The professional mathematician is a Platonist with regard to the existence of mathematical entities, but, if pressed to tell what kind of existence they have, he hides behind a formalist approach. In order to take both attitudes into account in a possibly serious way, the concept of suprasubjective existence is proposed. It involves intersubjective existence, plus a stress on objectivity devoid of actual objects. The idea is illustrated, following William Byers, by the phenomenon of the rainbow: it is not an (...) object but can be said to possess a subjective objectivity. (shrink)