Mathematics need some source of new ideas, and any other fields are very good sources of ideas: art, philosophy, physics, music, even sport probably. From the other side may important discoveries are made by means not-so-precise formulations of solutions known problems. Feynman path integral is good example, distributions theory as well etc. Forgive me, but math without such external source of ideas, without any contact with reality ( not exactly direct contact) in my opinion is not very productive nor interesting.
In fact it shares this property with any other kind of art.