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I read up on the theory of forms and I understand it like this:
When I see a cat I am seeing the particular of the true form a of a cat. All of the cats I see are just imperfect copies of the true form.
Why would we want to make this distinction? A cat is a cat. What value does this theory add to these observations?
The value in Plato's Forms for how we relate to objects requires that you change the formulation a bit:
(1) You see something, A, small, striped, and with weird eyes
(2) You identify the something, A, you see with Cat [capitalization intended]
(3) You see something else, B, fat, beige-colored, with a tail.
(4) You identify B with Cat
Work of the Form: Cat is what enables you to unify the disparate phenomenon that you see under a single idea.
You can disagree with Plato -- several understandings of knowledge and perception do, but the challenge that the forms answer, at least in part, is: "how do I compare things and put them in categories when everything I see is different?"
The distinction is between an abstract concept and a physical thing. That there is a purer idea of "car" than the one you perceive in the physical world.
When you look at Platos allegory of the cave, you can see he has an idea that whats normally perceived are but shadows of what is truly "real".
The idea of these "Forms" is they are the true reality, the higher "truth"
What we gain when we see five individual cats and think instead about a single ideal CAT is a sense of what it means to be a cat independent of any of the specific accidental details of any individual cat's existence.
In practical terms, we're talking here about conceptualization, a crucially useful process under which specific individual experiences are united into larger concepts of the world.
In metaphysical terms, the theory of the Forms is best understood as Plato's way of explaining his claim that the world we think we live in is just a transitory and illusionary copy of a more foundational and eternal Reality underneath the surface.
A cat is a cat. By why are this cat and that cat equally A Cat?
Indeed, there is a natural inclination to be impatient with philosophy, and certainly Plato's concept of the Forms does take some mental grappling. Why bother? First, so you will know what other people are talking about when they talk about Plato. Second, by wrestling with such ideas you can expand not only your knowledge but your imagination, the ways you see things and the points-of-view your brain can assume.
As was pointed out by others, the Form Cat is that which makes the specific cats the same sort of thing and not some other thing. All the instantiated cats are "part of" or, to use Plato's peculiar term, "participate in" Catness. How and why this concept becomes possible is not quite as simple as just "naming things." It may seem physically obvious with cats, but, as Plato explores in The Republic, what about "acts of justice"? Or things we call "good"? What is the Form of "The Good"? How can so many entirely different things assume or "participate in" this... Form?
And as noted previously, Plato draws on geometry as his paradigm. The triangle is not what you actually draw. Nor does it depend what you draw it on, or its scale, or how many times you draw it. It remains "triangular" or within the Form of the triangle. Plato is struggling to grasp the same kinds of truths we grasp in geometry, though in all the "roughly drawn" and sensed objects of the world.
Another metaphor employed by Plato is light. Today, we say that light enables us to see things. What we "see" is not neurons nor is it the "cat." We see the light bouncing off the cat and exciting our neurons, or so we put it. But in fact we never see Light. We only see the partial refractions, bending, and distortion of Light that we call "things." When you see the cat, you see a partial, temporary rupture in Light. The Forms also have something of this quality of "being partially revealed" by matter.
I think you have to put the problem within a historical context and the development of philosophy at the time. The motivation comes from the question, "What types of things exist?". This was the question the Presocratics were interested in.
If one answers this very simply without thinking about the implications of the word "exist", one might say something like, "Matter is all that exists" (followers of Democritus), or "Fire is all that exists" (followers of Heraclitus).
But these types of answers seem to leave out things like properties. Blue, smooth, round, heavy... properties associated with multiple objects at the same time... or all the concepts of mathematics. So it's a sort of unsophisticated/incomplete use of the word "exist" or "being".
My feeling is that historically... Plato was the first major philosopher to notice this issue with the word "existence". He saw that there were obviously things that existed other than material things. But instead of dealing with the language issue (use of the word existence or being), he kept that simple meaning of "existence" associated with material objects... but said that abstract concepts "existed" in another realm. Realm of forms.
"Whatever exists has to exist somewhere in some form" - something problematic about this way of thinking.
I mean, to me it's clear that numbers exist, just as moral truths exist. But to say they exist as an entity in some form. There's a leap being taken there. I think it's a naive extrapolation from our view of physical objects. Physical objects "exist out there", so abstract objects also "exist out there" is not a reasonable implication.
This has relevance in philosophy today. People still ask about the status of abstract objects within materialism.
