Dangerous eggs

Among the dangerous contraband items U.S. border officials look for when they search your car are Kinder Surprise candy eggs. The CBC has an account of one seizure. Punchline:

As trivial as the border seizure may seem, Bird said the U.S. government has sent her a seven-page letter asking her to formally authorize the destruction of her seized Kinder egg.

“I thought it was a joke. I had to read it twice. But they are serious,” she said.

The letter states if Bird wishes to contest the seizure, she’ll have to pay $250 for it to be stored as the two sides wrangle over it.

The overbearing threats, the lawyerly paperwork, the sheer pointlessness of it all—it’s like a portrait in miniature of U.S. law enforcement. It’s so poetic I could cry.

I think my view of government is getting warped. Every senseless indignity is starting to look less like random incompetence and more like a dominance display.

Wikileaks, journalism, and espionage

WOLF: Jeff, can I talk about the Espionage Act because that's really what's at stake now that they've invoked it. I predicted in my book The End of America that sooner or later, journalists would be targeted with the Espionage Act in an effort to close down free speech and (INAUDIBLE) of government. And we have a precedent for that. In 1917, the Espionage Act was invoked to go after people like us who were criticizing the first World War. Publishers, educators, editors. Wait, and people were put in prison. They were beaten. One guy got a 10-year sentence for reading the First Amendment. And that intimidation effectively closed down dissent for a decade in the United States of America.

The Espionage Act has a very dark and dirty history. And when you start to use the Espionage Act, to criminalize what—I'm sure you've handled classified documents in your time as a serious journalist, you know perfectly well that every serious journalist has seen or heard about classified information and repeated it. When you start to use the Espionage Act to say reporting is treachery, reporting is spying, it's espionage, you criminalize journalism. And that's the history that our country has shown.

TOOBIN: I recognize there is that history. And I'm familiar with the red scare, too. America is different now.

WOLF: Oh, it's worse in some ways.

TOOBIN: Well, I would disagree.

SPITZER: I want to ask Jeff a question though, because I want to come back to this Woodward distinction. You would agree with Clay and Naomi, I think, that Julian Assange would be precisely Bob Woodward if he had been the recipient of these documents, is that correct?

TOOBIN: I'd have to know a lot more.

SPITZER: But it might be the case.

TOOBIN: It well might be the case.

SPITZER: OK. So your sort of clear articulation of the beginning that he clearly violated something—maybe not so much.

TOOBIN: I'm not sure. Certainly the attorney general of the United States seems to think criminal—criminal activity was involved here. But I think the wholesale taking of enormous quantities of classified information and putting it on the Internet, even if you don't put all 250,000 documents on, I think that is a meaningful distinction from what Bob Woodward does.

SPITZER: It seems to me that Bob Woodward arguably did something much more egregious. He took real-time decisions about why we were going to war in Afghanistan, the discussions are rationale, where we would go, spoke to the most senior political and military officials in the nation and blasted that out in the book. A clear distinction.

TOOBIN: Well, again, there is a distinction in part because the president of the United States and the vice president are allowed to declassify anything they want at any time for any reason. So if the president declassified—

SPITZER: A lot of people who didn't have that power were sourced in that book. Seemed to be speaking in clear violation. They, in fact, should be subject to criminal investigations.

TOOBIN: I always wondered why—why Woodward gets away with it. It's an interesting question.

(CROSSTALK)

—Naomi Wolf, Jeff Toobin, and Eliot Spitzer, talking on CNN's Parker Spitzer December 23, 2010. (transcript)

(Sidebar: Toobin was either indulging in a little hyperbole, or he was under the impression that all the cables were released unredacted. Not that it makes a huge difference, but this is not what happened. Wikileaks has been working with five newspapers, including the New York Times, to redact the cables before they are released publicly. As of January 10, 2011, Wikileaks has released only 2,028 of the 251,287 cables. They have been releasing a few each day.)

When Dianne Feinstein called for Julian Assange's prosecution under the Espionage Act, she wrote:

Mr. Assange claims to be a journalist and would no doubt rely on the First Amendment to defend his actions. But he is no journalist: He is an agitator intent on damaging our government, whose policies he happens to disagree with, regardless of who gets hurt.

But what exactly is Assange or Wikileaks doing that investigative journalists do not do? Cultivating contacts with privileged access? Obtaining classified documents? Reading them? Publishing them? Refusing to reveal sources? Pushing an agenda? Trying to make a difference? Those are all things journalists do. It is arguably impossible to do the job right without doing those things.

Why does Bob Woodward get away with it?

Why indeed.

Things I learned recently

Handmade paper does not have a grain. The fibers stretch out in random directions. But factory-made paper uses a process where water flows across the surface of the paper as it forms. Take any sheet of printer paper and you'll find that you can curl up the long vertical edge over using less force than the shorter horizontal edge. Even though you're bending more paper the long way, you're bending fewer of the paper's fibers by going with the grain.

According to the Wikipedia article on shmoon, “The word ‘shmoo’ has appeared in nearly 700 science publications since 1974; it is used in labs studying the bread-and beer-making species Saccharomyces cerevisiae.”

The geoid is the shape of the mean sea level of the earth. It is not an ellipsoid. Gravitational forces due to variations in the earth's density subtly distort the geoid. This is why a GPS device can tell you you're 20 meters underwater when you're at the beach: GPS measures your altitude above a reference ellipsoid, not the true geoid. The difference can be much greater in magnitude than the tides.

The Aztec merchants

Trade in the Aztec empire was mostly barter, but I'm surprised to read that a few commodities were plentiful, portable, durable, interchangeable, and stable enough to serve as money. “Cacao seems to have been the most common form of money, and it did, indeed, grow on trees. It was widely accepted as payment for both merchandise and labor.” There were even, this book claims, cacao counterfeiters.

The merchants of the Aztec empire had a guild which set its own laws, enforced them, judged and even had some control over prices. Merchants “entered enemy territory as spies, they could declare and engage in wars, and they could conquer communities.” They became very, very rich.

What struck me about the Aztec merchants, though, was what they ultimately used their wealth for. The vast majority they spent on lavish feasts for other merchants. The guests gorged on tamales of corn and chile, drank chocolate, and ate hallucinogenic mushrooms. Throwing a feast was how you converted riches into an even more valuable currency: social status. “Through this elaborate system of potlatch-like feasts, the merchants were obligated by the guilds to use their accumulated wealth for socially prescribed ends.”

Hmm.

The book is The Aztecs of Central Mexico: An Imperial Society, by Frances F. Berdan.

Easier said than done

Epictetus wrote:

15. Remember, you must behave as you do at a banquet. Something is passed around and comes to you: reach out your hand politely and take some. It goes by: do not hold it back. It has not arrived yet: do not stretch your desire out toward it, but wait until it comes to you. In the same way toward your children, in the same way toward your wife, in the same way toward public office, in the same way toward wealth, and you will be fit to share a banquet with the gods.

I am thinking about friends who are far away and I am not a very good Stoic.

In prose these days, the style is to be plain.

Use common words; be forthright and succinct;

Avoid formality; use active voice:

In short, say what you mean and nothing more.

These rules apply whatever you may write,

To traffic signs and novels just the same.

And in return for hewing to these rules,

The writer gets free rein, a blank white box

In which to dump his brain without a care

For form or structure. Sonnets? Thank you, no,

Though every now and then we condescend

To post a wry haiku on someone's Wall.

We'll count, it's fun and shows how smart we are.

But from restrictions richness sometimes comes,

Or thoughtfulness, or creativity,

Or beauty. Or mere elegance of form,

If all the other virtues of our words

Should come to nothing.

Perhaps we set our words too many tasks?

We have to write so many words each day

Each word must be disposable and cheap,

Like coffee filters (this the most polite

Of several metaphors that spring to mind)?

The style is just a moral cop-out, then,

To make a virtue of our verbal cheapness?

Our age lacks all ambition. When we make

a simple thing, we make it quick and plain,

and if it works, we're pleased. Is that the way?

These explanations miss the mark, I think.

I don't know how it happened. But it's dumb.

Good music isn't formless and austere.

Good writing doesn't have to be that way.

If fail we must then let us fail in ways

No self-respecting writer fails these days

And striving, let us win what gains we might

By doing something hard each time we write.

Problems that are too hard

I am self-conscious about homeschooling. It's not something I would have thought to attempt, if it were just me. I believe teaching, like any skill, improves with practice and study; and I have neither practiced nor studied teaching young children. I think most kids learn more when they spend more time studying; and my six-year-old spends a lot less time “at school” than I did at his age—by a factor of five or more. (On the other hand, when he is at school, he is studying.)

But it has been fun noticing things that I can do as a homeschooling parent that wouldn't work at all in an ordinary school.

My favorite is that I am free to pose problems that are too hard.

The kids have school when I'm at work, but I write out a lot of their schoolwork in advance. One of the problems I recently posed for J. is this: I have a box that does some kind of arithmetic. If I put in 7, out comes 3. Similarly 3↦7, 2↦8, and 1↦9. What if I put in 5? (Of course, anything or nothing might come out, but we can learn that lesson another time.) This turned out to be baffling—but that's OK. It will sit in J.'s binder for days, weeks, or months, until one day he cracks it. I don't think he'll crack it by accident but rather because he tries harder or because he develops a better understanding of how addition and subtraction behave. I think it'll be pretty gratifying, and he'll have earned it.

J. likes puzzles. In that regard, at least, his childhood will be a little like mine. Only without the answers.

Homework

Earlier this year I suddenly remembered being taught in school to answer questions of the form “What is the difference between a delta and a wetland?” by copying the definitions of the two terms out of the book and putting the word while between them. Imagine writing this out in cursive on notebook paper:

A delta is a low triangular area of alluvial deposits where a river divides before entering a larger body of water, while wetlands are lands where saturation with water is the dominant factor determining the nature of soil development and the types of plant and animal communities living in the soil and on its surface.

...times twelve or so.

Looking back on it I have to wonder how on earth this happened. The event cries out for an explanation. What would make someone do this to a roomful of kids? Was it a case of perverse incentives? Incompetence? Or straight-up cruelty?

Another obvious question (if you're in my shoes) is if such a thing could possibly ever happen to a home-schooled kid. Sure it could. Parents can be incompetent too. Or cruel.

I try to be humane—I won't be committing this particular atrocity—but there are several things about homeschooling that give me pause. There are no good ways to measure whether things are going well. Finding out what I could be doing better is hard. As far as I can tell by searching the Web, not many people like me are doing this; or else they are all as strapped for time as I am. Most fundamentally, I don't know what I'm doing.

Near the beginning of The Port-Royal Logic there is a brief and somewhat odd discussion of the Pyrrhonists and the Academics, and I guess Michel de Montaigne:

We may indeed easily say outwardly with the lips that we doubt of all these things, because it is possible for us to lie ; but we cannot say this in our hearts. Thus Pyrrhonism is not a sect composed of men who are persuaded of what they say, but a sect of liars. Hence they often contradict themselves in uttering their opinion, since it is impossible for their hearts to agree with their language. We see this in Montaigne, who attempted to revive this sect in the last century ; for, after having said that the Academics were different from the Pyrrhonists, inasmuch as the Academics maintained that some things were more probable than others, which the Pyrrhonists would not allow, he declares himself on the side of the Pyrrhonists in the following terms : “The opinion,” says he, “of the Pyrrhonists is bolder, and much more probable.” There are, therefore, some things which are more probable than others. Nor was it for the sake of effect that he spoke thus : these are words which escaped him without thinking of them, springing from the depths of nature, which no illusion of opinions can destroy.

Antoine Arnauld, Pierre Nicole, The Port-Royal Logic, 1662, translated by Thomas Spencer Baines, 1861.

I love the last sentence here: “And don't you try to get out of it by claiming a sense of humor, either.” To me it hardly seems probable that Montaigne was not just saying that for effect. It's too perfect.

Questions about net neutrality

(I originally wrote the following in October 2009, but did not publish it because it seemed likely that I just didn't know what I was talking about, and everything was OK. That's still likely, but recent events suggest I could be wrong about that.)

This video starts out great. It's about what makes fertile ground for innovation. After about 2 minutes, it goes off into “I have a right to Internet access” territory, and it never really comes back.

The open Web means a lot to me personally. Everything this video starts out saying is true. The openness of the technology and the current benign behavior of the network owners means people can try stuff on an awesome scale. Life-changing stuff. Funny stuff. Stupid stuff. It's not hard; it's not expensive; anyone who has the hardware, software, network access, experience, and free time can do it. And once you have the first four, it's like magic. All you need is more free time. It's romantic and wonderful, and it's all true. (I know because once I had a lot of free time.)

Mozilla came out in favor of net neutrality last year after a long silence. I still have a lot of unanswered questions.

• Do we really have to invite the FCC to regulate the Internet? The video invokes the threat of censorship. But um, the FCC is the country's foremost censor. This is the agency that mandated the broadcast flag, that fined Clear Channel for carrying Howard Stern, that maintained a three-network TV oligopoly for decades. This is where the Parents Television Council sends letters when someone says something vulgar on network TV. The FCC is the antithesis of the Internet.

  Of course the Internet is protected by the First Amendment in ways broadcast media aren't, and the FCC knows that. The plan is that the FCC will work in the opposite direction, preventing carriers from filtering while resisting political pressures to indulge in unconstitutional censorship of its own. I'm skeptical. Astute observers may have noticed that the First Amendment is not really a guarantee of good behavior.

  (Before you blow me off, read the next point. I really believe the bigger a role we ask the FCC to play in Internet the more it will find itself facing very difficult questions about what content should be allowed.)

(Since I originally wrote this, a court ruled that as it stands, the FCC does not have the authority to enforce its new net neutrality rules. The FCC responded by saying it will “move to partially reclassify broadband as a common-carrier service” and at the same time “try to establish that it will not regulate many areas of broadband”.)

• There are questions of where to draw the line.

  Google has already gotten itself into the awkward spot of having to argue that, while the FCC should impose net neutrality rules on broadband carriers, the rules should not apply to applications Google builds on top of those networks:

  The FCC's open Internet principles apply only to the behavior of broadband carriers -- not the creators of Web-based software applications.

  I think what we want is a neutral infrastructure and vibrant content. Well, is a web browser infrastructure or content? How about a search engine? You need both to use the Internet effectively these days. Google offers products in both those markets. Should they be subject to neutrality rules? AT&T says yes, the FCC should police the entire Internet, but that's just AT&T being evil for effect. Is there a good answer to the question?

• Net neutrality advocates want ISPs to charge by the bit, not by content. Don't actual humans hate being charged by the bit? Do you like being charged by the minute for cell phone service? I always found it kind of annoying.

  I guess humans hate being charged by content too, but honestly I am happy with having to choose basic cable vs. various channel packages and I'm glad I don't have a meter on my TV.

• So I saw this picture on Twitter, and the story it tells is, “Your ISP wants to double your monthly bill for access to the whole Internet. Net neutrality is about saving you money.” Do they really? And is that really what it's about?

  Incidentally, why do I need a picture of a world without net neutrality rules? Am I not already living in that world?

• As I understand it, the U.S. doesn't really have net neutrality regulations yet, and ISPs are not in fact doing any of the things I'm supposed to be worried about. What am I missing?

  I guess Comcast was blocking Bittorrent for a while. I don't know much about that case, but I suspect Comcast just (cluelessly) took Bittorrent as a proxy for “this customer is going to soak up a ton of bandwidth and then get us sued”. Is that wrong?

  Most ISPs don't let you run a Web server or a mail server out of your home, either, and strictly speaking that's a violation of net neutrality, right? Now to the extent that they're just segmenting the market, I don't really care either way. But to the extent that those types of content actually cost more per bit to carry (security risk, legal liability, tech support, etc.), net neutrality would be bad, right?

• I suspect the real issue is that Google, Facebook, and Twitter don't want to pay my ISP for the privilege of sending me ads. I'm sure my ISP would love to be able to charge them for that, and that prospect probably terrifies Google in particular. Am I being too cynical? Would this be a bad thing? Is it right for Google to lobby the federal government to protect their profits?

I realize this probably reads like so much FUD, but it's meant as a collection of honest questions and I really do appreciate any answers you can provide. I still haven't made up my mind about net neutrality. The questions that intuitively seem important to me don't seem to have been part of the discourse.

A story with some scary parts

My four-year-old and I decided to collaborate on a book. I was surprised when she decided the story should have some scary parts in it. Like a dragon, she said. Or the dark, I said. Like a dark cave, she said.

Here's what I ended up writing.

The snow fell harder and harder. I remembered that polar bears dig dens in the snow. So to escape the biting wind, I begin to dig.

Suddenly the ice and snow collapsed. I slid. I fell.

Where was I? It was very dark. I was in a cave of ice.

I thought I saw two large, pale, gleaming eyes. Trembling, I crept closer. What was it?

It was a dragon. It saw me. I was afraid. Then I saw that the dragon was trapped in the ice, frozen in place.

The dragon's scales over its heart were warm. I could feel the heart beating, very faintly, very slowly. Thump. Thump.

I stayed there a long time.

The storm passed. I came home safely. That was one year ago. Now I am returning to find the cave again. The zoo wants the dragon for their collection.

As we got near the end, I asked her if the heroine should set the dragon free from the ice. Her eyes grew wide and she smiled and said no in an small tense voice.

Later J. read it and said it was pretty good. I wonder if he felt it was scary. The uneasy feeling I get from this story comes from all the questions it leaves open. Is the dragon a person, an animal, or a monster? Is it safe to take it to live in a zoo? Is it humane? To me those are the scary parts.

Infinity, part 2: Zeno's paradox

(An ongoing series. See part 1.)

In this capricious world nothing is more capricious than posthumous fame. One of the most notable victims of posterity's lack of judgement is the Eleatic Zeno. Having invented four arguments all immeasurably subtle and profound, the grossness of subsequent philosophers pronounced him to be a mere ingenious juggler, and his arguments to be one and all sophisms. After two thousand years of continual refutation, these sophisms were reinstated, and made the foundation of a mathematical renaissance...

—Bertrand Russell, The Principles of Mathematics (1903).

The previous post involved two different kinds of infinity. There's the infinite on, off, on, off... of Thomson's lamp. And there's the infinite division of time: one minute, then half a minute, then a quarter of a minute, etc.

Maybe your reaction to the paradox was, “Oh, that's impossible, there's no such thing as infinity in the real world.” Perhaps not. Perhaps once you get down to tiny enough fractions of a second, you see that light is emitted in tiny, discrete quanta, and to emit even one quantum of light each time the lamp turned on would require more than enough energy to burn it out.

But math refuses to make that excuse. Math deals with abstracts, ideals. Math must deal with infinity somehow. From the moment you start counting, it is always staring you in the face.

Shortly after I posted part 1, I ran across this old chat log. Oddly enough, it discusses the same two infinities, but in a different guise.

Eudoxus: Hey, are you there?

Plato: yeah

is this about that 0.999... = 1 thing, because I'm kind of busy

Eudoxus: no no

check this out, I ran across this paper about teaching math to children, and the example they were using was infinite series.

The question was, 1 - 1 + 1 - 1 + 1 - 1 + ... = ?

Care to guess?

Plato: 0

Eudoxus: Right, obviously

because all the terms cancel out

only there's another way to see it...

you start with 1, and then all the terms *after* that cancel out with each other. see?

So the answer is 1.

Contradictory.

Plato: hmm...

Eudoxus: You with me so far?

Plato: yup

Eudoxus: So here's the snippet from this paper that jumped out at me.

"It is important to point out that it is not enough to consider at the same time two conflicting statements in order to develop in pupils' minds the awareness of an inconsistency and the necessity of second thoughts (Schoenfeld, 1985): the perception of some mutually conflicting elements does not always imply the perception of the situation as a problematic one (Tirosh, 1990)."

[Infinite series: from history to mathematics education (PDF) by Giorgio T. Bagni. —Ed.]

This surprised me because it seems so obvious that inconsistency is a sign something is wrong.

Plato: yes

kids are dumb

from a contradiction, anything follows. Everyone knows that

Can I say something?

Eudoxus: shoot

Plato: this thing about the infinite series

Maybe this is dumb

but I don't see how you can get an answer

i mean it keeps going on and on

where would you put the equals sign?

Eudoxus: :)

Plato: which is a joke

Eudoxus: sure

Plato: but I think that, obviously, you are going to come up with weird answers when you start assuming that if an infinite series "stopped" and you could make an equation out of it, etc.

Eudoxus: But the answers work out just fine, and it's hard to avoid. I don't think you can do calculus, for example, without infinite sums.

Plato: that's why calculus is dumb

Eudoxus: Pfft. Calculus is going to be huge. I'm going to write a book about it, as soon as I get some technical issues resolved. I just hope no one else gets to print first.

Plato: you're all talk

Eudoxus: I am going to try and convince you that infinite sums can work, because if I'm wrong then all my work is contradictory and useless.

Have you heard of Zeno's paradox? The one with Achilles and the hare?

I mean, tortoise

Plato: Is that the half of half of half one?

Eudoxus: Yes.

Zeno says that for Achilles to catch the tortoise would require an infinite number of moves. Clearly impossible. Therefore motion must be an illusion--because there's no way to make sense of it.

Plato: Right. Well, it makes sense...if you assume that the dude is infinitely small...

I mean, in real life you wouldn't be able to do it

because at some point you would just be too big

to go that small a distance and have it mean anythign

anything

and zeno is such a jerk anyway

Eudoxus: Well, hang on.

Would you accept, arguendo, that very small distances exist, even if they are way too small for Achilles to see?

Plato: sigh, yes, Socrates

Eudoxus: :) Then suppose I have two marks on the ground, exactly 1 stadion apart.

Euclid could construct the line between them. And the midpoint. OK?

Plato: yes

Eudoxus: And as many points as you like, successively closer to point B. Right?

Plato: of course

Eudoxus: So you accept that there are (at least in principle) infinitely many points there, getting closer and closer to B?

Plato: yes, of course

Eudoxus: (thinking)

And between every point and its successor, there's some finite distance. That is, between A and the midpoint is 1/2 a stadion; between that point and the next is 1/4 stadion, and so on?

Plato: yes, a measurable distance, agree

Eudoxus: Now, none of the distances overlap. And taken together, they cover the entire line segment AB. Right?

Plato: yes

Eudoxus: So doesn't it make sense to say the sum of all these lengths is 1 stadion?

Plato: you couldn't take a sum

I mean, yes, together they do equal one stadion

But, um, you couldn't really measure each part and add them all up

Eudoxus: I couldn't construct all the points, either, as a practical matter. But that doesn't stop them from existing.

Plato: okay, true

So, yes, all of them together equal one stadion; I guess that's a sum

Eudoxus: I say it's a sum by analogy to the finite case. Other than a fear of infinity, I see no reason not to call this a sum, and say that 1/2 + 1/4 + 1/8 + 1/16 + ... = 1.

Plato: yes, yes, it's a sum

Eudoxus: Ah, then an infinite sum is possible.

Plato: See, here's the thing

You are being tricky

You are using infinity two different ways

On the one hand, infinity is just blah plus blah plus blah, etc.

On the other hand, you are saying blah/this finite thing + blah/this finite thing + blah/this finite thing...

Listen every finite thing in the world could be cut into infinite "parts"

Right?

Eudoxus: Sure.

Plato: Okay, but

that doesn't mean that you can just start with an infinite number of things and decide to multiply and add them and assume that in the end you'll get a finite thing

Eudoxus: Hmm.

I need to think about that.

Incidentally, what would you say about the sum 9/10 + 9/100 + 9/1000 + ...?

Plato: do you mean, an infinite series, the next being 9/10000?

Eudoxus: Yes

Plato: Well, that's the "other" sort of infinity

the type that is just an infinite series of numbers, added, that you are assuming will result in a finite sum

Eudoxus: I don't mean to "assume" that it will or won't. I want to find out if it will or not.

Plato: oh, okay

it won't

Eudoxus: No? But Euclid can also construct a point, call it C, that's exactly 90% of the way from A to B.

OK?

Plato: grr...

yes

Eudoxus: By design, AC is 9/10 of the whole.

What's left, CB, is the other 1/10, right?

Plato: yes

Eudoxus: So repeat with CB. You'll draw a point D that is very close to B.

CD is 9/100 of the whole. What's left, DB, is the other 1/100.

Plato: okay

Argh, so then, they equal a finite thing?

Eudoxus: You see where I'm going? You're constructing a picture of the statement "9/10 + 9/100 + 9/1000 + 9/10000 + ... = 1"

Plato: yes

but, okay, yes, I see that

Eudoxus: And what's another way to write 9/10?

Plato: no, I won't do it; because it is dumb

Eudoxus: (laugh)

when did you see it coming?

Plato: when you said "no no"

liar

Eudoxus: OK, the point of all that mess was of course that 0.999... = 1.

Plato: yes, I know

Eudoxus: I wanted to make an argument in a way that would appeal to a Platonist.

Plato: no name calling :)

Eudoxus: Hey, that's what you are. You should be proud. :)

Plato: I am, sort of. It's just that...okay, I still don't understand exactly. I mean, I get the proofs, but I still feel like we are adding apples and then adding oranges and then saying, voila, apples are oranges.

Eudoxus: Yes, yes, I know...

I'll think about it some more later...

You are right to say that not every infinite sum is as... easy to deal with as 0.999...

Plato: yes, exactly

But it's scary, isn't it

Eudoxus: ?

Plato: I mean, even just mathematically in general

Just the idea that you can get two contradictory answers to an equation

Eudoxus: Uh, yes, it's very troubling.

Plato: I mean, mathematics deals in absolutes

There's no room for contradictions, right?

From a contradiction, everything follows

Eudoxus: Yes.

Plato: Right, so if these infinity things are correct as is, what does that mean?

Eudoxus: Oh, you mean 1 - 1 + 1 - 1 + ...?

Plato: yes

it proves that 0=1, right?

Eudoxus: Well...

that sort of problem doesn't really arise in practice though.

Plato: o rly?

i can prove 0=1 and it doesn't affect your work? what kind of logic is this?

the consequences, man1`

Eudoxus: Sorry, I have to go. I think someone's calling me.

Plato: what???

come back here you coward

Eudoxus is offline.

You might be relieved to know that the paradox has been resolved.

The cure was transformative.

Perfect cinnamon twists

This is a great recipe for beginning bakers. They come out light and fluffy. If I can make them, so can you.

Bring to boil in large saucepan:

• 1 cup sour cream

Remove from heat. Stir in until well blended:

• 3 tbsp shortening
• 1/4 cup sugar
• 1/8 tsp baking soda
• 1 tsp salt

Cool to lukewarm. Add:

• 1 large unbeaten egg
• 1 package of dry yeast

Stir until yeast is dissolved. (At this point you could also throw in a dash of vanilla, almond, or orange extract, but I didn't and they were great. The sour cream alone gives them plenty of flavor.) Mix in with spoon:

• 3 cups sifted plain flour

Turn out onto a lightly floured board. Knead slightly a few seconds to form a smooth ball. Cover with a damp cloth and let stand five minutes to firm up. Meanwhile, in a small bowl mix:

• 1/3 cup brown sugar
• 1 tsp cinnamon
• 1/2 cup finely chopped nuts (pecans work fine)

Roll dough 1/4 inch thick in a rectangle 6"×24". Spread the entire surface with

• 2 tbsp soft butter or margarine

Sprinkle half of the dough (the long way) with the sugar-cinnamon mixture. Bring the unsugared half of the dough over the sugared half, pressing the top surface lightly to seal in the sugar mixture. Cut in strips 1/2 to 1 inch.

Grease 2 cookie sheets. Take each strip of dough, give each end a half twist (in opposite directions of course), and place on the cookie sheets about 2" apart, pressing both ends firmly and flatly to the baking sheet. Cover with a damp cloth and let rise at 85° or so until very light, about 1 hour 15 minutes.

Bake 12-15 minutes at 375°. While baking, mix in another bowl:

• 1-2 tbsp milk
• 1/2 cup powdered sugar

to make a fairly thick icing. Drizzle over the twists.

Makes about 2 dozen.

Mathematicians in training

First off, if you haven't played Set, you really should, because you're just the sort of person who would love it. It's a clever idea elegantly executed, and it just happens to be great competitive fun. I taught the kids to play, and the four-year-old has a definite edge on the six-year-old. I couldn't be more pleased.

We had a bit of a drive yesterday, and along the way we gave the kids some analogies to puzzle over. You know the sort of thing: “Ice is to water as rock is to what?” This turns out to be engrossing and surprisingly fun. (I might try it with adults sometime. I sense opportunities for nerd humor.) Here the kids were not evenly matched at all. The four-year-old would get the easy ones (cow : moo :: pig : x) but would guess random related words on the harder ones, apparently with equal confidence. The six-year-old saw more clearly what the game was about, so he was able to bring his greater general knowledge to bear.

Why is this post titled “Mathematicians in training”? Well, Set is a transparently mathematical game. There are 81 cards because 81 is 3⁴. The deck is the Cartesian product of four three-element sets. They form some kind of algebraic structure with extraordinary symmetry (of a kind I don't really know anything about—it's not a group—such that I'm tempted to get completely sidetracked here). But the kicker here is, the gameplay itself is mathematical. As far as I can tell, the only good strategy is to try to prove there are no sets.

Analogies are just little homomorphisms, which is to say, structure-preserving transformations. The idea that deep sameness is more interesting than superficial differences is more important to mathematics than numbers.

On the surface it seems like analogies are less mathematical than Set. Appearances can be deceiving.

(P.S. Figured it out. The sets in Set are the cosets of cyclic subgroups of (Z/3Z)⁴. The symmetry I was referring to above was that after you erase the underlying group operation, there's no privileged element. There are isomorphisms on the deck of cards, preserving the sets, mapping any given card to any other given card.)

Milestone

I probably learned about variables from playing around with a Commodore 64 when I was about the age you are now. But I didn't see them used in mathematics for many years, until they were finally introduced, in about 7th grade, as a tool for solving problems. Take a problem, write down the equation, putting variables for the unknown quantities, and then you have something you can solve.

A little while ago I realized that this isn't the only way, or even the most important way, that variables are used in math. Variables are used to write laws.

The other day you were getting a shower, and I told you that letters could stand for numbers, that you can use letters to write rules about numbers. The letter could stand for any number, and the rule would always be true. I wrote in the condensation on the glass:

A + 0 =

Then I stopped and said, well, A plus zero equals what? You said zero. I said I didn't think that was right, because what if A was seven? Seven plus zero equals zero? So then you said A. And I couldn't be sure but I thought you really got it. That's right, I said. I'm sure you could tell I was very pleased. Actually I was surprised and excited.

Later I wrote some math pages for you to solve. The first one said,

Here is a rule:

0 < A

Do all the numbers follow this rule?

If there is a number that doesn't, that's called a counterexample. It means the rule is false.

In math, a true rule is always true, for all numbers.

The other one had some mathematical statements on it and asked you which ones were true.

You surprised me.

(Click for a larger image.)

You got them all right. I asked you about the rule on the first page, A < 0, and you had to look up what the < symbol meant, but then you told me right away that it was false, because zero isn't less than zero.

I was amazed. I asked your mother, “Did you see those pages J. did today? What does this mean?” She wasn't surprised. “It means first-graders can learn pre-algebra,” I said, insistent.

“Some can,” she said.

She is half right: you are special; you are bright; and you are interested. But I know there are millions of special, bright, curious kids like you in this country, and I think by and large their schools are selling them short. You sure are lucky you've got me, kid. But not as lucky as I am to have you.