What are some applications of mathematics whose objectives are not computations?

Question

In mathematics education, sometimes a teacher may stress that mathematics is not all about computations (and this is probably the main reason why so many people think that plane geometry shall not be removed from high school syllabus), but I find it hard to name an application of mathematics in other research disciplines whose end goal isn't to calculate something (a number, a shape, an image, a solution etc.).

What are some applications of mathematics --- in other disciplines than mathematics --- that don't mean to compute something? Here are some examples that immediately come to mind :

• Arrow's impossibility theorem.
• Euler's Seven Bridge problem, but this is more like a puzzle than a real, serious application, and in some sense it is a computational problem --- Euler wanted to compute a Hamiltonian path. It just happened that the path did not exist.
• Category theory in computer science. This is actually hearsay and I don't understand a bit of it. Apparently programmers may learn from the theory how to structure their programs in a more composable way.

Answer 1

Would you count these sculptures by Bathsheba Grossman as non-computational? Maths for the sake of beauty. (Also they include a Klein Bottle Opener!)

A nice but technical example I remember from electronics electronics at university was the proof that a filter which perfectly blocks a particular frequency range but lets everything else through can't exist. The reason is that its response to a step input begins before the step is applied. Though I'm not sure whether to count this one since it does involve working out the step response via a Fourier transform.

Answer 2

What do you mean by computing? Is solving a Rubik's cube considered a computation? Is drawing a $17$-gon with straight edge and compass considered a computation? What about winning a game of NIM? How about general logical thinking problems?

If you define computing as finding the unique solution to a well-defined problem, then I think you can say that mathematics only has applications in computing. If you define computing to necessarily involve a number system, then it depends on your interpretation of certain problems. For example solving a Rubik's cube becomes related to numbers if you want to solve it with as few moves as possible.

Answer 3

Every proof that something cannot be done is a good example of what you're asking. Take, for instance, Abel's impossibility theorem.

Also, proving that something can be done is also often a good example. Take the Four color theorem, for instance. We are computing nothing here. The idea is to prove that you can clor any map, now matter how complex, using only four colors.