Another note on the theme of my bewilderment with pure mathematicians.

The eminent Fields Medallist Alain Connes has published a 3-page letter to the great Riemann for which he set himself the task: “What could I tell to Riemann that would surprise him and that would give him confidence that his hypothesis is true?” The famous Riemann Hypothesis, one of the million-dollar Millennium Prize Problems, is that all the complex zeros of the Riemann zeta function lie on what is known as the critical line.

Working by hand around 1859, Riemann managed to compute the first five or six zeros. In the computer era, by a heroic combination of algorithms and analysis and hardware, more than twelve trillion of them have been computed! They all lie exactly on the critical line, just as conjectured.

Connes does not mention these computations in his letter.

[12 February 2026]

We all know that Aristotle made the mistake of trying to deduce scientific facts from first principles, whereas Galileo had the right idea and relied on experiments. And so, modern science was born. Aristotle may have been brilliant, but he got it wrong, as did the followers who trusted his authority for a thousand years.

I agree with this summary. The telescope beats the armchair. Yet I can’t help noticing that most of us in our intellectual lives, including me, operate more like Aristotles than Galileos. We read, we think, we analyze connections and implications, and (in my case) we lecture about our conclusions to our students. For example, in preparation for today’s Applied Math 202 class about the heat equation, I did some careful thinking about how the mathematics and the physics of bouncing particles and Brownian motion and partial derivatives and Gaussian diffusions fit together. I even did a few calculations on my computer. It’s fascinating, it’s beautiful, and I believe it’s largely true. But I can’t help noting, no thermometer was involved. How often do I actually measure something? Maybe once a year?

[27 January 2026]