第459页
淇厚生 (士为悦己者读书)
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pvii Almost everyone knows that mathematics serves the very practical purpose of dictating engineering design. Fewer people seem to be aware that mathematics carries the main burden of scientific reasoning and is the core of the major theories of physical science. It is even less widely known that mathematics has determined the direction and content of much philosophic thought, has destroyed and rebuilt religious doctrines, has supplied substance to economics and political theories, has fashioned major painting, musical, architectural, and literary styles, has fathered our logic, and has furnished the best answers we have to fundamental questions about the nature of man and his universe. As the embodiment and most powerful advocate of the rational spirit, mathematics had invaded domains ruled by authority, custom, and habit, and supplanted them as the arbiter of thought and action. Finally, as an incomparably fine human achievement mathematics offers saticfactions and aesthetic values at least equal to those offered by any other branch of our culture. 按:按Hardy的观点,这种说法夸大了数学的重要和影响,见《自白》27节。 School courses and books have presented 'mathematics' as a series of apparently meaningless technical procedures. Such material is as representative of the subject as an account of the name, position, and function of every bone in the human skeleton is representative of the living, thinking, and emotional being called man. pix Confidence in man's power to solve his problems and indications of the method he has employed most successfully thus far can be gained by a study of his greatest and most enduring intellectual accomplishment-mathematics. p4 The subject is not a series of techniques. These are indeed the least important aspect, and they fall as far short of representing mathematics as color mixing does of painting. To describe mathematics as only a method of inquiry iss to describe da Vinci's 'Last Supper' as an organization of paint on canvas. Mathematics is, also, a field for creative endeavor. In divining what can be proved, as well as in constructing methods of proof, mathematicians employ a high order of intuition and imagination. p5 If mathematics is indeed a creative activity, what driving force causes men to pursue it. The most obvious, though not necessarily the most important, motive for mathematical investigations has been to answer questions arising directly out of social needs. Another basic use of mathematics, indeed one that is especially prominent in modern times, has been to provide a rational organization of natural phenomena. Intellectual curiosity and zest for pure thought have started many mathematicians in pursuit of properties of numbers and geometric figures and have produced some of the most original contributions. Over and above all other drives to create is the search for beauty. p6 In addition to the beauty of the completed structure, the indispensable use of imagination and intuition in the creation of proofs and conclusions affords high aesthetic satisfaction to the creator. Despite the clear indications of history that all of the factors above have motivated the creation of mathematics there have been many erroneous pronouncements. There are the charges-often made to excuse neglect of the subject-that mathematicians like to indulge in pointless speculations or that they are silly and useless dreamers. To these charges a crushing reply can readily be made. Even purely abstract studies, let alone those motivated by scientific and engineering needs, have proved immensely useful. 按:将Kline的回应与他的另一段话对比。Morris Kline, Mathematics: The Loss of Certainty, Oxford University Press, p285: Some pure mathematicians argue that there is potential usefulness in any mathematical development and no one can foresee its actual future application. Nevertheless, a mathematical theme is like a piece of oil-bearing land. Dark puddles on the surface may suggest that a particular spot be explored for oil and, if this is discovered, the value of the land is established. The proven worth of the land warrants further drilling in the expectation that more oil will be found if the drilling sites are not too far removed from the location of the original strike. Of course, one might choose a very distant site because the drilling is easier there and still claim that he will strike oil. But human effort and ingenuity are limited and should therefore be devoted to good risks. 对于MLC这段话表达的观点,Timothy Gowers的演讲The Importance of Mathematics中有一个非常好的反驳。 p7 Practical, scientific, aesthetic, and philosophical interests have all shaped mathematics. It would be impossible to separate the contributions and influences of any one of these forces and weigh it against the others, much less assert sweeping claims to its relative importance. p8 Mathematical language is precise, so precise that it is often confusing to people unaccustomed to its forms....This precision of mathematics appears as pedantry or stiltedness to one who does not yet appreciate that it is essential to exact thinking, for exact thinking and exact language go hand in hand. Mathematical style aims at brevity formal perfection. It sometimes succeeds two well and sacrifices the clarity its precision seeks to guarantee. p9 Mathematics is more than a method, an art, and a language. It is a body of language with content that serves the physical and social scientist, the philosopher, the logician, and the artist; content that influences the doctrines of statesman and theologians; content that sacrifices the curiosity of the man who surveys the heavens and the man who muses on the sweetness of musical sounds; and content that has undeniably, if sometimes imperceptibly, shaped the course of modern history. p10 In its broadest aspect mathematics is a spirit, the spirit of rationality. It is this spirit that challenges, stimulates, invigorates, and drives human minds to exercise themselves to the fullnest. p11 This sketch of the life of mathematics, however brief, may nevertheless indicate that its vitality has been very much depentent on the cultural life of the civilazation which nourished it. In fact, mathematics has been so much a part of civilizations and cultures that many historians see mirrored in the mathematics of an age the characteristics of the other chief works of that age. It is also true that the absense of mathematics creations is indicative of the culture of a civilization. p178 The difficulty most people experience in accepting a four-dimensional geometry and the corresponding equations is due to fact that they confuse mental constructions and visualization. All of geometry, including two- and three-dimensional Euclidean geometry, deals, as Plato emphasized, with ideas that exist in the mind only. Fortunately we can visualize or picture the two- and three-dimensional ideas by means of drawings on paper, and these drawings help us to remember and to organize our thoughts. But the pictures are not the subject matter of geometry and we are not permitted to reason from them. 按:Timothy Gowers, Mathematics VSI, Oxford University Press, p77: Although visualizing an object feels rather like looking at it, there are important differences between the two experiences. For example, if I am asked to visualize a room with which I am familiar, but not very familiar, I have no difficulty in doing so. If I am then asked simple questions about it, such as how many chairs it contains or what colour the floor is, I am often unable to answer them. This shows that, whatever a mental image is, it is not a photographic representation. p206 The major values of the Newtonian laws lies, as we have just seen, in the fact that they apply to so many varied situations on heaven and Earth. p232 It was a very fortunate circumstence that mathematics and science were closely linked in the Newtonian era and that physical reasoning could guide the mathematician and keep them on the right track. p258 Newton, along with Descartes, credited God with the act of creation but restricted His daily functions. p278 Dryden declared: 'A man should be learned in several sciences, and should have a reasonable, philosophical and in some measure, a mathematical head to be a complete and excellent poet...' p279 The principles of form in poetry were likened to mathematical axioms because the axioms determined the form as well as the content of the theorems. p367 Mathematics begins after the probabilities are known and is concerned with reasoning about the numbers so obtained. p379 yet physicists believe that each molecure follows the same physical laws that the earth follows in its path around the sun....,but that these procedures are too intricate to be grasped by our understandings. The same could be said of economic phenomena, the incidence of death, and other seemingly unlawful affairs. Thus phenomena that appear to be disorderly may be completely determined, and the mathematical laws obtained from statistical studies may merely reflect the existence of these underlying orderly physical processes. 按:The dialogue in XXIV is very interesting and inspiring. p459 The description of mathematics as tautology says that the choice of a set of axioms is like the purchase of a piece of mining land-the riches are all there. This description omits, however, the patient, hard digging which must be performed, the careful sifting of the precious metal from the base rock, the value and beauty of the treasure obtained, and the pleasure and exhilaration of accomplishment.
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