Hilbert says that we can introduce a new type of mathematical object when it is not only (a) consistent (meaning “can be introduced without generating contradictions, but also (b) useful in fulfilling a certain purpose. Hilbert says: “[S]uccess is necessary; here too, it is the highest tribunal, to which everyone submits” (Hilbert p.370). It looks like Hilbert is appealing to pragmatics, saying that we are justified in introducing a new notion insofar as it furthers and facilitates our human understanding. When talking about the infinite (something that he says exists nowhere in nature), Hilbert claims that we are justified in introducing the ideal elements “at infinity” insofar as they “have the advantage of making the system of the laws of connection as simple and perspicuous as is at all possible” (372-373).

-- AnnieS - 24 Oct 2006