Why history of mathematics matters

Since writing Approximation Theory and Approximation Practice a decade ago, I’ve found myself determined to know where each idea comes from. No matter what I am working on I need to know, who did this first, and when?

Lately I have realized that this predilection of mine springs from two strongly held views. One (moral) was expressed in “The diameter of intellectual space” in 2004. The other (intellectual) is my growing concern that mathematics has lost its way. Over and over again we see the pattern that one mathematician introduces a fundamental idea, and then others advance the topic. Obsessively, at the expense of all other developments, they focus on bringing the idea to its most general and technically difficult limit. Thus Runge’s theorem of 1885, which shows you can approximate certain functions, ends up viewed as a special case of Mergelyan’s theorem of 1951. What is Mergelyan’s theorem? Well, it’s the same as Runge’s! — but with weaker smoothness assumptions. Impressive and important, yes. Yet how sad it is that if you try to look up this subject, you’ll probably finding yourself reading about the technicalities, not about the actual point of it all.

I believe that because of mathematicians’ habit of making topics ever more technical, the original version of a mathematical idea is often more to the point than what followed. That’s why the originals mean so much to me.

[13 May 2019]

Square root of the population

27 years ago I wrote,

There are some 1010 people in the world. Roughly speaking, I am among the top 105 most successful. Accounting for this improbably high position in the hierarchy is a troubling problem for me. A tantalizing idea suggests itself: can one argue somehow that in a random population of n people, the expected position to find an introspective person of my sort is on the order of √n from the top?

Well, here’s an argument that gets exactly that answer. The person in position #1 will be pretty interested in his or her good fortune. Suppose we imagine that person #2 is half as interested, person #3 is 1/3 as interested, and so on: essentially, each person’s interest in their luck is proportional to their luck. Then the total amount of human interest in positional luck, integrated over the whole of humanity, is about log n.

Now pick a human being at random, weighted by their interest in their luck. The middle position is ½log n, which is the logarithm of √n. So there you have it. If I am person number 105, then half the population’s interest in their luck is to be found in people higher up than I am, and half in people lower down, making me quite entirely typical. Throwing in a few trillion dogs, squirrels, and mosquitoes, with their lower levels of introspection, won’t change the result.

[12 April 2011]

Exponentials give us randomness, and certainty

Here is a curious symmetry. To achieve randomness in science or technology, our best strategy is exponentials. You can toss a coin, but the outcome isn’t so random because it is sensitive only algebraically to the details of the throw. For truer randomness you need a chaotic system with exponential sensitivities, like a pinball machine or the Lorenz equations. Run such a system for a moment and your randomness might be 99%. If that’s not enough, run it a little longer to get 99.99%. The point is that with each new step, your knowledge about the system shrinks by a constant factor, soon reaching zero for practical purposes.

And to achieve certainty, our best strategy is exponentials again! At the level of fundamental physics, anything can happen because of quantum tunnelling. But some things “never” happen in practice, such as the radioactive decay of an iron-56 atom. Why? Because the frequency of quantum events shrinks exponentially with the width of a potential barrier. Thickening up that barrier in a physics experiment is like adding another level of error correction in an electronic circuit or taking another step of a random algorithm or making another compressed sensing measurement. With each new step, your uncertainty about the system shrinks by a constant factor, soon reaching zero for practical purposes.

[1 September 2011]

Protecting one’s woman

Men being bigger and stronger than women, I was musing how in the old days, a man used his strength to protect his woman. However evolved you may be, it’s hard not to feel a tinge of pride in being associated with this classic role.

These fine feelings were dimmed a little by the thought that, realistically speaking, probably about 80% of that protection was against other men.

[15 June 2018]

Science publishing then and now

Abi Gopal and I have a new idea we want to announce to the world, a new method for solving the Laplace and Helmholtz equations. We sent a Letter to Nature, which was rejected in five days. Then we sent a Brief Report to Science, and it was rejected in 23 hours.  Neither journal sent our contribution to referees.

But we want to put our names on this! So what did we do? We posted the announcement online as an e-print at arXiv.  Now we can move on and develop our idea at leisure.

Here is what tickles me about this story. You might think journals are the classic medium and arXiv is the new kid on the block. But in a sense, it’s just the reverse. In the old days, a scientist could send an idea to a journal and get it published quickly. A classic example was Gibbs’ announcement of what is now called the Gibbs phenomenon, which he sent as a letter to Nature back in 1899. These days, however, journals have evolved into armored tanks. Nature of 1899 is arXiv of 2019.

[6 February 2019]

Forty years of research talks

During my career so far, I’ve given around 800 talks on 150 subjects in 31 US states and 32 countries on 6 continents.

If an average audience is 30, it would seem around 25000 people have seen me speak, or maybe 10000 if you weed out repeats.  Such numbers are huge by an ordinary standard, but infinitesimal on the scale of politicians or celebrities.

[12 February 2019]

Identity politics and the principle of proportionality

In the old days the idea was that if A annoyed B, they sorted it out among themselves. Now, it is increasingly likely that B will accuse A of being a homophobe or a racist or some other kind of oppressor. Yesterday a woman I’d met only sixty seconds earlier accused me of talking like a man, adding that at her age, she didn’t have to put up with that any more. It hurt.

I may have been guilty as charged, but still, I think her response was inappropriate. The issue is what lawyers call the principle of proportionality. When B brings identity groups into the discussion, a tussle of one against another becomes a battle of millions against millions. In cases of genuine oppression, the big battle may be needed, but most of the time, day to day, that scale of response is disproportionate. It’s like sentencing Jean Valjean to five years for stealing a loaf of bread. It’s like pulling out a gun to settle a fistfight.

[7 February 2019]