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

12.1. ABEL’s critical attitude 223
from a manuscript entitled Sur les séries which ABEL worked on and hoped to present
in A. L. CRELLE’S (1780–1855) Journal. However, the publication never materialized
and ABEL’S manuscript was left in the form of an interesting draft. It was eventu-
ally published in the Œuvres. 11 In the draft, ABEL expanded an otherwise unspecified
function
and rearranged its terms to find
f (x + ω) = a0 + a1 (x + ω) + a2 (x + ω) 2 + . . . (12.1)
f (x + ω) = a0 + a1x + a2x 2 + · · · + (a1 + 2a2x + . . . ) ω + . . . .
From this, ABEL concluded
f (x + ω) = f (x) + f ′ (x)
1 ω + f ′′ (x)
2 ω2 + . . . (12.2)
“if this series is convergent”. 12 The argument was followed by remarks to the effect
that the series of (12.2) was indeed always convergent! This serves to illustrate that de-
spite the extensive criticism which ABEL raised against the unrigorous reasoning with
series, his own reasoning was constantly at risk of making the same mistakes. Fur-
thermore, the example shows how the reordering of terms was an unrealized problem
in the 1820s. This becomes interesting when we consider the emergence of a concept
of absolute convergence (see below).
12.1 ABEL’s critical attitude
ABEL’S name is frequently mentioned in the same sentence as CAUCHY and K. T. W.
WEIERSTRASS (1815–1897) when historians of mathematics attempt to pin-point the
movement within mathematics known as rigorization or — more specifically — arith-
metization. 13 And certainly, after an almost religious conversion, ABEL became an ar-
dent follower of a version of CAUCHY’S new rigor; a version which ABEL to a large
extent helped form, himself. On the other hand, rigorizing the calculus meant re-
founding the entire domain of analysis on a completely new system, and ABEL’S
mathematical contribution to the rigorization was limited to a single sub-discipline,
the theory of infinite series. But of equal importance, ABEL’S written testimony of his
conversion to Cauchyism and his hearted, public interpretation of some of its doctrines
helped shape the movement in the nineteenth century. In this and the following chap-
ter, ABEL’S critical attitude as well as his contributions to the theory of series will be
investigated and analyzed.
11 (N. H. Abel, [1827] 1881).
12 (ibid., 204).
13 See e.g. (Kline, 1990, 948).