RePoSS #11: The Mathematics of Niels Henrik Abel: Continuation ...

5.2. LAGRANGE’s theory of equations 67
because both multiplication (and addition) were implicitly assumed to be commuta-
tive, associative, and distributive. This concept of formal equality was intertwined
with LAGRANGE’S focus on the formal appearance of expressions which made the
form and not the value the important aspect of expressions. Denoting the roots of the
general µ th degree equation by x1, . . . , xµ, LAGRANGE considered the variables (roots)
to be independent symbols. For example, in LAGRANGE’S view, the two expressions
x1 − x2 and x2 − x1 were always (formally) different, although particular values could
be given to x1 and x2 such that the values of the two expressions were equal. The
independence of the symbols x1, . . . , xµ reflected the fact that in a general equation, the
coefficients were considered independent; to treat special, e.g. numerical equations, a
modified approach had to be adapted.
LAGRANGE’S formal approach reflects a general eighteenth century conception of
polynomials not as functional mappings but as expressions combined of various sym-
bols: variables and constants, either known or unknown. LAGRANGE was not par-
ticularly explicit about this notion of formal equality which occurs throughout his
investigations; however, he emphasized that
“it is only a matter of the form of these values and not their absolute [numerical]
quantities.” 31
The focus on formal values was lifted when GALOIS saw that in order to address
special equations in which the coefficients were not completely general — some or all
of them might be restricted to certain numerical values — he had to consider the nu-
merical equality of the symbols in place of LAGRANGE’S formal equality.
5.2.2 The emergence of permutation theory
An important part of LAGRANGE’S approach was the introduction of symbols denot-
ing the roots which enabled him to treat them as if they had been known. 32 This al-
lowed him to focus his attention on the action of permutations on formal expressions
in the roots. LAGRANGE set up a system of notation in which
f �� x ′� � x ′′� � x ′′′��
meant that the function f was (formally) altered by any (non-identity) permutation
of x ′ , x ′′ , x ′′′ . 33 For instance, the expression x ′ + αx ′′ + α 2 x ′′′ would be altered by any
non-identity permutation if α was an independent symbol (or a number, say, α = 2).
If the function remained unaltered when x ′ and x ′′ were interchanged, LAGRANGE
wrote it as
f �� x ′ , x ′′� � x ′′′�� .
31 “il s’agit ici uniquement de la forme de ces valeurs et non de leur quantité absolute.” (Lagrange,
1770–1771, 385).
32 (Kiernan, 1971, 45). Reminiscences of this can also be found with EULER.
33 (Lagrange, 1770–1771, 358).