Pure vs. Applied Mathematics

There is an ongoing debate amongst mathematicians about whether applied mathematics has value, and should be considered a part of math. As an applied mathematician, I follow this debate with interest, and have definite ideas on the subject. Applied folks and pure folks share many important attributes, such as the quest for solutions to problems, and an affinity for the simplest and most elegant of such solutions. Typically, the training supplied to each group shares a common core of rigorous mathematics. Where the two groups differ is in the type of problem that they want to solve, and this has important consequences. The pure mathematician’s primary tool is the theorem/proof methodology. In applied mathematics, there are many times that we can only gather evidence and develop working hypotheses, because the problems we’re solving don’t have strictly defined terms, and so a theorem and accompanying proof is not always possible. In this case, we look for salient features and their persistence in our computations, and we examine why this persistence might occur. In the case of simple models, we can often make specific claims about behavior and prove them. We then look for this behavior to persist as the complexity of the model increases.

My point is that there is a real and visible divide between mathematicians that solve pure and applied problems. It’s not imaginary, as some claim. But it’s also natural and necessary, given the nature of the problems that interest us. But we share a common framework for thinking, a common training, and in many cases a common set of tools. Certainly there is a need for both groups, and for all of the brilliant folks that do both.