Can I become a tenure-track prof in one dept (biology) and teach in a different dept (math) with only one PhD?

Can I become a tenure-track prof in one dept (biology) and teach in a different dept (math) with only one PhD?

TL;DR: no.

Long answer:

I was for several years the chair of a mathematics department and during that time oversaw teaching assignments for the 200-odd courses offered yearly by my department. I can recall only one case in which my department allowed a faculty member from a biology-related department who did not have an advanced degree in math or a closely related area (eg physics) to teach a math class (I believe it was calculus for biology students). Even this arrangement raised some eyebrows — mine, and possibly other people’s — but the circumstances were a bit unusual so it was allowed to happen, for reasons I won’t go into.

With this background, I can say with reasonable confidence that what you are proposing is impossible. Perhaps you can reasonably expect to be allowed to teach a math class once or twice in your career if that’s something you have your mind set on doing, but as a regular part of your job, at a normal university? No way. Even setting aside your lack of qualifications to teach math (an objection which one can imagine you overcoming under some hypothetical, if rare, circumstances), the fact remains that departments hire faculty among other important reasons to fulfill their own teaching needs. If you are in department X, you will be teaching the classes that department X offers - that’s why they are hiring you; so it simply doesn’t make sense for a neuroscience department to allow you to regularly teach classes of another department, except under some extremely rare and unusual circumstances.

Now, if you get a second PhD in math, or publish several years’ worth of postdoctoral research work in neuroscience that can be plausibly described as serious applied math so that you can credibly start calling yourself a mathematician, then your plan might start making a bit more sense. In that case you would probably want to look for positions that involve a joint appointment between a neuroscience department and a mathematics department — such things are not common, but they exist. Your teaching load will then be split between the two departments, likely in roughly the same proportion as the proportions of your position that are assigned to each of the departments. It’s still not likely that you will be allowed to teach only math classes, but it may be not too far from what you have in mind.

Edit: you also wrote:

I would ideally like to have a research career in neuroscience and teach lower-division math courses.

Another thought that occurs to me is that your expectation that you will only teach lower-division classes is also unreasonable, independently of the discipline. Professors are expected to teach at all levels: lower division, upper division, and (where applicable) graduate; again, that’s sort of why you’re hired in the first place, and that’s what sets you apart from an adjunct or lecturer. You seem to want to do the high-level parts of a professor’s job - doing research, running a lab etc - when it comes to research, but only the most low-level parts of the job - teaching the sort of beginning math classes that your former university allowed you to teach with only a BS degree - when it comes to teaching. That’s simply not how it works: no university will waste a professor’s position on job duties that can be performed by someone with much lower qualifications than a professor. That would very obviously not be an efficient arrangement for the university, either economically or pedagogically.

If you don’t want to teach anything more challenging than the least challenging thing that there is to teach, you may want to ask yourself if you actually want to be a professor at all.

What are your research interests within neuroscience?

Dan Romnik's answer presupposes that you want to keep the two fields completely separate: you'll run a neuroscience wet lab as your day job and moonlight as a calculus instructor at night. For the reasons he outlines, this would be very difficult indeed, verging on impossible.

However, neuroscience is a giant, sprawling field and the edge between theoretical/computational neuroscience, physics, and math is nebulous (if that). People study Ising models and manifolds and dynamical systems for many different reasons and from many different directions. If you were interested (and successful) in the more mathematical parts of biology, a joint--or even primary appointment--in a math department would not be out of the question. For example:

• Nancy Kopell at BU
• Bard Ermentrout and the rest of the Mathematical Biology group at Pitt
• Sue Ann Campbell at Western's Center for Theoretical Neuroscience
• Andrey Shilnikov at Georgia State's Computational and Dynamical Systems Unit
• Konrad Kording at UPenn
• Larry Abbott, Liam Paninski, Ken Miller, and other at NeuroNex Columbia.
• the late David MacKay, whose book you should read if going down this road.

I don't know that you would necessarily need a formal mathematical credentials, especially for joint appointments (how would that even work?!). However, you would need to demonstrate your mathematical chops through suitable publications and reference letters. Depending on your current math skills, a postdoc, perhaps with someone like those listed above, might be one way to go about getting those.

Having only one doctorate isn't the problem. But you would find it difficult to find a university that had compatible needs. If you are tenure-track in Biology, that department will want you to do research in Biology (of course), but also to serve the students in that field. If you aren't doing that, you will have problems, and if you ask for such a position, they would most likely say no.

On the other hand, once you are tenured, you might be able to teach the occasional math course and still have everyone happy.

An exception to the above might occur in a small Liberal Arts College (in the US) in which both math and biology are under the same Dean and the teaching requirements were seen as more important than the research. But there will still be few such positions open. And having a neuroscience lab in such a place would be difficult to manage as the funding would be pretty much all on you to arrange. You'd be unlikely to have any time to spend on other things.

Nice to dream, though.