Sidewalk tables of Paris and New York

In Paris as in New York, on a nice evening, you’ll find people sitting at tables along the edges of restaurants, half indoors and half outdoors. Legally speaking, the tables in Paris are outdoors — because smoking indoors is against the law. By the same principle, the tables in New York are legally indoors — because drinking outdoors is against the law.

[23 September 2017]



Everybody seems to be wired these days, and yesterday I stood on Broadway and took a tally as 100 people walked by. 45 were using a phone or at least had wires in their ears, and 18 were holding a phone without apparently using it. Just 37 showed no sign of an electronic device.

My first thought was, how awful! How sad that we are so wrapped up in our wired worlds! My second thought was, just imagine twenty years from now. We’ll be more networked than ever, but the phones and the wires won’t be visible any more. Our hands will be free and our minds will be who knows where.

[22 September 2017]

Beethoven, the Beatles, and Homo Deus

Kate and I listened to Beethoven’s masterpiece today, the Ninth Symphony. This last of his symphonies explores one theme, then another, growing and growing, and finally giving us that amazing choral finale which everyone agrees is his greatest work.

Immersing myself in this magnificent music, I felt the parallel with the Beatles’ masterpiece, the second side of Abbey Road, their last album. So we listened to that next. Like Beethoven, the Beatles here go beyond their usual tracks, weaving a great work that explores one theme, then another, building and building to the big drum solo and the final catharsis.

Meanwhile I have just read Harari’s book Homo Deus. Harari shows us how for hundreds of years in the West we located meaning and morals externally, in an all-powerful God, but now we have moved to locating them in ourselves, in humanity and its needs and feelings.

And I realized that Harari’s point is perfectly reflected in the stirring final lyrics of Beethoven and the Beatles. Beethoven (Schiller) finishes with a triumph of religion: “Above the starry canopy must dwell a loving father”  (Über’m Sternenzelt muß ein lieber Vater wohnen). The Beatles finish with a triumph of humanism: “And in the end, the love you take is equal to the love you make.”

[15 August 2017]

Crouzeix’s conjecture

I am on BA 285, one of thirty mathematicians flying to San Jose, California, to spend a week together at the American Institute of Mathematics trying to prove Crouzeix’s conjecture. The conjecture asserts that a certain quantity that arises in linear algebra is ≤2. So far, it’s known to be ≤2.41 (1+√2). This week, a man-year of time and a man-year of money will be spent trying to prune away that last 20%.

Direct consequences if we pull it off? Next to none. Nothing really depends on 2.41 being improved to 2.

The point is the indirect, the intellectual consequences. With luck, a week from now the theorem will be proved and the proof will have made use of a new idea or two that may lead on to further advances is the future. This is how mathematics — science — grows. If we succeed this week, the time and money will have been well spent.

[30 July 2017]

Being left-handed, being gay

The Wimbledon match between Federer and Zverev yesterday got me thinking. Federer is right-handed and Zverev is left-handed, and that’s all there is to it. Nobody wonders what made Zverev left-handed, or if he could be talked out of it. The tabloids do not print rumors that Federer has left-handed tendencies he keeps darkly to himself.

The analogy of left-handedness with gayness goes pretty deep. So far as I can tell, both are understood to have complex causes mixing genes, development in utero, and other factors hard to disentangle. Like so many analogies, this one draws you in with a complex skein of similarities and differences. A similarity is that not everybody is perfectly left-handed or right-handed, just as not everybody is perfectly gay or straight. A difference is that gayness comes with an obvious evolutionary cost, whereas left-handedness seems on the face of it to be evolutionarily neutral. And what about our cultural responses to these syndromes? My opening paragraph may suggest that left-handedness is accepted without a ripple, but it is not so simple in Saudi Arabia, India, or China.

[9 July 2017]

Two white noise paradoxes and Einstein 1905

There’s a paradox of white noise: in its standard idealization it has equal energy at all frequencies, hence infinite energy. Curiously this paradox is linked to two of Einstein’s great papers from his annus mirabilis.

The ultraviolet catastrophe. The first paradox was a big problem implicit in 19th century physics. Statistical mechanics predicted that a cavity should support electromagnetic waves of all frequencies, each with an equal amount of energy. So at each instant a cavity should radiate infinite energy! Planck’s quantization, it was later realized, resolved this difficulty by positing that in fact, higher frequencies have less energy in them. Einstein investigated more deeply the implications of quantization for the behavior of light photons, for which he won the Nobel Prize.

Brownian motion.  Meanwhile another work of Einstein’s that year eventually led to today’s standard idealization of Brownian motion, which has become a central subject in mathematics. The idealization is this: Brownian motion is the indefinite integral of white noise. But white noise has infinite variance! This time the physicists are not concerned, since bouncing molecules don’t go down to the infinitesimal scale, but the paradox still needs a mathematical resolution if we are to work with these ideas rigorously. In effect mathematicians take the view that white noise itself is never measured, only integrals of it, and these are finite because of sign cancellations. The higher the frequency, the greater the cancellation. But the details make stochastic analysis technical and forbidding.

Did Einstein notice that blackbody radiation and Brownian motion are linked in this way?

[6 May 2017]