Substance is form. – Eduardo Dubuc (from the Introduction to Kan Extensions in Enriched Category Theory).

A form in mathematics might refer to one of the following:

etc., with some special versions being notable enough to have entries of their own in the nLab, e.g., topological modular form, Maurer-Cartan form, Killing form, etc.

“Form” has also of course philosophical meanings, which sometimes come to the fore when speaking of the relation of category theory and its abstracting principles to the rest of mathematics. For example, the quote of Dubuc at the top of this page is part of a larger remark: “The author hopes that it will not be totally incorrect to say that this paper is a testimony of two basic mathematico-philosophical principles. First, ‘the relevant properties of mathematical objects are those which can be stated in terms of their abstract structure rather than in terms of the elements which the objects were thought to be made of (Lawvere)’ coupled with ‘the relevant facts of category theory hold because of formal interconnections between the concepts involved rather than because of their substantial content (which is none)’. This, because of that peculiar characteristic of the mind which leads every human being to the convinction that abstract ideas are real, can be pushed forward (extrapolated) into a simple purely philosophical principle, namely, ‘substance is form’. Second, ‘everything in mathematics that can be categorized is trivial (Freyd)’ which should be understood. ‘Category Theory is good ideas rather than complicated techniques’.” See also nPOV.

Last revised on January 9, 2015 at 15:18:47. See the history of this page for a list of all contributions to it.