{ 1 } - a mere quibble about how to use words like "exists" and "entity."
{ 2 } - a conflict that makes a big difference to mathematical practice.

The dispute is over whether and how we can use bound variables (like "some") to refer to abstract entities (like numbers and sets). Mathematics is full of such references -- for example:

For every prime number x, there is some greater prime number y.

For every set A, there is some complement set B, which contains all and only those entities that aren't members of A.