The interesting thing about the question is that there really aren't any right or wrong answers.

People pointing out that maths is full of advancements that had no immediately identifiable use at the time, but that came to be useful later, is correct. Yet it doesn't even begin to answer the OP's question.

I doubt very much that many people choose to pour their lives into endeavours that they don't particularly enjoy just because some hypothetical person at some hypothetical future point in time might hypothetically find a hypothetical use (hypothetically ;P).

The answer is that "value" presupposes the question "valuable to who and why?"

Newton invented Calculus because he had an immediate use for it. Other mathematicians pour themselves into solving problems because they enjoy it and find a lot of reward in the prospect of solving a previous unsolved problem. Both are "valuable", just to different people for different reasons.

Yeah it's fine for mathematicians to amuse themselves, the problem is when they demand salaries to do that and taxpayers like OP rightfully ask "whats in it for me?" And when the answer is "IDK but maybe in a century we might have a problem this math is useful in solving" then it's not surprising that no one wants to fund pure math research. It's not the 20th century anymore when math research was going to meaningfully improve someone's life through the invention of things like electrical devices.

Yeah, if you force other people to pay for something then you had better offer them an attractive value proposition. Though public funding of mathematics and other sciences is not what I thought we were discussing :)

In which world do you think mathematicians are raking in taxpayer money? Mathematics requires very little funding: a blackboard, some chalk, a pen and paper, a desk, some coffee. That's it.

Thats a pretty myopic view of history there champ.

The pragmatic, practical perspective here is that funding the egg heads has had incredible outcomes (and its so cheap too), so dont let the simpletons shake the golden goose down just because they don't understand anything they cant fuck, fight, or eat.

You described my thinking very well, and much better than I. Thank you

Until it's not, because someone suddenly realizes that a seemingly completely unrelated problem in a completely different subfield that's holding up a major proof is actually analogous to a problem where this result removes a bunch of roadblocks.

That's the problem with math from a "so what is this good for?" perspective: we don't know yet, but we sure have a litany of instances where seemingly useless proofs had a profound impact anywhere from weeks to centuries later.

They were trying to solve a real world problem using this kind of math, and then decided it'd be easier to improve upon the math itself than to continue to pound at the problem - so by definition, I'd say you're absolutely wrong here, or they never would have started this proof in the first place.