I am taking a brief break from grading to make a few notes about something I've become increasingly interested in recently, which I've been labeling for myself as "thinking around" (to be contrasted with "thinking about").
I'm going to start with two examples, one from Hume and one from Berkeley.
On my reading of Hume, there is a sort of mental activity one can engage in towards that which is strictly and literally inconceivable. This activity is supposition. In one part of my dissertation, I attempt to show that Hume can embrace this form of mental engagement without abandoning his commitment to analyze all mental activity (of the understanding) in terms of conception (i.e. ideas). At any rate, there are a few passages which are naturally read as Hume allowing that some things can be supposed which cannot be conceived. This type of mental engagement, I argue, allows a response to a Reidian objection which charges Hume as unable to account for reductio ad absurdum reasoning. So, while you cannot, on my reading of Hume, think of or about an even prime greater than 2, for example, you can think around such a prime, allowing you to reason your way to its non-existence.
Berkeley, like Hume, has a view of conception bound up with what ideas one possesses. Consequently, Berkeley deploys arguments about the nature of ideas to show that certain things are inconceivable. But, as is somewhat explicit in the third Dialogue between Hylas and Philonous, and fully explicit in Alciphron VII, Berkeley introduces a way to defend the meaningfulness of discourse in which meaningful terms to not signify ideas (rejecting a straightforward Lockeanism about language), with something I'll call "notions" (though I don't know if Berkeley consistently uses the "idea"/"notion" terminology to track this distinction). Having a notion of something does not require having an idea of it. Thus, even though I cannot have an idea of immaterial susbtance, I still have a way to engage with propositions about immaterial substances (whether we are speaking of me or god). This too is a sort of thinking around, as I understand it.
Resources which allow a philosopher to permit our thinking around something which we cannot (on their view) properly think about or of are important elements of their views for two reasons. First, they can give us important insights about other aspects of their views. For instance, noting that Berkeley must appeal to some such resource in the third dialogue, to explain how we can believe in immaterial substance helps us exclude some (seemingly natural) interpretations of the first dialogue arguments against material substance. While it might appear that Berkeley is offering a straightforward inconceivability argument against belief in material substance there, it is clear from his own later admission that we cannot strictly conceive of immaterial substance that the dialogue one argument must be more complicated than it at first seemed.
Second, however, they are important for allowing us to see how powerful objections to those philosophers wind up being. Take Hume, who embraces the view that we cannot conceive of anything which is impossible. Given that various philosophers have appeared to sincerely defend views which, for Hume, turn out to be impossible, there is the objection that Hume cannot be right, because we could not then make sense of such apparently sincere defenses. A natural sort of reply is to invoke some sort of verbal confusion underlying the dispute. But that line of reply is not always satisfying, and does not always do a good job of addressing the behavior of his opponents. On the other hand, Hume's resource of supposition-without-conception permits him a more robust way to understand his opponents as engaging with these impossible views (apart from merely "mistakenly defending that the sentences which express those impossibilities actually express truths").
I'm sure that similar sorts of resources crop up in the views of other philosophers, but I don't want to just start casting around randomly. If anyone has suggestions of places to look (especially in terms of early modern figures other than the "canonical" British empiricists), please let me know.