"The natural development of this work soon led the geometers in their studies to embrace imaginary as well as real values of the variable. The theory of Taylor series, that of elliptic functions, the vast field of Cauchy analysis, caused a burst of productivity derived from this generalization. It came to appear that, between two truths of the real domain, the easiest and shortest path quite often passes through the complex domain." (Paul Painlevé, "Analyse des travaux scientifiques", 1900)
"Every generalisation is a hypothesis. Hypothesis therefore plays a necessary rôle, which no one has ever contested. Only, it should always be as soon as possible submitted to verification."
"Let us notice first of all, that every generalization implies in some measure the belief in the unity and simplicity of nature." (Jules H Poincaré, "Science and Hypothesis", 1901)
"If we study the history of science we see happen two inverse phenomena […] Sometimes simplicity hides under complex appearances; sometimes it is the simplicity which is apparent, and which disguises extremely complicated realities. […] No doubt, if our means of investigation should become more and more penetrating, we should discover the simple under the complex, then the complex under the simple, then again the simple under the complex, and so on, without our being able to foresee what will be the last term. We must stop somewhere, and that science may be possible, we must stop when we have found simplicity. This is the only ground on which we can rear the edifice of our generalizations." (Henri Poincaré, "Science and Hypothesis", 1901)
"A generalization is a mountain of observations; from the summit the outlook is broad. The great observer climbs to the outlook, while the mere thinker struggles to imagine it." (Charles Se Minot, "Fifty-first meeting, The Problem of Consciousness in Its Biological Aspects", Proceedings of the American Association for the Advancement of Science, 1902)
"It has been said that no science is established on a firm basis unless its generalisations can be expressed in terms of number, and it is the special province of mathematics to assist the investigator in finding numerical relations between phenomena. After experiment, then mathematics. While a science is in the experimental or observational stage, there is little scope for discerning numerical relations. It is only after the different workers have 'collected data' that the mathematician is able to deduce the required generalisation." (Joseph W Mellor, "Higher Mathematics for Students of Chemistry and Physics', 1902)
"In generalizing lies the difficulty of scientific map-making, for it no longer allows the cartographer to rely merely on objective facts but requires him to interpret them subjectively. To be sure the selection of the subject matter is controlled by considerations regarding its suitability and value, but the manner in which this material is to be rendered graphically depends on personal and subjective feeling. But the latter must not predominate: the dictates of science will prevent any erratic flight of the imagination and impart to the map a fundamentally objective character in spite of all subjective impulses. It is in this respect that maps are distinguished from fine products of art. Generalized maps and, in fact, all abstract maps should, therefore, be products of art clarified by science." (Max Eckert, "On the Nature of Maps and Map Logic", Bulletin of the American Geographical Society Vol. 40, 1908)
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