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.
The forms are supposed to exist in what is, for Plato, not inaccurately described as a "Platonic heaven." For Plato, when human beings die, their souls achieve some sort of reunion with the forms—reunion, because souls originate in and even, in life, have some recollection of, this Platonic heaven.
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.
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.
Medieval Greece / Byzantine Empire
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