In the Stanford Encyclopedia of Philosophy article, 'Platonism in the Philosophy of Mathematics', the following formalisation is given for the existence of a mathematical object:
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.
I am curious as to how these objects are rigorously defined and distinguished from other objects. Clearly there is some natural intuition which suggests a function is a mathematical object but a mountain is not. However, I'm not sure whether the distinction that a function is studied in pure maths whilst a mountain isn't is satisfactory: it seems that, were all humans to stop studying mathematics (or if we never begun in the first place), a function would remain a mathematical object and a mountain would never become one, irrespective of the actions of humans.