A good heuristic is:
If it doesn't cost anything, generalize.
In particular, if we have
- a theorem, and
- a proof thereof,
then we ought to look for a maximal generalization of this theorem, subject to the constraint: "the basic ideas of the original proof can be used to manufacture a proof of the generalization."
I do my best to abide by this heuristic, since I regard it as "best-practice."
However, its lately occurred to me that I've been failing to heed this heuristic. Bigtime. The basic reason is category theory / categorial semantics.
Let me explain. Its well known that we may consider models of a signature in any category possessing sufficient structure. Thus, set-theoretic models are just a special case. Furthermore, the internal logic of different categories is different. Therefore:
We should prove things not only under minimal hypotheses; but also, in the weakest possible logic.
However:
I tend to prove things by assuming an internal logic that is classical. For example, I will say something like the following. Let $G$ denote a model of Group. Then [insert argument here]. We conclude that $G$ satisfies $\varphi$. Thus, $\varphi$ is a classical theorem of Group.
The problem is that typically, only a weak fragment of classical first-order logic was actually needed to state and/or prove $\varphi$. Thus, the conclusion "$\varphi$ is a classical theorem of Group" is a far cry from being maximally general.
So here's, a little summary.
I want to produce well-conceptualized, maximally general, best-practice mathematics. My ignorance of categorial logic is getting in the way. In the short term, what should I do?
The medium-to-long term solution, of course, is to actually learn categorial logic! And frankly, I'm trying. But progress is slow, and I don't really have a good resource to learn it from. So, my next question is
2. Given my specific concerns and interests, what would be a good resource for learning this stuff?
Furthermore:
3. Should I just pick a book and work through it, problem-by-problem? Is there a better way to learn this material?
Thank you, and your time is appreciated.
I guess that the answer depend strongly on what do you already know.
If you already have a good confidence with lots of advanced category theory you could run through Johnstone "Sketches of an Elephant".
Then there's another text book: McLarty "Elementary categories, elementary toposes".
Another reference that I've used to learn topos theory is Barr and Wells' "Topos, Triples and Theories" which can be downloaded from the link.
All these references have as main aim toposes and so most of the material is about categorical logic in a topos. Nonetheless Johnstone and Barr&Well's book have some material about cartesian closed categories (and so algebraic theories), coherent categories and so on.
Of course these books require that the reader have a previous knowledge of category theory and feel quite confident with most of the basic construction of category theory. In case you lack of this knowledge I suggest you take a look to Awodey's book "Category Theory", which is one of the best introductory book in category theory which also have a logical flavour (at least in my personal opinion). Then you can take a look to Bourceux book to learn some more advanced category theory.
Hope this helps.