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

4.2. Mathematical change as a history of new questions 55
a reformulation of the problem to a question of the form “is this goal achievable?” In
the case of the quintic equation, the search for an algebraic solution was reformulated
to a question whether such a solution existed at all.
Such change of attitude toward mathematical goals signal — as JACOBI soon real-
ized — a change toward more general and abstract mathematics. In order to answer
questions concerning possibility of existence, ABEL used implicit quantification over
all possible solutions to the question. His approach was based upon the classification
and normalization of these objects which were therefore studied — not individually —
but as items belonging to a collection defined by a concept. Thus, a concept based ap-
proach to doing mathematics was intimately connected to the kinds of questions asked
and addressed.
In the theory of equations, having established the existence of both algebraically
solvable and unsolvable exemplars, ABEL raised the question of determining directly
whether a given equation would be solvable or not. In a notebook manuscript, ABEL
set out to address this question. For certain types of equations, he made some progress;
however, it was left to GALOIS to outline a theory, based on the same inspirations as
ABEL, which — when elaborated — was powerful enough to answer the question.
rules of mathematical reasoning and the system of primitive truth from which deductions are made.