(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.