What is the point of pure mathematics?

…are there any realistic benefits that actually help the world run?

(Original question details)

How is "actually helping the world run" synonymous with "having a point"? Does music help the world run? Cosmology? Cricket? Sculpture? Ballet? Philosophy? 3D Animation? Archaeology? Caber tossing?

Mathematics is a huge edifice of human thought, more beautiful and deep than anything else for those with the training and inclination to appreciate it. That is point enough.

That's not to say that pure mathematics doesn't also have practical applications - the point is that it doesn't need those in order to matter. But of course, it does have such applications. Time and time again we've learned that mathematical abstractions find unexpected applications, and are in fact completely necessary for us to understand the world around us, and to influence it.

Modern theories of physics - general relativity, quantum field theories, string theory - rely on very deep and abstract mathematical ideas. The study of algorithms and data structures, forming the foundation of computer science, requires a healthy dose of "pure math", and many extremely mundane technologies utilize those ideas (an often-quoted example is Google's original PageRank algorithm, which is an instance of an idea from graph theory). It is hard to imagine modern Economics or Earth sciences without statistics, another field that relies heavily on pure mathematics.

However, as I said, most mathematicians - even those working in combinatorics, statistics or theoretical computer science, fields that are very close to engineering - don't need those applications in order to enjoy their work, or justify it. In fact, some of them positively relish the fact that they operate in purely abstract serenity, untainted, as it were, by machineries and technologies and so-called "applications". Hardy famously expressed this view in A Mathematician's Apology.

Caring about why every natural number is the sum of four squares requires nothing more than intellectual curiosity, and satisfying our desire to pursue such questions in the name of sheer curiosity is the "point" of pure mathematics.

(This image shows the distribution of the roots of a certain family of polynomials, discussed here).