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.


(Endnotes.)

* 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.

Lost Blogging: from before there were blogs

Going through some old papers today I found this quotation written out on a scrap. I'm certain, from my weak recollection and from the strata in which I found it, that the paper well predates our journal here, so we understand from the first the title 'Lost Blogging' in a metaphorical or, even, existential sense. I was sure the quotation (which was unattributed) is from Aristotle, according to my recollection and to the style of writing, and, behold, I found it in the Ethics, at the place designated X.9.12, under the editorial heading "From Ethics to Politics: Moral Education." Aristotle is talking about how to ensure that people are "finely brought up and habituated" in order to be "someone who is good," and his thesis is that one must study politics to this end.

A father's instructions lack this power to prevail and compel; and so in general do the instructions of an individual man, unless he is a king or something like that. Law, however, has the power that compels; and law is reason that proceeds from a sort of prudence and understanding. Besides, people become hostile to an individual human being who opposes their impulses, even if he is correct in opposing them, whereas a law's prescription of what is decent is not burdensome.

Presumably one understands the first sentence to be speaking of the case when the child is not a prince or something like that.


And, under the same title, a maybe-related addition to our pantheon of plush or graven idols, icons, and hangers-on, a Hawaiian war deity:

Kū-ka-ili-moku, "Kū the seizer of land."

Tigers as capitalists

I was in the local university store, which is a kind of hybrid of a book/textbook shop and dorm appliance outlet and, well, other things, with my imaginary interlocuter.

"What," he said, "are these for?"

I verbalised a grunt. "What's what?"

"These," he said, and I saw him gesturing slightly, as though he felt he was about to be bitten by some rough texture, or that someone, maybe a salesperson, would see his shock-stopped interest.

"These," I said, gently picking one out, struggling in time pressure for the mots justes, "are for showing everyone how much you like tigers."

"By attaching a tiger's tail to yourself?"

"Yes," said I, "or maybe for twirling about at a football game."

"Only if our team wins. And -- good heaven!" he exclaimed on lifting his sight-line from the hypnotic illusion of writhing orange and black. "What is all that for?"

"Those," I said, "are table accents. They're for giving a bit of interest to your Ikea products."

"And the big ones?"

"For Afghan rugs? Are there tigers in Afghanistan? Well, near them, anyway. Maybe they're for people who don't want to feed cats, they can have these just lazing around, the mighty predator in your home, or if you have a kind of lobby or cloakroom at the front of your home you could put them here -- "

"And are those backpacks on the far wall? Good heavens, I wouldn't want to be that kid!"

I was becoming anxious. "Yes, you know, I think they are. Hey, didn't you want to see the books? Let's go downstairs and take a look."

My imaginary interlocuter can hardly resist looking at books. "I always say I won't buy any," he quipped in a quick about-face, "and then it's Kaufmann this and Heidegger that and, oh, look, one volume on the philosophy of mathematics, that would be a nice resource, and while I'm getting something anyway I've been meaning to read Aurelius' book and let's see what's new and cool in the world of Plato scholarship or what they're saying Nietzsche was arguing for this month and I walk out with three or four or five of them and -- will you look at that?!"

I hardly had a choice.

He had stopped in mid-stair, right foot forward a half-step, right hand suddenly tight on the rail. "How did that happen? What's with these people?"

What could I say? but that Princeton tigers paint it orange and black.

Metablogging Redux: About labels

Nota bene: I am now using this post in its capacity as a page-in-the-web, and leaving only the skeleton of the journal entry that was here. So things will be added, deleted, and changed to keep its new functionality. Pay it no mind: as I wrote in the original draft, "it turns out that you can make your chronometer an organ of goodfact around here," referring to the fact that I can give a post an arbitrary timestamp. Please let me know via comment if any links are broken or things like that.

The new purpose of this page is to explain the newly-implemented system of post tags. If you've never wanted to find an earlier entry but couldn't remember its date, this can't be of the slightest interest to you.

* * * * *

"I'm trying out the latest Google-given gadget," I wrote in the original draft, "the blog post label. They're tags you give to your weblog entries, like themes, that let readers easily find or catch up on the history of a recurring thought by using a built-in search for the tag. Since I'm very much for being able to have a discussion that lasts more than one post and doesn't have to constantly recap what came before, I've decided I need to embrace this little classifying tool." These tags show up at the bottom of each post, after the word "Labels:", and before the signature/timestamp line.

Here's a current list of the tags in use at Sun-Drunken. I break them into two classes. The first group is roughly "things that make posts," and consists, in alphabetical order, of:

• Day by day, a catch-all category for posts and reports about things that just happened to happen some day. These posts are the journal side of Sun-Drunken, rather than the soapbox side.
  
• Deeper than day had been aware, a series about the night and night-time photography. The label is a line in a poem of Nietzsche. See the second entry in this category for an introduction, and an explanation of the label's meaning.
  
• JoaLDG, "Journal of a Lower-Division Grader." These are the entries about my experiences grading linear algebra papers and my contemporaneous thinking about pedagogy. See the earliest post in this category for an introduction.
  
• KUCA, the Lord Kelvin Useless Creation Award, an annual (so far) citation of a person or group whose continued survival has never been of the slightest use to any creature. See the earliest post in this category for an introduction.
  
• Lost Blogging, a quasi-behind-the-scenes series in which fragmentary posts that got lost or just never came together are given for what they are. See the earliest post in this category for an introduction.
  
• Metablogging, mainly posts of an administrative about the weblog, and not blogging-about-blogging. That typically falls under "Shameless whinging" or "Wordsmithing," depending on the tone of the article.
  
• Photoblogging, a series of posts built around annotated photographs. This is a catch-all category for the heavily photographic posts, and they have no other common identity, although in most cases they are pictoral tours of someplace I went or thing I did.
  
• Schachblogging, posts about chess or involving chess. (Schach is German.)
  
• Tigers Paint it Orange, a record of Princeton University's fascination that became Sun-Drunken's as well, the tiger. Also includes a few other bits of Princetoniana that made it to these pages.
  
• An Underground Den, a series about the places I live in. Photoblogs about my rooms fall here, as do stories about locks. The full quotation is "Behold! human beings living in an underground den," the words Socrates uses to introduce the famous allegory of the cave in Plato's Republic.

The second group of tags is roughly of themes that sometimes occur in parts of posts, and consists in alphabetical order of:

• Allzumenschlich, signifying instances of human failure or by extension general moods of existential brooding. Allzumenschlich is usually rendered in English as "all-too-human." This is the melancholy counterpart to "Man is something that shall be overcome."
  
• Man is something that shall be overcome, signifying things about or exemplifying our Sun-Drunken spirit of "Nietzschean optimism." This is the hopeful counterpart to "Allzumenschlich." Indeed, the tag is a direct quotation of Walter Kaufmann's translation of Zarathustra's "highest hope."
  
• Shameless Whinging, which is just that, because it happens to everyone now and then.
  
• They Should Have Sent a Poet, meaning usually just what it says: some image, thought, or other bit of phenomenology that mandates an articulate sense of wonder. Sometimes this tag is used ironically, as in, "they should have ... instead of me." This quotation is from the film (book?) Contact, when Ellie sees during her voyage cosmic conjunctions she likens to visual poetry.
  
• Three Thousand Years, that being about the duration of human history to date, (not counting highly scattered records of earlier times). Sometimes these posts are literally about history, but more often they are about the story of man, our place in history, and especially the history of thought. The full quotation is from Goethe, "He who cannot draw on three thousand years is living hand-to-mouth."
  
• Truly You Have a Dizzying Irony, or, "shameful (as in self-conscious) whinging." This tag is to be read in the voice of Cary Elwes.
  
• Wir mussen wissen, wir werden wissen, indicating mathematical content, these being famous words of David Hilbert. They mean, "We must know, we will know," referring to his belief that "there are no absolutely unsolvable problems." He said this in a 1930 radio address, which in fact was recorded, so you can listen to this mathematical titan say them in his own voice: just listen.
  
• Wordsmithing, or, writing about writing.
  

Please note that some posts are unlabelled. Usually these are the "Into the West" and "Toward the Rising Sun" posts that signal I'm about to shift locations. A few others I was inclined to leave unlabelled for other reasons: sometimes because they don't seem to fit any of our tags very well, sometimes because I'd just as soon leave the entry in a more obscure position.

* * * * *

Let me close on my original joke:

If you're wondering why someone would waste their time going back to sort, catalogue, number, index, and file something that will likely never be of the slightest interest to anyone but for the completeness of the thing, please find the nearest male and ask him to explain it.

JoaLDG: Negative Reinforcement (or, Sin and Redemption in Mathematics)

In his book Innumeracy John Allen Paulos, the veteran professor and exegete of mathematics and statistics, recounts a story about some pilots and their instructors.

This phenomenon[, regression to the mean,] leads to nonsense when people attribute [the regression] to some particular scientific law, rather than to the natural behaviour of any random quantity. If a beginning pilot makes a very good landing, it's likely that his next one will not be as impressive. Likewise, if his landing is very bumpy, then, by chance alone, his next one will likely be better. Psychologists Amos Tversky and Daniel Kahneman studied one such situation in which, after good landings, pilots were praised, whereas after bumpy landings they were berated. The flight instructors mistakenly attributed the pilots' deterioration to their praise of them, and likewise the pilots' improvement to their criticism; both, however, were simply regressions to the more likely mean performance. Because this dynamic is quite general, Tversky and Kahneman write, "behaviour is most likely to improve after punishment and to deteriorate after reward. Consequently, the human condition is such that ... one is most often rewarded for punishing others, and most often punished for rewarding them." It's not necessarily the human condition, I would hope, but a remediable innumeracy which results in this unfortunate tendency.

He goes on to give two paragraphs' worth of further examples about movie sequels, music albums, baseball players (what is it with Americans and baseball?), and stock markets, but the lesson for would-be pedagogues is clear: don't treat your subject with statistical rigour and suffer the fate of all pseudosciences that came before.

The contrary view alluded to in the first quoted sentence, namely, the ascription of intentionality and significance where there are only the capricious mechanisms of probability, is, I believe, a symptom of a whole another and different ontology of the physical world than the scientist/naturalist's, one which is surely mistaken. And it is one which is far too vast to take on in the screed of a single evening. So we'll consider ourselves admonished on two levels, namely with regard to our evaluation of our methods, and with regard to our concepts of praise and blame, and press on.

(By the way, every living person in the developed world, I kid you not, should read one of Paulos' books. It doesn't really matter which one, they're all more or less the same. Innumeracy is good and A Mathematician Reads the Newspaper is too. Go read!)

Not to put it in too maudlin terms, but I think I believe in positive reinforcement. As a very general principle this is connected to the Nietzschean optimism that (on our weblog's happier days, if I have succeeded) is our spirit here. (I keep saying that's our spirit, anyway.) If one "doubts with well-founded suspicion" that good things are possible, I want the courage to build a world, speaking literally or of the world as a metaphor for my ontology, which those good possibilities populate. As a principle of pedagogy it is maybe a reaction to something, some grousey grouch in my past, or maybe a recognition that the undergrads of today are the colleagues of tomorrow -- put another way, that the aims of a class are largely but not entirely about the students' command of the syllabus material and I am judged not merely by the standard of review for an educator-as-mechanism.

On the face of it, there's something very suspicious about negative reinforcement. I don't insist on being loved or liked -- I'll settle for respected -- and I don't repudiate Burrhus Skinner (entirely). It's just unclear how this negativity is going to get any desired result, particularly when many students are already very anxious about math, or about their academic position. What intermediate steps have failed that we need to take out this big club with its dangerous and indeterminate consequences? I fear that its advocacy comes from exasperation, but my perverse optimism tells me it could still lead to good things.

Because there's exasperation to be had. I read an article, "Teaching Freshmen to Learn Mathematics," whose author, Steve Zucker, (a professor of mathematics), took the following extreme (?) position. "We shouldn't," he argues, "overlook the power of negative reinforcement." He goes on to describe two mathematical errors often committed (in the past -- he tells us that their frequency has since dropped off) by his Calculus II students which he calls the "ultimate sin" and the "penultimate sin."

(They are, for the interested and mathematically inclined, respectively (a) the belief that a series, say the harmonic series, converges if its terms vanish; and (b) the computation of a limit of a sequence whose terms are given by some expression in n, like (1 + 1/n)ⁿ, by selectively letting instances of n go to infinity -- in this case, inside and then outside, concluding that the limit is 1, when, as everyone knows, it is in fact e.)

He says that the demonstration of these sins by his students argues some very dire things about the relationship of the student to the course and even to the instructor. (It's clear at any rate that the students don't understand what they've been taught if they make such errors, but naively we might wonder about the value of introducing eschatological language to describe these mistakes.) He informs his students that commission of the penultimate sin on any submitted work will immediately earn zero credit for the problem, and commission of the ultimate sin will earn negative credit.

I have to admit that I'm a little envious. I'm pretty sure I couldn't get away with that.

Anyway, since we're all here to hear about linear algebra, and not about calculus, which no one understands anyway, I have a couple of candidates for confusions that drive me up the wall. Following precendent, I would like to propose, as an "opening bid," the:

• Penultimate linear algebraic sin -- phrases like "the vector is linearly independent" or "the vector is linearly dependent." I would also like to take on the concept of a "redundant vector," which to my view is simply pedagogical lemonade encouraging a nonexistent intentionality, but them lemons are in the textbook, with a footnote about how "redundant vector" isn't actually a linear algebraic concept but the author has found it "useful" in teaching, if you can believe that, so I'll save that one for another post.


• Ultimate linear algebraic sin -- any sentence of general type, "a basis for the kernel of matrix A (or another space) is the span of such-and-such vectors."

I'm not wedded by iron chains to these as my two least favourite things to read on students' papers, but it's hard to take a shot at things like co-ordinate chauvinism or implicit inner products which are, among other things, too far advanced errors to strenuously indict. (We'll see if I still feel the same way about co-ordinate chauvinism in the next two weeks, when eigenvectors and similarity of matrices are the topics of discussion.) The two cited transgressions show, roughly, that the student is confused about the relationship between vectors and vector spaces: the role of linear combinations in general and their significance to the concepts of independent sets and spanning sets (and bases) in particular.

And besides, these not-silly-mistakes (as Zucker would put it) -- especially the ultimate linear algebraic sin -- show up with depressing regularity. So I think I can summon a roughly analogous frustration to our poor calculus instructors -- though, mind you, I'm just grading these papers, not teaching the class. So how do I feel now about negative reinforcement?

I'm giving it a try. As I said I don't think I could get away with negative credit for problems, and anyway it would be a completely unfounded move for someone who doesn't even have any interaction with the students to tell them why. I mean, I can't even explain to them verbally what's wrong with what they've written, but commission of the ultimate linear algebraic sin is cause for loss of two points from five, no matter what the problem was about, and no appeals, damn it. It's not right for this nonsense to get written and for me to say nothing: that doesn't do anyone at all any favour.

How's it working? It's a little tough to evaluate, as I said, not being the instructor. I don't recall reading an instance of that sin lately. It drives my blood pressure up a notch to see so I'm pretty sure I'd remember if it had been there -- and, hey, I have been privately calling it the ultimate linear algebraic sin for a couple months now. Forgetting that would be like forgetting another fall of man. On the other hand, the students haven't been asked too many times to give a basis for the kernel of some matrix, and when they do, now that I think about it, they've been saying things like "the kernel of A is the span of such-and-such vectors," and not even using the word basis, as though they'd like for me to draw an inference, just answered like they would have in the first weeks of the class.

I...

I think my plan may have had some unintended consequences. Like apostasy.

Negative reinforcement, huh --

At this stage I'd like to remind everyone that it's not my fault, I didn't do anything, it's just regression to the mean. Somehow.

I'm going to abruptly end this post on the pretext that it's already long enough and, umm, further data must be collected to continue the discussion.

Happy Assyrian New Year

Tomorrow will be a wonderful day. I'll wake up and check the news and know that don't-believe-a-damn-word-you-read-online day (aka 1 April) is as far away as it possibly could be.

This day is painful. Yesterday FIDE (the world chess federation) did YAST (yet another silly thing -- I won't explain that thing, it doesn't have a good length-of-exposition to payoff ratio) and apparently said they weren't going to fix it, or something like that, and there was a protest from the Indian Chess Federation, advocates of the injured party, Vishy Anand, and now Chessbase is reporting that FIDE has reversed its position, except the article in question was published on 1 April, and these Chessbase guys are incorrigible pranksters, they really delight in it -- I mean, they're usually more obvious about it, but I read this thing, this news report, and I'm even doubting the factual basis -- and if I had that kind of doubt when reading a newspaper, why would I waste my time with that? Editorial indiscretion, what a bother.

But, you know, it's more like next week that I'm free, because today's debris will still be there tomorrow. For days yet I'll be checking the dateline of everything I read.


Postscriptum primus. Last entry updated today [1 April] with some pictures, because we haven't had any pictures in a while.

Postscriptum secundus. Turns out they were serious. "I never doubted you," said C-3P0, "much."

Postscriptum tertius. Turns out a lot of people doubted them too :

The day we published our April Fool's prank we also carried an important news item: in view of wide-spread protests FIDE had decided to correct their April 2007 rating list and to include the Morelia/Linares event in the calculations. This, many readers believed, was the April Fool's joke. In fact a colleague had told us on the previous evening that he intended to use it as his prank: "FIDE admits error, vows to correct it immediately" was the hilarious article he was working on. On April 1st we received irate messages from him accusing us of poaching his joke. It took a while to convince him that the FIDE decision was for real.

Well. There it is.

Postscriptum quartus, (since, hey, we've followed up on our story a couple times already.) Thanks to the nonflat nature of the Earth and the roughly Copernican character of our solar system, it turns out that Vishy Anand thought the first part was a prank, until he was informed likewise by a vast number of concerned well-wishers. (Take that, medievalism.) I had no idea they celebrated Assyrian New Year in India.