"Here, then, in mathematics we have a universal language, valid, useful, intelligible everywhere in place and in time - in banks and insurance companies, on the parchments of the architects who raised the Temple of Solomon, and on the blueprints of the engineers who, with their calculus of chaos, master the winds. Here is a discipline of a hundred branches, fabulously rich, literally without limit in its sphere of application, laden with honors for an unbroken record of magnificent accomplishment. Here is a creation of the mind, both mystic and pragmatic in appeal. Austere and imperious as logic, it is still sufficiently sensitive and flexible to meet each new need." (Edward Kasner & James R Newman, "Mathematics and the Imagination", 1940)
"The main thing geometry gives us is the ideal of a logical system and of precise thinking, and some acquaintance with the language in which logical arguments are usually expressed. The answer to a problem in actual life can often be obtained by further investigation of the actual facts, while in geometry it can always be obtained by reasoning alone." (George D Birkhoff & Ralph Beately, "Basic Geometry", 1940)
"It will probably be the new mathematical discoveries which are suggested through physics that will always be the most important, for, from the beginning Nature has led the way and established the pattern which mathematics, the Language of Nature, must follow." (George D Birkhoff, "Mathematical Nature of Physical Theories" American Scientific Vol. 31 (4), 1943)
"Thus the use of language itself is based on the principle that any symbolism which works has objective validity; and it is illegitimate to use words to contradict this principle." (Kenneth Craik, "The Nature of Explanation", 1943)
"Mathematizing may well be a creative activity of man, like language or music, of primary originality, whose historical decisions defy complete objective rationalizations." (Hermann Weyl,"Obituary for David Hilbert", Royal Society Biographies Vol. 4, 1944)
"Figures and symbols are closely connected with mathematical thinking, their use assists the mind. […] At any rate, the use of mathematical symbols is similar to the use of words. Mathematical notation appears as a sort of language, une langue bien faite, a language well adapted to its purpose, concise and precise, with rules which, unlike the rules of ordinary grammar, suffer no exception."
"The words of the language, as they are written or spoken, do not seem to play any role in any mechanism of thought. The physical entities which seem to serve as elements in thought are certain signs and more or less clear images which can be 'voluntarily' reproduced or combined. […] But taken from a psychological viewpoint, this combinatory play seems to be the essential feature in productive thought - before there is any connection with logical construction in words or other kinds of signs which can be communicated to others. The above-mentioned elements are, in my case, of visual and some of muscular type. Conventional words or other signs have to be sought for laboriously only in a secondary stage, when the mentioned associative play is sufficiently established and can be reproduced at will. " (Albert Einstein, [letter to Hadamard, in (Jacques Hadamard, "The Psychology of Invention in the Mathematical Field, 1945)])
"One of the difficulties which a mathematician has in describing his work to non-mathematicians is that the present day language of mathematics has become so esoteric that a well educated layman, or even a group of scientists, can comprehend essentially nothing of the discourse which mathematicians hold with each other, or of the accounts of their latest researches which are published in their professional journals." (Angus E Taylor," Some Aspects of Mathematical Research", American Scientist , Vol. 35, No. 2, 1947)
"Algebra reverses the relative importance of the factors in ordinary language. It is essentially a written language, and it endeavors to exemplify in its written structures the patterns which it is its purpose to convey. The pattern of the marks on paper is a particular instance of the pattern to be conveyed to thought. The algebraic method is our best approach to the expression of necessity, by reason of its reduction of accident to the ghost-like character of the real variable." (Alfred N Whitehead,"Essays in Science and Philosophy", 1948)
"The study of mathematics cultivates the reason; that of the languages, at the same time, the reason and the taste. The former gives the grasp and power to the mind; the latter both power and flexibility. The former by itself, would prepare us for a state of certainties, which nowhere exists; the later, for a state of probabilities which is that of common life. Each, by itself, does but an imperfect work: in the union of both, is the best discipline for the mind, and the best mental training for the world as it is." (Tyron Edwards,"The New Dictionary of Thoughts", 1948)
"Besides electrical engineering theory of the transmission of messages, there is a larger field [cybernetics] which includes not only the study of language but the study of messages as a means of controlling machinery and society, the development of computing machines and other such automata, certain reflections upon psychology and the nervous system, and a tentative new theory of scientific method." (Norbert Wiener, "Cybernetics", 1948)
"[Disorganized complexity] is a problem in which the number of variables is very large, and one in which each of the many variables has a behavior which is individually erratic, or perhaps totally unknown. However, in spite of this helter-skelter, or unknown, behavior of all the individual variables, the system as a whole possesses certain orderly and analyzable average properties. [...] [Organized complexity is] not problems of disorganized complexity, to which statistical methods hold the key. They are all problems which involve dealing simultaneously with a sizable number of factors which are interrelated into an organic whole. They are all, in the language here proposed, problems of organized complexity." (Warren Weaver, "Science and Complexity", American Scientist Vol. 36, 1948)
"Now when we speak of a machine that thinks, or a mechanical brain, what do we mean? Essentially, a mechanical brain is a machine that handles information, transfers information automatically from one part of the machine to another, and has a flexible control over the sequence of its operations. No human being is needed around such a machine to pick up a physical piece of information produced in one part of the machine, personally move it to another part of the machine, and there put it in again. Nor is any human being needed to give the machine instructions from minute to minute. Instead, we can write out the whole program to solve a problem, translate the program into machine language, and put the program into the machine." (Edmund C Berkeley, "Giant Brains or Machines that Think", 1949)
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