No, I don’t want to turn this into some sensational piece about declining school systems.
Instead, I want to talk about a very real feeling that I experienced while studying this semester.
First, let me set the scene. I’m a graduate student in theoretical physics. As this might suggest, I deal with mathematics on a daily basis. It’s just a part of the toolkit that makes me a scientist. It shouldn’t be much of a surprise then that I am comfortable with mathematics.
What might be a surprise is that I found myself dreading certain aspects of it.
One of my classes this semester was quantum field theory. It’s basically a course that takes the lessons of quantum mechanics and tries to make them play nice with special relativity. You learn about bosons and fermions, and the machinery (read: mathematics) needed to describe them.
Part of this machinery involves Lie algebras. Put succinctly, Lie algebras are examples of elements with an operation that doesn’t commute. For example, if we have the real numbers and the operation of multiplication, we know that 4 × 5 = 5 × 4 = 20. The numbers can be swapped around without screwing anything up.
If you only look at the usual settings for algebra (real numbers or complex numbers if you’re being fancy), it’s easy to think that all mathematical systems work this way. However, there is one setting where this can clearly be shown to be false: matrices. In that case, the elements don’t commute. If you have a matrix A and a matrix B, then in general AB ≠ BA. Matrices aren’t the only case of non-commuting objects, but they are the setting that is most familiar.
The Lie algebra theory needed for quantum field theory essentially boils down to this observation. We end up working with matrices, and this means we can’t move around our objects freely. We have to be careful with how things are written in our equations.
I’ve had a lot of experience with matrices. They are standard fare in the physics undergraduate curriculum, so it wasn’t like I was caught blind. However, the way these were introduced in the course (with these things called “structure constants”, Lie brackets, and representations) was confusing to me, so I wasn’t able to follow. There was too much that I was trying to “hold in my head”, and my mind refused to absorb more information regarding these ideas.
To give you a glimpse of what happened, my course in quantum field theory was four weeks long. The discussion about Lie algebras occupied week three. At the end of that week, I had basically no recollection of what happened in class. Sure, I worked through some problems and listened in the lectures, but I didn’t absorb any of it. Whenever the professor spoke of something that smelled of Lie algebras, I blocked it out. I didn’t want to do this, but my mind seemed to have a self-defense mechanism that kicked in.
This was so bad that I basically had to relearn the material in the final week of classes before I had my final assessment. During that time, I was able to figure things out much better. Once I could work through the details slowly, it helped. But during that week of Lie algebras, I got almost nothing out of the course.
I think this is a cautionary tale for everyone who is learning new ideas in mathematics or physics. Even I, a student who has been working with mathematical ideas for years, felt overwhelmed by a topic to the point that I just wasn’t absorbing it. I had what you might call “mathphobia”. I like mathematics, but during that period, I dreaded any discussion of Lie algebras.
The world isn’t neatly divided into those that are “good” at mathematics and those that are “bad”. Instead, we all have strengths and weaknesses. Furthermore, we need to be aware that there are different levels of abstraction in which we can think about a concept. Going to a high level right at the beginning is a recipe for frustration. It’s not that you won’t ever be able to understand the concept. It’s that you aren’t giving yourself the appropriate challenge.
For myself, I had to go back to the basic definitions of Lie algebras and their representations before I could even connect it to the non-Abelian gauge theories I was studying (and am still trying to understand). Doing this was annoying, but crucial in being able to move past this aversion.
It also makes me think that we need to tread with a lot more care when it comes to teaching students. How many students feel challenged by the material early on in their education and just decide to check out? Reducing that number is surely a worthwhile goal.
If anything, my hope here is to illustrate that anyone can feel overwhelmed by mathematics. It’s not like you study a lot of mathematics and one day you cross this threshold where everything then makes perfect sense for the rest of your life. Instead, there will always be more challenges ahead.
The key is in understanding that the aversion you’re feeling can be overcome. It’s not easy, but if you approach it with the right mindset, it’s entirely possible.
Remember that there are many levels of abstraction for a given mathematical topic. It might seem cool to jump to the most abstract incarnation of an idea, but that’s usually a quick way to get frustrated at the impenetrability of the topic. Instead, pick an avenue that looks interesting to you, don’t start too high at the beginning, and search for the right fit.
Feeling an aversion to a certain mathematical idea is normal, but don’t let it turn into full-blown mathphobia. Zero in on the cause, and find ways to work on clearing up the confusion. This is often by looking at the fundamentals again, but it can also be through consulting different sources.
If you want to reduce your chances of being put-off by mathematics, be willing to change the level of abstraction in which you engage the subject. Above all, give yourself grace when you’re confused, and remember that you can gain the understanding you want with enough work.