More topological reflections of arithmetic

For this part of the story I want a really-pretty diagram that even the mathjax still doesn't understand yet how to draw, so I've prepared a dead picture of it.

The doubled-arrows say what kinds of squares they cross --- what sort of equations. The whole thing is basically a fancy elaboration of the idea $ \Phi + X = Z + Y $ and $ \Psi Z = C \Phi $ and $ A Z = X C $ and $ A Y = X B $ and $ B \Phi = Y \Psi $; and it says that if what it means by those are true, then (that's the dotted double arrow) also what it means by $ \Psi + A = C + B $ is true. Note, however, that not all arithmetically true things are topologically true; in arithmetic we have: if $ A B = 1 $ and $ C B = 1 $, then $A = C$; but this isn't true in figures: one would need the right diagram to make it work.

The best known (if not best-known) generality described by the above diagram is known as “Mather's Second Cube Theorem”, and it's at the heart of a lovely little construction I mentioned earlier, which you get with $ \Psi = * $ . There's essentially only one way for a space to “be $ * $”, so these situations are rather special; on the other hand, as long as $ \Phi $ all comes in one piece, there's exactly one way to fit any $A,B,C$ as in the diagram over any compatible $X, Y, Z$. What it meant for the earlier story is that one doesn't need to worry about $ \Phi$ , because it might as well be $ * $. Making that change doesn't even change the ... well, I can't quite say what it didn't change, because I haven't spelled it, here. But when you can simplify the problem without changing the solution, you're making progress!

There are lots more nifty diagrams involved, probably the prettiest being

--- the $\gg$ here is doing what the $\Downarrow$ was above, never mind. Anyways, I was going to write a fun little paper about it, but then I found this fine gem, which I think describes about the same argument, but better! This happens to me a lot, as you might imagine. One day...

Anyway, if anyone wants more, just ask me!

A little less about that dullest of things

Dear Ludo,

After that long interruption, I had thought of returning to the earlier theme, but I find that Stilwell has been writing more articulately than I could a decent approximation to an Economics of Charity; I hesitate at some of his bolder sloganisms --- I hesitate at most slogans, for I am a timid creature by temperament. Particularly, it seems wrong to say that a true fiat issue is “not backed” and even by nature “unbackable”: rather, the backing of a true issue is its tradability, or under another aspect, the actual wealth of its market, or again, the productivity and honesty of its citizens.

Modulo such concerns, which are either natural or naturally bunk (I am not qualified to guess), I think I can recommend a cautious reading of his notes.

That is all.

from the armchair

Ignosce me

Non veterescentur nobiscum consenescentis;
Senectudino non delassantur neque damnabuntur aevis.
Occidens Solis, ac Oriens, ei commemorabimus.

It isn't really one of mine; the sense is of a verse they recite in these parts every Armistice Day, and the individual stem words are mostly out of Whitaker's. But we keep trying!

Well, that was ...

... that was something of an else.

Dear Mr. Returning,

I had limping hopes two years back that an hostile lower chamber might have called their king to account for transmuting and muting "creed and trust", in the Primordial Fix, to "quodcumquelatry", without consulting them or getting it officially rewritten. Yea, limpid limping hopes, but alas, it seems they did not much care.

In any case, we'll (or... they will... I'm a remote-dwelling foreigner, afterall) have four more Epiphanies and Easters and Christmases under your presiding, maybe you'll notice them? I'm afraid your managers won't let you notice them, properly; it'd be such a relief, though, if you could escape the Lower Downs for a little bit, and just soak-up the splendour and fear of some things "above your pay grade", or consider some of the things others built before anyone here was born, but are still using today. It's a sad thing for the princes, though, that princes are not by their handlers allowed that much contemplation, but only entertainment and diversion.

Anyway, as you boldly march forward with eyes firmly fixed on anything but where you're steppeing or standing, may I recommend a little book about a genuine (and successful!) search for common ground? Here's sufficient excerpt to find which one:

You hold that your heretics and sceptics have helped the world forward and handed on a lamp of progress. I deny it. Nothing is plainer from real history than that each of your heretics invented a complete cosmos of his own which the next heretic smashed entirely to pieces. Who knows now exactly what Nestorius taught? Who cares?

an extern