Our frequent and reliably insightful commenter Jonathan Kariv sends along this neat puzzle:
Your enemy chooses 10 points on an infinite tabletop and gives you 10 coins of the same size (let’s say U.S. quarters). Can you always place the coins on the tabletop in such a way that all 10 points are covered, but no two coins overlap?
In a radical departure from his previous expressions of dissillusionment, Paul Krugman has implicitly declared in his latest blog post that we are now living under the best of all policy regimes. I presume he will now be able to retire with satisfaction from his career as a gadfly.
The context is Eric Cantor’s demand that any federal disaster relief in the wake of Irene be offset by spending cuts elsewhere. Krugman thinks this is silly, and proves his point with an appeal to the standard Ricardian theory of public finance. According to that theory, which all economists understand and accept, if you’ve got to bear a cost, it’s best to spread that cost out over as many activities as possible. So ideally, you’d pay for disaster relief partly through spending cuts, partly through (current) tax increases, and partly through an increase in the deficit. Therefore says Krugman, “the bottom line is that basic, regular economics says that Cantor isn’t making sense.”
Since Krugman has carelessly neglected to spell out an important detail of his argument, let me fill in the gap for him: The Ricardian conclusion does not come from thin air; instead it follows logically from certain premises, key among which is that you’re starting from an ideal policy regime.
A few years ago, I discovered that reading on my Kindle is about 1000 times better than reading a book. This year, I discovered that reading on my iPhone is about 100 times better than reading on my Kindle. As a result (and also as a result of a lot of time spent on airplanes), I’ve been on a mad fiction-reading spree the past few months. Some mini-reviews:
Paul Krugman wisely reminds us that:
The odds are that one of these years the world’s greatest nation will find itself ruled by a party that is aggressively anti-science, indeed anti-knowledge. And, in a time of severe challenges — environmental, economic, and more — that’s a terrifying prospect.
Yes, a terrifying prospect — and an excellent reason to limit the powers of ruling parties, though Paul never seems to notice this.
For the next several days, I’ll be in at the Liberty and Democracy Forum in Brazil, and likely out of Internet range. I’ll
see you late next week.
By now you’ve probably encountered one of the various “do you push the fat man in front of the subway train to stop it from running over five innocent people trapped on the tracks?” puzzles that moral philosophers have been using lately to distinguish the consequentialists (who care only about ends) from the deontologists (who care also about means). (I invoked some of these puzzles, in a slightly non-conventional way, in Chapters 16 and 17 of The Big Questions.)
Now come psychologists Daniel Bartels and David Pizarro to report that choosing what they call the “utilitarian” solution (by which they mean the consequentialist solution, though one might have more faith in their scholarship if they’d used the accurate word) is correlated with higher scores on measures of psychopathy, machiavellianism, and “life meaninglessness”. Those who would push the fat man are more likely to agree with statements like “I like to see fistfights” or “The best way to handle people is to tell them what they want to hear” or “When you think about it, life is not worth the effort of getting up in the morning”. In other words, they are what we tend to think of as an unpleasant bunch.
But you see, here’s the thing: Bartels and Pizarro did the wrong test. Because in every one of the dozen or so moral dilemmas presented to the subjects, they asked: Would you shoot the crewmember? or would you execute one of the hostages? or would you flip off the switch to the ventilator? — which doesn’t get to the moral issues at all. The moral issue is this: In your opinion, should you shoot the crewmember or execute the hostge or flip the switch? The researchers never asked those questions, so we have no idea what the subjects’s opinions were.
Tell me a story about 20 children trapped in a burning building and ask me if I think I should rush in to save them. My answer is yes. Now ask me if I would. If I’m honest, I am very unsure I would do the right thing.
So what do we learn from this research? Here’s one interpretation: Maybe most people agree that you should push the fat man, and most people realize that they themselves would shrink from this moral duty out of squeamishness (just as most people agree that you should save the children from the fire, and most realize they might shirk this moral duty). Therefore they answer “No, I would not push the fat man.” But it’s only the psychopaths and Machiavellians who exaggerate their own moral virtues by claiming that they would rise to this particular occasion.
For the first time ever, I am deleting a post.
The numbers in this morning’s post (now missing) were completely wrong as were, therefore, the conclusions I drew from them.
For the record, all of the numbers concerned what happens if you save $1000 a month. I often show these numbers to my students, and when I do, I get the assumption right. But this time, I somehow a) convinced myself that the assumption was $100 a month, not $1000 a month and b) therefore concluded that saving is a whole lot easier than what I tell my students every year, and c) said a lot of nonsense that followed from this.
(I tell my students that for *them*, saving $1000 a month will soon probably be a plausible strategy, which is likely to be true. Having conflated $1000 with $100, I drew implausible conclusions in the blog post about what you could do on $25,000 a year.)
I could say things about the folly of posting at 2AM, but I think the wiser course is simply to apologize.
One of Paul Krugman’s favorite tactics is to assert that all he’s doing is channeling the time-honored lessons of Economics 101 — pre-empting dissent with the implication that any dissenter must be either an ignoramus or a radical. (Journalistic honesty compels me to acknowledge that I might have employed this rhetorical tactic once or twice myself over the years.)
It’s interesting, then, to take note of how very far his central arguments actually deviate from Economics 101. Here’s what he said last week on his blog:
Mulligan and others keep emphasizing examples of individual groups that have managed to gain jobs by cutting wages or offering other attractions to would-be employers. They then assert that these examples tell us what would be needed to expand overall employment.
The point, of course, is that all such arguments amount to committing the fallacy of composition…The essence of macroeconomics is understanding why such things are a fallacy, why what happens if one group does something is not at all what happens when everyone does it.
But you see, here’s the thing: According to the standard Economics 101 version of the sticky-wage Keynesian model, this is a case where what happens if one group does something is exactly the same as what happens when everyone does it. According to that model, as long as wages continue to fall, firms will continue to move along their labor demand curves until we reach full employment.
Re yesterday’s puzzle, you’ll find answers in the comments. (We are blessed with some very smart commenters here at The Big Questions!!)
Commenter Roger Schlafly pointed this Wikipedia article where I was surprised and delighted to see a reference to a paper co-written by my old friend Dave Rusin. I did not remember that Dave had anything to do with this problem, but in retrospect I bet I knew this at one time.
I managed to dig out some notes I jotted down on this subject many many years ago. I have not doublechecked these results, and I can’t completely vouch for the careful accuracy of my younger self, so take these for what they’re worth. But here’s what I once claimed to have proved:
The reason there is exactly one pair of nonstandard six-sided dice is that six is the product of two distinct primes. For the same reason, there is exactly one pair of nonstandard n-sided dice when n is 10, or 15, or 21, or …. For any product of three distinct primes, there are at most 40 nonstandard pairs.
I also found (in what appears to be my handwriting) this chart, which I reproduce with the same caveats:
I love this problem, which I found on the Internet many years ago. I suppose you could find a solution by Googling, but that’s of course no fair.
A standard pair of six-sided dice induce a probability distribution on the outcomes 1 through 12: The probability of rolling a 1 is 0, of rolling a 2 is 1/36, of rolling a 3 is 1/18, etc. Is there any nonstandard pair of six-sided dice that induces exactly the same probability distribution? If so, how many such pairs are there?
(A non-standard pair of six-sided dice might have, say, the numbers 1,2,2,3,8,9 on one cube and the numbers 2,3,4,4,4,4 on the other.)
 |
 |
Each generation wishes it could go back fifty years and shake some sense into those people who were so bound by unnecessary customs, and so blind to the options they could have chosen and the changes that loomed on the horizon. As I said on Tuesday, this was Edith Wharton’s theme when she wrote in 1920 about the 1870’s, and it’s the theme of Mad Men, written in 2010 about the 1960s.
I invited you on Tuesday to speculate about which of our own quirks will trigger this sort of bittersweet nostalgic frustration among our descendants fifty years from now. There were some great responses in the comments.
Here are some predictions of my own that I think are least plausible — some moreso than others, but I’ll throw them out in no particular order.
 |
 |
 |
Kathleen Kennedy Townsend, writing in the Atlantic, has figured out that Jesus Christ wants you to be a Democrat. There are, you see, 2500 passages in the New Testament that call on us to care about other people. Rick Perry (and presumably others of his Republican ilk) ignores those passages, according to Ms. Townsend, when he voices “concerted opposition to government social programs”.
Now, of course Rick Perry is no more concertedly opposed to government social programs than is Kathleen Kennedy Townsend; instead, they disagree about how those programs should be structured and how extensive they should be. Not even Ms. Townsend (unless she is an even greater lunatic than she appears to be) believes that such programs should be unlimited, so her disagreement with Rick Perry is largely over where to draw the lines. Somewhere in those 2500 New Testament passages, she’s managed to discern an endorsement for her own preferred lines over Governor Perry’s. Quite a discerning reader she must be.
But it gets worse: According to Ms. Townsend’s reading of the Bible, we ought to “use all the tools we have at hand to help the poor, the sick and the hungry” — and I’m guessing that’s not someplace Kathleen Kennedy Townsend wants to go. That’s because using all the tools at hand to help the poor, the sick and the hungry means unleashing the power of capitalism. Regarding the poor and hungry, it means eliminating barriers to trade and immigration, reducing or eliminating capital taxation, and eliminating or drastically restructuring most federal regulations. Regarding the sick, it means curbing the power of the FDA, eliminating the tax deduction for employer-provided health insurance, and committing ourselves not to regulate the prices of prescription drugs. As a general rule, it means diminishing the power of the political class that Kathleen Kennedy Townsend has devoted her life to serving.
Reading Edith Wharton’s Age of Innocence, it strikes me that this must have been the Mad Men of 1920. That was the publication date, but the story is set 50 years earlier, in a world poised on the edge of cultural upheaval. The characters are blind to how dramatically the world is about to change, and to how much better their lives might be if only they could break out of the social strictures of their time. They manage to be both charmingly quaint and tragically foolish. We care about them, but we also want to take them by the shoulders and shake them into something more like ourselves.
As Edith Wharton viewed the 1870s, and as Mad Men views the 1960s, so the fiction of the mid-to-late 21st century will probably view us. Which of our quaint but tragically foolish ways do you think it will emphasize?
Putting aside all issues of who I did or did not agree with, and putting aside all issues of who I would or would not like to see in the White House, and ranking them solely on the criterion of “Demonstrated Ability to Think on Their Feet”, I score it this way:
1) Gingrich
2) Santorum
3) Bachmann
4) Cain
5) Paul
6) Romney
7) Pawlenty
8) Huntsman
You?
The most fun you can have on the Internet is to find a beautiful, succinct argument with a conclusion so unexpected it seems like magic. For today’s fun, I am indebted to Michael Lugo, at God Plays Dice.
Lugo’s original post is so good it seems almost superflous to paraphrase it, but I can’t resist the temptation.
Drill a tunnel through the earth, from anywhere to anywhere — New York to Maine, or New York to Australia, or wherever else you like. Like so:
Now drop the object of your choice (Lugo suggests a burrito, but you might prefer a gravity-driven train) into the tunnel entrance and wait till it comes out the other side. It’s a standard calculus problem to calculate how long you’ll have to wait: The answer is 42 minutes, regardless of the length of the tunnel. I’m sure I once found it surprising that the tunnel length doesn’t matter, but I’ve known it long enough that I now take it in stride. So that’s not how Lugo surprised me.
The surprise is that if you change the size of the earth (while maintaining its density), the answer is still 42 minutes. Whether the earth is the size of a pea or the size of the solar system, it’s a 42 minute trip from one end of the tunnel to the other. (We’re — quite reasonably — ignoring the effects of relativity here. For an earth that was half the size of the universe, we’d have to make some corrections.)
Why so? You could, of course, discover this through a direct calculation. But Lugo provides a much slicker argument, namely:
The EPA announced yesterday that new regulations mandating fuel efficiency standards for heavy trucks will cost vehicle buyers $8 billion, but that will be paid for in fuel savings over a year or two.
Oh. Sounds like the mandate is quite unnecessary then, no? With numbers like that, consumers will demand high efficiency vehicles with or without the EPA. Unless, of course, the EPA is, umm….lying.
In a research paper with seemingly tragic consequences for science fiction fans, researchers in Hong Kong have confirmed that individual photons can’t move faster than light.
(Hopes had been raised a few years back, when a pair of German physicists claimed to have broken the light speed barrier by propagating waves that effectively arrived before they departed. I said at the time that I could do just as well without all the fancy lab equipment — all I need is a yardstick and an axe. If I chop off the last 12 inches, then the center of the yardstick moves from the 12″ mark to the 18″ mark — a six inch advance in exactly zero time. Of course a sane person might argue that this six inch “advance” involves no actual forward motion, but if I correctly understood the German paper, the same sort of objection would apply there as well.)
It’s being reported that the results from Hong Kong doom all hope for time travel. That’s true or false depending on how restrictively you interpret the phrase “time travel”. Back in 1938, Kurt Goedel (yes, the great logician Kurt Godel — though this particular work has nothing to do with his work in mathematical logic) constructed an example of a universe — that is, a structure that obeys all the laws of physics as laid down by Einstein — that is completely filled with closed timelike curves. If you lived in that universe, you’d be able to travel only forward in time, but still eventually come back to your starting point in both time and space — just as a bug can go in just one direction around a circle and still come back to its starting point. Just like Groundhog Day.
From Air Canada’s inflight magazine:
And since I’m sure someone will ask, here is, apparently, the only solution they could think of:
Three logicians walk into a bar.
The bartender says: “Would any of you guys like a drink?”
The first logician says: “I don’t know.”
The second logician says: “I don’t know.”
The third logician says: “No.”
Hat tip to Adam Merberg, who isn’t sure of the source.
I spent most of July in the UK, with limited Internet access (exacerbated by my Verizon iPhone’s inability to communicate with the European cellular services), and (I’ve only just realized this now) without buying even a single newspaper. So I know almost nothing of what’s gone on in the U.S. over the past several weeks, except for some vague sense that there was a brouhaha over raising the debt ceiling. Even over the few days I’ve been back, I’ve felt no urgency about catching up, though I’m sure that will kick in any day now.
There was, of course, never any need to raise the debt ceiling; there was only a need to prioritize debt service over other stuff the government shouldn’t be doing anyway. To a very rough approximation, the annual budgets of the Departments of Commerce, Agriculture and Labor add up to the annual interest payments on the national debt.
Be that as it may, it will be interesting to catch up on the news and see what got cut. For now, I’ll just say that if we still have a Department of Commerce, then they didn’t cut enough. If we still have a Corporation for Public Broadcasting, or a National Endowment for the Arts or Humanities, then they didn’t even try.
But now that I’m back, I’m unlikely to dwell on what’s become old news. My plan is to ramp back up to regular blogging, starting with a few things that struck me as noteworthy while I was traveling in Britain. I hope you’ll forgive my long absence. It’s good to be back.
Obviously, blogging has slowed pretty much to a crawl around here the past several weeks, due largely to a big writing project that I’ll tell you more about in the fall. That project is done, but now I’ll be traveling pretty much non-stop for the entire month of July. I’ll try to file an occasional foreign dispatch, but it will be sometime in August before we get back to our regular daily schedule. I’m aiming to ramp up to that schedule by the beginning of the academic year at the latest. Thank you all for bearing with me.
Our reader Jonathan Campbell has an excellent post on a bizarre example of legal “reasoning”. A defendant fires 95 shots into the woods; his friend fires 5. A man in the woods is hit by a bullet and dies. The defendant is 95% certain to be the killer (in other words, he is the killer beyond a reasonable doubt), but he is acquitted because the nature of the evidence is “statistical” and therefore cannot be the basis of a conviction.
This tends to confirm my suspicion that a legal “education” consists primarily of having your capacity for logic beaten entirely out of you. As Jonathan points out, all evidence is statistical in exactly the same sense that this evidence is. I put a bullet through your heart and you take five minutes to die. Did the bullet cause your death? Well, not certainly — there’s some small chance that you died of a heart attack that would have killed you anyway, thirty seconds before the bullet was able to finish its job. Is there a high probability I caused your death? Yes. Did I do so beyond a reasonable doubt? Yes. Is the remaining doubt of a statistical nature? Yes. So if there were any consistency in the law, of course I’d be acquitted. And of course I would not be. Which means, as has been noted in this space before, that the law is an ass. So, perhaps, are a substantial fraction of its practitioners.
For God’s sake, don’t let your children subscribe to Sirius/XM.
Since May 4, when Sirius rearranged all its channel numbers, my radio has been badly confused. If I punch in station 23, it goes to the station that’s currently 23 for a while, then jumps to the station that used to be 23, etc. And certain stations, which according to the Sirius website are part of my standard package, are completely inaccessible.
Given my past experience with XM customer service, I knew this was not going to be an easy fix, so I’ve been putting off making the call. Today I had some spare time. Sure enough, I’ve spent over TWO HOURS on the phone with these people being alternately put on hold, lied to, put on hold, lied to some more, and put on hold again.
They claim the missing channels are missing because they’re “premium” channels not included in my package. Except that their website clearly identifies these channels as standard channels that *are* part of my package. They tell me that they’re instituting a fix at their end which requires me to leave my radio on for fifteen minutes before it takes effect; this gives them a convenient excuse to hang up and not be there fifteen minutes down the line when nothing has changed. When I complain about how long I’ve been on hold (the automated system always says the wait time is “about eight minutes” before stranding you for half an hour), they give me a direct number to call to bypass the queue. I call that number and am told that no, this number is only for radios installed on airlines or boats. I complain that I’ve just waited twenty minutes to get this message. They give me a *different* number to call, promising me that there is currently no wait at that number. Thirty five minutes later, I’m still waiting.
Ah, but what about just using the form on their web site? Well, you see, that form will not allow me to submit a query unless I give it the serial number of my radio — a serial number that it insists is wrong, even though I have *copied and pasted* it from the “My Account” section of their own damned website. Therefore my query cannot be submitted.
|
 |
Here is Senator John Thune (R-SD), speaking on the floor of the United States Senate:
Ethanol producers have been ripping us off for a long time, and they’ve come to rely on that for a source of income. So it’s only fair to let them rip us off a little longer.
I’m quoting from memory, so I might have the wording slightly off, but that was the gist of it. Oh, wait, here’s the exact quote:
We have a lot of folks who made investments, you have people across the country whose livelihoods depend upon this. I think it makes sense, when we put policy in place and we say it is going to be in place for a certain period of time, that it be honored.
As you can see, my parapharase was accurate.
Senator Thune speaks in the great tradition of his institution. Back in 1848, senators by the score made exactly the same argument for preserving slavery. A lot of folks had invested in slaves, you know. And their livelihoods depended on it.
The other night at dinner, I was asked whether, when the Beatles came to the US in 1963, I had had any sense that something really big had happened.
Well, I was pretty young in 1963, probably too young to think about such matters. I remember having little interest in the Beatles, but being being very aware that they were something very big. Everyone was aware of that. But unless I am mistaken, pretty much nobody realized that we were witnessing something really big and lasting. More generally, I doubt that anyone at the time had any inkling of the long-term significance of rock ‘n’ roll. We knew it was popular, but we had no idea it would change the world. I’m not sure that in 1963 anyone knew that it was possible for music to change the world.
This led to the more general question: How quickly are great cultural watersheds recognized for what they are? In the few areas I know something about, I think the answer is “usually pretty quickly”. I remember 1910 even less vividly than I remember 1963, but I am pretty sure that it wasn’t long between the appearance of The Love Song of J. Alfred Prufrock and the realization (at least among people who care about this sort of thing) that poetry had changed forever. In mathematics, at least in the past century (and I’m pretty sure for several centuries, or even millenia, before that), major paradigm shifts have generally been recognized very quickly. When a Serre or a Grothendieck upends the mathematical world, the mathematical world quickly knows it’s been upended.
Vladimir Voevodsky, one of the world’s best and most influential mathematicians, has stirred up a bit of a hornet’s nest with a video lecture suggesting the possibility that the Peano Axioms — the standard axioms for arithmetic — might be inconsistent.
Since the Peano Axioms are known to be consistent, it’s tempting to dismiss the whole lecture as either a prank or a shocking display of ignorance. The latter temptation is buttressed somewhat by Voevodsky’s bold misstatement of Godel’s Incompleteness Theorem, which plays a central role in the lecture. On the other hand, Voevodsky is smarter than almost anyone else on earth, which earns him the benefit of the doubt — maybe what he’s saying is subtler than it seems. On the other hand, some of those in the “shocking display of ignorance” camp are among the few people in the world who might be as smart as Voevodsky.
To believe that the Peano Axioms are inconsistent, Voevodsky must reject all of the known proofs that they are consistent. In particular, he must reject the simplest and most convincing of all those proofs, which goes like this:
On May 17 of this year, these fifteen World War II veterans were awarded the French Legion of Honor medal for their service in France. They were cited for their courage and their contributions toward the French liberation. Third from the left, in the light blue jacket, is my Dad.
Words like “awe” and “gratitude” cannot begin to describe what I feel toward these people, whose sacrifices secured the unprecedented safety, prosperity and freedom that have graced my life and so many others of my generation. In the world they created, those sacrifices have become (for people like me) unimaginable.
These are the giants who cleared my path through life. I’m glad to see them honored, though no honor can ever be enough.
One of my weaker columns when I wrote for Slate was a highly unsatisfactory stab at why some hotels charge for wireless and others don’t. Today my wife, who had never seen that column, asked me the same question, and I think my off-the-cuff answer was probably better than anything I said in Slate.
So here’s my new stab at this:
I’m on the road in beautiful Bellingham, Washington, with very little opportunity to blog. Otherwise, I’d have chimed in a few times in the ongoing discussion of the “Who Owes Whom” post that immediately precedes this one. As it is, I’ll save all my comments till I return next week.
PDF