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.

Prehistory, continued

I guess I always thought of technology as strictly cumulative, but reading a little about the Stone Age dispelled that notion. Archaeologists have had to reverse-engineer, from the artifacts they left behind, the toolmaking skills and technology of extinct cultures. How did Cro-Magnon mammoth hunters store meat? Wouldn't it spoil? Well, we really don't know that sort of thing anymore, but there's a way to find out:

…[U. Michigan researcher Dan] Fisher butchered a draft horse using stone tools he'd knapped himself, then cached the meat in a stock pond. Naturally preserved by microbes called lactobacilli in the water, the flesh emitted a faintly sour, pickled odor that put off scavengers even when it floated to the surface. To test its palatability, Fisher cut and ate steaks from the meat every two weeks from February until high summer, demonstrating that mammoth hunters might have stored their kills in the same way.

–Tom Mueller, “Ice Baby”. National Geographic, May 2009.

People showed up on Australia forty thousand years ago. Even though the sea level was lower then, they would have had to cross a channel 55 miles wide to get there. It's a little mysterious to me, as the oldest boats ever excavated anywhere are dugout canoes at most nine thousand years old.

So the first real agriculture (planted fields) happened about ten thousand years ago, dogs were domesticated about fourteen thousand years ago, but apparently people were exploring the South Pacific in logboats, miles from shore, forty thousand years ago. (The oldest stone tools predate H. sapiens and appeared 2.5 million years ago.)

This week I learned...

It's prehistory week at the jorendorff household. This week I learned:

• The fossilized skeletons of a 14-foot Xiphactinus (a mean-looking Late Cretaceous fish) and its last meal, a merely 6-foot bony fish which it swallowed whole, are on display at the Sternberg Museum of Natural History in Hays, Kansas.

• Woolly mammoths had a three-inch-thick layer of fat underneath thick skin, fur, and long shaggy hair. (Yet their environment was such that occasionally one would be flash-frozen, without spoiling the meat, to be eaten by modern dogs ten thousand years later.)

Last week:

• The original Nintendo GameBoy had an 8-bit processor with a HALT instruction which games were supposed to use to wait for interrupts. At least one game would sometimes busy-wait instead (yuck!).

• There's a tool, dwarfdump, that dumps DWARF debug info from an executable or object file.

• On Mac there's a lazy debug-info-linking scheme that causes dwarfdump not to see any DWARF in compiled executables. The Mac tools that come with Xcode are aware of this magic, but dwarfdump isn't.

Earlier still:

• The Spiral of Theodorus shows that the square roots of integers can be constructed with straight edge and compass.

• According to Tim Sweeney, quoted in this DDJ article:

  Any loop written in a traditional programming language can be vectorized, to execute 16 iterations of the loop in parallel on Larrabee vector units, provided the loop body meets the following criteria:

  • Its call graph is statically known.
  • There are no data dependencies between iterations.

  So compilers will be able to do a lot more vectorization.

Steampunk 4 life

“But surely (said I) that would make us Slaves to these Automata, if indeed they did not destroy us altogether.”

“Nay (quoth Doctor Albertus), it is not the Extinction or Enslavement of our Race that I see when I gaze into the Future of Mankind. On the contrary, I see naught but Liberty. There are some among us destined to be Monarchs, but how many are they? Each Country admits of but one Monarch, for that is the very Meaning of the Word. The Rest of us, and thou and I, Sir George, are in that Number;–the Rest of us, I say, are destined not to rule, but to be ruled; and in such Circumstances, our Happiness depends upon the Virtue of the Ruler. Now, who would not chuse rather to be guided by Reason, than to be subject to arbitrary Tyranny? Therefore I proclaim the Manumission of the Race of Man: For now we are Slaves to the Whims of Tyrants; but soon, when the Automata take their Place as Heirs of the whole Earth, we shall be guided only by Reason, and live under Rulers which cannot hate, or persecute, or lie, or sin in any Way.

“But if we shall be ruled by Automata, why should we not also be served by Automata? Machines have always served Men, tho’ in a limited and primitive Capacity; but what great Accomplishments lie within our Grasp, when we shall have Machines of greater Capability to serve us!–Machines that shall build, or dig, or plough the Earth; Machines that shall row our Ships faster than the Wind, or push our Carriages; Machines that shall fly through the Air like Birds, and carry us away with ’em on Wings like those Daedalus once dreamed of. Famine shall be unknown; the most impossible and artistic Constructions shall be put up in a week; the most distant Climes shall be brought near, and the most distant Peoples made our proximate Neighbors. In short, the Want, Misery, Ugliness, and Hatred of our current Existence shall give way to an Age of Plenty, Happiness, Beauty, and Peace.”

—The Wonderfull Automaton, a gothic novel in letters from (as literary conceit would have it) the eighteenth century

Dr. Boli's online magazine is, for lack of a better word, curious.

Lockhart's Lament

Lockhart's Lament (PDF, 25 pages) starts out like this:

Everyone knows that something is wrong. The politicians say, “we need higher standards.” The schools say, “we need more money and equipment.“ Educators say one thing, and teachers say another. They are all wrong. The only people who understand what is going on are the ones most often blamed and least often heard: the students. They say, “math class is stupid and boring,” and they are right.

and ends up like this:

How sad that fifth-graders are taught to say “quadrilateral” instead of “four-sided shape”, but are never given a reason to use words like “conjecture”, and “counterexample”. ...

Mathematics is about problems, and problems must be made the focus of a student's mathematical life. Painful and creatively frustrating as it may be, students and their teachers should at all times be engaged in the process—having ideas, not having ideas, discovering patterns, making conjectures, constructing examples and counterexamples, devising arguments, and critiquing each other's work.

The author is a bit crazed, but that just makes it more fun to read. In the unlikely case that you somehow got here while thinking math is stupid and boring, or if you've ever found yourself teaching a stupid, boring math class, take a look.

P.S. As the previous post maybe suggests, I've only recently discovered how to learn math by making conjectures and trying stuff, which is what Lockhart recommends.

In unrelated news, J. is pretty sharp at finding lines of symmetry. I need to give him a circle to play with and see what he says. (evil chuckle)

P.P.S. I got this link from humph, who is also a one-of-a-kind teacher (but not crazed).

The ring Z[i]

This is a little self-portrait, "The Artist Trying to Learn Abstract Algebra", probably of no interest to anyone else.

I read an introduction to rings (in Gallian, fifth edition, which I enthusiastically recommend). Now I'm trying to come up with some conjectures and prove or disprove them before I start on the exercises. (This book has great exercises, and doesn't bother teaching anything in the text that it can teach in an exercise.)

I figured maybe every ideal of the ring Z[i] is a principal ideal generated by some element of the ring. This morning I think I have the proof. It's a consequence of Z[i] being enough like the integers to support Euclid's algorithm.

That in turn is a consequence of Z[i] having something like integer division. You can define a well-ordered metric M on Z[i] such that M(0) < M(a) where a is any other element; and for any a and nonzero b, there exist a quotient q and remainder r such that a = bq + r and M(r) < M(b). That the domain of M is well-ordered implies that Euclid's algorithm terminates.

Z[i] also has something like prime and composite elements. For example, 5+i can be factored into (1-i)(2+3i). I wonder if these two properties are actually the same thing.

I think the ideals of Z[i] generated by "prime" elements are prime ideals.

A storytelling game

I've played this game a few times now and have really enjoyed it. I've only tried it with two players. It might work with more.

Rules

Each player starts by jotting down a very brief story outline: just five lines. Each line should be seven words or less. It's OK to steal the outline of a familiar story, as in the example below.

1. a girl in red
2. a wolf with a plan
3. the wolf eats grandma
4. the wolf's disguise is unconvincing
5. someone saves the day with an axe

Or of course you can make up your own. You do not have to stick to fairy tales.

Each player passes their story outline face-down to another player. (No peeking!) Then the game begins.

From here on out, it's very simple. All the players are going to cooperate to tell one big story including all the elements in all the story outlines. On your turn, look at the first line of the story outline that someone placed in front of you. Suppose it says ‘a lonely house on a hill’. Tell the first little bit of the story. Keep it brief, and be sure to work in a lonely house on a hill.

Then it will be someone else's turn. Perhaps they will read: ‘a girl in red’. They will pick up the story where you left off, adding a girl in red.

Take turns for five rounds.

The end.

A few game design notes

I haven't played many storytelling games. This game mostly tries to avoid the mistakes of Once Upon a Time, which I greatly enjoyed back in the day, but which has some flaws. It features actual competitive gameplay and a winner. But you don't win by telling the best story. You win by getting all the cards out of your hand.

The fixed deck of character, setting, and item cards in Once Upon a Time got boring after just a few games. At some point, the cards stopped helping. Instead I would find myself trying to scrape together yet another storyline involving a shepherdess, a ring, and a disguise. (I have not played Nanofictionary, but it seems like it might have the same problem.) It was usually fun anyway. But I think having the players instead supply fresh material for each game might help. And story outlines seem better than cards in other ways. They naturally provide characters and exposition early in the game, plot developments in the midgame, and endings at the end. They can revisit previously introduced elements instead of constantly adding more folderol to the story. And they're fun to make.

The endgame in Once Upon a Time was especially unsatisfying: as soon as a player could play all the cards from his hand, he would turn over his Happily Ever After card and crash-land the story into the overly specific, predetermined ending printed on it. This would be something like, ‘...and for all I know, they may be dancing still.’ Constraining the last words of the story just seems like a mistake. In this game, everyone gets to tell an ending, and when it's your last turn of the game, you know it. So every player gets their crowning moment, and loose ends tend to get tied off.

Stoicism, Christianity, and Mother Goose

I read the Handbook of Epictetus. It's very brief, just a few pages really. I'll quote a few paragraphs that should make it clear what Stoicism is about. (I'm quoting a recent translation by Nicholas P. White which I really like. The translations I found on the Web seem stilted, or florid, by comparison; though Higginson isn't bad. Of course you can try the original Greek.)

Some things are up to us and some are not up to us. Our opinions are up to us, and our impulses, desires, aversions—in short, whatever is our own doing. Our bodies are not up to us, nor are our possessions, our reputations, or our public offices, or, that is, whatever is not our own doing. (1.)

You are foolish if you want your children and your wife and your friends to live forever, since you are wanting things to be up to you that are not up to you, and things to be yours that are not yours. (14.)

If you are fond of a jug, say “I am fond of a jug!” For then when it is broken you will not be upset. If you kiss your child or your wife, say that you are kissing a human being; for when it dies you will not be upset. (3.)

Do not seek to have events happen as you want them to, but instead want them to happen as they do happen, and your life will go well. (8.)

The best of it is extremely well said, but the content is troublesome. If the sample above does not convince you that you should prepare yourself to be unperturbed when your wife and children die, then the rest won't either. Still, given that you're reading this, something of Stoicism is very likely in you. It's in Western culture.

I guess none of the quotes I picked addresses the evident problem of whether a Stoic may act, or whether he must be distant and docile at all times. The Handbook doesn't seem to offer a head-on answer. Modern Christianity's interpretation of Stoicism does, though.

God, grant me the serenity

to accept the things I cannot change;

the courage to change the things I can;

and the wisdom to know the difference.

—Reinhold Niebuhr, around 1944.

The wisdom, that is, to know what is up to us and what is not up to us. It is exactly this wisdom that is on offer in Epictetus: nothing is up to us except how we see things and how we comport ourselves. I think I like the modern Christian philosophy, demanding as it is of courage and wisdom, better. But then, I tend to like messy, perilous things in principle even when they are not so enjoyable in practice.

The Wikipedia article on the serenity prayer offers this lovely postscript. I have no idea why this rhyme is not a widely-known classic.

The philosopher W.W. Bartley juxtaposes Niebuhr's prayer with a Mother Goose rhyme (1695) expressing a similar sentiment, but without comment:

For every ailment under the sun

There is a remedy, or there is none;

If there be one, try to find it;

If there be none, never mind it.

Sonnets

Once I wrote two lines of what would have been an awesome sonnet.

Shall I compare you to my friend Matt Jones?

You are more lovely and not half so drunk.

In San Francisco, while everyone else was napping, I managed to sneak out to City Lights Books. There I stumbled on Sonnets by Giuseppe Gioacchino Belli, translated by Mike Stocks. Translated. Imagine translating sonnets. I opened it up and read one, and it was pretty good, so I bought the book.

I emerged unconvinced that translating sonnets is possible or sane. I don't know what I expected. The translations are real sonnets that rhyme and scan, an amazing technical achievement already, and they have a nice spontaneous feel. Not everything works. Many lines, inevitably, are awkward. The English slang Stocks uses is so different from what I grew up with as to sound inauthentic (a real shame).

Oh but what a pleasure to be introduced to Belli. He wrote these sonnets in Rome in the 1830s. If half of them are as true to life as they feel, it was a city alive with outrageous characters, illicit sex, and poor anger management. The best ones are little candid character sketches, slightly or heavily satirical, of harried mothers, old men, annoyed girlfriends, corrupt priests, and so on. There's one from a furious stutterer. Reading them is like eating cupcakes while falling in love.

I'll post one of the translations that I like, with the warning that this one is atypically clean.

A Very Roman Pastime

The treat that us lot liked the most when small,

the biggest thrill, the real McCoy, the biz,

was finding new-built homes and palaces

and using lumps of coal to trash the walls.

So here we'd doodle numbers, little sums,

and Gordon knots and those of Sollymom,

and there some Lotto stuff; and then move on

to filthy words and pricks and twats and bums.

Or else we'd take a stone or nail or stick

to gouge the plaster out, and draw a pic

so deep we'd hit the bricks and stuff below.

Those were the days all right, my God. Although,

that said, I like to dabble still, it's true...

and when I see a nice white wall, I do.

At the moment, a few more translations are posted on the book's web site.

Recently I learned...

• Your browser uses the public suffix list to determine whether two web sites may share cookies. This is not very robust but better than the previous strategy.

• If you take a long strip of paper, fold it in half as many times as you can, and unfold it, it'll make an approximation of the fractal shape called the Heighway dragon.

• If you take two fractions, say 1/2 and 1/3, and add the numerators and denominators, you get the mediant, in this case 2/5. I don't know much about the mediant, but it is linked to Ford circles in a way I don't really understand, and I'm told mediants give a startling way of approximating the value of continued fractions.

• glibc's qsort only does an actual quicksort as a last resort. If there's enough memory, it does a merge sort.

  This is old news, and I kind of figured it was the case, but I never looked at the source code before.

And I was reminded that complex numbers are key to quantum mechanics, something I missed when I was writing about complex numbers a year or two ago.

We've been unpacking books that have been in boxes for a year and a half. It's like meeting old friends. I read the epic of Gilgamesh, the book of Joshua, and Six Easy Pieces. I love small books.