What’s formalism? I guess there are many ways you can define this term, and it varies across different subjects. In mathematics, I would like to describe it as mathematical rigor, writing impeccable proofs that are universally and logically sound.
In mathematics, we (I) really like to achieve formalism. Even when things do seem easily understandable, we continue to formalize them. In fact, I enjoy doing so, even though I often make mistakes; not that it seems more professional after formalizing them (you probably need strange symbols and complex expressions for that), but because it just feels right.
I mean, it’s definitely a very useful tool in theoretical mathematics, where we often try to prove “big” things with little known information, and we have to make use of every possible characteristic of the problem statement available to us. But is this still necessary in applied mathematics?
I realized that when approaching applied mathematics problems, I formalize problems to a seemingly unnecessary extent; sometimes, I feel stupid looking back at what I wrote, using an extensive amount of mathematics to describe what is actually an extremely simple problem.
I also don’t think I’m the only one doing this (although I do not have particularly strong evidence to back up this claim). When we immerse ourselves in the seemingly rigorous and nuanced mathematical language, it’s natural to lose track of the physical essence of the original problem.
Maybe it’s because my understanding of mathematics is too shallow, or maybe the abstract nature of modern mathematics easily detaches one from reality. It seems to be evidence of the unreasonable ineffectiveness of mathematics.
To be honest, there may be some value in doing so; by creating unnecessary formalism, one does decrease the chances of producing contradictory arguments. Even in practical situations where the conditions (or even the answer) are incredibly obvious, sometimes conflicting understandings are inevitable without formalism.
Also, there’s an interesting point that formalism “invents” mathematics; sometimes, an unnecessary piece of formalism can make a mathematical tool easily generalizable to other difficult problems (which require complex mathematical tools). In these situations, such a generalization/solution may not be found without the help of these overly rigorous mathematical expressions.
Anyway, that’s just my personal opinion; many people find mathematical formalism difficult to understand, generally impractical, and annoying. Of course, there’s always that bit of formalism that’s strictly required to use a mathematical tool properly, and there really isn’t much to debate about in such cases. Regardless, I enjoy the formalism, and it will probably require some effort for me to change such a habit.
