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

3.3. The state of mathematics 43
traditions. The new French approach to rigor and the German focus on pure math-
ematics (i.e. mathematics without the justification of applicability) were important
backgrounds for ABEL’S mathematics; no expressed concerns for applicability can be
found in ABEL’S writings; he was — in every respect — a pure mathematician. On
the other hand, as will become evident in chapters 16 and 19, ABEL was no dogmatic
rigorist when he aimed at producing new mathematical results.
3.3 The state of mathematics
Some tendencies in the mathematicians’ thoughts about mathematics just prior to the
period of main interest merit attention. In particular, prominent mathematicians to-
ward the end of the eighteenth century expressed the belief that mathematics was just
about to reach its pinnacle of cultivation. From the eighteenth century perspective,
where mathematical praxis was to a large part made up of formal and explicit ma-
nipulations of known algebraic or analytic dependencies, these methods seemed of
limited scope. For instance, J. L. LAGRANGE (1736–1813) — a few years after his in-
novative paper on polynomial equations from which our story will commence in the
next chapter — wrote to J. LE R. D’ALEMBERT (1717–1783) in 1781,
“It appears to me also that the mine [of mathematics] is already very deep and
that unless one discovers new veins it will be necessary sooner or later to abandon
it.” 16
Similar dark visions seem to be recurring at intervals — often in the form of fin-
de-siècle pessimism. Even into the nineteenth century, a similar view was expressed
by J.-B. J. DELAMBRE (1749–1822). In 1810, DELAMBRE delivered a commissioned
review of the progress made in the mathematical sciences after the French Revolution.
He expressed his concern over the future of mathematics in the following way:
“It would be difficult and rash to analyze the chances which the future offers
to the advancement of mathematics; in almost all its branches one is blocked by
insurmountable difficulties; perfection of detail seems to be the only thing which
remains to be done. [. . . ] All these difficulties appear to announce that the power
of our analysis is practically exhausted in the same way as the power of the ordinary
algebra was with respect to the geometry of transcendentals at the time of
Leibniz and Newton, and it is required that combinations are made which open a
new field in the calculus of transcendentals and in the solution of equations which
these [transcendentals] contain.” 17
16 “Il me semble aussi que la mine est presque déjà trop profonde, et qu’à moins qu’on ne découvre
de nouveaux filons il faudra tôt ou tard l’abandonner.” (Lagrange→d’Alembert, Berlin, 1781. Lagrange,
1867–1892, vol. 13, 368); English translation from (Kline, 1990, 623).
17 “Il seroit difficile et peut-être téméraire d’analyser les chances que l’avenir offre à l’avancement des
mathématiques: dans presque toutes les parties, on est arrêté par des difficultés insurmontables;
des perfectionnemens de détail semblent la seule chose qui reste à faire; [. . . ] Toutes ces difficultés
semblent annoncer que la puissance de notre analyse est à-peu-près épuisée, comme celle de