mathematical knowledge is defined by its computation-granting abilities. Knowledge exists if and only if you can compute answers to questions exhaustively in the domain of that knowledge.

It could be said that no human yet has a “full understanding” of integration, or that Liouville’s theorem shows the impossibility of solving this problem completely. And so to add to “can you compute” you must allow a response, “impossible given the constraints.”

Once a field that doesn’t have strong computational roots becomes computational (see, topology becoming algebraic topology), you get mathematicians calling algebra the “devil” and complaining that a subject is no longer beautiful.

why are some algebraic structures more useful than others in software applications?

A “strong” structure admits constructive characterization theorems, and efficient algorithms to convert objects into various canonical forms.

Very nice post. Explains the immense power of linear algebra and graph theory.