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.

* * * * *

...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.

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.

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.


Endnotes.

* 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.

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.

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:

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:

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.

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:

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.

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.