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

4.2. Mathematical change as a history of new questions 53
4.2 Mathematical change as a history of new questions
A permeating theme of the present work is the emergence of new questions in the
early nineteenth century. The description and analyses of ABEL’S algebraic works
serve to illustrate three aspects of this process:
1. New questions may have unexpected answers which push mathematics for-
ward.
2. New and fertile questions may arise from importing methods or inspirations
from one theoretical complex into another; entirely new theories may develop.
3. A deliberate reformulation of hard but improperly formulated questions may
transform them into forms more open to mathematical treatment. The process of
reformulating the question may involve a process of scrutinizing mathematical
intuitions.
New questions with unexpected answers. Ever since procedures to algebraically
compute the roots of cubic and bi-quadratic equations were discovered in the mid-
dle of the sixteenth century, the search had been on for a generalization to quin-
tic equations. Once R. DU P. DESCARTES’ (1596–1650) new notational system trans-
lated the problem into purely algebraic manipulations of symbols, the belief became
widespread that such a generalization had to be obtainable. Although the goal defied
even the greatest mathematicians for centuries, the belief remained intact as late as
the second half of the eighteenth century. EULER, for instance, felt assured enough
about the general algebraic solubility of equations to utilize it as the basis for proofs
of another almost self-evident result: the fundamental theorem of algebra.
In 1770, LAGRANGE decided to study carefully the reasons behind the solubility
of equations of degrees 1,2,3, and 4 with the hope of obtaining some kind of general
procedure which could subsequently be applied to the fifth degree equation. LA-
GRANGE’S investigations were important in two respects: firstly, they provided a the-
orization of the problem into problems of permutations of the roots — a mathemat-
ical tool which would become immensely important for the problem, and secondly,
LAGRANGE envisioned that the powers of his analysis were not powerful enough to
deduce the desired result. This second observation can be taken as the first hint that
such solutions were beyond the reach of humans.
In the last years of the eighteenth century, the full consequences of the failure to
obtain algebraic solutions to the general quintic were realized and published indepen-
dently by two mathematicians located at opposite ends of the professional spectrum:
the German “prince of mathematics” GAUSS and the much lesser known Italian RUF-
FINI. In 1799, GAUSS remarked as a criticism of EULER that the algebraic solubility
of equations should not be taken for granted. The same year, RUFFINI published the