How can a mathematician handle the pressure to discover something new? Suppose I'm an aspiring mathematician-to-be, who started doing research. Although this is really what I love doing, I found that one disturbing point is that there's always the pressure of discovering something new. When I'm doing mathematical research, there's always the fear in the back of my mind, that maybe, I don't get new results. In the past, I could think freely about mathematics, without the pressure, but now that it's "my job", I have these problems.
How to handle this? Since this site is for mathematicians from the graduate level onwards, maybe somebody has a good suggestion. Note that, although this is a mathematics forum, I think this question is appropriate here, since it perfectly matches with the description of the "soft question"-tag.
 A: A friend of mine once told me quite brightly about mathematics that "when you search, you find". As naive as it sounds, the more the times passes, the more I believe it.
Moreover, mathematics is not only about finding. Part of the job is also made of acquiring a deeper understanding of concepts, relearning your field, sharing your knowledge and interests, and many "new" ideas come from this crossed points of view and interactive community. Take any topic, spend time into it as a whole, and you will get used enough to it to naturally understand what is missing, what could be asked, what could be tried, etc. In this spirit, "failure" (not finding as easily and what expected) is also part of the job: failing to do something is underlining what is missing, being able to share it, and this will provoke ideas, will lead you to ask specific researchers for the missing steps, and will plant seeds that will eventually blossom. 
What is not easy to handle is that it takes time, and hence it requires confidence. This is an important psychological aspect of doing mathematics.  When you share with other researchers, you realize that they are just as human as you, and not the apparently perfect spirits they seem behind their articles :) I believe it is necessary to mourn the will to be perfect and embrace everything, we are all doing part of the effort, this is essential to understand and believe.
Maybe could I suggest reading Netz's article on "deuteronomic texts". It is a famous work on history of mathematics that shows how much of the advances and efforts in (greek) mathematics have been made after the one who we credit with the discovery, by "digestion": treating examples, spreading the ideas, rewriting, unifying, choosing brighter notations, etc. I always thought it gives a good taste of how a community should work. And mathematicians are a community, it always worked through sharing, attempts and failures (look even at Gauss' or Serre's letters).
A: I think this is a common question in almost all fields and not just math. Hence although I am not a mathematician (not even remotely), I feel the urge to share my thoughts.
As a general advice, you might be interested in reading some materials about Goal-setting theory. In a glance

Goal-setting theory is based on the notion that individuals sometimes have a drive to reach a clearly defined end state. Often, this end state is a reward in itself. A goal's efficiency is affected by three features: proximity, difficulty and specificity. One common goal setting methodology incorporates the SMART criteria, in which goals are: specific, measurable, attainable/achievable, relevant, and time-bound.
An ideal goal should present a situation where the time between the initiation of behavior and the end state is close. With an overly restricting time restraint, the subject could potentially feel overwhelmed, which could deter the subject from achieving the goal because the amount of time provided is not sufficient or rational.
Most people are not optimally motivated, as many want a challenge (which assumes some kind of insecurity of success). At the same time people want to feel that there is a substantial probability that they will succeed. Specificity concerns the description of the goal in their class.
The goal should be objectively defined and intelligible for the individual. Similarly to Maslow's Hierarchy of Needs, a larger end goal is easier to achieve if the subject has smaller, more attainable yet still challenging goals to achieve first in order to advance over a period of time. A classic example of a poorly specified goal is trying to motivate oneself to run a marathon when s/he has not had proper training. A smaller, more attainable goal is to first motivate oneself to take the stairs instead of an elevator or to replace a stagnant activity.

IMHO, while the question is clear, but it points to a surprisingly complex issue.
A: The trite but true answer is to discover new things but how does one do that? What you really should be asking is how to discover what needs to be discovered? The framing of questions or ideas that require inspection or further examination can, sometimes, lead to a discovery. All of the advice you receive on how to discover is not very helpful if you don't know first where to look and then recognize that there is something either missing, incomplete, or plain incorrect that needs to be addressed. This is why so many students end up studying/researching in the field(s) of their mentors and not necessarily being very successful as a result. (OK-it is my opinion that some folks believe that there aren't that many new, good, ideas to go around and so when they find one that is productive they will work it until it is no longer fun) 
What has worked the most frequently for me is the "why the hell is that?" or the "well, that makes no damn sense at all!" method of question framing. You'll note that neither of these actually are suited to making a discovery but to identifying where to start exploring. You will quickly learn what other people think that they have found if you are familiar with the area (have read and digested all of the relevant literature, chatted up all of the major and minor players, and know the history of how those discoveries came about). The process of identifying what needs to be done or what remains to be addressed in a particular subject will, frequently, be clear if not immediately obvious. The discoveries that you seek typically end up being the mechanics of the solution used to address those questions. Feynman's and Newton's discoveries hinged on the formation/assembly of the question(s) to be answered. (It is also my opinion that the more practice one gets in making discoveries that the easier it becomes to make new discoveries - that is at least what I have discovered for myself)
A: I consider my job of mathematician not as intended to produce new mathematics, but to explore and spread mathematics. Then exploring mathematics consists in


*

*hearing/reading mathematics (which you can find great, great but badly written/explained, uninteresting for you, uninteresting at all, etc),

*thinking about some questions without really knowing their "status"

*rethinking about mathematics that are written in a way you've initially thought as confusing

*solve problems, that you've formulated or read, or a mixture

*fail to solve problems


[etc...! and the distinction between all this is fuzzy of course; here I put aside from this list the various mathematical extra-activities of professional mathematicians, notably in the evaluation of other's works.]
This is "personal work" and the counterpart is to communicate your work (in the large sense suggested above):


*

*talk math with your colleagues

*give talks or lectures about things you've learnt

*give talks/lectures and possibly write surveys or books about things you've learnt and reprocessed

*write papers with results you consider as new.

*ask mathematical questions (in private discussions, in talks, in written papers, in web sites such as here...)


This is not comprehensive: for instance the above list restricts to spreading maths among somewhat specialized mathematicians; it could include spreading towards a non-specialist or non-mathematicians, or trying to develop interactions and applications of pure mathematics... the latter are important activities of a number of mathematicians and these are essential roles; not so far in my personal experience.
Nevertheless, there's a growing tendency (especially in some countries), towards a job exclusively intended in the purpose of publishing "new research" and this is indeed the most visible, and in particular, for a young mathematician, the way considered as most accurate to judge the ability to work as a mathematician. That it tends to remain, all along the carrier, the main/exclusive way a mathematician is now judged, is more of a problem.
Last and not least, the meaning of "new" (new results, new research, new ideas, new maths)... is very subjective!
A: I can share my own experience: similar considerations always disturbed me too.
To deal with this, I always considered mathematics research as a hobby. From the very beginning I thought that my main job (from which I make a living) must be teaching. And I do research as a hobby in the time which remains from my teaching
duties. When I was a student, I knew that if my research is successful, I will probably qualify for some university teaching job, if not, I will go to high school. In fact, after graduation, I worked for 12 years on a pure research job
(without teaching duties) but I always tried to have some connection with a university (did part-time teaching job) in the hope that if my research results are not good enough I can always switch to teaching. Then I switched to a full time teaching, as a professor at a university, and feel comfortable with this job.
Actually few countries have many pure research positions in mathematics. Most mathematicians are affiliated with universities. Of course, in many universities
there is a pressure to publish. But in most cases it is not so strong, after you are promoted to a permanent position, and one can write textbooks, for example, survey papers, etc. or do some other useful things, after all:-)
Additional remarks. By "teaching" I mean teaching on all levels. I know some very good mathematicians whose main contribution to mathematics is their graduate students, rather than own research results. Another activity somewhat close to teaching is writing books. There are many very respectable mathematicians whose main contribution to mathematics consists of several good books.
A: There’s nothing particular to mathematics with this problem - performers (actors, musicians, comedians) have a much stronger version of it and writers are in much the same position as us.
A: I think the only way to handle this is just to carry on. Then, after a while, you may become confident enough in your ability and desire to do mathematics at a level acceptable to you. If that does not happen, you may/should consider other occupations. 
A: From the way your opening sentence is phrased ("an aspiring mathematician-to-be, who started doing research"), I am guessing that you are at stage of being a graduate student (or not far from that). The answer I am giving is based on this assumption.
Basically, for an "aspiring mathematician-to-be", a matter which is as important as producing new results, or maybe much more important than that, is to identify areas in which you enjoy trying solving problems, and spend some time learning ideas, methods, and interesting questions in those areas. I use the plural form "areas" intentionally: it often helps to (a) have a backup problem to think about when you are stuck and (b) have knowledge of something outside of your primary research area because that's how you sometimes stumble upon unexpected interesting connections. 
And, well, if your immediate environment creates a constant pressure to discover something new, do keep in mind that it is a feature of your environment rather than of life of a mathematician in general, and that great things rarely happen under pressure. In fact, from my personal experience, it is pretty much the opposite: one keeps working on a question for hours, days, weeks, months, maybe years, often without tangible progress, and then, on some occasions, breakthrough moments happen and you produce something new. Eventually, one either learns to be comfortable with working as a research mathematician, or finds a profession that has a mathematical component but slightly more instant gratification. 
A: It's fun. 
It's like being a hunter, pursuing the prey through the forest, looking for its tracks, the scraps of fur it left behind, piecing together its movements, building up to the moment when you have it in the sights of your gun. 
Or like being a miner, digging deep into the mysterious earth, using your axe to get through intervening layers of rock, giving up on one tunnel, trying another one, hoping for that moment when you suddenly happen upon the sparkle of a vein of rare gems. With, along the way, a lot of sweat.
Or like being a creative artist, an artisan, or perhaps even just a house painter (which I was in college). You have a vague idea in your head of what you want the final outcome to be. There are many small details that have to be worked out, that are hard to anticipate, but are fun and interesting to ponder as you work through them. Eventually, you're done, you can stand back and admire your work. Be it a painted house or a theorem.
A: This is ancient history, and considering my age, I may have told this story here before. I started at Harvard graduate school in 1957, the same year that Hironaka arrived there to work with Zariski. He was already an accomplished mathematician, even if he didn’t yet have a PhD. Early that year, I must have said to him that I couldn’t imagine ever doing research, and he said, in essence, Oh, you learn about some subject, think about it in depth, and before you know it, you’re proving Theorems. I thought, This guy must be in Cloud Cuckoo Land, I’ll never do that. But of course, that’s exactly what happens.
A: Although I'm not a professional mathematician, I am a professional scientist who relies heavily on mathematics, and have made a few small research contributions to mathematics (combinatorics, statistics, linear algebra, ...) proper along the way.  Perhaps my experiences would be of some help.
To excel or even work in any advanced and valuable field (physics, mathematics, neuroscience, ...), you will be under pressure.  There is competition for employment, tenure, grants, recognition, and so on.  At base, the only way you'll thrive is if the day-to-day act of working brings you some joy—even if it can (and will) be often frustrating.  Taking breaks, especially for vacations and exercise, can reduce stress and indirectly help your research.
Perhaps overlooked is this:  Avoid stress elsewhere in your life.  Avoid relationships (romantic or otherwise), living situations, financial arrangements (getting into debt), and so on that add to stress and burden.  We all have different tolerances for stress, so if you can expend your "stress budget" on mathematics, you'll likely be happier and more productive.
Try to tackle problems at the "right" level of difficulty.  As David Hilbert stated, the best problems are not too hard and not too easy.  Spend time judging the apparent difficulty of a problem before devoting yourself to it.  For instance, I would not recommend approaching the Collatz Conjecture (unless you're a Fields Medalist) because every mathematician who has looked into it recognizes it as extraordinarily difficult.  (Erdös said:  "Mathematics isn't ready for problems such as that.")
And play with mathematics!  John Conway (and others) have brought a sense of play to a number of areas of mathematics... not worrying about whether something deep arises.  (And yet, in his hands, it so often did!) 
A: There is a quotation I once heard that perhaps someone could provide a reference to. It may even have been an MO comment...
The gist of it was that as one gets along in one's mathematical career one's research often starts to feel less like tackling some impossible obstacle than preparing a meal for friends. 
As I understand this, we mathematicians find a corner of the subject where we want to resolve some problems that are not so flashy as to draw international attention and gain us a Fields Medal or a tenure-track job at Princeton, but are interesting and important for the area of mathematics we love and are aiding in the growth of. The former sort of goal may drive one's early career, the scratching, biting and clawing that seems necessary to secure the time to think with a decent academic job. Later on, post-tenure, one can try to focus a bit more (with equal or greater intensity) on figuring out what is actually going on with a bit of mathematics. Usually, this corner is interesting to a small community of mathematicians who really will care about the work. It really feels that research is a conversation with these people, and that the work is pursued in a spirit of mutual appreciation of the beauty of that subject and a shared, deep desire for increased clarity...although the realities of promotion and pay still drive the occasional flare up to attempt to become famous...
The fewer of these flare-ups, the better, I think. They are driven by economic forces, rather than honest mathematics. The greatest of us probably think of these sorts of economic forces the least, either because they developed their mathematics early enough to head off these forces early on and now have coveted jobs, or simply they are used to living a spartan lifestyle that shields them from typical economic concerns.
This answer, by the way, is a comment mainly intended for the OP, and may be too riddled with opinion to be appropriate. If down voted sufficiently, I will remove it.
A: Many other answers make what seem to me important points. Emanuele Tron’s observation that novelty is rather fuzzy and relative, and YCor’s suggestion that good research is not just about “new results”, particularly resonated with me.
Implicit in those answers is the fact that your "job" can be at most to produce work which is new enough to be publishable. This is a highly specific and technical sense of “new”! For example bringing together ingredients from two different published papers, to prove a result which does not obviously follow from either, might absolutely be “new” in this sense. 
Of course it is good to be more ambitious than this. In fact it would be nice to come up with some utterly unprecedented new ideas! But that would be extraordinarily rare and even if you did do it, you might not realise that you had. Most new mathematical ideas seem to me to come from understanding existing work more deeply, so as to extend it to new cases or prove stronger results with it.
In practice a lot of the struggle can be to work out what results of a certain type have appeared in, or are well known to follow from, published work. Then you will be able to see the gaps where new results could fit. Because every person has a slightly different perspective, other people may be surprised by the work that results!
If you follow the approach in the last paragraph, there might be a few false starts proving theorems that others do not find exciting or which are not publishable, but this is also part of the learning process. Remember that suggestions for alterations, extensions, alternatives or improvements can all be positive responses which show confidence in your ability.
A: The theoretical physicist Richard Feynman was in a similar state of mind, he referred to it as a "burn-out" feeling: Now that he had landed the University professorship he had strived for, he felt the obligation to do something "important", but he had lost the joy of doing science just "for fun". In his autobiography he describes how he recovered:

Then I had another thought: Physics disgusts me a little bit now, but
  I used to enjoy doing physics. Why did I enjoy it? I used to play with
  it. I used to do whatever I felt like doing - it didn't have to do
  with whether it was important for the development of nuclear physics,
  but whether it was interesting and amusing for me to play with. So I
  got this new attitude. Now that I am burned out and I'll never
  accomplish anything, I've got this nice position at the university
  teaching classes which I rather enjoy, and just like I read the
  Arabian Nights for pleasure, I'm going to play with physics, whenever
  I want to, without worrying about any importance whatsoever.
Within a week I was in the cafeteria and some guy, fooling around,
  throws a plate in the air. As the plate went up in the air I saw it
  wobble, and I noticed the red medallion of Cornell on the plate going
  around. It was pretty obvious to me that the medallion went around
  faster than the wobbling. I had nothing to do, so I start figuring out
  the motion of the rotating plate. [...] Then I thought about how the
  electron orbits start to move in relativity. Then there's the Dirac
  equation in electrodynamics. And then quantum electrodynamics. [...]
  The whole business that I got the Nobel prize for came from that
  fiddling around with the wobbling plate.

A: A small practical suggestion on how I sometimes handle the pressure you mention. 
Something new is really a fuzzy concept: maybe something you think to be known has a yet undiscovered subtle mistake in the original paper (and thus is technically not known to be true), maybe something apparently brand new already appears as a lemma in some article you've never heard of in another field, maybe it is an almost trivial variation on a well known idea and it does not deserve to be called new, maybe every expert in the field kind of knows it but no one ever wrote it down in full detail and rigour. Novelty is really, because of the way we do mathematics in this age and time, a continuum ranging between the "published many times over in different guises" and the "no one has any clue even about where to start", rather than a binary distinction between known and unknown. 
(The only distinction where you can precisely draw a line is between what is known and unknown to you. Whence the other answers stressing the importance of learning known facts as a preparation to and a diversion from research.) 
So, if it helps, trick yourself into thinking of the open problem you're working on as if it were something whose truth is already known to humanity but whose proof is buried in a very old journal issue which is not to be found anywhere, or which needs a simpler proof than the long and complicated known one, or whose solution already is in the air in the mathematical community and which just needs a keen young researcher to connect the dots.
Sometimes a bit of reductionism can ease the psychological pressure that's behind the novelty part of a new result; do not think of yourself as a daring explorer who ventures in the vast uncharted lands of the unknown, but as a modest worker in a familiar place who is putting together pieces of several familiar puzzles in a moderately new arrangement. (I think someone once said as a hyperbole that not a single original idea has ever been had in mathematics, to emphasize the aspect of combining known facts in new ways and seeing them in new lights.)
A: Unless you work on an important open problem and someone solves it before you, not much harm is done. Very often your solution provides a different perspective on the problem and you can prove a slightly different or stronger result with your approach. That happened to me a few times. Also, if your solutions is more elegant and easier to read, people will be interested in your paper as it will clarify and popularize the field.
A: Others had said it better than I will, but let me share my own experience.
In the past, when I was working on my PhD, I felt that pressure quite hard. But it was not a “moral” pressure like yours: it was a “material” one. If I couldn’t find something enough interesting to be named a PhD thesis, I wouldn’t be a mathematician at all. 
That was an awful feeling: working with that feeling in the back of my mind was the worst thing I could have done. In the end, after some years of suffering, and changing my thesis advisor, the thesis was written. I have to credit my second advisor for it very much. But also, I guess, because without noticing it, I just forgot my fears and began to enjoy the maths I was doing for the sake of it. Because the problems I was facing –and eventually solving, some of them- where great fun!
My career as a math researcher has had its ups and downs after that, but the best moments are always when and I have fun doing it: reading papers from other colleagues, studying books to learn a new subject, writing some papers, making a few speeches now and then… Eventually, reading, studying, writing… you eventually ask yourself questions: “Really? –I don’t understand this.” Or, “I don’t understand this THIS way. Let’s try it to write it the other way around…” And, maybe, sometimes, this try becomes a paper –most of the times it doesn’t, though. 
Either way, the most important thing, I insist, I’ve been doing this because I enjoy it: it’s a challenge to understand someone’s other paper, a new technique, a new subject… It it’s not, if it’s boring, if I’m doing it because “I have to” –and not because it’s fun-, then I just quit: let’s forget this s***.
Turns out that, now and then, some colleagues found some of my papers deserved to be published in some not too bad journals. Great! –This is good for my status and, in time I get some extra pay because of it: not too bad. Others enjoy some of my speeches. Good: you feel good when others like what you’re doing. I have interesting conversations about maths with colleagues here and there and, since I’ve been very lucky, I’ve found funny people around the world to have some laughing and share beers talking about spectral sequences that don’t want to converge when you need them to do it: silly, isn’t it?
Most of the time, though, research is a lonely business: you, your sheet of paper, and your pen.
Do you like it? Do you keep soiling sheets and sheets of papers, from which you’re going to save just 10% -or less? Do you like it, or do you do it because “you have to”?
Well, I couldn’t survive doing it because “I have to”. But because I like it. Because it’s great fun.
A: I will address the non mathematical aspects of this.  Even though I understand it from the perspective of a graduate student of mathematics, I agree with Chris Godsil that it happens in many different areas apart from mathematics.
The key points in your post that strike me are pressure and "my job".  Can you reliably identify the pressure and its source?  Is your conception of "my job" accurate enough that the pressure it might generate reasonable and appropriate?  I recommend using not only your advisor, counseling services, colleagues, friends, and other face-to-face resources to help you consider these issues, but also to reexamine your value system, and see if what you are doing needs to change to fit your values.
I'm going to guess that your position and situation are similar to mine when I started graduate school.  I had a teaching assistant fellowship and enough resources to support a four year course of study.  I thought of it as a way to occupy myself, with the major pressure being to graduate.  This meant looking at many things of interest and writing a dissertation on something "new".  There were other duties too, but I took the perspective that this was how I wanted to spend my time, and that the requirements should exert only a reasonable amount of pressure.
One suggestion that might help is to change your idea of "my job" to adjust the pressure.  Maybe your job is to try different avenues of investigation and write a daily or weekly report of your findings, whether good or bad or neither.  This has the advantage in that you use and develop your perspective, and the report you produce is always new. The impact on others may not be, but out of 52 weekly reports, only one or two really need to have a great impact on others so the others will also consider it new.  For example, John Baez does a community service by his blog, even if he never does anything highly original other than present thoughts he sees or hears from other people.  He might even get some new research out of it.  Would you like that kind of job?
(For another example, my four years turned to seven, the dissertation while somewhat new was minor and had contributions from my advisors, and never got formally submitted.  I also found the pressure to graduate greatly reduced right after I presented my results in a conference.  So my "my job" got done, but in a different way than I had originally thought.  And I understood more about my values in the process.)
Gather advice from many sources, but seek from within to find an answer.
Gerhard "That's What Research Is, Right?" Paseman, 2018.03.11.
A: When someone discovers "new mathematics", they have in essence discovered how to think precisely about a concept genuinely foreign to the human race. 
An essential piece of this process is the formalization of your intuitions regarding these new thoughts; the nature of your true goal as a mathematician precludes the possibility of anyone really 'holding your hand' for the final steps in the process. 
There is no way better to hone your formalization skills (and your intuition) than to do math all day, every day, regardless of whether it is new or not. This is what keeps me going -- if it ends up being the case that I am too simple to really discover anything of novelty so be it, that is ultimately irrelevant.
Doubt is somewhat intrinsic to the process, and with good reason -- what mathematicians seek to do is not trivial, and is (in my personal opinion) one of the highest callings possible for a sentient being.
A: Being a student is a stage of learning. Being a research student is also a learning stage. The difference is mind-shift. The latter demands your focus on a single thing. You devote most of your time on this focused work. Nobody expects you to find new things. On the other hand, when you focus all your energy on a single thing the chances are more that you might find something new.
I'm not a research student. I had never been. However, I have never stopped learning things. Teaching is my profession. Students ask me questions. I answer them If I know. Otherwise, I study them and then answer them. This process help me to learn new things. Also, it helps me to find something new.
Sometime back, a student asked me a question that could have been answered using an existing formula (Derangement formula). Having not come across this formula, I tried my own way of solving the question. In the process, I found two new formulas that are different from derangement formula. Though the approach to the problem is the same, my processes were different. That is how my mind worked out. When I realized that a formula was already existing, it did not discourage me. Instead, I felt the 'aha' feeling.
There was a pressure that I had to answer the question. But I took my own time to solve it. Once I put my whole mind on that, It was just a matter of few minutes to get the result.
