So for Plato, forms exist in some other "divine" realm similar to the way concrete objects exist in our ordinary world. So the existence of forms is comparable to the existence of concrete objects... except not in the same realm.
But what exactly do contemporary philosophers mean when they say abstract objects exist. I doubt they mean they exist in some other dimension as perfect versions of the ones we have in our world.
Eg: Putnam and Quine are called mathematical platonists:
They believe abstract mathematical objects exist... but what exactly does this mean? I really don't understand what exist means other than the way concrete objects exist, except in some other realm/dimension. Is that what they mean by existence of mathematical objects?
So what are nominalists and realists actually debating about when they say abstract objects don't exist vs exist?