Why are set theory and numbers important to philosophy?

Question

I'm reading David Papineau's Philosophical Devices, and there's a section on numbers and set theory. But there's not deeper hint on why it's important to philosophy.

I guess that in mathematics we can think of formulas for building the elements of some set, such as the set of even numbers, the set of odd numbers, the set of all the prime numbers (which I guess that until this date, have no formula) so perhaps it's important to make formulas for some objects in philosophy?

I guess I can see the importance of the knowledge of the axiom of comprehension (and the Russell set), which states that for any condition C, there exists a set A such that (for any x)(x is an element of A iff x satisfies C). It is there to remind us that such axiom can't hold without some exceptions. Although I fail to see where one would create an object (in philosophy) and then notice a similarity of this object and Russell's set.

Note: If you know better tags, please edit. I've tried to use the tag set theory, but It does not exist.

Answer 1

I think that the relevance of logic for philosophy is high if you are interested mainly in the so-called analytic philosophy (started with Frege and Russell).

For me is difficult to think to the relevance of set theory for philosophy outside the branch of philosophy of mathematics.

Of course, if you are interested into ontology or philosophy of language and you think that those branch of philosophy are involved with (the existence of) abstract objects, then mathematics is a paradigmatic case for the reflection around the nature and our possibility of knowledge of abstract objects.

Answer 2

Set theory is a bridge between logic, which is traditionally considered a branch of philosophy, and mathematics.

Thanks largely to the philosopher/mathematician/logician Bertrand Russell and his mentor, Alfred Whitehead, the idea that all mathematics could be reduced to logic was at one point in time a major topic of debate in philosophy, until Kurt Godel decisively disproved it with the incompleteness theorem.

Answer 3

It's very hard for me to imagine how you could possibly theorize usefully about ontology or epistemology without testing your theories to see whether they say plausible things about mathematical entitities and our knowledge thereof. And it's very hard for me to imagine that you'd do a very good job of this unless you had a pretty clear understanding of the foundations of mathematics, which is to say a clear understanding of things like numbers and sets.

Answer 4

Set Theory is essential in the philosophy of mathematics -and perhaps Univalent foundations/category theory will be too one day when they are slightly more approachable to the untrained.

The reason for this is that set theory offers an incredibly elegant and effective model for the whole of mathematics; in other words, all of mathematics, it seems, can be reduced to set theory.

Therefore, if one is interested in what numbers are, what mathematics is about etc. set theory is of enormous interest both in the way it represents certain things - the natural numbers, for example, are just deadkind algebras in infinite sets, therefore within a standard model of set theory (ZFC) there are an infinite number of different models for the natural numbers.

In addition to the way set theory represents mathematics, the axioms of set theory are the subject of much controversy: Mathematics can be reduced to them, but should we believe that they are true and why? The subject matter is also quite controversial.

Finally, there are questions about which logic should be used to capture our conception of the set-theoretic universe and why.

This is a very, very brief overview, for more detail Id recommend the stanford encyclopaedia page and for a book length treatment M.Potter's Set Theory and its philosophy. Its fantastic, very clearly written and he develops all of arithmetic in the book alongside some excellent, and very subtle, philosophical commentary.