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Today the word form can refer to (at least) three different kinds of mathematical object:

  1. A homogeneous polynomial. This was apparently started by Gauss (1801), renaming what others had called formulasa. (See e.g. Bachmann 1922, p. 17.)

  2. A scalar-valued linear or multilinear map. Apparently started by Kronecker (1866) / Weierstrass (1868), rather out of the blue.

  3. A field of forms in the sense 1 or 2. Apparently started by Christoffel (1869) / Lipschitz (1869), renaming what others called differential formulasb or expressionsc. (See e.g. Weitzenböck 1922, p. 29.)

Question: Has anyone anywhere ever discussed these choices and switches in terminology?

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References: e.g.
a) Euler (1770, 1774, 1827), Lagrange (1773, 1774), Liouville (1852).
b) Bernoulli (1712), Euler (1755, 1768), Agnesi (1775), Cousin (1777), Lagrange (1786), Bossut (1798), Poisson (1811), Abel (1826), Liouville (1852, 1856).
c) Gauss (1815), Jacobi (1845), Riemann (1867), Sturm (1877), Frobenius (1879), Darboux (1882), Cartan (1899).

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    $\begingroup$ On an $n$-dimensional vector space, a linear form $\varphi$ is the same thing as a homogeneous polynomial in $n$ variables (indeterminates) of degree 1 once you express $\varphi$ in terms of a basis. Thus your second and third mathematical objects are closely related. $\endgroup$
    – KConrad
    Oct 28, 2017 at 4:34
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    $\begingroup$ You can also think of forms of of an algebraic group (groups defined over the same field and isomorphic to it over some extension), or to (automorphic) form as particular elements in an automorphic representation. $\endgroup$
    – Desiderius Severus
    Oct 28, 2017 at 6:40
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    $\begingroup$ @DesideriusSeverus it's not specific to algebraic groups, but to plenty of structures defined over a base ring or field (scheme, variety, algebra, vector space with a quadratic form, etc). Serre's book "Galois cohomology", Chap 3, starts with a paragraph "forms" in this general sense (albeit with no general definition). (quoth: Let $K/k$ be a field extension, and $X$ an "object" defined over $k$. We say that a object $Y$, defined over $k$, is a $K/k$-form of $X$ it $Y$ becomes isomorphic to $X$ after extending scalars to $K$) $\endgroup$
    – YCor
    Oct 28, 2017 at 14:04
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    $\begingroup$ Not to be confused with the question "Forming the word 'trace'." $\endgroup$
    – R. van Dobben de Bruyn
    Oct 29, 2017 at 22:12
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    $\begingroup$ Formally, "formula" is the diminutive form of "forma". $\endgroup$
    – Pietro Majer
    Oct 29, 2017 at 22:14

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The evolution of the concept of a form from arithmetic to algebra is discussed on pp. 20, 21, 27 of F. Brechenmacher (arXiv:0712.2566; revised version published in 2016):

Whereas such terms as “forms” and “transformations” had been given an explicit mathematical definition in the arithmetic of quadratic forms in relation to the notion of equivalence relation that had been introduced by Gauss’ 1801 Disquisitiones arithmeticae, they pointed to various and mostly implicit meanings within the algebraic framework of the discussion. (...)

Kronecker had been implicitly referring to the legacy of the works of Gauss and Hermite on the arithmetic of quadratic forms in 1866 — [when] he preferred to make use of the term “form” to name what others would designate as a function ([Weierstrass, 1858]) or as a “polynom” ([Jordan, 1873]) (...)

Kronecker blamed algebraic methods [notably by Jordan] for their tendency to think in term of the “general” case with little attention given to the arithmetic difficulties that might be caused by assigning specific values to the symbols (...) [He] appealed to the tradition of Gauss on behalf of his claim that the theory of forms should be considered as belonging to arithmetic and should consequently focus on the characterisation of equivalence classes in establishing arithmetical invariants thanks to some effective procedures such as g.c.d.s computations.

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  • $\begingroup$ Thank you. (Your text seems to differ significantly from both the arXiv and the published version?!) $\endgroup$
    – Francois Ziegler
    Oct 28, 2017 at 16:09
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    $\begingroup$ indeed, this is not a literal quote but my attempt to summarize several pages of text into something that would be appropriate for the MO answer box; I have removed the "quote" formatting to avoid suggesting that this is quoted verbatim. $\endgroup$
    – Carlo Beenakker
    Oct 28, 2017 at 16:55
  • $\begingroup$ The quote (now literal) nicely explains Kronecker’s motives — but he must have been “blaming” someone else in 1866, before Jordan appeared. Elsewhere Brechenmacher (2007, footnote 3) describes Jordan’s 1874 switch to “form” as almost political; I wonder why he (and Jahrbuch, Kronecker) keep misquoting Christoffel’s 1868 Theorie der bilinearen Functionen as bilinearen Formen? $\endgroup$
    – Francois Ziegler
    Oct 29, 2017 at 13:49

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