  The first two claims are tolerably clear for present purposes. Existence can be formalized as ‘∃xMx’, where ‘Mx’ abbreviates the predicate ‘x is a mathematical object’ which is true of all and only the objects studied by pure mathematics, such as numbers, sets, and functions. Abstractness says that every mathematical object is abstract, where an object is said to be abstract just in case it is non-spatiotemporal and (therefore) causally inefficacious. (For further discussion, see the entry on  abstract objects.)