Since writing Approximation Theory and Approximation Practice a decade ago, I’ve found myself determined to know where each idea comes from. No matter what I am working on I need to know, who did this first, and when?
Lately I have realized that this predilection of mine springs from two strongly held views. One (moral) was expressed in “The diameter of intellectual space” in 2004. The other (intellectual) is my growing concern that mathematics has lost its way. Over and over again we see the pattern that one mathematician introduces a fundamental idea, and then others advance the topic. Obsessively, at the expense of all other developments, they focus on bringing the idea to its most general and technically difficult limit. Thus Runge’s theorem of 1885, which shows you can approximate certain functions, ends up viewed as a special case of Mergelyan’s theorem of 1951. What is Mergelyan’s theorem? Well, it’s the same as Runge’s! — but with weaker smoothness assumptions. Impressive and important, yes. Yet how sad it is that if you try to look up this subject, you’ll probably finding yourself reading about the technicalities, not about the actual point of it all.
I believe that because of mathematicians’ habit of making topics ever more technical, the original version of a mathematical idea is often more to the point than what followed. That’s why the originals mean so much to me.
[13 May 2019]