Many-worlds interpretation of rounding errors

Every floating-point rounding error, Wilkinson teaches us, can be interpreted by backward error analysis: it’s the exact result, but for slightly perturbed data.

Every quantum mechanical measurement, Everett teaches us, can be interpreted as the splitting of two universes: one where this choice happens, the other with the other.

I like to imagine a kind of Everett–Wilkinson interpretation. Every floating-point operation is exact, hooray! — but each time one happens, we are jiggled ever so slightly, just one part in 1016, into a neighboring universe.

[17 February 2015]

The Universe Speaks in Numbers

Graham Farmelo gave a lecture here at Oxford yesterday about his book The Universe Speaks in Numbers. His theme, to quote from the web site, is that physicists these days seek the laws of nature “with the help of cutting-edge mathematics.”

This situation is controversial. Everyone knows that Newton and Gauss did real physics, but what about the two modern heroes Farmelo focused on, Atiyah and Witten? Is string theory real physics?

An unexpected twist in the discussion turned up at coffee just now with André Weideman, Yuji Nakatsukasa, and Trond Steihaug. Steihaug is studying Newton’s notebooks, and he was telling us how they show Newton carrying out systematic numerical calculations to many digits of accuracy. Gauss, too, was known for his extensive calculations.

Suddenly we noticed the oddity in Farmelo’s title, displayed on the poster behind us. The universe speaks in “numbers”? Of course, the word is intended as shorthand for mathematics. But that shorthand is long obsolete, for in fact, most mathematicians of recent generations have very little interest in the real numbers Newton and Gauss were calculating! I just checked and found that the only numbers that appear in the 106-page joint paper by Atiyah and Witten are the integers 0, 1,…, 10, 11, 24, 27, 36, 48, 72, and 144, together with i, π, and e.

All this suggests a possibility Farmelo cannot have intended. Maybe the universe indeed speaks in numbers; and maybe physics lost its way when it came under the spell of a kind of mathematics that is number-free?

[17 May 2019]

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]