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

Chapter 9
The nineteenth-century change in
epistemic techniques
Of the numerous transitions in epistemic techniques — changes in how mathematics
was conducted — which took place in the 1820s, few were as far reaching as the ini-
tiation of the movement aiming at rigorizing analysis through arithmetization. The
rigorization of analysis involved fundamental changes in the basic concepts of the
discipline and also manifested itself on the technical level. The causal events leading
to the rigorization are varied and span both external and internal factors. However,
it is no coincidence that the rigorization was originally promoted in textbooks which
were needed for the large-scale instruction in mathematics brought about by external
events.
Critical revision: A change in epistemic techniques. Central to the replacement
of existing practice was the prominence given by leading research mathematicians
to the rigorization program and the critical revision. J. L. LAGRANGE’S (1736–1813)
textbooks marked a new awareness concerning the foundations of the calculus — later
rigorization built upon the Lagrangian program.
In the 1820s, A.-L. CAUCHY (1789–1857) presented his revision of the foundations
of analysis which meant ingeniously revising the basic notions of the discipline. At
the core of the change, CAUCHY discarded the eighteenth century conception of for-
mal equality between expressions in favor of a new concept of arithmetical equality
between functions. This change had implications for most of the other basic notions:
limits, convergence, continuity, and differentiability to name but a few. For instance,
CAUCHY was led by his new rigor to abandon attributing meaning to sums of di-
vergent series and to promote tests of convergence into central positions within his
theoretical framework.
By the mid-1820s, N. H. ABEL (1802–1829) expressed severe concerns for the con-
temporary state of the calculus: he felt that it lacked system and rigor. Simultaneously,
ABEL revealed his interest in finding out how the previous generations could have ob-
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