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Disclaimer: I'm an undergrad with very limited knowledge of how academia really works and what the atmosphere and culture is like.

I'm very interested in pure mathematics, and I think I'm pretty good at it. I'm currently an undergraduate in my university's Pure Math program. However, I'm also incredibly interested in mechanical engineering and entrepreneurship, and want the skills to be able to build things, to be involved in and lead engineering projects and design systems and products that will one day (hopefully) make the world a better place.

So I was thinking, maybe I should finish my pure math degree, then enter mechanical engineering and possibly work towards a PhD in that field. However, I would still love to be an avid contributor and active member of the pure mathematics research community.

In a hypothetical situation where I get a PhD in mechanical engineering, what would be the best way to get involved in the pure math academic community, to the extent of:

  • writing and publishing papers
  • contributing to the works of other established pure math researchers
  • potentially being a part of research groups or committees
  • etc.

Basically, I want to have a PhD in one field, and somehow be able to participate in the lifestyle of a PhD in another field sometime later in life. How might I accomplish this? Is there a generally accepted path academics take to achieve this, or is it frowned upon by academics to attempt to be an active member of two, divergent academic fields?

Edit: I wanted to clarify that I wouldn't get the PhD in mechanical engineering, only to then be an active member in a different academic community. Ideally, I would participate for a large portion of my life in mechanical engineering, and another large portion of my life in pure math.

I also wanted to point out that I recognize that getting two PhDs isn't a valid option, nor is it one I necessarily want to pursue. I also don't want to do a bachelor's degree and a PhD at the same time.

Finally, I would like to add that I am not in any form attempting to prognosticate my life; maybe in two years I won't be interested in mechanical engineering and entrepreneurship anymore. I'm just curious about the nuances of such a path.

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    Ambition's awesome! I love it when people wanna do crazy stuff. Just, it gets really personal too. If you're going to blaze your own path, exactly where you're going and why becomes a huge issue. So, exactly what do you want from all of this? What's driving you, and where do you ultimately want to end up? – Nat Jul 25 '17 at 18:46
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    My answer is too short for an answer: Isaac Asimov – pojo-guy Jul 25 '17 at 19:26
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    Not an answer since not general enough. But just due to the fields involved: a Masters (/professional bachelors depending on country) in Engineering lets you work as an engineer and that is all anyone normally wants from an Engineer. Where as because of the immense depth of knowledge for pure math, often it takes a PhD to become actually well grounded in enough information to really work in the field (i.e. even undergrad or masters math is just scraping the surface). So possibly a better direction is to get your professional engineering qualification then Phd in Pure math. – Lyndon White Jul 26 '17 at 3:13
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    +1 for Lyndon White: Engineering is somewhat special because it's a strictly regulated field; It's a legal issue rather than an academic one. – errantlinguist Jul 26 '17 at 9:38
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    I know mathematicians in mechanical engineering departments, and they all work (worked?) on things like finite element analysis (FEA). However, that is not quite pure mathematics, is it? Then there is stuff like discrete mechanics and variational integrators. – Rodrigo de Azevedo Jul 26 '17 at 9:50

10 Answers 10

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In principle there's no problem with that. Many researchers change their fields over the course of their career -- sometimes slightly, sometimes drastically. No one cares what your PhD says on it, just the quality of your work.

There are a couple of caveats. If you need funding for the new field, you may have trouble convincing funding agencies that you're competent to perform the work. And of course it can be hard enough to do good work even when you devote all your time to one field, let alone two (though on the flip side, using insights from Field A may give you a new view on Field B).

But the field your PhD is in, by itself, doesn't limit your activities.

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    To this excellent general answer I would add that, in particular, pure math is nowadays of tremendous importance in mechanical engineering. It is not the case of completely different topics as, say, Latin Literature and Mathematics. – Miguel Jul 25 '17 at 20:56
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    @Miguel can you name some specific examples of specialized topics in pure math (as opposed to standard things like differential equations) being important in mechanical engineering? This sounds quite interesting to me as a pure mathematician who is also interested in engineering and other applied areas. – Dan Romik Jul 26 '17 at 2:49
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    @Dan Romik This would be a question by itself :) but I would aim at Lie groups, Hamiltonian mechanics, and mathematical control theory... not easy to set the frontier between theory and applications. The numerical counterpart (arguably applied math) would go from symplectic methods to robotics. – Miguel Jul 26 '17 at 14:11
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    Generally I agree, but people may care what your PhD says on it if you're applying to be an assistant professor. – Ellen Spertus Jul 26 '17 at 15:38
  • @EllenSpertus Certainly it might, but in my experience the opposite was true. My PhD said Field A, I had spent several years as a post-doc doing Field B, and when I applied for faculty positions in Field A the feedback was often "Yes, nice work, but we're looking for a Field A and you're really a Field B." But faculty positions are so scarce nowadays that I can see being excluded for minor things like your PhD field just to reduce the number of candidates. – iayork Jul 27 '17 at 10:25
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The good news is that what you are thinking of doing is technically possible: if you try to publish a paper in pure math, no one will care if you have a PhD in math, mechanical engineering, Egyptology, or any other subject, or no PhD at all. The only thing that will matter is how good your work is. The PhD itself, or lack thereof, will not be an obstacle in any meaningful sense.

The bad news is that your question reflects a certain naïveté, in the sense that what you are thinking of doing will be extremely difficult to accomplish in practice, to the extent that only very few and rare individuals are talented enough to successfully develop and become successful at two parallel and unrelated academic interests. More specifically, if you don't get a PhD in math and spend the time instead developing a career in mechanical engineering research, it's quite likely that you'll lack both the time and the access to training resources (an adviser, graduate classes) that will enable you to reach a high level as a pure mathematics researcher.

The bottom line is that most people already find it challenging enough to become very successful at one academic discipline, that having the same ambitions with regards to two separate disciplines is a somewhat far fetched notion. It's great to want to pursue multiple interests, and I'd encourage you to keep studying pure math and doing your best to make a contribution to this area for as long as you have the time and passion to do so. But it's best to be realistic about how difficult it would be to do that as a side hobby.

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    Mostly agree, except for the first paragraph: "[...] no one will care if you have a PhD in math, mechanical engineering, Egyptology, or any other subject, or no PhD at all. The only thing that will matter is how good your work is." On the off chance that you actually believe this, I think you are far too confident in the honesty of peer review in practice. – enthdegree Jul 26 '17 at 2:19
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    @SSimon it's a matter of degree, not a binary, yes/no thing. Obviously in a purely hypothetical sense, anyone with an internet connection and a library subscription has access to all the world's information. However at a practical level, almost no one becomes a successful pure math researcher without going through the rigor and structure of a formal PhD program. This site has numerous questions in the flavor of "can I publish pure math research as an amateur?" and people are always jumping to offer the idealistic answer "yes, of course you can". While that's true in theory, I think ... – Dan Romik Jul 26 '17 at 2:57
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    ... it's important to be realistic about how difficult that would be, considering that even with formal training it's already very difficult to do this kind of research and be successful at it. – Dan Romik Jul 26 '17 at 2:57
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    @enthdegree I've reviewed many papers in my career and never checked or cared whether their author has a PhD, let alone a math PhD. No one I know feels any differently. So I stand by my statement, at least in the context of pure math publication. – Dan Romik Jul 26 '17 at 3:02
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    @SSimon you are reading many things into my answer and comments that simply aren't there. Where did I "tell him that pure math isn't necessary"? (And "necessary" for what?) Where did I say that publishers ask me to disclose my PhD field? In what way did I "misunderstand him"? Also, what on earth does quantum computing have to do with mechanical engineering? I simply don't understand at all what you're talking about here, sorry. – Dan Romik Jul 26 '17 at 10:10
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Speaking as someone who is an academic in mechanical engineering and who has had a fairly rigorous training in mathematics, I must say that although it is technically possible to switch fields to pure math, that's a very strenuous path to follow, and I honestly cannot see a more difficult career shift for a MechE PhD than moving from MechE to pure mathematics.

I know several people on a personal level who have done the opposite -- moved from pure math to MechE -- and that's a much easier and smoother transition. But I don't recall having met anyone ever who has done what you suggest, and honestly I don't even see how that would be manageable, even for someone who deals with pure math on a frequent basis, such as those working on dynamical systems and fluid dynamics. Meaningful contributions to modern pure mathematics requires a level of profound knowledge that is far beyond what a MechE graduate student has been trained for, and more importantly, you would have to relearn most, if not all, of the mathematics that you are familiar with, which honestly is harder than learning the "correct" way the first time.

Is it possible? Yes, Ed Witten moved from being a history major to a leading physicist to a Fields medalist. Is likely? If you were to move to any other field, I would say yes, but as for pure math, it is very unlikely to be a successful plan. Research in pure math is very different from research in mechanical engineering.

  • Maybe you are being too restrictive about what is pure mathematics? From the point of view of an Engineer, numerical analysis is pure enough and, yes, there are meaningful contributions from engineers, as long as they are willing to learn the needed mathematics. – Miguel Jul 26 '17 at 14:17
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It's entirely possible to hold multiple, diverse interests. The real skill would be for you to find a particular research niche that you're interested in and then find a way to combine your knowledge of maths with mechanical engineering to carve out your own little area of research.

There is no pre-designed route that academics have to travel down based on previous qualifications et cetera, but rather it's a matter of following your nose and doing what you want to do.

6

Find multidisciplinary studies or research. My graduate program had a large number of professors in it and I do not believe any of them held a PhD in the exact degree program, because it was multidisciplinary.

Luckily you are interested Mechanical Engineering, which has ties to many other research areas.

If you are intentional about the types of research you do (for example, how does pure math apply within mechanical engineering?) you will find many opportunities to be involved in both subjects.

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One can have a PhD in physics and publish engineering or chemistry papers. But, they need to have access to resources: library, peers, grants, equipment, etc. If their primary research field takes up too much time, their research in the secondary field can reach only the hobby level.

There is also a difficulty with being accepted in the other field if you are new and unaffiliated. If you try to publish something slightly controversial, or unusual, the reviewers might dismiss your work too soon with the comment that you don't know what you're talking about. It also happens to PhD students sometimes, and it happened to me. Sometimes the issue might be simply your different way of stating the problem.

On the other hand, I know a few cases of professors contributing in more than two subfields such as glasses and astrophysics, or condensed matter and string theory. And then there are exceptional people like John von Neumann. I have no idea what you would have to do to reach that level, though I think Napoleon said that every grad student carries a virtual Fields Medal in his pocket.

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In many fields related to social sciences, interdisciplinary collaboration is very viable and even encouraged. This is particularly true in newer, inherently interdisciplinary fields--education, public policy, international studies, information science.

But even in the stodgiest of social science disciplines, economics, there is a growing appreciation for interdisciplinarity. I know of multiple development economists who get research ideas from anthropology, from religion, from psychology. Kahneman even won a Nobel prize in economics, and coauthored prospect theory, which might be the most important econ theoretical development in the last half-century.

So in the social sciences there is definitely value to having an awareness of multiple disciplines, with some important caveats.

  • Conventional measures of academic success in the US (faculty job, tenure, promotion) largely depend on finding one research community, saturating it with your work, and becoming well-regarded by more established scholars in that field. You need one single "intellectual home." Always focus on speaking your "first language."

  • You should publish as much as possible in your intellectual home. However, if it is less established, then publishing in more prestigious outlets in more traditional disciplines could be OK, even preferable.

  • Your methods and theoretical approach might be driven by your primary field, particularly in the more established disciplines (econ, soc, anthro). Econ journals don't publish ethnographies; rarely would they publish a theory without some math-based model behind it. It's just how that discipline works.

  • Bridging your disciplines of interest is much much better than doing scattershot work in each. So, the corollary is, only take on projects that support your research trajectory in your primary field.

  • As @axsvl77 mentioned, collaboration is a good way to participate in your secondary field[s] without having to do too much of the intellectual heavy lifting. Unless you're a genius, you probably have time to become expert in one area while the tenure clock is running.

  • Reading lit in your secondary field[s] is great, though. It will help you do more meaningful work in your primary field and avoid the disciplinary siloing/reinventing the wheel problem that pervades academia.

IMPORTANT DISCLAIMER: Social sciences are what I know reasonably well. Other answers are much better for the original questioner's fields of math and engineering. I write this in the hope it's useful to those in non-STEM fields who may see this Q&A.

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What I read in your question is that you are interested in many thing. This is good! And you worry that a career in pure math will cause you to "miss out" in another thing that you are interested. What to do?

I think the other answers have neglected to mention on of the most common ways to expand your area of research: collaboration. It is very common for academic researchers to reach into other fields through collaboration - there are even interdisciplinary research groups and grants.

So a way to have your cake and eat it too, is that you start a pure math career, and then once you have tenure, you can branch out into new areas of interest. How would you do this? Maybe attend conferences about topics that interest you and meet someone who is researching something you are interested in, someone who needs help from a mathematician. This is extremely common - there is surely someone in your department who does this.

I have seen Physics faculty working on art projects, and engineering faculty in collaboration with a philosophy project.

(Note: I am still a grad student, so I have not done this yet. I'd like a comment confirming from faculty about this)

  • Exactly. Collaboration helps get over the "spreading yourself too thin" problem. Trying to lead projects in multiple disciplines as a young academic is probably a bit much, but forging interdisciplinary collaborations can be OK. In my field (a social sciencey inherently interdisciplinary field) it's actually highly valued. – Philip Jul 27 '17 at 18:01
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There are very good research career possibilities in multidisciplinary research. As an (applied) mathematician working in the field of biophysics, I am convinced that modern problems of natural sciences and engineering will need even more of novel mathematical developments than what we can presently develop.

One example: man-made structures still rely a lot on simple geometries, whereas living systems make a tremendous use of curved and fractal surfaces (think of lungs and brains). Mechanical engineering lacks efficient approaches to predict the properties of these structures. Mathematicians with a better than usual skill at differential geometry will have a great input for this.

Another example: inverse problems are very often encountered when focusing on natural/life science systems, ranging from weather to cancer. And this might also be useful for a posteriori understanding of the function of man-made structures. There is still a lot to do in this field.

So, as several other answers, I recommend a PhD in pure/applied maths on a topic for which you can foresee applications on current issues in natural sciences/engineering.

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You might just have to. Science is evolving fast nowadays. Many new fields appeared in 20th century and scientists had to switch because there was great demand (and riches and fame to be had!). There are many but I can think of all the new medical fields, game theory in social sciences, and, of course, computer science. They say 65% of children entering primary school today will ultimately end up working in completely new job types that don’t yet exist. Think of all the fields of science that 21st century will give rise to. Even if you don't switch, you will probably cross into several other fields. Fasten your seat belt, you will have a fun ride.

protected by eykanal Jul 28 '17 at 0:55

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