Abel's famous "devil quote"

Question

There is a famous quote of Abel's:

Divergent series are in general something fatal, and it is a disgrace to base any proof on them.

The first part is often translated freely as "Divergent series are an invention of the devil". It is frequently presented as an introduction quote in elementary texts on general summability techniques. Just a random example. And another one.

But what exactly did Abel mean by this? Was he indeed so short-sighted in this instance that he did not conceive of the more general interpretation of series as a map (with optional regularity constraints) from partial sum sequences to (complex) numbers? Is it a philosophical interpretation? Or is it taken out of (historical) context? I am mainly confused because one of the most well-known elementary summability methods is derived directly from one of his own theorems, and is in fact named after him: Abel summability.

Answer 1

The original french quotation of Abel, contained in a letter to his former teacher Holmboe (January 16, 1826), is as follows:

Les séries divergentes sont en général quelque chose de bien fatal et c’est une honte qu’on ose y fonder aucune démonstration. On peut démontrer tout ce qu’on veut en les employant, et ce sont elles qui ont fait tant de malheurs et qui ont enfanté tant de paradoxes...Enfin mes yeux se sont dessillés d’une manière frappante, car à l’exception des cas les plus simples, par exemple les séries géométriques, il ne se trouve dans les mathématiques presque aucune série infinie dont la somme soit déterminée d’une manière rigoureuse, c’est-à-dire que la partie la plus essentielle des mathématiques est sans fondement. Pour la plus grande partie les résultats sont justes il est vrai, mais c’est là une chose bien étrange. Je m’occupe à en chercher la raison, problème très int´eressant.

The english translation is

Divergent series are, in general, something terrible and it is a shame to base any proof on them. We can prove anything by using them and they have caused so much misery and created so many paradoxes. . . Finally my eyes were suddenly opened since, with the exception of the simplest cases, for instance the geometric series, we hardly find, in mathematics, any infinite series whose sum may be determined in a rigorous fashion, which means the most essential part of mathematics has no foundation. For the most part, it is true that the results are correct, which is very strange. I am working to find out why, a very interesting problem.

So, Abel was mainly concerned with the paradoxical nature of divergent series, such as $$1-1+1-1 \ldots = \begin{cases} (1-1)+(1-1)+(1-1)+ \ldots =0 \\ 1+ (-1+1)+ (-1+1)+ \ldots =1 \end{cases}$$ since in his time concepts such as the Cesaro summation or the Riemann's theorem on rearrangement of conditionally convergent series were not yet discovered (in fact, Abel died in 1829, whereas Riemann was born in 1826 and Cesaro in 1859). This preoccupation pushed him in investigating the nature of convergence and led to his celebrated summability methods.

An interesting discussion of divergent series, starting precisely with Abel's quote, is contained in C. Rousseau's preprint Divergent Series: past, present, future, arXiv:1312.5712.