Musings on Theoretical Mathematics

Math is useful and important because it’s generalizable. The general ideal behind discovering a mathematical subject is that someone assumes some properties about a mathematical structure and then tries to see what other properties emerge as a consequence of those initial assumptions. Someone else can then apply the discovered theory to an existing problem by showing that the problem fits the proposed structure of the theory. Once one shows that the problem fits with the theory, all emergent properties of the theory follow in the problem or application as a result of the exploration of the mathematical subject that the theorist has already done.

This is why often times, many mathematical theories exist well before we’ve discovered anything to which we can apply the theory. Imaginary numbers were discovered in the 1500s and didn’t see much real-world application until electrical engineers in the 1800s and 1900s started employing them to solve complex differential equations related to how electrons flow through a circuit. Rather than needing to develop the theory of imaginary numbers in order to further the field of electrical engineering, the theory was already there and ready to be applied once an application arose.

It’s easy for one to think about a branch of theoretical mathematics and feel uninspired due to the lack of immediate application, but it’s exciting almost because of that lack of application. The entire focus of the discipline is to abstract the subject beyond any one immediate application. Without decoupling the theory from the application, there is no room for future innovation as a result of an existing theoretical framework. This is because at any given time, the only applications one can think to develop are those that exist at that point in time.

No one in 1500 could have predicted that electrical engineering would exist one day. The scientific basis wasn’t there yet. But one could wonder about what would happen if the square root of a negative number existed. So they thought about that instead and then the theoretical framework that furthered the advances of electrical engineering already existed once the scientific basis was present for the subject to come to fruition.