Working towards an academic career that lets me do research in two (largely) unconnected fields—is this even possible?

Question

In undergrad, I did a double major in pure mathematics and linguistics. I ended up loving both fields, and wanting to pursue an academic career. The only problem is that I couldn't decide between them. There is certainly some overlap between the two fields, but not especially much, and this makes interdisciplinary work somewhat rarer and the choice between the two fields feel much more discrete. After a few years of thought, I started to feel like the only option that would really make me feel satisfied is to at least try to seriously pursue research in both fields, even if this project turns out to be a total failure. I have gotten cautions from just about every source telling me this is not very plausible, but I am somewhat dedicated to attempting it anyway, in at least some capacity.

After graduating, I applied to grad programs in both fields. I got into a PhD in linguistics at a top university, and better yet, it offered the chance to pursue some math-heavy research topics. I ended up accepting this offer after much deliberation (I made a post about it here a few days ago!) and it looks like that's where I'll be headed in the fall. However, I've been second guessing my choice, because I feel like there is more room for someone with a math background to find their way into linguistics research than vice-versa, and I am really deeply loath to give up on the possibility of doing any pure math research.

With all that said, I'm asking everyone here a very open-ended question: what steps would I go about taking to make this pipe dream a reality? Or at least, to approximate it as closely as possible within the constraints of time and institutional pathways? Any thoughts are appreciated.

Answer 1

Warning: this answer is primarily just my opinion, based on my own experiences.

In general, if you're interested in doing research that consists of "applying mathematics to X", I believe it is almost always better to focus on studying mathematics first and X second. I will give two (certainly debatable!) reasons for this:

1. The most difficult part of understanding "mathematics applied to X" is almost always mathematics, not X. So getting a formal education in mathematics and then studying X on your own is more helpful than the reverse.
2. A mathematics degree is often more versatile, so if you don't find a job in the narrow subfield you hoped for, it is likely to provide you with more alternatives.

I received this same advice when I was considering options for graduate school, and in my experience it turned out to be correct (though it is impossible to know where the alternatives would have led). Of course, I may be biased as I am an applied mathematician.

Answer 2

I would be very wary of trying to pursue an academic career in two fields unless you can find a way to leverage both in some interdisciplinary way. The amount of work required in the current academic market (especially at the PhD and postdoc stage) simply doesn't allow you to split your efforts between two subjects, even with "exceptional dedication" - there are only so many hours in a day, and everyone around you will be exceptionally dedicated as well as fully focussed on one field.

However this needs not mean that you abandon one field. Things you can do:

1. Cultivate "consultant collaborations" - contacts with people in the other field whose work has interdisciplinary leanings but lacks the contribution of an expert. This would allow you to maintain a network and some academic footprint in the other field while you concentrate your primary research programme in a single field. I've seen this work well for statisticians, programmers, physicists (my own field was too far from linguistics so I don't have any direct experience in that direction)
2. Remain active in a non-academic role. I knew several biologists whose academic work had turned entirely molecular, but who kept their interest in ecology/nature (often the thing that had drawn them to biology in the first place) through involvement with environmental and conservation efforts. I still do some science outreach, although I've not done anything to do with the research I talk about in years.
3. Park it for a while and go back in a few years. One of the functions of a PhD is to allow you to sharpen your focus into a topic to the point that you can add something genuinely new to existing knowledge. Some degree of single-mindedness is part of the package. Past the PhD stage and especially once you start establishing your own research trajectory, identifying where novelty may lie becomes more important. At this point, having a few years of focus on your primary field under your belt, you can turn to your other field for inspiration.
4. Interdisciplinary work. I've kept this last because, although it's the obvious choice, I would only recommend it if you find the actual interdisciplinary questions interesting, rather than a way to keep sitting on the fence indefinitely. Otherwise you risk committing to work that is neither fish nor fowl and not quite satisfactory in two directions.

Answer 3

Why not apply to a computer science department and do research in natural language processing. This is math + linguistics, and you'll have plenty of options afterwards in either research or industry.

Answer 4

As a linguist with computational interests, I think that the possibilities before you are endless. There are many directions you could go in applying mathematics to language, both in the applied and theoretical realms.

Pace David Ketcheson's answer, I tend to think that linguistic expertise would run out before mathematical expertise would. That's because computational linguistics is still in its relative infancy. The math doesn't have to be very advanced before you'll find that no one has applied it to linguistic problems yet—or at least not in a compelling way.

(Now, if by doing computational linguistics one means applying a bunch of machine learning algorithms to data that happen to come from speech corpora, then yes, in that case mathematical ability might become the limiting factor.)

Answer 5

I'm writing this from the perspective of ecology, which is a very interdisciplinary field, and with complete and total ignorance about linguistics, let alone how math plays into it.

In ecology, we have a lot of people whose research primarily entails taking existing tools and theorems and applying them to answer questions about different systems. But we also have a lot of people that do less applied work and more theoretical research (in general with the hope that their work will hopefully lead to new tools for the applied folks). I enjoy mathematics as well and have been involved in at least one project that took a fairly old mathematical topic that was basically unknown in my field and developed a whole new set of tools for ecologists based on it. Presumably, there are tons of other mathematical theorems unknown to ecologists just waiting to be applied in some way.

As a soon-to-be Ph.D. student, you're in a good position to do the same type of thing for linguistics; expanding the applications of mathematics for linguists. Now, that's not quite what you seem to be hoping for, which appears to be researching new mathematical theorems, but here's the thing, eventually you could end up in a situation where you start to hit the limits of the mathematical topics you've been applying. This means you could be in a situation where you have to start developing new mathematical theorems to answer new questions about linguistics (or maybe just improving on existing answers). It wouldn't be the first time mathematical theorems have been inspired by real-world questions.

Now, you're getting to do what you want, and you have whatever funding opportunities linguistics offers to back you. The caveat is that it may take a long time to reach this point. But, as I mentioned before, I'm also completely ignorant about linguistic research, so maybe this isn't as realistic as it may be for other fields.

Answer 6

There certainly are some people who have made outstanding contributions in one area after having done a PhD in some other. For example, Harish-Chandra did his PhD in Physics and then went on to do outstanding research in Mathematics which is of some interest to Physicists as well. So there are exceptions to David Ketcheson's answer and you need not "give up" on the possibility of learning/doing Mathematics after joining a PhD programme in linguistics!

The principal difficulty is "keeping up" with Mathematics while doing a PhD in linguistics. Others (like Ottie) have pointed out various ways of achieving this.

A juggling analogy may help you plan your approach. The first task with a ball that one learns is to throw one ball up and catch it. Most people stop at that point. However, some people learn to throw and catch two balls by keeping one in the air at all times. A yet smaller number manage with three, and so on. You are trying to be in the smaller set. It requires practice, but is, for many people, not impossible to achieve. You need to juggle life, linguistics and mathematics which should be possible once you gain some confidence with the first two!

Most active Mathematicians are aware that there are areas of mathematics that they will never be able to study properly. So, do not worry too much about the fact that you will not do graduate courses in Mathematics. There is no harm in picking one topic in Mathematics that particularly interests you and then following up on it. One of the big advantages of the way mathematics texts are written is that self-study is possible. You will, in addition, be at a good university. Thus:

• You will have access to a good library and good internet services.
• You may be able to get some help when you get "stuck" (as does happen to all of us!) from people on campus.

Be aware that there will be naysayers.

Answer 7

I think the answer to what graduate degree to pursue is down to whether you ultimately want to be a mathematician or a linguist -- what is the focus of your interest, the mathematics or the linguistics?

Yes, I am suggesting that you will have to have a primary focus, a subject "hat", if you will, that determines the lens, perspective and methodology of your research, even when looking at the same research questions.

I am a pure mathematician now, but I studied both theoretical physics and mathematics and for the longest time could not decide what to ultimately focus on. Now, these fields are (from what I can tell) much more closely interwoven than maths and linguistics, but even then, there is a pretty clear distinction between the mathematicians and the physicists interested in the same things (even though the mathematicians will know a lot of physics, and the physicists will know a lot of maths). The two communities interact quite frequently, but have different ways of thinking about problems and are often interested in different aspects of the same questions.

I became a pure mathematician (geometry, with some string theory inspiration) after my first year of PhD, and I think that is about the latest feasible time to make this switch (without "losing" time). Realistically, it is very hard to do mathematics research that is recognised as such by other mathematicians if this is not what you do full-time (as always, there are exceptions.) And if you want to do pure maths research specifically, I do not think you will be able to do that if you are not fully in that field. (This may be less true in applied maths; I cannot really judge.) Modern maths research is extremely specialised, and at the start of your career, you'll have a hard time getting up to speed in one area, let alone multiple at once.

In your PhD, you will (by the nature of how research works as a beginner) focus on a pretty specific question in a pretty limited area. Yes, you'll learn a lot of general things, but your research will be specialised. And how this specialisation looks will, I think, be very different depending on whether the project is set and supervised by a mathematician or a linguist, even if they ostensibly deal with the same subject matter. People who make fundamental contributions to different areas of research usually do so later, broadening their expertise after having first specialised.

What you do not want to happen is that mathematicians think you're a linguist and linguists think you're a mathematician and ultimately neither field really engages with your research (or neither department/neither mentor feels you belong to them, as a student). (This is advice I was given when I was choosing PhDs, and even though I did not listen to it right away, it became clear that it's a valid warning.)

This all is not to say that you should abandon linguistics -- mathematics is not the ultimate abstract uber-science that everyone has to do. There is obviously huge value in specialising in a different science that merely uses maths as a tool, rather than building the toolbox.

So maybe the question is: Do you primarily want to

• build new tools for a broadly applicable toolbox and try them out a little on specific applications, or
• take tools and use them to really get into the weeds of a science question, with the focus on the question rather than the tool?

Ultimately, there is no one right career path, and every decision you make comes with the cost of of something else you do not do; that's simply how it is.

Practical considerations Re: funding & degree: Why don't you give the linguistics PhD a shot, if you do not have offers for funded maths PhDs? That way, you can see how you like it, think things over, and if next year you decide that you actually want to be a mathematician, you apply to maths grad school again?

Answer 8

I don’t have the reputation to upvote the other response or can’t put a comment on here, but someone I went to undergrad with did undergrad and master’s in linguistics than built a company around semantic search. I have no idea if he was a Ph.D student that dropped out. Then the company got sold to Microsoft and he became a mid/upper manager in Microsoft over several other data science projects.

I am also reminded of some science fiction (Asimov?) that described creating a way of rating language automatically to detect bias, misleading content or something like that. I would really like to be able to tell Facebook or Google to stop giving me search results or posts that represent highly political, fraudulent or controversial information. I want to see kittens and babies, not conspiracy theories.

Any system that quantifies language could potentially leverage math on the back end.