You know what's bothering me today? Well, Joe Lieberman and his enablers, for one thing. But on a more personal level, the current convention for writing functions. Yes, functions, mathematical functions like you learned in high school.
Your average computer scientist researcher might be incapable of de-crudding your Windows Registry or making Bluetooth work on your Dell whatever. What your average computer scientist researcher is apt at is contemplating languages, specifically the languages of computing. We all know at least a handful of computing languages, and we are sometimes obliged to concoct new ones. We might have fond favorites languages, but no real fidelity to one or another. This promiscuity with respect to language means that we tend to become connoiseurs, sometimes rather opinionated ones. And in fact anytime someone goes expressing thought via written notation, we CS people are liable to turn our jaundiced eye thither and offer our punditry. Consider chess notation. There's algebraic chess notation, such as "Bb5 ax6." There's descriptive notation, such as "Queen's bishop to queen's knight 3." In each case you have the signification of some chess move, but almost as interesting to us are the properties of the notation itself: the first is compact, one may say terse, but fairly self-contained; the second is quite readable, but its meaning depends essentially on its context. Further it is bulky, almost flabby, yet has a romantic texture (with notes of chocolate and berry revealing an English terroir; although the finish is somewhat metallic).
The point is, we CS folk are something like language gourmands, bloated and bloviating on the subject of language, full of opinion, and liable to tell you about it. Don't get us started. Still, the fact is, we really do know something about what makes a good notation, and why. This goes beyond a mere sense of taste, to a well-developed theory, you may call it an aesthetic theory, of language design. There are numerous books on this subject.
Math is also written in a symbolic language, to state the obvious,
and mathematicians are just as capable of being persnickety about
this language -- we call it notation -- as the CS folk. In fact
the CS folk are mathematicians, but even the mathematicians who
are not of the CS tribe have firm opinions about how to turn math
into symbols. If you draw a little squiggle of a curve and mark
the two ends, respectively, with
(x, y) and
(x + dx, y + dy)
some mathematician is liable to grumble at you about how
the "d" should be a delta, and how only "in the limit"
may you replace the Greek D with a Latin D.
Another mathematician might magnanimously overlook this particular
transgression of the guild rules, and say it's OK.
To those not in the know, the point of contention would, I think, seem arcane.
Indulging or forgoing such pecksniffian critiques
is a personal choice for a mathematician, but a typically pinched one:
I daresay any question of significantly bigger scope has for
the most part already been decided, hardened into orthodoxy, and
all guild members are called upon to defend it.
Just as the monolingual citizen born into the Roman empire might never
appreciate that
there are other principalities, and that the Mediterranean is not
the center of the earth, so too is the modern mathematician
monolingual with respect to signifying mathematics.
When you
take high school calculus, they teach you the way to write
down an integral. (Possibly when you get to university, you might learn that
some radicals put the "dx" on the inside.
Po-tay-to, po-tah-to.)
The fact is that there is really only one language of math.
It is worth noting that "the" math language is pretty difficult, actually. Not to the insiders, of course -- we wouldn't be in the guild if we found the language difficult -- but, it has been proved, to legions of nonprofessionals. We lose sight of the profound gap between signification and the signified, as it relates to math. Of course, professional mathematicians try not to spend too much time ruminating on this gap, just as good astronomers try not to spend too much time meditating over their telescopes -- but those who research math education, like Keith Devlin of Stanford, know that for the average math learner, our notation is hard. Very hard. A major source of confusion. Devlin cites an extremely interesting study done with children in Brazil, who could work out complex arithmetic at over 90% accuracy when they were freed from the burdens of mathematical notation; but when the same questions were posed using traditional notation, their success sunk below 40%.
Ah, but to those in the know, math notation is fine. Well, that's what I said, but I want to backtrack from that assertion now, since it's too panglossian. Conventional notation is fine for a lot of things. But it really starts to seem threadbare and underpowered when we come to probability and statistics. I'm losing steam for writing more more on this theme, so I'll come to the point, and it revolves around functions. The usual way we write functions in math is like this: "f(x)" is a function. Well, already I anticipate static. Many will nod, but others I daresay will protest that "f" is the function, and "f(x)" is the function evaluated with input x. Or still others might wave away the difference by saying that although f(x) literally denotes a particular value, it does so just as much as x denotes a particular value, and since x is unspecified, f(x) figuratively represents any and all values of this function. It's a synechdoche. It's poetry, you clod. What, can't you handle symbolism in a symbolic notation? It's like x is everyman, and f is his shoe size, so f(x) represents not any one shoe size but all shoe sizes, of all mankind. Get it? Of course the part about f being shoe size is also symbolic. Just some symbolism inside the symbolism of the symbolism.
Thus f(x) literally denotes a particular value, but figuratively denotes any and all values, except when context tells you otherwise. It's up to you, the reader, to know which interpretation is correct. This layering of interpretations is, sadly, the root of much evil when it comes to probability and statistics. You can't get away from the figurative notation there. In fact, it just gets worse. We use p(x) to denote a probability density function, yet we use p(y) and p(z) to denote totally different density functions, and the only clue is that the letter inside differs. What's worse is that we write expectations using a notation like, for example, E[X|Y] where we want the expected value of X given a particular value for Y, that is, X is really just a label, not a value at all, not "every-value," but just a name. Whereas Y is here really meant to signify a value, probably in an "every-value" sense, though it depends on context. In point of fact, "X | Y" is a new random variable, but as such it's a new construction, what in computer science we would call an "anonymous function," literally an anonymous random variable mapping a sample space to some real number space. I would like to suggest that any notational language that supports such anonymous constructions is very complex. And when one is speaking a very complex language, then I think one must be very careful to express just what one means. And having a multiplicity of interpretive possibilities for a simple scrap of everyday math language like p(x|y) is, I think, a bad thing. A very bad thing. Abominable maybe. At least I abominate it: I invite you to join me. I love the poetry of Keats and Shakespeare, but I also want to being able to say things cloddishly and unambiguously when I will. No one wants to be forced to speak in symbolist verse at all times. But I feel like we have no literal language in math for some of these things -- we are thrown back on our "symbolic" notation (ha ha) and can't even nail down what we mean. Ok, I'm too tired to develop this thought further. Discuss amongst yourselves.
(I'm well aware that there's a fully formalized, axiomatic way to write down the basic molecules of meaning in probability theory. If you know this, then you surely also know that we don't typically employ that body of knowledge to workaday statistics like trying to describe what detailed balance is in MCMC sampling. The result is that the usual math notation for expressing interesting probability theory is, in my opinion, a mess.)
Obviously it's been awhile since I updated this page. It's been a full semester, overflowing actually. Anyway, one result of my studies is the following picture of caffeine, inspired in part by the funny kitteh pictures at www.lolcats.com and icanhascheezburger.com. No, it's not sophisticated humor.
I've been taking the last two weeks off. Finals are over and my teaching assistantship has ended for the summer, but I have not started work yet. It's nice to decompress a bit from the hectic year: I'm reading for pleasure, snapping some photos, and waking up late. Alas, it shall soon end.
I first wolfed down Over the Edge: Death in Grand Canyon, after having heard of the book in the always-classy Outside. The great thing I learned is that insiders omit the definite article before Grand Canyon. It sounds irritatingly English but I guess we'll all just have to get used to it. ("Oi, oyve fallen in Grand Canyon and bashed me Uncle Ned! Tayk me to hospital!") Apparently the snakes, scorpions, spiders, and centipedes never really kill anyone there. The major killers are airplanes and the Colorado river. This book was interesting but I can tell that the young manuscript lacked the guiding hand of a loving-yet-firm editor.
Now I've just begun Alan Turing: The Enigma, and I'm struck by author Andrew Hodges's skill at writing economical prose that tells the story, both facts and mood, of the young Alan Mathison's early years. The mood is depressing: at turns lingeringly genteel, threadbare, choking, lonely, and doomed.
It was at Chatrapur, in the autumn of 1911, that their second son, the future Alan Turing, was conceived. At this obscure imperial station, a port on the eastern coast, the first cells divided, broke their symmetry, and separated head from heart. But he was not to be born in British India. His father arranged his second period of leave in 1912, and the Turings sailed en famille for England.
This passage from India was a journey into a world of crisis. Strikes, suffragettes, and near civil war in Ireland had changed political Britain. * * *
But this conception of the modern world was not shared by the Turings, who were no dreamers of the World-City. Well insulated from the twentieth century, and unfamiliar even with modern Britain, they were content to make the best of what the nineteenth had offered them. Their second son, launched into an age of conflicts with which he could become helplessly entangled, was likewise to be sheltered for twenty years from the consequences of the world crisis.
What do you make of the use of both conceived and conception above?
In April I went to Tuscaloosa, Alabama to race in the Collegiate National Championships for triathlon. You can read more about it here.
You know you like them. It's like a solitaire version of Scrabble, except it's fun. Hmm. Ok, maybe they are kind of lame, but I know I like them.
Mr. Leopold Bloom ate with relish the inner organs of beasts and fowls. He liked thick giblet soup, nutty gizzards, a stuffed roast heart, liver slices fried with crustcrumbs, fried hencod's roes. Most of all he liked grilled mutton kidneys which gave to his palate a fine tang of faintly scented urine.
Kidneys were in his mind as he moved about the kitchen softly, righting her breakfast things on the humpy tray. Gelid light and air were in the kitchen but out of doors gentle summer morning everywhere. Made him feel a bit peckish.
The coals were reddening.
Another slice of bread and butter: three, four: right. She didn't like her plate full. Right. He turned from the tray, lifted the kettle off the hob and set it sideways on the fire. It sat there, dull and squat, its spout stuck out. Cup of tea soon. Good. Mouth dry. The cat walked stiffly round a leg of the table with tail on high.
-- Mkgnao!
-- O, there you are, Mr Bloom said, turning from the fire.
The cat mewed in answer and stalked again stiffly round a leg of the table, mewing. Just how she stalks over my writing-table. Prr. Scratch meh head. Prr. Lol.
