Applied math has various definitions depending on the person. We will list and study the prominent definitions, as understanding what applied math is—or should be—directly informs the vision and mission of an applied math community.
Applied Math as a Lubricant for Science and Engineering
Some view applied math as a lubricant that facilitates the progress of other sciences. In this perspective, a biologist might provide experimental data to an applied mathematician in order to add resolution and certainty to their observations.
For example, in a 2022 Open Biology paper, we studied the motion reversal behavior of algae called diatoms. Using stochastic differential equations and asymptotic methods, we explained how this behavior enhances the diatom’s diffusion and spreading. While biologists could have empirically reached this conclusion by comparing diatoms that reverse direction more often to those that reverse less frequently, we helped them reach the conclusion more quickly and robustly.
This role of applied math is undoubtedly useful, but it is also limited. It is bound by the horizon of the science or engineering it supports. If applied math is confined to this role, it cannot transcend the goals of the science it serves and, at best, merely accomplishes those goals. Therefore, it is essential to look beyond this definition.
Applied Math as a Rigorous Exploration of Theoretical and Speculative Aspects of Other Sciences
Mathematics has a rich and diverse history, with one branch rooted in ancient Greece, particularly in Platonism. In the Platonic view, mathematics prides itself in its ability to study and discover timeless truths valid in every corner of the universe. This capacity contrasts with the empirical sciences who can not help but be tentative and limited in scope by what can be properly observed. For example, cosmologists have a very limited grasp in what life could be out there in the universe but axiomatic mathematical thinking can contribute powerfully to that subject.
In a more detailed example consider evolutionary biology. Evolutionary biologists having gathered copious amounts of data globally, assert that life became what it is now on the planet through mutation and natural selection. Living things reproduce making copies that are slightly different from them and nature picks the copies that survive. Ultimately this leads to a particular shape/pattern for life. Many know of this explanation. However, there are a few big questions that this common sense explanation does not quite account for. One of these questions is how reproducing life arose in the first place and another is how it managed to jump from unicellularity to multicellularity. We notice that biologists can only do so much empirical science to answer the second question because the jump happened many many years ago far from the inquisition of the biologists. The remaining option is to look at current living unicellular organisms that seem to attempt or be in the process of becoming multicellular. As applied mathematicians we carefully model this kind of organisms' motion and study the role of multicellularity to their ability to move around and navigate. This sheds some light on why the happenstance of multicellularity might be favored and grow into a pattern.
Beyond truth seeking science, applied math can imagine and theoretically explore futuristic but pragmatic projects. From space travel to terraforming planets to longevity economics to cosmological sociology (The Dark Forest by Liu Cixin), applied math can boldly but clearly think about next generation ambitions.
However, one criticism of this definition is that it overlaps with pre-existing fields. Some might argue that applied math would be redundant under this definition. This is ultimately a semantic and trivial conflict. Moreover, the applied math community we aim to build is in a position where this conflict would likely not arise. If it does occur, it is best to cooperate and coordinate.
Applied Math as Outreach for Pure Math
In Western and other developed nations, pure math often has a niche and incentive structure. There are national and international competitions to be won, medals to be collected, and even millions in prize money to be awarded. Top schools bring together the brightest youth to compete for meaningful compensation. However, many of these incentives are lacking in developing countries like Ethiopia.
In Ethiopia, some progress is being made in fostering mathematical competition (Ethiopian Mathematics Professionals Association (EMPA)), and there are modest prizes available. However, pure math remains largely unappealing to most. It is not worth pursuing because it does not pay off in a country where "payoff" can ultimately be a matter of life and death and because many assume that pure math is not for them or for others around them. Finally and more pertinently, pure math's quickly but justifiably growing esoterism only makes things worse.
A great way to combat the lack of appeal of pure math is by using applied math as the Aaron to pure math’s Moses. Applied math can clearly demonstrate its relevance even in developing countries, while also advocating for the value of pure math. For example, when I present on applied math research, I always dedicate a slide to discussing the pure math I used or struggled with.
In conclusion, a robust applied math community serves as:
- A catalyst for science and engineering
- A pioneering explorer for science and engineering
- An evangelist for pure mathematics
