As a student in both mathematics and physics, I often see the differences in mindset between the two fields, and how these mindsets change the way classes are taught. The former is usually about structure and patters, while the latter is about modeling the world using mathematics. The problem is that belonging only in the camp of physics seems to be a dangerous thing to do, in terms of building one’s foundational understanding.
For a small example, take quantum mechanics. There’s no way to get around it: quantum mechanics requires an understanding of mathematical probability. From finding expectation values to normalizing the wavefunction $\Psi$, it’s important to understand how probability relates to ideas of qunatum mechanics. Unfortunately, due to the progression of courses that students take, some students are seeing these ideas of probability density functions and other aspects of probability for the first time within the quantum mechanics course. And, since the course is primarily physics related and not mathematical in nature, the ideas behind probability are skipped in favour of the results.
This isn’t always a bad thing, but what it does sacrifice is the knowledge of the structure behind the physics. Why do these ideas of probability work? Why does the wavefunction have to asymptotically approach zero at infinity? Why is the probability of finding the particle at a specific point zero? These kinds of questions make a lot more sense when thinking of them through the lens of probability. Therefore, those without that kind of background are at a disadvantage.
Structure and Purpose
I sometimes wonder about what my classmates (who are only in physics) think about mathematics. Is mathematics simply a tool to get to the physics behind systems? Is mathematics more of a nuissance than anything else?
I don’t simply make these claims out of the blue. I’ve heard my fair share of people within physics talk about mathematics in a way that suggests that they only use it because they have to. It’s this mindset that partially worries me, because it completely misses the point of what mathematics is about, and only serves to further the dogma that mathematics is about calculating quantities.
Mathematics has been an important part of my education precisely because it is the opposite of doing calculations. If one only takes science or other quantitative courses, one misses the side of mathematics which is why mathematicians continue to work in the first place. To them, it’s not about calculating an inner product or evaluating an integral. It’s about noticing structure, finding patterns, and forming convincing arguments as to why things logically have to be this way. This is the hidden veil behind mathematics that students don’t see until much too late, and by then only a handful have stuck around.
The bottom line is that we have mathematical tools and we have a mathematical mindset. The former is what we use to serve all of science and other quantitative fields. This is computational in nature, and comes from the results of mathematics. But the latter is whole different paradigm. It’s about producing results (but not necessarily ones that are deemed immediately useful). The latter is about studying structure and generalizing, which requires a playful mind that is open to new possibilities.
The problem is that this other part is difficult to train, and it’s not immediately productive. It’s easy to know if you’re making progress while evaluating an integral. At one point, you have nothing, and then after some work, you get an answer. But cultivating a mathematical mindset is more about struggling for hours on end on a problem, only to realize that you were in a dead end. It’s about trying things that you aren’t sure will work, and having that rush of adrenaline when things do work.
Of course, I’m well aware that this happens on the front lines of all of science, but this isn’t the aspect that is apparent to students either while they are studying. As a student, I don’t see that much work on what we would call the front lines. Instead, I’m still behind, learning about concepts that were devloped many years ago. This is the natural progression of science students. What it also means is that most problems I complete are computational in nature. However, in my mathematics courses, I’ve moved away from computation and more towards abstraction. As I mentioned above, it’s a different mindset, but one that is very helpful to have as I work through problems both in physics and mathematics. It’s something that I wish more science students would engage with, because it’s a fantastic skill to have.