There is an interesting paper by Scott Aaronson about the intersection of those two disciplines and how the more CS-y approach can help solve philosophical problems:
While what they teach at the university today for a philosophy degree may let you focus on things that look imprecise, I find your statement is rather ironic. The difference between someone, especially coming out of school, who is well versed as a logician (hint they may have a math degree, but it's still philosophy) versus some one who has been exposed only to computer science, often parallels what I have seen interviewing people with a physics background versus an engineering background: almost all of the time the engineer might be able to give you enough to solve many of the problems, and have a little background, but the physicist will typically have a far stronger "toolkit" to analyze the problem, especially when the problem is novel. So goes it with a CS grad versus a math grad, typically around here, though YMMV.
And wasn't Wittgenstein essentially out to destroy traditional philosophy?
Philosophy is the "science of everything", and as such, all fields of (western, academic) knowledge are considered parts of it. Math in particular is formally equivalent to logic, which is the universal language on which philosophy is written.
Edit: to be clear, I didn't mean to imply the philosophy dept isnt rigorous, but instead, merely, that nearly the same classes would be offered in each dept and math, unlike philosophy, suffers from the perception of being too formal if anything.
That isnt to say there are no uses of them in CS or that CS ignores them all. But many systems were invented to deal with explicitly philosophical problems (eg. modal logic for metaphysical necessity, para-consistency for paradoxes, etc.).
In some very specialized areas non-standard logics do get used (theorem proving uses intuitionist logic, AI/Machine Learning might use fuzzy, etc.) but its still a limited subset over all.
No doubt, in heavily mathematical CS departments there might be courses on logic in a more complete sense.
CS researchers do develop logical systems of a kind: in particular they have developed ways of writing logic systems programatically (and usually end up having to ditch a few things and thus invent a new system).
However if you bought any major books on logic (as a subject) or looked into any contemporary research in logic (as a subject, not "doing logic on an ARM, etc.") you'd find the authors were philosophers. Occasionally mathematicians, and esp. mathematicians who then became philosophers.
I agree that philosophy often deals with topics that have not yet been properly formalized - after all, this is the discipline that's on the frontlines of important problems, asking questions that later get answered with more formal approach.