Augmenting Your Toolkit
As a physics student, I use a lot of mathematical techniques to solve problems. This isn’t surprising, seeing as though mathematics is the language of physics. I’ve learned how to use complex numbers, how an inner product isn’t only the usual dot product, and have seen how we can use Fourier series to solve more general boundary value problems that arise from differential equations governing electromagnetism or heat flow.
However, I think it’s fair to say that often this comes across as learning how to do a “neat trick” to get the answer we are looking for. If I think about the times I’ve learned new mathematics in a physics class, it was always in service of some problem we were trying to solve. As such, the professor didn’t spend a whole lot of time going through the mathematical theory behind what we were doing. Instead, the focus was on introducing the basics so that we could finish a problem.
The reason for this is simple: time is limited, and physics professors can’t spend a ton of time introducing the general theory and then covering the physics. There’s already a bunch of material they need to cover, so going over the mathematics is often glossed over in favour of giving students a “crash course”.
I understand why this is done, but it does mean I find myself wielding mathematical tools that I don’t fully understand. I can solve some specific physics problems with the tools, but if you change the setting just a bit, things get confusing. Therefore, I find the mathematics you learn in a physics course is good, but not great for broad understanding.
This is one of the reasons I enjoy taking mathematics classes. For some of my physics classmates, they see a mathematics class and wonder why in the world they should care about the theory apart from solving an actual physics problem. I can sympathize with this view because I know there are people who don’t really care about the mathematical foundations behind the physics. I just don’t happen to hold that view.
When you take a mathematics class, your first reaction might be that everything is so abstract and far-removed from the physics you’re used to. That’s by design, and it will actually help in the long run. The added abstraction means you can strip away the veneer of a particular physical problem and consider the idea in its purest form. You can then adapt the idea for various applications.
The power of this approach is that it gives you a sense of where a mathematical technique or tool comes from. It doesn’t just pop out of thin air when solving a physics problem. Rather, you start to see the roots of where this tool came from. (Sometimes, this turns out to be in service of a physics problem!) This helps me gain a better understanding of what is going on. Personally, I hate having to grab a random mathematical technique that I don’t understand but know it “just works”. I’m a student because I want to understand every part of the process, including the mathematics.
I know that physics students work with mathematics throughout their whole education, but I think more is even better. This is part of the reason why I decided to also get a major in mathematics. I wanted to learn more about the techniques and tools I used without being constrained by specific physical problems.
In my mind, this is just the next step in an attitude that many students in physics have. If you ask them why they enjoy physics, a lot of them will respond by saying that you can take a few simple principles and gain a lot of explanatory power from them. One only needs to point to the phenomenon of simple harmonic motion to see the truth of my statement. Harmonic motion occurs everywhere in physics, in quite different scenarios. By understanding how simple harmonic motion works, you can understand a variety of systems.
Similarly, once you start entering the more abstract realm of mathematics, you start to see that the whole field of physics is only one area in which mathematical tools can be used. This is what I mean by the “next step”. Just like a few physical principles can explain a whole bunch of physical situations, a few mathematical principles can be used to explain ideas throughout our world (not limited to physics).
I think this is a solid argument for getting physics students to take more mathematics classes. Granted, I realize that some students aren’t interested in mathematics, and that’s fine. However, for those that claim they want an intimate understanding of the physics they learn, I think taking more classes in mathematics can only help.
I’m not saying you need to start taking mathematics classes that are completely removed from physics (or very far away). What I’m saying is that it’s worth taking classes that expand and go in depth on concepts you see in your physics classes. Yes, you might be able to get by with the basics that your physics professor teaches you within the course, but I would argue that this isn’t enough. It’s a good start, but you only start understanding the ideas when you get the full mathematical treatment.
As physics students, we have the tendency to jump right in with new mathematical tools. We push the details away from our minds, mostly because we don’t have time to ponder them while solving complicated physics problem. While this might let you pass a class, it doesn’t help you understand the concepts. By taking mathematics classes that “fill in” those details, the tools themselves will make more sense.
Think of it this way. You study physics in order to explain a bunch of phenomena with a few concepts. Studying mathematics lets you see where the mathematical concepts come from, giving you even greater explanatory power.