When I make a claim in a proof that a mathematical entity exists, is this no more than saying that the theory I'm working within is consistent, and that all the steps upto that point in the proof are allowed moves in the theory?
By mathematical entity I mean a number, say 3; or an algebraic system such as the group of rotations of the square or the ring of natural numbers; or the category of varieties (a variety is an algebraic system with a fixed number of operations). It could also be a circle, or the category of all manifolds. It could be set theory axiomatized in ZFC.
Not only do these entities relate to each other (for example, after 2 is 3; the group of rotations of a square is related to that of a cube) but also they have structure (3=2+1 in the ring of natural numbers; or the circle, a 1-dimensional manifold is made up of 0-dimensional points)
Existence is a loaded word, we experience in our everyday lives with the existence of chairs and tables or the colour red. It seems very odd that the same word is used for entities that are unavailable directly to the senses, but at the same time it is 'oddly' appropriate.
Taking the platonic perspective - that these entities exist in their own world - and that we access this world through an effort of will. Then supposing that our own physical universe does have a complete mathematical theory. Then the entities in this theory, as well as the theory itself appear to exist in this other separate world, and our own physical world (with us in it as we are physical beings) appears to be incarnated in someway in this other world. This seems odd. Perhaps this suggests that the universe cannot have an underlying mathematical theory?
On the other hand, taking an entirely formal point of view seems (to me) deeply unsatisying. Its an interesting/entertaining point of view, granted.
For one thing, a formalisation appears to be only a perspective or an approximation. For example I have a circle in mind, I could choose to formalise this as a figure of plane geometry as Euclid originally did or a 1-dimensional smooth manifold. It is approximate, as formalisations have changed over time, and are likely to keep on doing so.