I've graduated high school and I am joining college soon. The problem with me is that I'm not finding less tangible math interesting at all.

Some people find abstract math to be very beautiful, and I'm exactly the polar opposite. I am turned off by seemingly pointless abstract mathematical structures with no use whatsoever. For example, some people find group theory to be very interesting after learning the definition of a nilpotent group; I couldn't care less as I never got why would someone make those weird definitions in the first place.

I have found that I will find a topic to be (very) interesting iff that has mathematical "real-life"/concrete/tangible/physics related applications (e.g., proving isoperimetric inequality using Fourier Analysis or Prime number theorem with complex analysis). But to be able to appreciate how that theory relates to the more concrete applications (e.g., understanding the proof of PNT with complex analysis), I need to navigate through the material I find "boring", which turns me off since if an exposition of the topic doesn't routinely provide examples of such concrete applications, I get bored very quickly.

For example, currently I'm now working my way through Stein and Shakarchi, Complex Analysis and Stein and Shakarchi, Fourier Analysis. What really intrigued me at first is that you can prove really cool number theoretic result with complex + fourier analysis and motivate the Olympiad coloring proofs (see this link for the details of what I mean by this) with discrete fourier analysis. So when I started reading I was very interested and breezed through the first few chapters (Chapters 1,2,3 in both books). But now I'm on the section "The action of Fourier transform on F" and I find the chapter to be too much boring so I am now feeling disinterested.

So, mathematicians here: What are some tricks/strategies to keep students like me motivated with the material even when it feels less tangible to me?

PS: It's not that I have problem with understanding the material. In the chapters I read in SS, I didn't face any difficulty with any of the exercises/unstarred problems (though I did face lots of difficulty with starred problems.)


closed as primarily opinion-based by Somos, Morgan Rodgers, The Count, Will Jagy, Lee David Chung Lin Jul 14 at 2:44

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    $\begingroup$ The fact that you are not enthralled by abstraction for the sake of abstraction is not a bad thing. Maybe that's a good thing. I think the essence of mathematical beauty is that math tends to work out more nicely than we had any right to hope for, and when that happens it's as if we've glimpsed some deep and mysterious perfection in the universe. "God's thoughts", as Einstein would say. But also, have you considered other fields such as biology? Biology has become an extremely exciting field these days. $\endgroup$ – littleO Jul 13 at 23:08
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    $\begingroup$ You're not alone. Do what you are good at. $\endgroup$ – Somos Jul 13 at 23:13
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    $\begingroup$ To some extent, you have to study what is immediately interesting to you. But you also need to have some trust in the mathematical community. Many abstract topics are widely taught precisely because they have diverse applications, but you have to force yourself to be patient. (You mentioned complex analysis and the PNT. One of my favorites was proving the fundamental theorem of algebra via Rouché's theorem.) But I have also encountered what appears to be formalism for its own sake, or for the sake of ends that I did not find interesting. Discerning between these two cases is difficult. $\endgroup$ – sasquires Jul 13 at 23:26
  • $\begingroup$ @littleO I have read a bit of physics/CS and found them to be very interesting for the very same reason (they are very concrete stuff). It would have been nice if I got the option to do Math+Physics course (since then I could have read the math with a physical example in mind) but the college I'm going to (the only good math college in my country) don't offer phy+math but math+cs course. Anyway do you have some advice on how to keep myself motivated on the abstract stuff in first reading ? This is making my progress to be really slow... $\endgroup$ – phi-man Jul 14 at 6:17
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    $\begingroup$ @phi-man I don't know, but one idea to ponder is taking a backtracking approach to learning math, where you dive straight into whatever advanced material mosts interest you, and then backtrack to fill in any knowledge you're missing. Peter Scholze reports learning this way, for example. More info: math.stackexchange.com/a/1894348/40119 $\endgroup$ – littleO Jul 14 at 7:28

My guess is the majority on this site enjoy the abstraction quite a bit, and even go out of their way to try to generalize their understanding of a concept independent of any tangible application.

You shouldn't be discouraged by this though. Take a breadth of courses when you get to college, talk to some student organizations, find an applied stream that strikes the right balance of technical minutia and applicability.

In terms of motivation, my experience at university has been that the utility of a mathematical concept isn't always apparent when it's being instructed in a class setting, or even from a textbook for that matter. Sometimes you have to go out of your way to find that information, which I would suggest if this is paramount to you.


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