David Lewis was one of the great philosophers of our time. He was a genuine philosopher, his focus was on theoretical metaphysics. And he had something to say about mathematics. His last book - he only wrote four - was Parts of Classes (1991). It had a strong set theoretical - and as you may guess mereological - impact.
I wonder if there is a thorough mathematical analysis of - some of - Lewis' concepts and arguments, subsequently or unknowingly (see Michael's answer below).
I am especially interested in his concept of natural properties which he introduced in New Work for a Theory of Universals (1983), and whether and how this concept might be applicable to mathematics.
Very brief summary (just a teaser): To each set corresponds a property. Such properties are abundant. To the definable sets correspond a restricted but still very large family of properties. Natural properties in turn are very sparse, play a prominent role (at least in Lewis' metaphysics), have to be grasped somehow intuitively, and it's not clear, how they and their corresponding sets can be characterized.
PS: I found an announcement of a talk on Natural Properties in Mathematics. Does anyone have a transcript of this talk or something like that?
PS 2: Here is an older question of mine asking for natural properties in arithmetics.