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Whether this is "impostor syndrome" or not, you should realize that you can always quit later. Just because you start a Ph.D. program, doesn't mean you have to labor away for 5 or 7 years at something you suck at. Try it for a year and see how things go. If you still love math and still get A's, carry on. If you can't stand it, then drop out. At least then you'll know for sure. You won't spend the rest of your life wondering if you should have given the Ph.D. a shot.

If you don't drop out. Swell. If you do, the year is not wasted. You still learn some valuable math that you can put to some use somewhere. Some graduate credit might transfer to the philosophy department. If you like teaching, then you will have taken a few more courses which will enhance that skill.

Win-win.

I think you should give the math PhD a try. I am almost exactly like you in terms of strengths and weaknesses, yet doing a stats PhD was the best few years of my life, and now I am smarter and better-rounded than if I'd done a PhD in a personally "easier" subject.

Math has always been my hardest subject in school; all the way through elementary school, high school, and college, I disliked math classes and excelled at the other subjects. I'm quite error-prone when handling messy formulas, and I have a hard time manipulating abstract mathematical objects in working memory. But, like you, I was a good student and managed good grades in math, even though it was definitely my weakest subject.

Then somehow I fell in love with statistics and, rather to my surprise, found myself at a very selective statistics PhD program. I had the time of my life and graduated early. Here is what I'd recommend:

1. Know your own intellectual strengths and weaknesses. Imposter syndrome is a real thing, but don't let others tell you to automatically attribute your perceived weaknesses to imposter syndrome. Rather, calibrate your strategies and expectations to your own strengths and weaknesses. For example, like you, I easily forget useful math facts, which is a real pain on timed, closed-notes exams. As mentioned above, I'm also pretty slow at actually doing math accurately. Therefore, for high-stakes exams like the quals, my study strategy focused on training my weaknesses aggressively through assiduously memorizing math facts. During the exams themselves, as you'd do at, for example, an athletic event, my strategy switched to playing to my strengths in order to earn as many points per unit time as possible. For me, that meant using my strong understanding of the concepts to earn lots of points on that front, and quickly bailing on any integral that seemed like a pain to simplify (because the probability that I'd mess it up is pretty high, and it's just not a good points:time ratio). Again, this kind of attitude is not imposter syndrome: you are actively working to improve your weaknesses, while finding ways around them on tests.



2. 
   

   Math research is tremendously different from "school math". With no false modesty, I have many classmates who are much better than me at math. However, many students who love and excel at school math neither love nor excel at research math. Among the key differences are:

   
   
   
   
   

   • The role of high-level creativity and conceptual understanding is much higher in research math than school math. Major research advances don't always come from breathtaking mathematical shrewdness. Sometimes they arise from out-of-the-box thinking or the creativity to recognize and repurpose a useful analog in a disparate subdiscipline. Like me, you may find yourself quite adept at the latter.

   
   

   • Research math requires many other skills besides muscling through integrals. Given that you excel at humanities classes, I suspect that, like me, you're an excellent writer. You will spend a lot of time writing when doing a PhD and subsequently as a researcher. If you write, say, 30% better and faster than the average math PhD student, you will find yourself flying through your dissertation and paper submissions. I wrote each of my 3 dissertation papers in 1 week. This more than made up for the time lost along the way to asking Wikipedia for the 844th time how to do a Taylor series expansion.

   
   

   • Research math is also the ultimate "open-book exam". There are no time limits, and you can Google Scholar to your heart's content. This also means that, as discussed above, you can play to your strengths. For example, whereas others with perhaps more mathematical agility develop new research directions from the bottom up by playing with formulas and combining things in clever ways to see what happens, this does not work for me: I simply get stuck in an algebraic morass. Instead, I work from intuition and conceptual understanding first, then reach for simulation tools, and only then put pencil to paper.

   
   
   
   

Good luck with your decision.

You should first think about what you plan to do with your math PhD, should you stay the course and complete it. It's surprisingly common for people to not consider this.

There are two common reasons for doing a math PhD. One is to do research. A PhD is required to do research, and a math PhD is required to do math research.

A math PhD would also help you get a good teaching job, possibly in a math related field.

There are other jobs for which a math PhD isn't required, but is helpful. E.g. finance. But you don't need a math PhD to go into finance.

However, you should be aware, if you are not already, that doing a math PhD is hard, and math research is harder, and commonly only done by people who are really, really into math, consider math among their primary interests in life, and possibly the main professional one. Also, these days math research is extremely competitive, and it's hard to get a decent math research job.

The bottom line is, try to be aware of where you are going, and whether it is somewhere you want to go.

I only mention these obvious facts, because it is my experience that graduate schools and math departments do a very poor job of educating graduate students about these issues. Unfortunately in some cases there is also a conflict of interest. The department needs cheap labor to teach their classes. Often this cheap labor is obtained from graduate students. This is true in the United States, at least. You don't say where your location is.

I also happen to be Indian, and was in math departments for a while. I was better than most people I knew at math, usually by a long way, but it was difficult for me to perform at the level that would have been required for a successful research career. Also, I don't know if, at the end of the day, whether I was interested enough.

I don't think anyone here can advise you whether you should be doing a math PhD or not, but again, consider carefully whether what you are doing is something that you really want to do, and whether the direction it is taking you in is a direction in which you want to be going.

Don't quit.

When I began my Ph.D studies (in computer science, not mathematics) I'd been out of undergrad for five years. I was certain I'd forgotten everything I once knew and I was terrified I'd be sharing a classroom with people who had just finished undergrad and had all their learning fresh in their minds.

On the very first day of class, in my very first class — Advanced Algorithmic Analysis — the prof stood at the whiteboard. "We're going to start right where we left off in undergrad," he said, and began littering the whiteboard in really dense math that I didn't understand at all. I looked around the room and everyone was just nodding: they clearly understood it perfectly.

After ten minutes the professor concluded his derivation. "Is that all clear?" he asked us. "Does anyone have any questions?"

I was too scared to put my hand up. I had decided I was going to drop out that very day. I could not compete with my classmates; they all understood this, and I had no idea what was going on.

"In that case, could one of you explain to me what I just did? Because I made all that up. It's nonsense."

For the next ten minutes we received a kind lecture about how all of us were in the program because we were smart people with a demonstrated track record of understanding things. If we did not understand what was going on in a class presentation, that was a sign something was wrong with the class, not that something was wrong with us.

I stayed in the program. I'm glad I did.

You are absolutely not alone. Other people have already mentioned Impostor Syndrome. It's real and a lot of people suffer from it. The best advice I can give you is to be open with people that you have it, and — this part is very important — ask questions, even the 'dumb' ones.

Graduate school success belongs to the people who are willing to ask the 'dumb' questions. And very often there's nothing dumb at all about the questions!

Remember, the admissions committee aren't a charity organization. They'll only accept you if they think you're adequate enough to attend the program. And if they think you're adequate enough, you should certainly not think you "suck at math".

Have some confidence, and think positive! You've been accepted at a school when many others who want to attend were rejected. You're a success story! Relax, work hard, and enjoy the program.

First of all, math classes are hard. I struggled in them, my peers struggled in them, everyone struggles in them. Its new material and not entirely intuitive (e.g. Ito Calculus). As for not remembering everything, that is pretty normal as well. This seems to be more common as you further and further specialize. Within my area of expertise, I could have a good grasp of knowledge, outside of that bubble, its a little vague.

You might be interested in a more applied field that can take advantage of your linguistic abilities. Applied statistics is a massive field with lots of low hanging fruit. This can range from anything to health outcomes based on social media posts to how well students perform after a policy change. You take the best parts of math and then you apply it to real world problems. That said, this does likely mean more probability and linear algebra, though you can have a surface level understanding of both those areas (like myself) and do good work in statistics.

I think you should follow your instincts on this one. Some people may give you the "don't worry" message because it makes you feel better. But I get the impression that you are putting down rational inferences, not just irrational self doubts.

Admissions committees have to fill slots and bring people in all the time who don't make it through. Of course they prefer not to get weaker candidates or those who won't make it through from persistence/desire/love (and the two are correlated). But it doesn't really hurt them that much and they have statistics in mind that some percent of students won't fit. The real tragedy is to the STUDENT who wastes productive years and is miserable.

So follow your gut on this one. Look for something that uses some of your verbal skills and is not so, so, soooo hard core math as a Ph.D. in math. Go work for McKinsey or Microsoft or the like.

Often the people who excel are not just those who have played to their strengths, but also those who have overcome their weaknesses. The skills and talents you have in other realms may well help you outdo your peers if you can only make it past the obstacles that are stopping you now.

With all that said, there's real evidence that being a negatively stereotyped minority in a given field can significantly lower your performance below your natural abilities, especially if you dwell on the stereotypes. Claude Steele (provost at Columbia University) has extensively studied this topic --you may find his book, Whistling Vivaldi valuable, it includes some practical strategies to combat the stereotypes.

You should also not be afraid to ask for extra help. One of Steele's findings is that low performing students and negatively stereotyped minorities are much less likely to seek out help because of embarrassment. They are unaware their high-performing classmates are often quietly taking advantage of a wide range of extra help, including peer study groups and one-on-one sessions with their professors.

I finished a Physics PhD and I felt the way you're describing for most of the time. My adviser complemented me about 3 times if I remember correctly.

For me, teaching (as a lab TA) was real boost to my confidence and understanding of Physics (which is pretty close to math). Teaching was also where I started to see interconnections between subfields that I didn't grasp as an undergrad.

But I suck at math. I think I only was accepted because I'm a woman and Indian. Everyone is so smart in all my classes. During my undergraduate degree I was amongst the top of the class but once I got to a master's, I was at the bottom of the barrel. I can't understand things the way everyone else does. Everyone is always able to engage in the lesson and it takes me 4 hours just to understand the smallest thing, it's depressing.

This isn't necessarily a problem. Different people have different ways of learning. Me, as soon as somebody starts presenting equations on a slide, my brain switches off and I don't get much out of the lecture - I need to read through that kind of material by myself, in my own time. Sometimes it takes me several readings to get through it, or I have to try out an example before the abstractions click for me.

It might be worth your while to read about different learning styles and see if you can identify what strategies work best for you. I would also strongly suggest talking to somebody at your university about your situation (adviser, support staff, whoever you're most comfortable talking to).

Anyways, even the most basic math facts I will forget. It's embarrassing to be a PhD student and forget things so easily. I don't even want to make friends because I'm scared they will start talking about math and I won't know what they're saying

You might be surprised at how many of your fellow students are concealing the same worries. I still have to look up "basic" facts quite often, but that's okay; real life isn't a closed-book exam.

Nobody here can say for sure whether you should persevere with the PhD. That depends on too many factors - some people manage to complete doctorates but at too much personal cost, or they find that the doctorate doesn't position them for a satisfying career. But I wouldn't quit at this point just on the basis of "not being smart enough".

IMHO, the link between "PhD" and "clever" is often overemphasised. Yes, you need some degree of intelligence to do a PhD, but other factors are also important - material support, self-discipline, resilience, enthusiasm, etc. etc. I can pretty much guarantee that some of the smart students you're envying right now will drop out of their PhD programs because they can't handle those other challenges.

I agree with others that you should try if you love it. Two things to consider -

• Think about the alternative. You will never know if you could have
  done it and you will probably regret quitting before you start.


• It is very likely that the people think the same about you, where you are the one who is smarter and more talented. They have reason to think so! You were at the top of your class, you are in the 95% percentile. Imagine how the other 95 feel about you =P

I understand how you feel. I also love mathematics/statistics and computer science but consistently score better in the humanities with less effort. That does not mean that I have no talent, I still score better than many of my classmates in STEM fields, and I do not love the humanities in the same way. In the end you become good at what you spend time on and you want to spend time in mathematics. Do your best, decide later.

I want to posit a few things that you indicated:

• You are very good at mathematics, but you need to work hard at it.


• You are very good at english, history, and philosphy and feel that those proficiencies come much more naturally to you.

I will clarify that I do not have a post-graduate degree, but I was recommended to a program because I apparently am very good at bridging gaps between disparate fields (psychology, education, and engineering). In my professional career I have found this to be very handy. It can be a boon to be exceptional at one thing, it can be a source of great opportunity and wealth to be able to understand several things very well and bridge the gaps between them.

Your opportunity at a PhD program will open doors for you to explore how you can connect your various proficiencies. You will absolutely have to work hard and the fact that you've done so in the past bodes well for you. I don't know how one could bridge mathematics with philosophy, but I don't know if I'm particular exceptional in either of those fields. You, however, might be. And you might be able to forge those bridges and advance our collective understanding in both fields forward.

As you seem to have a great opportunity before you, I strongly urge you to pursue it if you have the means to. You may find it's a dead end, but there is considerable education to be gained in failure.