Paving the cowpath in research within pure mathematics: A medium level model based on text driven variations.

ABSTRACT

In this paper we show how simple text-driven variations of given statements in mathematics can lead to interesting new problems and push forward a whole theory around simple initial questions. We exemplify this in two cases. Case 1 deals with problem-posing activities suitable for pupils and case 2 is a rational reconstruction of the organisation of mathematical knowledge within problems of graph colorings. Mathematicians learn to systematically look for subsequent problems around a given problem. We argue that this toy-model captures a nontrivial part of professional mathematical research within the pure fields and conjecture that it even grasps high level developments in mathematics. By doing this, we implicitly encourage a very simplistic view on criteria, so to speak a "cowpath" approach to progress in mathematics. The term "cowpath" is borrowed from architecture and software design, where it is commonly used. While we can contemplate which pathways are ideal, we may also just plant grass and see where people choose to walk. Those pathways are also self-enforcing, since we are less hesitant to walk on those rather than criss-cross the landscape.