I'm interested in the development of mathematical definitions for the sake of engineering, and what makes a particular definition better suited for a problem than another in any particular context.
Ideally, I aim to compile a list of examples of definitions, their historical context and a list of inadequate definitions which were proposed, along with a rationale of what was missing from the inadequate definition.
A concrete example is Bell Labs being contracted to create secure military communications, and Claude Shannon developing his theory of mutual information as a result. A failed attempt would be Karl Pearson's suggestion of mutual information as P(x,y)^2.
Here's a blog post that goes a little bit into that history, and points at the kind of comparitive research I'm interested in: https://www.lesswrong.com/posts/GhFoAxG49RXFzze5Y/what-s-so-bad-about-ad-hoc-mathematical-definitions