I had several questions about my majoring in pure math and applied math. To people not acquainted with the different branches of mathematics, this is the picture they have in mind:
Here, pure math is thought to be the kind of math that has no foreseeable application, and applied math consists of everything else. Fortunately (for me), the actual picture looks more like this:
Applied math is about modelling systems (e.g. physical systems) using calculus and tools derived from calculus. Pure math can be thought of as a combination of two sub-fields: algebra and analysis. Algebra deals with properties of sets objects that can be operated on (i.e. added/subtracted/composed), and analysis provides a rigorous basis for calculus (i.e. a lot of epsilon-delta proofs). Since applied math relies on calculus, analysis is a topic of interest to applied mathematicians as well. Thus pure and applied math intersects.
A lot of times, we consider other branches of math that are "applied" as their own branches. In that case, the picture looks more like this:
Moral of the story is: pure math and applied math do intersect. In fact, if you pick any two branches of math, there is probably an overlap.
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All math can have applications, and you never know where they might come from. There are many examples of surprising applications for things that were thought completely pure at the time they were developed (e.g. group theory).
ReplyDeleteAlso, your definition of applied math in terms of calculus doesn't sit well with me because the application of math to genetics and genomics (where continuous functions are rare, irrelevant, or oversimplifications) often involves math that is not really calculus.
I find the term "applied math" problematic for the reasons Steve outlined but Lisa's use of the term is pretty conventional. Perhaps it's misleading, with math now being use extensively beyond physics and engineering, but it's inevitable that names and classifications will reflect historical links rather than current developments. Lisa's point about the branches intersecting holds regardless of what these specializations are called.
ReplyDeleteI enjoyed running into differential equations when taking some postgrad probability/stochastic modeling courses - as an undergrad they were very firmly in the "applied math" camp. Multivariate probability/stats uses some linear algebra of course, that's something I'd like to look into in more depth - when I studied it I seemed to be multiplying a lot of matrices but without any link to the underlying theory.