Nature has recently published a fascinating article (paywall) developing the argument that theoretical work in mathematics that has no apparent application can prove to be really useful in the future. These quotes summarise the argument:
The mathematician develops topics that no one else can see any point in pursuing, or pushes ideas far into the abstract, well beyond where others would stop.
There is no way to guarantee in advance what pure mathematics will later find application. We can only let the process of curiosity and abstraction take place, let mathematicians obsessively take results to their logical extremes, leaving relevance far behind, and wait to see which topics turn out to be extremely useful. If not, when the challenges of the future arrive, we won’t have the right piece of seemingly pointless mathematics to hand.
These points are then illustrated with seven examples where advances in mathematics precede, sometimes by centuries, their use in new innovations or products. One of the examples explains that the mathematics of quaternions, which were first described in the nineteenth century, turns out to be really useful in computer game programming.
The examples provide evidence that abstract developments can prove useful, but I was left with a question. If the new understanding hadn’t happened first, would the application itself have driven the new mathematics? This is a hypothetical question, and there is no doubt that having the maths in place already will have speeded up the application. In order to get a better picture, though, it would be interesting to know how easy it is to find examples where new advances in maths have been catalysed because of a pressing need to solve a practical problem. If can think of examples like this please add them to the comments.