"It is true that of far the greater part of things, we must content ourselves with such knowledge as description may exhibit, or analogy supply; but it is true likewise, that these ideas are always incomplete, and that at least, till we have compared them with realities, we do not know them to be just. As we see more, we become possessed of more certainties, and consequently gain more principles of reasoning, and found a wider base of analogy." (Samuel Johnson, 1825)
"Geometry is a true natural science: - only more simple, and therefore more perfect than any other. We must not suppose that, because it admits the application of mathematical analysis, it is therefore a purely logical science, independent of observation. Everybody studied by geometers presents some primitive phenomena which, not being discoverable by reasoning, must be due to observation alone." (Auguste Comte,"Course of Positive Philosophy", 1830) "
"The errors which arise from the absence of facts are far more numerous and more durable than those which result from unsound reasoning respecting true data." (Charles Babbage, "On the Economy of Machinery and Manufactures", 1832)
"Every syllogism, then, consists of three propositions; the two first of which are called the premises and the third the conclusion. Now, the advantage of the all these [valid] forms to direct our reasoning is this, that if the premises are both true, the conclusion infallibly is so.This is likewise the only method of discovering unknown truths. Every truth must always be the conclusion of a syllogism, whose premises are indubitably true." (Leonhard Euler, "Letters of Euler on Different Subjects in Natural Philosophy Addressed to a German Princess" Vol. 1, 1833)
"Theories usually result from the precipitate reasoning of an impatient mind which would like to be rid of phenomena and replace them with images, concepts, indeed often with mere words." (Johann Wolfgang von Goethe, "Maxims and Reflections", 1833)
"Geometry is that of mathematical science which is devoted to consideration of form and size, and may be said to be the best and surest guide to study of all sciences in which ideas of dimension or space are involved. Almost all the knowledge required by navigators, architects, surveyors, engineers, and opticians, in their respective occupations, is deduced from geometry and branches of mathematics. All works of art are constructed according to the rules which geometry involves; and we find the same laws observed in the works of nature. The study of mathematics, generally, is also of great importance in cultivating habits of exact reasoning; and in this respect it forms a useful auxiliary to logic." (William Chambers & Robert Chambers, "Chambers's Information for the People" Vol. 2, 1835)
"Scientific Ideas can often be adequately exhibited for all the purposes of reasoning, by means of Definitions and Axioms; all attempts to reason by means of Definitions from common Notions, lead to empty forms or entire confusion." (William Whewell, "History of the Inductive Sciences from the Earliest to the Present Time", 1837)
"The connected course of reasoning by which any Geometrical truth is established is called a demonstration." (Robert Potts, "Euclid's Elements of Geometry", 1845)
No comments:
Post a Comment