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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.

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This is by the person that gave the talk you mentioned: www-personal.umich.edu/~tappen/… It discusses natural properties a bit, but does not provide something like a formalization. –  Michael Greinecker Apr 23 '12 at 22:04
    
Off-topic, but that poll you link to is a hilarious beauty contest, no more. Kripke one place behind Hegel, and one ahead of Nietzsche? Come now. And who's idea was "200 years" so that Kant is excluded? –  Charles Matthews Apr 24 '12 at 11:56
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I briefly consider mathematical naturalness in section 9.8 of my book books.google.co.uk/books?id=a0U98qfx4TQC –  David Corfield Apr 24 '12 at 12:34
    
@Charles: I found it so obvious that the poll wasn't to be taken seriously that I didn't mention it. –  Hans Stricker Apr 24 '12 at 12:40
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1 Answer

Lewis introduced the concept of common knowledge (I know that you know that I know that you know that...) in his book Convention. The concept has been formalized in a partitional model of knowledge in the simple and elegant paper Agreeing to Disagree by Robert Aumann. Aumann wasn't aware of the prior work of Lewis. The concept of common knowledge became one of the building blocks of modern non-cooperative game theory and has been extensively studied and generalized. A survey can be found here.

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Thanks, Michael. That's the kind of answer I hoped for. (Now I hope for an answer concerning natural properties.) –  Hans Stricker Apr 23 '12 at 21:50
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