"Everything in physical science, from the law of gravitation to the building of bridges, from the spectroscope to the art of navigation, would be profoundly modified by any considerable inaccuracy in the hypothesis that our actual space is Euclidean. The observed truth of physical science, therefore, constitutes overwhelming empirical evidence that this hypothesis is very approximately correct, even if not rigidly true." (Bertrand Russell, "Foundations of Geometry", 1897)
"A collection of terms is infinite when it contains as parts other collections which have just as many terms in it as it has. If you can take away some of the terms of a collection, without diminishing the number of terms, then there is an infinite number of terms in the collection." (Bertrand Russell. International Monthly, Vol. 4, 1901)
"Mathematics, rightly viewed, possesses not only truth, but supreme beauty - a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show." (Bertrand Russell, 'The Study of Mathematics", 1902)
"Arithmetic must be discovered in just the same sense in which Columbus discovered the West Indies, and we no more create numbers than he created the Indians." (Bertrand Russell, "The Principles of Mathematics", 1903)
"It is one of the chief merits of proofs that they instill a certain skepticism as to the result proved." (Bertrand Russell, "The Principles of Mathematics", 1903)
"Motion consists merely in the occupation of different places at different times, subject to continuity. [...] There is no transition from place to place, no consecutive moment or consecutive position, no such thing as velocity except in the sense of a real number, which is the limit of a certain set of quotients." (Bertrand Russell's, "Principles of Mathematics", 1903)
"In the beginning of algebra, even the most intelligent child finds, as a rule, very great difficulty. The use of letters is a mystery, which seems to have no purpose except mystification. It is almost impossible, at first, not to think that every letter stands for some particular number, if only the teacher would reveal what number it stands for." (Bertrand Russell, "Mysticism and Logic: And Other Essays", 1910)
"In fact, the opposition of instinct and reason is mainly illusory. Instinct, intuition, or insight is what first leads to the beliefs which subsequent reason confirms or confutes; but the confirmation, where it is possible, consists, in the last analysis, of agreement with other beliefs no less instinctive. Reason is a harmonising, controlling force rather than a creative one. Even in the most purely logical realms, it is insight that first arrives at what is new." (Bertrand Russell, "Our Knowledge of the External World", 1914)
"The true spirit of delight, the exaltation...which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry." (Bertrand Russell, "Mysticism and Logic", 1917)
"[...] instead of asking what can be defined and deduced from what is assumed to begin with, we ask instead what more general ideas and principles can be found, in terms of which what was our starting-point can be defined or deduced. It is the fact of pursuing this opposite direction that characterizes mathematical philosophy as opposed to ordinary mathematics." (Bertrand Russell, "Introduction to Mathematical Philosophy", 1919)
"Mathematics, rightly viewed, possesses not only truth, but supreme beauty - a beauty, cold and austere, like that of a sculpture. without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry." (Bertrand Russell, "The Study of Mathematics", 1919)
"Mathematics is a study which, when we start from its most familiar portions, may be pursued in either of two opposite directions. The more familiar direction is constructive, towards gradually increasing complexity: from integers to fractions, real numbers, complex numbers; from addition and multiplication to differentiation and integration, and on to higher mathematics. The other direction, which is less familiar, proceeds, by analyzing, to greater and greater abstractness and logical simplicity." (Bertrand Russell, "Introduction to Mathematical Philosophy", 1919)
"Metaphysics, or the attempt to conceive the world as a whole by means of thought, has been developed, from the first, by the union and conflict of two very different human impulses, the one urging men towards mysticism, the other urging them towards science." (Bertrand Russell, "Mysticism and Logic: And Other Essays", 1919)
"The essence of language lies, not in the use of this or that special means of communication, but in the employment of fixed associations (however these may have originated) in order that something now sensible - a spoken word, a picture, a gesture, or what not — may call up the 'idea' of something else. Whenever this is done, what is now sensible may be called a 'sign' or 'symbol', and that of which it is intended to call up the 'idea' may be called its 'meaning'." This is a rough outline of what constitutes 'meaning'." (Bertrand Russell, "Analysis of Mind", 1921)
"Science does not aim at establishing immutable truths and eternal dogmas; its aim is to approach the truth by successive approximations, without claiming that at any stage final and complete accuracy has been achieved." (Bertrand Russell, "The ABC of Relativity", 1925)
"Scientific laws, when we have reason to think them accurate, are different in form from the common-sense rules which have exceptions: they are always, at least in physics, either differential equations, or statistical averages." (Bertrand A Russell, "The Analysis of Matter", 1927)
"Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means." (Bertrand Russell, 1929)
"Although this may seem a paradox, all exact science is dominated by the idea of approximation. When a man tells you that he knows the exact truth about anything, you are safe in inferring that he is an inexact man." (Bertrand Russell, "The Scientific Outlook", 1931)
"To create a healthy philosophy you should renounce metaphysics but be a good mathematician." (Bertrand Russell, [lecture] 1935)
"In every writer on philosophy there is a concealed metaphysic, usually unconscious; even if his subject is metaphysics, he is almost certain to have an uncritically believed system which underlies his specific arguments." (Bertrand Russell, "Dewey’s New Logic" [in "The Philosophy of John Dewey", ed. by Paul A Schilpp & Lewis E Hahn, 1939])
"In various ways, methods of approaching the mathematician's ideal were sought, and the resulting suggestions were the source of much that was mistaken in metaphysics and theory of knowledge." (Bertrand Russell, "A History of Western Philosophy", 1945)
"It seems to me now that mathematics is capable of an artistic excellence as great as that of any music, perhaps greater; not because the pleasure it gives (although very pure) is comparable, either in intensity or in the number of people who feel it, to that of music, but because it gives in absolute perfection that combination, characteristic of great art, of godlike freedom, with the sense of inevitable destiny; because, in fact, it constructs an ideal world where everything is perfect and yet true." (Bertrand Russell, "Autobiography", 1967)
"But it is just this characteristic of simplicity in the laws of nature hitherto discovered which it would be fallacious to generalize, for it is obvious that simplicity has been a part cause of their discovery, and can, therefore, give no ground for the supposition that other undiscovered laws are equally simple." (Bertrand Russell)
"The greatest challenge to any thinker is stating the problem in a way that will allow a solution." (Bertrand Russell)
"When you want to teach children to think, you begin by treating them seriously when they are little, giving them responsibilities, talking to them candidly, providing privacy and solitude for them, and making them readers and thinkers of significant thoughts from the beginning. That’s if you want to teach them to think." (Bertrand Russell)
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