Author Archive for Steve Landsburg

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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Diversification

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?

(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.

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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?
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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.

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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?

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.

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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:

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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?

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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.

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An Immense Discovery

bastiat.smallA month ago, I prematurely celebrated the 209th birthday of the great economic communicator Frederic Bastiat. Today (unless I’ve screwed up a second time) is actually his birthday. A good way to celebrate is to read (or reread!) Bastiat’s little masterpiece Economic Sophisms; you will never see the world the same way again. Here’s a taste:

At a time when everyone is trying to find a way of reducing the costs of transportation; when, in order to realize these economies, highways are being graded, rivers are being canalized, steamboats are being improved, and Paris is being connected with all our frontiers by a network of railroads and by atmospheric, hydraulic, pneumatic, electric, and other traction systems; when, in short, I believe that everyone is zealously and sincerely seeking the solution of the problem of reducing as much as possible the difference between the prices of commodities in the places where they are produced and their prices in the places where they are consumed; I should consider myself failing in my duty toward my country, toward my age, and toward myself, if I any longer kept secret the wonderful discovery I have just made.

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For Heaven’s Sake

HuckabeeMAHere’s Mike Huckabee, quoted in The New Yorker:

If somebody asked me, How do I get to Heaven, I would tell them that the only way I personally am aware of is faith in Christ, because I believe the New Testament. That’s the only map I got. Somebody says, Well, I got a different map. O.K.! You know what? If it works, I’m not going to argue with you.

Well, that makes sense. If somebody asked me, How do I get to Mount Rushmore, I would tell them that the only way I personally am aware of is Route 90, because I believe in Google Maps. Somebody says, Well, I got a different map. O.K.! You know what, if it works, I’m not going to argue with you. Unless, of course, I actually care whether you make it to Mount Rushmore or not, in which case I might take the trouble to defend my map.

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Frolicking

I’m taking four days off for camping, ice cream making and general frolicking in the woods. See you all on Tuesday!

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The Machinery of Freedom

machineryDavid D. Friedman’s The Machinery of Freedom, a classic of libertarian thought, has long been out of print and hard to find. (Well, it’s easy to find, actually. But hard to find for less than about a hundred bucks.) It is therefore a very good thing that David’s gotten his publisher’s permission to post the entire book on the World Wide Web, for free.

What does David get out of this? Well first, of course, he wants you to read his book. But second, he’s about to start preparing a third edition and welcomes reader feedback. If you post your comments here, I’ll make sure he sees them.

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Nonsense 4.0

powerballThe most recent winning Powerball numbers were 9,30,31,50,54,39. But a month ago, nobody would have placed any significant probability on those exact numbers coming up. What better illustration that questions about the future cannot be answered, even in the probabilistic sense?

If that made you scratch your head, your scalp will probably be rubbed raw before you’re finished reading Anatole Koletsky‘s Wall Street Journal essay, excerpted from his book Capitalism 4.0. (Caveat: I have not read the book, so I’m not sure how much danger the rest of it poses to your scalp, or to your sanity.) Mr. Koletsky’s “proof” that some questions “cannot be answered, even in a probabilistic sense” is this:

In 1980, nobody would have put any significant probability on computer sales exceeding car sales by a factor of 10 to 1.”

But that’s not all! There’s also this:

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You Can’t Keep a Good Straw Man Down

The artwork above is courtesy of Jodi Beggs, proprietress of the lively Economists Do It With Models site, who graced us with a visit in yesterday‘s comments and expanded on those comments on her own page. (That’s me kicking Paul Krugman in the gut.)

Jodi objects to the tone, and in part to the substance, of my response to Paul’s recent attacks on the “deficit hawks” who oppose various spending programs that Paul happens to favor. I’d summarized his rhetorical technique as follows:

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There He Goes Again

krugmanPaul Krugman sinks to a new low with this passage:

In America, many self-described deficit hawks are hypocrites, pure and simple. They’re eager to slash benefits for those in need but their concerns about red ink vanish when it comes to tax breaks for the wealthy. Thus, Senator Ben Nelson, who sanctimoniously declared that we can’t afford $77 billion in aid to the unemployed, was instrumental in passing the first Bush tax cut, which cost a cool $1.3 trillion.

Where to begin?

First, no economist—let me repeat that—NO economist, not even Paul Krugman on the days when he’s being an economist—would count a tax cut as a cost for purposes of policy analysis. A cost is something that consumes resources, not something that changes the ownership of resources. My Principles of Economics students all understand this; so, presumably, does the Nobel-prize winning author of a prominent Principles textbook. (A possible exception: You could call a present-day tax cut costly if it necessitates a future tax increase which, for some reason, is costlier to collect than the present-day tax. I guarantee you this is not what Krugman has in mind. If it were,the $1.3 trillion number that he highlights would be totally irrelevant to the actual cost.)

Next, unemployment benefits are costly, both insofar as they discourage recipients from seeking work and insofar as they necessitate taxes that discourage productive activity. The cost of $77 billion worth of benefits is not $77 billion, but it’s not zero either.

So unemployment benefits are costly and tax cuts are not. Which doesn’t mean that all unemployment benefits are bad or that all tax cuts are good, but it’s plenty adequate to absolve the hypocrisy charge.

But Krugman, as is his wont lately, appears committed to the following flat-out dishonest rhetorical agenda:

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What Even Google Can’t Tell You

How far is it from Seattle to Vancouver? About 142 miles, according to Google maps. But a classic paper from about 15 years ago estimates the distance as 75,000 “economic miles”, meaning that the cost of transporting goods across the border is equivalent to the cost of adding an extra 75,000 miles to the trip. A subsequent paper estimates the economic distance from the United States to Japan as roughly 7 light years, raising the question of why we trade with Japan when it would be so much cheaper to trade with Alpha Centauri, a mere 4 light years down the road.

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Environmental Economics

oilspillWhen big companies (like, say, British Petroleum) wreak great havoc (like, say, by spilling millions of gallons of oil into the Gulf of Mexico), it can be good policy to make them compensate their victims (like, say, with a $20 billion claim fund). It can also be bad policy.

A.C. Pigou taught us that we get better outcomes when decisionmakers bear the costs of their actions. Ronald Coase taught us that Pigou’s lesson cuts two ways. The shrimp boats that are sitting idle today are sitting idle partly because BP decided to drill in the gulf, but also partly because the shrimpers chose to operate in the vicinity of an oil rig. In this case, making BP feel the costs of its own decisions entails insulating the shrimpers from the costs of theirs.

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Alternative History

I’m visiting the Atlanta Fed this week, and had planned to arrive in my hotel room Tuesday evening in time to compose something interesting for you to read Wednesday morning. But the Atlanta airport was “closed for weather” as the airlines choose to word it, so I flew to Greenville, SC instead, drove a rental car to Atlanta, and arrived too late to say anything thoughtful.

I will leave you instead with this seven-year-old quote, and let you contemplate how history might have been different if anyone had listened:

Today, I will introduce the Free Housing Market Enhancement Act, which removes government subsidies from the Federal National Mortgage Association (Fannie Mae), the Federal Home Loan Mortgage Corporation (Freddie Mac), and the National Home Loan Bank Board. . . . Congress should act to remove taxpayer support from the housing GSEs before the bubble bursts and taxpayers are once again forced to bail out investors who were misled by foolish government interference in the market.

—Ron Paul, 2003

(Hat tip to Patri Friedman.)

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