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

208 Chapter 11. CAUCHY’s new foundation for analysis
1821 Cours d’analyse d’École Royale Polytechnique.
Première partie. Analyse algébrique
1823 Résumé des leçons données a l’École
Royale Polytechnique sur le calcul infinitésimal
1829 Leçons sur le calcul différentielle
Table 11.1: CAUCHY’s textbooks on the calculus
the truth but should never be allowed to act as exact proofs. In particular, CAUCHY
mentioned how arguments by the generality of algebra had been leading mathemati-
cians into unfounded passages “from convergent to divergent series, from real quan-
tities to imaginary expressions”. 3 CAUCHY continued,
“Similarly, one should realize that they [arguments by the generality of algebra]
tend to attribute to algebraic formulae an indefinite extension whereas in
reality, the majority of these formulae only subsists under certain conditions and
for certain values of the quantities which they contain.” 4
Important examples of the problems which this requirement addressed was the
relationship between formulae such as
1
1 − x and
∞
∑ x
n=0
n
(11.1)
which have been described above. Mathematicians unknowingly adhering to the for-
mal concept of equality had been aware that counter-intuitive results could emerge
if numerical values were inserted into the two expressions and their equality was ex-
tended to cover numerical values. For instance,
1
1 − (−1)
1
= , but
2
∞
∑ (−1)
n=0
n = 1 − 1 + 1 − 1 + . . .
and the sum did certainly not represent the value 1 2 in any numerical sense.
CAUCHY removed these anomalies by dismissing the concept of formal equality
and carefully analyzing the conditions under which a numerical equality between ex-
pressions such as (11.1) would hold. Thus, he found and emphasized that for numer-
ical equality it was required that |x| < 1, and consequently the peculiar results were
all explained away.
3 (A.-L. Cauchy, 1821a, iii).
4 “On doit même observer qu’elles tendent à faire attribuer aux formules algébriques une étendue
indéfinie, tandis que, dans la réalité, la plupart de ces formules subsistent uniquement sous certaines
conditions, et pour certaines valeurs des quantités qu’elles renferment.” (ibid., iii).