From time to time Mathoverflow allows soft questions because they are arguably best answered by active mathematicians and they can benefit other mathematicians/PhD students/math undergraduates. I think this is such a question.


I'm a mathematics student planning to enroll in a good math PhD program this Fall. I have always been extremely disciplined in math and my goal has always been to pursue a math PhD. However, I've had the opportunity to work in computer science, and this has caused some doubts about the significance of my future work in mathematics. I imagine such doubts are nonunique to myself and that the best place to ask is here, from people who've been through a PhD themselves, who are wiser, and who may possibly have had these same thoughts. (I hope it is clear I am asking this out of good nature and that this is not dismissed as a cynical thing to ask.)


My main question: **Is pure mathematics useful, specifically, outside of mathematics itself?** Instead of giving a definition of "useful," perhaps I can share some doubts I have about the significance of pure mathematics research.


1. It seems to me that in all honesty, pure mathematics does not immediately benefit the population at large in a direct and obvious way. At best benefits are usually theoretical (e.g., "These methods *could*...").
2. I think that very, very few people actually read and care about the average published pure mathematics paper. I think it's because **math papers are hard and it's not clear that they are interesting or useful to math as a whole or to the future of humanity**. There are very obvious exceptions, for example, for papers like Fermat's Last Theorem, which are arguably achievements for humanity. But most papers are objectively not of this level of significance and may not always contribute to major problems.
3. It seems that the only reason we, as a population, care about mathematics, is because of the "cool" open problems which are simple to understand but difficult to prove. But this account for only a very small portion of active and successful mathematical work (since math papers don't always try to solve such problems because they're very hard). So doesn't this imply that my work as a future research mathematician is actually not useful for the future of humanity?
4. It seems that pure mathematics was originally created to solve practical and interesting problems, and that as we turned to use abstraction as a tool to solve things (because abstraction is a very useful problem solving tool), we have arrived many years later to nested layers of subproblems of subproblems, whose depth is so deep that such problems of these areas are hard to understand and are not obviously useful for the world or for anything outside of that area of mathematics itself. It seems that mathematics is a science that studies itself, and so at a certain point, it does not have an immediate practical use outside of itself.


I can't be the only math person to have every had these thoughts. As a hardcore pure math person it almost feels like a sin to have such doubts (not literally of course). I would very much like to be wrong, to learn from anyone's objections, and to do my PhD as I planned (although I obviously can't enroll with these doubts and will just continue working in CS). This leads to my secondary questions: **Have any mathematicians ever had these thoughts? How did they reconcile these thoughts with their career choice?**



Is pure mathematics useful outside of mathematics itself?
This is not really an answer to the question as asked, but I believe it's important and relevant to your problem, and too long for a comment.


I will not here express any opinion about the validity or importance of your doubts, or share any of my own beliefs about them. Instead, the point I want to make at the moment is that, in my opinion, it is possible to pursue a PhD and a career in mathematics, and believe that one is benefiting the world thereby, while also believing that one's own research in pure mathematics is completely useless (regardless of the validity, or lack thereof, of the latter belief).


The point is that the majority of mathematicians in academia do not spend all of their time doing research; most of them also spend time teaching undergraduates. If they work at a liberal arts college, they may spend more time teaching than doing research. I believe it's inarguable that mathematics education is important for students, and those of us who teach them are benefiting the world.


One might say, then, why do research at all? Aside from the obvious answers that we enjoy it, I believe our research benefits our students as well (and many universities also believe this). This is particularly true when we are able to create opportunities for students to research with us (an experience from which they can learn a lot, independently of the value or lack thereof of the research they do -- like perseverence, problem-solving skills, etc.). It also makes us better teachers, by keeping us excited about the subject, giving us new ideas for ways to improve our classes, keeping us connected to a wider community of mathematicians, and giving us ways to convey our excitement about mathematics to our students.


Of course, this varies somewhat by university. At some research-focused universities, teaching undergraduates is regarded as something to get out of the way as quickly as possible to focus on research. Someone who approaches teaching with that attitude is probably not benefiting the world by their teaching very much. But there are plenty of colleges and universities where teaching is valued and supported by the administration and the community, and if you are worried about the possible uselessness of your research I would recommend that, in *addition* to reassuring yourself about the usefulness of pure mathematics, you put some effort into becoming a good teacher, and consider jobs at more teaching-focused schools.


