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What I had in mind with the "mathematical thinking" comment was that either formal or informal methods of proof, as taught in an undergraduate intro logic class, don't really go far enough to competently follow some philosophical arguments. As an example, suppose someone defines a concept and then claim that all instances of it have some property. You need to know how to prove that, and you need to know what a counterexample would be like, to engage with this. Drawing out consequences and coming up with counterexamples are two important things philosophers do. Naive set theory is a good way to train this: how to apply a definition, how to prove things using definitions and various methods of proof, how to find a counterexample, or combinations thereof, e.g., proving by contradiction by assuming there is a counterexample etc. That skill is important even if you don't do logic or philosophy of math. (That you come away knowing some set theory of course also is.)

 Joshua Knobe

Hi Eric,

Thanks for your engagement with this issue. I really appreciate this thoughtful response, and I'd love to have a chance to hear a little bit more about how you think we ought to go about dealing with these issues.

To begin with, consider an especially extreme option. A department could say that at least one course in logic is required for all graduate students but that courses in statistics do not count toward requirements in any way. My sense is that this extreme option does not make any real sense as a way to prepare students for philosophy as it is currently practiced. To put it in the terms Richard introduced, philosophers these days rely heavily not only on mathematical thinking but also on empirical thinking, and a proper graduate education needs to prepare them for the kind of thinking they are actually going to use.

However, as the discipline moves away from this extreme option, there are still plenty of other options to choose from. Do you feel like any of the remaining options would address the concerns you lay out in your post?

Eric Schliesser

Well, I raise different kind of concerns, but yes keeping the logic requirement in addresses some important ones. There is another concern that I did not mention, but Eric Brown did in response to my post on facebook: "Quite apart from its utility as a tool, formal logic is not just a traditional field of philosophical inquiry but also, arguably, the one in which philosophers have made the most progress"

On how to prepare philosophy graduate students in empirical thinking, that's a very interesting and important question. One could demand that one take courses in probability/statistical/empirical methods as an undergraduate before they are allowed to enter a PhD program. That is, not unlike some the prerequisites in medicine, you just stipulate they need to be taken somewhere during undergraduate life or remedial work before grad school. (Even in our rather non-technical MA program in political science, we demand something like that--of course, students can take some of our own poli sci methods courses, but they can also do these in other fields.)
One could also create hybrid degrees (not unlike HPS), where one demands a MA in an empirical field alongside the PhD.

The nice thing about the two options just mentioned is that you can keep the philosophy curriculum in tact without sacrificing some of the things that make it worthy.

But my own sense is that things are moving in the direction that you want within philosophy, but that as a consequence there will be increasing divergence of expectations and standards within it.

Robert Gressis

Departments could require that graduate students who don't meet certain formal requirements have to take an intensive summer course in formal methods before starting the program.

Eric Steinhart

Hi Eric, I've long supported having more math in philosophy curricula. As you may know, I wrote a book on math that philosophers can use:

More Precisely: The Math You Need to do Philosophy

It might be useful!
- Eric

Graham White

Well, I have a background in mathematics, but I don't really do philosophy of mathematics (or, indeed, philosophical logic); mostly I do metaphysics and the philosophy of action. Looking at the use of logic in philosophy from the outside, it seems to be really rather strange as a selection of logical topics: no proof theory, no higher order logic, no type theory, no category theory (all of these are areas of logic that are used a lot in computer science, for example, and that aren't obviously irrelevant to philosophy). Now one can, of course, say the usual things: finite time in a PhD course, too many prerequisites for quite a lot of this stuff, and so on. What this amounts to is the limitations of finiteness, and what follows from it: the essential contingency of any selection of technical topics to teach to philosophers. But what's strange about the philosophical logic done in the Good Old Days (the 50s to the 70s, or so) is that it looked, from the inside, as if it was based on a natural and obvious selection of technical material: that philosophers could be happy that they knew the correct technical material, and that no more was needed. This was a very strange set of affairs, and what might be an advantage in the present setup is that the selection of topics at least looks contingent from the inside. (The question about the fragmentation of the profession is, of course, another matter, and the present situation might not be advantageous from that point of view).

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