What makes a good mathematical theory?

Question

I recently read about Galois connections, and that they show up in a lot of different places in mathematics. Given there apparent ubiquity, I thought they might have a rich theory. However, when asking other people in my department, I got varying answers as to why they were not particularly interesting and/or were not worth studying. It is hard to tell if this is due to a bias or whether there could be an objective argument made as to why mathematicians invest a lot of effort in characterizing and classifying one class of objects such as groups or topological spaces and not another.

I am interested in answers that either address the particular theory of Galois connections (and possibly arguing that they are in fact worth studying) or attempt to give a general criteria for what makes a good mathematical theory.

Answer 1

A general answer which does not consider Galois conenctions:

See Terrence Tao's: What is good Mathematics.

Also from General Topology (Kelley pg 111):

The development of general topology has followed an evolutionary patttern which occurs frequently in mathematics. One begins by observing similarities and recurring arguments in several situations which superficially seem to bear little resemblance to each other. We then attempt to isolate the concepts and methods which are common to the various examples, and if the analysis has been sufficiently penetrating we may find a theory containing many or all of our examples, which in itself seems worthy of study. It is in precisely this way, after much experimentation, that the notion of a topological space was developed. It is a natural product of a continuing consolidation, abstraction and extension process. Each such abstraction, if it is to contain the examples from which it was derived in more than a formal way, must be tested to find whether we have really found the central ideas involved. This testing is usually done by comparing the abstractly constructed object with the objects from which it is derived. In this case we want to find whether a topological space, at least under some reasonable restrictions, must necessarily be one of the particular concrete spaces from which the notion is derived. The "standard'' examples with which we compare spaces are cartesian products of unit intervals and metric spaces.

Answer 2

In my opinion, a good mathematical theory should move a particular area one step forward (overcome established traditions, provide further conceptual clarity). This is a given. It should extend what has already been proven, perhaps, provide more light on a subject which was previously in the dark.

Yet, I also think a great or good mathematical theory should spark one's imagination. This is crucial for me. It is important that a mathematical theory, once it is generally accepted - triggers a general collective consciousness into action, in order for the area under inspection to progress. This can only be done if the theory presented contains a certain amount of rigor and divine creativity, (what in Germany, they would call a Geist i.e. invoking some sort of creative spirit, so to speak).

I would like to reference a passage from both Frege and Hilbert's correspondence:

In a letter to Frege, Hilbert wrote:

...every theory is only a scaffolding or schema of concepts together with their necessary relations to one another, and the basic elements can be thought of in any way one likes.

Frege, G. 1980. Philosophical and mathematical correspondence, ed. B. McGuinness.