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In particular, besides the more obvious fact that your courses and qualifying exams are going to be in different fields, are there any subtler reasons to prefer one department over another? Some possible refinements of my question are:

  1. Is there a significant difference in resources available to logicians in one department over another? (Perhaps philosophy departments tend to be more well-funded than mathematics departments, or perhaps there are more fellowship opportunities available to students in math departments.)
  2. Are there any significant differences in academic expectations? For instance, are there perhaps greater expectations to publish more frequently when in one department versus the other, or does perhaps one department take its language requirement more seriously than the other?
  3. Related to the previous question, are there any significant cultural differences between the two departments?
  4. Does choice of department have any impact on postgraduate academic job prospects?
  5. Is it possible at all, once you are in one department as a graduate student, to switch to the other?

I ask these as general questions, though here's some information about my personal circumstances if they make any particular answer more apparent:

I am currently a third-year undergraduate math major interested in graduate study in logic. I have the dubious and unsavory distinction of having a departmental GPA lower than my overall GPA, but outside of a course in ethics and a (spectacular) course in philosophical logic, my philosophy-related coursework has tended to focus more on continental philosophy. In short, it is not clear to me that I do better in one field than another (I have done well in all my philosophy/philosophy-related classes, while I have done either spectacularly well or spectacularly horribly in my math classes), and I deeply enjoy learning about both. I have also heard rumors that philosophy graduate admissions tend to be, by virtue of self-selection and small entering class sizes, much more competitive than math graduate admissions, but given my own record, it's not clear that admissions to one department will be easier than admissions to another (though this has little to do with whether one type of program is better than the other).

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    One point (a little short for an answer): from what I have seen, maths departments are typically reluctant to hire people whose PhD’s are from philosophy departments (even when the candidate has strong and clearly-documented mathematical background), while philosophy departments are more often willing to consider people with maths PhD’s for jobs in logic. – PLL May 17 '16 at 23:49
  • Berkeley has an interdisciplinary logic program where you can either be advised by a mathematician or a philosopher. You still need to pick a side eventually, but you have a bit more time and it's easier to switch. – Noah Snyder May 18 '16 at 13:03
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I am a math professor, and from what (very) little I know about philosophy, I would expect the academic cultures to be very different. Once (as an undergraduate) I went to a lecture given by a leading contemporary philosopher. During the Q&A period, one audience member remarked: "I think your point of view is obviously wrong... " and proceeded to elaborate at length! I've never seen this in mathematics. Despite what you've seen in movies, I've found academic mathematics to be quite friendly and relatively nonconfrontational.

My understanding is that (perhaps unfortunately) philosophical logic and mathematical logic are considered entirely separate academic disciplines. I went to graduate school at a university with a strong mathematical logic program, and what my friends were doing was recognizably mathematics. I don't think I heard them talk about philosophy, at least any more than it came up during random conversations. I would recommend to you that you learn something about model theory, recursion theory, axiomatic set theory, the incompleteness theorem, etc. and see if this is something you enjoy.

I doubt very much that you could switch from one department to the other, although I don't have any firsthand knowledge of this.

I would certainly expect graduate admissions and the job market to be very different. My understanding is that the job market in mathematical logic is very difficult; indeed, I only know of one of my logician friends that is now in a permanent academic position. (That said, the others found good non-academic jobs.) However, my guess is that the job market in philosophy is still bleaker.

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    On the job market: I'm in philosophy, and to my understanding the market in the mathematical sub-branches of philosophy (philosophy of math, mathematical logic, etc.) is exceptionally poor. – commando May 17 '16 at 21:40
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    @commando The market's not great in the logical disciplines in (or even most pure branches of) mathematics now either, though it is possibly worse in philosophy. – Kimball May 17 '16 at 22:03
  • For what it's worth, "something about model theory, recursion theory, axiomatic set theory, the incompleteness theorem, etc." are what we covered in our philosophy department's logic courses. – Joshua Taylor May 18 '16 at 12:36
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    While we're talking about logic-related disciplines and employability, it's worth mentioning theoretical computer science (eg complexity theory) as a closely related discipline with relatively strong employment prospects. – Noah Snyder May 18 '16 at 13:13
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I'm a Philosophy post-doc in the US. Here's my two cents on the issue.

Are there any subtler reasons to prefer one department over another? . . . Does choice of department have any impact on postgraduate academic job prospects?

The primary reason to prefer one department to another would have to do with your plans post-graduation. If you want an academic job and your PhD is in philosophy, then it would be hard to be hired by a mathematics department. (I've seen it happen with philosophers of mathematics, but it's rare.) Since most professors end up teaching lots of introductory courses, which sounds more appealing: teaching lots of introductory calculus sections, or teaching lots of introductory philosophy? That's one useful way of thinking of it. If you want a non-academic career, the questions are a little different. My sense is that a math PhD is probably more "marketable" to industry than a philosophy PhD, but your mileage may vary.

Is there a significant difference in resources available to logicians in one department over another? (Perhaps philosophy departments tend to be more well-funded than mathematics departments, or perhaps there are more fellowship opportunities available to students in math departments.)

I would expect there to be more money for math than philosophy. Mathematics in the US is part of the prestigious STEM group of disciplines that attract lots of external funding. There is very little external money (comparatively) for research in the humanities.

Are there any significant differences in academic expectations? For instance, are there perhaps greater expectations to publish more frequently when in one department versus the other, or does perhaps one department take its language requirement more seriously than the other?

I don't think so. For a newly minted PhD in philosophy, you wouldn't necessarily be expected to have already published a lot. It's not uncommon for a new philosophy PhD to have one or two things already published or at least under review, however. My sense is that that is comparable with math, but perhaps somebody here can correct me.

Related to the previous question, are there any significant cultural differences between the two departments?

Philosophers can be very argumentative, and you'll be expected to know a whole range of stuff beyond just logic. There's a tendency among some philosophers to treat logic as basically trivia and not as properly philosophy. That's an unfortunate attitude. Likewise, I suspect that if you went to a math department, they'd probably regard the philosophical implications of logic as wooly-headed, poorly-defined problems that any reasonably confident mathematician could sort out in a long weekend, and that won't have any bearing on the real problems like . . .

Is it possible at all, once you are in one department as a graduate student, to switch to the other?

This would be administratively impossible at my university. You'd have to reapply to the different program, because of how funds for PhD students are apportioned by the university. There'd be nothing stopping you from taking a few courses at the other department, provided your chair and the prof whose class you were attending approved. But I think it'd be hard to do more than just a few classes.

. . . philosophy-related coursework has tended to focus more on continental philosophy . . . I have also heard rumors that philosophy graduate admissions tend to be, by virtue of self-selection and small entering class sizes, much more competitive than math graduate admissions . . .

Admissions to philosophy programs right now are really competitive. Let me also say that this is because of the terrible job market in academic philosophy. Prospective graduate students who want an academic career are advised only to apply to very selective programs in the US. Something like 50% of the tenure track jobs in philosophy the last few years have tended to go to the top ten ranked programs in the field. (Look at the Philosophical Gourmet Report for information on these.) Be very careful about attending any philosophy PhD program outside the very top! The result of this is that admissions at these programs is extremely competitive.

A bit of bad news though is that all of those programs are resolutely "analytic" programs rather than "continental" ones. There are good programs in the US where you might be able to find a faculty member or two with interests in continental figures, such as UChicago, UCBerkeley, and Columbia. If you're interested in admissions to one of these programs, then you'll need to make sure that at least one of your letter of recommendation writers is a famous enough philosopher to be well-known by the faculty of those institutions. There aren't a whole lot of "continental" philosophers who would be well known at those places, which may make your path a lot tougher.

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    As a mathematican, I would not regard the philosophical implications of logic as wooly-headed, and I would certainly not claim to be able to say anything about it in a long weekend. "Poorly defined", perhaps yes in a way --- but only in the sense that you can't make it absolutely precise. That doesn't mean that we don't respect it, but it does mean we don't consider it part of our discipline. – Anonymous May 17 '16 at 19:36
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    @Anonymous, I kid, I kid . . . – shane May 17 '16 at 19:37
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    Sorry, maybe I take things too seriously. :) That said, maybe I should add that (unfortunately) many mathematicians also regard logic as being a not-very-important part of their discipline. When mathematics job ads specify what specialty they are looking for (which not all of them do), it is almost never logic. – Anonymous May 17 '16 at 20:52
  • @Anonymous, Interesting. How often would you say do mathematics ads specify some field of specialization? In philosophy, it is reasonably common for jobs to be "open" in the sense that in principle everyone can apply for them. My sense is that those are the jobs most philosopher-logicians would apply for, since we have very few ads specifically for philosophy of logic (on the order of 1-2 per year, I would estimate). – shane May 17 '16 at 20:54
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    I'd say roughly half of them call for some particular specialization or range of specializations. – Anonymous May 17 '16 at 21:13

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