The goal of math is perhaps the answer but also the understanding. One can't use the answers without understanding them, which is a fundamental difference between math and a program. Most modern math isn't applicable in the sense that you just take some result and it gives something practical. The value of math is in understanding it so that it can be properly applied which requires a deep knowledge of why it is so in the first place.

> If they had no useful purpose, why did we think about them for years?

A lot of mathematics is mostly useless, and probably will be for hundreds of years. Some of it ends up being useful, but it's unlikely that much of math like work on geometric langlands will have any applications whatsoever. Math is a service department and the way people teach math best is to be experts in it. In turn, the best way to do that is to develop new math, at least within the context of academia.

Yes, there is some pure math that is used eventually, but I'd say (as someone who worked in pure math for over a decade) that as the results become more specialized and abstract over time, fewer and fewer of them will ever be used.

Every once in a while you see a news article about some math related to physics or something but that's 1/1000 results, if that. The fact is, most modern math research is likely never to be "used".

That doesn't make it useless because maybe we just want to know, but we have to be realistic here...