What is the true generalization of a notion?

Question

In mathematics we usually can generalize a particular notion in many different ways. Some of these generalizations could be contradictory. When I teach maths/logic to my students I usually encourage them to think about possible generalizations. Sometimes their generalizations are very strange. In such circumstances I usually discuss with them on "truth", "usefulness" or "naturality" of such generalizations.

Question: What are our philosophical, mathematical or meta-mathematical criterion to decide on the most natural or most useful generalization of a particular mathematical notion? Are the most natural and the most useful generalizations of a notion necessarily same? Which is the best generalization?

Answer 1

I can't help thinking of the idea of a most general unifier. Indeed, maybe the analogy isn't entirely specious; mgu's provide a satisfactory first-order theory of unification, and most of the examples one would think of for difficult notions to generalise are second order. Well, that's just philosophical babbling.

Perhaps more usefully, this kind of question has been considered before. An old Monthly article that I enjoyed discussed how a search for generalisation could lead to mathematical discovery: Stolarsky - Searching for common generalizations: The case of hyperbolic functions. Although I haven't read it, I discovered in searching that Zeilberger wrote a similarly themed article: Zeilberger - The method of undetermined generalization and specialization. Other authors have also considered the process of generalisation in its own right; for example, Blumberg - On the technique of generalization, and, of course, perhaps most famously, Pólya (Pólya - Generalization, specialization, analogy).

None of these answers the question, and I think that it is more or less obvious that there is no answer; but each of them seems to have a useful perspective to add to the pile of thoughts.

Answer 2

Mathematics grows not only through the process of generalization but also through the process of specialization. If one has a definition and one can generalize or specialize that definition so that where before one could not tell X from Y in the original case but the "refined" definition allows one to tell X from Y mathematical progress can be made.

A favorite example of mine of this type is the idea of a polygon - usually thought of as something in the plane which is non-self-intersecting but "closed.". The essence of being a polygon was the "flatness" of its edges and that there are only a finite number of edges. So when one invents the idea of convexity one gets lots of new developments. One can talk about regularity of polygons whether or not they are convex. One can allow vertices of polygons to coincide (as Branko Grünbaum does). Many types of polygons are now studied: star shaped, rectilinear, those with infinitely many edges, closed, open, non-planar, knotted, etc. The resulting world of polygons is so rich that geometers will be kept busy for a long time investigating "named" kinds of polygons, and almost certainly there will be new types to catch our attention.