocular vision for tracking distances.  Let's say they have to pay one labor... one silver piece each time they want to move to an object."

"In worlds where H is true, observers who pay to measure a few red things will find, that of all the red things they have measured, every one of those red things was large."

"Now let me ask again, in case anyone has seen the point before I speak it:  How can knowing necessary truths save you money?"

"Well, if you know that triangles are red, and that red things are large, then you don't have to go check the size of triangles."

"To state it precisely, some observers may have guessed the unnecessary truth that all triangles are red, observing the redness after the delay.  They may have separately guessed the unnecessary truth that all red things are large, after paying to measure some red things.  Maybe they've never measured any of the red things that were triangles! we can suppose for the sake of clarity.  Then the necessary truth, 'Q is true in all worlds where Z and H is true', can allow them to guess the unnecessary truth 'All triangles are large', which necessarily follows from other unnecessary truths they've guessed.  And even if they've never measured the size of a single triangle before, they can guess - though not know for certain - that every triangle they've seen was large, and that the next triangle they see will be large.  If it's the kind of knowledge that matters, but not enough that you need to be very sure of it, they could use that guess in place of paying a silver piece to measure it."

"Of course, it isn't a necessary truth that the observers are capable of figuring this all out - that they can operate the necessity, 'Z and H yield Q'.  We could draw an illusion of a world where the observers totally fail to figure that out.  It would still be true across all planes and all illusions that could ever be drawn in full detail, but the people in that illusion wouldn't know it."

"It isn't necessary that entities successfully operate universal necessities in order to see which new guesses must follow from old guesses, which means that some entities do better or worse at this than others.  This is true when considering all possible worlds as a whole, and also happens to be true within my homeworld, and almost certainly in this one."

"So now we shall turn to the question: suppose you were constructing a new entity from scratch.  How would you go about embedding in them an internal reflection of the interuniversal Law, the ability to operate necessary truths correctly...  No, sorry, that's probably too much of a leap to ask in one go.  Strike that, restart.  Suppose you were comparing two entities: how would you say that one was doing better or worse than the other at being Lawful in this exact sense?"

- Keltham hasn't noticed but his teaching style clearly has half the class extremely panicked. They are concealing it very well.

...it really seems bizarre, that you could teach Law this way, with trick questions and guessing games and strange rules about how you're supposed to volunteer wrong answers if you aren't sure you know the right one. It seems like the habits of mind that would teach are - well, does she actually think that it'd teach unLawful habits of mind, or just horrendously ill-advised ones, there is a difference -

- if you built a military out of Kelthams it would not be a very good military, which is a perfectly serviceable definition of Law, the discipline and coordination required to win wars. The Kelthams -- and, plausibly, the people taught like Keltham - would be wrong, a lot, out loud and cheerfully, they'd consider everything their business, they'd ask questions they shouldn't ask -

- he did behave differently with Contessa Lliratha, maybe there's a kind of distinction the mind can successfully maintain, irreverent in most contexts but deferential where it actually matters - but it seems like it would be hard to tell if someone will be deferential when it actually matters, if they've spent their entire life in contexts where it doesn't, not being sufficiently deferential at all -