The Forms
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Plato spoke of forms (sometimes capitalized: The Forms) in formulating his solution to the problem of universals. The forms, according to Plato, are roughly speaking archetypes or abstract representations of the many types and properties (that is, of universals) of things we see all around us. There are, therefore, on Plato's view, forms of dogs, of human beings, of mountains, as well as of the color red, of courage, of love, and of goodness. Indeed, for Plato, God is identical to the Form of the Good.
For Plato, the forms exist in what is known as a "Platonic heaven," and when human beings die, their souls achieve reunion with the forms. Plato makes clear that souls originate in this "Platonic heaven" and have recollection of it even in life.
Form and idea are terms used to translate the Greek word eidos (plural eide). "Idea" is a misleading translation, because for Plato, the eide do not exist in the mind.
Several of Plato's dialogues make use of the Forms, including Plato's Parmenides, which outline several of Plato's own objections to his Theory of Forms.
A serious problem for the Theory of Forms is similar to that for Cartesian Dualism, if these Forms exist in 'another world', or are not of the same order of reality as matter, how can they interact with matter to 'inform' it. One standard response was to argue that Forms are part of the same Cosmos and so can interact with matter. But in this case we have the so-called Third Man Problem. If all things (men for instance) have Forms, and things are components of the Cosmos, then Forms are things too and so must have their own Forms ad infinitum... This is not a total refutation of the theory, but its remedy is difficult.
For more information about Plato's theory of universals (forms, ideas), see Platonic realism. See also the divided line of Plato. It is interesting to note that al-Farabi, an excellent student of Plato and Aristotle, didn't even mention the Forms. (cf. "The Philosophy of Plato and Aristotle" by al-Farabi.)
Plato's concept of the Forms found visual representation in the work of Conceptual artist Joseph Kosuth in his work "One and Three Chairs" and other similar works.