William Thurston passed away a little over a month ago, and while I have never had the occasion to read any of his work, this article of his, entitled “On Proof and Progress in Mathematics” has been reposted, and I think it’s worth a read for those who think about how mathematical knowledge progresses. For those who do theoretical engineering, I think Thurston offers an interesting outside perspective that is a refreshing antidote to the style of research that we do now. His first point is that we should ask the question:
How do mathematicians advance human understanding of mathematics?
I think we could also ask the question in our own fields, and we can do a similar breakdown to what he does in the article : how do we understand information theory, and how is that communicated to others? Lav Varshney had a nice paper (though I can’t seem to find it) about the role of block diagrams as a mode of communicating our models and results to each other — this is a visual way of understanding. By contrast, I find that machine learning papers rarely have block diagrams or schematics to illustrate the geometric intuition behind a proof. Instead, the visual illustrations are plots of experimental results.
Thurston goes through a number of questions that interrogate the motives, methods, and outcomes of mathematical research, but I think it’s relevant for everyone, even non-mathematical researchers. In the end, research is about communication, and understanding the what, how, and why of that is always a valuable exercise.