Path from humanities undergrad to math PhD

Question

This is out of left field. I have recently been realizing that, since my favorite classes are math, my best friends have all been future mathematicians, and I love nothing more than thinking about math, perhaps I should look into a PhD or career in mathematics. I just finished my second year at St. John’s College in Maryland, a fairly selective liberal arts college with an integrated Great Books curriculum. I will graduate with a double major in Philosophy / History of Science and Mathematics. I was wondering what it would take for me to get into, and succeed in, a math PhD program, or even a well-regarded one. I have a GPA of 3.9 as it stands, with all A’s in my math courses, but there are no opportunities for modern mathematical research at the College.

What would it take if I wanted to pursue a career in math? What could I do, starting now? I think this might be a crazy question, so I’m okay with hearing crazy answers. Thanks.

Answer 1

What you need to be admitted to a mathematics program is to meet the admission requirements of a PhD program. These will vary from program to program. Some might require math degree, some might have set of courses that are required. But you just have to look.

I would caution you about drawing the line from "I enjoyed math courses" to "I should do a math PhD."

What level were the courses at? Were they computational classes (e.g. calculate this, find the derivative of that) or proof classes? Did you get to abstract concepts like vector spaces, groups theory, topology, etc? Junior math classes are radically different from senior ones, especially ones intended for math majors.

More importantly, are you interested in inventing new math? There's more to a PhD than learning a bunch of math.

Math is a field where my impression is you are expected to have a fairly specific technical background. You might be able to survive without thos background, but you'll end up having to speedrun a full undergraduate course in the first couple weeks of a graduate class.

Something to consider might be a second degree program or a post-baccelaurate certificate that would let you get all the required math courses you need without doing an entire second 4-year bachelor's.

Answer 2

I'm an SJCA graduate with a PhD in physics. I graduated with your dean.

The SJC education does not provide the math education you'll need to enter a PhD program. You won't know the language, you won't know the tricks, you won't know how to do problem sets (n.b. put a box around your answer), or even how to take notes in a lecture (n.b. take notes). But you can learn all of these. As others have mentioned, you can transfer to another school, you can take classes somewhere else, you can do a post-bacc (https://www.mathalliance.org/our-partners/post-baccalaureate-programs.html). No one knows the cost/benefit ratio of these options better than you.

I had an outstanding high school education and knew enough integral calculus, physics, and programming to jump into advanced undergrad classes in diff eq., computer science, etc. This gave me a massive advantage over the other SJC students I know who tried advanced STEM degrees. Plenty try.

Committing to a PhD is a big decision. Have you met a math PhD? Have you talked to a university math professor? Get out there and network in person. The post-bacc link above looks like a great place to start if you're committed to staying at SJCA.

Answer 3

I graduated from St John’s College (Santa Fe campus) in 2013 and got my PhD in math (statistics and data science specialization) without any other coursework in between. It is absolutely doable, just uncomfortable! Prepare yourself to feel unprepared, and recognize that although the SJC math program doesn’t teach you how to do ODEs or linear algebra (this one was probably the biggest gap, catching up on linear algebra) it does teach you a lot of the fundamental number and set theory that you need to succeed, to say nothing of the critical thinking skills you’ll also need. You can do this if it’s what you want to do, and you’ll never be or feel totally prepared for the next step, just don’t quit after a few bombed exams (par for the course) and eventually you’ll look back and realize you figured it out even though it may currently feel like a long shot.

Answer 4

To address OP's situation directly: it will be very difficult for you to go straight into a math PhD program directly after completing the St. John's program.

If you were at any other liberal arts college, the solution would be to start taking a few math classes, and then eventually changing your major. But this doesn't seem to be possible in St. John's, where everyone basically studies the great books curriculum.

Of course, the St. Johns curriculum has its own merits, and you would learn lots of things outside of math. If you decide to finish your current program, potential routes to a math PhD might be (1) do a second undergraduate degree or (2) do some math-related masters program.

I would suggest you spend some time this summer studying mathematics, either through self-study, or by taking classes at a nearby university or community college. Also do some research (e.g by reading this website!) to learn about what a PhD in math is actually like, what the typical career pathway of a mathematician is like, and what career options exist for math PhDs (spoiler alert: not everyone becomes a math professor).

At that point, if you are still interested, you should seriously consider transferring to a different institution to finish your undergrad studies.

Answer 5

In the United States, most graduate programs don't really care what your bachelors degree is in, as long as you meet whatever prerequisites are required for entry into their program. The assumption is, I think, that you have already demonstrated (by earning a bachelors degree) that you are capable of learning, and that you are likely to be a competent self-learner. Whatever background you lack, you can make up quickly.

That being said, there are prerequisites for admission into graduate programs in mathematics. These are not universal, and there are lots of places where you might be able to get away with less, but I would expect that most places will want you to have done upper-division coursework in

• Real Analysis: This is essentially a recapitulation of introductory calculus, but with an emphasis on proving those results. When you took calculus, did you talk about epsilon-delta arguments? Well, you will now! For a lot of students who think they like mathematics through high school and the first couple of years of college, this is where they hit a wall and choose to pursue other things.

• Topology: Historically, topology developed as a way of abstractifying notions of "closeness" which show up in analysis. The modern field of topology does a lot more than that, however, and is concerned with a wide variety of questions which end up touching on a lot of different areas of mathematics.

• Abstract or Modern Algebra: This is the study of abstract structures on sets. The objects introduced in such classes are things like groups (a set with an additive structure), rings (a set with compatible additive and multiplicative structures), and fields (rings with a little extra structure).

If you haven't taken at least one semester of each of these courses, I do not think that it is likely that you will have much of a chance of being successful in applying to graduate programs in mathematics. On the one hand, you would be starting a year behind anyone else in your cohort; and on the other hand, without these courses, you have not demonstrated that you know what research level mathematics might look like, nor have you really shown that you are interested in that kind of mathematics.

As generic advice to someone who has just finished their second year of college (at an American institution), I would suggest that your best bet is to plan on taking whatever your institution considers their "bridge course" to higher mathematics (if you have not already). This might be a course in "discrete mathematics", or it might actually be called "transition to higher math" or something similar. Generally speaking, the intention of such a course is to take students who are planning on majoring in mathematics, and preparing them for the rigour of upper-division classes. Presumably, you have an advisor who can help you to figure out what this class is at your institution.

If no such class is offered, the courses I listed above are likely where you need to go next but, again, talk to an advisor or someone in the mathematics department who can give you advice for your institution.

Finally, I assume that your institution allows you to choose a major and a minor. If you are already heading into your third year, you may not really have time to complete a mathematics major (depending on what coursework you have already completed), but you likely still have time to pursue a minor in mathematics. If you are serious about graduate school in mathematics, I would suggest that you consider this option.

Answer 6

To be honest, I don't have a very positive opinion of SJC's Great Books curriculum, but that's out of the point, so I'll not discuss that. What is on point is that, from the perspective of doing STEM as a profession, SJC's purported "science education" pretty much misses the point. No one really cares if you've studied Euclid's Elements from cover to cover, or have read Newton's original writings on calculus; those are really only of interest to historians of mathematics and not mathematicians, so the math & science component of SJC's curriculum is really about the history thereof.

Mathematics and science is all about understanding the important ideas and, more importantly, how to use them. (Unlike in the humanities, the "original text" is usually a terrible source to learn these ideas.) It seems that SJC's curriculum fails miserably in this aspect. There is no modern mathematics in the curriculum! (modern analysis, linear algebra, abstract algebra, topology, etc.) There is a bit of problem solving involved, but on such a low level (roughly equivalent to what I learned in junior high and high school) that it could by no means be sufficient preparation for graduate study in the sciences. The SJC's curriculum seems to foster a very specific skill set and a very specific type of person, and it doesn't align at all with the skills and traits required for a professional mathematician or scientist or engineer or so on.

As others said, you can dual enroll in nearby institutions like the USNA or UMD. But I don't think this is worth the hassle and the extra effort (the SJC curriculum entails a lot of work already).

Since you know that you are into mathematics, it is unlikely that SJC's education is exactly the one you need: if you want a liberal arts education (which I had and loved), there are hundreds of SLACs and/or smaller, teaching-focused universities, with excellent math departments! Transferring into one of those seems like a very good idea.

And, if you like the rest of your curriculum, it's not that you can't pursue these subjects elsewhere. While those institutions might not follow the exact same "Great Books" curriculum, their course offerings will generally cover the material taught at SJC, with similarly small class sizes, close-knit discussion based classes, and highly qualified professors.

Answer 7

Granted this was decades ago, but I had a friend who graduated from St. John's in Annapolis with a degree in mathematics, and though he did not, he certainly could have gone on to a PhD program in mathematics. Looking at the program (https://www.sjc.edu/academic-programs/undergraduate/subjects/mathematics), the emphasis is certainly more classical (geometry, set theory, number theory, mathematical physics) than most other programs, but ultimately if you're pursuing a PhD in mathematics, the undergraduate degree is more about learning how to address and solve problems...how to work your way from "huh?" to rigorous proof. Very few students enter graduate school without what some would consider "holes" in their undergraduate mathematics resumé. My undergraduate program did not include classes in number theory or topology; other students in my graduate classes had different deficits.

The lack of "modern mathematical research" opportunities can be remedied in multiple ways. At St. John's you are less than a mile from the US Naval Academy, which has a large, active mathematics department. Understanding the security situation is different there from most institutions, you may be able to attend their research seminars, and get to know people there...a faculty advisor at St. John's may be able to facilitate. University of Maryland and UMBC are both within a 45 minute drive, as well. You may be able to hook into these institutions for activities like the Putnam Exam and the Mathematical Contest in Modelling, if St. John's does not host.

I'm not saying it will be easy, but as St. John's is the environment in which your interest in mathematics seems to be blossoming, I'd be loath to recommend you leave just for a potential future that time may steer you away from.

UPDATE

I'd like to address several criticisms of my thoughts that have popped up in the comments. The first is that the St. John's curriculum does not prepare a student for modern mathematics. The world may have changed in the decades since I went to graduate school, but I don't remember jumping into cutting edge research during my first year classes; rather it was about strengthening skills and deepening knowledge in core disciplines. One can learn to study mathematical texts and write proofs from 400 year old mathematics as well as from 100 year old mathematics. I agree that this is a harder way to go about it, and the student will enter with knowledge deficits. But that doesn't mean impossible, and there will be skills that students learn in the St. John's curriculum that will put them in a better position than many others to close their knowledge gap.

The second criticism is that the student will need to leverage outside resources. Well, yes, but isn't that true for most other mathematics programs? At how many schools is a student who simply takes the classes required for a mathematics degree prepared for graduate school without some level of external preparation? Out-of-class research, enrichment activities like the Putnam exam, and attending seminars are always encouraged for students looking to attend graduate school. Again, this will be harder at St. John's, but with at least three active research departments within an hour's drive, it is not impossible.

Several commenters, and other answers, suggest that the OP should just change schools. And if the OP were attending a smaller state institution or liberal arts school I would likely agree with that advice. But St. John's is such a different environment from other schools that such a transition is not risk-free. My friend who attended St. John's told me several stories of some of his eccentric fellow students, and admitted that he himself probably would not have been successful at a more traditional institution. My answer is attempting to meet the OP where the OP is, and offer strategies that the OP can use to increase engagement with mathematics, without second guessing a personal choice to attend an unusual institution. I freely acknowledge that this will be a significantly harder path. But it is a path that can absolutely lead to a fulfilling career in mathematics, because I know someone for whom it worked.

Answer 8

You would not the first person to do this. This has been done before. One example If know of is a professor I knew in grad school. She was an English major and working as a technical writer. She realized she liked math so she took a bunch of leveling courses through a local college. Then, applied to several programs and got into an Ivy league school. She realized she liked applied math rather than pure math and went with that program. After getting her PhD, she was at a national lab and is now a professor.

Answer 9

You have time remaining in your degree? You can take mathematics courses if you wish?

My advice: Take the math course in abstract algebra; take the math course in real analysis. After you do that, if you still want to do graduate work in math, then go for it!

Those course are what math admissions committees will like to see in your record. If you have not done those courses, I do not see how you can "know" that you want to go into math.

Answer 10

You do not say exactly what is to your mathematical taste, but you might consider computational mathematics. This might be regarded with scorn by some purists, but you would be following in the footsteps of Newton, Euler, Lagrange, Legendre, Fourier and numerous other giants. There is still ample scope for imagination and creativity, employment and funding opportunities are good, and access to the Naval Academy could provide great connections to departments that treat the topic seriously.