Monthly Archive for July, 2010

Weekend Roundup

roundupWe led off the week with a diversion that might have been premature in the sense that I can do more tricks now than I could then, and a bit more gracefully too, I think. More video when I feel ready to graduate to actual fire.

Next a post on the different kinds of logic, and a related post on what it all means.

Sadly, the latter continued to draw comments from readers who want to “define the natural numbers via axioms”, whereas the whole point of these posts is that nothing of the sort is possible.

On Thursday I took issue with Robin Hanson’s take on polygamy; Robin responds here.

And on Friday I pointed to an unconventional high school valedictory speech.

Note to RSS readers: Friday’s “high school” post was originally scheduled for Thursday. But when I read Robin’s polygamy post on Wednesday night, I wanted to respond to it, so I scheduled that post for Thursday and rescheduled the high school post for Friday. For some reason the rescheduling didn’t take, so that the high school post was briefly posted Thursday morning before I realized what had happened and took it down. By then, though, the RSS feeds had it. So that’s why many of you saw the same post two days in a row.

Back on Monday of course.

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How to Succeed in High School

High school valedictorian Erica Goldson explains the secret of her success:

I have successfully shown that I was the best slave. I did what I was told to the extreme. While others sat in class and doodled to later become great artists, I sat in class to take notes and become a great test-taker. While others would come to class without their homework done because they were reading about an interest of theirs, I never missed an assignment. While others were creating music and writing lyrics, I decided to do extra credit, even though I never needed it.

This is from her valedictory address to her fellow graduates; you can read the entire speech on her blog.

What do you think?

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More Wives are Unsafe Wives?

One argument that’s often made against legalized polygamy is that rich old men will marry lots of women, leaving lots of poor young men both single and sexually frustrated—-and that’s bad, because poor young single sexually frustrated men are prone to criminal acts of violence.

Over at Overcoming Bias, Robin Hanson objects that if people really believed this argument, they’d want to criminalize lesbianism and extramarital affairs, both of which also contribute to the problem of men-without-partners.

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Godel, Fermat, Hercules

HerculesAndHydraYesterday I answered one of Coupon Clipper‘s questions about Godel’s Theorem. Today I’ll tackle the other: Does Godel’s Theorem matter on a day-to-day basis to practicing mathematicians?

And the answer is: Of course not. Mathematicians care about what’s true, not about what’s provable from some arbitrary set of axioms. (Of course this is an overgeneralization; some mathematicians have built distinguished careers on worrying about what’s provable from various sets of axioms. But they are a small minority.) Godel’s Theorem says that not all true things are provable. But for the most part, we’re happy just to know they’re true.

The flashiest example I can give you—and one I’ve used on this blog before—is Fermat’s Last Theorem, which says that no equation of the form xn + yn = zn has any solutions, as long as n is at least 3 and x, y and z are non-zero. Proving this was the was most famous unsolved problem in mathematics for 350 years until it was solved (to much public fanfare) by Frey, Serre, Ribet and Wiles in the 1980′s and 1990′s.

We know from that work that Fermat’s Last Theorem is true. However, we still don’t know whether Fermat’s Last Theorem follows from the standard axioms for arithmetic. And—this is the point—very few mathematicians care very much, at least by comparison to how much they care about the theorem itself. (Here is one of my favorite papers on the subject. Tellingly, the author is a philosopher.)

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First Things and Second Things

The occasional commenter who goes by the name Coupon Clipper has emailed me some interesting questions about Godel’s Theorem. I think I’ll answer them here.

The first question is about first-order versus second-order logic, so let me explain the distinction. When we reason formally about arithmetic, we need to clearly specify the ground rules. This means, among other things, specifying the language and grammar we’re allowed to use. A very simple language might allow us to say things like “2 + 3 = 5″ or “8 is an even number”. With a language like that, you could talk about a lot of sixth grade arithmetic, but you wouldn’t be able to say anything very interesting beyond that. To talk about the questions mathematicians care about, you need a language that contains words like “every”, as in Every number can be factored into primes or Every number can be written as a sum of four squares or Every choice of positiive numbers x, y, and z will yield a non-solution to the equation x3+y3=z3 . That language is called first-order logic.

With first order logic we can specify a set of axioms, and then follow a prescribed set of rules to deduce consequences. For example, if you’ve already established that every number is a sum of four squares, then you’re allowed to conclude that 1,245,783 is a sum of four squares. (The general rule is that if you’ve proved that every number has some particular property, then you’re allowed to conclude that any particular number has that property.)

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With the economy still faltering and economists increasingly in disrepute, I’ve decided that prudence dictates the acquisition of a new marketable skill. How am I doing?

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(Larger version here.)

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Weekend Roundup

roundup2Before we get to the roundup, here’s the latest chapter in the ongoing intellectual suicide of Paul Krugman:

  • Economists Carmen Reinhart and Ken Rogoff write a scholarly paper purporting to show that high levels of government debt lead to slow economic growth. For the record, I have not read this paper.
  • Krugman, while praising the authors’ previous work, asserts that this time, there’s no there there. Specifically, he says that most of the Reinhart-Rogoff evidence comes from four episodes. According to Krugman, none of these four episodes counts. One could certainly well imagine a reasoned argument along these lines.
  • Krugman’s, however, is not that reasoned argument. Here is how he dismisses the episode labeled “Canada in the 90s”:

advocates of austerity have been using Canada in the mid-90s as an example of a success story; surely they can’t have it both ways.

The problem, of course, is that there is no “they” who are trying to have it both ways. Reinhart and Rogoff have made an argument about Canada in the 90′s. That argument stands or falls on its own. It is no refutation to observe that somebody else might have made some other (correct or incorrect) argument about Canada in the 90′s.

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Pop Answers

Yesterday’s pop quiz posed this question:

Suppose that an acre of land in Iowa can yield either 50 bushels of wheat or 100 bushels of corn, while an acre of land in Oklahoma can yield either 20 bushels of wheat or 30 bushels of corn.

Which state has the comparative advantage in growing wheat? Which state has the comparative advantage in growing corn?

Suppose the residents of each state consume 200 bushels of wheat and 360 bushels of corn. If, instead of pursuing policies of self-sufficiency, each state specializes in its area of comparative advantage, how many acres of Iowa and Oklahoma farmland are freed up for other uses?

Quite a few people got this right in comments. In Iowa, the opportunity cost of a bushel of wheat is 2 bushels of corn. In Oklahoma, the opportunity cost of a bushel of wheat is 3/2 bushels of corn. Becauses 3/2 is less than 2, Oklahoma is the low-cost wheat producer, which is the same thing as saying that Oklahoma has the comparative advantage in wheat.

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Pop Quiz

Commenting on this essay by former Intel chief Andy Grove, Tyler Cowen writes that “Only he who first shows he understands comparative advantage has license to partially reject it.”

Hear hear. When someone says “I understand comparative advantage, but in this case it doesn’t apply”, or “I understand comparative advantage but in this case it is overridden by other considerations”, my experience tells me that you can be nearly sure you’re talking to someone who does not in fact understand comparative advantage.

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Play At The Plate

If you haven’t seen this, you should see it. You don’t need to know or care much about baseball to be delighted. It’s the bottom of the eighth, tie score, bases loaded.

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Ultimately Simple

Stop me if you’ve heard this one. A subject (called the proposer) is placed in an isolation booth and given ten dollars to divide between himself and the stranger in the booth next door. The stranger (called the responder) can accept or reject the division. If he accepts, they each take their shares and go home. If he rejects, they each go home with nothing.

In experimental plays of this ultimatum game, responders tend to reject splits that are substantially worse than 50-50. This is offered as some kind of reproof to the principles of economics. After all, the responder is turning down free money.

But so what? Continue reading ‘Ultimately Simple’

A Pencil in the Eye

Okay, if Paul Krugman is going to keep on writing the same column twice a week every week forever, then I am going to keeping on objecting to it forever, though not, I promise, twice every week.

A couple of bullet points from his latest:

  • In response to the priorities of Senator John Kyl, Krugman writes: “So $30 billion in aid to the unemployed is unaffordable, but 20 times that much in tax cuts for the rich doesn’t count.” Oh, for goodness’s sake. $30 billion in aid to the unemployed might or might not be good policy and 20 times that much in tax cuts might or might not be good policy; that’s beside the point here. The point is that these are quite entirely separate issues and one’s position on the first need not dictate one’s position on the second. Aid to the unemployed is costly. Tax cuts are not. Didn’t I just say this?
  • Continue reading ‘A Pencil in the Eye’

Weekend Roundup

Somehow we’ve gone a month since the last weekend roundup. So this will reach back a little further than usual in time.

Riddles. We tackled some riddles: Why do guys with deep pockets take on risky ventures instead of selling them off to someone with nothing to lose? Why, when a plane headed for Atlanta is diverted to Greenville, does everyone else choose to stand for an hour at the ticket counter while I (and only I) saunter over to the Hertz counter and grab one of many available cars? And why does Jet Blue, after investing $800 million in its new terminal at JFK, choose to make that terminal so hellish a place that I for one will never travel through it again if I can possibly avoid it?

Paul Krugman. Yes, I know, I can’t seem to let this topic go. I was at it here, and then here, and here and finally here.

Let me summarize my complaint in a paragraph: Krugman has some policies he’d like to see enacted. Some people oppose those policies for silly reasons and others oppose them for sensible reasons. Krugman habitually ridicules the silly reasons and pretends that he has therefore dispensed with the sensible reasons.

More specifically, Krugman attacks “deficit hawks” but ignores the “spending hawks” who present a much stronger case for fiscal restraint. He’s right to attack the deficit hawks, who make the silly mistake of conflating spending (which is costly) with tax cuts (which are not)—but then he makes the same mistake himself when it suits his purposes.

Incidentally, my Toy Stories post contains a link to a toy model intended to highlight the key questions that Krugman willfully ignores. At the end of that post I added an addendum confessing to arithmetic errors in the model and inviting readers to correct them. On a second reading, I realized there are no arithmetic errors—just one typo in an equation. Because some comments refer to that typo, I’ve chosen not to correct it, but it’s explained in the current addendum to the original post.

Books. Our book posts covered everything from the ridiculous to the sublime to the magnificent.

Math. The music of the primes gives a glimpse of the glorious intricacy of arithmetic, and our post on Fermat’s Last Theorem gives a small taste of how to tackle a particularly vexing problem.

Videos We had videos on cruel and unusual punishment, on the end of racism, and on how to fix everything.

Miscellaneous. Can Mike Huckabee possibly believe the things he says about religion? Does anyone still subscribe to the superstition of dollar cost averaging? And why the disproportionate outrage about an oil spill in the Gulf when there’s so much more to be outraged about?

Okay, we’re more or less caught up now! See you Monday.

The Girl Who Played With Numbers

girlfireI’ve just finished reading The Girl Who Played with Fire, the second book in the series that begins with The Girl with the Dragon Tattoo. I’m not giving away any significant plot point when I tell you that there’s a character who works on Fermat’s Last Theorem as a hobby, or that the book was clearly written (or perhaps translated) by somebody with no clue how mathematics works or what Fermat’s Last Theorem is about. I particularly liked the reference to Andrew Wiles using the “world’s most complicated computer program” to solve the problem. It’s my understanding that Andrew barely even uses email. And certainly if you understood anything about the nature of the problem and/or the solution, you’d recognize the absurdity of trying to tackle it with a complicated computer program.

Be that as it may, I finished the novel with a few hours left to spare, so of course I was inspired to work on Fermat’s Last Theorem, or at least on the simplest cases. The problem, if you’ll recall, is to show that there are no integer solutions to any of the equations x3+y3=z3 , x4+y4=z4 and so on, except for the so-called trivial solutions in which one or more variables take the value zero.

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The End of Racism

I was delighted last month to learn that racism in America has been thoroughly vanquished, as evidenced by the NAACP’s having nothing better to do than complain about a greeting card that shows cartoon characters encountering black holes as they hurtle through space. (“It’s very demeaning to African American women”. See if you can guess why, then watch the video below to check your answer.)

I realize that some will criticize the NAACP for over-reacting here, or for mis-reacting. But cut them a break. You don’t see them doing anything truly loonytunes, like, say, commanding the amorphous Tea Party movement to “expel the bigots and racists in your ranks or take full responsibility for all of their actions.” Right?

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A hat tip to our frequent commenter Ken B. for pointing me to the video.

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Eggs and Baskets

eggsOver at Marginal Revolution, Tyler Cowen puzzles over Ian Ayres‘s take on investment strategy:

In our risk-reducing implementation, we want people to borrow to invest more when young and then invest less when older. The lifetime exposure to stocks is held constant. Compare the following two investment paths:

Option 1:

  • Year 1 Invest $1
  • Year 2 Invest $2
  • Year 3 Invest $3

Option 2:

  • Year 1 Invest $2
  • Year 2 Invest $2
  • Year 3 Invest $2

Our view is that option 2 is the safer bet.

(Note that when Ayres says “invest $2″ he does not mean “Add $2 to your investment”. He means “Have a total of $2 invested.” So under Option 1 you add a dollar a year to your investment. Under Option 2 you do all your investment up front and then scoop out all the profits every year (or scoop replacement funds in if you’ve taken losses). Option 1, in other words, is the widely touted but thoroughly ridiculous strategy often called Dollar Cost Averaging.

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All in the Details

jetblueIs there a name for this phenomenon? A firm sinks vast resources into an immensely complicated engineering project and gets most of it right, but gets one detail so glaringly wrong that it seems like they might just as well not have bothered. Lexus and Jet Blue come to mind.

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Back Tomorrow….

After several days in 105 degree desert heat, followed by 12 hours of travel, I think I am going to collapse now rather than blog. If all goes according to plan, I’ll be back with something substantive Tuesday morning.

Leavin’ On a Jet Plane

bellI’m off to Las Vegas and FreedomFest, where I’ll be speaking (probably) on why More Sex is Safer Sex, and debating Dinesh D’Souza in a specatacle billed as Religion on Trial. The religion debate will take place Friday July 9 at 5PM at Bally’s Casino and we expect it to be carried on both C-Span and C-Span 2, though I’m unclear on whether the coverage will be live. (Quite possibly it depends on what else that’s newsworthy is going on.)

All of which is my excuse for taking the rest of the week off from blogging. I’ll see you next week, and I’ll let you know how things went.

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Toy Stories

toysPaul Krugman is at it again, casting aspersions on everyone who opposes extended unemployment benefits while offering absolutely no positive argument for those benefits. Let me explain what would count, to an economist, as a positive argument.

There’s no question that extending benefits would be good for the currently unemployed, and no question that it would be bad for those who are called on to foot the bill. Economists usually deal with that kind of conflict by asking what policy you’d prefer if you had amnesia, and and didn’t know your own employment status. (You can read a lot more about this approach to policy analysis in Chapter 16 of The Big Questions.) The amnesiac is an impartial judge who is forced to care about everyone, because he/she might be anyone.

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How To Fix Everything

A couple of weeks ago, I visited the Atlanta Federal Reserve Bank and gave a talk on “How to Fix Everything”. Here is the video:

(Edited to add: The video I originally posted was hard for several of you to view due to the large file size. I’ve shrunk the filesize by 2/3 and converted from m4v to flash, which I believe should solve the problem.)

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Cruel and Unusual?

Michelle Lyn Taylor, age 34, got drunk one night and tried to seduce a 13 year old boy by taking his hand and putting it on her breast. This was definitely Not Cool. Nevada prosecutors thought it was so uncool that they charged her with a crime (“lewdness with a child under the age of 14″) that carries a mandatory life sentence and then refused to plea bargain. (Two years earlier, a woman who had sexually abused two boys repeatedly over the course of a year was offered a plea bargain and served ten months in jail.)

In the videotaped sentencing hearing, which you can see in its entirety below, the judge seems bemused by the prosecutors’ choices but unmoved by the defense attorney’s attempt to raise a constitutional objection. The defense attorney, not entirely unreasonably, pretty much loses it.

Michelle Lyn Taylor is now serving a life sentence and will have her first shot at parole ten years from now. Does this strike any of you as reasonable?

The Music of the Primes

If there is a God, this is the closest you’ll ever come to hearing Him sing. Let me explain.

Continue reading ‘The Music of the Primes’