Mathematics: Journey & Reflection

This is a personal post about my journey of studying mathematics, formally and informally. I also talk about my favorite topics, reflect on my studies, and think deeper on math learning as a whole at the end.

A Tale of Homework “Math Sets”

In middle school, I showed promise of moving up a grade in mathematics, and my teachers at the time decided to have me do two math sets per night instead of just one - therefore putting me on the speed track to the next level. Mind you, math sets were normally around 30 problems a piece - the entirety of the problems listed at the ends of the textbook’s chapters. So for some time, I was doing 60 problems total each night, out of two different books.

Inspiration to Continue

When I applied to college, I applied mostly to physics programs simply because I was both interested in learning more science, and had some natural talent in mathematics, so it wasn’t entirely hell to deal with. I was also highly competitive. From reading on the internet, I knew that a number of people squandered the opportunities in university (and their full potential) by picking easy majors, and I didn’t want to make the same mistake and regret it.

As my application results came back, my gut feeling shifted to do math instead physics, but at the time I didn’t know if I actually had real interest in it, or if I just generally thought it was “cool”. Math seemed like a world of its own - an exclusive club full of esoteric tools and knowledge. I was curious about it, but never looked much into Calculus or any higher-level math between all my schoolwork at the time.

What really piqued my curiosity was how mathematics presented a completely unknown space to me at the time. It was full of “unknown unknowns”, and I was aware of this and wanted to go exploring down the rabbit hole. Math’s very own concepts are essentially impossible to even guess about as a regular person. While this exists in other fields, this characteristic is particularly intense in mathematics. What are “partial differential equations”, anyways? It’s certainly much more difficult to take a guess on that, compared to something, say, in physics like “thermodynamics”, which can be somewhat guessed about what it deals with off the bat just from hearing the word.

Also, the study of math can be a fulfilling independent endeavor. I spent a lot of my time tinkering in my workshop, and I saw parallels to how I could enjoy “figuring out” higher levels of math by myself through its concrete methods and rigid rules. Physics, however, seemed to take more teacher explaining for me to be able to do it, and was never quite as clear from just the textbook itself when compared to math. Maybe I am just lazy, who knows.

All mathematicians are lazy.

Into the Collegiate Abyss

As I started my first few semesters and got a feel for some subjects, I became more confident that math was the best option for me, that would not only fulfill my interest in the subject, but also stretch me to reach my potential.

The math courses were the ones I was most comfortable with. I knew what to expect of the workload, and I could sit down after the lectures, focus on figuring it out and solving the problems myself, feel like I “got” what I was supposed to “get”, and then do it again for the exam. I had the curiosity, and was motivated by it to reach those “esoteric” levels and see what sorts of incredible knowledge was hidden deep in the field.

In the end, I switched majors a few times between other STEM fields but decided to stick to my initial gut feeling to do applied mathematics as a major. Regardless, I took courses in all sorts of other subjects, some of which were physics, meteorology, psychology, and computer science.

I spent very little time writing proofs and instead dealt with applications regularly. The formal math education I received can be mostly summed up by the following list of courses and subjects:

About half way through, I went through a slump in motivation, got a few C’s and then rebounded. It was at this point I knew I had made the right decision to choose math over physics, because I hated the physics course I was in. Towards the end, I basically had a choice between continuing in school full time, or doing finance or data science as a job. I picked data science, did a capstone research project, and got a tangentially related job in data engineering.

Favorites

Numerical Analysis & Scientific Computing

I particularly enjoyed numerical analysis because of the iterative algorithms and multistep methods which could be coded into small (“small”) scripts and functions (root-finding, ODE-estimation, etc.). Much of what I learned of numerical analysis I also programmed in Python, and this was a great start into scientific computing (along with an intro course to it) before I started doing finite difference methods and handling more complex systems with code.

Numerical solutions of differential equations mixed both of these subjects and also incorporated some PDEs and interesting physics applications (like the heat equation and diffusion problems). I most enjoyed handling the algorithmic nature of finite-difference methods with programming. Numerical analysis is much more code-able for a beginner programmer than other math concepts, and I was just at the brink of becoming “good” at Python scripting, so my aptitude flourished in both subjects as I used one to learn the other, and vice versa.

Calculus III & IV

Vector fields and really anything to do with vectors is awesome, especially in 3D spaces. I particularly liked once Calculus concepts started to compound onto themselves to reach things like triple integrals in Calc III in fancy spherical coordinate systems. Understanding line and surface integrals was also intriguing - I especially liked solving gradients, flux (of fields), and using Stoke’s/Green’s Theorem. These theorems tied together a few concepts like line integrals, surface integrals, and curl, and they helped me understand these math pieces more holistically.

Linear Algebra

Since I love vectors, of course I enjoyed Linear Algebra. Taking this course was probably the most pure math I dealt within a single semester. I wasn’t sure what to expect from it, but coming out of it, I realized it made me think higher of studying pure math. Through the abstraction it really felt like I had learned a new subset of the language of math. To this day I think fondly of my time studying it, and I wonder about exploring pure math deeper due to it.

Notes

Interestingly, I took both Linear Algebra and Calculus III off-campus, and during summer semesters. Back then, I wondered if I would just be speeding through it and unable to get much out of the courses, but now looking back, it seems that it actually helped that I took these outside of the grind of regular semesters. I was able to apply my entire headspace to just the work of a single course, and these two courses consistently stick out to me as ones that I got a lot out of.

Post-Bachelor’s

I still want to continue learning mathematics.

More recently I have been following Quanta, and researching / refreshing in numerical analysis while improving the related code repositories that I built in college about various methods and algorithms.

I want to dive a bit deeper into pure math, and this is something that’s been slowly happening alongside my computing endeavors. Most of all, what has personally happened post-bachelor’s, is that I slowly have become less neurotic as I made progress in paying off my student loans. Getting a degree in the USA is high-risk, high-reward if done right, and I nailed it, but suffered with regards to stress while I pushed myself to achieve, out of my own competitive nature (and ego).

Reflections & Deep Thoughts

It seems to me that part of the struggle in studying mathematics is overcoming the anxiety behind it. Similar to the idea that procrastination is due to fear of failure, there is some anxiety in dealing with mathematics that “getting” the concept will never be achieved. Almost - not wanting to move forward into a problem or new topic because of worry that it’ll reduce confidence if not immediately understood. It takes time to digest the concepts, and connect the structure of them to other pieces of the field. There is a certain confidence in knowing that you have the capability to figure out any sort of mathematics with the right curiosity, motivation, and time invested.

Higher math concepts are not always difficult to grasp, but both the endless jargon and the complexity involved in formualization (the math “language” itself) makes them esoteric and tedious. Furthermore, explaining direct applications is always difficult no matter what because applications are so vast and so deeply rooted into the problems of other fields. It takes some security of mind to move forwards into the abyss without knowing when it’ll all “click” together.

In reality, the “way that it’ll click” changes shape as more is learned (there is more context inside the brain), so the initial “wanting it all to click” is the wrong inquiry into what actually happens – it all never entirely feels like it clicks together, it is too multifaceted to do so.

Instead, knowledge builds upon itself and becomes more nuanced and less easily explainable in short form without feeling like you’re lying. This is why academics are so unconfident. The “clicking” is less of a harmonious feeling and more of a deep, tedious awareness of concepts.

A course in college about “how applications of advanced concepts arise” may have been useful. It would have been great to have a professor teach a course simply about understanding what happens as math advances, wherein the whole concept of building something, generalizing it, and then having it be used down the road is explained with clear examples of the advanced concepts. The field of mathematics benefits intensely from the compounding returns of past mathematicians' work – but seeing this in action at the higher levels, instead of, say, trigonometry, is much more involved. I was lucky that I got some insight into this by studying other STEM fields alongside math.

Much of what I’m explaining is a part of getting an applied math degree and incorporated into many courses, but as an ingredient in the soup that’s already mixed in, which makes it harder to isolate and consider the characteristics of directly. Albeit, it literally is hard to explain, and my hypothetical course here would never be able to be comprehensive, so maybe it’s best we just leave things as they are. Such a course, if it existed, I’d surely complain about if I went further in my math studies and started to question the approach of the course. Who knows. Maybe I’m just being lazy and don’t want to do the process of synthesis myself. All mathematicians are lazy.

To be honest, it seems that what I’m explaining is more of a core component of advanced education than a flaw in the system: you’re told less directly what to think, firstly because the level of complexity makes it difficult for a teacher to make broad statements about fields that are appropriately backed by hard evidence (and not opinion), and secondly because you’re expected to do more thinking independently.

A New Journey of Synthesis

My first steps into the world of advanced mathematics are long gone. I would like to continue my studies to gain more breadth and go deeper to discover more about the world of mathematics. I want to understand the cutting edge clearer too, but that will take time. Insights beyond simple understanding take time to synthesize after the process of discovery, and tend to be less available for direct reading, since everyone does it differently.

There is no finish line, just tinkering and chipping away. Such is the fun of math.

Best,

Daniel