![In mathematics, as in any scientific research, we find two tendencies... [T]he tendency toward abstraction seeks to crystallize the logical relations inherent in the maze of material in a systematic and orderly manner. On the other hand, the tendency toward intuitive understanding fosters a more immediate grasp of the objects... a live rapport with them... which stresses the concrete meaning of their relations. ...[I]ntuitive understanding plays a major role in geometry. ...[S]uch concrete intuition is of great value not only for the research worker, but... for anyone who wishes to study and appreciate the results of research in geometry. (David Hilbert)](https://cdn.quotesdtb.com/img/quotes_images_webp/37/david-hilbert-abstraction-anyone-291937.webp)
In mathematics, as in any scientific research, we find two tendencies... [T]he tendency toward abstraction seeks to crystallize the logical relations inherent in the maze of material in a systematic and orderly manner. On the other hand, the tendency toward intuitive understanding fosters a more immediate grasp of the objects... a live rapport with them... which stresses the concrete meaning of their relations. ...[I]ntuitive understanding plays a major role in geometry. ...[S]uch concrete intuition is of great value not only for the research worker, but... for anyone who wishes to study and appreciate the results of research in geometry.
David HilbertRelated topics
abstraction anyone concrete find geometry grasp great hand intuition intuitive live major manner material mathematics maze meaning orderly rapport research role study tendency understanding value worker relationsRelated quotes
I think that one of the things that is needed is the fact that I don't think enough people in high positions in our country accept the importance of our world role with sufficient gravity. In other words, I think the tendency, because of politics and getting elected, is to stress local issues; and of course, they are important. But, I would like to see more of the leaders of our country spend more time traveling for one thing, getting to know the world, and studying history. To me, one of the sad things about our country is that our leadership - to a greater extent than I would like - is more concerned about very domestic issues than they are about our relations in the world.
David Rockefeller
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Robert Rauschenberg
In a country where feeling for art is not one of the national characteristics, it is particularly difficult to arrive at a common understanding of the term "art". There are many obstacles in the way, and some of them are imposed by the mere existence of such a body as the Royal Academy, for lacking any intuition in this matter of art, the majority of people turn to that institution for guidance, and naturally assume that any work receiving its blessing must be a work of art in the strict meaning of the word, and consequently the artistic quality of a painting is, in the popular estimation, measured in ratio to its degree of photographic realism.
James Bolivar Manson
We are told by physicians and scientists that thousands of cells in the human brain are still dormant and, consequently, that the average human being uses only a small part of his equipment. The area of the brain which is found around the pineal gland is that connected with the intuition, and it is these cells which must be roused into activity before there can be any real intuitive perception which, when aroused, will manifest soul control, spiritual illumination, true psychological understanding of one's fellowmen, and a development of the true esoteric sense, which is the objective before you at this time.
Alice Bailey
Modern empiricism has been conditioned in large part by two dogmas. One is a belief in some fundamental cleavage between truths which are analytic, or grounded in meanings independently of matters of fact, and truths which are synthetic, or grounded in fact. The other dogma is reductionism: the belief that each meaningful statement is equivalent to some logical construct upon terms which refer to immediate experience. Both dogmas, I shall argue, are ill-founded. One effect of abandoning them is, as we shall see, a blurring of the supposed boundary between speculative metaphysics and natural science. Another effect is a shift toward pragmatism.
Willard van Orman Quine
