Why doesn math make sense anymore?
Feelings of self-doubt and inadequacy are extremely common amongst math graduate students, math Post Docs, and new math faculty. (They may also be common in other fields, but I’m in math myself, so I happen to be a little more well-informed here than I am elsewhere.) While some of them might actually be inadequate, many are not. When the feelings aren’t accurate, this is sometimes referred to as “impostor syndrome”. (I mention this not because I feel it needs psychiatric help, but to illustrate how common and well-known the phenomenon is.) The upshot of this is that you should probably find some other way of deciding whether this is for you than your gut feeling, which is probably going to tell you that you’re nowhere near smart enough regardless of how true it is. Ask your professors if they think you’ve got what it takes. Don’t ask them what they think is a good career path in general (well you can, but that should be a separate question). If you’re doing well on exams, you probably
A pure math background is not a waste. I went through / am going through something similar: I started off university in pure math because I was good at and enjoyed math. A couple of years and many math courses later I found that some of this stuff is really kind of hard, that I’m not enthralled by most of the material, and that even if I think it’s a beautiful subject, I probably don’t actually want to do it full time. Somewhere around this point I looked a bit more seriously at the psychology courses I was taking for fun, and decided to make a (second) major of it. I’ll be doing grad school in some kind of neuroscience. No regrets here — I think a math background will come in handy: an ability to understand statistics is superior to an ability to simply use it, models are mathematical, mathematical thinking is useful for issues of research design, etc. But most importantly, mathematics trains your mind for whatever you may wish to apply it to. I suspect the ratio of insight available
That time you spent thinking about math can be put to use in any number of academic subjects–computer science, physics, chemistry, biology, or economics to name just a few. Or you could go further afield and use your skills in abstract thinking and problem-solving in business. Or you can give your brain a break and travel around the world for a year, maybe take up painting or creative writing. No matter what you want to do, you will very likely find a way to apply your training to it. I encourage you not to rush into the decision. Don’t go to graduate school if it’s not what you want to do, and especially don’t go into a field you’re not going to enjoy. And especially especially don’t study something just because of macho posturing. A friend of mine told me that the way to pick a subject of study in graduate school is to figure out something you really like doing, then come up with a course of study that lets you do a lot of that. This is how I picked my graduate program, and although
Ah, one last thing. When I was an undergrad, I had a vague sensation that applied math was somehow viewed as “dirty” or inferior. After I got to grad school, that feeling slowly seemed to disappear. Now, I’d be totally weirded out to hear a pure mathematician say something bad about applied math. If you want to dip into applied math, don’t let anything stop you. You probably won’t even have to decide before you go to grad school (although it may influence your decision about which schools to apply to).
As an undergrad math major, I was advised by a professor (of applied math) to go to grad school in anything but math, which he described as “stuffy.” I don’t know how true that is, but having the same kind of math anxiety you describe after four years of abstract math, it offered me a great “out.” I’m terrible at biology, chemistry and physics, so I picked computer science, and it was a great choice. I think economics would have been a good fit for me too. The job market for PhD’s in the sciences is pretty bad right now, so you may want to consider that, but just because you go to grad school doesn’t mean it has to be math grad school — it doesn’t even have to be anything in which you have any real background! (I had two programming classes in college, and that was it.) Consider a field where you can use your math abilities at a less abstract level while solving more applied problems where you can get your hands dirty.
