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

Anticipating the Commemoration of This Friday

A longish excerpt (I can never seem to trim them at all... ), from a rather sad thread from a novel by one of the most sympathetic writers Her Britannic Majesty Victoria might have read in the first printing.

By many devious ways, reeking with offence of many kinds, they come to the little tunnel of a court, and to the gas-lamp (lighted now), and to the iron gate.

"He was put there," says Jo, holding to the bars and looking in.

"Where? Oh, what a scene of horror!"

"There!" says Jo, pointing. "Over yinder. Among them piles of bones, and close to that there kitchin winder! They put him wery nigh the top. They was obliged to stamp upon it to git it in. I could unkiver it for you with my broom if the gate was open. That's why they locks it, I s'pose," giving it a shake. "It's always locked. Look at the rat!" cries Jo, excited. "Hi! Look! There he goes! Ho! Into the ground!"

The servant shrinks into a corner, into a corner of that hideous archway, with its deadly stains contaminating her dress; and putting out her two hands and passionately telling him to keep away from her, for he is loathsome to her, so remains for some moments. Jo stands staring and is still staring when she recovers herself.

"Is this place of abomination consecrated ground?"

"I don't know nothink of consequential ground," says Jo, still staring.

"Is it blessed?"

"Which?" says Jo, in the last degree amazed.

"Is it blessed?"

"I'm blest if I know," says Jo, staring more than ever; "but I shouldn't think it warn't. Blest?" repeats Jo, something troubled in his mind. "It an't done it much good if it is. Blest? I should think it was t'othered myself. But I don't know nothink!"

Bleak House, Charles Dickens

Thus Jo. This isn't quite the first time we've seen this little land behind the iron gate, nor is it the last we will see of it; but it's probably the best showing the sorry spot has. We may gather from some details that this particular "berryin' ground" is rather overused, in a pauperish way, for its size. It is, in way, a geographic distillation of all the sadness that grows about the venerable institution not at all far off, where the learned gentlemen discuss every side of "equity" that will do no one any good untill there's nothing left to be equitable about.

For locus focus
a reader

I was sort-of daydreaming...

so I don't quite recall how much there is...

but, well, Here is a lovely recording

what struck me most about it is the drums. They can particularly be heard around the final Amen, but there's somewhat more.

Here's another I just can't get out of my head at the moment

I blame Robert Reilly.

---

Over the summer, I was contemplating an arrangement of things like the following: start with a special square
$$ \begin{array}{ccc}
A & \to & B \\
\downarrow & \ulcorner & \downarrow \\
C & \to & P
\end{array}
$$
There is then some delightful nonsense which guarantees the following special squares
$$
\begin{array}{cccccc}
& E & \to & F & \to & *\\
& \downarrow & \lrcorner & \downarrow & \lrcorner & \downarrow \\
& A & \to & B & \to & P \\
E & \to & G & \to & * \\
\downarrow & \lrcorner & \downarrow & \lrcorner & \downarrow \\
A & \to & C & \to & P \\
& & & E & \to & F \\
& & & \downarrow & \ulcorner & \downarrow \\
& & & G & \to & *
\end{array}$$
That last one is the really-special one, because of where the $*$ sits in it. It makes $E$ rather simpler than it otherwise might have been... But that's a story for another day.

A microcosm of true liberty

Dear Carmen,

For your consideration.

It is not often that the music director at our parish asks us to attempt polyphony --- understandably, for it is rather a lot more work for him, and there aren't really that many of us singing for him.

But in consequence it is not often that (and hence it is notable when) I get to enjoy this particular curious experience: when a singer is able to sing his part near-automatically, he can actually hear the whole music around him even as he is singing it. I am convinced that this --- to hear the whole even while attentively making part of it --- is the true purpose of polyphony.¹ It is, I would like to say, a microcosm of true liberty.

What I mean by this: what each voice has to do is well-delineated, and it is necessary that each attend to both the governor and the other voices so far as not to overwhelm or misclash with them; yet at the same time, assignment of parts is done with each singer's particular voice in mind, and singers assigned to the same part do not, must not sing identically --- it is usually necessary to breathe at inopportune moments, and so we support eachother, taking turns. Within the stringently prescribed form, there are deeps to navigate and intelligence to be applied. But, when each knows well-enough how he is to sing his part, he isn't trapped within it, but by the whole is lifted above his own line, and... it's hard to find words for it. If folk could live like that, it should have been a happy shire indeed.

Anyways, that's what I did this Sunday between about 10:30 and 1:30.

vox clamantis in urbis

1) This is not to say this is the true purpose of the art of polyphonic music or its performance --- that were God's Glory; but the particular means chosen by polyphony as opposed to plainchant, for instance.