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

21.3. The role of counter examples 397
Arguments carried out “in general”. In the formula based paradigm, a situation
sometimes arose in which the formula carrying the mathematical argument did not
apply in all (numerical) cases. A number of such examples have been described above
starting with EULER’S awareness that peculiar numerical results could emerge if specific
values were inserted in expressions which were formally equal. 17
As J. V. GRABINER describes, 18 J. L. LAGRANGE (1736–1813) held a strong and
lifelong belief in the concept of “the general” in mathematics. Not only could formulae
which were valid “in general” be of high importance — a general approach and system
in mathematics was also strived for. When LAGRANGE presented his argument that
“all” functions could be expanded in Taylor series, he was also aware that this might
indeed fail to be true for particular functions at particular points. 19 However, these
instances where the general results failed to be true were particular, peculiar, and of
little interest to mathematicians ascribing to the formula based paradigm.
In connection with ABEL’S Paris memoir, an even more elaborate case was pre-
sented. At a crucial point in his argument to determine the number µ of independent
integrals, ABEL employed a generalized degree operator called h. 20 Just as is the case
for the ordinary degree operator of polynomials deg P (which ABEL also used), the
degree of a sum may fail to be the maximum of the two degrees,
deg (P1 + P2) ? = max {deg P1, deg P2} , (21.1)
if deg P1 = deg P2. However, in the Paris memoir, ABEL was not interested in peculiar-
ities and he simply argued that the equality corresponding to (21.1) was true “in gen-
eral”, i.e. with the exception of some particular cases of little interest (see page 362).
Once the paradigms had shifted, the precise determination of the number µ (called
the genus) and the investigation and exposure of the necessary assumptions became
a hot topic of mathematics.
In a similar situation, ABEL concluded his summary of well known properties of
elliptic functions in the Précis by the statement:
“The formulae which have been presented above uphold with certain restrictions
if the modulus c is arbitrary, real or imaginary.” 21
ABEL’S way of obtaining the important formulae — often through tedious manipula-
tions of infinite representations — could result in particular cases for which the formu-
lae degenerated or produced false results. However, these cases were few and did not
constitute an obstacle to presenting the formulae.
17 See section 10.1.
18 See (Grabiner, 1981a, 317) or (Grabiner, 1981b, 39).
19 See e.g. (Lagrange, 1813, 29–30).
20 See section 19.3.
21 “Les formules présentées dans ce qui précède ont lieu avec quelques restrictions, si le module c est
quelconque, réel ou imaginaire.” (N. H. Abel, 1829d, 245).