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In popular culture, mathematics are often split between applied mathematics, which provide tools for science and engineering, and pure mathematics, that are somewhat useless. Although this distinction is dubious, this might be a real issue when searching for funding. This distinction is often replied with the insight that "pure math" find their applications decades, or centuries afterwards.

And yet, there has to be examples where pure maths stayed pure and without applications until now, and we can't really foresee how they could be useful for anything other than themselves. What are some of the oldest or most iconic examples of this ?

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  • $\begingroup$ Really cool question!! $\endgroup$ Mar 22, 2021 at 10:25
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    $\begingroup$ Maybe "useless" is not the best word... With no applications. $\endgroup$ Mar 22, 2021 at 10:25

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There is a clay tablet that is roughly 3800 years old, called Plimpton 322. It contains a list of Pythagorean triplets. One of them is the triplet $$12709^2+13500^2=18541^2.$$ I don't think there is or was any practical application of this particular identity.

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One possibility is the computation of $\pi$ with many digits. In the sixteenth century, Ludolph van Ceulen computed $\pi$ with $32$ digits and, of course, now we now how to compute it (and have computed it) with billions of digits. But if you want to compute the perimeter $p$ of a circle whose radius $r$ is the distance from the Sun to Neptune with an error inferior to the thickness of a hair (let's say, $0.1$ millimeters) using the formula $p=2\pi r$ (that is, in such a way that difference, in absolute value, between the real value of $p$ and $2\pi r$ is smaller than $10^{-7}$, if $p$ and $r$ are measured on kilometers), you only need $15$ digits.

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  • $\begingroup$ Really cool fact! Would you edit your post to what 'error inferior to the thickness of a hair' entails? $\endgroup$ Mar 22, 2021 at 10:36
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    $\begingroup$ @LordCommander Done. $\endgroup$ Mar 22, 2021 at 10:51
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    $\begingroup$ @Servaes does any of those applications need higher than 30 digits? $\endgroup$ Mar 22, 2021 at 11:56
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    $\begingroup$ @HansLundmark The choice of $\pi$ for that is kinda arbitrary, you could compute other number for that. I wouldn't consider that as an application of $\pi$ more like an application of computable irrational numbers maybe. $\endgroup$ Mar 22, 2021 at 13:19
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    $\begingroup$ Actually i wouldn't consider it as an application of computable irrational numbers either, as all you need is some computer-intensive algorithm, doesn't need to be something that approximates a computable irrational number. $\endgroup$ Mar 22, 2021 at 13:31

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