Competing at the Limit

I participate from time to time at a site called Math StackExchange, where users ask and answer questions about mathematics.  Most often, the questions relate to a student's coursework, but there are some deeper questions as well.  It's one of a family of similar StackExchange sites devoted to a wide variety of topics, only some of which are academically inclined.

One question that comes up every now and then is the definition of a limit.  It looks like this:

limx→af(x)=L⇔∀ε>0,∃δ>0,∀x,0<|x−a|<δ⇒|f(x)−L|<ε

And it reads like this:

The limit of f(x) as x approaches a equals L, if and only if for every positive ε, there exists a positive δ such that whenever x is within δ of a (except possibly exactly at a), f(x) is within ε of L.

Understandably, to many math students starting introductory analysis, this looks like so much gobbledygook.  Textbooks typically try to aid understanding by drawing a picture of a function f(x) in the vicinity of some value x = a, showing that as x gets closer to a, f(x) in turn gets closer to its limiting value L (which might not in fact be f(a) itself, if that value even exists).

But what if the sticking point for students isn't always that notion of better and better approximations (central as that is to the definition of a limit)?  What if the sticking point is the interplay between the "for every" (symbolized by the upside-down A: ∀) and the "there exists" (symbolized by the upside-down E: ∃)?  The intent of this definition, first conceived of by the French mathematician Augustin-Louis Cauchy (1789–1857) and formalized by the Bohemian mathematician/philosopher Bernard Bolzano (1781–1848), is to ensure that we can always get as close as we want to the limiting value (without necessarily hitting it), simply by being as close as we need to be to the argument x = a.

We can represent this as a sort of (almost irredeemably nerdy) game between two players, the Verifier and the Falsifier.  The Verifier is trying to prove the limit is right by showing that everything near x = a maps to an f(x) that's close to L, while the Falsifier tries to disprove the limit by challenging the Verifier to get even closer to L.  For instance, if the function f(x) = 2x+3, the Verifier might be trying to demonstrate that the limit of f(x), as x approaches 5, is 13:

Falsifier.  I don't think it's true; I think the limit is not 13.

Verifier.  Well, if that's so, then you must think there's some neighborhood of 13 that I can't force f(x) to lie in.

Falsifier.  Right.  OK, I challenge you to get within 0.1 of 13.

Verifier.  Sure.  If x is within 0.05 of 5, then f(x) will be within 0.1 of 13: f(4.95) = 2×4.95+3 = 12.9, which is within 0.1 of 13, and f(5.05) = 2×5.05+3 = 13.1, which is also within 0.1 of 13.  [There is more to it than that, such as that f(x) is monotonically increasing, but we'll leave these details out for now.]

Falsifier.  All right, but can you get within 0.01 of 13?

Verifier.  Yes.  All I have to do is force x to be within 0.005 of 5: f(4.995) = 12.99 and f(5.005) = 13.01.  In fact, I can answer any neighborhood of 13 you challenge me with, simply by halving it to obtain my vicinity of x = 5.  If you want me to be within ε of 13, then all I have to do is be within δ = ε/2 of 5.  Then f(5–ε/2) = 2×(5–ε/2)+3 = 13–ε, and f(5+ε/2) = 2×(5+ε/2)+3 = 13+ε.  It's foolproof.

Falsifier.  Hmm, I guess you're right.  I'll have to concede that the limit is 13.

The exchange would have gone quite differently if Verifier had claimed that the limit was 12.  Then, for instance, when Falsifier challenged Verifier to get within, say, 0.1 of 12, Verifier would have been unable to choose a vicinity of x = 5 such that f(x) is between 11.9 and 12.1 over that entire vicinity, because any value of x very close to 5—close as we like—always has f(x) very close to 13, and that clearly doesn't fall between 11.9 and 12.1.  But if Verifier can always figure out the right vicinity to force the function to fall in Falsifier's neighborhood, then they can prove the limit to be correct.

This approach to proofs has much broader applicability; in game semantics, and in a kind of logic called independence-friendly logic, many demonstrations rely on this kind of interplay between a Falsifying universal quantifier (the "for every" ∀) and an existential quantifier (the "there exists" ∃).

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Now for a digression to something that will seem totally unrelated at first.

In the late 11th century, into the 12th, there lived a Breton named Pierre le Pallet who was a precocious philosopher.  He was initially trained by William of Champeaux, but quickly grew capable of duelling wits with his teacher, and ended by starting a school of his own, against the advice of William.  By all accounts, he was a self-proud man, convinced simultaneously that he was brighter than anyone else and that no one else was giving him proper credit for this.  In his defense, he was generally regarded as one of the leading philosophers of his time, his specialty being logic, a tool that he wielded in an almost competitive spirit in defense of positions that were then considered heretical.  It was during his late adolescence that he took on the name that we know him by today, Peter Abelard.

As Abelard, his fame grew considerably, and people from all around sought his counsel.  One of these was a canon in Notre Dame named Fulbert, who wanted Abelard as a tutor for his niece.  She was then in her early twenties (we think—there is significant uncertainty about her birthdate), and had demonstrated herself to be remarkably capable in classical letters.  She had mastered Latin, and Greek, and Hebrew, and had applied these to a study of Christianity, to which she was devoutly dedicated.

Her name was Heloise d'Argenteuil, and she and her relationship with Abelard were in time to become famous.  Both of them found the other attractive, and in or around 1115, they started an affair just out of the watchful eye of her uncle.  Ostensibly, Abelard was tutoring her, but this would be interrupted periodically by a bout of lovemaking.  When they were separated, they would exchange personal messages on wax slate (parchment being too expensive even for billet doux that would have to be discarded or hidden).  A message would be incised on a layer of wax mounted to a wooden back; this message could then be read and the wax melted and smoothed over to be used again and again.

The two lovers could not necessarily deliver the messages personally without incurring Fulbert's suspicion, and so would have to rely on the discretion of messengers.  But as the messages were typically written in Latin or Greek, which the messengers couldn't read, teacher and pupil could exchange their letters under the apparent guise of lessons.  Abelard and Heloise apparently exchanged over a hundred letters this way, letters we have access to only because Heloise seems to have transcribed them onto a scroll (now lost) which was found centuries later by a French monk named Johannes de Vepria.

The affair progressed as far as Heloise bearing a son by Abelard, whom she called Astrolabe, after the astronomical instrument, and about whom we know almost nothing at all.  Around this time, Fulbert caught wind of it, and managed to force them to marry, although Abelard extracted a promise from Fulbert not to publicize the marriage, so as to protect Abelard's reputation.

Fulbert, however, had had his own reputation damaged by Abelard over other matters, and so he began spreading rumors of the marriage.  Abelard had Heloise installed at an abbey for her own protection, a gesture that Fulbert misunderstood as Abelard trying to wash his hands of her.  So Fulbert hired some henchmen, and one night, they went to Abelard's sleeping quarters, and castrated him.

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Abelard went into seclusion, and it is unclear that he ever saw Heloise again after this time.  However, about a decade or two later, they exchanged a sequence of seven or so longer letters, instigated when Heloise somehow got her hands on a letter that Abelard had written to a monk about his life story.  That letter included a retelling of her own story, and the two lovers were reintroduced to one another in this way.

Except that by this time, Abelard had decided to impose a sort of pious asceticism on himself that extended to any romantic feelings he might have had for his one-time wife.  Heloise, in turn, wrote him back, entreating him to concede those feelings, feelings she was sure he still retained.  In the last pair of letters, Heloise appears to have relented, and buried herself in her religious life, and Abelard seems to have praised and encouraged this.  But these letters are permeated through and through with an almost overwrought subtext.

So who convinced whom?  As if in honor of these two, whose story has become synonymous with medieval romance, the roles of the Falsifier and the Verifier are often personified by the love-denying Abelard, whose initial is a convenient mnemonic for the universal quantifier ∀, and by the love-asserting Heloise, whose name is sometimes spelled Eloise, whereby her initial is a convenient mnemonic for the existential quantifier ∃—symbols ineluctably entwined in the cherished logic of Abelard's youth.

i saw in yesterday your pretty when

I almost called this "in crude homage to edward estlin," but I thought maybe that would be too predictable.

Most people know about E.E. Cummings's free verse.  I first came into contact with his name, if not his poetry, from a poster in my seventh-grade English classroom.  (Does anyone remember Mr. Clancy from Redwood Junior High?  No?)  I don't think I actually read any of his poems until rather much later.  I did hear an exquisite (and in context, wholly inappropriate) love poem of his in Woody Allen's Hannah and Her Sisters, entitled "somewhere i have never travelled,gladly beyond."

I may as well say that although his deconstructive approach to grammar is refreshing, I find some of his poems orthographically grotesque.  Not for the reasons most frequently cited; I have no problem with his lack of capitalization (I do that myself in chats), or his exuberantly nested parentheticals, or anything pedestrian like that.  No, what bothers me are the superlatively trivial things, like not having a space after commas (see above, you have no idea how that killed me to accurately reproduce his title), or before parentheses, and that sort of thing.

Anyway, because of the renown of his free verse, not many people know that he wrote sonnets, too, and intensely romantic ones at that.  Sonnet XCII of his 95 Poems is one of his better known ones; it goes

i carry your heart with me(i carry it in

my heart)i am never without it(anywhere

i go you go,my dear;and whatever is done

by only me is your doing,my darling)

                                                                    i fear

no fate(for you are my fate,my sweet)i want

no world(for beautiful you are my world,my true)

and it’s you are whatever a moon has always meant

and whatever a sun will always sing is you

here is the deepest secret nobody knows

(here is the root of the root and the bud of the bud

and the sky of the sky of a tree called life;which grows

higher than soul can hope or mind can hide)

and this is the wonder that’s keeping the stars apart

i carry your heart(i carry it in my heart)

(It's a good thing that all I had to do was cut and paste; I don't know that I could have elided all those spaces otherwise.)  Anyway, here's my tyro's try at the same kind of thing, and at least it's honest, it's a thing I feel (and doggone it, I shall put spaces where I will):

i saw in yesterday your pretty when

i saw in yesterday your pretty when
and past a rise your beautifully where
(i do lose during you my now and then,
and inside you(r inside) my here and there).
since draw me to your captivating why
(a finger may mislead, i have no who
that cries the how you tear), i heard them sigh
your fragile yes or maybe noes to do.
with you i have no ask or answer (no
inquire or wonder, neither no believe,
no yet or still, no if (or so, or so)
for(giving life, where is no is to grieve))
but breath demanding breath, each every day
in death for(little death) you to replay.

Copyright © 2013 Brian Tung

The Slowness of the Post

Once again, it's National Poetry Month here in the States.  (Do I have any international readers?  Actually, do I have any readers?  Sometimes it's hard to tell, hint hint.)

I can't just be posting old ones, though, so this one is kind of new.  I started this and had the opening quatrain and the ending couplet a couple of years ago, but then gave it up.  This is my rather indifferent attempt, I'm afraid, at filling up the inside.  Don't worry, I'm not giving up my day job; I like that one a bit too much.

[EDIT: And I just noticed that this poem has extra bonus enjambment.  So, umm, yeah.]

the slowness of the post

When lovers in years past took quill in hand
to add to their epistolary chain
the latest, best-wrought link, they might complain
about the slowness of the post.  They planned
their thoughts for days, while trains traversed the land
with bundled hearts and holes, delayed by rain,
or frailty, or the smugness of the sane.
(But, yes, this did make love more tragic.)  And
now, though we write in liquid crystals, though
we fancy we eliminate the tragic,
anticipation, overnight, is no
less puzzling, no less vexing—no less magic.
    Look to the sky—yes, look, her answer, soon—
    look for it ere the waning of the Moon!

Copyright © 2013 Brian Tung

In Crude Homage to Edna St Vincent Millay

[From my Facebook page, originally written in February 2010, and here lightly edited.  I've been thinking about posting poetry here a bit, especially as it's National Poetry Month and all.]

Sonnets are convenient lunchtime reading material: short, yet dense with rhythm and sense (when reasonably well written). I've been working my way haphazardly through a collection of sonnets by Edna St Vincent Millay, one of the great romantic poets—at least, so say those who would know. Not to put too fine a point on it, I like them. What's more, I've been informed that it is a prominent chick-lit marker for the protagonist taking herself (over) seriously when she reads or even quotes poetry by Millay.

Well, I am nothing if not self-aggrandizing (often in the guise of self-deprecation), so despite my obvious gender challenge—being a guy—I have taken it upon myself to attempt a quasi-imitation of Millay. ("Aping" might be more apt a word.) I have used, as she does, the sonnet form (of a fittingly quasi-Petrarchan variety), and I have also taken as my subject unrequited affection, a common enough Millay theme. I have not, however, tried generally to affect the effortless facility with which she, Yoda-like, twists normal English word order like a pretzel. (I believe that the subject is permitted to precede the predicate at most once per stanza.)

By the way, Millay's name being two and a third dactyls lends itself conveniently to the limerick form. So as a kind of appetizer, and by way of introduction:

To Ms Edna St Vincent Millay,
I now offer this humble assay.
    For her sonnets are kings
    Of romantical things
And just what they're about none can say.

And perhaps you'll find that you like the appetizer better than the main course anyway.

in crude homage to edna st vincent millay

Your lips not once did tender mine, and yet
I loved you—no, and never once your hand
grasped mine in supplicating fever, and
I loved you still—nor even did you let
your eyebrows knit, or mouth to trembling set,
and still I loved you. (Once, perhaps, unplanned,
to quell persistent pity's keen demand,
you touched my head, as one would with a pet.)
Thus singularly blessed I count those days
where in a wondrous haze I hoped (or guessed)
that glances cast in jest were courting plays
to clasp in rapt amazement to my breast.
    And so I say—in seeking Love's mad thrall—
    you loved me best who loved me not at all.

Copyright © 2010 Brian Tung

EDIT: This sonnet can be considered a kind of lame reply to this one by Millay.

Matching Up in Hyperspace (or, Thirty Dancing)

Maybe it's because I've been writing about basketball a lot, but I thought today I'd do something a little different before continuing on, as promised, with a second game theory post.

A while ago, I remember reading an analogy about why it is that oil and water don't mix. (I don't remember where I read it, though, so if you recognize it, please tell me.) Is it that water molecules only "like" water molecules, and oil molecules only "like" oil molecules? Not at all—they all like water molecules!

A water molecule is often drawn as H-O-H, but that drawing is a bit misleading. The hydrogen atoms are actually attached at an angle, as below.

This one looks a bit like a Japanese cartoon character, if you ask me. At any rate, this asymmetry, top to bottom (as drawn here), means that we can speak of an oxygen end (the bottom) and a hydrogen end (the top). What's more, because of the way that electrons are arranged in each atom, the oxygen atom tends to draw electrons away from the hydrogen atoms. The oxygen end, so to speak, has more electrons hanging around it than the hydrogen end. Since electrons are negatively charged, the water molecule has a positive pole (the hydrogen end) and a negative pole (the oxygen end), and we say that the water is a polar molecule.

Water molecules attract each other because they are polar. The positively charged hydrogen end of one attracts the negatively charged oxygen end of another. In steam, the gaseous form, this is almost impossible to make out, because the molecules are too far apart and energetic, bouncing around far too wildly to show any mutual attraction. However, in ice, the solid form, the attraction is much more obvious.



It's a bit hard to tell which hydrogen atoms are associated with each oxygen atoms, but that's because in ice, the bonds are a bit confused. Even so, however, it's clear that we don't have water molecules bonding together oxygen-to-oxygen, or hydrogen-to-hydrogen. They only attach oxygen-to-hydrogen (in the hexagonal arrangement that yields those lovely snowflakes), because the molecules are polar that way. That's the way water molecules "like" each other.

Liquid water is intermediate between ice and steam. The molecules aren't fixed in place to each other as they are in ice, but neither are they bouncing wildly as they are in steam. Instead, they wander amongst each other, like people milling about in a crowd. And as they wander around, they stick to each other a bit, on account of their polarity. They attach and cohere, which makes water bead up, among other things.

What about oil molecules? Oil molecules tend to be symmetric in such a way that there is no clear polar end as there is in water. As a result, they are much less polar than water molecules are. Nonetheless, being weakly polar (under appropriate circumstances), they "like" other polar molecules, too. So why don't they attach to the water molecules, too?

The reason is that there is only so much room for molecules to attract each other. And here's where the analogy I mentioned earlier comes into play. You often find, at a school, that the most popular kids date other most popular kids (when they date), and the least popular kids date other least popular kids (again, when they date). Why is that? Is it that the least popular kids aren't attracted to the most popular kids? Well, it might sometimes be because of that, but often, they are attracted to the most popular kids; that is, after all, part of what makes someone most popular.

What gets in the way, however, is that the most popular kids, like most others perhaps, are also attracted to the most popular kids, and since such pairings satisfy both attractions, they get paired first. Then the next most popular kids pair up with other next most popular kids, they get paired next. And so on down the line. Or so the story goes.

Of course, it isn't quite that neat and clean with kids, but it is a reasonable approximation with what happens when you combine oil and water. They don't mix because the most popular water molecules hook up with other most popular water molecules, while the least attractive oil molecules are left hooking up with each other.

So much for oil and water. But now let's go back to that analogy, which as it so happens is what I really wanted to talk about. (The rest of that science was just for show?!) It doesn't ring true because we all know couples where we think, "Wow, she paired up with him?" How does that happen? It happens because people aren't one-dimensional.

Suppose all people were one-dimensional. Then you could rate each person with a number x—say, from 0 to 100. (I hate it when things are rated from 1 to 100. What's middle-of-the-road on such ratings? 50.5?) In such a case, if you have two 100's, wouldn't they choose each other above all others? You couldn't easily see a 100 pairing with a 25, if there's another 100 to choose from. Under such circumstances, the nth highest-rated male would always match up with the nth highest-rated female. Just like the kids at our hypothetical school.

Note: For reasons I despise (expositional convenience, basically), I'm writing this out heterosexually. Let it be clear that this isn't mandated in any way, and I'm aware of that. This treatment unfortunately makes it easiest for me to separate out two groups and draw what amounts to a bipartite graph between them. Sorry!

We might say, callously, that only one pair of people would say they feel completely satisfied with the pairing; everyone else is "envious" in the sense that there's someone else with whom they would rather have paired up. That's inevitable with one-dimensional people.

So let's give people another dimension: Let them now be rated with two numbers (x, y). Now, there is no universal and complete ordering on people. We might agree that if someone has both numbers higher than someone else, they are more appealing, but there is no universally accepted way to compare two people with one number higher and one number lower. This is akin to the problem with PER. It's entirely possible that everyone could be envy-free.

Here's what I mean. Suppose you have three males and three females. The three males are (60, 30), (50, 50), and (30, 60). So are the three females. Now there's no way you can say that the (60, 30) male is inherently superior to the (50, 50) male, or vice versa. The same is true of any other two males, or any two females. To decide amongst the alternatives, one needs a discriminating function of some sort. Let's say your function is 2x+2y. Then you would rank your three choices 180, 200, and 180, and you would choose the (50, 50) over either the (60, 30) or the (30, 60). If, on the other hand, your function was 3x+y, you would rank your choices 210, 200, and 150, and you'd choose the (60, 30) over the other two. Finally, if your function was x+3y, you'd pick the (30, 60) first. So it's possible for each of the alternatives to be first in someone's eyes.

Of course, to be a completely satisfactory pairing, both sides of the pairing must feel they got the best catch. But consider the (60, 30) male. Being a high-x kind of guy, he naturally values x more than y, perhaps, and his discriminating function will reflect that. (Some people, all they care about is x.) He might be exactly the sort of guy with a function like 3x+y, and would therefore pick the (60, 30) female. She, thinking likewise, would pick the (60, 30) male back. Likewise, the (50, 50) people might pair up with each other as mutually optimal choices, and the (30, 60) people too. It doesn't have to match that way, of course; it just has to match one-to-one. Maybe the (60, 30)'s love the (30, 60)'s, for instance, and vice versa.

On the other hand, this matching leaves someone who's (40, 40) out in the cold, because no discriminating function will rate them ahead of everybody else. Whoever matched up with them would always be upset that they didn't at least match up with ol' (50, 50).

It boils down to who's on the Pareto front. The Pareto front is made up of everyone who isn't universally worse than some other option. An illustration of this in two dimensions should hopefully make it clear:

Everyone on the Pareto front could be someone's optimal choice; everyone else would be a consolation prize. It's possible that everyone would be on the front, but it's unlikely, given a random selection of people.

Let's not be too hasty, though. There's an interesting dependency between dimensionality and being on the hull. In one dimension, exactly one person is on the front (barring ties); everyone else is beneath him or her. In two dimensions, it's a bit more complex, but suppose you had a hundred people, evenly spread out between (0, 0) and (100, 100). On average, maybe five people would be on the front. (The actual average is the sum 1 + 1/2 + 1/3 + ... + 1/100.)

Now let's increase it to three dimensions. If you have a hundred people spread out between (0, 0, 0) and (100, 100, 100), on average about 14 people would be on the front. All 14 could be the optimal choice for some prospective mate. As the number of dimensions goes up (and the number of possible discriminating functions, too!), the percentage of people on the front also goes up. With four dimensions, the average number of people on the front is 28; with five, it's 44; with six, 59—more than half! Ten dimensions are sufficient to push it up to 94, and by the time you have, oh, let's say thirty dimensions, the odds are about ten million to one in favor of every last person being on the front. Remember, it isn't necessary to have a highest value in any of the dimensions to be on the front; all you need is to not be lower than anyone else in all of the dimensions. As the number of dimensions goes up, it becomes awfully unlikely that you'll be lower than anyone else in every single dimension. We can have an entirely envy-free matching, all with the help of increased dimensionality.

OK, this may seem completely crazy, and I wouldn't blame you for calling shenanigans. Who would actually go and rank people using a set of thirty numbers? But this is exactly what one of those on-line dating sites advertises it does. Well, not exactly; it actually claims to use 29 dimensions. Why 29? I would imagine because it sounds somewhat more scientific than thirty. But beyond that, I think that they use as many as 29 because it makes it almost inevitable that you'll be on the front, that there'll be someone who you find optimal (or very nearly so), for whom you will likewise be optimal (or very nearly so). And although I think that's partly a marketing gimmick, I think there's some truth to it, too; if there weren't, the human race would have died out long ago.

I mean, how else does Ric Ocasek land Paulina Porizkova? For real, I mean!

Seventh Night

Last night was Seventh Night (七夕), the seventh night of the seventh month in the lunisolar calendar followed traditionally by the Chinese. Because the Chinese calendar usually starts with the second new moon after the winter solstice, Seventh Night usually falls sometime in August in the western calendar.

Seventh Night is associated in Chinese tradition with the story of the Cowherd and the Weaver Girl. In one common telling of the story, a young cowherd by the name of Niulang (牛郎) came across a fairy girl bathing in a lake—a girl named Zhinü (織女). Fascinated by her beauty, and emboldened by his companion, an ox, he stole her clothes and waited by the side of the lake. When she came out looking for her clothes, Niulang swept her up and took her back home. In time, they were happily married with two children. But when the Goddess of Heaven found out that a fairy girl had married a mere mortal, she grew furious and sent Zhinü into the sky, where she became the bright star Vega, in the constellation of Lyra the Lyre. (Watercolor by Robin Street-Morris, 2007.)

When Niulang discovered that his wife had disappeared, he searched high and low for her, but was unable to find her. Eventually, the ox told Niulang that if he killed him and wore his hide, he would be able to ascend the heavens to find Zhinü. Niulang did as the ox suggested, and took his two children with him to find his wife, becoming as he did the star Altair. Find her he did, but the Goddess of Heaven, angered once more by Niulang's impertinence, drew a river of stars—the Milky Way—forever separating Niulang (the star Altair) from Zhinü. Their two children became Tarazed and Alshain, the two dimmer (but still bright) stars that flank Altair in the constellation of Aquila the Eagle. But apparently the Goddess of Heaven was not entirely heartless, for once a year, on the seventh night of the seventh month, she sends a bridge of magpies (鵲橋) to connect the two lovers, for just one evening. And so Seventh Night is associated with romance (and also, interestingly, with domestic skills).

The celestial setting for the entire tale can be found in the Summer Triangle, which is bounded by three stars: Altair, Vega, and Deneb (in the constellation of Cygnus the Swan, also known as the Northern Cross). The Summer Triangle can be found in the night sky throughout summer and autumn; at this time of year, it passes nearly overhead at about ten in the evening. (Photograph by Bill Rogers of the Sa-sa-na Loft Astronomical Society, 2009; click to enlarge.)