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

124 Chapter 6. Algebraic insolubility of the quintic
on RUFFINI’S research was, of course, CAUCHY whom ABEL met in Paris without any
traceable interaction taking place. 54
Although the primary information on how ABEL came to know of RUFFINI’S proofs
is rather sparse, I find further support for the assumption of independence in the
mathematical technicalities as documented in the preceding sections. RUFFINI’S and
ABEL’S differences in notation and approach to permutations, ABEL’S definition of
algebraic expressions and his careful proof of the auxiliary theorems describing them
all suggest to me that ABEL’S deduction was a tailored argument for the impossibility,
independent of any earlier such proofs. The common inspiration from LAGRANGE,
which permeated both the works of RUFFINI and ABEL, should be evident enough to
account for similarities in studying the blend of equations and permutations.
6.8 Limiting the class of solvable equations
At a conceptual level, ABEL’S proof that the general quintic could not be solved al-
gebraically was more than just another proof added to the body of mathematics. For
centuries, mathematical intuition had suggested that an algebraic solution to the fifth
degree equation should exist but probably be difficult to find. ABEL had demonstrated
that any supposed solution carried with it an internal contradiction and thus the re-
sult not only made the belief in general algebraic solubility tremble, it completely
destroyed it.
In negating the existence of an algebraic solution, ABEL provided an instance of
a negative result — negative in the sense that it contradicted contemporary intuition.
Similar counter intuitive results abound in mathematics in the period as a result of a
fundamental transition toward concept based mathematics. 55
The outspoken reactions of the mathematical community to ABEL’S impossibility
proofs can be divided in three. Some mathematicians, often belonging to the older
generation or the loosely institutionalized amateur mathematicians, protested against
the result and held both the statements and their proofs to be flawed. To these math-
ematicians, the break with their established intuition forced them into their rejection.
Others accepted the result but provided refinements of the proofs and their assump-
tions. And yet others not only accepted the results but saw that the quintic only con-
stituted one example of an unsolvable equation. Thereby, the more general problem
of algebraic solubility could be formulated.
From the perspective of investigating the concept of solubility, the quintic helped
distinguish the class of algebraically solvable equations within the class of all polyno-
mial equations (see figure 6.1). On the other hand, in his research on the division of the
circle, GAUSS had demonstrated that infinitely many equations existed which could
54 For a discussion of the relationship between CAUCHY and ABEL, see chapter 12.
55 See chapter 21.