Monthly Archives: December 2011

Paul Krugman likes math but sells it short

From Krugman’s blog today, via Deane Yang’s FB feed:

Math is a friend of mine. There have been a number of occasions in my life when doing the math on an economic model has led me to conclusions very different from my preconceptions.

But I have always been able, after the fact, to find a way to express in plain English what the math is telling me. If you resort to math to justify what looks like a very foolish claim, and you can’t find a plausible way to express that justification in plain English, something is wrong.

I disagree.  The reason we use mathematical formalism is exactly because it expresses things that can’t be said precisely in English, or any other natural language.

We can, should, and do utter English sentences that paraphrase mathematical assertions; but that’s not the same thing.

Possibly useful analogy: “Music is a friend of mine.  There have been a number of occasions in my life when a piece of music has conveyed to me a powerful emotion or sensation.  But I have always been able, after the fact, to express in plain English the way the music sounded and the way it made me feel.”

If someone told you this, you would say “NUH UH,” and I think the same response is due Krugman here.

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Four Tucson notes

  • As far as I can tell, Cafe Poca Cosa is still the best restaurant in the city.
  • Front page news during our stay:  the state has outlawed the Tucson school system’s Mexican American Studies program.
  • Downtown Tucson is an interesting case study for people interested in urban cores.  Grand old theaters, lots of vacant buildings, Cafe Poca Cosa, and strangely specific yet apparently operational businesses (Ace Rubber Stamps, Wig-O-Rama.)  Almost nobody on the street at 5pm on a Wednesday.  On the other hand, no sense of blight.  Is Tucson considered a successful downtown renewal, a failed one, or something in between?
  • The only Republican campaign signs I saw in Tucson were for Ron Paul, and there were quite a few of them.  Remark:  this is also true of Madison.
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Movies I cried at in 2011

  • Bridesmaids
  • The Muppets

What has become of me?  It has something to do with having kids, I think.  Some people say “I became a totally different person when my children are born” but for me it’s been almost the opposite.  In this one way, though, I’ve changed.  Before children I used to be impervious to sentimental scenes.  Now I choke up because one puppet misses another.  Mysterious.

 

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Reader survey: do you know your credit card number by heart?

I don’t know mine.  I have to look at my card whenever I purchase something online.  Why?  It seems to me that I type or say my credit card number as much as I type or say my phone number, and I would consider it totally weird not to have my own phone number committed to memory.

On the other hand, my youth was spent in an environment where you had to recall your own phone number all the time, because it wasn’t programmed into your phone and you had to dial it every time you wanted to use it.  So the followup question for those readers who grew up in the cellphone era is:  do you know your own phone number by heart?

 

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Are Wisconsin Democrats blowing the recalls?

That might seem like a strange question, given that United Wisconsin claims it already has the 500,000+ signatures they need to force a recall election this spring, and are aiming for a million by the end of the 60-day petition period next month.

But the Walker-Kleefisch recall isn’t the only one going.  Petitions are circulating on five state senators — four Republicans and one Democrat.  Democratic gains in the last recall left the GOP with a precarious 17-16 majority in the upper chamber.  So if Democrats gain one seat, they take over Senate control.

There’s a big difference between this recall election and the previous one.  The state senators recalled last year were elected in 2008, a year of Democratic dominance;  the Republicans who managed to get elected that year were strong candidates in Republican-leaning districts.  And even so, two lost their seats.  This time around, it’s the opposite.  Van Wanggard, Pam Galloway, and Terry Moulton all knocked off Democratic incumbents in the 2010 Republican sweep; and even with that wind at their back, Wanggard and Galloway each won by modest 5-point margins.  There’s every reason to think those two, at least, would be vulnerable to Democratic challengers.

So why aren’t Wisconsin Democrats putting more resources into these races?  The Van Wanggard recall has just about reached the required number of signatures, but will need a lot more to be safe from legal challenges.  Pam Galloway’s petition is only 70% there.  And with Christmas and New Year’s coming, the second month isn’t likely to be as productive as the first.  Per the linked Isthmus article, none of the recall committees has more than $7,000 on hand.  Why?

I get that the Walker recall is the main event.  But with no obvious candidate to oppose the governor, the recall election is at best a coinflip for Democrats.  Maybe the state senate recalls are a coinflip, too:  but changing control of either one would effectively halt Wisconsin’s ability to make meaningful legislative changes.  For Wisconsin Democrats, that would be a huge success.  And one coinflip out of two  is a much easier game to win than one out of one.

And some good news for people who like poll news:  my colleague Charles Franklin, who knows more about polling data than anyone I’ve ever met, is taking a year’s leave of absence from UW to poll the hell out of Wisconsin in 2012 as head of a new project at Marquette Law.  Expect lots of posts here about his sweet crunchy data.

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David Foster Wallace was not famously depressive

The LA Review of Books, reviewing Hubert Dreyfus and Sean Kelly’s All Things Shining:

It may seem strange for a book about the good life to make such an extended example of Wallace, given that he was famously depressive and hanged himself.

No!  David Foster Wallace was not famously depressive.   Lots of people who read him very, very thoroughly, including me, didn’t know he suffered from depression until after his death.  His depression is only intermittently present in his writing and never governs it.  To read his books as a warm-up to his suicide is to waste them.

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The experts list, or: how can a journalist find out how to compute pi to high precision?

A reporter at the Wisconsin State Journal called me the other day with a really good question.  He had heard that pi had been computed to ten trillion decimal places.  And he wanted to know:  how could you possibly measure a circle that precisely?

So how did he know to call me?  Because I’m on the experts list, which UW-Madison’s public relations office set up to give journalists the opportunity to consult a Wisconsin professor on just about any subject.  Topics of current news interest get promoted to the front: on today’s front page we’ve got the professor who can talk about Kim Jong-Il, the professor who can talk about the Supreme Court’s decision to take up the Arizona immigration law, and the professor who can talk about the Scott Walker recall.  (I have a feeling that last guy is going to be on the front page for a while.)

Such a simple idea, but such a good one!  The UW-Madison ought to be a resource for Wisconsin journalists — and everybody else in Wisconsin, for that matter.  Good for the PR office for making it as easy as possible to reach faculty who want to face the public and share what they know.

Oh:  here’s the article about pi in the WSJ, by Dave Tenenbaum.  I thought it came out well!

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“Kakeya sets over non-archimedean local rings,” by Dummit and Hablicsek

A new paper posted this week on the arXiv this week by UW grad students Evan Dummit and Márton Hablicsek answers a question left open in a paper of mine with Richard Oberlin and Terry Tao.  Let me explain why I was interested in this question and why I like Evan and Marci’s answer so much!

Recall:  a Kakeya set in an n-dimensional vector space over a field k is a set containing a line (or, in the case k = R, a unit line segment) in every direction.  The “Kakeya problem,” phrased loosely, is to prove that Kakeya sets cannot be too small.

But what does “small” mean?  You might want it to mean “measure 0″ but for the small but important fact that in this interpretation the problem has a negative answer:  as Besicovitch discovered in 1919, there are Kakeya sets in R^2 with measure 0!  So Kakeya’s conjecture concerns a stronger notion of “small”  — he conjectures that a Kakeya set in R^n cannot have Hausdorff or Minkowski dimension strictly smaller than n.

(At this point, if you haven’t thought about the Kakeya conjecture before, you might want to read Terry’s long expository post about the Kakeya conjecture and Dvir’s theorem; I cannot do it any better here.)

The big recent news in this area, of course, is Dvir’s theorem that that the Kakeya conjecture is true when k is a finite field.

Of course one hopes that Dvir’s argument will give some ideas for an attack on the original problem in R^n.  And that hasn’t happened yet; though the “polynomial method,” as the main idea of Dvir’s theorem is now called, has found lots of applications to other problems in real combinatorial geometry (e.g. Guth and Katz’s proof of the joints conjecture.)

Why not Kakeya?  Well, here’s one clue.  Dvir actually proves more than the Kakeya conjecture!  He proves that a Kakeya set in F_q^n has positive measure.

(Note:  F_q^n is a finite set, so of course any nonempty subset has positive measure; so “positive measure” here is shorthand for “there’s a lower bound for the measure which is bounded away from 0 as q grows with n fixed.”)

What this tells you is that R really is different from F_q with respect to this problem; if Dvir’s proof “worked” over R, it would prove that a Kakeya set in R^n had positive measure, which is false.

So what’s the difference between R and F_q?  In my view, it’s that R has multiple scales, while F_q only has one.  Two elements in F_q are either the same or distinct, but there is nothing else going on metrically, while distinct real lines can be very close together or very far apart.  The interaction between distances at different scales is your constant companion when working on these problems in the real setting; so maybe it’s not so shocking that a one-scale field like F_q is not a perfect model for the phenomena we’re trying to study.

Which leads us to the ring F_q[[t]] — the “non-archimedean local ring” which Dummit and Hablicsek write about.  This ring is somehow “in between” finite fields and real numbers.  On the one hand, it is “profinite,” which is to say it is approximated by a sequence of larger and larger finite rings F_q[[t]]/t^k.  On the other hand, it has infinitely many scales, like R.  From the point of view of Kakeya sets, is it more like a finite field, or more like the real numbers?  In particular, does it have Kakeya sets of measure 0, making it potentially a good model for the real Kakeya problem?

This is the question Richard, Terry, and I asked, and Evan and Marci show that the answer is yes; they construct explicitly a Kakeya set in F_q[[t]]^2 with measure 0.

Now when we asked this question in our paper, I thought maybe you could do this by imitating Besicovitch’s argument in a straightforward way.  I did not succeed in doing this.  Evan and Marci tried too, and they told me that this just plain doesn’t work.  The construction they came up with is (at least as far as I can see) completely different from anything that makes sense over R.  And the way they prove measure 0 is extremely charming; they define a Markov process such for which the complement of their Kakeya set is the set of points that eventually hit 0, and then show by standard methods that their Markov process goes to 0 with probability 1!

Of course you ask:  does their Kakeya set have Minkowski dimension 2?  Yep — and indeed, they prove that any Kakeya set in F_q[[t]]^2 has Minkowski dimension 2, thus proving the Kakeya conjecture in this setting, up to the distinction between Hausdorff and Minkowski dimension.  (Experts should feel free to weigh in an tell me how much we should worry about this distinction.)  Note that dimension 2 is special:  the Kakeya conjecture in R^2 is known as well.  For every n > 2 we’re in the dark, over F_q[[t]] as well as over R.

To sum up:  what Dummit and Hablicsek prove makes me feel like the Kakeya problem over  F_q[[t]] is, at least potentially, a pretty good model for the Kakeya problem over R!  Not that we know how to solve the Kakeya problem over F_q[[t]]…..

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We are the 81%

Strange column in the Isthmus this week by conservative columnist Larry Kaufmann, who says people are wrong to think about inequality as a problem when the great purring engine of American productivity is lifting all boats.  (Not a mixed metaphor — in the world of this column the engine is so awesomely strong that it actually lifts up millions of boats on some kind of mechanized platform.)  Oh, and also, Occupy Madison is a bunch of smelly hippies who should shut up already.  Kaufmann:

So is the American dream still alive?  In terms of absolute mobility, the answer is yes.  Between 1968 and 2006, 81% of children had a higher inflation-adjusted family income than their parents did…”

Now, let’s be fair — 81% is pretty good!  And that figure doesn’t sound so implausible:  after all, America is a richer country than it was in 1968, so why wouldn’t most individual Americans be richer?

Still, I wanted to check up.  So I went to the source; Kaufmann ascribes it only to the Pew Economic Mobility Project, which publishes a lot of papers, but after a few misses I found the 81% figure in “Family Structure and the Economic Mobility of Children,” whose lead author is UW-Madison economist Thomas DeLiere.

And the 81% number is right there on page 11.  But there’s a footnote, reminding us that these numbers are “adjusted for family size.”  That is: for the purposes of this computation,  a family with four children counts as lower-income than a family with two children and the same household income; the bigger family has to divvy up those dollars between more people.  Without this adjustment, the proportion of children whose household income as adults exceeds the income of their childhood household drops to 66%.

That’s still a large majority!  But there’s more — you’ve still got to ask why household incomes went up so much between 1968 and 2006.  Another paper from the Economics Mobility Project reveals of a big chunk of the reason; the proportion of women in the workforce went from 40% to 60% over that period.  Median individual income for men actually dropped over this period.  (And no, the figures in DeLiere’s paper aren’t adjusted for this; I asked him.)

So yes:  almost all present adults have more money than their parents did.  And how did they accomplish this?  By having one or two kids instead of three or four, and by sending both parents to work outside the home.  Now it can’t be denied that a society in which most familes have two income-earning parents, and the business-hours care of young children is outsourced to daycare and preschool, is more productive from the economic point of view.   And I, who grew up with a single sibling and two working parents and went to plenty of preschool, find it downright wholesome.  But it is not the kind of development political conservatives typically celebrate.

Further reading:  My guess is that Kaufmann learned about the EMP study from research manager Scott Winship’s article on the research in National Review, since he quotes Winship:  “The finding of pervasive upward absolute mobility flies in the face of liberal accounts of a stagnant middle class.”  Winship’s piece is longer, better written, and more careful than Kaufmann’s;  he doesn’t dodge the fact that the flow of women into the workforce drives a great deal of household income growth, but he doesn’t place a lot of importance on this.  Winship is a Ph.D. economist who does this stuff for a living, so his view must be given a lot of weight.  But I can’t make out what the argument is from the single paragraph in NR.  Is he saying that men’s earnings are decreasing because they’re voluntarily taking on fewer hours of work?

Winship also emphasizes the finding that children in Canada and Western Europe have an easier time moving out of poverty than Americans do.  This part is absent from Kaufmann’s piece.  Maybe he didn’t have the space.  Maybe it’s because a comparison with higher-tax economies would make some trouble for his confident conclusion: “the punitive redistribution policies favored by Occupy Madison will divert capital away from productive initiatives that enhance growth and earnings opportunities for all, while doing nothing to build the stable families and “bottom-up” capabilities that are particularly important for helping the poorest Americans escape poverty.”

When the Isthmus is running a more doctrinaire GOP line on poverty than the National Review, the alternative press has arrived at a very strange place indeed.

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Best Writing on Mathematics 2011, and Nathanson on massive collaboration

In my mail:  The Best Writing On Mathematics 2011 (Mircea Pitici, ed.) from Princeton University Press.  Just to get this out of the way:  I’m in here!  They reprinted my compressed sensing article from Wired.

You might now be wondering:  are there really enough popular math articles published in a given calendar year to fill up an anthology?  No.  There are not.  But this is part of the charm of what Pitici has done.  His very broad definition of “writing on mathematics” allows him to include useful professional advice for young mathematicians from Andrew Schultz, reflections on a career in math education from John Mason,  and academic-yet-readable philosophy (“What Makes Math Math?”) from Ian Hacking, whose The Emergence of Probability is my favorite book in history of mathematics.

I especially like Mel Nathanson’s pessimistic take on massive collaboration in mathematics — because it is a forcefully written, carefully argued case for a position with which I mostly disagree.  “I would guess that even in the already interactive twentieth century,” he writes, “most of the new ideas in mathematics originated in papers written by a single author.”  I would guess otherwise — at least if you restrict to the second half of the century, when joint papers started to become really common.   Mel calls me out for writing about Tim Gowers’ Polymath Project in the New York Times with “journalistic hyperbole” — and here he is right!  It is very hard, in the genre of 300-word this-year-in-science snippet, to keep the “gee whiz!” knob turned down and the “jury is still out” knob turned up.

Gowers claims the classification of finite simple groups as a pre-Internet example of massively collaborative mathematics.  Nathanson agrees, but characterizes the classification as fundamentally uninteresting, “more engineering than art.”  What would he say, I wonder, about recent progress towards modularity of Galois representations?  It’s very hard to imagine him, or anyone, seeing everything that’s happened in the last 15 years as a mere footnote to Wiles.  (But maybe some of the experts who read this blog would like to weigh in.)

Nathanson concludes:

Recalling Mark Kac’s famous division of mathematical geniuses into two classes, ordinary geniuses and magicians, one can imagine that massive collaboration will produce ordinary work and, possibly, in the future, even work of ordinary genius, but not magic.  Work of ordinary genius is not a minor accomplishment, but magic is better.

Yes, but:  magic can only happen in the already-enchanted environment created by the hard work of many minds, alone and in teams.  Math is like earthball.

 

 

 

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