Saturday, March 20, 2021

Bias, Unseen But Not Unfelt

 [This post is adapted lightly from a Facebook post I just made.]

<tl;dr> The Atlanta crime doesn't need to have been racially or sexually motivated, per se, for race or sex to have been a factor. </tl;dr>
So the other day, about a week and a half ago, I was accosted at the drug store where I was picking up some medications for the family. Harassed, really. Some fellow had come in line behind me, rather closer than six feet away. So (as is my wont) I moved forward a bit so I was halfway between him and the person in front of me. Something about that set him off and there followed a fairly dreary 30 to 45 seconds of him pointing in my face and accusing me of racism. He was Black, you see.
I'm not in need of any kind of support over this; it didn't last long, and I'm afraid I have too high an opinion of myself to be too upset by it. Mostly, I was anxious that he was breathing at me pretty heavily from close up (12 to 18 inches?). Only now, as the usual incubation period has more or less passed without any symptoms, have I gradually relaxed—about that, anyway. (And yeah, I'm aware the chances of my contracting COVID from him were pretty low—I figure about one in a thousand, something on that order. We were both masked. But I'm the sort to obsess about it a little.)

I've been thinking about that episode the last few days, though, in light of recent events in Atlanta. And it's not just about the fact that it happened, or that a police captain characterized the shooter as "having a bad day." These things are bad enough, but I think it's clear that they're bad. Few people are having trouble understanding that.
What got to me was a side line after the captain was taken out of his spokesperson role. Officials were quoted as saying that although the shooter denied a racial motive, they weren't ruling one out. And though that's not a bad thing as far as it goes, I'm concerned that it focuses on what is really a small percentage of a very large problem.
See, as I say, most people understand that racism is a bad thing, or at the very least, they understand that it's generally viewed as a bad thing. So as far as overt expressions of racism go, they know not to do it, or if they do do it, they keep it among like-minded people. But that's just the tip of the racist (or sexist, or any otherist in general) iceberg. Underneath all of that is a much larger mass of subliminal prejudicial behavior that mostly goes unnoticed.
Maybe that fellow would have harassed me anyway, but I think he was just that little bit more likely to do it because I was Asian. Or maybe that shooter would have been up for shooting someone who wasn't Asian, or wasn't a woman, but I think he was that much more likely to do it because they were. And a million other things that happen every day, of lesser consequence, but are just a bit more likely to have happened to the people they did in fact happen to.
And what makes them so insidious is the spectre of plausible deniability, that in any individual situation, one can defend oneself sincerely and successfully against charges of bias. Only in the large, statistically, can these biases be seen.
Most of these are not racist or sexist motives per se. Most of the time, the person is not actively (consciously or otherwise) seeking out someone who fits a particular profile. But by the same token, when the situation hits them, the voice inside them that says, hey, maybe let's not escalate this—that voice is just a little bit softer when it's someone they don't sympathize with for those reasons.
That voice is inside us all (mostly). But I don't believe that this voice speaks equally in response to all people of all creeds, colors, and sexes. I certainly don't believe mine does. Oh, I don't think I'm exceptionally biased or anything; I'm quite ordinary.
But part of the reason I wasn't more upset about being called racist, I think, is that I deeply believe bias exists in us all, and it's not possible to eliminate it. We can reduce it, but there's a part of being human that makes kneejerk classification a bit too automatic. The only real way to address that irreducible core of bias, I feel, is to explicitly bend over backward to counteract it; it's just too easy, otherwise—too human—to believe, honestly, that one is free of bias. And maybe I gave this man a pass for that reason.
Or maybe I just don't like to think of myself as being upset by it. Who knows?

Friday, December 13, 2019

High-Dimensional Weirdness

At work, I run a mathematics colloquium that meets every other Thursday.  I don't always present—I probably present about 20 to 25 percent of the time—but I did a recent one on the behavior of high-dimensional spaces.  I then came upon an oddity that I thought was worth sharing, for those three or four of you who might like that kind of thing.

In this presentation, I made reference to some dimensional weirdnesses.  While making the point that additional dimensions make room for more stuff (as I put it), I pointed out that if you put four unit circles in the corners of a square of side 4, you have room for a central circle of radius r = 0.414.  (Approximately.  It's actually one less than the square root of 2.)


Correspondingly, if you put eight unit spheres in the corners of a cube of side 4, you have enough space for a central sphere of radius r = 0.732 (one less than the square root of 3), because the third dimension makes extra room for the central sphere.

If you were to put a sphere exactly in the middle of the front four spheres, or in the middle of the back four spheres, it would have a radius of r = 0.414, just as in two dimensions, but by pushing it in between those two layers of spheres, we make room for a larger sphere.

Finally (and rather more awkwardly, visually speaking), applying the same principle in four dimensions makes room for a central hypersphere of radius r = 1 (one less than the square root of 4).

The situation for general dimension d (which you've probably guessed by now) can be worked out as follows.  Consider any pair of diametrically opposed unit hyperspheres within the hypercube (drawn in orange below).  Those two hyperspheres are both tangent to the central green hypersphere, and they are also tangent to the sides of the blue hypercube.

We can figure out the distances from the center of any unit hypersphere to its corner of the hypercube, as well as to the central hypersphere.  Since we also know the distance between opposite corners of the hypercube, we can obtain the radius of the central hypersphere:

One interesting consequence is that at dimension d = 4, the central green hypersphere is now as large as any of the orange unit hyperspheres, and above dimension d = 9, the central hypersphere is actually large enough to poke out of the faces of the hypercube.  Keep that in mind for what follows.

One other oddity had to do with the absolute hypervolume, or measure, of unit hyperspheres in dimension d.  A one-dimensional "hypersphere" of radius 1 is just a line segment with length 2.  In two dimensions, a circle of radius 1 has area π = 3.14159; in three dimensions, the unit sphere has volume 4π/3 = 4.18879....  The measure of a unit hypersphere in dimension d is given by

For odd dimensions, this requires us to take a fractional factorial, which we can do by making use of the gamma function, and knowing that

With that in mind (and also knowing that n! = n (n – 1)! for all n), we can complete the following table for hyperspace measures:

That last entry may come as a bit of a surprise, but it is simply a consequence of the fact that as a number n grows without bound, πn grows at a constant pace (logarithmically speaking), while n! grows at an ever increasing rate.  As a result, the denominator of Vd totally outstrips its numerator, and its value goes to zero.

But what if we combine the two, and ask how the measure of the central green hypersphere, expressed as a proportion of the measure of the blue hypercube, evolves as the number of dimensions goes up?  On the one hand, we've seen that the measure of a unit hypersphere goes to 0 as the number of dimensions increases, but on the other hand, the central green hypersphere isn't a unit hypersphere; rather, its radius goes up roughly as the square root of the number of dimensions.  How do these two trends interact with increasing dimensionality?  In case it helps your intuition, here's a table for the ratios for small values of d.

Those of you who want to work it out for yourself may wish to stop reading here for the moment.  Steven Landsburg, who is a professor of economics at the University of Rochester but earned his Ph.D. in mathematics at the University of Chicago, told a story of attending a K-theory conference in the early 1980s, in which attendees were asked this very question.  Actually, they were specifically asked not to calculate the limiting ratio, but rather to guess what it might be, from the following choices:

  • –1
  • 0
  • 1/2
  • 1
  • 10
  • infinity

Attendees were invited to choose three of the six answers, and place a bet on whether the correct answer was among those three.  Apparently, most of the K-theorists reasoned as follows: Obviously, the measure can't be negative, so –1 can safely be eliminated.  Then, too, the central green hypersphere "obviously" fits within the blue hypercube, so its volume can't be greater than that of the hypercube, so the ratio of the two can't be greater than 1, so 10 and infinity can likewise safely be eliminated.

Well, "obviously," you know that the hypersphere can in fact go outside the hypercube, so 10 and infty can't actually be eliminated.  So what is the right answer?

At the risk of giving the game away so soon after offering it, I'll mention that the answer hinges on, of all things, whether the product of π and e is greater or less than 8.  Here's how that comes about: We know that the measure of a unit hypersphere in dimension d is given by

But that's just the unit hypersphere.  If we take into account the fact that the radius of the central green hypersphere is

then the question becomes one of the evolution of the measure Gd of the central green hypersphere:

To figure out how this behaves as d goes to infinity, we first rewrite it as

Next, we make use of Stirling's approximation to the factorial function:

Applying this to n = d/2 gives us

and when expressing it as a proportion of the measure of the hypercube of side 4, we get

Finally, we observe that we can write (by taking into account one extra higher-order term in the usual limit for 1/e)

and we see that

The right-hand side is eventually dominated by the factor involving πe/8 = 1.06746..., which drives the ratio Gd/4d to infinity as d increases without bound—but it takes a long time.  A more precise calculation shows that the fraction first exceeds 1 at dimension d = 1206.  A plot of the ratio as a function of dimension looks like this:

Notice that the ratio reaches a minimum of very nearly 0.00001 at 264 dimensions; the exact value is something like 0.00001000428.  As far as I know, that's just a coincidence.

Friday, September 20, 2019

Misunderstood Rules in Sports, Part One of a Trillion

Because I apparently don't have enough random crap on my plate, I occasionally participate on Quora.  I'm there as Brian Tung; I'm not hard to find, other than you actually have to want to find me, and so far, that's not a very common thing.

Anyway, I often find myself embroiled in various debates (generally well-mannered, if not always good-natured) about various sports rules.  Most recently, the question was about passes or shots that go over the backboard.  For example, should this shot from 2009 by Kobe Bryant count?

Or how about this one from Jamal Murray, in 2019?

The common feeling is that these should not count, because the ball goes over the backboard, and everyone knows that a ball that goes over the backboard is out of bounds, right?


Well, it's complicated.  Complicated enough that I'm just going to drop this here for the next time this comes up.  Here's Rule 8, Sections II.a and II.b from the official NBA site:

a. The ball is out-of-bounds when it touches a player who is out-of-bounds or any other person, the floor, or any object on, above or outside of a boundary or the supports or back of the backboard.

This part of the rule is about what the ball touches, not where it goes.  There's a bit of excitement in that it uses the word "above," but in context, I think it's pretty clear that it refers to the ball touching something or someone above the boundary (the out-of-bounds line).

b. Any ball that rebounds or passes directly behind the backboard, in any direction, or enters the cylinder from below is considered out-of-bounds.

This is the relevant part.  Note that it uses the wording "directly behind the backboard."  To me, that means you take the backboard, and project it back away from the court; anytime the ball passes through that imaginary three-dimensional box, it's out of bounds.  It says nothing about the ball passing over the backboard.  If it meant that, I think it would have said that.

In both cases, the ball clearly goes over the backboard, but it never goes directly behind the backboard.  In the case of Kobe's shot, the best angle in this video (pretty poor resolution, but it was the best I could find) is found at about 0:48.  As for Murray's shot, well, read on.

I think the phrase "directly behind" is crucial.  It isn't enough that the ball go behind the plane of the backboard (which is four feet inside the baseline, so that would happen all the time).  It has to go somewhere where, if you were to look from the opposite baseline, you would see the ball through the backboard, not around it.

If you go online, you will see a majority of the web sites that discuss this question insist, quite authoritatively, that such shots are not to be counted.  As irritating as I sometimes find this, it's sort of understandable, because the wording of the rule is a bit terse, and also because the rules vary from governing body to governing body, as well as era to era.  For instance, these shots would be illegal in the NCAA:

Rule 7-1-3.  The ball shall be out of bounds when any part of the ball passes over the backboard from any direction.

This rule is stated again, almost verbatim, as Rule 9-2-2.

On the other hand, they're legal in FIBA:

Rule 23.1.2.  The ball is out-of-bounds when it touches:
  • A player or any other person who is out-of-bounds.
  • The floor or any object above, on or outside the boundary line.
  • The backboard supports, the back of the backboards or any object above the playing court.
So there's some excuse for getting this wrong (plus they eschew the Oxford comma, but that's another blog post for another time).  If that's not enough, the rule in the NBA has changed—see the postscript below.

Fortunately, we have an approved ruling, from none other than Joe Borgia, NBA Senior Vice President of Replay and Referee Operations (I'll bet you already knew that):

Jamal Murray's shot is discussed as the third case, at about 1:38 of the video.

"...When you look at this angle, our rule is the ball cannot pass directly behind the backboard.  So when you saw that replay, you saw the ball went up, and it went over, but it never went directly behind it.  Otherwise, we would have seen it through the glass; that would have been illegal.  But up and over is fine, so that is a good basket."

I think that should settle the matter fairly nicely.


Here's more from Borgia:

"The old rule stated it was illegal when the ball went over the backboard (either direction). So imagine the backboard extending up to the roof—if the ball bounced off the rim and hit any part of the imaginary backboard a violation was assessed. We had too many game stoppages when the ball bounced over the edge so we changed the rule to say the ball cannot go directly behind the backboard. That is why I said the backboard is now an imaginary ‘tunnel’ that goes back, not up to the roof like in the old rule."

Saturday, July 27, 2019

Postmodernism and a Classic of Chinese Literature

Bottom line up front: This is probably going to end up long, longer than it is now.  That might be true no matter when you're reading this.

A couple of years ago, I detailed on this blog a series of Chinese novel reading projects: 西遊記 Journey to the West by 吳承恩 Wú Chéng'ēn, 生死疲勞 Life and Death Are Wearing Me Out by 莫言 Mò Yán, 邊城 Border Town by 沈從文 Shěn Cóngwén, and 圍城 Fortress Besieged by 錢鐘書 Qián Zhōngshū.

After that, I took a bit of a break.  I had intended to continue on to 紅樓夢 A Dream of Red Mansions by 曹雪芹 Cáo Xuěqín, and had even read a couple of pages, but my father warned me against that one, suggesting instead 三體 The Three-Body Problem by 劉慈欣 Liú Cíxīn.  Well, I read a couple of pages of that too, but put it aside, probably because I read the Wikipedia plot summary and I decided  I didn't like the conspiracy-theory angle.

Then sometime in the spring of 2018, I restarted Red Mansions once again, this time in (relative) earnest.  I had bought David Hawkes's English translation around the time of my first abortive attempt, and I now followed along in both languages, more or less as I had with my previous projects.

Red Mansions (more commonly translated as The Dream of the Red Chamber, but Hawkes suggests this is misleading, and I tend to agree) is unusual—perhaps even unique—in Chinese literature for persistently and insistently asserting its own fictionality.  Other Chinese novels exhibit an array of the magical and the mystical, more so than Red Mansions, but even with that wink and nod to the reader, the novels themselves typically present the events as though they really happened, usually tying the events to a specific epoch in Chinese history (for example, such-and-such a year in so-and-so's reign).  Historicity is a big deal in Chinese fiction, ironically enough.

Not so Red Mansions.  After Cao motivates his novel with the desire to commemorate the young girls he knew as a well-to-do boy, the rest of the novel is said to be a story engraved on a consciousness-endowed, polymorphic jade stone, whose own story frames the central story, and who is brought down to earth to experience life by a Daoist priest and a Buddhist monk.  Echoes of all three (or perhaps it is they themselves) reverberate throughout the book, pushing the plot—engraved on the stone, remember!—this way and that.  Such adumbrations seem familiar to those of us looking back at the evolution of 20th-century Western literature; see James Joyce's Finnegans Wake for a notable, if rather denser, English analogue.  But for a novel written in 18th-century China (manuscripts were circulating at the time of Cao's death in 1763 or 1764, and the first printed edition arrived in 1791), it was positively revolutionary.

Perhaps because of that, perhaps because of the iconic love triangle in the central story, or perhaps it is supposed to be revered in the annals of Chinese literature, Red Mansions occupies a central position in the Chinese collective literary consciousness.  (My mother started reading it when she was younger, and never finished it.  She found it fairly ordinary, but in addition, she has a tendency to mistrust any hyperbolic criticism, positive or negative, and the mountains of praise heaped on the story, amounting almost to hysteria, turned her off to reading it.)  When I went to Taiwan earlier this year, I stopped in a bookstore, and there were no fewer than a dozen different editions of Red Mansions, along with at least as many critical studies and examinations.

And Red Mansions is enormous.  I read a version I had found online, cobbling it together and having to fix occasional typos, and in one case, replacing three pages that had strangely gone missing.  At a normal font size, it occupied nearly 1400 pages; this is typical of printed editions too.  The English translation by Hawkes and John Minford (Hawkes's student) runs about 2500 pages, in five volumes.  (This kind of expansion is typical of translations from Chinese to English, and there's plenty of speculation as to why that is.)  This is something you have to commit to.

Speaking of the translation, Hawkes and Minford are meticulous, translating Cao's versatile prose and poetry more or less word for word.  As is typical, the author makes assumptions of his readership, assumptions that are still reasonable-ish for well-read modern Chinese, but which native English readers have no hope of meeting.  Hawkes and Minford choose the expedient of weaving context into the main text, resorting to footnotes and appendices only when absolutely necessary to avoid an abrupt dump of history.  Some appendices also explain some editorial choices in the translation.

Some of the word choices are oddly obscure, opting for 75-cent words (accounting for inflation) when a nickle will do without interrupting the tone.  And when I say 75-cent words, I mean words that I had never heard of in my entire life until now.  I'll try to collect a selected list of them so you know what I mean.  But by and large, the text fits what I read in the original Chinese.  There is another complete English translation, by the husband-and-wife team of 楊憲益 Yáng Xiànyì and 戴乃迭 Gladys Tayler Yang, that is also supposed to be good, and a bit more literally faithful.

The Story

At the center of the story that occupies the vast majority of Red Mansions' 120 chapters is the 賈 Jiǎ family.  Attached to the emperor by virtue of the service of past family members, long since dead, they are wealthy and extravagant.  People dress up to have tea, to move from one house to another in the compound, to go to bed.  They live a life of leisure, eating rare delicacies and drinking fine wine.  Even when they fall ill, their medicines (Chinese traditional, naturally) are the most exquisite available.  Their ginseng has to be picked at just the right time, with just the right shape to it.

The young scion of the family is 賈寶玉 Jiǎ Bǎoyù, a precocious and willful boy of about 13 at the start of the novel, who is pressured by his father to study the Confucian classics, but who mostly only has eyes for the girls of the family.  His name means "treasured jade," because he was born with a jade stone in his mouth—the magical stone from the frame story.  (An alternate title for the novel in both Chinese and English is 石頭記 Shítoujì The Story of the Stone.)  The two principal girls in the story are 薛寶釵 Xuē Bǎochāi, the only daughter of Baoyu's mother's sister, and 林黛玉 Lín Dàiyù, the only daughter of his father's sister.

Daiyu and Baochai are complementary yin and yang.  Daiyu is artistic, mercurial, and consumptive; Baochai is sensitive, compassionate, and robust.  A combination of dream sequences and wordplay implies that Baoyu's ideal woman would be a combination of the two: Both Daiyu and Baochai share one character of their given name with Baoyu.

But most of the family's younger generation is girls—a circumstance that exerts multiple forces on the main characters.  Baoyu is the only proper male member of the Jia family in his generation; he has only a half-brother Huan who is miserably jealous of Baoyu and who spends most of the novel plotting against him and otherwise acting like a dog who has been kicked to the curb rather too often.  As a result, tremendous pressure is brought to bear on Baoyu to continue the line and to sustain the emperor's favor.  As the family holdings slowly dwindle as the combined result of extravagance, bad luck, and traitorous servants, the family feels with greater urgency every ebb and flow in the affairs of Baoyu.

It is not only Baoyu who feels the effect of the gender imbalance in the household.  Daiyu comes to the family grounds when her mother dies and her father, who cannot bring her up, sends her to his in-laws.  From the beginning, she feels like an outsider with almost all of her relatives, despite their best efforts—all, that is, except Baoyu, to whom she feels an almost instant connection and affinity (and vice versa).  Otherwise, she is in constant fear of being left out on her own in the cold.

It is their romance, suppressed and sublimated by the strictures of Chinese tradition (in which marriage is a matter of parental prerogative), that forms the backbone of the novel, and which plays against the backdrop of the slowly declining Jia family fortunes.  Daiyu yearns with all of her heart to marry Baoyu, both for survival and because she loves him, but it is not up to her.  And because there are no other eligible Jia boys, any other girl—meaning Baochai, first and foremost—represents potential competition for a prize that only one of them can win.  In the end, the resolution of this emotional struggle also serves to drive the resolution both of Baoyu's psychological development and, at a larger scale, of the Jia family's fate.

The Authorship Question

It almost wouldn't be a classic Chinese novel if there weren't some question about its provenance.  Journey to the West, for instance, is merely attributed to 吳承恩 Wú Chéng'ēn; it is not actually known with certainty that he wrote it.  He is known to have written something by that name, but because there are in fact many writings of various lengths and degrees of historical accuracy by that name (it is rather generic, after all), and it was not found in his possession after his death, the attribution is only probable.

In the case of Red Mansions, there is no such question regarding Cao and the first two-thirds of the novel.  Though there are a dozen or so different manuscripts, the differences are generally minor and betoken no substantial variance on plot or characterization.  Nor is there nowadays any question that Cao is responsible for them.

The problem arises with the remaining 40 chapters.  There seem to be no fair copies that date back to Cao's day that contain anything past Chapter 80, at all.  And the plot moves along with sufficient leisure—the leisure that eventually dissuaded my mother from finishing the book—that by Chapter 80, things only then seem to begin to climb toward a climax.

Nevertheless, in 1791, for the first printed edition, 高鶚 Gāo È, along with his friend 程偉元 Chéng Wěiyuán, cobbled together a collection of manuscript drafts that together appeared to comprise the 40-chapter conclusion of the novel.  By this time, the authorship of the novel had been forgotten and would have to await future literary investigation to rediscover.

But there would be other, thornier questions to resolve almost immediately.  The general public had been clamoring for the end of Red Mansions, and Gao's completion served to satisfy their needs.  The more dedicated aficionados of the book were another matter.  At issue are an array of intimations and premonitions in the first part of the book, notably a series of poems in Chapter 5, which seem to impose quite clear restrictions on the eventual fate of many of the main characters (including the "big three").  These are further reinforced by a series of well-known annotations by anonymous commenters who are nevertheless clearly intimate friends or relations of Cao.  But Chapters 81 through 120 in Gao's edition seem to contravene much of this material, some of it quite severely.

For example, in Chapter 5, Baoyu dreams that he sees a book that depicts, in pictorial and textual riddle form, the fates of the girls in the family.  One of them is 香菱 Xiānglíng, which Hawkes renders as Caltrop.  The picture associated with Caltrop makes it clear that she will die at the hands of the jealous stepwife of her master.  But in Gao's ending, it is the stepwife who dies, accidentally poisoned by her own hand when she tries to murder Caltrop.  What's more, it seems likely, in the light of various suggestive passages, that Cao originally had planned a much more harrowing ending for the Jia family than what was eventually presented in Gao's ending.

There are lesser inconsistencies, different manners of death from what seems preordained.  Together, they seemed to indicate to the increasing number of close students of the novel that the completion that Gao edited was not Cao's.  Either Gao edited material that was written by someone else, or (it was suggested increasingly often as decades passed) Gao wrote it himself.  This is still the orthodox position.  In recent years, statistical stylometry has even been employed to show that there is a substantial discontinuity in style between the first 80 chapters and the last 40.

On the other side of the ledger are troubling inconsistencies of the same sort, which already appear in the first 80 chapters that are universally acknowledged to be Cao's.  The root of the problem is that Cao was an inveterate reviser, who by his own admission (in the body of the novel itself, naturally) had already rewritten various parts of the entire story several times.  Over time, he must have changed the fates of many characters across the entire breadth of the book.  He was not, however, the most careful reviser, however, and scattered in the thousand-plus pages are numerous continuity errors.  Chief among these were the various poems.  They could not be rewritten nearly as easily or as transparently as prose, so in many cases, Cao merely left them the way they were (possibly intending to return to rewrite them, should the opportunity arise), preserving the older versions of characters (in Hawkes's words) "like flies in amber."

Such observations have led Hawkes, Minford, and Anthony Yu (who authored the tremendously literate translation of Journey to the West, remember) to conclude that despite the questions raised by some of the unfulfilled prophecies, the last 40 chapters in Gao's edition appear to complete Cao's general intent, if not his exact wording, and that Gao likely did just edit some collected fragments, rather than creating the completion out of whole cloth, as used to be the prevailing opinion.  But the debate continues.

Its Place in Chinese Literature

All of these needlesome questions notwithstanding, Red Mansions engrosses more of the Chinese reading public than ever.  What accounts for its endless fascination?

Some of it is surely what my mother complained about: a kind of worship cult that has grown up around it.  Because it is continually written about, readers conclude, there must be something for people to be writing about.  We always want to know what all the fuss is about.

But it seems to me that there is more to it than mere reputation.  There is an air of mystery pervading it, both in the story itself and in the story of its creation.  And despite its occasionally glacial pace and fascination with 18th-century Chinese high-class culture, it confronts questions about the meaning of life and reality more directly than any other prominent piece of Chinese literature.  To read Red Mansions is to expose oneself to contradictions of experience and truth.  One can decide that they are merely a matter of perspective, but I think it is hard to argue that they are immaterial—fictional or otherwise.

Thursday, October 18, 2018

Mookie Betts's Glove Was in the Field of Play

I got the tl;dr out of the way in the title.

I've written previously about the value of multiple points of view (literal points of view in this case, but I think it's valuable for figurative points of view, too).  Last night, in Game 4 between the Boston Red Sox and the Houston Astros, was another example.

Here's the situation as it was in Houston (the location is kind of interesting, though not really important to the ruling).  It's the bottom of the first, and the Astros are already down 2–0, but they have George Springer on first after a one-out single, and Jose Altuve up to bat.  Altuve hits a deep fly to right, and Red Sox right-fielder Mookie Betts reaches up and seems about to make the play, when his glove is closed shut by a fan's hand.  The ball bounces back into right field, where Betts retrieves it and fires it back into the infield.  Altuve ends up on second, and Springer (who presumably had to wait to see if Betts made the catch) stands on third.

Umpire Joe West initially calls a home run, and then appears to indicate interference.  The umpires collectively go to the replay, and after a delay of a few minutes, they call Altuve out, and order Springer to return to first.  After Marwin González is hit by a pitch, Yuli Gurriel flies out more conventionally to right and the Red Sox escape without further damage.

In the aftermath of the Red Sox' 8–6 victory, however, there was considerable controversy over whether the interference call was the right one.  The ruling was that because Betts's glove did not exit the field of play—that is, it did not cross the imaginary plane of the outfield fence—he was interfered with.  Had the glove been beyond the fence, then any contact with the fans would not have been considered interference.

The problem is that it's far from obvious where Betts's glove was at the moment of contact.  The Red Sox observed (as did some others) that Betts's body had yet to reach the fence, but the Astros pointed out that Betts was reaching backward for the ball.  Both sides agreed that the ball would have gone into the stands were it not for Betts, and both sides agreed that Betts had a good chance of catching the ball.  (I've seen a few fans claiming that Betts simply closed his glove early, but neither I nor any professional commentator seems to find that credible.)

Nevertheless, both sides also thought the replays confirmed their conclusion, each perhaps pretending to a greater certainty than they really felt.  They're really not that conclusive either way, at first glance, and it was important, probably, that the call on the field was interference.  Here's a shot from one angle, for instance (the left-field camera, I think):

Can you tell where Betts's glove is in relation to the fence?  I can't.

Well, we don't have to tell from that shot alone.  Here's a second shot from another angle (maybe the first-base camera):

Hmm, it's not obvious from that shot either.

Once again, though, we don't have to rely on either shot in isolation; fortunately, the two images together will tell us what we need to know.  Both shots show the play a split-second after the fan had made contact with the glove, and with the ball just about to strike the outside of the glove.  The fans are still looking up because they're not trained to follow the ball into the glove, and because that baseball is moving fast, but that white blur is the ball in both photos.

How does this help us?  Well, let's take a look at where the glove is in relation to the wall.   Here are the same two shots, but with the same location marked on the outfield wall padding:

Notice where the glove is in relation to that mark in the two images.  It's to the right of that mark from the point of view of the left-field camera, but it's just about in line with the mark (or maybe a little to the left) from the point of view of the first-base camera.  It's simple triangulation: If the glove is directly above the fence, then it should be in the same position with respect to the mark from both views.  If it's in front of the fence, it should appear further to the right in the first view (from left field), and if it's beyond the fence, it should appear further to the left in the first view.

Since it's further to the right in the first view, the glove must have been in front of the fence at that moment, and the interference call is the right one.  (I was mildly surprised to discover this, by the way.  If I had to guess, I would have guessed that the glove was beyond the fence—but I would have been pretty loathe to guess.)  Without knowing more about the location of the cameras relative to the wall, we can't be sure how much in front it was, but at any rate, the contact was made in the field of play.

ETA: Here's a third, intermediate view (from behind home plate?), further confirming the findings:

Monday, April 23, 2018

Cicada Recurrence and the Allee Effect

One of the best-known phenomena in the insect world is the unusual recurrence of various populations of cicada.  There aren't any cicadas out here on the West Coast, where I live, but they are endemic to the Northeast.  The periodical cicadas (there are non-periodical cicadas, apparently) are notorious for having life cycles that are synchronized to one of two (relatively) large primes: 13 years and 17 years.  The big question, of course, is why: Why do cicadas have life cycles that are synchronized in this fashion?

One could divide the 13-year cicadas into 13 distinct subgroups, depending on which year they emerged, and divide the 17-year cicadas into 17 subgroups along the same principle.  Physical observation of cicadas, as shown in the Wikipedia plot summary, reveals that only about half of the 13+17 = 30 subgroups actually manifest in the United States (where the cicada is native), however, with two subgroups becoming extinct within the last century or two.  Nonetheless, the periodicity is well enough established that there should be a rational explanation of this phenomenon.

 One historically proposed reason for the synchronization has been that the long recurrence time limits exposure of the species above ground to predators, and that when they are exposed, there are so many of them that predators cannot possibly decimate them (a fact well attested by the unfortunate farmers who have to deal with them), thereby ensuring the continued existence of the population.  Although this is surely part of the answer, it only explains why the period is long; it doesn't explain why the period isn't 12 or 15, for instance, rather than 13 or 17.  These latter periods would only provide additional benefit if the likely predators of the cicada likewise had a life cycle punctuated by years of inactivity, which turns out not to be so.

A more successful explanation involves hybridization.  It is hypothesized that whatever mechanism governs the return of the population after however many years is based on a biological clock that is adjusted to activate periodically, and that if a 13-year cicada were to mate with a 17-year cicada, the result would be a substantial number of cicadas with unpredictable, but likely shorter, periods.  (Too long, and the individuals would die of old age, anyway.)  Such offspring would be more vulnerable to predation, so there is an evolutionary premium placed against hybridization.  Computer simulation studies show, however, that if we assume an initial species-wide distribution of a variety of periods—some prime-numbered, some composite—the prime-numbered periods remain, but so do some of the composite periods.

This 2009 paper, by Tanaka et al., explains away the remaining composite periods by means of something called the Allee effect.  In many population dynamics analyses, it is assumed that the fewer instances of a species exist, the more likely any instance is to survive—it being presumed that there is no disadvantage owing to an excess of resources.  There may be no such disadvantage, but it is nonetheless the case that there are situations where the reverse is true, for small populations: the greater the population, the more likely any individual is to survive to reproduce, because it benefits from the increased support and robustness of the larger population, up until the point where that larger population represents more competition than cooperation.  This reverse but very natural-seeming tendency constitutes the Allee effect.

Tanaka and company simulated the cicada species under a very simple hybridization model, both with and without the Allee effect, starting with subgroups with a range of periods varying from 10 through 20 years.  They found that without the Allee effect, there was broad survival of all of the cicada subgroups, with the 16-year subgroup thriving the best.  But with the Allee effect, the result was startlingly different: Only those cicada subgroups with periods of 13, 17, or 19 years survived, depending on some of the initial parameters.

Since the actual mechanism of the periodicity is not well understood yet, this study is more suggestive than dispositive, but the results are provocative.

Tuesday, March 7, 2017

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:

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×(5ε/2)+3 = 13ε, 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" ∃).

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.

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.