"As for methods, I have sought to give them all the rigour that one demands in geometry, in such a way as never to revert to reasoning drawn from the generality of algebra. Reasoning of this kind, although commonly admitted, particularly in the passage from convergent to divergent series and from real quantities to imaginary expressions, can, it seems to me, only occasionally be considered as inductions suitable for presenting the truth, since they accord so little with the precision so esteemed in the mathematical sciences. We must at the same time observe that they tend to attribute an indefinite extension to algebraic formulas, whereas in reality the larger part of these formulas exist only under certain conditions and for certain values of the quantities that they contain. In determining these conditions and these values, I have abolished all uncertainty." (Augustin-Louis Cauchy," Cours d’analyse de l’École Royale Polytechnique", 1821)
"On the whole, Divergent series are the work of the Devil and it's a Shame that one dares base any Demonstration on them. You can get whatever result you want when you use them, and they have given rise to so many Disasters and so many Paradoxes." (Niels H Abel, [letter to Bernt M Holmboe, a former teacher] 1826)
"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." (Niels H Abel, [letter to Bernt M Holmboe, a former teacher] 1826) [alternative translation]
"Divergent series are in general very mischievous affairs, and it is shameful that any one should have founded a demonstration upon them. You can demonstrate anything you please by employing them, and it is they who have caused so much misfortune, and given birth to so many paradoxes." (Martin Ohm, "The Spirit of Mathematical Analysis and its Relation to a Logical System", 1842)
"If we then compare the position in which we stand with respect to divergent series, with that in which we stood a few years ago with respect to impossible quantities [that is, complex numbers], we shall find a perfect similarity […] It became notorious that such use [of complex numbers] generally led to true results, with now and then an apparent exception. […] But at last came the complete explanation of the impossible quantity, showing that all the difficulty had arisen from too great limitation of definitions." (Augustus de Morgan, Penny Cyclopaedia, cca. 1833-1843)
"A great discovery is not a terminus, but an avenue leading to regions hitherto unknown. We climb to the top of the peak and find that it reveals to us another higher than any we have yet seen, and so it goes on. The additions to our knowledge of physics made in a generation do not get smaller or less fundamental or less revolutionary, as one generation succeeds another. The sum of our knowledge is not like what mathematicians call a convergent series […] where the study of a few terms may give the general properties of the whole. Physics corresponds rather to the other type of series called divergent, where the terms which are added one after another do not get smaller and smaller, and where the conclusions we draw from the few terms we know, cannot be trusted to be those we should draw if further knowledge were at our disposal." (Sir Joseph J Thomson, [letter to G P Thomson], 1930)
"[...] the only characteristic property that continuous functions have is that near objects are sent to corresponding near objects, that is, a convergent sequence is mapped to the corresponding convergent sequence. It is reasonable to say that we cannot expect to extract from that property neither numerical consequences, nor a method to extensively study continuity. On the contrary, analytic functions can be represented by equations (precisely speaking, by infinite series). Compared to analytic functions, continuous functions, in general, are difficult to represent explicitly, although they exist as a concept." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)
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