My original skeletal note on form dates back to the first months of my writing here. This is intended to be the beginning of a better treatment.
When I speak of form, I have in mind first of all the various uses of the term in Aristotle, but secondly a family of ways of looking at the world largely in terms of what we call form, as one might broadly say that both Plato and Aristotle did. Then there is a very different but also interesting family of uses in Kant. There are also important 20th century notions of “structure”.
Form in its Platonic and Aristotelian senses is closely related to what we might call essence, provided we recognize that essence is not something obvious or pre-given. At the most superficial level it may refer to a kind of shape, but it may involve much more.
Plato was classically understood to assert the existence of self-subsistent intelligible “forms” that do not depend on any mind or body. I prefer to emphasize that he put a notion of form first in the order of explanation — ahead of any notion of something standing under something else, ahead of notions of force or action, ahead of particular instances of things. Related to this, he put the contents of thought before the thinker, and used the figure of Socrates to argue that a thing is not good because God wills it to be so, but rather that God wills a thing because it is good.
Aristotle identified form with the “what it is” of a thing. He put form and things like it first in the order of explanation, but explicitly argued that form is not self-subsistent. At the same time, he made the notion of form much more lively. While Plato had already suggested that form has an active character and that the soul is a kind of form, most of his examples of form were static, like the form of a triangle or the form of a chair. Aristotle on the other hand was very interested in the forms of the apparent motions of the stars; the marvelous variety of the forms of animals, considering not only their anatomy but patterns of activity and ways of life; and the diverse forms of human communities, their ways of life and institutionalized concepts of good. Form figures prominently in the development of the notion of ousia (“what it was to have been” a thing) into potentiality, actualization, and prior actuality in Aristotle’s Metaphysics. Aristotelian form is interdependent with logical “matter” in such a way that I think the distinction is only relative. It is also inseparable from a consideration of ends. (See also Form as Value; Form, Substance.)
At first glance, Kant’s notion of form seems like the “mere form” of formalism, contrasted with something substantive called “content”. A certain notion of formalism is so strongly identified with Kant that in some contexts it has become a name for whatever was Kant’s position. I think some of Hegel’s criticisms of Kantian formalism are legitimate, and some overstated. In any case, the categorical imperative and its consequences of respect for others and the value of seeking to universalize ethical precepts — perhaps the first really original constellation of ethical ideas since Aristotle — are deeply tied to Kant’s so-called ethical formalism. Kant seeks a formalist path to the highest good, and argues that only a formalist path can truly reach it. The fact that it is a path to the highest good has deep implications for the meaning of this kind of “formalism”, and sets it apart from what is referred to as formalism in mathematics, logic, or law. This could also be related to Kant’s idea that ethical reason comes before tool-like reason in the order of explanation.
The 20th century notion of “structure” — to hazard a simplifying generalization — is about understanding each thing in terms of its relations to other things — principally how things are distinguished from one another, and how one thing entails another. Structure is form interpreted in a relational way that transcends fixed objects and properties. Objects and properties can be defined by relations of distinction and entailment.