Imposter Syndrome and depression: I feel pretty depressed that I am not as "qualified" as a close family member. In particular, this family member has undergraduate and graduate (PhD) degrees from Ivy League schools in mathematics. Whenever I try to solve a problem, this family member does it in half the time. Does this mean that I should try and find another field to specialize in?
Research in math (and probably in general) is hard. It doesn't get better over time, but rather we seem to get used to the accompanying depression and angst.
When I feel down, I try to remind myself that I do math because I enjoy it, not because I want to be the best at it, and for some reason people continue to pay me money to do it. If they stop, I'll figure out something else. In the meantime, I get to think about problems that I love, even if the process of that thinking isn't always fun.
The above is also hard. There are still rough days. It helps me to know that other people, including really smart people doing great math, experience the same feelings.
If you want to switch fields, go ahead. But in general, any field will have someone (seemingly) faster than you - this is not personal, we all feel like this.
I would recommend reflecting upon why you are in mathematics in the first place. If you're in it because you love doing it, does it matter if someone else is a bit faster? There is plenty of mathematics to go around.
Do you love research and/or math itself? Do you love solving problems and learning about new things and finding out how things work? If not, then I'm afraid your motivation is misdirected.
As you progress through life, you will meet many more people that are much smarter than you. I teach students who are much smarter than me and I wish I had their brain. Some colleagues/peers are simply out of this world.
However, there is a place for you. There are many research areas and countless opportunities to contribute. If it's not in math, then it's in other areas; there are many areas/problems waiting for new tools and perspectives. This means it is all hands on deck. The scientific community is small relative to the world's population. We are motivated to generate and verify new knowledge. If not us, who else?
Go back to basics, and follow your heart and interests. Keep moving and try to stop comparing; otherwise, it's like spending too much time in an art gallery and not doing any work :)
Math can be brutal to your self-esteem. If you are a poet of average skills you could (perhaps) convince yourself that you are an under-appreciated genius with as much raw talent as a winner of the Nobel Prize in literature. In mathematics, if you are mathematician of average talent, you can't plausibly convince yourself that you are even remotely as talented as a winner of the Fields Medal.
If you get to that stage you have to ask yourself what is more important -- your ego or your love of mathematics. If the former, by all means switch to another field, perhaps one with more subjective criteria where you can convince yourself that you are the best. If the latter -- keep with what you are doing.
Many mathematicians end up teaching undergraduates. If that is the career path you end up following, the knowledge that you are not a world-class research mathematician (if in fact you are not) can help give you the humility that you need to e.g. not lose patience with students who can't seem to understand the chain rule. What you are feeling now about your family member is similar to what some of your students will feel about you. This can help you empathize with such students. If you know what it is like to struggle with a concept that some people find obvious you will be in a better position to help others who are in a similar state.
Although my field of study is not Math, I can relate to you because I had a experience that was very similar to yours. Just like you, I felt unqualified in my studies and just like you, I had thought about changing to another field and even giving up on trying hard. Eventually, I changed my field of study but not because I wanted to be perfect or better than others in it but because I came to the conclusion that if I want to spend my time and life on something, it should be something that I enjoy doing. I decided to focus on enjoying what I do and trying to improve myself over time rather than comparing myself to other people. For me, it is a fact that almost in any aspect of life, there are people that are better than me. As a matter of fact, there are people in my new field that are better than me (I have close interactions with some of them on a daily basis) and well, that's life.
Also, it is good to remember a well-known quote from Thomas Edison:
Genius is 1% inspiration and 99% perspiration
I have studied biographies of many prominent scientists and artists and two personality traits that I have found very common among them are that they enjoyed their work and they worked hard. In my opinion, these are the qualities that you should concentrate on. If you truly enjoy Mathematics, then just enjoy it. For a good biography, I encourage you to read the wonderful memoir of Issac Asimov, I, Asimov.
By the way, I hope that by 'feeling depressed', you do not mean that you are suffering from clinical depression (depression is always an alarming word for me). It is quite normal to feel disappointed about your qualities sometimes but, if you feel depressed about other significant aspects of your life too, please also consider seeking clinical help.
Does this mean that I should try and find another field to specialize in?
In any field you choose to specialize in you will encounter someone -- probably multiple people -- like your family member who have better training and probably more talent than you. That makes me wonder why you assume the premise that it is especially bad to have a family member who is that super-talented person? Won't it be just as frustrating to have such a person as your supervisor or your colleague? In which case, isn't the conclusion that your only hope is to retire to a desert island and not attempt to do anything meaningful with your life, so that you never have to compare yourself with someone who is better than you at what you do?
The answer of course is that the premise is incorrect. You should aspire to choose a profession that brings you the highest level of personal satisfaction. If math is that area, it would be a shame to choose a different area for the sole purpose of avoiding professional jealousy between you and a family member, when in all likelihood you will encounter many other people likely to arouse similar feelings of jealousy in you. Better to learn to deal with the jealousy, accept that you are not the smartest person on earth, and find joy and meaning in your ability to make humble but useful contributions to mathematical knowledge. Good luck!
you don't say how far along you are. If you haven't finished a first degree in maths and he has a PhD from a good university then he ought be at least twice as fast as you. In fact, if he is only twice as fast as you, you are doing rather well.
A more interesting question to ask, is how is his career going? I have seen plenty of people with PhDs from top name institutions have trouble making a career in maths. If you really truly believe that you have less talent and he is having trouble in the job market, it's probably a sign that you should specialize in something more employable.
Whenever I try to solve a problem, this family member does it in half the time. Does this mean that I should try and find another field to specialize in?
If you and this relative are unable to put a bit of academic distance between yourselves, for example by refraining from tackling the same math problem (and then comparing your results and your speed), then I imagine you would probably be happier not studying in the same field. It sounds as though you may be setting yourself up repeatedly for this type of comparison, which you then get discouraged about.
In a way I can relate to this experience. My siblings are both in your situation. I am and have been the "smart" guy in my family. My siblings were very often compared to me by teachers, friends and family members and used to be asked why they can't be as "good" as I was. That got them depressed and made them give up on things they wanted to achieve because they believed what they were told, that they "have no talent". If I ever regret something in life, is that I didn't pay attention to that and didn't as much as tell them what exactly talent was.
Because, here is the thing. I have no particular talent in anything. I only know that if I work hard enough, smart enough, and I keep myself healthy, I can acquire many talents. How to learn something is a whole science, and I don't plan to discuss it here. All I'm saying, you can get a PhD, a successful career, even if you seem much less competent at the beginning than your peers. In fact, I was surprised myself how many of my PhD colleagues who seemed to struggle with just about everything, ended up making important contributions to the research fields they happened to specialize in.
I also think I should mention that even if you don't know whether you'd like or not a certain research field, the beginning is not the moment to decide on that. It's as if you'd decide not to marry someone because you don't like their nose. But, if you stay long enough to know them and you realize they treat you well and have many qualities, I doubt you'd care so much about the nose. Same with the research. Once you reach a certain level of understanding, you begin to like the field.
If I were to do a PhD again, I'd choose a field that gives me enough job opportunities, enough challenge, and problems worth solving. I'd not factor other's talents in this decision.
Einstein struggled a lot more with mathematics than a whole lot of his contemporaries, particularly mathematicians. David Hilbert was able to retrace (and do the finishing touches Einstein was still working on) a significant amount of Einstein's General Relativity work in a small fraction of the time Einstein needed and made a paper which he later retracted since he saw it as credit-stealing.
Einstein needed a lot of time to get where he wanted to but he had the advantage of knowing for sure where he wanted to get. That produced more tangible results than brilliant minds without his aim.
So no, the mere speed and skill with which one of your relatives (what does it matter that he is a relative anyway?) works with mathematics does not tell much about what mathematics can do with and in your life.
I mean, if he is a relative, that should be a good omen, shouldn't it? That makes it likely that your genes are not entirely remote from your interests.
On the contrary this should encourage you to go into the same field as your family member. The best way to improve at something is to surround yourself with people who are more talented than you are.
When you have a problem that you are working on (or even better after you have solved it) you can discuss the problem with your family member about how they would have solved and how you could solve it better in the future. You also have the benefit of having casual conversations about things tangentially related to your professional interests.
I think that key thing here is that you already have a network of highly skilled people in your field. This is an extraordinary advantage and you should harness it.
This may be a cause to move into a different sub-field than him... Many people want the glory and to feel they have done something. There is no reason to think this cannot be done within mathematics.
But on the thing of comparision, do you have better questions than him? Do you have novel approaches? These are the things that matter. Also when you talk of his ability to solve problems more quickly, are you talking about elementary problems that take mere minutes? Does he betray a deeper understanding than you?
You should only consider a change in a specialisation if you choose to, not due to a feeling of being forced - the latter, in all likelihood, would not be sustainable. Explore your specialisation to see if you can identify and capture a niche area that grabs your interest.
When I feel a similar feeling, I look at a list I prepared of what I have accomplished within the topic, what I have going at the moment and the exciting prospects that I have planned. Doesn't always work, but does help most times - there is no magic solution though.
One thing to consider is engaging your family member in a collaborative project, work together and even learn from them. Good chances are that he'll learn off you at the same time.
Perhaps, there is an opportunity to be mentored by your family member - benefit from his knowledge and skills.
As a saying goes "chase the dream, not the opposition" - but without the opposition, as you are not in competition with your family member.