Undergraduate 3rd Year - Need some Academic / Career Advice

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Hi,
I needed some help/advice.

I am a Math Major at Dickinson College. I am also taking lots of courses in Physics and Philosophy. At this point, which is the fall semester of my junior year, I have:

- 3 more courses left to complete my Math major. I still need to take Abstract Algebra, a Senior Seminar on Topology/Complex Analysis, and a 300 level elective - I'm thinking of a course on Computability and Complexity. So far in addition to the basic requirements for the major up to Real Analysis, I've also taken some additional electives. I love most of the courses so far.

- 3 more courses left to complete a major in Philosophy. I kept taking Philosophy courses just out of interest - things like Philosophy of Mind, Modern Philosophy, and Logic, and turns out at this stage I only have 3 more left to major. I'm interested in two more courses: Metaphysics and Epistemology, so I'd only have the Senior Seminar left to complete a major - so I'm strongly considering that possibility.

- 5 more courses left to complete the Physics Major. I've taken classes up to the Upper-level Dynamics and Chaos class after the Sophomore year classes in Modern Physics and Mathematical Methods in Physics. I plan to take the Upper-level QM course next semester, the ElectroDyn. course next fall and probably the Thermodynamics course the spring after. That way I'd have taken all the upper-level courses - all that'd remain is a two-semester senior year project-based seminar.

That's where my dilemma is. I can technically fit in all of these and do a triple major - while still having 1 more room for an elective. But I could also choose not to add the Physics major, while still completing all the upper-level courses.

I never intentionally aimed for a triple major or multiple minors. It just happened in the process of selecting the most interesting classes each semester. I've since discovered that I like understanding the general principles and abstractions by which we can describe the world at a fundamental level - hence the love for Math, Philosophy, Physics (I also hold some interest in Computer science in which I'm currently taking a class since it's so powerful and useful in thinking about the world). I like understanding the theorems we apply to problems at a deep level so I like my Math classes. I like seeing them applied to spit out facts about the world in my Physics classes. Then I like seeing the broader picture in all these in my Philosophy classes.

So I'd describe myself as a theorist. As far as Physics is concerned, I'm happy to take the upper-level courses. But I definitely don't want to go to Pure Physics grad school or experimental Physics grad school -- if I do it'll either be in Theoretical Physics or Mathematical Physics. I do like conceiving of and reading about experiments, but haven't really liked my labs so far - it's just a lot of tedious handiwork and teamwork.

So I'm thinking - should I really do the senior, faculty-supervised, lab-based seminar? The senior seminar will add extra work to my schedule (although manageable - I'm used to taking many and difficult classes). Plus, as more of a theoretically inclined person does it matter? Does it matter to Theoretical/Mathematical Physics grad school programs, given that I would have taken all the important course? Plus, if I don't do so I'll have the opportunity to do my own independent study - perhaps on a topic I am interested in. Or I could do research in Math, or take extra electives like Dynamical Systems, or Combinatorics or even take a few more computer science courses such as in Data Structures and Algorithms. I personally think that these additional electives are really interesting. Perhaps I should focus more on these Mathematical topics, and differentiate myself and strengthen my application in that regard. Plus I like having a balance of courses where I am taking one Physics course per semester, rather than two.

The con of that would be that I do think that the senior seminar still does offer something -- some more lab project-based research experience, which does teach or hone in certain skills. Might offer a new dimension. Might also be impressive for grad school applications. Maybe these skills are crucial even for theoretical schools. And lastly, I'm not sure if not completing the seminar, and hence not completing the major, will cause regret in the future? What kind of regret? Well, first, that of not completing the major with only 2 seminar courses left and after having completed all the upper-level courses. I'll be saying that I'm a Physics minor despite having taking pretty much all the upper-level courses. Will that stop me from Physics-related jobs in the future, or will it make me less credible for talking about Physics? I've always been interested in Physics since my childhood. A major seems like an identity of sorts. Completing a major now would be easier than say, having to find a way to do it later in case there's regret.

In an ideal world, I'd take an extra year to complete it, or maybe in an ideal world I wouldn't need to bother about it, but given reality, I wonder how I should think about this potential additional major. Maybe I should push for an extra year to graduate? idk.
 
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  • #2
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Your dilemma is not unlike the dilemma we all face in college to some degree about whether we've chosen the correct major, taken the courses we need vs those we want to take vs those we took but hated afterward.

The danger here is becoming ADD in your pursuit of multiple degrees and not spending enough time on the degree you will eventually fall into. Regrets will be everywhere if you fail to identify this one true passion of passions.

It reminds me of the poem by Robert Frost that was based on his observation of a friend who struggled to decide things:

The Road Not Taken
Two roads diverged in a yellow wood,
And sorry I could not travel both
And be one traveler, long I stood
And looked down one as far as I could
To where it bent in the undergrowth;

Then took the other, just as fair,
And having perhaps the better claim,
Because it was grassy and wanted wear;
Though as for that the passing there
Had worn them really about the same,

And both that morning equally lay
In leaves, no step had trodden black.
Oh, I kept the first for another day!
Yet knowing how way leads on to way,
I doubted if I should ever come back.

I shall be telling this with a sigh
Somewhere ages and ages hence:
Two roads diverged in a wood, and I—
I took the one less traveled by,
https://en.wikipedia.org/wiki/The_Road_Not_Taken

It's considered one of the most misunderstood poems of all time. I think it captures the dilemma we all face and how we regret our choices later on. no matter which path we take. Sadly, for Frost's friend (Edward Thomas) he decided to enter WWI and died a short time later.

https://en.wikipedia.org/wiki/Edward_Thomas_(poet)

I was a Physics major who loved physics theory notably SR and GR and math but struggled with the more abstract math courses even as I tried to do both majors. I was also a computer fan like you but it was in a time when they were mainframes and you needed access to an expensive terminal just to say hello. Most games of the time were of the order of tic-tac-toe and Gunner IV so the allure of gaming wasn't there. And at the same time, I was working 30 hrs a week and commuting so while I made enough to pay for college, I missed out on the social aspects.

Back to your case, I believe you may try to spread yourself too thin in trying to accomplish so much. Your interests appear very wide and divergent. You may need to focus on one path first (choose one).

When we are faced with these dilemmas we often resort to a pros and cons list with weighted factors. Do this for each course of action you're considering and it should become clear which path would be the best to take.

Personally, I would choose the physics path and augment it with math courses that way you can become either a physicist or applied mathematician or even a computer scientist specializing in computer modeling. Choosing one path will give you focus and a way forward.

However, continue to evaluate your choices and make smaller pivots as needed until you graduate. The real goal here is to graduate with a major in hand.
 
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  • #3
CrysPhys
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OP: Have you made a definite decision as to whether you plan to continue on to grad school immediately upon completion of your undergrad degree? If so, what field?
 
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  • #4
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First you need to decide whether or not grad school is in your future and if it is, what the requirements are to be admitted to the graduate programs you would potentially be applying to. The problem with triple or even double majoring is that you end up taking fewer courses in the major than you would if you'd single majored. What courses in each major are you sacrificing in order to make room for the other 2 majors? You say you've covered all the important courses for Theoretical/Mathematical Physics grad school programs, but are you certain? While I do know that students typically take fewer courses in their major in the US than we do here in Canada, a 4 year program leading to graduate school here would typically consist of approximately 24 mandatory core and support courses (for Physics that would be Physics & Math) plus a further 6 program specific elective courses bringing the total to 30 out 40 courses. In addition, a minimum of 6 courses in the major must be 4th year courses and of which the equivalent of 2 would be a full year senior thesis course. So even with overlap between majors, it would certainly be difficult to meet the requirements for admission to graduate school if you were to double major, let alone triple major without exceeding the requirements for graduation. As to that, it's totally up to you whether or not you think it important to be able to declare a major in all 3 subjects and whether or not a 5th year of undergraduate studies would be worthwhile.
 
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  • #5
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OP: Have you made a definite decision as to whether you plan to continue on to grad school immediately upon completion of your undergrad degree? If so, what field?
Yes, I intend to go to grad school either immediately after graduation or 1 or 2 yafter. Ideally, I hope to work for 1-2 yrs after graduation, using that time to both earn some money, while also plugging in knowledge gaps here and there so that I’m even readier for grad school (basically I want to look back and connect and solidify what I’ve learned in the previous 4 yrs before taking the next step.
As for field, I’m looking to narrow it down within Math or Math-related topics really; given my interests, classroom experience, learning habits, etc that makes the most sense right now - I want to understand the world at a deep level - from the ground up- and be able to communicate that understanding to other people. But I don’t know if I should narrow down to something like Mathematical Physics or Applied Math or Mathematical Philosophy/Logic and Computation or just Pure Math. I’m still trying to figure that out.
 
  • #6
hutchphd
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It reminds me of the poem by Robert Frost that was based on his observation of a friend who struggled to decide things:
Your posting truncated the last line!!

And that has made all the difference.
 
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  • #7
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The danger here is becoming ADD in your pursuit of multiple degrees and not spending enough time on the degree you will eventually fall into.
I agree that there’s that danger. That’s definitely something for me to strongly think about.

Regrets will be everywhere if you fail to identify this one true passion of passions
I do feel like I have narrowed down my passion to “that of understanding the fundamental nature of the world from the ground up and communicating that to other people”. I just need to find a way to narrow it down even more.

The thing is from my POV these subjects seem so inescapably interlinked in terms of how they borrow from one another and in terms of their ultimate goals that at this stage its hard to eliminate one. I personally feel very sure about my Math major - as far as that is concerned I feel very relaxed about my decision - I wouldn’t drop this major for anything. And that’s because I feel like Math is more from the “ground up” in a sense , you literally start from the simplest parts and build all these elaborate theorems. Philosophy does the same thing, but its not as rigorous without math; doesn’t have that efficient symbolic language. That being said, Philosophers do take a more holistic view of things imo and ask interesting questions. Lastly math helps describe lots of stuff.
Part of my dissatisfaction with Physics classes comes from the fact that lots of things are stated without a rigorous build up of the underlying math. For example, while I like seeing my Professors apply differential equations to solve problems, I just can’t stand not being able to understand the essence of what’s going on.
That being said, I like seeing how the math plays out in the Physics classes because I’m interested in understanding what that symbolic language says or could say about the world. There’s something really fascinating about that correspondence and the power it gives us. It simultanously reveals something about the nature of reality and our underlying thought processes.

Thanks for the Robert frost poem. I never viewed the poem that way and it’s truly new and revealing to me.

Back to your case, I believe you may try to spread yourself too thin in trying to accomplish so much. Your interests appear very wide and divergent. You may need to focus on one path first (choose one).

When we are faced with these dilemmas we often resort to a pros and cons list with weighted factors. Do this for each course of action you're considering and it should become clear which path would be the best to take.

Personally, I would choose the physics path and augment it with math courses that way you can become either a physicist or applied mathematician or even a computer scientist specializing in computer modeling. Choosing one path will give you focus and a way forward.

However, continue to evaluate your choices and make smaller pivots as needed until you graduate. The real goal here is to graduate with a major in hand.
Thanks for the advice. I’ll certainly see how much I can handle - I agree its an important point. And yeah, perhaps I should do that pros and cons thing - but I’d want those “pros and cons” statements to be backed up by some sense of empirical knowledge hence asking questions on this forum.
I’ve considered the idea of doing Physics only using the perspective that you mentioned - but I do want to complete the Math major; for else I’ll feel a sense of dissatisfaction in my Physics classes for not understanding the underlying theorems in the math. I will continue to make those small pivots tho, and hopefully things will narrow down even more within the next year.

I was a Physics major who loved physics theory notably SR and GR and math but struggled with the more abstract math courses even as I tried to do both majors. I was also a computer fan like you but it was in a time when they were mainframes and you needed access to an expensive terminal just to say hello. Most games of the time were of the order of tic-tac-toe and Gunner IV so the allure of gaming wasn't there. And at the same time, I was working 30 hrs a week and commuting so while I made enough to pay for college, I missed out on the social aspects.
Thanks for sharing your story, it’s interesting and also makes your points relatable. I loved SR too! I had a course on it last semester - it’s probably been my favorite thing to study in Physics so far. I haven’t studied GR yet; we don’t have a course on it so I might try to learn it on my own. It’s impressive that you worked 30 hrs while going to college , I can understand that that made doing a double major extra difficult. I personally don’t have major obligations outside my classes( I do work a few hours) , except for a relationship I’m in , which has been rather difficult to handle.
Btw Did you continue with Physics afterwards? Were you happy with your decisions?
 
  • #8
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First you need to decide whether or not grad school is in your future and if it is, what the requirements are to be admitted to the graduate programs you would potentially be applying to. The problem with triple or even double majoring is that you end up taking fewer courses in the major than you would if you'd single majored. What courses in each major are you sacrificing in order to make room for the other 2 majors? You say you've covered all the important courses for Theoretical/Mathematical Physics grad school programs, but are you certain? While I do know that students typically take fewer courses in their major in the US than we do here in Canada, a 4 year program leading to graduate school here would typically consist of approximately 24 mandatory core and support courses (for Physics that would be Physics & Math) plus a further 6 program specific elective courses bringing the total to 30 out 40 courses. In addition, a minimum of 6 courses in the major must be 4th year courses and of which the equivalent of 2 would be a full year senior thesis course. So even with overlap between majors, it would certainly be difficult to meet the requirements for admission to graduate school if you were to double major, let alone triple major without exceeding the requirements for graduation. As to that, it's totally up to you whether or not you think it important to be able to declare a major in all 3 subjects and whether or not a 5th year of undergraduate studies would be worthwhile.
Thanks for the reply. I do think that, based on what I’ve talked with my Math and Physics professors, that I shouldn’t have requirement problems for grad schools. Of course, not majoring in Physics might open up space for Independent research and stuff while majoring in Physics will provide lab research so the decision does influence the strength of the grad school application in that sense. Two of the Professors I talked to did say that my breath of courses in Math is enough for a strong grad school application while not taking the lab research in Physics will not harm a Theoretical/Mathematical Physics application either . Most of my online search through grad school application requirements has given me the same impression, although a research seems important for some of the top schools. I agree that I need to look even more into that ; its part of the reason I asked the question on the forum really. Thanks for your reply though — it informs me that the requirements in colleges outside of the US might be very different — and so I must look into programs outside and see if I’m prepared for those options. I haven’t looked as much into Unis outside the US , but I definitely should.
 
  • #9
mathwonk
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Yours may be the most thoughtfully posed questions of this nature I have seen. You seem to clearly understand your options and to have pursued faithfully your goals. It seems to me however that you have not quite decided what your next step will be after college, in terms of just what grad school direction to take. Thus it makes sense to me to continue to explore and learn more about the options that still seem attractive to you, in order to be more sure you choose the one that appeals most. good luck, and it seems clear that you will do well in what ever you choose. Just suggesting you try to give yourself as much data to make that choice as feasible. You still have over 3 semesters left, so are somewhat early to start closing off options, in my opinion. E.g. one of the courses still to be taken is apparently abstract algebra. My friends in physics used to tell me that a primary mathematical tool used in theoretical physics was group representations, namely study of homomorphisms from abstract groups into certain groups of linear transformations. This is a combination of abstract algebra and linear algebra, and is strongly influenced also by differential geometry and manifold theory, so there are still some significant math topics out there for you.
 
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  • #10
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Yours may be the most thoughtfully posed questions of this nature I have seen. .... E.g. one of the courses still to be taken is apparently abstract algebra. My friends in physics used to tell me that a primary mathematical tool used in theoretical physics was group representations, namely study of homomorphisms from abstract groups into certain groups of linear transformations. This is a combination of abstract algebra and linear algebra, and is strongly influenced also by differential geometry and manifold theory, so there are still some significant math topics out there for you.
Thank you for the advice. Hopefully next semester will clarify things for me. And your point about linear algebra is really interesting, makes me curious.
 
  • #12
I feel compelled to post because I'm in a somewhat similar boat as OP. I just graduated with a double major in Math and Philosophy.

I agree with the guy who said you should pick one field and focus on it as soon as you can if you want to go to grad school. The best grad schools are hyper-competitive and the sooner you can lock into a path and pursue it with relentless drive (get as much undergrad research experience as possible) the better. I see having spread myself too thin as a major mistake. Note that from what research I've done, tenure-track professorship positions in your (our) fields of interest are extremely competitive. You basically have to be willing to move anywhere in the country for one. This may not be what you want. If I'm wrong on this, someone please correct me.

If you go on ArXiV and look at some math papers, you'll see that math research is about going very deep into the weeds of a very specific and niche field rather than primarily building "from the ground up" with rigor like in an analysis class which seems to be one of the things you've identified as liking about math. Check out these pages on Terry Tao's blog:
https://terrytao.wordpress.com/career-advice/
https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/

You say philosophy lacks the rigor of math. I agree to some degree. Though contemporary research in analytic philosophy involves rigor and often symbolic manipulation. One thing that may interest you if you absolutely can't narrow down an option in one of these fields is Notre Dame's PhD program in Logic and Foundations of Math.
https://philosophy.nd.edu/graduate-program/program-in-logic-and-foundations-of-mathematics/
Students fulfill breadth requirements involving both graduate math and philosophy which is cool. I assume you'd have to be the kind of person to really enjoy books like "Godel's Proof".

Alternative paths: Have you considered law school? It uses many of the same skills as philosophy and math and you can actually get a job at the end of it (provided you go to top 14 or top regional). I'm looking into it. However, note law is a rat race. If you get the physics degree, you might be able to go into patent law. You can't with a math degree unless you have certain specific science courses as well so beware & google the undergrad patent law requirements. People in the corporate world seem to view physics majors as "geniuses". MBA could also be an option. Many famous executives with philosophy degrees, also a rat race. High school teaching is also an option. In short you might also find you get tired of the endless abstraction of philosophy and math and want to do something practical. Probably my worst mistake in college was failing to realize that at some point I have to get a job...
 
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  • #13
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Thank you for the reply. This is very informative. Thank you especially for your advice, and the link you provided on Terence tao's blog and the Notre Dame program, both of which I found to be very useful and interesting.

I would have loved to narrow down even more on my interests. To be honest, I think I know why I have not been able to narrow down more. I have been in a very difficult relationship for the last year and half, plus this pandemic which has made classes difficult - they've really impeded my ability to focus on classes, hence the feeling of not having put in enough effort and thus feeling confused and lost. I won't go more into it since this is something personal. But I agree that I need to find a way past that and find a way to narrow down.

I like your advice on doing research. I was actually talking with a Professor a few days ago about that, and he kind of opened up my eyes to how valuable it can be. He does research on non-linear oscillators, so I might be able to work on the theoretical side of physics -- doing applied math/physics research -- that'd complete both my physics major and give research work that'd be useful for either math or physics grad school. I'm also talking now with Math professors to see what's available in the math department, to see if there's an interesting pure math research. I also plan to talk with Philosophy professors to see what they might be able to supervise. It might also be possible to do research in something like logic, which would involve both math and philosophy. I might even be able to pursue multiple research senior year and the summer. Ultimately, I'll need to figure it out.

Thanks for the info about how competitive it is for tenure track jobs. It's good to know those things. I personally don't think I'd have problems moving across parts of the country. Establishing a family or staying in one spot definitely isn't one of my priorities. Of course, there's no knowing if that might change in the future. Please let me know if this should be a bigger concern.

About Law: I used to do a lot of debating in high school, and even until freshman year I was doing civic action work on campus, and took interest in civic and legal matters. I still enjoy intellectual discussions. But, I've figured out that I have no interest anymore in taking classes in the social sciences or arts or humanities --those that deal with understanding society (except the parts of Philosophy that deals with logic, epistemology, metaphysics -- basically those that directly complement math and physics). I don't mean to say that I am asocial or don't take interest in politics -- just that I don't see myself doing any academic work in these areas. To that extent college has definitely helped me narrow down my interests. If it turns out that I cannot find a job by virtue of studying math, physics or philosophy, I'd much rather go into applied math, or computation or data science or even engineering if job finding seems absolutely impossible otherwise.
 
  • #14
vela
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So I'd describe myself as a theorist. As far as Physics is concerned, I'm happy to take the upper-level courses. But I definitely don't want to go to Pure Physics grad school or experimental Physics grad school -- if I do it'll either be in Theoretical Physics or Mathematical Physics. I do like conceiving of and reading about experiments, but haven't really liked my labs so far - it's just a lot of tedious handiwork and teamwork.
Most physics students feel this way, but the reality is that most who go onto graduate work will do experimental physics. The lab courses you've had are likely not representative of doing experimental research. Doing some research will give you a better idea of what you can expect from working in a lab.

I'll take a different view than the others here. I've always felt it's a bit of a shame that many undergraduate students don't take the opportunity afforded to them to explore different areas, particularly subjects outside their major. In graduate school, that's where you'll need to focus on one particular topic. I don't feel you should limit satisfying your intellectual curiosity at the stage you're at. As Steve Jobs once noted, you don't know how the knowledge gained from a seemingly irrelevant course at the time might prove quite relevant in the future.

On the flip side, I don't think you'll regret not getting a degree because you're a class or two short. What's important is the education you received, not what's on a piece of paper. That said, you shouldn't completely ignore the practical consequences of your choices. If you want to go to physics grad school, for example, getting a BS in physics will likely smooth the path to getting accepted at many schools.
 

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