Two views of mathematical functions

The 19th century view was that a function was analytic. Of course, some functions were less smooth than that, but analyticity was the starting assumption.

The 20th century view is that a function is continuous. Of course, some functions are smoother than that, but continuity is the starting assumption. (Have you ever seen a merely continuous function? I think there is one main example: a random Brownian path.)

The 20th century view has utterly taken over among pure mathematicians. One can understand this on the basis that analytic functions are too easy to offer research challenges. Unfortunately, the same view has been copied by the numerical analysts, who ought to have been guided by what applications look like. One sees this in their edifice of results expressed in the language of Sobolev spaces. The point of Sobolev spaces is to make precise distinctions between, say, a function with one derivative and a function with one-and-a-half derivatives. Nineteenth-century mathematicians did not spend their time on such distinctions.

And so it is that numerical analysts prove technical, rigorous theorems about the convergence of algorithms whose convergence is nowhere near optimal for most applied problems. Meanwhile pure mathematics continues its drift away from science.

[1 December 2018]

Flypaper and the Faraday cage

For the first time in my life this summer, I had to buy flypaper. The fruit flies in our kitchen in Lyon were annoying. It became fascinating to watch them dart around seemingly at random, and occasionally light on the sticky paper.

Suppose they fly truly in a path of Brownian motion (in fact it’s not so simple), and suppose I had hung a ring of flypapers (in fact I hung just one). Then this would have been the same mathematics as the Faraday cage! For electric fields are governed by Laplace’s equation, just like Brownian motion.

Our analysis of the Faraday cage translates as follows to fruit flies getting through a ring of flypapers to a banana. (a) As long as there are good-sized gaps between the papers, the number of flies getting through will not be negligible (this was Feynman’s error). (b) If the papers themselves are made narrower, even down to a scale of millimeters, the effectiveness of the cage diminishes only slightly (in proportion to an inverse-logarithm of the radius). This is one of the paradoxes of Brownian paths, going back to Berg and Purcell in 1977.

[24 November 2018]

Fake News

In this awful Trump era we’ve learned that whether people consider something to be true or not may not depend very much on, well, whether it is true or not. Deeper than truth, we have learned, lies allegiance.

But this is not our first disobedience! Wise societies have long accepted that while it is appropriate to reason from facts in other areas, rationality becomes boorish and simplistic when the subject is religion. In that department, one must respect a deeper source of knowledge called “faith”. So, long before the current era, we’ve all had plenty of practice with the intellectual dishonesty needed for a world of fake news. Learning to shut off our minds when discussing religion was our original sin as rational citizens, the tasting of the tree of disknowledge.

[28 November 2018]

Macron is more unpopular than Trump

In my year in France, I never once heard anyone speak with enthusiasm about Emmanuel Macron.

For those of you who’ve forgotten the bitterness of late 2018, let me summarize. The leading figure in the world trying to tear down the postwar liberal order is Donald Trump, the most destructive force the world has seen in my lifetime. Against him, the leading figure trying to sustain the postwar liberal order is Emmanuel Macron, the first truly international leader Europe has produced since Angela Merkel.

Yet as of this month, in their home countries, Macron’s popularity ratings are lower than Trump’s (low thirties vs. mid-thirties). “Well you know,” is the typical assessment of educated thoughtful French, “he’s a man of the Right.”

What this means in effect is that rather than stand up for the world we believe in, we prefer to wait for somebody better than Macron to come along, and then we’ll think again. I hope you see that this note is not about the French, but about human nature.

[16 October 2018]

False friends and sunny surprises

Getting deeper into French, for an English speaker, is a rich journey. I knew of the notion of false friends, but I had dimly imagined this was a universal effect to be found between any two languages. But no, the relationship of French and English is special.

First let me record a few false friends I hadn’t noticed before this year. I love the way navigation means sailing, and clairvoyant means seeing clearly, and a comedien is an actor, and apprecier means to assess, and sanitaire means health-related, and acquiescer means to agree. All like the English, yet not.

And let me mention some of the sunny surprises, discoveries that THAT’s where a certain English word came from! Today’s Le Monde reports a couvre-feu — a curfew. The participle aisé is a French word for easy. Represailles gives us reprisals, and effrayé afraid, and ennuyé annoyed, and nuisé noisy. A tailleur is one who cuts and measures, a tailor, and a pair is an equal, a peer, and farouche turns into ferocious. The word dûment puzzled me until I realized that ment here is just the adverbial suffix as usual and can be translated into “ly”. And in a biography of Napoleon I got a kick out of a general who had to se rendre. Surrender!

[15 September 2018]

Nuclear power and the flu

The public hates nuclear power, and one reason may be that the name reminds us of nuclear weapons. Would history have turned out differently if a less alarming name had been adopted early on? The model is magnetic resonance imaging, a clever branding of the technology of nuclear magnetic resonance.

I’ve noticed an example in the other direction, a case where we might have been better off if a more alarming name had been adopted. The disease we call “the flu” is lethal, but vaccination rates are low. Perhaps part of the problem is that the flu sounds so familiar, like a family pet. It might be better if we called it influenza.

[1 October 2018]

Good, closer, best

It’s funny how the good-better-best sequence misbehaves if you apply it to the word “friend”. A best friend is a perfectly good superlative of a good friend. But if you say Joe is a “better friend” than John, that introduces a negative tone. The right thing is to say that Joe is a “closer friend” than John.

As my mother always told me and my sister, comparisons are odious.

[20 September 2018]