I'm talking about the division outlined by MacIntyre in Whose Rationality? Whose Justice? between the Classical-Christian tradition, the Encyclopedists who claim to follow the Enlightenment but don't actually believe it, and the Nietzscheans.
Ok, there has been plenty of discussion about the nature of rationalism since those philosophers too.
I don't see why it isn't obvious that the idea of a circle is just as real as anything else you know. The implications of that not being the case is absurd.
I guess I should expand. There are usually two senses of the word 'real' people use in these sort of conversations. The first is 'something that has an effect', or actually people tend to use it more in its negative sense, not real, 'something I can discount'. I'm not even going to bother with that one, I don't think anybody will argue passionately against the utility and impact of the idea of the circle.
The other sense is having to do with whether something has a presence in the universe. Most people I know qualify this as some sort of fuzzy concept of being something made out of atoms, or something. That's a slipperly slope. The laws of how physical things in the universe operate seem as 'concrete' in their own way as material objects, but don't exist as objects in the universe. We know them by their effects, quite often manifested in circles. So their reality lies solely in whether they 'matter' to us. Money has a very dubious existence, yet I'd say it's one of man's most pressing realities, just because of how much it matters.
I submit that reality itself is subjective and only the first definition of 'real' exists. The word in any sense implies a value of something 'mattering' and for something to 'matter' there has to be someone for it to matter to. I think it's safer to stop using the word 'real' and frame things more in the way of domains; areas of how things matter organized by their impact to each other.
I remember looking up the definition and finding that perfection wasn't inherent in the description... So yes I suppose they do exist, but of course the shape we use only corresponds to a symbol so it only exists because we exist.
Is a perfect circle a form that exists beyond space and time, and the imperfect circles that we see illustrated in the world around us are just participating in "circleness"? Or, are circles abstracted concepts derived from our examination of the things we sense around us (In other words: without material things that appear circular, circles do not exist)?
Without the forms of space and time, especially space, the external manifestation of a circle does not exist. Our minds lend the object the appearance of circularity fashioned from our minds' spatial inner representation of a circle. Where does this representation come from? It is a mental capacity to form such representations, but that's not to say that the material world does not lend its matter to our inner spatial idea of a circle. Its creation is based out of the need to understand the external world.
The circle exists as an idea, or ideal, but it is a real idea as others have pointed out.
"I absorb energy like a sponge everywhere I go. It allows me to see the world and my purpose in it." Zak Bagans, Ghost Adventures (INFJ)