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

10.3. Early rigorization of theory of series 201
To summarize the debate, we have to emphasize three dates. In 1812, GAUSS’
presented his research on hypergeometric series in which his concept of convergence
remains undefined; in 1821, A.-L. CAUCHY (1789–1857) promoted the convergence
of the partial sums into the only acceptable definition of convergence; but as late as
1831, GAUSS employed a D’ALEMBERT-like concept of convergence which entailed the
vanishing of the terms and did not provide convergence of the series. To believe that
GAUSS had anticipated CAUCHY’S notion of convergence and the ratio test in 1812
thus seems to be the least efficient interpretation. GAUSS may very well have held the
same conceptions about convergence in 1812 as he evidently did in 1831. Instead, it
seems that until CAUCHY’S work, different notions of convergence were co-existing
and the position of definitions and tests of convergence within the structure of the
theory of series floated.
10.3.2 BOLZANO’s rigorization of the binomial theorem
Contrary to GAUSS, the Czech priest and mathematician BOLZANO did not have the
ear of the international mathematical community although his ideas and visions for
the foundation of the calculus reached even further than GAUSS’. To promote inter-
est in his work, BOLZANO published critical investigations and new proofs of key
theorems of analysis. He hoped that mathematicians would pay more attention to a
broader philosophical program which he was developing.
In 1816 and in Prague, BOLZANO published a book entitled Der binomische Lehrsatz
which is of particular relevance to the current purpose. 28 In that book, BOLZANO
scrutinized existing derivations of the binomial theorem before going on to present his
own proof. As noted, N. H. ABEL (1802–1829) once praised BOLZANO’S cleverness
(see p. 42); important aspects of ABEL’S criticism may well have their origins with
BOLZANO.
BOLZANO’S critical attitude. In the introduction of his book, BOLZANO reviewed
the structures of previous proofs of the binomial theorem. In the process, BOLZANO
developed a penetrating criticism of the accepted methods of reasoning with infinite
series. Soon, others would repeat BOLZANO’S criticism — at least, ABEL’S judgement
of eighteenth century epistemic techniques in analysis resembled some of BOLZANO’S
points.
A number of interesting themes were raised in BOLZANO’S introduction. BOLZANO
observed that the foundation of the entire “higher analysis” (calculus) rested on Tay-
lor’s Theorem and that this theorem in turn relied on the binomial theorem. Conse-
quently, the obscure status of the proof of the latter theorem had severe implications
for the entire discipline.
28 (Bolzano, 1816). BOLZANO’S titles are often very precise and very long; here the abridged version is
used throughout.