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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.
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  • $\begingroup$ How do you stress mathematics is not all about computation by removing plane geometry from high school? $\endgroup$
    – Faustus
    Jan 10, 2019 at 15:13
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    $\begingroup$ @Faustus I think the point is that keeping plane geometry shows that math is not all about computation. $\endgroup$
    – Arnaud D.
    Jan 10, 2019 at 15:15
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    $\begingroup$ @Faustus I meant what Amaud D. says. High school plane geometry is more about proofs. And people think this is a good reason to keep plane geometry in high school. $\endgroup$ Jan 10, 2019 at 15:17

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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.

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  • $\begingroup$ I think the sculptures you linked to are inspired by the mathematics behind them rather than applications of mathematics, but I think the example in your second paragraph is valid. Thanks and +1 $\endgroup$ Jan 11, 2019 at 17:58
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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.

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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.

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  • $\begingroup$ What qualifies as a computation is of course a bit ambiguous, but I would say not all applications of math are about computations. In the category theory example, it seems that the use of the theory is to help programmers to make their programs more composable. $\endgroup$ Jan 10, 2019 at 15:31

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