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David Hilbert
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„One can measure the importance of a scientific work by the number of earlier publications rendered superfluous by it.“
„We must not believe those, who today, with philosophical bearing and deliberative tone, prophesy the fall of culture and accept the ignorabimus. For us there is no ignorabimus, and in my opinion none whatever in natural science. In opposition to the foolish ignorabimus our slogan shall be:We must know — we will know!“
„The various branches of geometry are all interrelated closely and quite often unexpectedly. This shows up in many places in the book. Even so… it was necessary to make each chapter…self-contained… We hope that… we have rendered each chapter taken by itself… understandable and interesting. We want to take the reader on a leisurely walk… in the big garden that is geometry, so that each may pick for himself a bouquet to his liking.“
„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.“
„[O]ur purpose is to give a presentation of geometry… in its visual, intuitive aspects. With the aid of visual imagination we can illuminate the manifold facts and problems… beyond this, it is possible… to depict the geometric outline of the methods of investigation and proof, without… entering into the details… In this manner, geometry being as many-faceted as it is and being related to the most diverse branches of mathematics, we may even obtain a summarizing survey of mathematics as a whole, and a valid idea of the variety of problems and the wealth of ideas it contains. Thus a presentation of geometry in large brushstrokes… and based on the approach through visual intuition, should contribute to a more just appreciation of mathematics by a wider range of people than just the specialists.“
„[M]athematics is not a popular subject… The reason for this is to be found in the common superstition that [it] is but a continuation… of the fine art of arithmetic, of juggling with numbers. [We] combat that superstition, by offering, instead of formulas, figures that may be looked at and that may easily be supplemented by models which the reader may construct. This book… bring[s] about a greater enjoyment of mathematics, by making it easier… to penetrate the essence of mathematics without… a laborious course of studies.“
„Only an idiot could believe that scientific truth needs martyrdom“
„A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street.“
„To new concepts correspond, necessarily, new signs.“
„Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts.“
„History teaches the continuity of the development of science. We know that every age has its own problems, which the following age either solves or casts aside as profitless and replaces by new ones.“
„Mathematics is a presuppositionless science. To found it I do not need God, as does Kronecker, or the assumption of a special faculty of our understanding attuned to the principle of mathematical induction, as does Poincaré, or the primal intuition of Brouwer, or, finally, as do Russell and Whitehead, axioms of infinity, reducibility, or completeness, which in fact are actual, contentual assumptions that cannot be compensated for by consistency proofs.“
„If one were to bring ten of the wisest men in the world together and ask them what was the most stupid thing in existence, they would not be able to discover anything so stupid as astrology.“
„Sometimes it happens that a man’s circle of horizon becomes smaller and smaller, and as the radius approaches zero it concentrates on one point. And then that becomes his point of view.“
„This conviction of the solvability of every mathematical problem is a powerful incentive to the worker. We hear within us the perpetual call: There is the problem. Seek its solution. You can find it by pure reason, for in mathematics there is no ignorabimus.“
„Mathematics knows no races or geographic boundaries; for mathematics, the cultural world is one country.“
„Every kind of science, if it has only reached a certain degree of maturity, automatically becomes a part of mathematics.“
„Begin with the simplest examples.“
„If we do not succeed in solving a mathematical problem, the reason frequently consists in our failure to recognize the more general standpoint from which the problem before us appears only as a single link in a chain of related problems. After finding this standpoint, not only is this problem frequently more accessible to our investigation, but at the same time we come into possession of a method which is applicable also to related problems.“
„Good, he did not have enough imagination to become a mathematician.“
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