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.

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.

Trapped in a Menger Sponge

Saw something amusing today on a math blog. Under the heading of "weird things math/CS/physical science students build," this entry from the Cornell math club:

"Math Happens"

It's a Menger sponge, an object like a Sierpinski carpet, but in dimension 3 and with cubes, (he said like that instantly explained everything). You start with a big, solid cube and divide it into a 3-by-3-by-3 grid of smaller cubes. Then you knock out the centre piece of each face and the centre piece of the whole thing, leaving 20 pieces left, like a cube with a jack hollowed out in the middle. Repeat with each remaining piece, "ad infinitum," as they say. The Menger sponge is the limit of this process (more precisely, the intersection of the intermediary constructions). It has the really cool property that every curve (topological space of [Lebesgue covering] dimension one) embeds homeomorphically into the sponge, according to the Wiki article, making it a universal object for (these, topological) curves.

Needless to say, the first thing that flashed through my mind was half a script and the byline for Cube 3. One small problem: the thing is Lebesgue null. On the other hand, the reasonably high Hausdorff dimension (about 2.73) suggests something can be worked out, either by holding our noses and going or switching to the four-dimensional analogue.

I'm afraid the ol' UBC Math Club hasn't constructed anything quite so, well, intimidating. There's just this fun guy:

Pumpy the Irrational Pumpkin

I can't recall seeing any curiously intriguing math-themed objects at Princeton, although I haven't been keeping my eyes open. It's more of a tigers-and-abstract-statuary sort of place. Next time I'm there I'll snap a shot of the bust of David Hilbert in the common room.

Resurrection: "aufersteh'n, ja aufersteh'n wirst du"

Hi, everyone. Long time no see. Qu'est-ce qui se passe?

The big news at the beginning of the semester is and must be the new classes, but we're still easing into that (one hasn't yet begun, even), so I'll hold off for the moment. Suffice to say that this year's crop looks to surpass that from the fall, with two number theory classes, including one taught by Andrew Wiles (which hasn't begun, so who can say if it will be beyond the horizon), and a geometry-flavoured class. Indeed, all would be perfect except for a foul scheduling accident that I had to watch happen, in person, with all the dread of watching two trains ram, from a distance, slow-motion-like. The other number theory class was originally at 1630-1830 Monday evening, so it was no surprise to me when the first order of business there yesterday was to negotiate a new time. Alas! Miserere, misero me, the time picked was 1400 Tuesday, consuming, like Jormungandr, the likewise-located discrete math class of Paul Seymour that I had so anticipated. An embarrassment of riches, verily. If this is the mightiest problem we encounter --

-- but it is not; finding the damn'd local chess club is far thornier. It is far more difficult than checking their website. However, I heard today that Ed Witten (yes, that Ed Witten) plays chess at such-and-such a location Fridays at nine o'clock in the evening or so. Needless to say I was struck profoundly by this remarkable confluence, and will have to investigate this rumour.

I should say where I heard it: over dinner. The dean of the graduate school, Bill Russell, hosts monthly gatherings for a small number of random invitees. Tonight my ticket came up. Despite having several days to think about it, though, I didn't manage to ask for a suggestion on dress. This kills me every time. I end up coming with the most mongrel compromises. Today I interpolated my blue sweater between the lavender shirt (and tie) and my suit jacket, with the light pants (and the good shoes, needless to say). I think this worked quite well; in fact the assistant dean had a red sweater-vest type under his suit jacket, and I managed not to be better dressed than the host, which would presumably be unforgiveable. (That was my fear, anyway; you can set me straight on protocol, a subject which didn't quite come up in the past.)

Anyway, dinner was quite the usual story: I managed to definitely not impress before and during dinner but relaxed into the setting afterward. I had a lovely chat with variously an historian, an architect, an economist, and a neural microbiologist. I asked our economist whether he feels (as I had read in a paper recently) that his field has gotten out of touch with the real world, and he told me that he likes the elegance of his theories. I suggested he move to Fine Hall. Architects, it turns out, have a huge breadth of knowledge that they draw on professionally, and we found a mutual interest in Greek philosophy and Bach, inter alia. Finally our historian friend won my heart by enthusiastic interest in my mathematical anecdotes and asking to keep my demonstration napkin.

I'll spare us all the meditations on the nature of happiness and our relationship to the Platonic Realm and instead present a preliminary trifecta of anecdotes.

1. Birthday Surprise. This is a fun trick to warm up with that consistently astonishes people who've never heard it. It has a good moral, too, that statistics does not come intuitively to people in general. The question is: how many people do you need in a room before it's more likely than not that two of them share a birthday? I observe, trying out some showmanship, that of course you'd need 367 to be certain of it. Now no one ever ventures a guess about number, but on first impression 23, the correct answer, is rather small!
   
2. Bottle Imp Paradox. This is one of my favourite problems, but you might want to replace it with something else, because I've never yet met someone who thought the same way. The set-up is this. You are offered a chance to purchase, for any price you care to name, a bottle imp. This bottle imp grants an unlimited number of wishes for you, with the sole condition that you must in turn sell (not discard or gift) the bottle imp to someone else after some finite time span, say twenty years. The condition of sale is that you must sell the imp for a strictly smaller amount of money. (Now, this is a game theory problem, not a gedankenexperiment in the value of a thing, so what this means is that if I pay one hundred cents then I sell it for at most ninety-nine cents. There is no such thing as a half-cent or a peso or inflation or whatever.) Failure to sell (by you, or by the next person, who inherits all the conditions) results in a terrifically gruesome, unspecified penalty exacted by Mephistopheles, from whom you cannot hide. (Or maybe Samiel, the Dark Huntsman.) The question: Do you buy the bottle imp? And if so, for how much? The rational answer is that you do not buy it, for any price: for clearly you would not buy it at one cent, for then no one would be able to buy it; but then not at two cents, for then no one would be willing to buy it; or if not n cents, then also not n+1. But, and I think I'm not alone here, I definitely would buy the damn thing, for as much as I had in my piggy bank. (The bottle imp can make currency, so it's all funny-money anyway.) I'm not saying the bottle imp's wishes would make me happy, technically speaking, but it's got to be worth it; and after all if I'm being irrational in buying it, I can bet there's an irrational person to sell it to....
   
3. Angle trisection. Save the best for last, assuming they're still paying attention. Many people (most, frankly, considering the audience you've got to have even to consider telling these stories) have heard that it's impossible to trisect an (arbitrary) angle using straight-edge and compass, or at least can think back to their high-school geometry and what it means. (Bonus points for me: mention that one uses Galois theory to show this, giving a beautiful application of the theory which I tell everyone I study when they ask.) Regular listeners will of course recollect that this can be done by origami, and the construction is wonderfully simple and easy to demonstrate with a pen that writes well on napkins.

So, a very pleasant evening. Moving on, more miscellany. Of course the Olympics have started. I hoped to catch highlights on cbc.ca, as they stream their daily newsprograms -- but the perverse Olympic broadcast regulations force them exactly not to stream their shows for the duration. So I'm completely blanked out, with the sole consolation that the curling scores are updated more-or-less in real time. It would have been nice to see the figure skating program (heck, to watch the curling!) although it seems my favourite guy, Alexei Yagudin, has bowed out (he was having knee problems last I heard, which was years ago... it's been a while without television, really).

Also: you may have heard there was a lot of snow here recently. We're relatively south and still got quite a bit. I'm told it's expected to warm up very shortly, and then it will all be gone. Already much of it seems to have melted. This meant only that I had to act fast, of course. So I took a quick jaunt to get the shot I needed, and picked up a few incidental ones along the way. My batteries, let it be known, did not fail me.

The view almost immediately outside my room. Continue past an archway hidden by the tree:

For those worrried I would catch my death of chill going to eat breakfast and dinner, the far building opens into the dining hall, so splice this one with the last and you'll see about how far I walk. Speaking of which, time for a little indulgence in an old passtime.

A dragon and a monkey and some other things guard the door. And look up a bit?

More. (Click through for the original.) And -- what's that? -- how can it be?

For he is the Kwisatz Haderach! Oh, yes, more tigers.

Lastly, what we were waiting for:

Click through and compare and compare.

It was a little hard to get the shot, for this reason, but we did all right.

Which brings us about to the end, or a good enough place for one. Good night, all. Magnificent dreams.

Math jokes: "your lie when you said to me, 'I did this only as a game'"

Famous math-joke punchlines: "Ah-ha, a solution exists!," "thus reducing it to a problem previously solved," "totally true but completely useless," "assume we have a jabberwock," where a jabberwock makes the problem trivial, "there is at least one borogrove at least one side of which is mimsy," and so on. If you've never heard them, you've not been listening to math jokes, you sensible fellow.

But there are others, call them sporadic math jokes (that's one of them), which don't fit into the mold. Those punchlines above are jokes about mathematicians more than math; the closest I can think of to the latter is "let epsilon be less than zero," which presumably is completely cryptic to non-mathematicians (that's the whole joke, not the punchline), or maybe one of the waitress jokes ("one-half x squared,... plus a constant," "in characteristic two"). Here's another one that bridges the gap:

A mathematician keeps a diary. It reads:

• Monday: Tried to prove theorem.
• Tuesday: Tried to prove theorem.
• Wednesday: Tried to prove theorem.
• Thursday: Tried to prove theorem.
• Friday: Theorem false.

Sometimes it's just one of those weeks.

It cuts even more when the "mathematician" is a student and the "theorem" is an assigned exercise. Sometimes this happens, quite by accident, even on an exam (oops! -- I've seen a couple of these), say if a little hypothesis gets left out or if the problem is copied without discrimination from another source.

On the opposite side, firmly in "not funny," are Stiller's monsters. Computer scientists who play chess and have too much computing power sitting idle engage in the following project: to enumerate and evaluate all legal chess positions with some small number of pieces, say, five or six or seven. You can download complete six-man tablebases, as they're called; that's the two kings plus four other pieces. They'll only cost you several gigabytes. It will also be extremely boring to blindly wander through them, although in normal chess praxis from time to time it would be helpful to have a computer program capable of not just crunching moves with grandmaster vision but of perfectly evaluating every six-man endgame.

In the famous rematch Deep Blue v. Kasparov the machine had six-man tablebases, and it must have weighed on Mr. Kasparov's mind that if the board got too light with material the computer would begin to play not just mortal chess but mathematical perfection, as though announced on the trumpet from the throne of god, or if you prefer, gleaned from the immodest prostitution of the Platonic Form of Chess itself.

And god and Plato can be inscrutable: this is where Stiller's monsters come in. They're a couple of six-man endgames, winning for one side, but where the shortest forced win is around 250 moves; they're named for the man who first enumerated the six-man endings and found them while looking for long forced wins. In the linked article, Tim Krabbé writes:

They are beyond comprehension. A grandmaster wouldn't be better at these endgames than someone who had learned chess yesterday. It's a sort of chess that has nothing to do with chess, a chess that we could never have imagined without computers. The Stiller moves are awesome, almost scary, because you know they are the truth, God's Algorithm - it's like being revealed the Meaning of Life, but you don't understand one word.

The seven-man tablebases of course will be downright huge, but a couple of endgames, like KRRN v. KRR (king-rook-rook-knight versus...) have already been worked out. There is a position in this class which is winning for the superior side but it takes 290 moves to prove it. The last ten or so, mind you, are pretty easy; but the first ninety have to be perfect.

It's the dark side of discrete math. Sometime's there's an obscure obstruction to a general claim ("theorem false") and sometimes ("file under Stiller's monsters") it's true but you have to beat the devil to prove it.

On 37

Today's a big day. Later this evening, in just a few hours, Papa and I will hop down to the local courier concern and send my boxes off their way. Today's the day where we must be absolutely certain: if it doesn't go today and it can't fit into the luggage than it's not going. I think it's all in order. Thanks to the kind Party the Third the luggage recently got bigger. I think that's set my mind easier, but I'm still highly anxious about things I know not what.

I'm packing my speakers off, too, so I thought I'd take today to transfer all the data I need to the shiny new laptop. While this transfer proceeds, I thought I'd take a moment out to tell you about 37.

37 is a remarkable integer, and my favourite one. It's prime, of course. Other than that it doesn't seem on the face any more or less remarkable than, say, 47. Another principal candidate for Favourite Integer is 26, which besides figuring in my birth date is the only number between a square (25) and a cube (27), (a proposition dating to Fermat).

You may think that there are so many integers it would be impossible to have a favourite one. This is not so. First, the Law of Small Numbers suggests that small numbers are the really staggeringly remarkable ones; they just get more boring as they get bigger. Paradoxically, of course, this makes it all the more interesting when the smallest example of something turns out to be 37; but 78,557 being the smallest Sierpinski number doesn't really make it more endearing. It's a delicate balance. Someone who thinks Graham's number or Skewe's number or something like that is the best integer is, I'm sorry to say, lacking in taste.

Yesterday I happened to mention over dinner somewhat provocatively that 37 is my favourite integer, which I shouldn't have done, because it was not a good time to explain why, since it's a rather long story that will require a detour through a lot of elementary algebraic number theory. But having stepped in it, I had to say why; and it's an interesting story, to me, anyway; so here we are. Don't let the words distract you from the music. If it gets too bad, just imagine Kosh is saying it. (This works especially well if you have questions. Query: What does this mean? Answer: Yes.) For the mathematicians in the audience, I will beg your forbearance with the simplifications, beginning with my second definition.

A (rational) integer is a counting number, 0, 1, 2, ..., or the negative of one. The set of such is labelled Z (for zahlen, German "number"). Prime integers, as we learn in grade school, are those which are divided only by themselves and 1. Otherwise a number is called composite. Every composite number can be expressed as a product of prime numbers. In fact this prime factorisation is unique.

There are other kinds of numbers, which are not integers, like the square root of two, say. We can do a kind of generalised arithmetic with these numbers, too. The set ("ring") Z[α] consists of all the things that look like a + bα + ... + cαⁿ, where a, b, ..., c, n are some integers. We can ask about what sorts of properties of integer arithmetic carry over to these new kinds of arithmetic. One question is: is there also unique factorisation into primes in a ring Z[α]?

It turns out that the answer is usually no. An easy example is to take α equal to the square root of -5. Now we can write 6 = 2*3 = (1 + α)*(1 - α), and it turns out that all of 2, 3, 1 + α, and 1 - α are irreducible in this ring, so that these factorisations are "essentially different." (Irreducible is related to but not quite the same as prime in a way which I will not say.)

However, there is a very deep theorem due to Dedekind on this subject. The first idea is to introduce so-called ideal numbers, not all of which exist in the ring Z[α], but which are related to the numbers in this ring. With these numbers, unique factorisation can be restored. This is an excellent achievement because unique factorisation is a very strong property and a lot of consequences follow from it purely formally. It turns out, to give you an idea that this is not so strange, that the only ideal numbers you need can be denoted (a), which kind we identify with just a, or (a, b), which we can think of as the greatest common divisor of a and b, for a, b elements of Z[α]. (All of this is true only for certain α. I won't say which, but all the α I mention in this post are of this kind. An example of something that doesn't work is π. Another example is the square root of 5. It's tricky.) There is an object called the class group which one can define from these ideal numbers. Very roughly speaking, it tells us how many essentially different kinds of factorisations (into non-ideal numbers) there are in the given ring. The very deep theorem of Dedekind which I mentioned is the statement that the class group is a finite set. Its size is usually denoted h (and depends on α, of course). We have unique factorisation if and only if h = 1.

With this behind us, we are going to specialise to the case where α = ζ_p is a (primitive, complex) p^th root of unity. This means that α^p = 1, but α is not equal to one. For example, the number i which solves the equation x² = -1 is a fourth root of unity. Now let h be the class number of the ring Z[ζ_p]. We say that the prime p is regular if p does not divide h, and irregular otherwise.

37 is the smallest irregular prime.

You may wonder why anyone would be so daft as to make this definition in the first place: why is this property of any interest at all? The answer, as it is with so many things in elementary algebraic number theory, is Fermat's Last Theorem. It is possible to give an "elementary" proof of FLT for the case where the exponent is a regular prime.

(For the math guys who haven't read it. The idea is to start with x^p + y^p = z^p and factor the LHS into a product of terms looking like x + ζ^ky where ζ = ζ_p is as before. Now if we had unique factorisation in Z[ζ], a product of relatively prime factors being a p^th power means that each factor is itself a p^th power, a very strong condition, which we can get our hands on by passing to ideals. For the other case we note that the gcd of two of those factors divides also their sum and difference. In any case, there is a long and difficult calculation ahead, which takes a few pages after the preparatory lemmas are stated. At a critical juncture we have some ideal I whose p^th power is principal. If p is regular, then it is coprime to h, and this implies from the definition of the class group as fractional ideals modulo principal ideals that I itself is principal. That's the only place where it matters that p is regular. I think there is also some assumption that p doesn't divide xyz in the argument I remember.)

Frankly, I find it remarkable enough that there are any irregular primes at all. There are three less than 100. It turns out that there are even infinitely many irregular primes. I guess when I understand this fact I won't find it so astonishing that there are any at all. But the other surprising thing is that we don't even know if there are infinitely many regular primes, although it's conjectured (and there seem to be more regular than irregular in the ranges where we know). You can take a look around some of the links on the right column to read some strange facts and more technical descriptions if you like.

Postscript, immediately after. That ζ looks really ugly in this font. Shame, it was a favourite Greek letter of mine. Also, tthere's a post from "last night" [early this morning] just below here, too.