quivering through sun-drunken delight

Saturday, June 02, 2007

Tigers as social animals

I found out a little more about what this book is about today.

As I walked the narrow road down to campus last night I began to feel like I was in the middle of a 1920's Broadway musical. There were all these people, walking the other way, for some unclear reason (no one comes to visit us). They were all dressed quite similarly: straw hats, like from Citizen Kane or a carnival act, with bits of cloth or a ribbon tied to it, in orange or black or both, and flimsy-threaded suit jackets cut in matching, antiquated style, with checkered diamonds of orange, black, and white, or vests with the same pattern, over white or black shirts and white pants. Everyone is wearing these buttons over their breast with class numerals in large font above their name. Later I determined that they had some kind of event going on in the courtyard one over from my room, but I didn't stop to ask questions.

Later still I saw a paper posted with some kind of schedule on it (yes, it was orange paper), and from this I gleaned that there is something going on called Reunions 2007.

Since yesterday it's been all tigers, all the time. Blouses of orange, or mottled orange and black and white. T-shirts of orange, of course, and t-shirts with the university name in that ubiquitous arched font, or t-shirts in black with an enormous orange letter “P” on the back, as though everyone knows what that means. Kids dressed up like their parents, the poor guys, or in coloured shirts and caps. A sea of orange shirts and khaki shorts under carnivalesque tarps (good planning: between spells of awful heat it rained something fierce here last night, thunder and lightning, for a few hours). Suit jackets in orange and black stripes, of variable width, for some; slick sporty rain jackets in black and orange, patterned after the university crest, with class numerals on the back, for others. A few groups in what might have been track suits in orange with black piping, but the impression given was more somewhere between astronaut and construction zone traffic warden. Hats and caps of all styles for the sunny weather, though I didn't see any tiger-themed parasols: not just straw hats or American hats but Texan hats and cloth hats and English bonnets. For the gents, ties, solid orange or striped or a black field with pattern of orange crests or shields with black relief. Handbags in a plurality of styles, if not colours, for the ladies. Cloth belts of orange and black stripes – well, it would be better if no one at all wore those. All this and still more besides, but not all that ornaments is orange.

I saw three kids riding the back of a bronzed Bengal, getting their picture taken. I think I want my picture taken, and you know where.

Lawn signs were posted for the mayoral election in orange and black and white. Actually, signs for the Democratic primary for the mayoral election.

I saw an elderly man on a scooter with an orange pennant flag. No stripes that I could see. You might think that would be brash, but I got the impression that the older alumni were enjoying things rather more than their newer brethren.

Shops with Princeton pennants in the windows, shops with plush tigers reclining on rails, shops with signs offering a fifteen-percent discount to anyone wearing orange and black (meaning they gouge the locals but not the tourists? – counterintuitive.)

An enormous inflated triangle, in orange, half again as tall as I, with black lettering promising "Princeton's Famous Triangle Show!" Elsewhere, balloons, in orange or black, of course, or orange with black strokes, but not as many as you might guess. It may be that the main balloon-using events occurred before I came by, or are yet to come.

And, heavens guard us, I spotted no less than four tiger tails, two apparently attached to people, so that they became human-tiger hybrids. (Science has gone too far, I say.) No, I couldn't quite see how they were being worn. One kid was sitting with his back to me. The first one I saw was trailing a woman as she walked. I did a double-take, but she quickly vanished from sight. It seemed to be hanging from a back left pocket, or possibly from a belt loop on that side.

Quite a show. Makes me want to buy that book. Je suis désolé, but I've no photos to show you. In truth, I was too frightened to try. Well, who can say how a tiger might startle?


Thursday, May 24, 2007

Dandelions: thick as goblin arrows in the sky

This entry goes out to all the weed-haters of the world. Ignore the dubious photoshopping around the fringes; or, for fun (or not), spot where the car used to be (dratted things always getting in the way). For best viewing effect, click through, imagine you live in a castle twenty meters this side of the road, pan across field slowly from left to right while noting differential composition of flower heads and clocks, play music of horns, strings, and male chorus tinged with dread and heroism in the dark, and read passage in the voice of Ian McKellen.

* * * * *

A horde of dandelions approaches. Command?
...they are coming.

They have taken the bridge and the second hall. We have barred the gates but cannot hold them for long. The ground shakes, drums... drums in the deep. We cannot get out. A shadow lurks in the dark. We can not get out... they are coming.

* * * * *

Naturally when researching this entry I also came across the equivalent passage from the book, which perhaps you can try to play with an Ian McKellen synthesiser in your head, if you can figure out what those diacritcal marks mean.
We cannot get out. We cannot get out. They have taken the Bridge and second hall. Frár and Lóni and Náli fell there... went five days ago... the pool is up to the wall at Westgate. The Watcher in the Water took Óin. We cannot get out. The end comes... drums, drums in the deep... they are coming.

* * * * *

I remember one time that I looked down and saw a dandelion in the grass. It was dead, and it looked awful, an organic crater in the ground. I thought: here's an absolutely useless plant. It gets everywhere, doesn't do anything, looks wretched, leaves a mess behind. But they always keep coming back. And I wondered: what's it for? How did it get here? And I knew the answer: a dandelion is a machine for making dandelions. That's enough.

Thursday, May 17, 2007

JoaLDG: no coin flipping in the exam hall!

Leave your calculators and quarters at home: yesterday was the linear algebra final exam. So today it was graded. It took a little under five hours, from nine-thirty to a quarter past two, for maybe a hundred and fifty papers. I had one problem to grade. For five hours. A little numbing, but not so fatiguing as you might think from it being doing math for so long at a stretch: it consumes much more energy to shift to thinking about a new problem than to continue looking at the same kinds of solutions to the same problem. Grading in parallel is easier as well as more consistent.

Actually this one problem was in two parts, a and b, and the two parts didn't have anything to do with one another, so really it is two problems. They were couched as “true or false” questions, but when they add “Explain.” to the end that actually means: “prove or disprove.” Since this isn't the math majors' linear algebra class, it will perhaps be not too surprising to you to read that generally speaking the option taken was “not prove.” I was not unduly burdened by doing “really two problems.” It turned out that the composer of part b badly mistook the difficulty of his problem: the average over the entire class for that problem was about one-and-a-half percent. There were earned a total of twelve points in all those hundred-fifty papers, eleven points from seven hundred and fifty. (After a vast, vast number of zeroes were awarded, after I began to feel that I was back in kindergarten learning fine motor skills by forming the numeral 0 over and over again, well, I started counting.) And of those four people sharing twelve points only one earned full credit, so congratulations to MJA.

This is, I confess, a fair disgrace. (I should mention that part a was rather better done – I wasn't counting but I'd guess the average was somewhere around forty percent, plus or minus ten. So I wasn't just a zero-scribbling bit of broccoli for those hours. Maybe you find that a fair disgrace too, but at least it's a spread.) The other one is really just a waste of everyone's time. I can't blame the problem-poser for this disaster; the problem was merely a great blind-spot for almost everyone.

Something is shameful if it's counter to some law – what law do I mean here, calling this a disgraceful situation? Really just what I said: no spread of grades.

Grading problem sets I am acutely conscious of their existence as a pedagogical tool. Ultimately everything I do is predicated on the need to indicate to the students the deficiencies and successes of their technique. So for example if the problem is to compute some certain numbers or vectors and the bottom line is incorrect and the reason why is exactly that there was an error in carrying out one step of an algorithm a little while earlier then substantial partial credit is earned. This is only natural: I am not grading the answer but rather the mastery of the material and the techniques of the class. This mandates more nuance than up-or-down.

There is, of course, a different point of view, that overly discretised up-or-down view, which we'll call the “either the building is going to stay up or it's going to fall down” vantage. I think we can all agree this is the standard to which we'd like to hold our civil engineers, but it seems a little draconian for first-year math students, and moreover, as I said, unpedagogical. I must even object: what do I care for the right answer? If I wanted the right answer then I'd do the problem myself. And if some miss the nuance in my nuanced feedback, all the worse for them, but I don't mean to pander nor panic.

And yet there is a well-known and popular case of this discrete approach. The Putnam contest for undergraduate math students applies a similar rubric: “not a solution” gets in the range of zero to two points (from ten); “solution” gets in the range of eight to ten. The problems are, to be sure, not easy, but the easiest problem on the Putnam paper isn't so hard that you couldn't manage it in three hours (if you were an energetic young undergraduate math student), and yet the median score, the score which half the contestants don't exceed, is perennially zero or one point, out of one hundred and twenty. (One year the median score was three, and some wag remarked that this was a reflection of how “ridiculously easy” the contest was that year.)

Well! If I graded linear algebra problem sets in this style, I can only think I would quickly discover how expendable I am – just after all but a handful of students from forty drop out in fear of failing the class. And if this seems draconian even in a prize exam, well, in the first place, it is the rule of the game, and everyone comes to play the game. But moreover it has after a fashion a certain logic: every problem in our linear algebra textbook may be all-too-easy for me to solve three different ways – there's a reason I'm qualified to grade this stuff – but if you give an incorrect solution to a problem no one knows how to solve, how can we really say how close you are? Every false theorem is one mistake away from being proven. It's not a hopeless problem to say how near or far a proof is – there's a reason they pay those research mathematicians so well – but the iron prison of the idol called Rigour lets no one free who hasn't really a complete and correct solution.

So when I grade prove-or-disprove problems on a final exam, what am I doing? Really my goal cannot be pedagogical. The students are permitted to look at their papers in the sequel but my understanding is that this is not typically done, and in truth I don't expect it to be otherwise. I sympathise with those who, coming to the end of a difficult course which they have not yet totally arranged into comfortable, familiar parlor room furniture in their head, would rather take a long, quiet, contemplative silence by the koi pond. So I have no belief that I am still teaching anyone anything.

Rather I am evaluating: who has it and who hasn't it? We need our grades to give a spread, to differentiate the students, and form them into some groups, the men and the boys and those in between, linear algebraically speaking. This does inform one's granting of partial credit. For example, it is very far from my mind that a simple response of “true” or “false” should beckon my pen hand to form anything other than a zero. (No doubt you have already guessed this, since it really beggars belief that only four people should have their coins come up tails, even if your name is Rosencrantz.) If there is no evidence that a proof could be in the offing, there is no reason to say that anything good has happened.

This problem of credit for the right single word has been a serious challenge to me: how could it be that any credit at all is earned by an utterly confused answer merely because one key word happened to be correct? Yet at the same time I want to encourage students to make guesses, develop their intuition, and so forth, even if they can't finish it off. After all, a good conjecture is the first step to a theorem. So on problem sets a one-word answer should worth a little bit of partial credit – typically one or two points from five, depending on the depth of the problem in question and the credibility of the notion that the student just didn't want to bother justifying their answer. (That unmathematical practice I strenuously discourage, even to economics students.) But to avoid giving the slightest encouraging word (or numeral) to errant nonsense I have developed what I have been privately (now publicly) calling the egregious weirdness doctrine. (I like to give names to these sorts of things, as you may have gathered from my discussion of the ultimate linear algebraic sin, but typically I'd rather terminologically paint myself in a more temporally-based judicial role.) This doctrine states that if I am considering a range of possible grades for a submission then if there is egregious weirdness present I will err to the low side. To clarify, some silly examples of egregious weirdness; pretend I've added parenthetically “(good grief!)” after every last one of them: claiming a matrix with a row of zeroes is invertible; writing down two vectors that are parallel after applying Gram-Schmidt; citing a theorem's converse despite having disproven it in, well, every single problem for the last two weeks; and so forth. They're the kinds of mistakes that are manifest and impossible for me to believe derive from anything but the coupling of linear algebraic ennui and the sort of deep confusion I called “the fog of the Nothing,” which “roam[s] between the sky and the space which is beyond, fantastic, undreambound,” in the first JoaLDG entry.

If the egregious weirdness doctrine explains how I can give credit for one-word answers on problem sets without seeming to punish people for just “writing a little bit more” (shouldn't it be a principle that writing a few more lines oughtn't worth an answer less credit?), perhaps its dual explains how I can't do the same on the final exam: only if there's something to praise can I say it worths something on the high end.

I just wish we hadn't ended up with a Putnamesque lack of hosannas on that problem.

* * * * *

Though it's the end of linear algebra for the school year, this is not, I should assure the gentle reader, the end of Journal of a Lower-Division Grader, nor the beginning of a hiatus. At my current rate, and at their tendency to bifurcate themselves into two and three parts, I have stories enough percolating on note cards to keep you and me over the summer.

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Saturday, May 12, 2007

JoaLDG: Artin on matrices

Preface. This entry is a first part of two planned about the mathematician Emil Artin. It's a little heavy on the math, (no surprise!), so let's heed in advance some advice we're about to quote and remember to pass gently over the oppressive parts without letting ourselves be burdened by their gravity. Listen to the music and not the song -- or was it the other way around? -- never mind, they're both light and airy.

Emil Artin (1892-1968) was one of our most formidable expositors of mathematics for mathematicians. To give just a most obvious and striking example of this talent, there is a reason why all introductory texts on Galois theory sound the same, and that reason is that they all borrow very, very heavily from Artin's book on the same. Artin on this subject was original: it was he who reformulated the work of Evariste Galois (1811-1832) from a theory of the symmetries of roots of polynomials into a theory of the symmetries of field extensions. Considering how this view now completely dominates it is a little surprising to learn that it was only so recently developed – Artin's book Galois theory was published in 1942, from his lecture notes, being fruit of work from the preceding years.

(But maybe not too strange. A long parenthetical digression giving context could be placed here. Let it suffice to say that even the notion of a quotient group was only formalised in the 1920's, and one can hardly state the “fundamental theorem of Galois theory” as we know it today without understanding group-theoretically what a normal subgroup is.)

I don't know what I'm doing in the fall but there's a certain chance I'll be teaching a section of this linear algebra class I've been grading – and I surely will be doing so sometime before I graduate – so some things Artin has written, and one passage in particular which I'll quote presently, have been a little on my mind. How do you tell people about linear algebra? At heart all I can answer is: Really, the same way as you do for anything else. Karl Jaspers thought that the problem of communication was one of the fundamental problems of philosophy. But we needn't feel abstractly pessimistic or overburdened: there are plenty of fundamental problems we manage willy-nilly to cope with every day. We have twenty thousand purely practical facts to draw on. And in this case, one of them is Artin's legacy.

Artin's book Geometric algebra is curiously organised: he deposits in the first chapter, prior to the main subjects of the book, all the external tools and apparatuses he'll need in the sequel. ("Curiously?" Well, normal people would put this in an appendix.) In the very thoughtful short preface labelled “Suggestions for the use of this book,” he explains:
The most important point to keep in mind is the fact that Chapter I should be used mainly as a reference chapter for the proofs of certain isolated algebraic theorems. These proofs have been collected so as not to interrupt the main line of thought in later chapters.
He goes on to say that “the inexperienced reader should start right away with Chapter II,” which to me reads like an agreement that Chapter I ought to be adjacent to the other cover. (Is he saying that the experienced reader shouldn't start right away with Chapter II?) He continues on, to give some of the best advice possible for reading mathematics, namely,
This skipping [of “a few harder algebraic theorems” in “a first reading”] is another important point. It should be done whenever a proof seems too hard or whenever a theorem or a whole paragraph does not appeal [!] to the reader. In most cases he will be able to go on and later on he may return to the parts which were skipped.*
Probably the students will object: there is hardly time for all this, to first skip and then to come back. Perhaps so. It is certainly unrealistic to think that the students will understand something that's unlike anything they've ever seen before in time to give clear and concise solutions on the weekly problem sets. But by the end of the class there should no longer be any mystery about the material of the first week, and the gap between the end of lectures and the beginning of the exam period (in Princeton's fall term, this is a gap of a whole month!) is enough time to start to put the entire course into perspective. On this time scale, the advice is not only reasonable, it is the only sound thing to do, if one operates according to the principle that no one ever learned a thing the first time he saw it. (Well, how could he?)

I want to convince you that Artin is super-cool. For that purpose there is at bottom only one thing to do: namely, show you that he's a rebel. A wild, wild rebel. Just thirteen pages into this book, not far into his appendix-at-the-beginning**, he has stated a theorem whose content is that when you fix a basis (of some vector space -- you're skipping that link, right?) there's a correspondence (“isomorphism,” in the vernacular) between linear transformations and matrices, (and change of the choice of basis corresponds to conjugation of matrices). He goes into a lamentation/screed for a page and a half, (emphasis added):
Mathematical education is still suffering from the enthusiasm which the discovery of this isomorphism has aroused. The result has been that geometry was eliminated and replaced by computations. Instead of the intuitive maps of a space preserving addition and multiplication by scalars (these maps have an immediate geometric meaning), matrices have been introduced. From the innumerable absurdities – from a pedagogical point of view – let me point out one example and contrast it with the direction description:

Matrix method: A product of a matrix A and a vector X (which is then an n-tuple of numbers) is defined; it is also a vector. Now the poor student has to swallow the following definition: A vector X is called an eigenvector if a number λ exists such that AX = λ X. Going through the formalism, the characteristic equation, one then ends up with theorems like: If a matrix A has n distinct eigenvalues, then a matrix S can be found such that S-1AS is a diagonal matrix. The student will of course learn all this since he will fail the course if he does not.

Instead one should argue like this: Given a linear transformation f of the space V into itself, does there exist a line which is kept fixed by f? In order to include the eigenvalue 0 one should then modify the question by asking whether a line is mapped into itself. This means of course for a vector X spanning the line that f(X) = λ X. Having thus motivated the problem, the matrix A describing f will enter only for a moment for the actual computation of λ. It should disappear again. Then one proves all the customary theorems without ever talking of matrices and asks the question: Suppose we can find a basis of V which consists of eigenvectors; what does this imply for the geometric description of f? Well, the space is stretched in the various directions of the basis by factors which are the eigenvalues. Only then does one ask what this means for the description of f by a matrix in terms of this basis. We have obviously the diagonal form.

I should of course soften my reproach since books have appeared lately which stress this point of view so that improvements are to be expected.

It is my experience that proofs involving matrices can be shortened by 50% if one throws the matrices out. Sometimes it cannot be done; sometimes a determinant must be computed.
He then re-enters the stream of the exposition. “Talking of determinants,” he says, “we assume that the reader is familiar with them.” And we're off.

But, by the by, and coming back to our concrete problem, is that prophecy correct, that future books will move toward Artin's geometrical view? I have a vast number of linear algebra books, and on inspection it's not so rosy. Some do exactly what Artin decries, without any apparent shame. Some even make linear transformations into second-class objects by introducing hideous notation for the matrix of a transformation with respect to such-and-such bases (not necessarily the same basis for the input as the output – good grief!). Some try to “motivate” the problem with differential equations, which from the point of view of an engineer may not be so ridiculous as it seems to us on the face of it (and anyway systems of first-order linear DE's are a classic application in such a course). Bretscher, the book they use here, is actually not so bad. It claims to emphasise geometry, and seems to do so pretty well, for the level of the class.

But there's the rub: Artin's approach is too difficult to teach to the students and still expect them to also master the matrix material they'll need to, you know, actually do some problems and not fail the course. In the end it is not a way to teach “something that's unlike anything [the students have] ever seen before,” because its emphasis on geometric character presupposes some geometric intuition to which one can appeal – in other words, some underlying familiarity not necessarily with linear algebra but, absent that, with some other and really more difficult mathematics. Bretscher's compromise, and it seems a reasonable one to me, is to give examples of matrices with special geometric meanings, transformations we've seen before (rotations, reflections, projections), and ask what their eigenvectors are. The students should be able to answer right from the geometry they already understand, without ever writing down a matrix, (although they could write it down if they wanted to, or felt the need to).

This balance between the conceptual and the formal the would-be instructor must maintain with care and deliberation. Maybe I won't stake my infant career on Artin's throwaway comments. I could just photocopy that page as a handout. If that handout wouldn't confuse anyone. But in that case I can always give it out to them, on my didactic authority if they don't feel comfortable judging it for themselves, that it's all right to skip it.


* I am reminded of an English teacher from high school who wondered why it was, or how it came to be, that everyone thinks they should read a book by starting from page one and continuing to read page-by-page. Clearly there are many more ways to read the book, although most make no sense. Probably this habit is out of respect to the author, who presumably (though this belief is often well-characterised by the negative, skeptical connotation of "presumption") has put his industry and his learning into crafting a well-structured book. A reader inexperienced in some subject hasn't necessarily the knowledge to know what parts he needs to read to do whatever. But if all this is so it merely makes us wonder instead (1) why a well-structured book means a linearly-structured book; and (2) why more authors couldn't write such helpful "suggestions for the use of this book."

It is a custom, I should mention, in many corners of the textbook world to outline a couple of different options for the use of the book in a one- or two-semester class: cover these chapters but not those sections, and so forth.

** We need a good archaeologism, but for that one needs good Latin. “Precedix” is tempting, coming fairly directly from Latin “praecedere,” but doesn't carry quite the right meaning: it is “a thing coming before,” whereas Artin's appendix-at-the-front is more like elementary material. We could try “fundix,” from fundere, (cognate with to found: “found a city,” “a foundation,” and such), but it sounds ridiculous. Maybe “precedix” is better; after all, an appendix in English doesn't literally mean “a thing hanging on,” either.

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Friday, May 04, 2007

Nachtblogging: "deeper than day had been aware"

I've been collecting night-time photos for a little while, since last summer, casually, whenever the fancy struck; and for a few weeks I've had it in mind to start posting some. At first it was going to be just one post, but I think I have too many of them for that. And besides, I tipped my hand with the last entry. So here's a first stab at it.

Each picture below is a thumbnail; click through for more.

* * * * *

The other day I was returning from my after-dinner walk to pick up this week's problem sets when I happened to look down.

Some flowers in grass at night

Just some flowers? This is a long-exposure shot (one second); I'm sitting on the stone path behind our viewpoint and using my bag to try to stabilise the camera. If you click through, you can see I wasn't wholly successful. On first appearance you can't really see too well what first caught my eye, which was the grouping of the purple bulbs around the one orange-red bulb. The blues are a little too strong to let the purple flowers stand out -- actually, the ones on the left are well-hidden. In the click-through the bulbs are a bit easier to pick out. But whatever my original intent was, this scene has taken on its own character.

This shot summarises the palette that attracts me to night-time settings: forest greens, deep blues, and burning reds. The set is suffused with an almost unholy dim glow, and the long exposure yields powerful juxtapositions of light and dark. The backlight brings up the gothic architecture beautifully, the foreground flower is curiously emphasised, and the whole thing takes on a dramatic energy. In short, it's interesting: no, this isn't a picture of flowers; it's a picture about flowers.

Compare a roughly equivalent day-time shot:

Some flowers in grass at day

This is the same plot, (a few days later, when I decided I wanted a comparison shot to show you: if you look carefully you can see the flowers are noticeably further along in bloom). The click-through is not as big and I haven't tried to crop it for composition. Nonetheless I think it's clear that even if the photo doesn't outright fail it is at best "just some flowers." The only thing close to interest is the upper-right corner, where the grass and the jagged shadow meet the wall. We can try to rescue it, like so:

Some grass and flowers by a wall at day

You can decide for yourself if you think this merits being called a "rescue." It has a few merits, yet at best "it is what it is," and that is nothing close to the evening scene.

Let me share something curious from the same set. It's a bit nerve-wrecking taking these photos at night because of course I don't know until I get home whether the photo is bright enough, or too blurred, or whatever other failures might have happened -- the camera's LCD is hardly good enough to tell me this, especially when the ambient no-light makes it difficult to judge the brightness of the image! When that happens I can (short of trying to retake the photo -- not always possible) only hope for a software solution.

The software I'm using has a button suspiciously labelled "Photo fix." This runs a half-dozen or so algorithms to try to correct some problems -- colour balance, contrast, saturation, others. (You can also run these algorithms individually at a strength you specify, in case you actually know what you're doing.) The result of running this thing, as you might guess, is usually more entertaining than usable: although it often does a good job of identifying perceptual objects in the frame that are fairly-well washed out to mortal eyes, the end product of this process is typically pretty far-removed from reality. I would be very reluctant to use such a thing unless I needed an illustrative photo and all of mine were useless.

However, sometimes it can surprise you.

Some flowers by a wall at faux-day

This came from a failed (too dark) shot of these flowers by running this "photo fix" algorithm. As a photograph, this obviously fails. The colours are wrong, too washed out; close-up, everything looks grainy, like it had been taken with a high-ISO film; and it's also blurred -- probably a consequence of the smoothing algorithm rather than my unsteadiness. Yet despite all this the effect is not altogether awful if I forget that it came from my camera and instead imagine it came from some novice impressionist painter's workshop. Looking at the click-through, the colours at the interface of the wall and the garden are still not good, but move away to center on the red flower, with just the green surrounding it:

A painted flower?

Very striking! Call it found art.

* * * * *

This night-time photo series gets its own tag, "Deeper than day had been aware." There's an explanation behind this which is too long to give in entirety on the "About labels" page, so I'll take a page from what I did before there were labels and introduce it here, on the second entry under this tag (I retconned the previous entry into this grouping).

The quotation is from a poem in Nietzsche's Zarathustra:
I was asleep --
From a deep dream I woke and swear:
The world is deep,
Deeper than day had been aware.
As always, the translation is Walter Kaufmann's. (The other day I was in the bookstore [ahem] and a new translation caught my eye -- I might pick it up one day -- aren't you proud of me, that I didn't the first time I saw it? -- but wait, maybe it won't be there when I go back!) In the original, it is:
Ich schlief, ich schlief --
Aus tiefem Traum bin ich erwacht:
Die Welt is tief,
Und tiefer als der Tag gedacht.
Kaufmann's translation is obviously fairly literal, but I like the fact that "Und tiefer als der Tag gedacht" sounds every bit as good as "Deeper than day had been aware."

This is a quatrain in a song that occurs several times in Zarathustra. It is, I would say, a rather important poem to the book. Unfortunately, I can't quite tell you its name. It first appears in Book III under the title "The other dancing song" ("Das endere Tanzlied") and in Book IV under the title "The drunken song" ("Das Nachtwandler-lied" -- more on that later). On the other hand, in that latter setting Zarathustra introduces it like so, in Section 12, after quoting pieces of it in the previous eleven parts:
Have you learned my song? Have you guessed its intent? Well then, you higher men, sing me now my round. Now you yourselves sing me the song whose name is "Once More" and whose meaning is "into all eternity" -- sing, you higher men, Zarathustra's round!
In the original, that last sentence is:
Singt mir nun selber das Lied, dess Name ist "Noch ein mal," dess Sinn ist "in alle Ewigkeit," singt, ihr hoeheren menschen, Zarathustra's Rundgesang!"
These German originals, by the way, are courtesy of Project Gutenberg's e-book. The declared name and meaning are given literally in Kaufmann's translation; a "Rundgesang" is a kind of chorus song (in the sense of a circle of people singing -- "runde" is cognate with our "round").

So what about that "Drunken song"? It will surely not surprise you to hear that this isn't exactly what "Das Nachtwandler-lied" means. A wandlung is a change or transformation (cognate with German "wandern," same as our "to wander"; so a change in the sense of a wandering away from the original), but according to my dictionary (thank you!) it also has a meaning in the German Ecclesiastical tradition -- it refers to the transubstantiation of Christ! Since Zarathustra is filled from cover to cover with Biblical allusions, it is not a difficult guess to make that this is the meaning intended. So I might guess at a rather more literal translation: "The night-consecrating song." All this just demands the question: what did Kaufmann have in mind?

Of course, "Night-consecrating song" is pretty good for our purposes here, too, even if it does miss some meaning there.

By the by, Mahler set this song-of-indeterminate-name to music in his Third Symphony, (Fourth Movement). The symphony is good but the movement in question is just eight minutes so I can't recommend it on that basis alone. Mahler's certainly isn't a drinking song, but a fairly ethereal piece with a light instrumental accompaniment (horn and clarinet solos with strings) to a soprano.

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Tuesday, May 01, 2007

Tech envy

There's been a thunderstorm going on here for the last few hours, with a lot of lightning casting blue across the sky. There was even something pretty close once, a few hours ago -- close enough that I heard thunder not as Zeus' far-off rumbling but as the angry, stabbing cry we best know from sound-effects shops. (That was a little creepy. I got a little surge protector from Belkin when I found myself running out of outlets but I'd rather not find out exactly what the asterisk next to the "ten thousand dollar guarantee!" was referring to.)

Anyway, having had a long-standing affinity for the mystery and romance and beautiful colours of the night, I thought I should try my luck at getting a nice blue-backlit shot of the tree outside my window. Alas, predictably it came to nothing. "If only I had a camera with a massive lens," I thought, "and a ten-frame-per-second continuous drive mode. Then I could cast my net wide as Orion's bow and gather up all the colourful shells of the sea."

But, no, there's nothing to do but sigh and think nighttime thoughts, about a future of digital rebellion against analog transience.


Saturday, April 28, 2007

The Future of Credibility: “everything is what it is because it got that way”

It's Pandaemonium. I came to praise it, not to bury it.

In his tendentiously-titled Consciousness Explained, Daniel Dennett suggests an outline of a theory of the generation of speech. His outline is expressly directed against the classical notion that when I say something it is because I mean it. More precisely, he is arguing against the claim that there is necessarily within me a Central Meaner, perhaps that being myself or else a kind of homunculus, who holds safe the semantic meanings that I intend, and against which all my attempts to frame utterances are compared for semantic fidelity.

Imagine a great horde of senseless daemons, he says to us, rather than a single central homunculus, each with a phrase or piece of a phrase to suggest. These pieces are mere “found objects” and most are senseless or irrelevant. Irrelevant to what? -- to some kind of goal being held, but, crucially, not necessarily a semantic goal. (Dennett gives an example caricature of a person responding to hostility, starting with the angry flush “Go on the offensive!” and passing through “Cast aspersions on some aspect of his body!” on the way to “Say: 'Your feet are too big!'”. He assures us that we can then go home and curse ourselves for not having thought of a wittier retort.) All these “word-daemons” compete to put their mark on each others' candidates for a verbal utterance, and the stream of language they generate is adjudicated over, “yes or no”-style, by a horde of equally senseless “content-daemons.” This chaos does not end when the daemons together have assembled something that meets a Central Meaner's review. Rather, the putative semantic intention is itself modified by the judgments of the content-daemons upon the candidate utterances of the word-daemons.* The intention is “drawn” through an abstract semantic space toward the candidates, just as the candidates are drawn to the intention. What one has is not a kind of bureaucracy but (in Dennett's terminology) Pandaemonium, and it is “a process that is largely undesigned and opportunistic.”

This, if I may remark, is not a model that would have been taken too seriously as recently as the mid-nineteenth century. It is a thoroughly post-Darwinian conception, this supposition that all this chaotic variation can with only subtle environment constraints and interactions nonetheless construct something complex, something with significance, an utterance with a Meaning. Dennett directs this schematic model against the suggestion that the need for a Central Meaner would give necessity to the opposing theory of consciousness that he calls the Cartesian Theatre, (which is, roughly speaking, the idea that there is a single [physical or abstract] place where consciousness “all comes together”). In doing so he links our naïve sense that there must be a Cartesian Theatre to explain our experiences with our naïve presumption that there must be an agent's design behind anything complicated. There is indeed a commonality between them: both are a kind of turtle-stacking. (Check out the graphic on that page!)

We therefore stand properly advised that in our new century we will brook no contention anywhere that sophistication mandates design. To now pass instantly from the ontologically profound to the perhaps merely interesting, and to keep up to the promise of the title and opening paragraph of this post, we in particular note that in the new society of the Internet there is no compulsion to subscribe to a Central Content Provider.

This is a change from the old model, the model of cable television and publishing houses, where the expense of broadcasting mandated a few powerful players. To be sure, Peter Mansbridge isn't going anywhere, but there's a reason why he's now reading out viewer e-mails on the air each night. It's spelled out in this article, which was first published in a magazine and only later reprinted in the author's blog – so don't hold it against it, that it didn't come from a Central Content Provider, that in fact you (maybe) first heard about it from my pointing it out.

For this is the basis for our Internet Pandaemonium: if you know me and trust me, if I have credibility, then my endorsement of content brings that content to your attention. If you agree it's view-worthy, you refer your own close neighbours to it. If not, it need not branch any more along that path, but remember that my other readers might feel differently. The more it's passed around, the more credence and significance it gains in the greater community: some memes gain tremendous attention, or tremendous notoriety, as any longtime Internet reader can verify. As for where that content came from before -- maybe I found it purely by chance, or from another blogger whom I read, who may himself just be another citizen marking out prose or maybe someone with a mission to find selected or special content, like a museum curator, or from a newspaper or other kind of dedicated, professional content provider. (As exciting as our new century is, we would be too reckless by half to lose our professional content providers: they still deserve our respect, even if their roles are changing.) In this particular case it's all-of-the-above: the article I just linked to isn't the link I followed to that blog. This fellow's blog is in my queue pending final decision on whether I should bookmark – add him to my trusted content providers. The ultimate origin of the article, for our purposes, is the author's blog, (strictly speaking, this origin is in the sense that “eukaryotes come from prokaryotes” rather than “eukaryotes come from primordial soup”, to anticipate parenthetically some ideas from the sequel); but every blogger is an author waiting for his work to be cited and to come to prominence, possibly as part of a greater content-complex (an organelle in a eukaryote). It doesn't happen often: the price, the scarce resource that demands differential survival (speaking so as to continue to anticipate), as always comes from the opportunity cost of a person's limited attention-bandwidth for consuming media. So most bits of content don't achieve total renown, but rather roam and hover between heaven and earth. (Mostly around earth: renown like flight is very expensive to support.)

Now let me make good on those anticipations. I used that word -- memes. If you've never heard it, maybe as always the Wikipedia summary can be helpful. In short, an idea is a meme, or more precisely an idea at the size that can be replicated. It might be as small as an idiom of language or the refrain of a song or as big as the text of a book. This is in analogy to the way that a gene is some span of codons in DNA, not of fixed length but rather defined in terms of its being able to be selected for or against. A gene is supposed to do something meaningful to the phenotype, which is either fit or not according to the environment, and this by backward-translation is a selection pressure against the various genes in the gene pool. The notion of a meme is, I believe, an idea whose time has come. Of the increasing number of contemporary popular works on the subject I can't sanction or sanction any, since I haven't read them. I can, however, (not to use my pulpit to promote too many more books) both on general and specific principles recommend Richard Dawkins' The Selfish Gene, which to my mathematician's eye reads not so much as a book about biology as an extended worked example to support a nearly-axiomatic theory about differential survival of replicators. (This description is not meant to put you off it: that's good praise!) The memes, the so-called New Replicators that “live” not in space but in peoples' minds (hence why they're new – not so long ago there weren't any minds), are first introduced by that name in Dawkins' 1976 book, although I imagine the notion was anticipated by other authors previously.

With the meme as part of our vocabulary, more or less everything we've described so far today takes a single shape. They're all stages of memetic evolution.

And while we're here, what about that Wikipedia? I cited it so casually to give a reference to memes, but isn't that where Wikipedia itself came from – some kind of memetic evolution? Take a listen to this talk by the Wikipedian progenitor himself and decide whether it sounds like the same thing. Not to be tendentious myself but I think it does: each Wikipedia article is itself a meme complex adapted under the pressure of its editors and its editors' minds – that last meaning, perhaps, the memes living in the editors' minds? So a Wikipedia article is a complex adapted to the memetic environment that the editorial community represents. If you think the situation is disqualified because the editors are agents with intentions, remember the parable of the Central Meaner: those intents themselves are memes, or perhaps more precisely certain products of memes (in the sense that the phenotype of an animal is a product of its genotype).

This is encouraging in the sense that it seems this is an example of memetic evolution, but, to get back to the original objection of the last paragraph, if we were inclined to be suspicious of the memes and the prospects for memetic evolution, the fact of a Wikipedia citation about them is not going to be terribly convincing. The mere fact of the citation is mere question-begging! On the contrary, it is good and well, we imagine, to say that animals rose from natural selection; but why is it that we suppose the conditions are present in our world of discourse to make possible a memetic evolution by “natural” selection, even if we believe (“in our new century”) that this is possible in principle?

For if the conditions are not there, a Wikipedia citation must surely be a deeply suspicious thing. Supposing that daemons can make wisdom just by nattering suggestions is no better than saying that monkeys can write Hamlet given typewriters nor than saying a mammal came about by accident. Even Plato knew, despite not having Darwin's idea as a counterexample, that sophistication does not imply an agent's design; but chance alone does not gain sophistication without selection pressures causing differential survival. (The “nonrandom survival of randomly varying replicators,” in Dawkins' one-sentence summary.) So rose the animals, and so rise the meme complexes – supposedly. But is there or can there be really such a thing as the so-called “wisdom of crowds”?

What, for example, to make of such a famous experiment as Kasparov v. The World? Let me recall the circumstances of this event to you. In Summer 1999, Microsoft sponsored a chess game played (at correspondence time control, about one move per day) on their website. Garry Kasparov, recently having regained his status as the invincible champion with spectacular triumphs at Wijk an Zee and Linares in the first quarter of the year, commanded the White pieces, (with the assistance of his usual seconds). Captaincy of the opposing Black forces was given to – everyone who showed up: any person could log on to the MSN Gaming Zone and submit a vote for their team's play. The move with a plurality of votes would be the one made. The match was not so uneven as it sounds: four strong junior players were enlisted to provide brief recommendations to The World team, and still more strong players volunteered, including Alexander Khalifman, who went to Las Vegas in Fall 1999, while the game was still being played, and there won the FIDE World Championship. (At this time, the world championship was still a divided title, and Kasparov refused to play in FIDE's events.) Or perhaps after I, and Dennett and Darwin, praise Pandaemonium you might think the game unfairly balanced in the other direction?

No, it was not so. After several months of tough play, during which at times all three outcomes (win, lose, draw) seemed equally plausible, Kasparov was victorious, (in a queen-and-pawn endgame). Is this the logical triumph of expertise? I would be quick, in my Platonic prejudice, to think it so: understanding trumps grasping because that's what understanding is. Similar one-versus-many contests have been held, albeit with less fanfare and publicity, and they have often been decided for the expert and not the putative wisdom of the mob, but far from always. More recently, Arno Nickel, who holds the ICCF (international correspondence chess federation's) grandmaster title, bested the computer Hydra, by a score 2.5-0.5, in correspondence games; and this unthinking computing beast the same creature of awful power that beat Mickey Adams 5.5-0.5 over the board! But Nickel too has lost a game against The World, in the form of the community on the website ChessGames.com; it concluded in January of this year.

So what to make of all this? In a phrase, it's the difference between a mob and a crowd. Fickle folly can be incendiary, but it knows not how to aim itself nor cares what it burns. If a man leads a mob, it is at most as wise as he, and often less. Irina Krush, one of the World Team coaches Microsoft hired in the match against Kasparov, expressed her displeasure when some of her mob's decisions decided against her and Khalifman. A mob can beat Kasparov if Khalifman can. After it gave such a tremendous fight it is ungenerous, to say a minimum, to put that World Team closer to the monkeys than to Shakespeare, but one idly wonders if simply the presence of better tools, like a ChessBase wiki to replace the primitive forum design of the MSN Gaming Zone, might have made the difference.** Anyway, it's quite possible that to beat Nickel in correspondence games is more impressive than to beat Kasparov, so the proper tools have surely been developed. But the lesson is clear: It is the networking of the crowd that sets the stage for the miracle of memetic evolution, and not the noisy, violent hub-bub of the mob.

And, to get back to our Wikipedia problem after an extended parable, a fringe article is only as good as its first and only editor-progenitor, but a well-eyed treatise is a tremendous thing – as long as Wikipedia itself is properly built to encourage the right selection pressures. Remembering the talk, the last pieces come into place. You have administrators, (Dennett's content-daemons, the judicial counterparts to the word-daemons striving to craft prose), and they, of course, don't need to be experts in the article subject areas, no, not in anything but encyclopaedia mediation, -- and then, more or less, the “undesigned, opportunistic” process of evolution can start. Alexander Khalifman is a tremendous “chess expert,” but that's a word-daemon: he isn't on the right ontological level to exploit a Pandaemonium to challenge Garry Kasparov.

After so much chatter, let us remind ourselves that – it is a fait accompli; Pandaemonium is already here! It came in the form of the blogosphere and YouTube and other networking sites. (Pre-Internet models suffer from limited connectivity and content distribution – somewhat deficient for good examples of a functioning Pandaemonium, and so every film studio executive dreams of that rare word-of-mouth buzz.) Bloggers write and post content and link to each other; YouTube users have schematic personal pages on which they can, in addition to posting their own videos, link to particular other videos or to other users' personal pages. This is the minimum requirement. It doesn't have to be one or two sites in particular, and may not be in future. Indeed the Internet crowd can sometimes be fickle in its endorsement (the various blog-hosting services all rise, compete, and fall among themselves). What's important is just the structure on which the community builds itself, and any sufficiently self-connecting framework could do. From there we little daemons, simultaneously both word- and content-, we take care of things themselves.

And quite well, too. So ends my speech of praise.


* For example, consider the "seductive turn of phrase." Where did the title "The future of credibility" for this entry come from, anyway? It's apparently a play against an earlier entry entitled "The end of credibility," (which, if you're looking for it, was the last of "Three Short Comments from Princeton"). Is this really the best I can do? The two entries don't seem to have a lot to do with each other, other than being about blogging. Maybe they're both about content distribution, from two different sides. Maybe it's just a weak play on words. But it has such a hold that the word "credibility" even gets a mention a little later, in the middle of a very important paragraph. There's supposedly an abstract semantic idea that this paragraph is to communicate, and that idea is independent of its instantiation in text, but in particular the "trust metaphor" to explain the low-level links between blogs is directly connected to this curious title. Can I say which came first in my thinking? For surely the meaning of the text would not be quite the same with a different metaphor.

In other examples, a "seductive turn of phrase" becomes its own justification, divorced from any external semantic concept. An easy way for this to happen is from grammatical ambiguity, say due to excessive editting for style over content.

Incidently, the titular quotation, "Everything is what it is because it got that way," is from D'Arcy Thompson's "On Growth and Form." (Peter Medawar called this "the finest work of literature in all the annals of science that have been recorded in the English tongue." I haven't read it. It's about 1200 pages, which is rather longer than this post.) It is a little more obviously connected to the technical side of the subject matter.

** Everyone who has ever spent a little time slumming on a forum whose theme yields to certain frequently-asked-questions knows that Sisyphus himself hadn't such a futile task. I have examples in mind, but they're such dangerously contagious and stultifying memes I don't dare quote them – they're the memetic equivalent of the flu.

Postscript. Sorry about all the broken links; I've fixed them. They were there because I was composing this entry in a word processor (for the obvious reason) which was automatically turning all the quotation marks into smart-quotes (that angle toward text, like “...”) rather than the (uniformly-oriented, like "...") quotation marks that one needs to put in HTML tags.

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