Is pure mathematics useful outside of mathematics itself? 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.

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*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...").


*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.


*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?


*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?
 A: 

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*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...").


Totally valid and arguably a fact; the trickle-down effect for pure mathematics research (when it exists at all) takes decades optimistically to reach the open mouths of the populace, but this could be argued as a virtue. In a branch like CS you could realistically see your research being 'used in the real world' in your lifetime, but how it's used wont necessarily be decided by you (a la Oppenheimer, Good Will Hunting scene, etc.).



*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.


The first part here is again pretty much a fact; mathematicians are a small subset of all humans, and even within that subset the average algebraic geometer won't care about the average paper in fields that are (arguably, relatively) closely related like category theory, and vise-verse. Where I vehemently disagree is why people don't care; there's just too much to even be aware of all of it, let alone care deeply for every result. This isn't anyones fault, but rather an apparent feature of knowledge -- there is a lot of it, too much for any one person.
People generally only care about a thing if it is relevant to something else they already care about, with the most primitive thing being ourselves/our loved ones. Some math/CS/science etc. is 'cared for' because people find it relevant to the things they care about, but most of it is too abstract for them to tell; even for other mathematicians there is often accidental interdisciplinary blindness to potentially relevant results between fields, simply because we can't tell or are unaware of what's happening over there.



*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?


I forcefully disagree here; I intend to spend my whole life doing math, and I haven't ever encountered an 'open problem' in the classical sense that I really care about more than any other theorem or lemma (I love them all equally). I care about mathematics because it allows me to think about things precisely, and there are things I want to think precisely about (black holes, origin of the universe, multiverse, etc.) which I have no idea how to think about at all without mathematics, except as armchair philosophy. Even with mathematics it can be a years long slog to get the thoughts correctly written out, but clarifying thought and making it precise is the primary purpose of mathematics in my opinion and a very important one for the species in general.



*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.


citation needed
But really, where did you get this impression? This sounds like a critique made by someone in a related branch who doesn't have a great opinion of pure mathematics, but as mentioned before what I believe math strives to study is pure, precise thought with no extra fat attached to it. As a Platonist I believe that an ideal realm of concepts underlies our reality and every possible reality in the multiverse, so I would go further and say that we're attempting to systematically explore and map out the abstract realm underpinning all possible realities, but this is getting a bit far afield for MO.
A: Alexander Grothendieck, 1966 Fields Medalist and considered by some as the greatest mathematician of the 20th century (see Wikipedia page, that cites this obituary), had similar doubts.
He certainly extended them to the point where he considered that scientific research as a whole "does not immediately benefit the population at large in a direct and obvious way", to say the least.
He finally discontinued his participation to the global research effort (although he may have continued to work outside the system, for his own pleasure).
A: We cannot tell where our work is going, but the huge payoff that applications of mathematics have had makes it worth while, even if almost all of our work is going nowhere, to keep digging. Mathematics becomes more unpredictable as it arises in more applications. Looking back at the history of mathematics, it would have been impossible to predict which directions of research would turn out to be the most useful. In the nineteenth century, one might have bet on Ceva's theorem, as a basic fact about the geometry of our world, and a fact which is not obviously true, but easy to remember, providing a unification of many basic results in geometry. Few would have bet on Boole's research into the nature of human thought. No one could see how to use it, and it sat outside of the mainstream of mathematics, with no connections to previous work. It did not unify. Today Boole's work plays such a foundational role in our world that electric tea kettles use 0 and 1 to mean "off" and "on". By studying human thought, Boole changed how all humans think about almost everything. I have often taught Ceva's theorem, but I have no idea how it could fit into applied mathematics. Apparently Ceva's theorem is related to some integrable systems in mathematical physics, so maybe ...
A: Adding beauty and joy to the world, contributing to humanity’s understanding: these are direct and immediate benefits from pure mathematics, even if they are not fiscal.
A: 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.
A: It's kind of funny that when I was a young silly undergrad my view on Math was completely opposite to yours: in my immature eyes the best Math to get involved with would be the most impractical one! I wasn't aware at the time of the Steve Jobs quote about making the dent in the Universe, but that was basically the idea. To go somewhere where no man has gone before. To study the Forms, not the Shadows. To discover something beautiful outside the mundane realm of practical applications.
Looking back, I was wrong. The most fruitful pure Math is not perfectly pure; it grows around applications like a pearl grows around a spec of dust. This applies to the most pure abstract things as well. People try to solve algebraic equations, that leads to algebraic varieties, and that leads to schemes. You don't start with schemes. Maybe that's the reason that pure Math tends to find applications eventually, sometimes decades after its development. Maybe that happens because pure Math ultimately grew from the real World, maybe that's just magic, but even the most impractical abstract subjects somehow find their applications.
A: Why do you want current work in pure math to "immediately benefit the population at large in a direct and obvious way"? Applications of pure math might take decades or centuries. As much as you may wish this process could be sped up, that's not how it typically happens, and when it does happen the underlying math might be building on concepts in pure math that were developed for no real-world purpose a long time ago. See the following pages:
Real-world applications of mathematics, by arxiv subject area?
Recent Applications of Mathematics
https://math.stackexchange.com/questions/280530/can-you-provide-me-historical-examples-of-pure-mathematics-becoming-useful
https://math.stackexchange.com/questions/486855/what-are-some-examples-of-mathematics-that-had-unintended-useful-applications-mu
Even in experimental sciences, where you might think people are trying to do things to help society now, research is often done just for the purpose of general understanding of that subject area rather than for an immediate and direct application.  Yet decades later the ideas can become useful. See this video about the scope of research needed that led to the covid vaccines: https://www.youtube.com/watch?v=XPeeCyJReZw.  And there is a famous real-world use of relativistic calculations for GPS: http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html. Einstein was not trying to help people navigate their vehicles when he was contemplating relations between space and time.
A: Is pure mathematics useful outside of mathematics? The other answers shows that yes, it can be very useful, either indirectly or directly. There is also pure mathematics which is only useful in mathematics. And then I suppose there is plenty of pure math which is not very useful even in mathematics.
Why do research in pure mathematics? Well, probably the real answer for most pure mathematicians is because it's fun and because they like it so much. That is maybe not a satisfactory answer for grant applications, or when a non-mathematician asks you this question at a party.
So maybe you should also ask why should people pay you to do pure mathematics?
You could justify this by the potential applications, e.g. the very cliche answer "they said number theory was useless too, but now we all rely on it to stay secure on the internet!". But that is not a very good answer, since it is very unlikely that your research in pure maths is the foundation for something as important as public key cryptography.
Other reason might be that you can teach students pure mathematics, who can then do a PhD in pure mathematics and teach more pure mathematics to people who want to study pure mathematics.
Maybe you talk about how you are "adding beauty and joy to the world" as in one of the answers. In that case I guess the reason for your funding is the same reason we fund say literary criticism or some niche art. Although to be honest for most pure mathematicians their audience will be very small, but also to be honest the same is true for most artists as well.
Actually, you don't really have to justify your pure maths study/research to anyone. One secret is that if you find a way to do a PhD in pure maths and later become a postdoc and later a professor, they just let you do it.
A: I do not know if they are "useful", but couldn't we consider that computers, and the software they run, are very concrete realizations of "pure" mathematics?
They are filled with concepts and results from number theory, functional analysis, algebra, geometry, graph theory, probability, and so on.
They would certainly not exist without many fundamental works conducted in these areas (and conversely).
A: How does one define "pure math"?  One could even argue that the answer to the title question must be No, on the grounds that once some part of mathematics finds a use "outside of mathematics itself" then by definition it is no longer pure math . . .
A: If you ask the question in more personal terms:

If I study pure mathematics, will that be useful
to me anywhere outside of mathematics itself?

...then the answer is likely yes.
Even with your going to "a good math PhD program", the odds are that you will be out of academia by 2030 or 2035. And by then:

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*you will have learned more math (including math you learned only partially in college), and will probably use some of that math professionally;

*you will have learned some programming skills, and will probably use the skills to devise algorithms for more practical problems;

*you will have learned how to present math in talks and present yourself in interviews, and will probably adapt that for other presentations;

*you will have learned techniques for dealing with your advisor, your peers, your source of funding, and will probably use some of those techniques with a spouse or friends or colleagues;

*you will have learned techniques for managing your time and your projects, and will probably use some of those techniques outside grad school too.

So you can evaluate the utility of grad school in those terms -- or evaluate the utility of other ways that you might spend the next years of your life.
A: The question you ask is commoner among would-be undergraduates of math who are trying to decide whether to do the special honor (all 'pure' math bar a mandatory C coding course)  or the general honor course (math plus modules in economics, physics, biology, genetics, statistics and computing) in mathematics. In such cases, doubtful students would meet their tutor alone or in a group where this was a shared concern and - hopefully - get convincing reassurance flush with many examples. Perhaps you didn't have this experience.
Hardly a generation (30 years) passes before a new "abstract" field of mathematics finds a crucial application in the real world. Check out the history of various branches of math in the 20th century for yourself. Now, the readiness and extent of application depends a lot on the local industrial culture however. For example, Chinese researchers working on metal-forging problems do not hesitate to deploy topologists to gain insights on the best forging sequences. In the west, such things were traditionally left to the black art (practical experience) of the actual forgers in the shop floor. Be assured that this will soon change however as the benefits of math involvement become vividly obvious in everyday goods !
Of course, the extent to which application of a 'pure' math concept will "jump out" into the mind of the person trying to solve the problem depends on his/her familiarity with both the math concept and its area of possible deployment: pure math people will generally not be familiar with the latter, applied math/scientists/engineers may not be familiar with the former. So having purists available for consultation by the applied math staff on say a team modelling epidemic spread might be sensible.
We don't know clearly how involved you want to be with the real world or what aspect of this involvement you feel is vital for your wholehearted commitment to further studies here.
Have you considered not doing your PhD straight through after your primary degree ? Any college course can be exhausting on a person's morale - there are few exceptions. Maybe doing something in the applied math arena but away from the campus might be personally beneficial at the present time as you pick through your own thoughts on the matter.
A: Yes many people have had these thoughts, including myself.  I do not think that  this means that you are insufficiently passionate about math.
I do think it is important to decouple your general question: "Is pure math research useful?" from your specific career decision. I am biased and not really qualified to answer the general question,  but my impression is that the answer is yes: our society invests very little into pure math research (relative to other areas) and math as a whole is highly interconnected, so even the purest research areas are often only a few degrees away from more useful ones.  And there is a vast ecosystem of mathematical sciences in engineering, applied math, statistics, CS, and operations research departments which interact with pure math in various ways.
On an individual level, though, it is true that most papers go unread and only have a small impact.  And most people who get pure math PhD's (even from elite institutions) do not work primarily as researchers--most of the productivity of an individual mathematician is through teaching and communicating mathematics.  If a precondition for you is that your main impact on the world to be through research, you should probably not do a pure math PhD.
For how/why pure mathematicians handle this situation: one answer is that we are a bit unusual (and sometimes slightly selfish) in that we tend to care deeply about our subject, and not so much about others' valuations of us.   Another answer is that for some people pure math is their "comparative advantage"-- they have a special talent and if they were in a different subject or profession, they would not be nearly as happy or effective.
A final answer is that as you learn more, subjects that appear cold and esoteric suddenly transform: they become rich and full of profound, challenging and fundamental questions. And as you acquire mathematical fluency, you get to see more of the connections between different areas.  On the other hand, you may find that as you get older  (this is my own experience) you have more of a desire to connect directly with the rest of society.  It is not unusual for older researchers to transition towards more "applied" areas.
To conclude,  I think that it is good that you are asking yourself these questions before making a career choice, especially before committing to the long process of earning a PhD. You need to weigh your personal values, strengths, and desires in order to make a good decision.
A: There are certainly examples of mathematical topics which were studied for their own interest and considered to have no practical applications but later turned out to have important applications. I don't know that the Greeks had any application for the theory of conic sections. A couple of thousand years later they turned out to be essential to understanding planetary motion. Maybe small primes are applicable but primes of hundreds of digits seem far from applicable. But they are integral to the widely used RSA encryption scheme.
So do what you love and in 2000 years someone will need it!
A: I think you're right to have these doubts, that is to say, I have had similar misgivings about the "usefulness" of my work in pure mathematics, and not only that, but the "usefulness" of pure mathematics itself, and having pondered them for some years they have slowly ceased to be misgivings and grown into convictions.
This is not to say that there aren't differences in "usefulness" between various areas of pure mathematics, and it is also not easy to narrow down what "usefulness" should mean. However, I have never seen the fact that I can't define a word as a reason not to use it. Thinking in terms of definitions and axioms comes at the heels of intuitive thinking, even in mathematics, so all the more in philosophizing about mathematics.
For me, the doubts about the "usefulness" of pure mathematics were ipso facto proof that the word "usefulness" has a meaning, even if I couldn't give a definition. Now some may take this as a rather grandiose form of self-confidence, but they would miss the point. If I can only express my doubts by using the word "usefulness", then the word must have a meaning, because it is the only possible way to describe the specific form of doubt that I was experiencing.
And I am actually saying this by way of advice. If you force yourself, as you do, to not use a word because you can't define it, even though the word accurately captures your sense of doubt or dissatisfaction, then you are in effect preventing yourself from accurately describing and understanding your own problem. (Again: not accurately in the sense of every word being susceptible of a mathematically precise definition, but in the sense of being a satisfactory expression of your own emotional/psychological/spiritual state.) I know it can be intimidating to be among mathematicians and use a word in a vague, philosophical way, because they could ask you to be more precise, and then you probably feel you couldn't, and it would be as if you lost the argument or something. But it's better to be thought a little less of by some, if it means that you retain the freedom to express your thoughts in the way that they occur to you. This is a lesson that I had to learn the hard way, but I am glad that I learnt it.
However, it might be (as I glean from your post) that your conception of the term "usefulness" is a more utilitarian one. But I don't think utilitarian considerations can ultimately help you decide whether your pursuits are meaningful in a more profound and personal sense: the latter has more to do with your sense of usefulness, with "that which makes you tick", and there seems little a priori reason that this personal sense would be somehow perfectly aligned with the "greatest good for the greatest number of people". (Regardless of your stance on utilitarianism -- personally I think it is simply bad philosophy -- this would be a heck of an assumption to make.) This is especially the case because this "greatest good" is most often not something that is "felt", but rather an abstract reality, so not personally meaningful to the person who is pondering the question, but rather a dry and impersonal summary of things that contribute to people's general sense of well-being, and only up to a certain point.) And conversely, no amount of proof that activity X is conducive to the general good or happiness or well-being of mankind, could force you to feel enthusiastic or passionate about activity X, or to experience happiness while engaging in it.
In regards to my own doubts, I could be a bit more precise on what I could have meant by "usefulness", or rather by my feeling that pure mathematics felt insufficiently "useful" to me, but it is more of an illustration than a definition. What bothered me about mathematics was mostly the financial side of it. My salary was being paid by the government, which meant that I felt some sort of responsibility to be able to explain to the average person in the street why what I was doing was useful (not in quotes this time, but actually useful according to the definition of the average person in the street!). I had never been able to do this without somehow having the feeling that I was stretching the argument, that I was somehow selling a lie. And I had tried a lot.
But then the question remains: do your doubts about mathematics only reveal something about you, i.e. that you are temperamentally unsuited to do mathematics or something, or do they say something about mathematics as well?
I think the right answer is that I can't tell, because I do not know you. But let me once more tell you about myself. I think I am temperamentally inclined to like mathematics, even to be passionate about it. I have had generally good results as a one-on-one tutor, I enjoy conveying the beauty and the fun of mathematics, and I have often had people say to me that they were jealous of me for being so passionate about something. I think this pretty much seals the case for my "temperamental aptitude" for mathematics.
Yet at the same time, here I was, having these doubts about the usefulness of mathematics.
So I can't see this as a purely subjective issue, and I don't see it as such. Let me be a little bold, and put forward the following thesis. I think that some, maybe most, pure mathematicians have created an idol out of pure mathematics. Which is to say: they have somehow set it up als the ultimate goal of life, the greatest pursuit a human being can engage in, the search for the ultimate form of truth, etc. Of course, they wouldn't phrase it exactly this way, but it was definitely the kind of rhetoric that first got me into studying mathematics when I was in high school. Mathematics is the "queen of the sciences", it is the language that the book of Nature is written in, etc. The so-called Proofs from the Book are even ascribed to God himself, if you please.
Personally, I have often experienced a similarity between a "cool" bit of mathematics, and a "cool" new gadget such as the latest iPhone. (Interestingly, you also use that same word.) As with all "cool" things, they quickly lose their glamour after some engagement with them. When you first encounter a new bit of mathematics, it can have a veneer of the miraculous about it, but this is always lost upon closer inspection (in my experience anyway). There is a saying by Schopenhauer that touches on precisely this (I can recall it only very roughly): There are three stages in encountering a new truth, the first one is where it is not understood, the second one is where it is beginning to be understood and seems the most exciting thing in the world, and the third and final one is where it is fully understood, and is considered to be trivial and uninteresting. End quote.
So no, I have grown totally disillusioned with the claim, often made by pure mathematicians, that the pursuit of pure mathematics can contribute anything towards the good life. Now you might again like to enclose that last term within scare quotes, the "good life"; however, I think the term catches rather precisely what I myself am after when I am wondering about the "usefulness" of this, that or the other.
All this is not to say that pure mathematics can't still be fun. And insofar as "fun" contributes to the good life, mathematics contributes to it as well. I mostly experience this fun side of mathematics when teaching mathematics to high school kids and students, which I still sometimes do in my spare time. And if I manage to convey some of my enthusiasm to them, I often find myself warning them not to become enamoured with the subject too much. The key to being good at mathematics lies far more in precision, intellectual rigour and discipline (imho), than in imagination and speculative fervour. An overemphasis on enthusiasm and passion could very well obscure this fact, and steer a person wrong. As it did me.
