On the wave equation and the beginnings of Fourier analysis:

“Very soon after that, people began…when I say people, I mean mathematicians.”

I only know Rudin from his analysis books, but come on, man! What, are you worried that someone will think mere people would be interested in this?

Another not great moment comes a few minutes later:

“D’Almbert raised a rather strange objection to that. This is a curious objection to my mind, because these were sophisticated people, and why d’Almbert couldn’t see…is hard to believe.”

It’s a lecture on history of mathematics. Rudin warns us at the start that mathematical history is understudied, and that mathematicians aren’t exactly qualified for it. I’m glad he had that awareness, because history starts right here, at that moment of disbelief. The historian has the tools and penchant to turn that disbelief into curiosity.

To be fair, Rudin offers an answer…

“Part of the problem was that the concepts were not well-defined.”

…though this is just another way of saying that they d’Almbert didn’t know modern math. But what was he thinking? Why did the objection seem natural to him? Why did it make sense? (There’s something sort of similar between some of the questions the math teacher and the math historian have to deal with.)