You mean who should pay? Whoever they can convince, of course. Historically it was wealthy patrons, royal courts, etc. Teaching indeed used to be the main thing

You mean who should pay? Whoever they can convince, of course. Historically it was wealthy patrons, royal courts, etc. Teaching indeed used to be the main thing a professor was expected to do, this changed in the 19th century with Humboldt's model of the research university, with the explicit purpose of not just passing on knowledge but constantly producing new knowledge as a core part of the operations. It could go back to the teaching model, if that's acceptable to them. Then they would have to do the research part on the side, as a hobby, like it was back in the day. In case of public universities, the mathematicians would ultimately have to convince the public, but of course in practice that means politicians and national grant agencies, who indeed report to politicians and the politicians chase popularity to the extent needed for reelection, so public sentiment has some relevance. Generally when you come to someone holding out your hand asking for money, being humble is a good first step, instead of getting angry at people for not sufficiently appreciating the big myterious value of what they are doing. Explain the spiritual value of the pursuit, they may or may not be convinced. This may be called a deficiency of democracy if your aesthetic values are different, but typically people don't like to fund abstract disconnected things, like expensive space travel unless it's embedded in some tribal jockeying like during the Cold War. In normal times people just want research that has life enhancing practical outputs. Nevertheless currently the West is structured as electoral democracy, so convincing the plebs is indeed necessary, no matter how lowly that feels from the ivory tower.