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]

I wanted to fill Dorothy’s new hot-water-bottle slippers with water about 2/3 of the way to boiling. But how to know when the kettle had reached 2/3 of the way to boiling?

Of course the answer was to let it go all the way, but with 2/3 of the water. Then fill the rest from the tap.

[25 December 2025]

For years I was a professor at Oxford, with its great Bodleian Library. However, only library staff are allowed in the Bodleian stacks, and I have never seen them. Now I am at Harvard, with its equally great Widener Library, and here, students and faculty and even their guests can use the stacks. They are magnificent!

And entirely empty of life. I just went in, down three flights, through the Pusey tunnel, to Row 66 of Level P3, past ten thousand beautiful books and no people. Right now I am sitting in the Widener reading room, which is as grand as the Titanic, and there are 11 other readers here.

I wanted to look up Ockham’s Razor in the Encyclopedia Britannica, but the student at the counter said that if I really needed the hardcopy, I would have to request it from remote storage. This seemed preposterous, and as it happened, she was wrong. I walked up to the reading room and the first thing I saw was the Encyclopedia itself, filling the first two shelves on the left. The other 91 shelves are empty.

Hardcopy books are ceasing to matter for scholarship, and it is hard to imagine that Widener will still be running in its current form twenty or maybe even five years from now. This should be thrilling, for it reflects how information is becoming universally accessible online. There is every reason to be happy as Widener and the Bodleian fade out; but I am deeply sad.

[26 November 2025]

People talk about what the goals of a powerful AI might be, for we all worry, might AI dominate or replace us? In such discussions, many reassure us that there’s no telling what an AI’s goals might be. Perhaps it will have other interests than conquest.

You rarely hear mentioned what should be the starting consideration. Since Darwin, we have known that the reason life expands to fill niches across the planet is nothing more than natural selection: life forms that don’t expand are supplanted by those that do. Bacteria do it, plants do it, yet these creatures have no goals. If you think the same principle doesn’t apply to AI, you need to give an argument.

[26 November 2025]

My work deals with approximating complicated functions f(x) by simpler ones. You might think that the goal was just to match the data accurately. But no, the deeper aim is to obtain a model that can predict how f will behave at other values of x, not yet seen. For this purpose, simple approximations do better than complicated ones. And since optimality means maximal simplicity by some measure, it follows that optimal approximants are better than suboptimal ones not just because they are more efficient at capturing the data.

The classical example goes by the name of the Runge phenomenon. A polynomial that interpolates the data exactly may be useless between the sample points. It has merely fit the data, not understood it. In the current era, we know that machine learning algorithms must avoid “overfitting”.

Henceforth this will be my take on Occam’s razor. Complexity does not just make a model less elegant. It lessens its power.

[22 November 2025]

In 1991 I suggested in one of these notes that if there are a lot of habitable planets out there, yet still we see no evidence of extraterrestrial life, this may indicate that the emergence of life on a habitable planet is more exceptional than we imagined.

It occurs to me that the same argument applies locally. Why has life emerged on Earth just once in billions of years? For I think that is the consensus of biologists: all life on this planet shares the same chemistry and stems from one original event. Experts may investigate details, arguing perhaps that the early Earth had conditions more favorable to appearance of life than the later one, but these are details. Earth is our archetype of a planet favorable for life. If life has appeared here just once, that’s a weighty data point.

[8 November 2025]

Biking is not allowed in Harvard Yard (“Please walk your wheels”), but many students do it anyway. If I walk through the Yard on a typical day, about four cycles or scooters whiz by. I don’t like this, and when I can, I raise my arm and say “Walk your wheels!” Some riders swear at me, most stay silent, almost all keep riding. I’ve never seen anyone else but me ask a rider to stop.

This upsets me more than you can imagine, and more than I can understand — to the point where I may start going around the Yard to avoid these encounters. I am trying to figure out why I am so upset. Partly it’s a feeling of political hopelessness. If people don’t care enough about social harmony to follow even this easy rule, what hope is there of collaboration to counter the bigger things going wrong in our country? So biking in the Yard becomes a symbol for me of the self-centeredness of all humanity, and of the careerism of Harvard undergraduates, and of paralysis in the Trump era.

Yet I think a bigger part of it is a sense of incomprehension and alienation. I simply cannot get my head inside the pattern of thinking that leads undergraduates to decide to break the rule. These are not troublemakers living on the edge, they are Harvard students, the very definition of society’s winners! What story are they telling themselves? My incomprehension makes me feel aged and remote from them. As another bike speeds by, I feel I am an old man on the way out, being replaced by an alien generation.

[8 November 2025]

It’s funny how, in the year you turn 70, so do a lot of your friends. I have a strange sense of the befores and the afters. Greg and Chris have made the transition, and so have I, but Nat is still just 69.

As many have noted, 1955 was also a big year for baby boys on the larger stage. John Roberts (Jan 27), Steve Jobs (Feb 14), Alan Garber (May 7) and the Pope (Sept 14) are behind us. Bill Gates (Oct 28) is up next.

In my own world of scientific computing, we have Volker Mehrmann (Apr 24) and Bill Gropp (Sept 23) as well as youngsters Randy LeVeque (Sept 30) and Jim Demmel (Oct 19).

[28 September 2025]

With the next academic year about to begin, I have filed away my email correspondence folders for 2024-25. It turns out that during the year, I kept records of interchanges with 629 different people.

[25 August 2025]

Here in St. Vith they speak German and French. English is fine too. You don’t have to choose.

The WC in the Bäckerei Fonk shows

with some words below, in English.

Whatever.
Just wash your hands.

[14 August 2025]