We started the week with a few words about estate taxes.

Then it was on to the justice system. On Tuesday I offered a little quiz on recognizing reasonable doubt (or its absence) in an exceptionally simple environment. Some readers thought that environment was too simple to be interesting. On the contrary, it’s simplicity is what makes it so interesting. If we can’t recognize reasonable doubt in such a simple environment, how can we ever recognize it in the courtroom?

On Wednesday, we talked about the appropriate numerical cutoff for reasonable doubt, and on Thursday we took a step back and asked what principles we should apply in choosing that cutoff. On Friday, I decried the dereliction of duty by judges and legislators who refuse to tell us what cutoff they have in mind when they use the word “reasonable”.

Some commenters thought that giving jurors a precise numerical standard was asking them to think more “mathematically” (whatever that means) than we can reasonably expect. But there’s no mathematics involved in telling a juror that he should convict if he believes that in 100 similar cases, at least 93 of the defendants will be guilty. No mathematics, that is, beyond the ability to count to 100, which is, I think, something we already expect of our jurors.

If I don’t tell you what “reasonable” means, then “beyond a reasonable doubt” makes as much sense as “beyond a gribzle doubt”. Judges could, if they wanted to, tell juries to convict if the evidence convinces them beyond a gribzle doubt, and then refuse to reveal what “gribzle” means. I don’t see how that system would differ substantially from the one we have now.

The Internet seems to have bred a peculiar subspecies of troll that cheerfully devotes enormous effort to refuting arguments nobody ever made. While they seem to have infinite time to construct these pointless rebuttals, these troll-types seem to have no time at all in which to actually digest the arguments they think they’re rebutting. They start with a guess as to what someone else might have said, and seem all but incapable of entertaining the notion that they might have guessed wrong. Is there a name for these people? “Crank” and “troll” are too general. If it were up to me, we’d reserve the word “Bozo” for this purpose, but it too is already in more general use. We need a new word! Give me your suggestions!

A title like More Sex is Safer Sex is like red meat to these folks and on Monday we took a moment to defend that argument against the latest Bozo Barrage (I’ll stick with this name till you give me a better one). Then on Tuesday we confronted a different subspecies of troll — the statistical obfuscator. Our reader Windypundit did some detective work and discovered that the offending graph was produced by an agency of the United States government. That, of course, is no excuse for perpetrating the deception.

Our graphical escapade led to a discussion of the gender gap in wages and whether it can be plausibly explained by employer discrimination. It’s often argued that it can’t, because that would require employers to sacrifice a profit opportunity. On Wednesday, I rejected that argument on the grounds that employers sacrifice profit opportunities all the time — but offered a (slightly) more sophisticated version that rejects the employer-discrimination hypothesis because it would require employers to sacrifice a very large profit opportunity. Of course, as our reader Patrick observes, this still doesn’t rule out the hypotheses of customer-discrimination or employee-discrimination. (In the latter case, male workers refuse to accept female colleagues. And again, the argument I gave can’t reject this. On the other hand, a different argument probably can — if the wage gap were driven by employee-discrimination, firms could profit not by hiring a few more females, but by hiring only females. In equilibrium, you’d expect half of all firms to be all female, half all male, and wages to be equalized across firms.

Thursday I reran a year-old post on how to add all the positive integers and get -1/12. This post generated just one comment! I’m not sure whether this was because none of you like this kind of thing, or whether you found it too awesome to remark on.

Continue reading ‘Weekend Roundup’

The topics of the week were daughters and divorce, capital taxation, capital taxation again, and intelligent childhood questions. I’ll be back with more, of course, on Monday.

We had substantive posts this week on two of our recurring topics — economic efficiency and the foundations or arithmetic.

The former brought us the honor of an extended visit from Uwe Reinhardt, who, as far as I can tell, objects not to the concept of efficiency or to its usefulness, but to its name. But any crusade to change a well-established technical term is, I think, doomed to failure.

Efficiency, of course, is only one of the normative criteria in the economist’s arsenal. I pointed, for example, to an earlier post where I’d outlined a toy framework for evaluating some of the normative claims made by one of Professor Reinhardt’s Princeton colleagues. That toy framework employs a utilitarian criterion that goes beyond efficiency. It evaluates policies on the basis of “what an amnesiac would prefer”, which is very different than a pure efficiency criterion. This kind of analysis is perfectly standard in economics, so any allegation that we fixate exclusively on efficiency is a bum rap.

On the other hand, some fixation on efficiency can be an extremely valuable exercise, for reasons that I hope this week’s post made clear.

Re the foundations of arithmetic, I posted to dismiss the view that the natural numbers are fictitious. As one commenter pointed out, this was largely an attack on a straw man, because almost nobody believes otherwise. Indeed it was. This was intended as an educational post, not a contentious one, and attacking straw men can be a very effective form of education. When I teach students about continuous functions, I ask them to imagine a hostile party who insists that the function f(x) = x is not continuous, and we talk about how you could most effectively convince him otherwise. The hostile party is imaginary, but there’s a lot to be learned from thinking about how you’d refute him.

We also speculated on the defining idea of the next decade and the ideal reading list for a course on how economists view the world.

And then there was the probability problem: A woman has two children, one of whom is a boy born on a Tuesday. What is the probability they’re both boys? Several commenters explained the answer very clearly. In case you haven’t read the comments and don’t want me to give away the answer, I’ll just say that it’s greater than 45% but less than 49%. See the comments on the original post for the reason why.

We’re coming up on a long weekend, and I’m taking Labor Day off. I’ll see you Tuesday.

It was a week of mathematics here at The Big Questions. I am still reeling from the momentous events that inspired Monday’s post; we now know that the Internet has changed mathematics forever. On Friday, we celebrated the momentous achievenments of the new Fields Medalists.

In between, we began what will be an occasional series on the foundations of arithmetic. In Part I, we distinguished truth from provability. In Part II, we distinguished theories (that is, systems of axioms) from models (that is, the mathematical structures that the theories are intended to describe). A theory is a map; a model is the territory. In Part III we talked about consistency and stressed that it applies only to theories, not to models. A purported map of Nebraska can be inconsistent; Nebraska itself can’t be.

It turns out (a little surprisingly) that any consistent map must describe multiple territories. (That is, any consistent set of axioms must describe many mathematical structures — or in other words, any consistent theory must have many models.) (This assumes the map has enough detail to let us talk about addition and multiplication.) These territories—i.e. these mathematical structures, all look very different, even though they all conform to the map. Conclusion: No map can fully describe the territory. No set of axioms can fully describe the natural numbers.

I’ll continue this series sporadically, and eventually we’ll get into some controversial philosophical questions. So far we haven’t.

Speaking of controversy, I’ve increased the default font size for this blog. Tell me if you like it.

We 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.

I jumped the gun on Tuesday, celebrating Frederic Bastiat’s birthday about a month early. Fortunately, our commenter Cloudesly Shovell saved me from embarrassment by noting that Bastiat is well worth an entire month of celebration.

On Wednesday, we had some biting words about math education from my colleague Ralph Raimi, whose web page I continue to recommend for amusement and edification.

And on Thursday and Friday, we took on current events, lamenting the President’s misleading suggestion that tax increases can be a cure, or even a palliative, for excessive spending, and lamenting the general lack of perspective that leads to more gnashing of teeth over a $10 billion oil spill than a $300 deadweight loss due to taxation.

Several commentators noted that with this last post, we’d come full circle right back to Bastiat, author of timeless That Which is Seen and That Which is Not Seen. In the words of commenter Seth, “A duck caked in oil is seen. The deadweight loss is unseen.” (ScottN and others made the same point.) Yes, that’s probably the explanation. How sad that after 200 years, Bastiat’s lesson (that the unseen is as important as the seen) has yet to sink in.

See you Monday.

I am amazed and delighted by the many excellent responses to my call for arguments about religion. These will be very helpful to me as I prepare for my debate with Dinesh D’Souza. Keep them coming!

Commenters also had a lot to say about the puzzle of the absent-minded driver. It seems to me that some of the analyses falter by being less than crystal clear about their assumptions: How much can the driver remember (e.g., if he updates his strategy at the first intersection and then arrives at the second, does he remember his original strategy or his updated strategy?), how much he can commit to (e.g. if he updates his strategy at the first intersection, can he commit to sticking to the new strategy at the second? can he commit to the method of updating he’ll use at the second?), how much he can anticipate (e.g. what does he believe about his future updates?), how smart he is (can he use his knowledge of his current strategy to figure out whether he’s already updated and hence what intersection he’s at?) and how sneaky he is (e.g. might he purposely adopt a bad strategy in order to trick his future self into updating to a good one?). I have what I think is a useful way of forcing puzzlers to be explicit about their assumptions, and had planned to post it on Monday, but I keep revising it, so it might be a few more days.

On Monday, we talked about what women want (hint: they prefer larger, and we’re not talking about wallets).

On Tuesday, we talked about beautiful folk songs; thanks to those who pointed me in new directions.

On Wednesday, we asked why job growth has been so sluggish compared to previous recessions.

On Thursday, I linked to a remarkably thoughtful and literate Fox News clip on immigration policy, featuring the fiery Don Boudreaux.

And on Friday, we lamented the economic illiteracy of local television reporters.

I’ll be back on Monday with more.

This week’s highlights:

• My list of sixteen most influential books, including the one that I had to read 95,146 times before I got the point.
• The problem with happiness research.
• The morality of counterfeiting
• The tragic side of Lent and Passover.

Enjoy your weekend and come back on Monday!

A sudden and mysterious fever left me unable to post my usual weekend roundup on time today. Now that I’m feeling human again, here it is, a few hours late:

We started the week with my best attempt to explain the intuition underlying the spectacular formula e^iπ = -1, frequently described as the most beautiful and astonishing equation in all of mathematics. Gauss reportedly once said that if this formula is not immediately obvious to you, you have no hope of being a mathematician—but I’ve heard more than one Fields Medalist say he’d been dumbstruck when he first encountered it.

We reviewed Yale professor Gary Gorton‘s account of the financial crisis; he says it was a bank run, and if you’re going to have banks, bank runs go with the territory.

On the lighter side, we talked about web comics; on the less-light side we talked about the advantages of genocide over other forms of mass murder, and about moral paradoxes.

Barring a relapse, I’ll see you Monday.

We started the week with a pointer to a hilarious recipe for salted water. If you didn’t follow the link then, you should follow it now. Click on the “reviews” tab and don’t be drinking anything when you read through these.

On Tuesday I made some snarky and cynical comments about the effects of health care reform on government spending. Fortunately, nobody at the Congressional Budget Office sued me for libel. Professor Joseph Weiler was not so lucky; when he posted a negative book review on a web site he edits, he was charged with criminal libel in France. Thursday’s post reviewed the astonishing story and Friday’s followed up with an account of the most devastating book review I know of (though commenters offered some good alternatives).

We paused midweek to acknowledge the birthday of Georg Cantor, and to summarize how he taught the world to think about infinity.

Next week: Commentary on health insurance premium hikes and much much more. Come back on Monday!