Since the perfect circle can not be drawn, Plato considered it an Ideal. The fact that we could understand ideals and work with them as reality led him to the belief that mathematics was real "out there" and had an existence all of its own apart from our own minds.
With Plato this comes up over the Allegory of the Cave, as found in Wikipedia: Plato imagines a group of people who have lived chained in a cave all of their lives, facing a blank wall. The people watch shadows projected on the wall by things passing in front of the cave entrance, and begin to ascribe forms to these shadows. According to Plato, the shadows are as close as the prisoners get to seeing reality. He then explains how the philosopher is like a prisoner who is freed from the cave and comes to understand that the shadows on the wall are not constitutive of reality at all, as he can perceive the true form of reality rather than the mere shadows seen by the prisoners.
The Allegory is related to Plato's Theory of Forms,[1] wherein Plato asserts that "Forms" (or "Ideas"), and not the material world of change known to us through sensation, possesses the highest and most fundamental kind of reality. Only knowledge of the Forms constitutes real knowledge.
Since the time of Plato, mathematicians have tended philosophically--at least those that bother to think about it--to divide into two distinct camps. Today we have the Formalists, such as Hilbert and the Platonists such as Godel. Godel once said that 'Mathematical objects are as real as physical objects and we have just as much evidence of that as of physical objects.'
Here's a quote: Platonism has always had a great appeal for mathematicians, because it grounds their sense that they're discovering rather than inventing truths. When Gödel fell in love with Platonism, it became, I think, the core of his life.
Platonism was an unpopular position in his day. Most mathematicians, such as David Hilbert, the towering figure of the previous generation of mathematicians, and still alive when Gödel was a young man, were formalists. To say that something is mathematically true is to say that it's provable in a formal system. Hilbert's Program was to formalize all branches of mathematics. Hilbert himself had already formalized geometry, contingent on arithmetic's being formalized. And what Gödel's famous proof shows is that arithmetic can't be formalized. Any formal system of arithmetic is either going to be inconsistent or
incomplete. Rebecca Goldstein, Godel and the Nature of Mathematical Truth.
i get the circle thing, but the numbers theory\neq me understanding what you're talking about.
