In symmetry, 1 + 1 = 1, where 1 (or c1) represents a 360 degree rotation around an axis. Also, 2 + 2 = 1, and 4 + 4 + 4 + 4 = 1 (180 degree rotation = 2, 90 degree = 4).
The platonic question is, is there a mathematical symmetry where the identity operation does not hold? Is there Euclidian mathematics where a 360 degree rotation does not return you to your original state, where one thing is not equal to itself, and so on.
An object is abstract if it is not spatiotemporal or causal. Asking what the number 4 weighs or does makes no sense. It is not mental, i.e. mind independent, in that it exists outside of any agent's thoughts. A Platonist would argue that even an alien civilization is going to discover, not invent, logic and natural arithmetic. Unlike say, Inglourious Basterds which is a mental product that did not exist before Tarantino thought it up.
In any case, just because someone calls themselves a platonist doesn't mean they've carefully studied Plato. In the same fashion as Kant, asserting time and space precede the apprehension of objects that are necessarily within time and space does not imply that time and space are necessarily part of the world itself, but only that one's means of judgement begins with the notions of time and space. In the same way, whether or not you agree with Kant, it is entirely plausible to assume instead that there is a shared faculty in the minds of humans that allows us to conceive of things like 4, and addition and subtraction and mathematical functions in general, and this faculty would neither be abstract nor independent but biological and empirically observable. If this latter case is true, then it would prove to be far more useful for scientific investigation than simply assuming that the reason why we can communicate about the number 4 is because it exists abstractly and independently, because that doesn't tell us anything more about why we have shared concepts of numbers in the first place.
The only idempotent element in a group is the identity. And in abelian groups where the binary operation is given by "+" the identity is denoted by "0". Yet you have "1" being idempotent, so conclude that this is not a group (or that this is a sleight of hand because you've made the identification that 1 = 0).