Metaphysics
Metaphysical debates about causation, consciousness, chance, change, mathematics, or modality have a lot in common. In all cases, metaphysical theories try to tells us what, if anything, makes a certain class of statements true. Among the possible answers, we usually find suggestions to reject the alleged phenomena, to declare them as primitive, and to reduce them in various ways to something else. But on closer inspection, there appear to be big differences, in particular with respect to what is required for a reduction.
In the philosophy of mind, saying that there are necessary connections between mental and physical facts counts as a metaphysical theory. In the philosophy of causation, at least a few philosophers are not satisfied with such an answer. What they seek is an answer to questions like this one:
As a matter of analytic necessity, across all possible worlds, what is the unified necessary and sufficient condition for causation? (Lewis, "Void and Object")
Finally, in the philosophy of modality and the philosophy of mathematics, it is generally agreed that necessary connections don't qualify as metaphysical accounts: almost nobody denies that mathematical facts supervene upon physical facts. But that's not what we want to know when we ask what makes it true that there are infinitely many prime numbers. What Platonism, fictionalism, structuralism, etc. offer are different analyses of such mathematical truths.
What is an analysis? Returning to causation, one might think that the desired relation is a priori entailment: what metaphysicians like Lewis seek is a condition such that it is a priori that if and only if that condition obtains, we have an instance of causation.
Then again, this isn't what we want in mathematics: if all mathematical truths are a priori, they are a priori entailed by anything, and a priori equivalent to any a priori truth. An adequate account of mathematical truths must provide analyses that are more than just a priori equivalent to the mathematical truths: they must qualify as "what we really mean by" the mathematical statements. The relation of "x is what we mean by y" is a hyperintensional relation akin to synonymy, but unlike synonymy it seems to have little to do with intensional isomorphism. I'm not sure what the relation is. Perhaps "all bachelors are unmarried" is not a good analysis of "there are infinitely many prime numbers" beause those statements have different conceptual roles (in some sense). Or perhaps the problem is that this analysis doesn't provide a general, simple, mechanical method for analysing other mathematical statements.
So, is there really something common to the various branches of metaphysics, or do the questions and possible answers only resemble each other superficially?