Monthly Archives: September 2007

They had not the habit of definition

The question of whether 0.9999…. is equal to 1 is, and probably always will be, a source of heated disagreement among people who know a certain amount of math, but not too much. High school math teacher Polymathematics delivered a magisterial series of posts on this question last year, which covers with admirable thoroughness every one of the many, many strange trails this argument likes to wander down. So I’ll leave that to him, and just use the question as an excuse to copy in one of my favorite quotes from G.H. Hardy, from his 1948 book Divergent Series:

“…it does not occur to a modern mathematician that a collection of mathematical symbols should have a ‘meaning’ until one has been assigned to it by definition. It was not a triviality even to the greatest mathematicians of the eighteenth century. They had not the habit of definition: it was not natural to them to say, in so many words, `by X we mean Y.’ … it is broadly true to say that mathematicians before Cauchy asked not ‘How shall we define

1 – 1 + 1 – 1 + …

but

‘What is 1 – 1 + 1 – 1 + …

and that this habit of mind led them into unnecessary perplexities and controversies which were often really verbal.”

The connection to 0.9999… is quite direct. To ask what 0.9999… is is to miss the point. Rather, you ought to ask “How can we choose a real number which deserves to be called ‘the value of 0.9999…?'” And once you have done this, you realize you are not at all sure what the definition of a real number is … and before long, you’ve learned the first few weeks’ material of a course in real analysis and all confusion has departed.

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Aziz Nesin, Ali Nesin, and education without permission.

Ali Nesin is a logician and group theorist at Istanbul Bilgi university; he’s the author of many popular math books in Turkish, including one with the intriguing title Mathematik ve Korku (“Mathematics and Fear.”) Many of his undergraduate students have gone on to excellent Ph.D. programs in Turkey and abroad, including a couple here at Madison. He also runs a summer school, called the Mathematical Village, in Sirince. This August, the Turkish government abruptly shut down the school, citing Nesin for “education without permission.” The Turkish press and mathematicians around the world worked quickly to publicize the shutdown, and within a week the school was reopened; though Prof. Nesin still faces charges.

For those of us outside Turkey, the immediate question was: why would the federal government be shutting down summer math camps in the countryside? The answer is that Ali Nesin’s father was Aziz Nesin, Turkey’s most popular satirist under many regimes unreceptive to satire, and as a consequence, an occasional convict. As a socialist and an atheist, he was one of the few people whom military nationalists and Islamists could enthusiastically agree to oppress. The Mathematical Village is built on land owned by the Nesin Foundation, whose income comes from ownership of all of Nesin’s works; and as such, it smells bad to the power structure in Turkey. Thus: “education without permission.”

(Warning: my knowledge of Turkish politics is very fragmentary; if I’ve gotten important elements of Nesin’s story wrong, please correct in comments.)

After learning all this, I borrowed Nesin’s Turkish Short Stories from Four Decades (trans. Louis Mitler) from the library to see what the commotion was all about. Lots of ironic little stories here — in fact, the irony is so classically and deftly deployed that the book could easily be taught in a high school English class to illustrate satire. “We Resemble You,” “The First Woman Ever to Understand Me,” and “Don’t You Have Any Donkeys In Your Country?” are especially good examples: let me boil this last one down to the underlying joke to give some sense of Nesin’s approach. A Turkish guide is traveling through the countryside with an American rug collector. They come across a peasant with an old, mangy donkey. The collector instantly recognizes the donkey’s filthy blanket as a spectacular antique carpet, and orders the guide to buy it. But don’t tell him we want the blanket, the collector warns the guide, tell him we need a donkey; otherwise he’ll realize the blanket is valuable and overcharge us! The peasant warns the guide that the donkey is lame, sick, and weak, but the guide insists he wants it. “Alright then,” the peasant says, “the price is $5000.” He explains to the shocked guide that if the American is so anxious to have such a bad donkey, there must be some reason for it.

The collector is disappointed, but $5000 is still much less than the value of the rug. So he agrees to buy the donkey. But when the peasant unties the donkey and leads it to the collector and guide, he takes off the blanket and slings it over his shoulder. Now the collector is really starting to get worried, and the guide tries to convince the peasant to include the blanket so that the donkey doesn’t catch cold and die.

“I can’t do it,” the peasant says, “absolutely not.”

“Why not?”

“Because this is my good luck donkey blanket.”

The guide asks the peasant what he could mean by this, and the peasant explains:

“I used to be a poor man who barely made a living selling donkeys. Then my father died and gave me this blanket. Now, two or three times a year, an American or European comes and buys one of my donkeys for $5000 apiece! At that rate I’m a rich man!”

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Two good Orioles endings

Good ending 1: The Orioles hang 3 runs on Mariano Rivera for the second time this year to tie the game in the bottom of the 9th, then win on a Melvin Mora bunt in the next frame, simultaneously clinching at least a tie in the season series and cementing the Yankees’ first second-place division finish since 1997. If that wasn’t enough, we got to beat up on Mike Mussina too. Rivera’s lifetime ERA against the O’s rises to a pedestrian 3.72. The three runs, by the way, were on a bases-clearing Jay Payton triple — his second triple of the night.

Good ending 2: Stat of the Day brings us the game of June 3, 1977; Royals catcher John Wathan comes up against the O’s Tippy Martinez, bottom of the 9th, Orioles up 7-5, but the Royals have the bases loaded and nobody out. Wathan hits a fly to right. Runner on third tags and scores. Runners on first and second tag too, but the throw from Pat Kelly comes in in time for Mark Belanger to tag the runner at second — now the runner on his way to third is caught in a rundown, and he’s out too. When the smoke clears, the Orioles have won the game 7-6, and Wathan has hit into a game-ending triple play and collected an RBI in the same at-bat!

There was something about Tippy that struck confusion into the minds of baserunners — let it never be forgotten that he is the only pitcher in the history of baseball to pick off the side. (The link is a beautiful account of this crazy game from Swing and a Miss– I implore you to follow the link and spend a little time with your 1983 World Champion Baltimore Orioles.)

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Show report: Andrew Bird at the WUT; ancient civilizations

I, like Em, went to see Andrew Bird last week. I appreciated all the pizzicato shredding and freaky whistling, but found I liked the more traditional songs best — “Fiery Crash,” “Plasticities,” and “Tables and Chairs,” this last a kind of homage to “Don’t Worry About the Government.” Which is good: we are well supplied with homages to the yelpy mid-period Talking Heads and need more homages to talky early-period Talking Heads.

Andrew Bird, though he might be too young to know it, has the exact same stage shtick as the great Steven Wright: a kind of mumbly I’m-saying-something-really-weird-and-pretending-not-to-know-it deadpan. Did not work for me in the rock context.

He played a song called “Scythian Empires,” which made me wonder why it’s so popular to write songs about ancient civilizations. Off the top of my head:

  • “Scythian Empires”
  • They Might Be Giants, “The Mesopotamians”
  • Mountain Goats, “The Anglo-Saxons”

The last of these wins by the virtue of the couplet

They used to paint their bodies blue

Some of them might be distantly related to you!

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Reading bad novels

When I was in graduate school I experimented with reading a lot of novels I chose at random off the shelf of the public library, novels by people I’d never heard of and which, as far as I knew, had never enjoyed any particular acclaim. I thought maybe there were things I could learn about the construction of novels that I couldn’t get from really successful examples. Maybe you can better see the device that joins the pieces when they’re not joined exactly square, so to speak. So I read these books in order to figure out what was wrong with them.

Anyway, what I learned was that most randomly chosen novels aren’t very good. What’s worse, the typical not-very-good novel doesn’t really have anything wrong with it. Its problem is an absence of things that are interesting — interesting sentences, interesting sounds, interesting ideas, interesting people. And “Don’t be uninteresting” is not a very helpful piece of advice. So I guess I’m just recording the fact that this experiment was a failure. In case you were thinking of trying this, I recommend reading good novels instead.

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Shared music naivete

Blogging from the College Library at Helen C. White Hall, where I like to work sometimes. One of its charms is that there’s always a wide selection of shared iTunes libraries, so I can listen to music selected by some brain other than my own. (Right now: “Dream On,” by Aerosmith, from a library heavy on the rap-metal. I may soon be hearing Hoobastank for, as far as I know, the first time.) Question: what exactly does this mean? Are the people whose music I can see right here with me in the cafe? Do they know I’m listening to their music? Is there some reason I shouldn’t share my own library when I’m working in here?

Update:  One annoying thing about listening to college students’ shared libraries on shuffle is the podcasts of their classes which pop up from time to time.  Also, Hoobastank is awful.

What is arithmetic geometry?

My colleague Timothy Gowers is very close to finishing a project of really immense ambition: the Princeton Companion to Mathematics, a gigantic book which aims to be a panorama of all of contemporary mathematics, presented at an undergraduate or even interested-amateur level. He has jokingly suggested that a good alternate title would be Mathematics: A Very Long Introduction. Some of the book consists of expository articles on the subfields in math — things you might take a course in, like analytic number theory, probability, or partial differential equations. Others treat notable theorems (Mostow Rigidity, Hilbert’s Nullstellensatz), notable mathematicians (charmingly alphabetized by first name), and notable applications to other fields. And some of the articles — to my mind the most ambitious of all — attempt to give some sense the nature of the mathematical project to outsiders. (“The general goals of mathematical research,” “The language and grammar of mathematics.”) The editors have made many sample articles available online (userid: Guest, pwd: PCM) — I encourage people to have a look! In particular, if you are wondering what I do all day, you can read my article on “Arithmetic Geometry.” If you want to start from the beginning of things, try Gowers himself on “Some Fundamental Mathematical Definitions” or “The Language and Grammar of Mathematics.” For an applied article, try Madhu Sudan’s “Reliable Transmission of Information.” Or if you just want inspiration, see Sir Michael Atiyah on “Advice to a Young Mathematician.”

Important biographical notes: Tim has a Fields Medal and a blog.

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Stroll me like you mean it

Via Jessie, here’s a promotional video of Bugaboo stroller stunts:

I, like Jessie, like to make fun of people with fancy strollers. But we all have our weaknesses, and mine is CJ’s giant Bob Revolution. If I were in a suburban dad punk rock band and we all had cool noms de punk rock, “Bob Revolution” would be mine.

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Letter from Belgium

One complains about the American political situation a lot, of course, and why not? But it’s good to keep in mind that other countries face structural dilemmas which are totally alien to us. In a long and brilliant post on Crooked Timber, Ingrid Robeyns explains the deadlock between the Flemings and the Walloons, and why Belgium has no government. Just to give a taste:

Governments in Belgium, both the federal and the regional ones, are always made up from coalitions. But these coalitions are not the same in all governments. In part this is due to the non-coinciding elections [Federal elections are held every 4 years, but elections for the regions and communities every 5 years, together with the European elections. Local elections are every 6 years.], in part this is also due to the fact that the parties do not have the same size at both sides of the language border. For example, the Flemish Christian democrats are the biggest party in Flanders, but are a rather small party within Francophone Belgium. Flanders also has a considerable extreme-right seperatist party, Vlaams Belang (which, ironically, is receiving some votes from Francophones in Brussels thanks to their security and anti-muslim agenda), whereas there is no such political factor in Wallonia. So these asymmetries create difficult situations. For examples, at the last regional elections the Christian Democratic Parties became part of the regional governments in Flanders and Wallonia, but they were part of the opposition in the Federal Government (which was made up from the (Flemish and French) Social Democratic Parties, and the (Flemish and French) Liberal Parties). This can lead to strange party-dynamics, which in the present crisis of the negotiations the federal level are also an explanatory factor why there is still no Belgian government. For example, the French Christian Democratic Party (CDH) is currently part of the government of the Walloon region, together with the French Social-Democratic Party (PS). But at the last federal elections, the PS (and its Flemish sisterparty SP) lost many seats, such that the current negotiations at the federal level are between the Christian Democratic parties and the liberal parties. This leads to the difficult situations for parties that are in different positions at the different levels. CDH is part of the center-left coalition at the Walloon regional level, but is negotiating to become part of a center-right government at the federal level. Since the voters are likely not to make a distinction between whether the acts of parties are made in their capacities as rulers at the federal versus the regional levels, it may be very difficult for any particular party to be in a center-right coalition at one level, and a center-left at another level.

There are some advantages to the calcified two-party system that Americans, to some extent, enjoy.

If you want to listen to the Mountain Goats song “Letter from Belgium” (and I really think you might!) you can listen to a live performance (19 Oct 2004, Mt. Pleasant, SC) here, via the remarkable Live Music Archive:

“That’s good, we can always use some more electrical equipment!”

Update:  If you listen to more of the linked concert, you hear John Darnielle getting the news of the Red Sox winning game 6 of the 2004 ALCS, right after “Going to Georgia.”

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My want more cheerios

CJ hasn’t learned the word “I” yet — rather, he’s learned that there’s such a thing as the 1st person singular personal pronoun, but he thinks it is “my.” Thus, “my want more cheerios,” “my have shirt on,” etc. Has anyone else ever heard of this substitution?

I do have a theory. I wonder if he could be hearing from our speech that “your” and “my” are symmetrical (i.e. if I say “Your shirt is dirty,” he would correctly say “My shirt is dirty”) and overgeneralizing to the belief that “you’re” and “my” are symmetrical (i.e. if I say “You’re having breakfast,” he would say “My having breakfast.”) Also, I don’t think we actually say “I” that much when we’re talking to him — for some reason we’re much more apt to say “Daddy’s going to change your diaper,” or “Mommy’s eating cereal” or whatever. We certainly do say “I” when we’re talking to each other, though. Do kids learn primarily from language directed at them or do they draw just as much from ambient conversation?

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