You might be correct in part or in whole. We'd have to discuss back and forth! My (1) and (2) will have to be at best only rough, and to save space I omitted

You might be correct in part or in whole. We'd have to discuss back and forth! My (1) and (2) will have to be at best only rough, and to save space I omitted the real examples I have in mind to formulate those two. Writing clearly, briefly, and precisely about big subjects is not always easy! The time I encountered Strum-Louville theory was a course from the fairly well known Hildebrand, Advanced Calculus for Applications from MIT and by a professor with a recent MIT Ph.D. By the way, that book is available in PDF on the Internet for free. To me the topic looked like the math of vibrating strings, etc., but the course did not go deeply into Sturm-Louville theory or anything else. So, from that course I have regarded Sturm-Louville theory now as in (2); but if there is some continuing, deep work there, maybe it belongs in (1). Maybe I touched on that subject once more; talking to M. Athans at MIT, he explained that the optimal control problem I was considering was a "two point boundary value problem with mixed end conditions". So, maybe there are connections with control theory. For Erdős, I have just regarded him as all in (1), but he seemed to have gotten problems from anywhere, including from outside math so might be in (2). Sooo, for the problems outside math, I meant relatively practical problems, ones where solutions would a lot of value, financial or scientific, for fields outside of math. I've always regarded Erdős as working a lot on interesting, tricky, puzzle problems. But I don't know Erdős's work at all well. Maybe I can summarize: Back when I had one foot in each of (1) and (2), my definition of applied math was like a recipe for rabbit stew -- "First catch a rabbit." So, first get an application, i.e., from outside math, where an application solves a problem that in some sense is pressing and practical. Examples: At FedEx, how to schedule the fleet? How to project revenue for years in advance? How best to climb, cruise, and descend an airplane? How to plan an airplane tour under uncertainty. Back in military work, how to use the Navier-Stokes equations to design ship propellers? How to process the data to do beam forming from arrays of passive sonar? How to evaluate the survivability of the US SSBN submarines? How to target missiles? How best to search for a submarine? In a simulation, how to generate ocean waves with a given power spectral density? How to respond to ill-conditioned matrices in regression problems? How to do least squares spline fitting, including for multidimensional data? For each of these problems, there were people who cared a lot about getting the solutions -- these were applications. Puzzle problems seem to be applications of a different kind. Maybe now there are some good applications to be made analysis of DNA.