[samus]

As you succinctly said it yourself, pure mathematics is indeed by definition unconcerned about whether their objects of study actually "exist". That's the business of physicists and certain subfields of philosophy.

Pure mathematics has indeed been challenged whether it actually makes sense to further fund it, but occasionally their objects of study turn out to be useful. For example number theory, which is the foundation of modern asymmetric cryptography.

But even if their research never turns out to be useful, pure mathematicians are unconcerned about it since they don't approach their research objects with any assumption regarding that.