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.