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.

ambivalence resolved

I visited Chapter's at Robson one time during the summer. I had mischronometrised and arrived quite early for a lunch appointment with Craig and Papa. It's very pleasant to visit all the old haunts: see what's new, what's old that's reprinted and new again, what never went away. I have a mental checklist: any new-old Silverberg? or Heinlein? or Herbert? or --

Alas, that there could be new-old Herbert; alas, there only is new-new Herbert. Instead of a re-issue of Destination: Void, which even Papa doesn't seem, to my frustration, to own, or of The White Plague or The Jesus Incident, we get, at best, The Road to Dune, which, well -- it styles itself thus, above the oversized "DUNE" in that characteristic font that now means another wretched volume has come upon us:

The companion to the New York Times bestsellers including never-before-published chapters from Dune and Dune Messiah, original stories, and a new novel by Brian Herbert and Kevin J. Anderson.

This book first came to my attention that summer day. It was sitting there in hardcover. There is barely a hundred and twenty pages of actual Frank Herbert material, but such is the hold of it over me that I was tempted to grab it right then. But my better senses prevailed, and I left it alone.

A few days ago I was again in the Chapter's at Robson. It was the day after Boxing Day. It was very obviously the day after Boxing Day. The shelves often contained gaps great and small and many books on them showed curled corners and scuffs on the cover and ruffled pages: surely they'd been bought and returned and reshelved this Christmas season. With two gift cards whispering in my pocket and no prospects to use them for many months anon I was eager to increase the burden of my space-crunch. So when I saw The Road to Dune in friendly paperback size I was certain I had to have it.

And at home as I started to read I was not disappointed. I read a summary of an article that Frank Herbert was thinking to write in the 50's, about how for the first time a system of plantings had been developed to stop the encroaching sand dunes from burying the little town of Florence, Oregon. The dunes are driven some twenty feet a year by wind, like water in tempo largo. More than ten thousand types of grass had come into the experiment to find one that would set down roots that would help fix the top sand in place. Reading this, reading Frank Herbert's excitement at the ecological solution to an historic problem and his agent's nonplussed replies, I thought of green plants in the desert, of ornithopters, of the opening of the film Koyaanisqatsi with Philip Glass's awed score over waves of water and clouds and cars, of (Mother of Muad'dib!) rain from the sky of Dune --

I decided that if I was ever in the area I should drop by to see the environs of the town of Florence, Oregon, this place that set in motion these thoughts in Frank Herbert's head. A quasi-hajj to Mecca-on-the-West-Coast. I discovered that my friend Daniel, a.k.a. Billy, formerly of UBC Math and now of Oregon Math, attends school just two miles from the town. That started me thinking.

From the reportedly many boxes of notes and text uncovered by Brian Herbert and Kevin Anderson in the pieces of Frank Herbert's estate they gave us some hundred pages of drafts and excised material from Dune. Some of it requires no explanation:

Paul stepped out of bed in his shorts, began dressing. "Is she [the Reverend Mother Gaius Helen Mohaim] your mother?" he asked.

"That's a fool's question, Paul," Jessica said. She turned. "Reverend Mother is merely a title. I never knew my mother. Few Bene Gesserits of the schools ever do; you know that."

There are so many things wrong with this passage about the only piece of interest is to know that even the pen of Frank Herbert scribbled the kind of things that I'd write and strike out instantly. There's no indication of where this fragment came from, and I imagine it was indeed quickly discarded, and survived by happenstance among the many pages of unused material.

Much of the discarded material is like this in character, almost pure exposition either dropped or strongly reworked into more natural settings. A little is reasonably polished work from earlier drafts of the plot, like a visit of Jessica and Duncan and twelve-year-old (!) Paul to Dr. Kynes. Some, a very little, is really literarily indistinguishable from the finished product; for example, a deleted scene from Dune Messiah wherein Paul receives Otheym's message from the Tleilaxu-made dwarf Bijaz, which ends:

"The spice, M'Lord," Stilgar said.

"Of course we'll stop all shipments of the spice," Paul said. "Let's see how they like that." [...]

Bijaz began to giggle.

Paul turned toward the dwarf, noting how the creature had the attention of everyone in the room.

"How they'll wish on the morrow they had no teeth," Bijaz sputtered between giggles.

"What in the name of the worm does he mean by that?" Stilgar demanded.

"Without teeth they'll be unable to gnash," Bijaz said, his voice reasonable.

Even Stilgar chuckled. Paul stood silent, watchful.

"Who do you mean by they?" Paul asked.

"Why, Sire," Bijaz said, "the ones who planted that stone-burner on you. Could it be they wanted you to stopple the spice?"

The prose is clean, smart, and beautiful, and paced like poetry. No coincidence: we are told that many passages in Dune began as Haiku and the like before being enlarged only as necessary to make grammatical English, and if Frank Herbert later discontinued this practice it's clear he'd mastered the lesson.

The Road to Dune contains a re-constructed short novel entitled Spice Planet, apparently based, how loosely one couldn't say, on original notes (characters and events) for a story about narcotics and feuding noblemen set principally on a planet known as Duneworld. This, it seems, was the first draft of the Dune concept, later expanded into a whole new universe shot through with themes of ecology, politics, leadership, history, religion, -- and so many more besides. It would be interesting to see the story of Duneworld written from Frank Herbert's hand, but mostly because it would be from Frank Herbert's hand.

The volume also contains a hundred pages or so of short fiction by Brian Herbert and Kevin Anderson. Most of it pertains to their books which they claim tell the story of the Butlerian Jihad. The last of them they describe in the introduction as in part a bridge between the end of Chapterhouse: Dune and their Hunters of Dune, the new first part of "Dune Seven" that they're writing. On the basis of this I sat down to read it. It's called "Sea Child," and it's a little over fifteen pages long. Someone please explain to me, then, why the first two are a summary of the current state of galactic politics. It's difficult to pick out something as representative of their prose, but this paragraph, towards the end of that summary, offends me particularly:

In moments of despair, Corysta [the Bene Gesserit protagonist of the short story] felt she had two sets of enemies, her own Sisters and the Honored Matres who sought supremacy over everything in the old Imperium. If the Bene Gesserits did not find a way to fight back -- here and on other planets -- their days would be numbered. With superior weaponry and vast armies, the Honored Matres would exterminate the Sisterhood. From her own position of disadvantage, Corysta could only hope that her Mother Superior was developing a plan on Chapterhouse that would enable the ancient organsiation to survive. The Sisterhood faced an immense challenge against an irrational enemy.

There's no sentence of this dreadful paragraph that doesn't make me cringe. The centerpiece of its naivete (as opposed to its rotten, cliche-ridden prose, which is a sample representative of the whole) is that little aside about Mother Superior. It reads like bad fan-fiction: the blunder is of assigning paramount significance to exactly what was in the text of the prior books, no less and no more. If we the reader spent two books following Mother Superior, surely Mother Superior is on everyone's mind. Truly she must have some awesome cult of personality to be solely responsible for the course of galactic politics. As to the details of the war, it's clear they don't understand that the weapons and battles are incidental. The motives of the foe, moreover, were revealed slowly and imperfectly to us, and I don't remember "supremacy over everything in the old Imperium" being anything more than a means; but this is the level of subtlety that Brian Herbert and Kevin Anderson will pitch to us.

For the lesson is that there will never be a moment in the books of these two like when Bellonda drops into the scene with an anguished cry: "Lampadas!" We know what she means: the training planet has been found and destroyed; all the Sisters and Reverend Mothers and the soldiers who defended them and their marshal, who we knew, died there. It is a crushing sentence for our protagonists and for us, and it's one the like of which we won't see anywhere in Dune Seven. They'd blow one word into a full scene and the content would be shot away, unrecoverable, just like they spammed us with two pages of irrelevant galactic simplifications in a fifteen-page short that barely gets off the beach before it delivers its metaphors with a heavy, Harkonnen-sweaty hand.

So what I'm saying is -- the spirit of restlessness that came years ago with the production of new books with "DUNE" in great letters upon them has fallen still. There is not now and will not be more old Herbert. The shlock released between those covers is puerile and clunky in equal measure. It is by me simply unreadable. There is no other thing to say about a book that can't pass half a page without dropping some dreadful nonsense, say:

In the floating image, the bristling ships opened fire, unleashing incineration waves with devices the Bene Gesserit had since named "Obliterators,"

to borrow a half of a single sentence from the first page of the excerpt of Hunters of Dune helpfully included at the back of Road.

So what I'm saying is -- I'm not buying any more of their damn'd books.

So what I'm saying is -- if anyone manages to get through "Dune Seven," let me know what's up with the Enemy. I'm all right with not knowing, but hey -- as with the transcendentality of π, if you could know, wouldn't you want to know?

That's all I'm saying.

Second Lord Kelvin Useless Creation Award

It's been a while. Let's get right to it: it's a little part of the story of the past week in the form of a glamorous and prestigious awards banquet.

Let me recall to you the award description:

Lord Kelvin, the famous British physicist, once wrote,

Quaternions came from Hamilton... and have been an unmixed evil to those who have touched them in any way. Vector is a useless survival... and has never been of the slightest use to any creature.

In honour of his getting one out of two, and being hilariously mistaken on count of the one he missed, the Lord Kelvin Useless Creation Award is given irregularly to "those entities that in the estimation of the award committee have never been of the slightest use to any creature."

You must be thinking: competition was tough this year. I haven't gone back and checked but I know I was publicly threatening at least one collection of miscreants crossing my path with a bit of perpetual disgrace. I can't remember if I said that it was Moving & Storage or Dining Services that was "in line for one, missing only the requisite irony." It was probably MS, because DS certainly has a lot of irony going for it. The awards committee had about made up its mind for an honourary citation one Saturday evening last February when they booked Procter Hall during the dinner hour for some random group who desired a big room for their random self-congratulatory shindig. And I swear that their failure to expand breakfast hours, instead tacking on an hour to dinner, missing the recommendation of a task force created to report on their operation, would be worth an honour by itself. But no: like a gift from the improbable child of Melpomene and Thalia came a late entry that stole the show. Dining Services will have to settle for a Lifetime Achievement Award around the time I graduate. The only trouble will be tracking down all their nonsense to stick on the citation.

But back to the darling of the hour, my own financial servitors here at Princeton, the PNC Bank. It's not just that they won't let me re-order cheques online until I've re-ordered at least once already, yet give me no instructions on how to do that. That's already doubly useless, but they're far more useless than that. Let me give the balance of the award citation in the form of a little story about how they got my mountain goat, and then tried to take away my other goats.

Last Wednesday I was out shopping for some things, starting with my glasses. I placed an order for those early in the afternoon, and after finishing up with the optometrist shining a light in my eyes had a few minutes to spare while the lenses were shaped. I had wanted to get some new shoes, so this seemed an opportune occasion. Surprisingly, I quickly found something I wanted. However, there was a slightly difficulty at the counter.

"Your card has been declined," the cashier told me.

"What?" said I, somewhat ineloquently, leaning down to read the transaction receipt before me, a formidable task with my pupils still widely dilated and adding to my disorientation. She repeated herself as I scanned to try to find the words that said this. "How can it be?" I asked myself. The card is not a true credit card, but my chequing account with a credit-card-number to access it; there is no chance something could have gone wrong, and the timing, say fourty minutes after purchasing my eyeglasses and paying for the eye exam before that, made me feel I had stepped into a Dali canvas.

I chalked it up to a most curious fault, until the next time I tried to use my card in this capacity. "Authorisation declined," I was informed enigmatically by a machine. "What?" said I, somewhat ineloquently, to no one at all.

I resolved to find out what was happening, soon, before Amazon tries to ship my books and discovers that my money is suddenly no good.

As it happens, to take a little digression, I had further reason to have a word or two with my bank. Back last fall they sent me a little Form W-9 in the mail, basically an information request, looking for my SSN. I, of course, neither had nor was eligible to obtain one. That's OK; I can give them an ITIN -- if I had an ITIN; if you recall the aftermath of last year's KUCA imbroglio, I discovered that none of the threatening letters were applicable to me since as a citizen of Canada my country's government has no tax treaty with the American state and I am permitted to apply for an ITIN only at the time of actually filing my return. So I filed away this request for a future time, like next year, after my spring tax return is processed.

Sometime over the summer my bank apparently became quite agitated with this situation, sending me duplicate forms with stern warnings about backup withholding. Their problem, you see, is that they didn't know how much tax I was meant to pay on the interest my account bears. Considering said amount was about seventy cents, I had been quite prepared to forego the whole amount and think myself the richer for having saved myself the anxiety of working out another stultifying form.

But now that I had a real ITIN, things seemed to have improved on the throwing-them-a-bone front. There was only one remaining problem: the last item of the statement about the signature line, the one I'm certifying under penalty of perjury, is that I'm a U.S. person. A little investigation (online; they didn't bother to include any instructions, the useless critters) revealed this is synonymous with citizen or resident alien. I am neither; I'm a nonresident alien, and will be for the duration of my stay as a graduate student.

Well, what to do? I had to speak to someone at the bank. So I went to do this.

We tackled the second problem first. After a little time, I was advised to merely cross out the offending part of the certification and sign the damn'd thing. "Isn't there a different form I'm supposed to submit?" I asked meekly. Well, yes, but no.

At this stage I broached the first problem. After a minute or two of data-hunting on the PC, my consultant informed me that this had been the trouble.

"What?" said I, somewhat ineloquently. She elaborated. My failure to submit a Form W-9 was at cause. "Oh, really," said I.

Presently I left the bank, hobbling my way down to Fine Hall for the next of the day's errands. And I thought: It's not just that this is the first chance I've had to possibly submit the damn'd thing. What are these useless creatures doing, declining my card because they don't know if I'm supposed to pay fourteen percent or thirty percent on seventy cents?

And that's what it'll say on the plaque. Congratulations, guys.

* * *

But at least she was right. Amazon has already started to send me some things.

Schachmaschinen: "a horse, a horse, my kingdom--"

An item on the list of Things Everyone Knows: that, without qualification, computers are better than humans at chess. Ever since Kasparov lost the first game of the original 1996 match against Deep Blue it's been downhill for chesspeoples looking to protect their mystique. It seems the Royal Game is fit just for calculators. Whereas in the 19th century it was held that a master's gift was for inspiration and fantasy, and optimism so reigned it was felt that a player could in the most seeming-rotten position discover some extraordinary resource to save the day --

-- or so the chess mystifiers and hagiographers tell us. I would doubt it, except that not too long before Everyone Knew that all functions are continuous and with one or two silly exceptions differentiable and so forth, which is a right bit of fantasy itself -- albeit abetted by ambiguity in the meaning of the operative terms.

Anyway, Kasparov lost the game, then stormed back to win the match 4-2 (+3=2-1), winning the sixth game in a particularly gruesome manner; he was in such high spirits as to spontaneously agree to a rematch during the press conference; and then next year came and crumbled in front of the world, losing the 1997 rematch 3.5-2.5 (+1=3-2) after another particularly ignoble sixth game, dual to the previous match; and that was it, chess is something you can calculate, and, incidently, all the hopes of the early AI pioneers that it would take real intellect to make a fine mechanical chess machine -- just vapour, like so much else.

But one didn't have to be a chess mystifier to know it just wasn't so. The computers were still weak relative to the best, demonstrably so, to those wanting to take enough time to demonstrate it, even if it was long past the time a humble patzer could hope to score a few points off his laptop any more than off a flesh-and-blood master, and Kasparov's loss was far from convincing.

So despite Deep Blue's "retirement" the experiment continued. The computers got better. Man still had some triumphs; for example, Ilia Smirin's 2002 match against four leading programs, which he won, 5-3 (+2=6-0). The game against "Gambit Tiger" was particularly curious and prompted one commentator to remark ironically that although computers were superior in strategy to humans, we could still overpower the machines tactically, (as Smirin did in a dubious position).

Fast-forward to 2004, with two high-profile matches between leading players and leading machines: Kramnik v. Fritz and Kasparov v. Junior. (You can always tell which side is computer by the goofy name, and which side human by the Russian name.) In the former Kramnik looked to be coasting to an easy victory before dropping two points in the second half to end 4-4 (+2=4-2), once simply hanging a piece and once playing a slightly speculative combination with a very deep refutation which the machine very impressively found. In the latter things looked up when Kasparov cooked the machine in a knife fight in the first round, then slightly down when in a later game he overlooked a mating attack from the machine, then slightly askew when in the final game he agreed to a draw in an unbalanced position -- in front of millions (maybe), since the game was televised on ESPN (!). Ending up at 3-3 (+1=4-1), one could be forgiven for having lost almost all interest in Man-Machine matches.

Then, of course, there was Hydra, which pretty much made it mandatory to lose all interest. ("Lasciate ogni interesse, voi ch'entrate"?) Finally the knowledge that the tail wags the dog came true with a painful demolition in Adams v. Hydra. (Finally not a Russian, but still, to be sure, a world-class talent. For the record, Smirin is Israeli.) This travesty took place just a few months ago and saw the bon vivant Englishman lose 5.5-0.5 (+0=1-5). Ugh. Even worse, the prize fund was structured so that money only came for scoring points (unusual: usually the purse is split in some specified way to the winner and loser of the match, irrespective of the score), so for enduring this drubbing poor Michael Adams picked up only 10 thousand. (For comparison, Kasparov and Kramnik pocketed mid-six-figure sums for their labours -- minus pay to their seconds, and so forth.)

Finally, today started the second annual (so far) Man v. Machine tournament in Bilbao, Spain. Three men, all former "world champions" (so to speak; they comprise an Uzbek, a Ukrainian, and a -- Russian), against three leading programs, two microprocessors you too could own (Fritz and Junior) and one supercomputer (Hydra, again). Care to guess the day's score? Yes, that's right.

Certain chess fans might note encouragingly that all is not yet completely lost: correspondence chess players, who play games over duration of months instead of hours, sending moves by post, still know how to take the machines' measure. But this wagging wolf will just give a desultory woof!. It's all true. If a chess titan spends months preparing for a match, then we'll watch it, for sports' sake. But these sideshows are dispiriting.