For example, quadratic form, differential form, multilinear form, ... I agree with the commenter Nicolo to this related question that usually "form" just means "function with codomain $\mathbb{R}$" -- but when did this convention become standard?

User Rob Arthan has an interesting explanation in a comment on the other question:

I think the usage is historical but deeply ingrained and comes from the days when functions were thought of as having algebraic representations as opposed to the arbitrary functional relations that we think of in the modern set-theoretic approach. So a quadratic, or bilinear or differential or ... form referred to the algebraic form of the representation of the function. (But this is really a mathematics history question and I am not qualified as a historian.)

This might also explain the usage of "form" in Fulton's Algebraic Curves (1969) to mean "homogeneous polynomial" (rather perplexingly to me).

Also in another comment on the same question above, user Lucian seems to suggest that the term might come from the expression "of the form". I don't know if that etymology applies here, but it does seem to describe the use of the word "formal" as in "formal sum".

Question: Does anyone have a source or reference explaining the origin of the use of the term "form" in mathematics?

The standard reference, "Earliest Known Uses of Some Terms in Mathematics" surprisingly does not have an entry (as of this writing), just for "formalism". http://jeff560.tripod.com/f.html


1 Answer 1


The word "forms" (or its equivalents in other languages) as applied to mathematics goes back over 2,000 years. Plato referred to "forms" as an abstract entity, including mathematical objects, to contrast to concrete objects. You can start reading about this here:


or here:


  • $\begingroup$ Wow this is a much older usage than I anticipated! Thank you for pointing this out $\endgroup$ Sep 28, 2016 at 8:21

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