"For those ultimate ratios with which quantities vanish are not truly the ratios of ultimate quantities, but limits toward which the ratios of quantities decreasing without limit do always converge." (Isaac Newton, "Philosophiæ Naturalis Principia Mathematica" ["Mathematical Principles of Natural Philosophy"] 1687)
"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)
"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 definite Integral always presupposes numeric values; consequently equations in which definite integrals occur are seldom or never correct as general (formal) equations, but can only be admitted as numeric equations; consequently the convergence of any infinite series which may occur in them is an indispensable condition, whereas the condition of convergence with respect to a general series in general investigations, such as must be necessarily first established as the foundation of the possibility of any calculation, is quite as absurd [...]" (Martin Ohm, "The Spirit of Mathematical Analysis and its Relation to a Logical System", 1842)
"Once more, an invariably-recurring lesson of geological history, at whatever point its study is taken up: the lesson of the almost infinite slowness of the modification of living forms. The lines of the pedigrees of living things break off almost before they begin to converge." (Thomas H Huxley, On the Formation of Coal, 1870)
"To the thought of considering the infinitely great not merely in the form of what grows without limits - and in the closely related form of the convergent infinite series first introduced in the seventeenth century-, but also fixing it mathematically by numbers in the determinate form of the completed-infinite, I have been logically compelled in the course of scientific exertions and attempts which have lasted many years, almost against my will, for it contradicts traditions which had become precious to me; and therefore I believe that no arguments can be made good against it which I would not know how to meet." (Georg Cantor, "Grundlagen einer allgemeinen Mannigfaltigkeitslehre", 1883)
"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)
"Analytic functions are those that can be represented by a power series, convergent within a certain region bounded by the so-called circle of convergence. Outside of this region the analytic function is not regarded as given a priori ; its continuation into wider regions remains a matter of special investigation and may give very different results, according to the particular case considered." (Felix Klein, "Sophus Lie", [lecture] 1893)
"Incidentally, naive intuition, which is in large part an inherited talent, emerges unconsciously from the in-depth study of this or that field of science. The word ‘Anschauung’ has not perhaps been suitably chosen. I would like to include here the motoric sensation with which an engineer assesses the distribution of forces in something he is designing, and even that vague feeling possessed by the experienced number cruncher about the convergence of infinite processes with which he is confronted. I am saying that, in its fields of application, mathematical intuition understood in this way rushes ahead of logical thinking and in each moment has a wider scope than the latter " (Felix Klein, "Über Arithmetisierung der Mathematik", Zeitschrift für mathematischen und naturwissen-schaftlichen Unterricht 27, 1896)
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