Having recently talked about the validity of pure mathematics as a scholarly field, I feel the need to defend my own territory of applied mathematics. In my last post, I discussed the joy that mathematicians achieve by exploring abstract problems that are logical constructions. The constructions themselves are the source of this joy. I argued that this work is valid in because it produces this enjoyment, whether or not the work has relevance to other disciplines, much like art, music, and literature. Applicability to another discipline is not required for a thing to have value. Nor is utility by anyone else for any reason. Aesthetics are completely and totally personal.
With that setup, let me talk about my passion. My interest in mathematics lies in the ability of mathematical tools to describe and sometimes explain different phenomena. The notion that the universe is consistent enough to describe with our tools is a pretty staggering one, and one that I contemplate with gratitude frequently. I have a memory from junior high physical science where we used a parabola to predict the motion of a ball, and the prediction was surprisingly accurate for such a simple model. This ability to capture the essence of the universe with simple mathematical tools is what I would call my passion. When I call myself an applied mathematician, it’s really just my way of saying that I love all things that can be described by mathematics. I’m not a meteorologist or a physicist because I am not just interested in the how, but more in our ability to quantify the how and to make predictions. This is the difference between an applied mathematician and a pure scientist, although the distinction is so fuzzy that many people like me are trained as engineers, meteorologists, physicists, computer scientists, statisticians, … You get the idea.
Sometimes this passion takes the form of inventing new methods for solving a particular class of problems in field, such as estimating the three dimensional wind velocity field from a set of radar observations using different sets of assumptions and different mathematical algorithms. Sometimes we just try to decide how to decide that one method is better than another. Sometimes we apply existing methods to new problems, which are often much larger than the original test cases or contain different types of uncertainty. I would consider each of these vague descriptions to be applied mathematics. Depending on how the work is packaged, you will find papers of these types in mathematics journals, journals in the field of application (in my case Monthly Weather Review or Journal of Atmospheric Sciences, etc), or in the in between journals like those published by the Society for Industrial and Applied Mathematics (SIAM). Doing the work I do allows me to learn about problems in almost every field that mathematics has a role in, since many of the techniques applied are similar in nature. For example, any field that uses partial differential equations as models has to solve them somehow, and the methods used for that are similar from field to field.
In short, I think that applied mathematics as a scholarly discipline is distinct from other disciplines, although it draws problems from other fields, and techniques from mathematics. The difference lies in the aesthetic, where the approach to solving complex problems and the ability of the mathematics to do so are what is appreciated.
Sean M. R. Crowell 