Mathematics is the art of explanation.
― Paul Lockhart
In projects I work on mathematics plays a key role. Too often the math doesn’t provide nearly enough impact because it can’t handle the complexity of applications. One of the big issues is the proper role of math in the computational projects. The more applied the project gets, the less capacity math has to impact it. Things simply shouldn’t be this way. Math should always be able to compliment a project.
This begs a set of questions to consider. For example what sort of proofs are useful? My contention is that proofs need to show explanatory or constructive power. What do I mean by this?
[…] provability is a weaker notion than truth
― Douglas R. Hofstadter
A proof that is explanatory gives conditions that describe the results achieved in computation. Convergence rates observed in computations are often well described by mathematical theory. When a code gives results of a certain convergence rate, a mathematical proof that explains why is welcome and beneficial. It is even better if it gives conditions where things break down, or get better. The key is we see something in actual computations, and math provides a structured, logical and defensible explanation of what we see.
How is it that there are so many minds that are incapable of understanding mathematics? … the skeleton of our understanding, …
― Henri Poincaré
Constructive power is similar, but even better. Here the mathematics gives us the power to build new methods, improved algorithms or better performance. It provides concrete direction to the code and the capacity to make well-structured decisions. With theory behind us we can define methods that can successfully improve our solutions. With mathematics behind us, codes can make huge strides forward. Without mathematics it is often a matter or trial and error.
Too often mathematics is done that simply assumes that others are “smart” enough to squeeze utility from the work. A darker interpretation of this attitude is that people who don’t care if it is useful, or used. I can’t tolerate that attitude. This isn’t to say that math without application shouldn’t be done, but rather it shouldn’t seek support from computational science.