Styles of mathematics
A lot of learning lies ahead.
It’s said that the heart of progress in science and math is coming up with the right question.
In graduate school, our astronomer had an idea. Well, many ideas, but the one we’re talking about came from a course he audited and an area of physics that he didn’t work in. It seemed to him that it might be possible to combine a technique from almost-pure mathematics and something from Quantum Field Theory to produce a really useful result. The drawback was that he didn’t work in either area, and didn’t understand them in anything but a general way. He didn’t have the background to even think about the question he wanted to ask.
The main barrier is that each field, while being highly mathematical, uses different mathematics and indeed approaches math in a different way. It may come as a surprise that there are such different styles of mathematics. Certainly our tutor’s Calculus students need to be reminded that the various details they’ve been taught along the way are all likely to be used: logarithms, polynomials, graphing, domains and ranges and the rest. Throughout secondary school mathematics maintains a unity in its diversity, and any student may have to solve a problem of any type.
This breaks down in college, where some students may not take any additional mathematics at all, while scientists and engineers face multivariable Calculus, differential equations and complex analysis. However, Physics majors and their kin do master a common set of material, so we can say that for them there remains something we might call a common style.
It all goes away in graduate school. Our astronomer learned how to deal with tensors and the apparatus of General Relativity, but still views the infinite integrals of Quantum Field Theory with concern. Likewise, the mathematical apparatus that comes under the heading of “analysis” seems quite foreign. He is aware that the rigor of modern math, including the areas he uses, depends on it; but how to deploy it to solve a problem is another matter.
But his idea remains, and he thinks he should really do something with it. He will need to master two distinctly different styles of mathematics, as well as a significant body of their results. And then (this is the irony) he may discover that the question he is looking for is trivial and has already been answered, or that it cannot even be asked. Either is more likely than the possibility that he’ll discover something new and useful. But he won’t know until he learns a lot more.
