Sunday, July 29, 2012

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 curaƧao (or maybe kirsch!) and go back to meringue mousse.

a disgruntled cook

Tuesday, July 17, 2012

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

Tuesday, July 3, 2012

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

Thursday, May 3, 2012

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.

Monday, April 23, 2012

On Being Wrong

It's often been brought to my attention, recently, that a graduate student --- indeed, any honest research professional --- spends most of his time being confused. Sometimes this confusion is consciously felt; sometimes it isn't. When it isn't consciously felt, it can lead to people declaring things like "Of Course the Riemann Hypothesis Ought to be True" (this is something most number theorists expect, and can't prove yet) or "Of Course You Can Square the Circle" (this is something actually true, in the right context, but not the context most people claiming it intend) or "Of Course All Widgets are Thingumy" (may or may not be true, depending on W and T, but usually not).

This happens to (honest) research professionals all the time, as well as to the more-normal core of Humanity; but there is a key difference between (honest) r.p.s and a particular subgrouping, not really quite fitting in the latter; which is that an h.r.p. is likely not to mind having been wrong, and will happily acknowledge it and receive correction. The weirdos are convinced that they Are Not Wrong, and no ammount of argument will convince them --- because the wrong conclusion derives from honest unknown confusion. Exposed to truth, honest unknown confusion learns only deceptive felt confusion, and prefers what it doesn't find confusing (even unknown confusion). I trust I make myself clear?

Anyways. I get confused a lot, and sometimes it shows. It's actually kind-of fun! Have a good day, everyone.

Monday, April 9, 2012

Monday Musings

Dear Herecomeseverybody,

If I may be allowed to imagine, I should imagine that today would be the liturgical anniversary of "Twin" Thomas' assertion "nisi videro ... non credam". Certainly, it wasn't yesterday, and it will certainly come before the Octave, when we shall hear "beati qui non viderunt et crediderunt".

At other times I've wondered why might Thomas not have been there that Easter Sunday; today, being in a mood for punning, it occurs to me that, in Vulgate and Douay, the Low Sunday Beatitude is, as all beatitudes, written in the perfect tense: "blessed are they who have not seen, and have believed". But it raises the curious question of whether this particular beatitude applies in that moment to anyone at all. The only candidates that spring to mind are Mary the Mother of Our Lord, Peter, and the disciple whom Jesus loved --- but that disciple ("he that saw these things... his testimony is true, and he knows his testimony is true...") was careful to write, earlier
4 ... they both ran together, and that other disciple did outrun Peter, and came first to the sepulchre.
5 And when he stooped down, he saw the linen cloths lying; but yet he went not in.
6 Then cometh Simon Peter, following him, and went into the sepulchre, and saw the linen cloths lying,
7 And the napkin that had been about his head, not lying with the linen cloths, but apart, wrapped up into one place.
8 Then that other disciple also went in, who came first to the sepulchre: and he saw, and believed.
Nonetheless, let us not think on these things into thinking of ourselves as being better than Peter or John were in those moments of great amaze. They had to live through the Gospel before telling it to anyone, whereas we grow up hearing about Easter as a thing accomplished, every year about this time --- we know the ending of this story long before we know what the story is even about, or where it really started.

Nor is Our Lord's visible revelation of his resurrection made to make up for what is wanting in his disciples' faith, but to make up for what is wanting in ours. It's one thing for the Eleven survivors among the Twelve to learn of the empty tomb and believe, but another for them to say so to those same priests and scribes and lawyers who, mere pages earlier were crying "surely thou hast a devil". That is, not only has He given them faith in His rising, he has also given them the power to say and we have seen Him. I have not seen, but I believe because Peter saw.

Let us think on these things with humble joy, and take every help to live as people new-raised from the death that is sin.

Happy Easter,
a simple one among everybody

Saturday, April 7, 2012

Tomorrow, we'll sing...

echo << eof >> /dev/null

II 1 So the heavens and the earth were finished, and all the furniture of them. 2 And on the seventh day God ended his work which he had made: and he rested on the seventh day from all his work which he had done.

eof