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.

But I think one could consistently take the position that while lesbianism contributes to the problem, it’s just not widespread enough to be worth stamping out, while polygamy (if legal) might well become so. And as far as extramarital affairs, I think Robin has it completely backward: When the wife of a 30 year old man (who is well past the prime age of violence) has an extramarital affair with an 18 year old, she is alleviating the problem, not contributing to it. Besides, most extramarital affairs do not deprive the husband of a long term sex partner.

I’m all for legalizing polygamy, because I’m all for legalizing almost everything. But I don’t think you can dismiss this argument—or the sincerity of its proponents—as easily as Robin seems to think you can.

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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.)

In Chapter 10 of The Big Questions, you’ll find a description of a game called “Hercules versus the Hydra”. It turns out that Hercules always wins the game, no matter how stupidly he plays. This fact (that Hercules always wins) is known to be true and known to be unprovable from the standard axioms. Fermat’s Last Theorem, by contrast, is known to be true and might or might not be unprovable from the standard axioms. If that question gets settled, most mathematicians will be interested enough to sit up and take notice. But unlike the Last Theorem itself, few are motivated to work on it.

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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.)

Second order logic expands the language we’re allowed to use, by allowing us to apply words like “Every” not just to numbers, but to sets of numbers. So in second order logic, we can say things like “Every set of numbers has a smallest element”. In first-order logic, that sentence would be ungrammatical and hence meaningless.

Now: Godel’s Theorem, as it’s sometimes stated, says that no first order logical system can prove all the truths of arithmetic. Start with any true axioms you want, and there will always be other true things you can’t prove—not just because you’re not smart enough but because there really are no proofs within your system.

Coupon Clipper’s first question is (I am paraphrasing, accurately I hope): So what? Why not just use second order logic instead? He also guesses accurately at the answer, which is that Godel’s Theorem applies just as well to second order logic as it does to first order logic. There will still be some true statements in arithmetic that your system can’t prove.

That answers the question, but it raises another question: Why do mathematicians prefer to avoid second order logic?

Second order logic certainly has its advantages, and here’s the big one: It let’s us nail down what we’re talking about. What we’re talking about, of course, are the natural numbers: 0,1,2,3, and the rest. And nothing more! We don’t want our system to contain numbers that are infinitely big or infinitely small or exotic in other ways. First order logic can’t rule that stuff out. Second order logic can.

In other words, in any system of first order logic, all the theorems you can prove are true statements about the natural numbers, but there will always be other more exotic systems of “numbers” of which your theorems are also true. That means there is no way, within the language, to distinguish between the honest-to-god natural numbers and some of these other systems. There is no grammatical way to say “No, no, I mean the *true* natural numbers, not those impostors!”

With second order logic, that problem goes away. The theorems you can prove are true statements about the natural numbers, and they’re not true statements about anything else. There’s no ambiguity about what you’re describing.

But the offsetting disadvantage is huge: In first order logic, I can tell you what all the rules are. (Remember, for example, the rule that says that if you’ve established that every number has some property, you’re allowed to conclude that any particular number has that property.) In second order logic, I can’t. Neither can anybody. Neither can any computer. It is a theorem that no computer program can generate all the valid rules of inference in second order logic. That’s in some sense a much bigger deal than Godel’s theorem. Godel’s theorem says that (in either first or second order logic) no computer can follow the rules and discover all the true statements of arithmetic. But now I’m telling you that in second order logic, no computer can even figure out what the rules are!

Hence the oft-repeated slogan that “second order logic is not logic”, and hence our reluctance to rely on it.

Coupon Clipper’s second question is “Does any of this matter for the actual practice of mathematics?”. That’s a much easier question with a much shorter answer, but I think I’ll save it for tomorrow.

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(Larger version here.)

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• 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.

The fallacy here is to lump together all of the people with whom Krugman disagrees on matters of policy, assume they all agree on matters of theory, and hold them responsible for each others’ views, even to the point of accusing A of inconsistency because he disagrees with B. For the first time ever, I went ahead and left a comment on Krugman’s blog to this effect. But it’s exactly the same intellectual crime I’ve been objecting to here on The Big Questions for weeks.

Most recently, I objected this past Monday, which brings us to the beginning of our roundup. Tuesday, we talked about the ultimatum game, in which experimental results indicate that people are hesitant to steal money. I’m not sure what that tells us beyond what we already knew. Wednesday we had video from one of the most amazing plays in the history of baseball. And on Thursday and Friday, we had a quiz question and its solution to test your understanding of comparative advantage—one of the deepest and most fruitful insights in the history of economics.

I’ll be back on Monday.

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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.

For a bushel of corn, the opportunity cost is 1/2 bushel of wheat in Iowa or 2/3 bushels of wheat in Oklahoma. Because 1/2 is less than 2/3, Iowa is the low-cost corn producer, which is the same thing as saying that Iowa has the comparative advantage in corn.

Knowing the comparative advantages tells us, without any further computation, that everyone can be richer if Oklahoma exports wheat and Iowa exports corn. But if we choose to do the extra computation, we see that with self-sufficiency, Iowa needs 7.6 acres to feed its people and Oklahoma needs 22 acres. whereas if Iowa grows enough corn for both states and Oklahoma grows enough wheat for both states, then Iowa needs only 7.2 acres and Oklahoma needs only 20 acres. The remaining acres are available to feed even more people, or to build NASCAR tracks.

(Edited to add: It was pointed out in comments that in the first couple paragraphs of this answer, I said “corn” when I meant wheat and vice versa. This is, I think, fixed now. Very sorry for not proofreading better.)

This example is ultra-simplistic in several ways of which here are two:

• It assumes that the yield in either state is a simple multiple of the number of acres under cultivation—in other words, we (unrealistically) assumed away diminishing returns.
• It assumes that food consumption in each state is fixed, as opposed to varying with price.

Econonerds might recognize that these assumptions are tantamount to assuming that all supply curves are horizontal and all demand curves are vertical.

But do not make the mistake of dismissing the example because of its simplicity. As I tell my students, you can’t understand the hard stuff till you’ve mastered the easy stuff. And as those students progress to more difficult and realistic exercises, they come to realize that the underlying logic remains pretty much inescapable—compute the comparative advantages, and you will discover an opportunity for everyone to gain from trade.

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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.

If I were permitted to write a licensing exam for economic commentators, I would surely incorporate a standard textbook exercise or two on this subject. Being lazy, I’d probably lift them from an existing textbook. For example:

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?

Anybody? Bueller? Grove?

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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? People turn down free money all the time. Just this morning, I saw a $20 bill fall out of a guy’s pocket and I returned it to him. I did that because I don’t like taking things that aren’t mine—a sentiment that’s easy to incorporate into orthodox theory. Likewise, if you put me in the ultimatum game (at least as it’s usually run), I’ll turn down whatever I’m offered, whether it’s 50/50 or 20/80 or 80/20. That’s because I don’t feel any better about taking money from the taxpayers who are presumably funding this experiment than I am taking money from the guy in front of me in the checkout line.

Now, not everyone is as moralistic as I am on this point (as I am not as moralistic as some others on some other points), but most people are not psychopaths, so most people are going to feel at least a little squeamish about taking money nonconsensually from others. Some might be willing to overcome that squeamishness in exchange for a reasonable share of the bounty (but not for a smaller share). That doesn’t mean they’re irrational. Quite the contrary. It means everyone has his price. Just like your Principles of Economics teacher taught you.

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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?
• And Krugman goes on: “On paper, solving America’s long-run fiscal problems is eminently doable: stronger cost control for Medicare plus a moderate rise in taxes would get us most of the way there.” In the words of a certain Nobel laureate, this passage makes me want to stick a pencil in my eye. It’s like saying “Getting my family budget under control is eminently doable: fewer caramel macchiatos plus more trips to the ATM would get us most of the way there”. Yes, cutting spending will help. No, moving money around won’t.

  Like it or not, the government can always raise taxes and therefore always has a claim on your wealth. You are the government’s ATM. Making a withdrawal today (by raising your taxes) means there will be less available to withdraw tomorrow. Raising taxes does not make the government richer. Didn’t I say this already?

• Yes, I know, a bunch of silly Republicans (along with a bunch of silly Democrats) have gone around saying the deficit is all that matters, which implies that a tax increase can offset a spending increase. Yes, silly people are silly. But the existence of silly people is not enough to justify additional spending. For that, you’ve got to, you know, attempt to justify the additional spending.

Sometimes a caramel macchiato is worth the cost and sometimes it’s not. Sometimes an unemployment benefit is worth the cost and sometimes it’s not. We have useful ways of thinking about these things. Some people say: “You know, that macchiato is expensive but it’s worth it.” Others say: “I don’t think that’s a very wise expenditure right now.” Paul Krugman says: “AHA! So a caramel macchiato is unaffordable, but you still haven’t visited the ATM!” And I say: How’s that again?

Stay tuned for more Godel, more Darwin, and a little less Krugman.

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