Dear Pen,

Thanks for pointing me to this. There are a lot of videos there, and I only so far looked at two: Jamie and Stillwell. Jamie talked about beauty and fruitfulness, and not much about depth. Stillwell (in math) put a very large amount of famous interesting mainstream mathematical material with interesting comments, and also put a lot of stuff up there dear to my heart – reverse math (even as the beginnings of a measure of depth), Kruskal’s tree theorem, graph minor theorem, and a little about large cardinals. Specifically, he said there was no fundamental theory of mathematical depth, but there is hope from the logic community, specifically citing Reverse Mathematics. I wouldn’t of course say that RM is anything close to a magic bullet for this.

Before delving into the many many other videos I thought I would ask you if you could recommend a few of them as particularly on focus about mathematical depth.

Actually, when I asked you about “what is good mathematics”, I more had in mind “what is an important mathematical question or mathematical issue or mathematical research project?” rather than “what is mathematical depth?”. But of course I am interested in all of it, including depth, beauty, fruitfulness, importance, and even difficulty.

I, for one, have much clearer ideas about “foundational importance” and “general intellectual interest”. Generally, but not always, one needs deep mathematics to do things of foundational important and general intellectual interest in mathematical areas.I would like to think that these is essentially the only criteria I use for my own research.

In fact, in a previous incarnation, I was a general foundational thinker who developedÂ major foundational schemes for all of the major subjects across the entire intellectual and artistic landscape (SMILE). That was a great life!

Harvey