Ockham sets and Mathematical sets

This diagram accompanies a comment made at Beyond Necessity.

  

These two diagrams may help explain the difference between Ockham sets and mathematical sets, as I understand it. It's very much a case of reference, which is depicted by the arrows. Linguistic terms are shown on the left, in quotes to emphasise their symbolic nature. The Ockham term 'a and b' refers directly to the two objects a and b. In contrast the mathematical term '{a,b}' refers to a 'set-object', depicted as a quiver containing two arrows pointing to the objects a and b. Thus the mset {a,b} is distinct from its members, a and b, though it refers to them. The empty set, {}, is not nothing. It's a quiver with no arrows. It's this indirection that enables msets not so much to contain other msets but to refer to other msets that give the mset language it's referential power. My view is that msets are an artificial extension to the referencing capabilities of natural language. This may explain why we feel that they have to be 'constructed'. Contrast with the Ockham world in which such intermediaries are absent.