Yesterday I found myself, once again, trying to explain what I do all day as a graduate student in mathematics to someone who has a maximum mathematical training of pre-calculus. I found myself describing the fact that until you are done with Calculus you are informed of “rules” of math which are you to believe without the “Why” (ie: the proof). When you finish Calculus you start the long and involved process of trying to figure out why calculus works (in greater and greater generality).
How do I explain how math generalizes? I mentioned the idea when we are younger we learn what a peanut butter and jelly sandwich is, then we learn to abstract this idea to understanding a PB&J is a *type* of something. It embodies the ideas we associate with “Sandwich”. Perhaps your working definition of a sandwich is two slices of bread with stuff in between. Then, many years later you may learn there exists objects known as open faced sandwiches. These only have one slice of bread and stuff. But they still embody something about a sandwich which is undeniable. (well you may disagree and that’s totally acceptable). But as you get older you can categorize and understand the generalities of a sandwich/ panini/ sub/ what have you.
In math we deal with the real line until we are old enough to comprehend that there may be other ways of looking at the world without the usual basic assumptions. (For example: You can draw a triangle with three 90 degree angles on a globe. try it!) Part of my work now is to comprehend these generalities- which are mostly sets of rules we require something to have. A little like “two pieces of bread and some stuff in between” only more math-y. I spend my days pondering strange situations where different rules hold then I logically deduce conclusions about these worlds. For example, in the Euclidean plane, a triangle’s angle measurements add to 180 degrees, but the example above proves that is not true for the space defined by the surface of a sphere.
Higher math is not trivial to explain to someone who doesn’t do it. But I think it’s important for non-math folks to understand what a mathematician does all day and that we do NOT sit around finding larger and larger integers. I would like folks to consider what I do helpful and necessary to society and not just something that’s “really hard”. If you are a mathematician- how do you explain your job?