# What is the true generalization of a notion?

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?

• Possibly related: matheducators.stackexchange.com/a/908/262 and, perhaps, the comments of mweiss on generative motives. Aug 2, 2014 at 5:36
• Sometimes it takes a lot of work to see that a generalization is "most natural" or even that it is reasonable at all. Consider, for example, the generalization from algebraic varieties to schemes. Would a typical mathematician (in constrast to a genius like Grothendieck) have imagined schemes as the answer to "what's a good generalization of algebraic varieties?" Aug 2, 2014 at 16:35
• Can you give an example of a student's strange generalization and how you applied "truth", "usefulness", and "naturality" to the discussion in order to make the educational applicability of this question more apparent? It would also simply be useful for people who have not had such discussions. Aug 3, 2014 at 8:56
• This question seems more suited to a blog, or some other platform which admits extended discussion. I have voted to close on these grounds. FWIW, I think concepts can be generalized in completely orthogonal directions, and there is rarely a "best" generalization. Aug 4, 2014 at 0:26