The Cogito was taken to task by many philosophers but the most thorough shellacking came from Friedrich Nietzsche in Beyond Good & Evil. There are intrinsic assumptions which were exposed quite fantastically in that work. The short version is that the Cogito fails fantastically to establish existence of self.
Your question seems less about existence and more about the resolution of the Platonism / Nominalism debate.
https://philpapers.org/surveys/results.pl
Whether or not abstract things exist is at the epicenter of that debate. Do numbers exist? Do shapes exist? Do bizarre creatures concocted by one's imagination exist?
I find that religious folks tend to side with Platonism; that numbers and imagined things exist.
Personally, that's ridiculous. Of course numbers do not exist, shapes do not exist. These are merely abstract constructs of one's mind. No more real than the boogeyman who stalks the dreams of children. It's electrical-chemical phenomenon of our brains. To say that something exists is to say that it is real, or to say that it is part of reality but our imaginations represent the imaginary or non-real. The position seems entirely bankrupt to me.
I think the issue Platonists have is thinking that properties are the same as objects. Objects have properties. An objects has properties such as shape, size, color, texture, yet Platonists think shapes are objects. That is their folly. An object can be [insert shape] but an object cannot be an [insert shape] because shapes don't exist. Objects which are circular exist, but circles do not exist. You cannot walk into the woods and pluck a circle from the weeds.
There is ample philosophy about this already and I think I've said enough to clarify my position. This forum is probably the worst place for this type of discussion. I attempted to describe the difference between intrinsic and extrinsic to somebody over several posts on another thread and the user failed miserably at comprehending this elementary distinction. I recommend reading actual philosophy (which I admit can be mentally exhausting) if you wish to further your own understanding of this debate.
