A House has an inside and an outside and a neighborhood

For the purpose of being well-understood, I was raised (though no-one ever mentioned it) not under the declaration that all men inherit the right to "life, liberty and pursuit of happiness", but rather in a realm which proclaimed "peace, order, and good government". Whether either of us now enjoys any of these goods is a question best left to future historians, perhaps, but never mind.

It would seem there is room for disagreement on the prudence of civil law recognizing the unique and preeminently worthy character of the free and total union of Man and Woman made concrete in the free and total union of one man and one woman coenduring with their common Earthly survival.

I recently highlighted differing opinions on this matter from two good and thoughtful Catholic fathers with whom I respectfully disagree.¹ The one asserts that "the state [has to] define marriage and know who is married in order to answer two questions: who owns what and whose kids are whose;" the other affirms his belief "that [one] should not be prohibited from pursuing a legal union with whomever [he?] like, according to the beliefs of another religion or in the eyes of government."

In prelude to some later-to-develope thoughts

Dear Moose,

Compare and contrast, two Catholic men who by their writing and thoughtfulness have much impressed me.

In the North corner, Darwin
In the South, John C.

Er... the cardinal directions cited here bear no relation to anything in the world of real; they're just meant to be different.

b.g. gruff II

Apocrypha Topologica II

Dear Mathematicelli,

If I may proceed,

The Cretan king Minos seems to have hit upon the psychological trick of bewildering his prisoners into imagining a topological obstruction where in fact the obstacle was only metric: that of the Labyrinth (with intimidating monster to keep you distracted). The solution that let Theseus escape the Labyrinth has found renewed popularity of late, and so is worth elaborating. Ariadne's reasoning might conceivably have run thus:

• If one enters the labyrinth and comes to its center, then one has got (and can get) from there to here

• Walking the same way backwards, one can get there from here

• What is wanted is some means to remember the path one took in getting here, and then to reverse it

• Since I don't know how long a path one might need to follow, the keeper of the memory had better be long!

In other words, we keep more information than just the fact that two points are connected by a path: we remember the whole path between them. Some recenter mathematicians more inclined to vandalism have suggested painting the walls of the labyrinth to remind yourself of where you have been --- which is sufficient data to escape, if you paratroop into the maze under cover of darkness; but the solution proposed by Ariadne and adopted by Theseus makes it easier to tidy-up afterwards: unroll a string as you walk along, and then follow it in reverse, winding the string again!

A point worth noting, which may have escaped Ariadne and Theseus in their flight, is that the re-winding and the follow-backwards phases of the solution can be performed in either order if only the string itself is slippery enough --- this could have been remarkably handy if the labyrinth had been under water, and had they wanted to get fish out of it without the fish seeing them and becoming suspicious. It also points out something special in the total path followed by Theseus through the Labyrinth: it is contractible!

Dungeons we may pass over, as well as the castle and siege warfare. Chain mail is about as old. But Somewhen between the Visigoths and Polyphony was discovered knitting. These, like most mechanical inventions, rely on metric phenomena to make their topology useful, but it is unquestionably their underlying topology that is used. (As a side-note, there's a lovely film-reference back to Ariadne in The Name of the Rose, where Adso returns his knit sweater to a trivial topology in order to escape a non-planar labyrinth! I'm curious how the Vandal painters would fare here!) If you'll forgive a jump-forward, the topologist Poincaré seems to have observed a woman knitting, and independently invented purling on the spot. I don't know what the full topological significance of that is, but the ubiquity of What Is certainly makes itself plain to those who can see it; for which give thanks to God, I think. About such other oddities as the Borromean rings we have remarked elsewhere.

I have no idea what's coming next; but this isn't a bad lot. We've more-or-less covered ±1800. Prof. Cauchemar

From a whisk-user

Dear Gelatinifers,

I wish to register a complaint; specifically, without endorsing the heresy sometimes wrongly known as "Luddite", it is nonetheless my firmly-held belief that recipes artificially biased in favour of electromechanical mixing apparatus should only be packaged with electromechanical mixing apparatus (and otherwise appear in books describing what they are), not with foodstuffs. I'm sure I shall, after one or two more trials, adapt the method so as not to melt the cream I've just whipped, but I don't see why I should have to use up the whole box of whatever it is you actually sold me, before it is properly useful, after you pretend to suggest a recipe.

As it is, I'm rather tempted to just get me a bottle of orange-infused curaçao (or maybe kirsch!) and go back to meringue mousse.

a disgruntled cook

Jigsaw

Dear Crowsfort,

I have a jigsaw puzzle. The pieces look like squares \[ \begin{array}{ccc} A & \overset{f}{\to} & B \\ g\downarrow & \Downarrow & \downarrow h\\ C & \underset{k}{\to} & D \end{array} \] ... actually, $f,g,h,k$ all know what their corners are, so we could leave out the $A,B,C,D$, but this gets distracting. Also, the $\Downarrow$ deserves to have a name, only I can't think of a good way to make it all fit. Which particular $\Downarrow$ a square has in it makes a difference, later!

Two pieces sharing an edge fit together, so that \[ \begin{array}{ccc} A & \overset{f}{\to} & B \\ g\downarrow & \Downarrow & \downarrow h\\ C & \underset{k}{\to} & D \\ C & \overset{k}{\to} & D \\ g'\downarrow & \Downarrow & \downarrow h'\\ E & \underset{l}{\to} & F \end{array} \] make a rectangle \[ \begin{array}{ccc} A & \overset{f}{\to} & B \\ g\downarrow & \Downarrow & \downarrow h\\ C & \overset{k}{\to} & D \\ g'\downarrow & \Downarrow & \downarrow h'\\ E & \underset{l}{\to} & F \end{array} \] or sometimes \[ \begin{array}{ccc} A & \overset{f}{\to} & B \\ g\downarrow & & \downarrow h\\ C & \Downarrow & D \\ g'\downarrow & & \downarrow h'\\ E & \underset{l}{\to} & F \end{array} \] and can even be squished down to a square \[ \begin{array}{ccc} A & \overset{f}{\to} & B \\ g'g\downarrow & \Downarrow & \downarrow h'h\\ E & \underset{l}{\to} & F \end{array} \] which is handy, though we don't often want to do that.

The good people who cut out my jigsaw puzzle were very nice, and provided an unlimited supply of various standard shapes, guaranteed to fit certain sorts of corners, so that if anywhere in the puzzle you find \[ \begin{array}{ccc} & & A \\ & & \downarrow f\\ B & \underset{g}{\to} & C \end{array} \] you can add in a square \[ \begin{array}{ccc} P_{f,g} & \to & A \\ \downarrow & \lrcorner & \downarrow f\\ B & \underset{g}{\to} & C \end{array} \] and in the same way, if you have \[ \begin{array}{ccc} A & \overset{f}{\to} & B \\ g \downarrow & & \\ C & & \end{array} \] you can fill it in \[ \begin{array}{ccc} A & \overset{f}{\to} & B \\ g \downarrow & \ulcorner & \downarrow \\ C & \to & Q_{f,g} \end{array} \] There is one other sort of handy square, looking like \[\begin{array}{ccc} A & \overset{f}{\to} & B \\ f\downarrow & = & \downarrow g \\ B & \underset{g}{\to} & C \end{array} \] which lets you go around corners, when it looks like a good idea. These have two further special types, \[\begin{array}{ccc} A & \overset{f}{\to} & B \\ f\downarrow & = & \downarrow = \\ B & \underset{=}{\to} & B \end{array} \] ... and there is another of the similar sort that I'm sure you can guess; and there's also vertical and horizontal versions of \[\begin{array}{ccc} A & = & A \\ f\downarrow & = & \downarrow f \\ B & \underset{=}{\to} & B \end{array} \] which also happens to be a $\lrcorner$ and a $\ulcorner$.

Actually, those last two squares are special cases of these two : \[\begin{array}{ccc} A & \overset{f}{\to} & B \\ =\downarrow & = & \downarrow g\\ A & \underset{g f}{\to} & C \end{array}\] and \[\begin{array}{ccc} A & \overset{f}{\to} & B \\ g f \downarrow & = & \downarrow g\\ C & \underset{=}{\to} & C \end{array}\] or reflections of them; but most of these two are neithert $\lrcorner$ nor $\ulcorner$.

They were also kind enough to suggest a few ways to get started, using a special corner called "$*$", or "the point", though it's not really the point of all this. Still, there's always exactly one edge $A\to *$, no matter what $A$ is, and you can also draw it vertically: \[ \begin{array}{c} A\\ \downarrow \\ * \end{array} \] The corner $*$ also has another nifty feature, that the collection of edges $ * \to A$ might as well be called $A$. There's only one corner, $\{\}$ to which you can't draw an arrow from $*$; but on the other hand, there's always exactly one arrow from $\{\}$ to any other corner $A$, including to $*$! So, for instance, there's a nice corner \[ \begin{array}{ccc} \{\} & \to & * \\ \downarrow & & \\ * \end{array} \] and because of the $\ulcorner$ pieces, this gets filled-in as \[ \begin{array}{ccc} \{\} & \to & * \\ \downarrow & \Box & \downarrow\\ * & \to & * + * \end{array} \] although it's more common, among my fellow puzzlers, to call that new thing $\mathbb{S}^0$. It has two points, as you can see. Oh! this one tile happens to be of *both* sorts: it's the standard tile to fill-in those two edges $*\to \mathbb{S}^0$ as well as the standard tile to fill-in the edge $\{\}\to*$ drawn twice from a single copy of $\{\}$. Sometimes it's fun just to look at the special pieces \[ \begin{array}{ccc} A & \to & * \\ \downarrow & \ulcorner & \downarrow \\ * & \to & \Sigma A \end{array} \] which highlight a fascinating sequence of corner labels $A, \Sigma A, \Sigma^2 A, \ldots$ --- the ones you get starting with $\mathbb{S}^0$ are called the spheres (or homotopy spheres) and have the special names $\mathbb{S}^n = \Sigma^n \mathbb{S}^0 $. Going in the other direction --- if you have a favourite arrow $ * \overset{a}{\to} A $, the special square you get is labelled \[ \begin{array}{ccc} \Omega_a A & \to & * \\ \downarrow & \lrcorner & \downarrow a\\ * & \underset{a}{\to} & A \end{array} \] ... to tell you how one is supposed to keep going after that, I have to tell you one last thing about the special squares labelled $\lrcorner$ and $\ulcorner$; given any square at all \[ \begin{array}{ccc} A & \overset{f}{\to} & B \\ g\downarrow & \Downarrow & \downarrow h\\ C & \underset{k}{\to} & D \end{array} \] there are of course the standard two squares \[ \begin{array}{ccc} P_{h,k} & \to & B \\ \downarrow & \lrcorner & \downarrow h\\ C & \underset{k}{\to} & D \end{array} \] and the other one to $Q_{f,g}$; in essence, what it means to be a $\lrcorner$ is that, there's essentially just one edge $ A \overset{w}\to P_{h,k} $ that fits into this puzzle \[ \begin{array}{ccccc} A & \overset{=}{\to} & A & \overset{f}{\to} & B \\ =\downarrow & = & w \downarrow & \Downarrow & \downarrow = \\ A & \underset{w}{\to} & P_{h,k} & \to & B \\ g\downarrow & \Downarrow & \downarrow & \lrcorner & \downarrow h \\ C & \underset{=}{\to} & C & \underset{k}{\to} & D \end{array} \] There's a similar story about unique edges $ Q_{f,g} \to D $ that fit in another puzzle --- try it and see! But particularly, since we always have this square \[\begin{array}{ccc} * & \to & * \\ \downarrow & = & \downarrow a\\ * & \underset{a}{\to} & A \end{array}\] there's exactly one $ * \to \Omega_a A$ that fits in all the necessary puzzles, and this is what lets us keep going to make new spaces $ \Omega^2_a A, \cdots $. Here's a puzzle for you: come up with a good edge $ A \to \Omega_{?} \Sigma A$! This entails finding a way to fill-in that $?$; you should be able to think of perhaps-two.

There are lots of things I haven't mentioned, but of course, that will always be true, even if I say all the things that should come first! You're welcome to play with the jigsaw, too; we'll never run out of pieces!

the joiner

To the credit of Mr. Bumble

or, a foreigner's comments.

Of course, the question of whether a given law is logically consistent with a narrow scope of (written, "foundational" or, paradoxically "superior") law is an important one, in that the police should be able to tell whether they should be arresting citizens or legislators; not that the latter is envisioned or practised much, according to a particular and narrow scope of law.

It is a sad thing, however, that this seems to be the only competence of various superior courts; elsewhere several people are remarking that more important than whether a document like Veritatis Splendor is infalible is whether it's true. Similarly, more important than whether a given law is legal ("constitutional") is whether it is just; or, under the maxim that an unjust law is no-law-at-all, whether a given piece of legislation is indeed a true law.

Alas...

no beadle me

Drip drip drip... (a lackadaisical rambling)

"Oh, Hi", as they say one says. I suppose I ought to write something.

As I begin to write in earnest, it is indeed starting to rain. The day has been brooding on this evening moment from early hours; but only now are lightnings and dropplets together falling. The heavens roar and pour forth to wash my dusty city: Vidi aquam egredientem...

*   *   *

The storm passes over with varying intensity, like lumpy oatmeal, or a battle waxing and waning as new troops weary and new forces drawn up. Sometimes the thunder recalls a lion purring --- not immediately violent, not quite safe. Sometimes the air in the house veritably tingles with a hunter's anticipation!

Here comes a thick bit, right now! Oh, what a noise! A hundred, a thousand snare drums without a Drum Major to coordinate them. They march off gradually, leaving a pancake-sizzle sound behind.

*   *   *

After prayer, what is one supposed to do, if one doesn't know what to do? Obviously, beyond thinking, too --- it's the thinking that seems to get me into trouble, you see. I think and think and usually come up with no decision. Most unhobbitlike am I, in that way. Do you remember, perhaps, when Bilbo said it?

"Go back? No good. Go sideways? Impossible. Go on? Only thing to do!

Somehow (perhaps I forget the eliminateds and return to them?), I neither reach a "yes, that sounds good" nor an "only thing to do!". Wandering about, getting dizzy... and forgetting why.

*   *   *

But that's OK. Gradually, we are walling-up the side-routes to traps and blinds; gradually, we learn to keep direction (that is, "of straightness") in the darkness we once took for seeing. True sight, in the true light, takes some getting-used-to.

Another period of dripping, of quiet, though it's all one rain. The same flood that purged the Earth also floated Noah along; the difference was a matter of disposition, of being inside or outside the Ark, the Barque... being inside, and it not being a flood, this quiet rainy bit is making me sleepy, so I'll turn off the drippy tap, now, and I'm sure you all won't mind. Maybe a good evening for a warm bath...

Good night.
God willing, some of us will see eachother in the morning.