How much mathematics and formal logic is needed to pursue graduate studies in Philosophy?

Question

On other online philosophy forums, this question got mixed responses. Some said that you could not get far in philosophy without a good background in mathematics. Others said that unless you venture into some areas of philosophy like philosophy of language, mathematics would not be that important. The remaining few adopted a more extreme position that philosophy could do without math.

I was wondering what Philosophy professors or people with and advanced degree in Philosophy think about this issue. Could you cite any peer-reviewed articles related to this question?

Answer 1

tl;dr- Philosophy departments don't tend to focus on math much, and Philosophy students have a mathematical proficiency much closer to an Art major's than to an Engineering major's. This answer doesn't comment on formal logic.

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Lack of math in Philosophy departments

You normally don't need much math to do well in a Philosophy department's graduate program.

For evidence, we can look at standardized test scores to see what students going into different fields tend to be proficient in. In these GRE scores by discipline, we can see that students planning to do Philosophy in grad school rank only somewhat better in Quantitative Reasoning than other Arts and Humanities students, but much worse than Engineering, Physical Sciences, and Economics students.

You might be able to do math in a Philosophy department

I'm not aware of any hard rule stating that you can't do something math-related in a Philosophy grad program. For an example, "Historical Development of the BFGS Secant Method and Its Characterization Properties" (2009) by Joanna Papakonstantinou is a great read. The author published this as their dissertation for a Ph.D. in Philosophy from Rice University.

The caveat here is that their advisory committee doesn't list anyone from the Philosophy department, so presumably they worked with people from other fields. Presumably if you want to do something Math-related in a Philosophy department, you may have to reach out to advisors from others fields to support you, as the Philosophy department's professors are unlikely to have enough background knowledge to evaluate your work.

But if you want to do math, why in a Philosophy department?

Part of the lack of mathematics in Philosophy may just be a definition thing. This is, "philosophy" may be the general field of knowledge, but "Philosophy" in the sense of "what Philosophy departments focus on" is a far narrower topic.

If you want to do something like hardcore math, you can - just, you'd probably prefer a Math Ph.D. program, which is likely to give you a better background and a more valuable degree. You can even do a math-heavy Ph.D. in fields like Engineering, Physical Science, or Economics if you focus on advancing the understanding of a type of math that's relevant to those fields.

For example, you may not usually think of a Business department as a place where math's advanced, right? But Business departments are really interested in Big Data and data science. So, it shouldn't be too hard a sell to get a Ph.D. in some sort of Business-related field, then just spend your whole time doing math or data analysis, as long as you somehow refer it back to some sort of practical business interest. That's the beauty of grad school!

Addendum: Philosophy students scored well on the rest of the GRE

Above, I'd noted that Philosophy students didn't score very well on Quantitative Reasoning, as that's relevant to the question. Looking back at it, I wanted to note that this may belie an interesting tidbit...

Despite middling performance in Quantitative Reasoning, Philosophy students achieved the highest scores across all disciplines on the other two sections of the GRE, i.e. Verbal Reasoning and Analytical Writing.

Answer 2

I'm a philosophy professor at a research university, with a PhD in philosophy. My experience, if you couldn't guess, is that it depends. For philosophy of math, physics, logic, and, yes, even language (as it's practiced today), you need to know some (but not too much) recondite math. You also need to be proficient with it. That is, having taken the courses won't suffice. In the case of philosophy of math, for instance, you really need to understand Compactness, Lowenheim-Skolem, Incompleteness, axiomatic set theory, Inner Models and Forcing, even if you don't contribute to logic yourself. You also want to be familiar with non-classical foundational frameworks, like Quine's NF or type theory, and non-classical logics, including modal, intuitionistic, and paraconsistent logics. Finally, you should have familiarity with formal theories of truth, especially Tarski's and Kripke's, and will want to know something of 'mainstream' mathematical programs, like Langlands.

Nat is right that if you do 'old-fashioned' epistemology or philosophy of mind, you won't need to know much more than formal logic. But a lot of contemporary philosophy, even historical, assumes competence with technical machinery. The situation is really not analogous to Art History or English. This is witnessed by the many interdisciplinary math/philosophy and related PhD programs, like Berkeley's Logic & Methodology of Science (founded by Tarski), Carnegie Mellon's department, Notre Dame's joint philosophy/mathematics PhD, Pittsburgh's HPS, Maryland's Philosophy of Physics program, etc. It is also witnessed by the many scholars in philosophy departments with PhDs in math, physics, computer science, or a cognate field, like Tim Bays (Notre Dame), Hugh Woodin (Harvard), Scott Weinstein (UPenn), Harold Hodes (Cornell), Haim Gaifman (Columbia), Michael Potter (Cambridge), Steve Awodey (Carnegie Mellon), John Burgess (Princeton), David Wallace (Pitt), David Albert (Columbia), Joel Hamkins (Notre Dame), Shaughan Lavine (Arizona), and so on.

Philosophy is a strange subject: it encompasses so much. It's not very useful, then, to look at GRE scores for "Philosophy" PhDs. That will lump together students at Carnegie Mellon with students at New School for Social Research. "Philosophy" in each department a distinctive thing. Even within a single department, some will ally with humanities, others with sciences. The best way to gauge what math you need is to read the philosophy that interests you. If you want to understand, e.g., Kit Fine's The Limits of Abstraction, a work of metaphysics, you're going to have to know a fair bit of logic and be able to follow long, technical proofs. But that's not the case with, say, Bernard William's Ethics and the Limits of Philosophy. Both are brilliant philosophical contributions, by the way, and plenty of students can mimic the first, but not the second, or vice versa.