~~Could you elaborate on the first paragraph of Section I? I don't recognize any of the Veda references and can't tell what you've picked up from real-world myth and what you've made up. How does this paragraph juxtapose with the next, in which you suggest the two statements are in opposition? Is this about the question of whether mathematics is made or discovered, with the Mathematicians claiming to answer it by making mathematics?~~

Honestly the Introduction is me just rambling on to get the juices flowing. The Veda references concern Prajapati, though I based that line on the writings of Calasso and his book Ka. So it's second hand information, for which I apologize - I was just typing away without thinking.. I'll try to hunt down the references. My thought here was that mathematics is part of the eternal, coming from a place like that of the Platonic ideals.

~~What do the irrational numbers have to do with defending against an army of moignos? It's true that they're a greater order of infinity than the rationals... how does this apply?~~

This is actually a simplistic notion on my part, in that I'm equating irrational numbers with Chaos. I know Dedikind's cuts place them neatly into the real number line, but I still can't help but feel even this is an admission of their unruliness.

~~I like the eies and their vulnerability to illusions. Normally illusions can't be cast on Mechanus at all, but perhaps an eie perceives even a failed illusion spell as if it were properly cast there...~~

This is again a little bit of silliness on my part, as there isn't anything truly 'imaginary' about complex numbers. But both complex numbers and illusions exist on a mental plane in my mind, though I realize that's a cop out on my part.

~~The ceirees make a cute plant for the plane. Graphical trees (graphs with no cycles) probably could as well! Perhaps most of them appear to be finite, but some very old colonies of ceirees are so long and entangled that no one is sure whether they're actually infinite or not?~~

Oh, I really like the idea of graph trees as plants! I can also see less experienced travelers getting lost in oceanic digraph and cyclic mazes. I'd love to have currency based on graphs where the inhabitants intuitively calculate the cost of a weighted graph.

~~The lifecycle of prufes and lemmas is fun, with the two species assisting each other in reproduction and all. Prufes seem like a sort of law elemental and thus don't need to eat, perhaps, but lemmas seem organic -- *do* lemmas eat, and if so, what? I would be interested to see a fuller writeup; a lemma could make a cute familiar for my Guvner wizard, who doesn't have one yet! :^)~~

I definitely want to explore lemmas and prufes, I see the former as the ratatosks of mechanus but I don't want their role to be redundant. Need to think about it.

~~Axioms are a concept already officially named. They are described in supplements pertaining to the Fraternity of Order and the Mathematicians; they are indeed principles, though on Mechanus perhaps they really can take human form. I would not give modrons or axiomatic creatures any particular knowledge of such beings. A PC can play an axiomatic creature, after all.~~

Who/What the Axioms are, if they can even be legitimately personified, is unclear. I like the idea that the most ordered place in the Multiverse has its own superstitions and where the gods walk its nice to have hyper-gods who must be taken on faith.

~~I'm not sure what's going on with the rod of induction. It doesn't seem to have any thematic connection with the proof technique of logical induction to me. Alternative proposition: if the wielder of the rod successfully casts a spell with a saving throw his target fails, then in the next round he may use the rod to recast the same spell with the same statistics on a new target, without expending one of his own spells or spell slots. This can continue until he runs out of targets, chooses to stop, or a target makes a saving throw or spell resistance roll.~~

Basically the rod works by assuming the "truth" of the spell. So you have the base case, the assumption, and the necessity of "proving" the n+1'th case. Is that still unclear? Again I have to admit this is somewhat tongue in cheek.

I like your idea, which seems to be based on the idea of the "Least Criminal" number (the minimal integer in the finite set for which P(n) is true?).

I really liked this article. It took me a while to respond since I wanted to digest it properly. I ended up with rather a lot of comments!

Could you elaborate on the first paragraph of Section I? I don't recognize any of the Veda references and can't tell what you've picked up from real-world myth and what you've made up. How does this paragraph juxtapose with the next, in which you suggest the two statements are in opposition? Is this about the question of whether mathematics is made or discovered, with the Mathematicians claiming to answer it by making mathematics?

What do the irrational numbers have to do with defending against an army of moignos? It's true that they're a greater order of infinity than the rationals... how does this apply?

I like the eies and their vulnerability to illusions. Normally illusions can't be cast on Mechanus at all, but perhaps an eie perceives even a failed illusion spell as if it were properly cast there...

The ceirees make a cute plant for the plane. Graphical trees (graphs with no cycles) probably could as well! Perhaps most of them appear to be finite, but some very old colonies of ceirees are so long and entangled that no one is sure whether they're actually infinite or not?

The lifecycle of prufes and lemmas is fun, with the two species assisting each other in reproduction and all. Prufes seem like a sort of law elemental and thus don't need to eat, perhaps, but lemmas seem organic -- *do* lemmas eat, and if so, what? I would be interested to see a fuller writeup; a lemma could make a cute familiar for my Guvner wizard, who doesn't have one yet! :^)

Axioms are a concept already officially named. They are described in supplements pertaining to the Fraternity of Order and the Mathematicians; they are indeed principles, though on Mechanus perhaps they really can take human form. I would not give modrons or axiomatic creatures any particular knowledge of such beings. A PC can play an axiomatic creature, after all.

I'm not sure what's going on with the rod of induction. It doesn't seem to have any thematic connection with the proof technique of logical induction to me. Alternative proposition: if the wielder of the rod successfully casts a spell with a saving throw his target fails, then in the next round he may use the rod to recast the same spell with the same statistics on a new target, without expending one of his own spells or spell slots. This can continue until he runs out of targets, chooses to stop, or a target makes a saving throw or spell resistance roll.