# How to stay interested in less-tangible math

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, math educators: 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.)

• This may simply be a case needing to gain a bit of mathematical maturity and mathematical breadth (different things), both of which very few high school students will have much of. Indeed, the fact that you're looking at nilpotent groups and complex analysis suggests that you might have leapt forward quite a bit in the last two or three years with your personal studies, and what you're feeling could be a by-product of this. To me, "motivate the Olympiad coloring proofs with discrete fourier analysis" is putting the cart before the horse, which seems to support my first sentence. Commented Jul 15, 2019 at 13:01
• I gather you enjoyed the first few Chapters in Vol 1 of the Princeton Lec. in Analysis. I felt the same. The discussion of the wave equation motivating the expectation of a product solution etc. is very nice. Generally speaking, that series (Vol. 1,2,3 and 4) are MUCH more interested in motivation than most analysis texts, so I'm afraid reading elsewhere is probably just going to increase your malaise. Anyway, you're not really interested in applications, I think you're interested in the big picture of things from your post. Be careful of thinking of the PNT as an "application". Commented Jul 15, 2019 at 14:47
• I love that you think of the PNT or the isoperimetric inequality as "tangible results". I hope that I have not misread your post. We need more students like you in this world. Commented Jul 15, 2019 at 14:51
• It's not obvious to me at all that this is off-topic. The self-learning tag doesn't automatically make a question off-topic. Commented Jul 15, 2019 at 21:49
• @Namaste I never intended to imply that I am the sole arbiter of whether something is on-topic. However, OP did make an effort to explain what they meant by "tangible" using multiple examples. Further, small edits to the question (I made them) easily make it more clearly on-topic. You are right that we can remove the attempt to poll users on whether they feel the same. Please feel free to make further edits. Commented Jul 15, 2019 at 22:47

Your remarks about group theory ("I couldn't care less as I never got why would someone make those weird definitions in the first place") suggest that you need to see the ultimate goal to appreciate the path to that goal.

(1) For nilpotent groups, a natural goal may be to understand which polynomial roots can be expressed in terms of radicals. Once you see that solvable groups are a key to this question, and that nilpotent groups are solvable, the pursuit makes more sense.

(2) Continuing with group theory, a natural goal is understanding the symmetric wallpaper patterns. Then studying the crystallographic groups, ultimately showing there are exactly $$17$$, is motivated.

(3) And just for fun, Group Theory in the Bedroom gives an interesting application of groups to turning a mattress to balance usage. In fact, this whole book is one example after another of interesting questions/goals that lead to substantive mathematics. The math is motivated by the questions in a way not always evident in textbooks. (I see this is along one of the lines suggested by @DaveLRenfro.)

Hayes, Brian. Group Theory in the Bedroom and other Mathematical Diversions. Hill and Wang, 2008. AMS review (PDF).

(expansion of a comment made a week ago)

When you get to college I recommend actively discussing math topics with undergraduate peers and graduate students (assuming a university with a graduate math program). This will help insert you into the mathematics community --- undergraduates will tell you about the Putnam Exam and which professors to approach (and which professors to actively avoid) and the like, while graduate students are useful for giving you an idea of what graduate school is like and advice on applying to graduate school and the hurdles for beginning research in math.

Also, to maintain interest, it helps to read books that could serve as inspiration, such as "What is Mathematics?" by Courant/Robbins, "A Course of Pure Mathematics" by Hardy, the 3-volume "Feynman Lectures on Physics" by Feynman (yes, it's physics, but despite Feynman's sometimes disdain for aspects of pure mathematics, I find his writing to be very VERY strong with what Terrance Tao calls the post-rigorous stage), "Visual Complex Analysis" by Needham, "A Concrete Approach to Abstract Algebra" by Sawyer, "Adventures of a Mathematician" by Ulam, etc.

Spend some time browsing university library shelves to get an idea of what exists in mathematics (besides what random googling provides) and to help regain interest in mathematics when you're too study-tired and jaded to do any formal work (working problems, studying texts, etc.). Look through back issues of expository and undergraduate level journals you find on library shelves for things that might catch your interest and which you otherwise would not have encountered. Here's a possibly dated list of such journals that I posted back in 2005. Probably the 3 MAA journals and The Mathematical Gazette are what I'd most strongly recommend browsing. Also, keep in mind that unlike the case with many other academic fields, math articles from 70 or 80 years ago can be just as useful as those written in the last 5 or 10 years.

I quite liked the comment (on the other site's closed version of this question), which said "You're not alone. Do what you're good at."

Consider, Richard Feynman:

Weiner:

Were you majoring in mathematics? Feynman:

Oh, yes. That’s interesting. At first I was in the mathematics course. It doesn’t make any difference what course you are in, really, very much. For the first year or so you take more or less the same thing. You take physics and chemistry and electrical engineering, mathematics, and so on. English. Somewhere around the first year I began to get upset. This wasn’t right. The mathematics, I looked at it, the mathematical things, were too abstract. They weren’t connected to anything, mathematics. And I went to the head of the mathematics department. This was in 1936, now, so you know it’s still in the Depression. I said to him, “Sir, what is the use of higher mathematics besides teaching more higher mathematics.” So he said, “Well, you could become an actuary,” calculating the insurance rates for an insurance company. This didn’t sit well with me, see. He also said that a man who asked that kind of a question is perhaps not right for mathematics. And I thought the thing I ought to do — I mean, I liked to get my hands dirty. I’d had a laboratory. The physical world was real, and the mathematics, I had become enthralled with, but not for itself, really — you know what I mean? It was fascinating, but my real heart was somewhere else. So I decided, I have to get my hands dirty, I can’t stand these abstract things. So I changed to electrical engineering, because there was something that was real. But then some few months later, I realized I’d gone too far, and that somewhere in between — that physics was the right place. So I moved around a little bit at the beginning, and ended up with the physics course."

https://www.aip.org/history-programs/niels-bohr-library/oral-histories/5020-2

This is not to say you should end up in physics, or EE. Math might be right for you. Or maybe you want to be even more far from math like in chem/bio/business/CS/English/etc. Nor that there aren't people who will enjoy math best. But the number of people doing abstract math is a small minority of those who learn math because it helps them with science or engineering or business or etc.

For a humorous take, see: https://xkcd.com/435/

You will even see some differentiation of math for majors and non-majors courses. For example, in statistics, where having intuitions and knowing how to work applied problems is more important to [everything else] and where understanding the justifications is more important to math majors. Another example is in group theory...where asymptotically approaching zero chem undergrads or grad students take abstract algebra. Instead, the minimal group theory needed for understanding molecular vibrations and IR spectra is taught in specialized "group theory for chemistry" texts. Another is engineering and physics where "engine math" or "math methods for physicists" sort of works as a stripped down PDE and linear algebra class (sometimes even a little complex analysis or the more practicial series manipulations and such in real analysis...I.e. actual computations and caring about the "Yo, Adrian" Bessel functions and the drum vibrations...but not having time for full courses, especially with emphasis on theory, and sometimes even shirking computational and applied problems, in math major courses.

Beware of answers on M(E)SE, Reddit, etc. from math majors or instructors who sometimes have a perplexing blind spot here...in recognizing the crushingly evident fact that the vast majority of math studied in the world is NOT for future Andrew Wiles wannabes, like uh...them.

• (+1) Technically not directly answering the question asked (possibly why there is one downvote?), but a useful commentary on what might be behind the question. Also, given that the OP (at question-posting date) just graduated high school, it's way too early to be worrying about finding group theory and the later chapters of complex variables and Fourier analysis texts to be too abstract, since the OP is about 3 years ahead and the problematic issues might simply be racing ahead too fast, something I was a champion of in HS and early college years and so I know what I'm talking about. Commented Jul 18, 2023 at 19:15

The trick is to start perceiving mathematical objects as "real" as the ones in the so called "real world". In fact, they are way more real than the physical objects, but I'd rather not digress on that now and concentrate on one more thing that is rarely taught or even mentioned.

You can develop a dual vision in which you see abstract mathematical setups as common situations and vice versa easily translating from one language to another. Here is an example that I recently ran over with my students (it was pure improvisation: they asked me how to solve some problems from Analysis and I tried to explain a bit how one could visualize the situation).

Problem 1 (abstract mathematics): Given a countable set of sequences $$a^{(n)}_1, a^{(n)}_2,\dots$$ increasing to infinity, construct a single sequence $$a_1,a_2,\dots$$ still increasing to infinity such that it is eventually smaller than each sequence in the given countable family, i.e., for every $$n$$, there exists $$K=K(n)$$ such that $$a_k\le a^{(n)}_k$$ for all $$k\ge K$$.

The solution for the finite set of sequences would be trivial: just take $$a_k$$ to be the minimum of $$a^{(n)}_k$$ over all $$n$$. The minimum of finitely many growing to infinity sequences still grows to infinity and we are done.

However the countable case was not so easy for them. And then I restated the problem in the following way.

Problem 2 (traveling on a railroad) You have a railroad going from your current location $$X$$ to infinity with infinitely many stations on the way. Every hour a train departs from $$X$$ and each train is slower than the previous one. You want to use these trains to go to infinity but you should eventually find yourself behind any particular train. How do you do that?

The answer came in about 3 minutes: you take the first train to the first station, disembark there, wait for the second train and take it to the second station, then wait there for the third train and take it to the third station, and so on. After that it took some time to convert it back into the abstract mathematical language (there are some pesky translation details that I'm intentionally skipping), but once the idea was there, it all was more or less routine. The main point, however, is that for a mathematician there is absolutely no difference between Problem 1 and Problem 2. Solving one of them is pretty much the same as solving the other and, because of that equivalence, they have equal "tangibility" and other properties, though Problem 2 uses next to no mathematical notions (the only one that is really there is forward infinity in time and space, which most people would, probably, consider "physical" too).

At more basic level of elementary calculus, there is no difference between a girl holding a dog on a leash with the dog holding a cat on a leash and a question how short the leashes need to be to ensure that the cat doesn't travel more than 10 yards away from the girl on the one hand, and the formal standard $$\varepsilon/2$$ argument that the limit of the sum of two convergent sequences is the sum of the limits (or that the sum of two continuous functions is continuous, or...) on the other hand.

At more advanced level, one of the proofs of the Ergodic Theorem is just the big yellow bulldozer pushing a huge sand pile that fills all holes in the road on its way. I really keep in memory nothing but that about the proof when I come to the board to teach it: the rest is just the translation of that image into a chain of formulas.

It won't be a big exaggeration to say that (almost) all mathematical concepts (definitely all basic ones and the vast majority of even the most sophisticated ones) can be translated into plain English and the underlying ideas explained on the setups having nothing to do with "mathematical abstractions".

Why isn't that done more often? The reason is the same that we don't normally accompany our typing in English by pictures except in the children books: we rely upon the idea that it is enough to type 5 letters "g-r-e-e-n" to bring in the reader's mind the pictures of the forest, the lizards, the traffic lights, the stale bread, the ecological party, etc., etc. What makes it "tangible" for us is just a huge association table in our brain that contains the "green" entry in many places. By itself, however, it is not even a certain wavelength range: all you saw when reading this passage was 5 fancy wiggles made out of a few dozen pixels on the screen: pure intangible meaningless abstraction, as it is. The same happens with mathematics: as long as the long association chains are not present, you just feel like the book author juggles the same 23 symbols over and over again in various combinations for no apparent reason: hdudfhwuio jddljk hjscf kkfelr.

What to do to develop those association chains? My advice would be to relax a bit on studying the theory and pick some problem solving books. I don't know what your favorite area of mathematics is so I'll abstain from naming any titles for now, but trying to figure things out and to navigate that parallel Universe by yourself is a lot of fun even if the solutions of some particular problems are already known to humanity and you can pretty easily take a sidestep at any point to find yourself on a totally uncharted territory.

Finally, before trying to answer the question "What is a pure mathematics good for except producing more pure mathematics?" try to answer the question "What are human beings good for except producing more human beings?" It is actually the same question in the sense I tried to describe. Once you find a satisfactory answer to either version, just translate it into the other setup.

Just my two cents.