(Click here to watch on YouTube.)

Sixteen years ago, Slate Magazine was launched, with Paul Krugman and me as the alternating economics columnists. At the time, Paul was fond of observing (with considerable dismay) that most of the time, highly educated and intelligent non-economists appear to be completely incapable of distinguishing between compelling arguments and utter nonsense in the field of economics. His essay on “Pop Internationalism” is a brilliant series of riffs on this theme — a guided tour of sheer balderdash that gets a respectable hearing even though no economist could possibly take it seriously. “Pop Internationalism” (the lead essay in the book of the same name) is high on my recommended reading list.

The lesson I took from this observation was that we (Krugman, I, and economic commentators in general) had a responsibility to explain not just what economists believe, but why we believe it — to help readers understand that there’s a rigorous underlying logic to the discipline, and that there are good reasons for insisting that people adhere to that logic. Nowadays, when he’s at his most obstreperous, I sometimes suspect Krugman of having drawn a very different lesson — that because nobody understands the real logic of economics, we can get away with saying any damned thing we want to. It’s a frustrating thing to watch, because when he’s good, he’s very very good. But when he is bad he is horrid. I won’t list examples here, but you can find quite a few by browsing my Paul Krugman archive.

There is, however, one area in which Krugman’s commentaries have my unequivocal approval, and that’s his habit of posting links to music videos. I don’t always agree with his choices, but nothing in economic logic precludes diversity in musical tastes. In fact, there are good reasons to applaud that diversity.

So in this one dimension, I aim to emulate Paul Krugman. I’m going to start posting occasional links to great (in my opinion) music videos. Also, occasionally to music videos that are not necessarily great, but at least greatly weird, because I like things that are greatly weird. Unlike Paul, I’m not going to wait until the music is somehow relevant to the news of the day. I’ll just post them when the mood strikes me.

Today’s video (see the top of this post) is from Danny Kaye and Louis Armstrong. Music doesn’t get much better than this.

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Why is the rate of teen childbearing is so unusually high in the United States as a whole, and in some U.S. states in particular? U.S. teens are two and a half times as likely to give birth as compared to teens in Canada, around four times as likely as teens in Germany or Norway, and almost ten times as likely as teens in Switzerland. A teenage girl in Mississippi is four times more likely to give birth than a teenage girl in New Hampshire—and 15 times more likely to give birth as a teen compared to a teenage girl in Switzerland. We examine teen birth rates alongside pregnancy, abortion, and “shotgun” marriage rates as well as the antecedent behaviors of sexual activity and contraceptive use. We demonstrate that variation in income inequality across U.S. states and developed countries can explain a sizable share of the geographic variation in teen childbearing. Our reading of the totality of evidence leads us to conclude that being on a low economic trajectory in life leads many teenage girls to have children while they are young and unmarried. Teen childbearing is explained by the low economic trajectory but is not an additional cause of later difficulties in life. Surprisingly, teen birth itself does not appear to have much direct economic consequence. Our view is that teen childbearing is so high in the United States because of underlying social and economic problems. It reflects a decision among a set of girls to “drop-out” of the economic mainstream; they choose nonmarital motherhood at a young age instead of investing in their own economic progress because they feel they have little chance of advancement.

The full paper is here.

Are you convinced?

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First, I am dismayed by the notion that anyone’s vote might be swayed by this issue, in either direction.

As I said earlier in the week, I do understand why this is an issue of major importance in some people’s lives. But the number of people affected, and the magnitude of the effects, are still meager compared to the effects of, say, trade or immigration policy. What ever happened to perspective?

Of course, Bryan Caplan will tell you that voters are systematically irrational in any event, so it might be just as well that they’re basing their votes on things that don’t matter very much, as opposed to basing their votes on things that really matter and getting them wrong. But it’s still disheartening to think about.

Second, I am dismayed by the President’s suggestion that he came to this viewpoint through observation of “incredibly committed monogamous” same-sex relationships among his staffers — suggesting (though not outright asserting) that monogamy ought to be somehow relevant to the legal status of one’s marriage. And here I’d thought the whole point of this gay marriage thing was that the way people have sex is not properly a concern of the legal authorities. If he’s continuing to deny this principle, then the President remains philosophically on the same side of this divide as Family Research Council.

I’ll go one step further: Any gay marriage activist who has taken the principled stand that people’s sex lives should be irrelevant to the legal status of their marriages, and who now embraces the president’s position, is, I think, engaged in some pretty serious hypocrisy. Because the President’s message to that gay marriage activist is not “You were right all along”. Instead it’s “I still think your position is wrong, but I’m going to give you what you want anyway.” I understand being joyful about getting what you want, but to follow that up with “Hooray, now the President’s on our side!” — when the President in fact continues to deny the primary principle on which you staked your claim — is pretty much equivalent to saying “I never really cared about the principle in the first place, as long as I got what I wanted.”

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For almost twenty years, Armchair has been widely recognized among economists as the book to give your mother when she wants to understand what you think about all day. In this new version, fully updated for the 21st century, I’ve completely rewritten several chapters to make them even clearer, livelier and more contemporary.

I (and you if you buy the book) am deeply indebted to Lisa Talpey who read every chapter multiple times, insisting that I keep rewriting until everything met her meticulous standards of clarity. Chapters I’d thought were pretty good are vastly improved thanks to Lisa; these include:

• Why Popcorn Costs More at the Movies (and Why the Obvious Answer is Wrong)
• Was Einstein Credible? (The Economics of Scientific Method)
• The Indifference Principle (Who Cares if the Air is Clean?)

and

• Why Taxes Are Bad (The Logic of Efficiency)

Others are almost completely rewritten to focus on issues that are in the news today; these include

• The Mythology of Deficits

and

• Unsound and Furious: Spurious Wisdom from the Media

By way of general housecleaning, I’ve excised all references to cassette tapes, Polaroid film, and Walter Mondale.

You can read the preface here. You can buy the book here. Here are direct links to the updated Kindle and Nook editions. (These editions are advertised as “published November 2007″, but don’t panic; that’s just the lingering shadow of last week’s glitch. They’re actually the brand-new 2012 edition.)

Dan Seligman at Fortune called the first edition of The Armchair Economist “enormous fun from its opening page”; Alfred Malabre of the Wall Street Journal called it “the most enjoyable and sensible book by an economist about economics that I’ve read in donkey’s years”; Milton Friedman called it “an ingenious and highly original presentation of some central principles of economics for the proverbial Everyman”; George Gilder called it “a crisp, lively, pungent display of the economist’s art”. This second edition is, I believe, all that and more.

The Armchair Economist is indeed the perfect gift for your mother, or for your father, or for the new college grad in your life, or even for yourself. Enjoy it, and come join the discussion right here.

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But Mitt Romney’s muzzling of foreign policy aide Richard Grenell (followed by Grenell’s departure from the campaign) seems to me to be a far more serious issue than mere homophobia. It indicates a lack of seriousness about foreign policy (and by extension about governance generally). Top-notch expertise is rare. To exclude expert advisors because of irrelevancies like sexual orientation (and it appears from news reports that this, not his Twitter postings, was Grenell’s key disqualification) is to handicap your administration from the outset. A week ago, Mitt Romney thought Richard Grenell was the best guy for this job. Nothing relevant has changed. I conclude that Mitt Romney doesn’t terribly much care whether a key foreign policy post is filled by the best guy for the job. That attitude is potentially dangerous on a scale where the Defense of Marriage Act registers only as a blip.

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As of this writing, the Amazon page still says you are buying the 2007 version, but that’s wrong. The version you’ll get is the new 2012 version.

Barnes and Noble still has the 1993 version for the Nook. This will be fixed in a day or two.

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There seem to be two kinds of people in the world: Those with enough techncal backgroud that they already know the answer, and those with less technical background, who have no hope — at least without a lot of work — of grasping the answer. I’m going to attempt to bridge that gap here. That means I’m going to throw a certain amount of precision to the winds, in hopes of being comprehensible to a wider audience.

As you might have heard, the set of real numbers is larger than the set of natural numbers, in the sense that there is no one-to-one correspondence between the two sets, and any attempt to construct such a correspondence will leave out some (in fact infinitely many) of the real numbers. We express this by saying that the reals are uncountable.

Now let’s be fanciful. Suppose the Devil comes along and prunes our mathematical universe. He throws away a whole lot of numbers, a whole lot of sets, and a whole lot of functions. When I say he “throws these away”, I mean he somehow arranges that these things will be erased from our mathematics; they will play no role in any calculations or proofs, and, though we will go on merrily doing mathematics as always, we will never notice they are missing.

In fact, he throws away so many real numbers that the remaining ones — the ones we have to work with — form a countable set.

The Devil’s problem is that we might notice that the reals are coutable, say by discovering an explicit one-to-one correspondence with the natural numbers. He solves this problem by insuring that, while he’s busy throwing away all those real numbers, he also throws away all the one-to-one correspondences between the natural numbers and the remaining reals. Those one-to-one correspondences exist, but they’ve been removed from our mathematics so we can’t discover them.

Then we’ll go on merrily believing that the reals are uncountable (in fact, we’ll still have our perfectly good proof of that fact) and there’s a sense in which we will be right. Indeed, when we say “the reals are uncountable”, what that means is: “There is no one-to-one correspondence between the natural numbers and the reals”. And in fact there is no such one-to-one correspondence. There used to be, but the Devil destroyed it.

But although we believe correctly that our set of reals is uncountable, the Devil believes correctly that it’s countable — because he kept copies of all those one-to-one correspondences he threw away. And though we believe correctly that our “set of all real numbers” is the set of all real numbers, the Devil believes correctly otherwise — because he kept copies of all those real numbers that he erased from our Universe.

Now comes the theorem: No matter what mathematical Universe you live in — that is, no matter what sets, functions, numbers, etc. you have access to — there is always a larger mathematical Universe, with more real numbers, and more one-to-one correspondences, whose denizens will believe correctly that our Universe was constructed by a Devil who started with the “true” real numbers, threw most of them away, left us only a countable set of them, and threw away enough one-to-one correspondences to keep us oblivious to that fact. (Of course, the Devil must worry that we’ll somehow discover that those one-to-one correspondences are missing, which means he’ll have to throw away a bunch of other stuff too, to prevent that discovery. And so on. But it turns out that he can always complete this herculean task.)

As we pass to larger and larger mathematical Universes, the set of natural numbers remains the same, while the set of real numbers keeps growing. No matter what Universe you inhabit, the denizens of the “next Universe up” will always agree that you’ve got the natural numbers exactly right, but they’ll sneer at your set of real numbers, which looks to them like a product of the Devil’s work — a mere countable set that doesn’t begin to account for all the “true” reals. Of course, there’s always another Universe where people are saying exactly the same things about them.

In that sense, there is only one true set of natural numbers, but there is no such thing as the “one true set” of real numbers. That makes it easy for me to believe that the natural numbers have a more solid kind of existience than those slippery reals.

That’s the main post. Now let me say a few things by way of partial penance for my (intentional) misprecision. What follows will be (slightly) less imprecise, but if you want true precision you should of course turn to the textbooks. (Try scouring the indexes for terms like forcing and Levy collapse.)

The standard axioms for arithmetic have many models — that is, there are many “number systems” that satisfy those axioms. The standard axioms for set theory also have many models — that is, there are many “mathematical Universes” that satisfy those axioms. The usual tools of first-order logic don’t allow us to distinguish among these models. On the other hand, almost all mathematicians believe that among the many models for arithmetic, there is exactly one, the so-called “standard model” that we’re actually talking about when we talk about arithmetic. (You can use the tools of second-order logic to specify this model uniquely, but I’d argue that that’s cheating. But while we might disagree about how to specify it, we pretty much all agree that the standard model exists.) On the other hand, when it comes to set theory, there is no clear way to point to one of the many models and say “that’s the one I’m talking about”. They all — or at least many of them — seem equally good.

In that sense, when we do arithemetic, we know exactly what we’re talking about. We’re talking about the good old standard natural numbers, and we pretty much all agree on exactly what those are (though we might disagree on how we know what those are). But when we do set theory, we’re in far murkier territory. I might be talking about one Universe, you might be talking about another, and we’d never know it. I’d say “the reals are uncountable” and you’d say “I agree”. But for all we know, we’re talking about different sets of reals, and either of us, if we knew what set the other was talking about, might say “but those aren’t the reals — and dammit, your set is countable”.

If you’ve got a solid advanced undergraduate math background, you might object that the reals are constructed from the naturals. We start with the naturals, form quotients, to get the rational numbers, then proceed to the reals by a process that involves taking limits (or something of the sort). Therefore, you might say, if I know what the naturals are, then I know what the reals are a fortiori. But the reals you construct depend not just on the rational numbers you’ve got available (which are the same in every Universe); they depend also on the sets of rationals you’ve got available — and the available sets differ from one Universe to another. No matter what Universe you live in, your construction of the reals at some point invokes the notion of “all” subsets of the rationals. And somewhere, in a higher Universe, the Devil is chortling — because he knows, and you don’t, that you’re missing a whole lot of subsets, and hence missing a whole lot of reals. But he too is missing a whole lot of reals, according to the next higher Devil.

I’m tempted to sum this up by saying that the natural numbers are real, but the real numbers are imaginary. Or, as Kronecker put it, “God created the natural numbers; all else is the work of Man”. (Note to Bob Murphy: My endorsement of Kronecker is contingent on a sufficiently metaphorical interpretation of the word “God”.)

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Today (May 1) was the official publication date for the all-new revised edition of The Armchair Economist. (Read the preface here.) Unfortunately, we had a major communication screw-up and purchasers of the electronic editions (Kindle, Nook, etc.) are still receiving the 1993 edition (which is labeled the “2007 edition” because that’s when it was converted to an e-book). This will be fixed in a day or so. I’ll post to let you know when it’s safe to buy.

Meanwhile, if you’ve recently purchased an electronic Armchair Economist, I encourage you to return it, wait just a day or two till you see the announcement that all systems are go, and then buy it all over again.

I cannot begin to tell you how sorry I am about this. There is at least one very particular head I’d like to see roll.

Thanks for bearing with the glitch, and watch this space for the big announcement that all systems are go.

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But if you were involved with mathematics in the second half of the twentieth century, sooner or later you were going to cross paths with Stanley, probably near the coffee machine in a math department. He’d proudly show you the little book he carried in his breast pocket, with the list of people to whom he owed money. Then he’d teach you something, or he’d tell you a good story.

Stanley had little tolerance for convention. His one permanent job, at the University of Rochester, came to an abrupt end during a faculty meeting where he spit on the shoes of the University president and walked out. Surely the same personality trait had something to do with his departure from the University of Chicago without a Ph.D., though the paper he wrote there (at age 22) has acquired fame and influence far beyond many of the doctoral theses of his more conventionally successful classmates. I’d like to tell you a little about that paper and what I think it means for the foundations of mathematics.

I write as one who believes (like most mathematicians) that the system of natural numbers (including the operations of addition and multiplication) exists in an objective sense. By that I mean precisely this: Statements about the set of natural numbers (such as “Every natural number is the sum of four squares”) have objective meanings; they are not just strings of words. I take it that a thing exists if one can speak meaningfully about its properties. The facts that every natural number is a sum of four squares, that every number can be factored into primes, and that an odd prime number is a sum of two squares if and only if it leaves a remainder of 1 when divided by 4, are all properties of the system of natural numbers. Because it has these properties, the natural number system exists.

It would be nice to give a succinct definition of the system of natural numbers. That turns out to be quite impossible. You can try writing down a list of axioms (Zero is a number; every number has a successor; no two numbers have the same successor; addition is commutative and associative and so forth), but it turns out that no matter what axioms you write down, there are always an infinite number of mathematical structures that satisfy those axioms but are not the natural numbers. Those structures — the ones that masquerade as the natural numbers by satisfying all the standard axioms, even though they behave very differently — are called nonstandard models of arithmetic.

This means, in effect, that there’s no way to uniquely specify the natural numbers. (Note to experts: You can of course uniquely specify the natural numbers if you’re willing to resort to a second-order theory, but that’s entirely unsatisfactory for reasons I’ve blogged about before.) Elsewhere, I’ve taken this remarkable fact as an indication of the natural numbers’ complexity — not only do they have no short description; they have no finite description at all.

Okay, then. What exactly are these non-standard models of arithmetic? What do they look like? What’s an example of a system that satisfies all the axioms of arithmetic but still isn’t ordinary arithmetic?

That’s where Tennebaum’s Theorem comes in. What Stanley proved (and of course I’m paraphrasing a bit here, because this is a nontechnical forum) is that in any nonstandard model of arithmetic, the rules of addition and multiplication are so complicated that no computer can be programmed to carry them out. In other words, you and I, whose brains are computers, have no hope of really understanding those addition and multiplication rules.

It’s been suggested that Tennenbaum’s Theorem creates a back door route to giving a precise and compact definition of the natural numbers. Namely: First we write down our axioms. Then we say, “Yes, I realize these axioms don’t nail down what I’m talking about — there are many different systems of `arithmetic’ that satisfy these axioms. But the one I have in mind is the one and only system of arithmetic that’s actually computable. That’s the natural numbers.”

Cute. But here’s why I’m quite sure it doesn’t work.

Here’s the problem. What does it mean to say that a computer can be programmed to add and multiply? It means that if I give that computer an arithmetic problem, it will return an answer in a finite amount of time. What does “in a finite amount of time” mean? It means “in a number of steps that corresponds to some natural number”. And what does that mean? Well, since my goal is to describe the natural numbers, I can’t just assume you already know what the natural numbers are. So it seems like my attempt at a description has come full circle right back to the beginning. Or to put this more succinctly, my attempt has failed.

I’ve been meaning to blog about this for ages, but have only just just discovered this paper by the philosophers Tim Button and Peter Smith that makes essentially the same point. They envision an interlocutor named Thoralf who is discomfited by the existence of non-standard models and worries that he has no idea which of these is the “true” set of natural numbers. We respond that it’s the one and only model where arithmetic problems can be solved by computation. Thoralf asks “What’s a computation?”. We reply that “A computation is a finite set of steps….”, whereupon Thoralf interrupts to ask “What does finite mean?”. We say “Something is finite if it can be measured by a standard natural number”. Thoralf asks: “And what are the standard natural numbers?”. And round and round we go.

Button and Smith conclude that “philosophical problems which are supposedly generated by mathematical results can rarely be tackled by offering more mathematics.”

It seems to me, then, that our only hope for picking out the honest natural numbers from among a sea of impostors is a direct appeal to intuition. Fortunately, almost all of us have that intuition. We’ve known what numbers are since we were three. We know what it means to say that every number is a sum of four squares. (Not all of us know how to prove this, but that’s beside the point. The point is that we all know what it means.)

So I contend that 1) the natural numbers exist because we can make meaningful statements about their properties, and that’s what existence consists of, and 2) the natural numbers are unfathomably complicated in the sense that there is no hope of pinning them down by any sort of description, even if we allow ourselves to incorporate sophisticated ideas like Tennenbaum’s into our description.

As many of you know, I’ve argued more than once (some of you might say more than necessary) that the existence of an unfathomably complex structure that was neither designed nor the product of evolution is a definitive counterexample both to the “intelligent design” argument that says a complex structure needs a designer and to Richard Dawkins’s position that all complexity is a product of evolution. It also settles (for me) the question of why the physical Universe exists — once you’ve explained the existence of something as complex as the natural numbers, explaining the existence of something as relatively simple as the Universe becomes a mere exercise.

Tennenbaum’s Theorem, on the face of it, presents a challenge to this way of thinking, but it is, I think, ultimately an unsuccessful challenge. It’s still a beautiful theorem, though. And Stanley was a beautiful guy. Sometime soon I’ll tell you more about him.

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Golden Balls is a British game show where players decide, in secret, whether to adopt a strategy of “Split” or “Steal”. In this episode, they face the following payoffs (in British pounds):

This is almost, but not quite a classic Prisoner’s Dilemma situation. (To make it a true Prisoner’s Dilemma, where stealing always beats splitting, you could change the lower-right hand box to “1 each” instead of “0 each”.) As in the Prisoner’s Dilemma, you can never go wrong by stealing — though you can go horribly wrong when the other guy steals, so it makes sense to reach a no-stealing agreement — and then to violate it.

In other words you’d pretty much expect homo economicus to steal every time. But this game is far more interesting than the usual textbook version of the Prisoner’s Dilemma, because it’s played by real people for real money and they negotiate in public for half a minute before they choose their strategies. In principle, the negotiation shouldn’t change anything (unless the players come to care about each other, or about the way they’re perceived by the audience). But in this episode, the negotiation took an unexpected turn.

What would you have done?

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