What my last post failed to address is the question of whether ideas "exist" in a meaningful sense. In one way, my explanation seems to support Plato's belief that ideas are the only thing that exist, and everything else is merely a poor imitation. The idea of a triangle, for example, is clear and precise; triangles in the real world, however, by definition are never as precise as the ideal triangle (remember, lines have no width in geometry). And if everything has only a relative existence, measured against an idea, it seems logical to say that ideas exist more completely than the material things of which they are copies.
This might be true, except for two things. First, we don't have such a precise definition of most things as we do of triangles or other geometric concepts (and geometry, of course, was central to Plato's philosophy -- the words over the Academy read, "Let none ignorant of geometry enter here"). For example, we know that a chair is something that we sit on, but the exact point at which a chair stops and a stool or a bench begin is something we conceive only vaguely. Suppose we came up with a precise definition of a chair, and then someone presented us an edge case that didn't fit our definition, but did meet most people's idea of what a chair is. In that case, we would surely revise the definition; or in any case, it would be the definition that was deficient, not our common understanding. We operate on the basis of our common understanding, not precise definitions. The concept of a chair is, in other words, a vague concept in itself. It doesn't make sense to say that a physical chair is an imperfect implementation of the idea of "chair" in the same way that a physical triangle is an imperfect representation of the idea of a triangle. The idea of a triangle exists whether anyone has thought of it or not; the idea of a chair only makes sense to the extent that a person has conceived of such a thing.
The other problem is that ideas don't exist in the same way that material things do. I have pointed out the limitations of "existence" insofar as the unity of what we would describe as objects is limited, but their material existence is not in doubt. (It is true that, at a the sub-atomic level, matter and energy become interchangeable to some extent, but energy itself has a physical existence comparable to matter. Although ideas might be instantiated by electrical impulses in the brain, the idea itself is independent of the firing of any particular neurons and has nothing comparable in the material world.) Whatever unity the particles that constitute my body may or may not have, they definitely exist and interact on an entirely different level than ideas. Each bit of matter is entirely unique and exclusive of every other bit of matter, whereas ideas are entirely abstract and "exist," to the extent that we may use that word, whether a person thinks them or not. It is true that each person thinks an idea in his own particular fashion, especially when we are dealing with complicated constructs such as "state" or "individual." Nevertheless, an idea qua idea is never muddled or heterogeneous. Our understanding may be muddled, but the concept exists cleanly whether we think it purely or not.
(This brings to mind Anselm's ontological proof of God: that we can imagine a perfect being, and since existing is more perfect than not existing, this perfect being must exist. The problem here is that the meaning of "exist" is not entirely clear. If we mean a physical existence, as I have been using that word, then we can't summon this God into existence via a mere intellectual exercise. As Kant argues, time and space are conveniences that our mind uses to understand the world. But the concept of "greatest" has no meaning in spatial terms; Anselm is mixing physical concepts with ideas that transcend the physical world, which results in a meaningless statement.)