Divergent Quotes (6 quotes)
I have not trodden through a conventional university course, but I am striking out a new path for myself. I have made a special investigation of divergent series in general and the results I get are termed by the local mathematicians as “startling.”
First letter to G.H. Hardy (16 Jan 1913). In Collected Papers of Srinivasa Ramanujan (1927), xxiii. Hardy notes he did “seem to remember his telling me that his friends had given him some assistance” in writing the letter because Ramanujan's “knowledge of English, at that stage of his life, could scarcely have been sufficient.”
It becomes the urgent duty of mathematicians, therefore, to meditate about the essence of mathematics, its motivations and goals and the ideas that must bind divergent interests together.
In 'Mathematics in the Modern World', Scientific American (Sep 1964) 211, No. 3, 42. Collected in Ronald J. Comer and Morris Kline, Mathematics in the Modern World: Readings from Scientific American (1988), 20.
Notable enough, however, are the controversies over the series 1 – 1 + 1 – 1 + 1 – … whose sum was given by Leibniz as 1/2, although others disagree. … Understanding of this question is to be sought in the word “sum”; this idea, if thus conceived—namely, the sum of a series is said to be that quantity to which it is brought closer as more terms of the series are taken—has relevance only for convergent series, and we should in general give up the idea of sum for divergent series.
…...
Since the seventeenth century, physical intuition has served as a vital source for mathematical porblems and methods. Recent trends and fashions have, however, weakened the connection between mathematics and physics; mathematicians, turning away from their roots of mathematics in intuition, have concentrated on refinement and emphasized the postulated side of mathematics, and at other times have overlooked the unity of their science with physics and other fields. In many cases, physicists have ceased to appreciate the attitudes of mathematicians. This rift is unquestionably a serious threat to science as a whole; the broad stream of scientific development may split into smaller and smaller rivulets and dry out. It seems therefore important to direct our efforts towards reuniting divergent trends by classifying the common features and interconnections of many distinct and diverse scientific facts.
As co-author with David Hilbert, in Methods of Mathematical Physics (1937, 1989), Preface, v.
The divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever. By using them, one may draw any conclusion he pleases and that is why these series have produced so many fallacies and so many paradoxes.
From letter (Jan 1828) to his former teacher Berndt Holmböe. In Morris Kline, Mathematics: The Loss of Certainty (1982), 170.
The series is divergent, therefore we may be able to do something with it.
Epigraph without citation, in Morris Kline (ed.), Mathematical Thought From Ancient to Modern Times (1990), Vol. 3, 107.
In science it often happens that scientists say, 'You know that's a really good argument; my position is mistaken,' and then they would actually change their minds and you never hear that old view from them again. They really do it. It doesn't happen as often as it should, because scientists are human and change is sometimes painful. But it happens every day. I cannot recall the last time something like that happened in politics or religion.
(1987) -- 
