How To Study Pure Mathematics

[Randell Magtoto]

Ive tried to find answers to my question on this community as well as several others,but couldn't find a satisfactory answer,so here I am. I thank in advance, to anyone who decides to give time to my question.

Background: I am trying to learn pure mathematics on my own from books and any other resources I can find online. I used to do contest mathematics an year or so ago,but have lost touch with almost everything over the past year. Now,I decided to study maths again and am hooked on the maths again but a little lost. I've read A short introduction to mathematics by Timothy gowers. And am currently planning to start the book what is mathematics? By courant and Robbins. I have the following plan for my studies after this book,

After that I pretty much have no idea of what things are and in what order am I supposed to study further. My plan is to study these books and continue further studies with the help of the further readings in these books,use online resources like Khan academy, mit open course ware,etc.(not a very good plan,I know).

If your plan is go from having a highschool level understanding of mathematics to an undergraduate degree level understanding from self-study, then 2 years is extremely optimistic. Depending on your work ethic I would say such a venture would take closer to 4-5 years or maybe even longer.

An undergraduate degree in mathematics covers a massive range of topics. By the time you finish an undergratuate degree you will have covered number theory, analysis (real and complex), group theory, linear algebra, differential geometry, topology, combinatorics, graph theory and more.

Now I don't want to discourage you because learning mathematics on your own is certainly do-able. In fact I think your current plan is good as a starting point. But realistically the resources you've listed here will only cover basic plane geometry, introductory calculus and introductory algebra. That's fine though, because these are the things you should be starting with. If you get a good understanding of these things then it will put you in a good position to start learning the more advanced stuff I listed above.

Like I said, don't take these steps as gospel, I've thrown them together based on my own experience learning mathematics, so some topics that others consider to be vital for understanding pure mathematics are probably missing. So following what I've suggested here will not give you an equivalent education to an undergraduate degree. This is what I think you should do if you want an approximation to an undergraduate understanding of pure mathematics. To be honest by the time you get to step 4 you almost certainly won't need this guide. You'll have a clearer picture of the landscape of mathematics, what you need to learn and also what you're interested in.

Other than that my only advice is similar @Alexey Burdin's comment. Don't give up, mathematics is hard so don't get frustrated if you don't understand something! There isn't a mathematician on earth that understands everything first time.

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.

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?

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.

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:

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 . . .

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.

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