Graduating with a Bachelor’s in (Applied) Mathematics is something I’m very proud of. I don’t remember being naturally talented in any capacity at mathematics (Abysmal grades in high school mathematics, failed pre-calculus several times, and got a ‘C’ in Calculus I) but somehow I stuck it out and got better at math, and got OK grades. Good enough to barely scrape by with an ‘Honors’ tag on my diploma, but likely not good enough to get into a PhD for Math. Oh well. I guess the mysteries of functors will remain mysteries to me.

Studying math was something that I thought was pretty hard. It was never a breeze for me, and many classes that my peers said were easy were classes that demanded my full attention for me to succeed in. If I slipped up, or failed to push myself to understand a concept that didn’t appear important, I would usually take a hit on my test score, or my final grade. I would say that I struggled more than my peers throughout my undergrad career. But struggle tends to teach you a think or two. After closing out my first academic quarter as a graduate student in a different area, I wanted to reflect a bit on what doing a bachelor’s in Math taught me. How it changed the way I think. The following list of ‘lessons’ is in no particular order. Some lessons are mathy, some are not. Some of these, you my disagree with. Well, they’re lessons that I think I learned, so here they are.

1. How to think carefully.

Part of mathematics is deriving logical conclusions from assumptions. Axioms. Whatever. When people say that they are ‘logical’ they usually don’t have a good sense of what ‘logic’ really means. Usually what they mean is ‘rational’. As in, “I carefully consider my options and the consequences of each option, then I choose the best one.” Most of us do this, one way or another – soaking up those utils – but I don’t think many people know what it means to explicitly state some assumptions you have, then argue towards a conclusion. Politics teaches us to argue towards conclusions, but without making mention of assumptions. Most people think this way. In mathematics, the assumptions are important, sometimes the most important part, of an argument. Studying mathematics taught me to chew on assumptions for a bit, before making statements. Of course, I’m not perfect, and sometimes I believe just whatever I want to believe, but at least in math I try to be careful.

2. How to deal with failure/doubt/insecurity.

This one is an ongoing process. Does anyone ever really figure this out? There is a benefit to being ‘bad at math’ the way I was. I was wrong a lot, and am still wrong a lot. I will argue with you about it, since I don’t want to be wrong, but it’s hard to look at a counter example to your proposition in the face and not admit defeat. That, coupled with the occasional bad grade, cast a lot of doubt on my intellectual aptitude. I often had to deal with failure, with subpar performance. I’m better at it now than I was, but it’s an ongoing process. I didn’t ever get straight A’s in undergrad. I didn’t get straight A’s my first quarter in graduate school – there’s nothing to do but keep trying. What else is there?

3. How to fill in gaps.

Math is interesting in that you never hear a direct answer, or give a direct answer. There are always a need to “fill in the gaps”. Even full solutions to an exercise often leave the conclusion for the reader, or hand-wave a necessary step, leaving you to supply the missing steps, or information. As far as I know, other fields don’t do this. My experience is limited, of course, but when I was taking an Organic Chemistry sequence in my brief stint as a chemistry major, I never saw a solution to an exercise of the form: “Try recognizing the fact that the compound is in alcohol” or something. In what world is a ‘solution’ given in the form of a hint? Mathematics. The truth is, there is always a gap to fill. You can always ask “why can we do this” when reading a proof. It’s turtles all the way down. You have to decide how deep to go, and when enough is a enough. A valuable skill. Especially when you need to learn how to do something you’ve never done before. Or code.

4. Math is many things, to many people.

A friend of mine likes to joke that combinatorics isn’t math. I think it is math, but I can see where the joke stems from. To him, Math seems to be a deep theory. Axioms lead to theorems, which lead to other theorems and consequences, and initiate connections between seemingly unrelated fields. The deep stuff is the math, the ability to generalize vast swathes of results, and contain them succinctly. To someone who struggled with fractions, math is fractions, or something physicists care about.

To me, math is a tool, and a generator of (interesting) puzzles which lead to more useful tools. I suppose the fact that ‘Applied’ appears in my transcript gives me away, but is there anything cooler than using probability theory to prove the existence of a certain structure? Actually, yes, the use of stochastic processes to model time series data, or predict cancer evolution is up there too. Currently, I’m learning about the use of math to rigor-ify algorithms. Very cool stuff. Maybe this isn’t deep enough, but it’s rigorous, it’s interesting.

On the other hand, I often have difficulty seeing math as art. I know some people think this, they’re out there, but I think the position is a little stuffy and elitist. But then again, I don’t know what a functor is, and there is the word “Applied” on my degree.

5. Doing exercises is the point.

What other reason is there to read and learn math if you will not use it to solve problems? What good is knowing math just to know it? If you buy a video game, do you play it? Or do you buy it to read the synopsis on its fan-made wiki, then talk about it with others? Do you do rock-climbing to get shredded forearms, or do you climb so that you can climb more, and higher, and experience the thrill of conquering? I think that doing things is better than knowing things, and it took me a long time to realize this. Math is about doing. You read the book so you know the definitions and the theorems, so you can solve the exercises, so you learn the subject more deeply, so you can solve more problems, so you learn more, so you can (eventually) solve a problem that no one has solved. And why did we solve the problem? Because it’s fun. Because you can conquer something. Because if you do it, then you will learn more, and you will teach others something, then you and they will solve more problems.

It took me a while to come to this conclusion. Reading math used to be really exhilarating for me – now I find it a little boring. I don’t think my passion has waned, but it has shifted. The joy now comes in doing exercises, in conquering. It took a while to get here – for about a year after I switched majors, I could solve almost no problems on my own. But, now I can. Sometimes I get stuck, or I need a hint, but is there anything sweeter than solving a problem, start to finish, by yourself?

I remember pretty clearly when I first started being able to solve math problems – real ones in Real Analysis. I was attempting to solve the chapter 7 exercises in Baby Rudin. Sequences and Series of Functions. I don’t really know what got into me, but I remember thinking that I was tired of relying on hints and office hours. I went to the library by myself, and I turned my phone off. I spent about 5 hours there, staring at the book, and writing on paper, trying examples and small cases of the problems. When I got stuck, I took a walk. Something clicked. I wrote down solutions to 4 of the 7 exercises we were assigned that week. I think it was the first group solutions that I really got entirely by myself. They were correct – I knew they were, but I checked with people anyway, and they were right. I felt pride. Before, I felt like a fraud, often having to rely on others to solve problems. That feeling of pride carried me the rest of the way through my major – I felt like quitting a lot before then. Eventually my skills improved, and I gained confidence. Now, I’m able to solve some problems out of a graduate level book in combinatorics – pretty much by myself. Maybe that’s a low bar from a math student, but it’s a nice feeling – much nicer than simply reading. Doing the math is the point of math. Paul Halmos was right – Math is not a spectator sport, and that’s what I like most about it.

***

Really, I think this last point is more generally applicable. There was a time that I did almost nothing, but I thought a lot about what I could do. I reveled in my potential but rarely exercised it. Rarely tried to actually test it. Those times were when I was most troubled, the most unhappy, and the most angry. Gradually, I began to resist against this. I realized that the interesting people around me were interesting precisely because they had actually accomplished something. Because they had actually put in the work to get the training, to get a cool job, or be useful to society. They could *do* things. I merely dreamed that I could do things. When that began to change for me, I began to change. There is a lot of satisfaction I get from a job well done, from a solved problem, to a good grade in a difficult class, to conquering a fear, or to getting a math degree despite lack of talent. There is almost no satisfaction to be had by reveling in your potential, but there is a lot in trying to realize it. Of course, failures come. But sometimes you win. And the more you win, the more you win.

I think math is beautiful for it’s power and applicability, but also the lessons it can teach you. Supposedly, to master a martial art requires inner tranquility. I like to think that to master mathematics would require the same. But maybe that’s too stuffy. If you don’t want to think about all this, just try doing some exercises out of a math book – alone. If they are too easy, try something a bit harder. The joy is in trying to win – you need a challenge for that.