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

8.2. Construction of the irreducible equation 165
As previously noted, ABEL’S belief that any problem could be converted into a
solvable one was held by most mathematicians throughout the 19 th century. It be-
came prominent in the so-called Hilbert Programme before the development of axiomat-
ics stressed that decidability could only be asked and answered relatively to the (ax-
iomatic) system in which the problem was embedded.
With ABEL’S new driving question, modified from the ones motivating the impos-
sibility proof and the study of Abelian equations, it was his intention to explore the
grey area between the entire set of equations and the ones known to be solvable (see
figures 6.1 and 7.3). ABEL’S hope was to delineate the border line between solvable
and unsolvable equations by some external characteristic.
8.2 The construction of the irreducible equation
satisfied by a given expression
The general program of ABEL’S research was to construct a list of all irreducible solv-
able equations and subsequently match any given equation against this list. His at-
tempt at implementing this scheme consisted of a construction of the irreducible equa-
tion satisfied by a given algebraic expression. After establishing certain properties of
this equation from the expression which satisfies it, ABEL returned to the problem of
determining whether a given equation was solvable or not.
The first part of ABEL’S notebook manuscript contained theorems and results pre-
sented in a clear and deductive manner. Their contents showed frequent similarities
with the opening studies of the form of algebraic expressions satisfying an equation as
carried out in the impossibility proof (see section 6.3.3). If anything, the 1828 notebook
lacked — by comparison to the impossibility proof — the clear, albeit defective, classi-
fication of algebraic expressions of which only reminiscences were given. The clas-
sification established in the notebook was insufficient to cover some of the required
deductions, and it is possible that ABEL, himself, had noticed this deficiency (see be-
low).
Basic concepts. In the opening section of the manuscript proper (following a lengthy
introduction), ABEL outlined his own characterization of algebraic expressions which
géomètres distingués on ne put parvenir au but proposé. Cela fit présumer que la résolution des
équations générales était impossible algébriquement; mais c’est ce qu’on ne pouvait pas décider,
attendu que la méthode adoptée n’aurait pu conduire à des conclusions certaines que dans le cas
où les équations étaient résolubles. En effet on se proposait de résoudre les équations, sans savoir
si cela était possible. Dans ce cas, on pourrait bien parvenir à la résolution, quoique cela ne fût
nullement certain; mais si par malheur la résolution était impossible, on aurait pu la chercher une
éternité, sans la trouver. Pour parvenir infailliblement à quelque chose dans cette matière, il faut
donc prendre une autre route. On doit donner au problème une forme telle qu’il soit toujours possible
de le résoudre, ce qu’on peut toujours faire d’une problème quelconque. Au lieu de demander
une relation dont on ne sait pas si elle existe ou non, il faut demander si une telle relation est en effet
possible.” (ibid., 217).