Eventually Keltham does succeed in deriving (this time using dubious infinitary arguments, instead of clear and simple combinatorics) that indeed:

If you start out thinking any fraction of LEFT and RIGHT between 0 and 1 is equally plausible on priors, and you see experimental results going LEFT on N occasions and going RIGHT on M occasions, the prediction for the next round is (N+1)/(N+M+2) for LEFT and (M+1)/(N+M+2) for RIGHT.

Worth pointing out explicitly:

You could see this whole fragment of mathematics, of Law, as something that learns.  If you show it 100 examples of balls with 70 going LEFT and 30 going RIGHT, it'll start to predict future probabilities of LEFT and RIGHT around 70% and 30%.

It only learns a very restricted class of things compared to entire people.  It can never learn, if you show it the sequence LEFT LEFT RIGHT LEFT LEFT RIGHT repeated 20 times, that it should predict LEFT with very high probability on the next two rounds, followed by RIGHT with very high probability.  It'll just predict LEFT with probability approaching 2/3 and RIGHT with probability approaching 1/3.

This, of course, is because people contain much more complicated and powerful fragments of Law within them, enabling people to learn much more complicated and powerful fragments of reality.

Even dath ilan doesn't know that much Law.

It's the knowledge a god would need to build a mortal from scratch.

(Certain parts of Civilization do, in fact, secretly know that much Law.  It's unfortunately only one piece of a much larger, insanely lethal challenge.)

- well now she wants to know whether the gods know how to build mortals from scratch. 

Not the point, probably. 

Keltham is having similar thoughts!  He'll wonder out loud about whether Golarion's gods know enough Law to do any significant editing?  If the newcomers haven't read the transcripts for this part yet - or it was less clear from the transcript than in-person - Golarion humans were obviously copied off some ancestors or cousins of dath ilan visible from Golarion's multiverse, the same way Keltham himself was able to appear here.  Keltham can eat Golarion food and that doesn't happen unless the proteins inside are the same and that requires common ancestry.  Similarly, the humans here birth half women and half men, so nobody tampered with that part.

If anything Keltham's sort of surprised that people here aren't more edited.  He wishes he knew what the gods' original edits to mortals were, whose reversion or failure was described as mortals gaining "free will".  That would provide a lot of hint about what level of innate-Law-editing the gods were capable of deploying.

He's guessing it was relatively shallow, though?  Not really anywhere near the level of 'make a mortal from scratch'?  Humans here wouldn't have shared ancestry with dath ilan if gods here could just make mortals from scratch.  The fact that the gods' edits eventually failed, and that people basically turned back into humans afterwards, says that it was probably much more like imposing extra structure on top, maybe magical structure.  Not really remaking and rebuilding things like a 'programmer' could do inside a 'computer', probably not even 'compiling' new 'genes'.

Well, that seems to square with the rest of reality as Keltham observes it?  None of the gods here seem powerful enough to be freely wielding Law on that level.

Korva is pretty sure that the gods did create mortals from scratch at some point - even with the claims about heredity in some of the past transcripts, which she has read, she's still extremely dubious about the idea that no higher power has ever shaped a lower one from nothingness or near-nothingness - but it does seem possible that no mortal has ever developed the power, and if it can be done with math at all, then it makes sense that it's the sort of thing a mortal could do. If Keltham does know some sort of building block towards mortals shaping mortals from scratch, that's the sort of thing that would go some way in explaining why the gods are so obsessed with him.

It's terribly inconvenient that this would actually be an important thing to study, given that Korva hasn't been following along for the past several minutes and is once again despairing of her future. Although if even dath ilan hasn't figured it out, then it seems pretty improbable that the collection of people here will.

...maybe she can at least study the history of beings whose creation might have been directly observed or heard of secondhand, and whatever is known of the changes that occur to humans after death. There. Not useless.

Thaaaat's going to have people quiet, nervous of heresy. "There are other races that are different from humans but similarly patterned, so on your theory I imagine the gods copied the shared-ancestry that also produced dath ilan and then made changes to produce, say, elves, or halflings."

Aaaaand probably don't list the very large number of other species that are people, lest that get Keltham thinking about fire elementals. 

"It's said that the First World is what the first attempt at creation was like, so you might study the fae if you were trying to understand how the gods shaped humans," Meritxell says. "Aside from the thing where you definitely should not study the fae."

- you know, actually, also, who's to say that dath ilani weren't copied off of Golarion humans and then sequestered in some place as some random guy's experiment, like OKAY YES SHE'S THINKING ABOUT HERMEA, given that Golarion obviously has more planar and possibly even more interstellar reach and there are various people who would absolutely do this, and that's why dath ilani apparently don't know anything about their history. Egotistical, much?

"Yeah, there's an awful lot of things in Golarion I'd run off and go look at, if it was safe for me to leave my fortress, and I didn't have more critical-pathy things to do with my time."

He does grok the concept of a critical path, though, that is also common wisdom out of Civilization for people building startups.  At some point he may start to develop cabin fever, but Day 20 is not that point.

"Your regular reminder that I will do anything, to go along with you at that time."

"Yes thank you I'm aware.  But as with sexual attraction of you towards me, my own desire to have you follow me around forever, telling me not to do things, may also take some time for me to develop."

"Okay, just saying, you will die in ten minutes if I'm not there and you will somehow manage to get your soul stuck in the process."

"Ione, I'm not saying no, I'm saying our 'moirallegience'* has not yet progressed to that point."

"Anyways."


(*)  Not the literal Baseline term.

By way of finishing up the morning's lecture, Keltham will also explicitly spell out the point that - outside of particular metahypotheses you're deliberately reasoning inside - it's generally considered a bad idea to assign Probability Absolutely Zero to anything.

Like the way that the Rule of Succession metahypothesis implicitly assigns Probability 0 that the ball just always goes LEFT or always goes RIGHT, because then you can never ever learn that thing, no matter how much evidence is presented you.  The Rule of Succession never figures that out for real, no matter how many cases of LEFT it sees.

If you assign Probability 0 that...

Good examples of somebody-might-think-they-were-probability-0 statements are harder for an outsider to come up with when they're relatively new to Golarion.

If you assign Probability 1 that grass is green, and then you go outside and suddenly all the grass looks blue to you, well, obviously you just conclude that your eyes are deceiving you.  Sure, you're vastly more likely to see the grass looking blue, if in fact the grass looks blue.  Maybe it's a million times more likely.  But it's not infinity times more likely.

If you say there's Probability 1 that grass is green, you're saying that's infinity times more likely than any hypothesis where it's not green.  And the evidence of your senses is always finite, they could always be mistaken.  So if you have an infinite prior, that's like being something that can't notice the truth, ever, no matter how hard and how many times you're hit over the head with it.

Thankfully, human beings are just more complicated things than the sorts of hypotheses that assign Probability 0 to the ball always going LEFT.  Even if your conscious deliberation somehow gets wedged into weird states where you claim to be infinitely sure of things - which, like, Keltham finds it hard to visualize anybody thinking that could possibly be a reasonable thing to do in the first place, like, how would you ever be able to Lawfully justify a claim to be able to guess something strongly enough that you could make guesses like that infinity times and be wrong zero times - but they did teach Keltham not to do that, so it's probably a sort of thing that people ever do, especially when they're kids -

- even then, a human being is a more complicated thing, and there'll be metahypotheses inside you that some part of you is still processing, still able to learn the thing that you claim to assign Probability 0.

If you say to yourself that you believe absolutely that the ball can only ever have some propensity to go LEFT or RIGHT, independently on each round, and any fraction between 0 and 1 is equally probable - and then reality shows you LEFT RIGHT LEFT RIGHT repeating forever - there will still be some part of you that notices.

Because you're not actually a kind of thing that can really assign Probability 0 or Probability 1 to anything, in all of your parts, just by saying that's what you're doing.

There'll still always be some part of you that can learn from sufficiently blatant evidence, that you're hit on the head with sufficiently repeatedly.  If it's a sort of thing that humans assign probability greater than 0 - equivalently, if it's a sort of thing that humans can learn - in the parts of you that your conscious decisions can't cripple enough.

This was also told to Keltham when he was a child, and emphasized to him.  He's not quite sure why it's important, no sane person would assign Probability 0 to any interesting proposition in the first place, but it sounded like a warning so he's repeating it.

That feels - disquieting, somehow. She's not sure it is something to be thankful for, and she's not at all sure it'd be true of devils.

"1 + 1 = 3?"

"Drugs, talk-control, hidden superbeings who created you from scratch to be mistaken about that.  In Golarion, Suggestions and more classified forms of mind-control magic, gods directly messing with you, hidden superbeings who could mess with gods."

"If one day you started adding 1 and 1 and getting 3 every time, including for things like physically adding an apple to an apple in a basket and getting three apples, you would eventually decide your old memories had been mistaken.  It's a kind of pattern you would be able to notice, if it were true.  So the probability that all of yourself assigns to it isn't 0."

You know, some of these warnings do actually make more sense as things to specifically emphasize to people if dath ilani education actually was created by some process that was specifically reacting to things that had been seen on Golarion. Just saying. Well, thinking.

Some of these emphases would make more sense if the dath ilani Keepers were specifically trying to counter Asmodeanism.  As taught in Golarion.

Maybe - faith in gods in general?  Pilar is less sure about that part.

"And on that note of irreducible epistemological existential terror, let's break for lunch!"

Well, that was one of the most unnerving mathematical experiences of Asmodia's life to date.

Otolmens was also unnerved by some of that so-called MATH.

The fact that it WORKED is even WORSE!

Mortals using INVALID reasoning to arrive at CORRECT answers may be all too close to using VALID reasoning to produce WRONG answers.

PL-timestamp:  Day 24 (20) / Afternoon

When it became obvious that Keltham would benefit from an alchemist Cheliax started reviewing candidates. Alchemists, of course, don't tend to advertise all their breakthroughs; that's a great way to inspire people to send spies and steal all your hard work and secret recipes. If you drop by every alchemists's workshop and ask them how much they have independently catalogued properties of gases and metals, the floor will fold under you and drop you into a pit of acid. 

If you have been urgently dispatched for this task by the Crown, you presumably have a Fly spell active, and you can remain in place and present your credentials and ask again, less politely. 

By the day after Keltham asks for an alchemist they've screened some, and picked one, and packed him off to Ostenso with a Bag of Holding stuffed with several thousand of his acid-damaged notebooks. 

He is carefully not looking at anything in particular about this place, in the hope that this'll make it less necessary to kill him afterwards.