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I'm currently finishing a PhD in mathematics from a university in Europe ranked top 10 in the world. I've always been torn between philosophy and mathematics, and I'd like to give philosophy a chance (now that I have the "safety net" of a PhD in mathematics). Also, for personal reasons, I'd like to move to the US. It turns out, at least according to the various admissions webpages, that doing a second undergraduate degree in the US is not as easy as I thought it'd be (even disregarding issues about funding, etc). Hence, my questions are:

1) Why are top US Philosophy departments (e.g. Harvard, UC Berkeley, ...) reticent to admit students already holding an undergrad degree in a different subject? In general, is the rule "no admission for students already holding a degree" as strict as it sounds, or can there be exceptions, depending on the context?

2) In said top US institutions, how common is "by-passing" the second undergrad degree and being admitted directly to a graduate program in philosophy?

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    To expand on a brief comment in Pete Clark's answer, you should be wary of viewing your PhD in mathematics as a safety net. In the US, you could probably always use it to get a job teaching low-level courses under poor conditions. But if you want a tenure-track job, you need to publish, teach, give talks, and participate in the mathematics community. If you set this aside for years to focus on philosophy, you'll damage your mathematics career in a way that would take further years to undo (while maintaining a viable math career would take a lot of time away from your studies of philosophy). – Anonymous Mathematician Jul 19 '15 at 21:04
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    @AnonymousMathematician I agree. The PhD being a "safety net" in my mind means that if I embark in this philosophy adventure and it doesn't turn out well (after, say, 1-2 years), I'd still have some chance of finding a decent job (not necessarily in academia nor in the US, though). – protagoras Jul 19 '15 at 21:10
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    Why do you feel you need for additional qualifications in philosophy? Why not just read and publish philosophy, as a person with a mathematics PhD? This sort of thing is not uncommon, especially if you are interested in publishing in a related area (i.e. philosophy of mathematics/science). A PhD is really just a ticket to an academic job, it doesn't force you to stay in the same narrow sub-field. – user1220 Jul 19 '15 at 22:59
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    No offense, but why do you need another university? It is like trying to re-do the high school years, and insisting to do it in a elite boarding school - at the age of 40. – Greg Jul 20 '15 at 6:57
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    @protagoras Your question is why top universities consider your career plan meaningless, and I answered. – Greg Jul 20 '15 at 22:46
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Maybe you do not know what an undergraduate degree at an American university means. You do not study, say, philosophy, you "major" in it. Depending on the university and your own choices, you may take anywhere between 1/3 and 2/3 of your courses in the area. You will also be taking basic courses on writing, social sciences, very broad courses in the sciences, and so forth, that to many well educated Europeans would feel like high school all over again. The idea of getting a PhD and then coming back for an undergraduate degree would be regarded as crazy by most American universities. I agree.

In general, yes, a PhD in subject X should get you out of having to do an undergraduate degree in Y and you should be thinking of enrolling directly in a PhD program in Y. But forgive me while I pick apart your question a bit.

I've always been torn between philosophy and mathematics, and I'd like to give philosophy a chance (now that I have the "safety net" of a PhD in mathematics).

The idea that someone regards a PhD in mathematics as a safety net is very amusing (or bemusing?) to me. If you have always been torn between mathematics and philosophy and got a PhD in mathematics, surely you must have devoted some serious study to philosophy, right? If the answer is "no" then I don't really understand the situation. If the answer is "yes" then I hope you have been exploring the connections between mathematics and philosophy, which is something appropriate for someone considering doing multiple PhDs.

If you want to study an area of philosophy with connections to mathematics (most obviously, philosophy of mathematics, but there are many other areas of modern philosophy which draw upon mathematics), then having a PhD in mathematics at a "world top 10 university" should be a tremendous advantage. The level of mathematical expertise you bring will probably be superior to that of some of the faculty you would be working with. Thus at the very least you should be looking for some kind of specially worked out, accelerated graduate program. I would encourage you to also look for postdoctoral and other temporary faculty positions: these can be places to transition from one field to another.

The desire to start up an undergraduate program after having received a PhD at a top place is frankly a bit worrisome to me. You've probably spent your entire adult life as a student, roughly 10 years of it. You now want to turn around and spend another roughly more 10 years as a student? That's about a third of your adult life. Don't you have other things that you want to do?

Added: Although I do not think you should consider enrolling in an undergraduate degree program, if you really haven't taken any courses in philosophy at the undergrad level [by the way, in the US it would be really screwy for someone to have a lifelong interest in X and Y, do a PhD in X and never take an undergraduate course in Y; but maybe less so in Europe] then I think you should enroll in -- or audit, or whatever -- at least one or two as a non-degree student. You just need some amount of assurance that academic philosophy is anything like what you think it is.

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    For that matter, my university (in U.S.) would not admit as a Ph.D. student anyone who already has a Ph.D., period. Doesn't matter about subjects. – paul garrett Jul 19 '15 at 20:45
  • @Pete L. Clark Thanks for the reply. I've indeed taken plenty of classes on philosophy (as an auditor) while an undergrad and grad student in maths, and I regularly go to seminars, so I have decent idea of what academic philosophy is like. Also, ideally I'd skip this all "do a second undegrad degree" business, but I have no clue on how admissions work for graduate study if one has a background in a different discipline, so I had to ask. – protagoras Jul 19 '15 at 20:54
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    @paul: That policy seems a bit much to me, but it's good to know that some places have it. (Mine doesn't, not even for a PhD in the same subject. I checked.) The OP should definitely ask around. protagoras: Good. Skip it. – Pete L. Clark Jul 19 '15 at 20:56
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    @Pete L. Clark Also, do you think doing a postdoc in a different discipline is actually feasible? (That'd be the best option.) – protagoras Jul 19 '15 at 20:57
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    @protagoras: I think it would depend upon the particulars of your situation and training, but it is not definitively unreasonable given the information you've provided. – Pete L. Clark Jul 19 '15 at 20:58
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The counter-question is, why do you want a second degree in philosophy?

Part of what a PhD teaches you is to teach yourself things. Many a PhD-holding scientist has branched into other related fields through self-study and connections/collaborations with other scholars, rather than doing a formal course of study. I know a neuroscientist who teaches in a biology department; I know several psychologists who have expanded into work in other disciplines, including applied statistics, political science, economics, business, and sociology. I've also come across the websites of philosophers who do research at the intersection of the physical sciences and philosophy, or at math and philosophy. Sometimes that expansion is formal - they take classes or earn a second degree (usually a master's) in the field, but usually it's more informal.

So I think first you need to figure out what you want to do with the philosophy degree. Is it just to satisfy your curiosity, or do you want to do significant research at the intersection of the fields? Both of those goals can probably be accomplished without getting a second bachelor's degree in philosophy. By your own admission you already have significant coursework in philosophy; you could continue to take classes as a non-degree student until you have the knowledge you need to do the research you want. You can start a research collaboration with philosophers interested in that intersection. I don't think philosophy does postdocs the same way that the sciences do, but you might find a humanistic postdoc that will take you on. An example are the Mellon postdoctoral fellowships that are at several universities; you'd have to make a really compelling case, though, because they are reserved for people with PhDs in the humanities and humanistic social sciences.

To more directly answer your original questions:

1) I doubt most people could answer that. The simplest reason is probably because they want to focus their resources on first-degree students. They may also reason that second-bachelor's students are less serious and invested (fair or not). There are also financial reasons; most schools and U.S. federal programs do not provide financial assistance for second-degree students. You'd probably have to ask the schools themselves for the exact answer. (It's not a departmental thing, either - the entirety of Harvard College doesn't admit second-degree students, not just philosophy). The rule is usually strict with no exceptions.

2) In very general, it's pretty common. To be more specific, though, you'd have to have an undergraduate preparation they deem sufficient to begin the graduate program. So if you didn't have an undergrad degree in philosophy but you'd taken 7 or 8 undergrad philosophy classes (and did well), and you also had a mathematics BA and indicated you wanted to study the intersection between mathematics and philosophy, an MA program in philosophy might admit you. However, very few PhD programs will take you, because most U.S. institutions have a policy whereby they don't admit people to do second PhDs. If you already have a PhD in any field, you typically can't do another one very easily.

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