Theoretical and applied math help - what and why?

Question

(Edit: If you wish to skip the prologue, you may go straight to the questions in the last few paragraphs.)

I'm not very far ahead at the moment (going to begin my undergraduate years after next summer), but there's been something lingering in my mind, and I was hoping someone would be able to help me out.

For most of my life, I've defended "theoretical" and "pure" work, holding a belief that knowledge is the most valuable item one can possibly maintain. I expected myself to grow up and go into theoretical math (as do my parents at the moment), which would entail completing graduate research, eventually earning a PhD, then moving into professorship. Recently, however, I have come to the realization that this isn't the lifestyle that fits my character best, and I've been wondering how worthwhile theoretical math actually is.

I have studied group theory, discrete (graph theory), linear algebra, real analysis, multivariable/vector calculus, probability/statistics, and advanced geometry. To be honest, my favorite bit has probably been discrete (I am also interested in computer science, if it helps), closely followed by multivariable calculus and linear algebra. While I have performed very well across all the topics - and do enjoy the interesting, incredible bits of group theory and geometry - I am not sure where they ultimately end up being used.

Let me translate the question such that it is not specific to me, but so that many can connect. I do acknowledge the fact that theoretical math is very important - I hold absolutely nothing against it - but more so now than before, I have begun asking myself "How does it matter if the Collatz (3n+1) conjecture is proven true? It is an interesting property of all numbers, but why does it matter?" Of course, when people would ask me similar questions in the past, I would defend my stance, making claims along the lines of "It is essential to further our knowledge," or "Just because it's interesting." I do in fact find such random things interesting; I have windows on multilinear algebra and n-ary group theory open right now. Recently, however, I've begun to wonder what we get out of solving these problems and studying these topics, and I've been sensing a stronger attraction towards "applied math." Not to say that this is in any way unfortunate; ultimately, math is math, whether theoretical or applied. But let me move on to the question(s).

What is the point of pursuing theoretical math? Fifty years down the line, how do you reflect on your life? What if, although unlikely, you haven't been able to make any sufficient contribution to the field?

And as for applied: what are the outcomes from pursuing applied math? What are the similarities and differences in topics between applied and theoretical, especially regarding the ones I mentioned before? Does applied math prepare you better than theoretical for fields such as economics/econometrics, physics, engineering, and computer science? Do you ever feel "limited" in your knowledge of math by branching into applied, or is it of equal rigor and level, simply less abstract?

Regarding education: for each of the following, would knowledge in applied math or theoretical math be more useful?

• Applied physics
• Theoretical physics (seems obvious, but worth asking)
• Astronomy and astrophysics
• Engineering: robotics and AI
• Engineering: aeronautical
• Computer science

To clarify, I am not worried about the difficulty of the material; my concern is being able to branch out to other fields (mainly within STEM, but also into my other interests in the arts and humanities), being able to implement the things I'm learning, and most importantly, feeling satisfied with myself at the end of the day.

Thank you very very much for reading the long post; if you can, please do try to answer any of the questions I asked, as it will help me reach a conclusion.

Answer 1

TLDR : if the result is more important to you, go applied. If the road-to-result is more important, go pure.

I'm sure you are aware that behind many world-changing inventions lie an accumulation of "pure math" results, that seemed useless at their time. Traditional examples include number theory results that are used in cryptography, or Turing's idea of a 'computer'. And I'm sure engineers would be able to bring up more examples.

The way I see it, applied mathematicians are the ones getting out the 'useful' results, but they must rely on the history of 'useless' pure math results to do so. So pure math is not just about 'gaining knowledge' or studying 'interesting stuff'. Pure math also participates in changing people's lives. The thing is, we never know which results are actually going to be useful, and if so, when.

So if you work in pure math, the probability of your results having a real-life use during your lifetime is pretty low (I'd say close to $0$). That's considering the ocean of math papers that are getting published these days. So if you're a result-oriented person, maybe you should indeed move to applied math.

Personally, I'm a 'road-to-results' oriented person. So what's important for me is the way I get to a Theorem, how it's proved, and essentially, the fun I have working on a problem. Whatever happens to the Theorem once it's out, I don't really care (exaggerating a bit here but you get the point). Why I'm more into 'pure math' is that there are less requirements on the applicability of the problems you choose. Meaning, you don't have to find a justification for everything you do as much as in applied math. Therefore, it is easier to find problems that are fun for you to work on.

As for your concerns on moving from pure to applied. Well, I'm pretty sure you'll have no problem with the 'math' part of applied math. But if you want to be a researcher, the hard part might be to know what to study, i.e. what exactly is useful. The big difference, I believe, is that you'll spend more time on figuring out what "science needs right now" than trying to devise proofs and finding problems. The fields you mention above all make sense, but I guess you'll have to pick and focus on one.

I'm in bioinformatics right now, moving to something more like theoretical computer science and graph theory. So I'm going the opposite way. But I'd just mention that if you like biology, the field always needs good mathematicians.

Oh and here's a difference between the fields that I've observed. In bioinformatics (applied), you can get a paper accepted if you introduction is buffed up correctly, and the problem of study is well-justified. The math has to "look correct", but I've seen papers pass acceptance despite serious flaws. It actually depends on the reviewers. In graph theory (say), the intro is the past results and definitions, and the proofs are checked extensively.

Answer 2

I would argue, as you say, that knowledge is worth something even in absence of practical applications. However, that is not to say that theoretical mathematics has no practical applications.

In my opinion the following is the main difference between applications of "theoretical" and "applied" mathematics:

Pure mathematics is an investment for the far future, not for an immediate application. You are not likely to see a big practical application of your work in your own lifetime. See this interesting question for some examples of unintended practical uses of theoretical mathematics.

Pure mathematicians' impact is more indirect in the sense that the mathematical results they come up with are used by applied mathematicians to create practical applications - and without the pure mathematicians we probably would not be where we are today.

You could maybe make an analogy to biology. When Darwin studied animals he probably had no idea what it would lead to; and many would probably say "What is the use of studying different kinds of finches?" But it eventually lead to an amazing discovery. Similarly, theoretical mathematicians ask "This is interesting, what is there more to know about this?" which eventually may lead to great applications, whereas applied mathematicians ask "This is the problem, how can we solve it?" and they try to solve it.

Regarding you, it seems like you want a more direct, immediate impact of your work, in which case I would say that you probably are more of an applied mathematican kind of person. But that is just my opinion.