"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] 1826)
"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)
"Between mathematicians and astronomers some misunderstanding exists with respect to the meaning of the term 'convergence'. Mathematicians [...] stipulate that a series is convergent if the sum of the terms tends to a predetermined limit even if the first terms decrease very slowly. Conversely, astronomers are in the habit of saying that a series converges whenever the first twenty terms, for example, decrease rapidly even if the following terms might increase indefinitely. [...] Both rules are legitimate; the first for theoretical research and the second for numerical applications. Both must prevail, but in two entirely separate domains of which the boundaries must be accurately defined. Astronomers do not always know these boundaries accurately but rarely exceed them; the approximation with which they are satisfied usually keeps them far on this side of the boundary. In addition, their instinct guides them and, if they are wrong, a check on the actual observation promptly reveals their error [...]" (Henri Poincaré, "New Methods in Celestial Mechanics" ["Les méthodes nouvelles de la mécanique céleste"], 1892)
"Mathematics has, of course, given the solution of the difficulties in terms of the abstract concept of converging infinite series. In a certain metaphysical sense this notion of convergence does not answer Zeno’s argument, in that it does not tell how one is to picture an infinite number of magnitudes as together making up only a finite magnitude; that is, it does not give an intuitively clear and satisfying picture, in terms of sense experience, of the relation subsisting between the infinite series and the limit of this series." (Carl B Boyer, "The History of the Calculus and Its Conceptual Development", 1959)
"[...] 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)
"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)
No comments:
Post a Comment