October 31, 2004

From politics to mathematics and back

Posted by shonk at 12:54 AM in Geek Talk , Politics | permalink | 5 comments

Last night I found myself with an unusually large chunk of time on my hands and, after doing some maintenance work around here that I’d been putting off, decided to catch up on some blog-reading. I read Colby Cosh’s excellent analysis of the ALCS from a week or so ago, enjoyed Billy Beck’s musings on book addiction and rantings on the justice system, caught up on the No Treason/Karen DeCoster/Thomas DiLorenzo shitstorm, uncovered the latest links that appear below in the “External Links” section, and enjoyed Scribbling’s pomegranate pictures. Somewhere along the way, I came across Cosmic Vortex’s “First political diatribe,” which suggests the notion of “political shock levels” as a complement to the future shock levels which extropians go on about. The author lays out a sort of political spectrum, ranging from communist to fascist, and then suggests the following:

Now, its very easy for a socialist and a progressive to discuss issues and come to an understanding, but try to get a socialist and a right wing republican together, and nothing will get accomplished except frustration and anger. Where does this leave us? Not in a good situation really - as theirs no real way to drag anyone more then 1 level away. Even if they did want to try to understand your position, they just couldnt map the concepts over if you jump too far. The cognitive differences would be un-breachable and it would require starting at the begining of their conceptual “tree”, validate every concept along the way, and maybe then something could be worked out.

Interesting idea and stated in a somewhat unique way. What really caught my attention in reading, though, was the sentence I’ve taken the liberty of italicizing. I have to admit, the very first thought that popped into my head upon reading that sentence was: “Sounds like a chain complex!” For those too lazy to click the link, a chain complex is basically a sequence of maps between objects such that moving two steps along always takes you to zero. They arise a lot in topology and homological algebra (for example, I first ran across them while learning about simplicial homology in an algebraic topology class). The connection with shock levels being that if you try to map more than one level down the line you can’t go anywhere but zero, just as the conversation between socialist and republican goes nowhere.

“A nice little analogy,” I though to myself, not quite realizing, for the moment, how loony it would have sounded had I tried to explain it (at this point tenses completely break down, given that I just have tried to explain it). Consider, in addition, how one of my office-mates and I had laughed earlier in the day when she described having just caught herself before asking two of her students (who are identical twins) if they were “isomorphic”. I know I’ve talked about this before (that time when a friend referred to this Strong Bad song as a “canonical techno song”), but I still find the way in which the accumulation of a new vocabulary shapes my outlook either amusing or frightening, depending on the time of day.

Of course, in a sense, the vocabulary is the least important part of what I’ve (hopefully) learned in the last year or so of grad school, but applying the vocabulary outside of its mathematical context is perhaps the most obvious outward sign. Well, one of the most obvious, anyway. Perhaps the other obvious sign of what might be called my increasing mathematical sophistication (or confusion, depending on your perspective) manifests itself in how I answer the questions of my students.

I’m currently teaching four recitation sections of a class innocuously called “Ideas in Mathematics” in the course catalogue, but of which the unofficial course title bestowed by the professor is “Mathematics and Politics”.1 A friend rather uncharitably characterizes it as “math for morons”, in that it’s the only freshman-level non-calculus course that fulfills the college’s math requirement. Anyway, the point is that I spend most of my time answering questions about the homework or the lectures, and, in answering, I often find myself engaging in impromptu monologues about how intuitionists would object to proofs by contradiction or how mathematics only describes the world insofar as it simplifies away the hard bits. And, most importantly, I have a very difficult time answering conceptual questions definitively.

Needless to say, I imagine my students find it frustrating when, for example, they ask “Why is a conditional true when its antecedent is false?” and I have to say something along the lines of the following:

Well, the short answer is because it’s defined that way, and the long answer is still because it’s defined that way, but it’s defined that way because that’s really the most reasonable way to define it. You see, when we’re thinking about whether a logical statement is “true” or “false”, it’s probably best not so much to think actually in terms of “true” and “false”, but rather in terms of compatibility with the world. In other words, can you believe the statement while also believing in reality without contradicting yourself? We only say the statement is “false” if not; otherwise we call it “true” even though it may be counterfactual, absurd, or completely irrelevant to reality.

At this point, I’m usually lucky if the looks I’m getting are merely quizzical. So I try again:

Well, let’s think in terms of an example. Suppose, back in 2002, a friend told you “If the U.N. approves a war in Iraq, France will go to war.” Now, we know that, in reality, the U.N. didn’t approve the war and that France didn’t go to war. So this is a situation where the antecedent and consequent are both false, so, if we’re thinking in terms of logic, we would say the conditional is true. Why? Well, because you can believe what your friend said and also believe in reality. That is to say, you can believe the statement without contradiction. So the assignment of “true” or “false” is more or less like how you treat a friend: because he’s your friend and you trust him not to mislead you, you assume he was telling the truth unless you can definitely prove that he was lying. In this case, the only way you could know he was lying would be if the U.N. had approved the war and France had stayed home (i.e. the antecedent is true and the consequent is false), since that’s not what actually happened, we would say that the statement is “true”.

Having given this explanation more than a half dozen times, it’s been mildly surprising that nobody has actually called me (and, by extension, math) out on it all being a bunch of convoluted bullshit, but I have to imagine some were thinking it. Usually, at this point, seeing the pained expressions on some of the faces staring out at me, I say something along the lines of “Of course, you could just think of this as the definition,” which seems to be a great relief to some. Which is ironic, given that, without the explanation, the notion that things could be this way just because that’s how they’re defined seems entirely unsatisfactory (which, by the way, I completely agree with. Definitions suck without context).

Having spent ten minutes writing about the conditional, I’m not sure it really illustrates the point I’m trying to make. Perhaps more appropriate would be the times that I’ve had to catch myself before I start ranting about epistemology, theories of logic, reductionism and how mathematics education is, essentially, a system of useful lies. Just as a calculus teacher extolls the virtues of the definite integral, talking about how useful it is and how many amazing physical properties it explains without mentioning that, in any actual application integration is not only difficult but usually impossibly difficult, I find myself teaching material which is useful in certain cases but usually too simplistic to be applicable to the real world. I try to point this out as much as possible, but I think it’s still probably misleading.

That having been said, the underlying concepts are, in fact, incredibly deep. It’s difficult, though, to emphasize that what’s important are the concepts, the fundamental ideas which lead us to particular formulas or computations, especially when midterms are looming and homework is due on Friday. I remember one student asking, the night before the midterm, if she ought to memorize a particular counter-example listed on the review sheet. My honest answer was “No, I don’t think you should memorize it; I think you should understand it,” which I don’t think she liked very much.

That question, though, lies at the heart of the topic that I’m apparently (finally) coming around to, which is that there seems to be a fundamental dichotomy in most people’s minds between, say, the humanities and mathematics. I doubt if anybody would ask an English professor, the night before a midterm, if he ought to memorize Joyce’s “The Dead” for the test, but in a math class it seems like a perfectly legitimate question (incidentally, I’m not trying to say that memorizing is completely worthless; in learning a foreign language, for example, unless one is lucky enough to be living in the country where the language being learned is spoken there’s really no way to make progress without memorizing verb conjugations, vocabulary, etc.). The fact that, for whatever reason, mathematics seems to be equated with rote memorization and plugging values into a formula seems to me to be one of the primary reasons that so many people have such a strong aversion to math.

Which is completely understandable, in a way. Memorizing is boring and almost completely lacking in cognitive content, which most people instinctively recognize, and the fact that math is equated with this boring activity is, I think, one of the primary reasons why an aversion to mathematics is considered acceptable even among people who would strongly decry stunted development in other intellectual pursuits. As John Allen Paulos puts it in Innumeracy: “In fact, unlike other failings which are hidden, mathematical illiteracy is often flaunted: ‘I can’t even balance my checkbook.’ ‘I’m a people person, not a numbers person.’ Or ‘I always hated math.’”

As I look back on the above, I hope I’m not giving the wrong impression about my students. They’ve been wonderful, certainly much more perceptive and good-natured than I had any reason to expect, and I hope they’re learning as much as I am. What it comes down to, I think, is that it’s virtually impossible to interact on a daily basis with people whose level of expertise in a given field is significantly less than one’s own without having to think quite a bit both about the nature of that expertise (imperfect though mine still is) and the misconceptions about the field that will inevitably come to light.

Anyway, I’ve now strayed quite far afield of what I originally intended to write, which was a self-deprecatory post about how I’ve become almost stereotypically geeky in grad school, but I guess the above sort of illustrates that point.

1 Actually, a very interesting class. Aside from learning some basic logic and doing some simple proofs, we’ve talked a lot about different voting systems, leading up to the proof of a simplified version of Arrow’s Impossibility Theorem, the full version of which says that there is no voting system (other than a dictatorship, which everybody pretty much agrees isn’t much of a voting system) which satisfies both the Pareto condition (which says that if everybody prefers candidate X to candidate Y, then Y will not win the election) and independence of irrelevant alternatives (i.e. there is no “spoiler effect”). Also, we’ve learned a bit about power indices, namely the Shapley-Shubik and the Banzhaf indices, and are now starting on some basic probability.

October 29, 2004

Yes, I'm lazy

Posted by Curt at 10:51 AM in Literature | permalink | 2 comments

Now that I finally have my own Internet connection, here's that sonnet I promised to post something like a month ago (it's #24, not 15):

Quand vous serez bien vieille, au soir, à la chandelle,
Assise auprès du feu, dévidant et filant,
Direz, chantant mes vers, en vous émerveillant
"Ronsard me célébrait du temps que j'étais belle."

Lors vous n'aurez servante oyant telle nouvelle,
Déjà sous le labeur à demi sommeillant,
Qui au bruit de Ronsard ne s'aille réveillant,
Bénissant votre nom de louange immortelle.

Je serai sous la terre, et fantôme sans os
Par les ombres myrteux je prendrai mon repos;
Vous serez au foyer une vieille accroupie,

Regrettant mon amour et votre fier dédain.
Vivez, si m'en croyez, n'attendez à demain:
Cueillez dès aujourd'hui les roses de la vie.

Pierre de Ronsard, "Sonnets pour Hélène" (II, 24)

My translation:

When you are old, in the evening, in the candle-light,
Seated by the fire, knitting and sowing,
You will say, while singing my verses, marvelling:
"Ronsard celebrated my beauty when I was young."

For you will have no servant who, hearing such words,
Tired and dozing from hard labor,
Who at the sound of Ronsard will not awake,
And bless your name with immortal praises.

I will be long buried, a phantom without bones
Who by the sombre myrtle-trees will myself repose;
You will be by the hearth, a stooped old woman,

Regretting my love and your pround disdain.
Live, if you believe me, and wait not for tomorrow:
Pluck today the roses of life.

Who says the Mongols didn't leave anything lasting behind?

Posted by Curt at 10:23 AM in Language | permalink | 6 comments

So I’m taking a class in Russian in Paris. Now, Russian is a very beautiful language, but it has one letter, one sound, which as my professor puts it, “makes foreigners cry.” It’s even worse than hearing French people trying to pronounce the letter “h”: the written letter appears approximately like “bl,” and it’s impossible to describe the sound, since there is no European-language equivalent, except to say that if someone punches you in the stomach while you are pronouncing the sound “ee” you will have a suprisingly similar result. Rather than bask in the impenetrable idiosyncrasy of such a linguistic quirk, my professor apologized while disclaiming responsibility for it. She claims that it’s not a Russian sound, that it is borrowed from the language of one of the Tartar or Mongol tribes that conquered Russia during the Middle Ages. I assume this is why the sound does not exist in many of the Southern Slavic languages. Well, now every time I pronounce the sound (which is pretty often—it’s the plural ending for most words) an image of burning villages and madly cackling horsemen always springs into my mind. Why, oh why, when Peter the Great was Westernizing Russia, could he not have taken time off from shaving off people’s beards to expel this barbaric Asiatic sound from the language?

October 21, 2004

Red Sox Nation

Posted by shonk at 01:31 AM in Sports | permalink | 1 comment

Wow. Incredible ALCS. Simply amazing. I’m not even really a Red Sox fan, but, during the last four days, I’ve found myself obsessed with this series, even to the point of acting out those little superstitions that sports fans inevitably obsess about when a team they care about is involved in an elimination game. The way they came back from being down 0-3 is, to me, simply mind-boggling, since their saving grace in the series was their bullpen which, not to put too fine a point on it, is pretty shaky except for Foulke (who absolutely played his heart out in this series; I’m still slightly amazed his arm didn’t just fall off on Tuesday night).

Francona is far from the sharpest manager in the game, but he did what he needed to in the last four games of the series, never looking ahead and preferring to overwork his best arms rather than saving them for a tomorrow that might never have come. Obviously, only time will tell if this strategy will ultimately cost them the World Series, but I have to believe that had he, for example, not inserted Foulke in the seventh inning of game 4, the World Series would have been a moot point. There are situations when being over-aggressive is the only way to win, and when you’re down 0-3 to the Yankees, that’s one of those situations. And Francona, dumb as he may be, recognized that fact (that having been said, I’m still a bit confused about his decision to use Pedro in the seventh inning tonight [though, admittedly, not Tim McCarver-level confused]).

Of course, it helped that the team he’s managing has absolutely the perfect personality for carrying off such an improbable comeback (which is, incidentally, why this year’s Sox are such an easy team to identify with). And the best exemplars of that personality might be Ortiz and Schilling, one of which came through again and again in the clutch while never seeming to take himself too seriously and the other of which pitched a gem in game 6 on sheer force of will. Bill Simmons compared Schilling’s game 6 to the Willis Reed game and the MJ flu game which might sound like hyperbole but makes sense given the juxtaposition between his performance and the recurring images of his bleeding ankle that Fox kept showing.

Okay, enough sports clichés and replay-announcer gushing. Back to your regularly scheduled programming.

October 09, 2004

The Song Remains the Same

Posted by shonk at 12:27 AM in Politics | permalink | 57 comments

Stumbled, quite by accident, across the second half of the presidential debate on the idiot box tonight, and I just have a few questions. First, has Kerry even read the Constitution? At one point, regarding a question about naming Supreme Court justices, he claimed that his priorities were on interpretation of the Constitution rather than on ideology, and then listed off three or four Constitutional “rights” that he would want an appointee to protect, like equal pay for women, abortion rights, etc. Admittedly, he may have been thrown off a bit by Bush’s wacky Dred Scott reference, but he followed up a question or two later by again suggesting that “a woman’s right to choose” is a Constitutional right, so I’m inclined to think it wasn’t a mistake. Now, no matter how one feels about any of these issues, the simple fact of the matter is that they’re not mentioned anywhere in any version of the Constitution that I’ve ever read.

On the other hand, what was up with that Dred Scott reference? It’s admittedly one of the most famous Supreme Court cases in history, so it may be the only one Bush has ever heard of, but it seemed like a particularly ham-handed way for him to try to dissociate himself from the Southern reactionary stereotype.

Speaking of stereotypes, what’s up with using “liberal” as if it were a dirty word? Now, admittedly, I’m violently opposed to the position identified with liberalism in the modern American political climate, but the day I accept “liberal”, a word the dictionary defines as meaning “Not limited to or by established, traditional, orthodox, or authoritarian attitudes, views, or dogmas; free from bigotry”, as a slur is the day you can officially pronounce my critical thinking capability dead.

If you’ll excuse the ranting dogmatism for a moment, both major parties in this country are conservative, not in the somewhat bogus sense that the word is currently used but rather in its original meaning: both are committed to propping up and sustaining the currently dominant power structures and institutions. Whether that’s a good or a bad thing is, of course, an entirely separate question, but the simple fact of the matter is that neither wants to fundamentally change much of anything.

Hypothetical questions aside, the debate convinced me of something that John T. Kennedy has been telling me for months: Bush is going to win this election. For better or for worse, his arguments are essentially positive: I will do X because I believe it is the right thing to do. Whether or not he believes his own arguments, and whether or not what he does exemplifies whatever beliefs he may actually have (and I’m skeptical that he has any), he’s arguing from an essentially stronger position than Kerry, all of whose arguments boil down to the following: I am not George W. Bush.

Admirable as that quality may be, Kerry doesn’t seem to even be pretending to believe in much of anything. Based on what he said (and I think my interpretation is relatively unbiased, given that I dislike both candidates), he’s not opposed to top bracket tax cuts because he has a strong belief in social justice or egalitarianism: he’s opposed to top bracket tax cuts because the beneficiaries only comprise 2% of the electorate. For God’s sake, he practically admitted as much when he tossed off that canard about he, Bush and the moderator being the only people in the room who would be negatively impacted by rolling back tax rates for the top brackets to Clinton-era levels.1 More fundamentally, at practically every stage he was reacting to what Bush said, in many cases legitimately pointing out inconsistencies or flawed reasoning, but still allowing Bush to dictate the terms of the debate. This became especially clear during the series of health care questions, in which Kerry spent more time explaining what his plan is not and how it was being misrepresented by G.W. than what it is, to the point of prefacing his answer to an entirely different question with a statement to the effect that his plan isn’t what Bush had been claiming it was. Despite the fact that I neither know nor want to know very much about Kerry’s health care plan, I’m certain that it was being misrepresented by Bush; nonetheless, getting defensive and allowing your opponent’s misrepresentations to fluster you is not a good way to win a debate.2

None of the above should be construed as an endorsement of Bush. Rather, it seems to me unlikely that someone running on an essentially negative platform, like Kerry, is going to defeat someone who at least pretends to be running on a positive platform. It didn’t work for Dole in ‘96 or McGovern in ‘72 and, if there’s one thing I’ve learned about politics, it’s that things don’t change all that much. Admittedly, Kerry’s more charismatic (has that been made into a newspaper pun yet? Kerrysmatic?) than Dole and (much) less radical than McGovern, so the race will undoubtedly be closer, but the old Zeppelin song title is still relevant.

1 A specious claim, by the way. Make fun of trickle-down economics all you like, but the rich generally don’t hoard their money: they invest it. And let’s just say there’s a relationship between investments and jobs.

2 Speaking of bad debating strategies, argumentum ad verecundiam seems to be another Kerry favorite.