Клайн, Морис цитаты
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Мо́рис Клайн — американский математик, известный своими работами по истории и философии математики, проблемам математического образования и научно-популярной тематике. Профессор Нью-Йоркского университета . Wikipedia

✵ 1. Май 1908 – 10. Июнь 1992

Клайн, Морис: 42 цитаты 0 Нравится

Клайн, Морис: Цитаты на английском языке

“The historical associations of the word algebra almost substantiate the sordid character of the subject. The word comes from the title of a book written by… Al Khowarizmi. In this title, al-jebr w' almuqabala, the word al-jebr meant transposing a quantity from one side of an equation to another and muqabala meant simplification of the resulting expressions. Figuratively, al-jebr meant restoring the balance of an equation… When the Moors reached Spain… algebrista… came to mean a bonesetter… and signs reading Algebrista y Sangrador (bonesetter and bloodletter) were found over Spanish barber shops. Thus it might be said that there is a good historical basis for the fact that the word algebra stirs up disagreeable thoughts.”

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Morris Kline

Источник: Mathematics and the Physical World (1959), p. 69

“Galileo had provided the methodology for the analysis of motions on and near the earth and had applied it successfully. Copernicus and Kepler had previously obtained the laws of motion of the planets and their satellites. …But Galileo had succeeded in deriving numerous laws from a few physical principles and… the axioms and theorems of mathematics. …The Keplerian laws …were not logically related to each other. Each was an independent inference from observations. …They seemed to be suspended in the same vacuum in which the planets moved.
Galileo's laws had the additional advantage of supplying physical insight. The first law of motion and the law that the force of graviation gives… a downward acceleration of 32 ft/sec2… explain the vertrical rise and fall of bodies, motion on slopes, and projectile motion. Kepler's laws… had no physical basis. …Kepler tried to introduce the idea of a magnetic force which the sun exerted… But he failed to related the behavior of the planets to the precise laws of planetary motion. …
The new astronomical theory was completely isolated from the theory of motion on earth. …it bothered mathematicians and scientists who believed that all the phenomena of the universe were governed by one master plan instituted by the master planner—God.”

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Morris Kline

Источник: Mathematics and the Physical World (1959), pp. 224-225

“The minimum principle that unified the knowledge of light, gravitation, and electricity of Hamilton's time no longer suffices to relate these fundamental branches of physics. Within fifty years of its creation, the belief that Hamilton's principle would outlive all other physical laws of physics was shattered. Minimum principles have since been created for separate branches of physics… but these are not only restricted… but seem to be contrived…
A single minimum principle, a universal law governing all processes in nature, is still the direction in which the search for simplicity is headed, with the price of simplicity now raised from a mastery of differential equations to a mastery of the calculus of variations.”

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Morris Kline

Источник: Mathematical Thought from Ancient to Modern Times (1972), p. 442.

“The famous sixteenth-century algebraist Jerome Cardan called negative roots fictitious, and the founder of modern symbolic algebra, François Viète, discarded negative roots entirely. Descartes, called them false on the ground that they represented numbers less than nothing and so were meaningless.”

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Morris Kline

Источник: Mathematics and the Physical World (1959), p. 52.

“The theory of perspective was taught in painting schools from the sixteenth century onward according to principles laid down by the masters… However, their treatises on perspective had on the whole been precept, rule, and ad hoc procedure; they lacked a solid mathematical basis. In the period from 1500 to 1600 artists and subsequently mathematicians put the subject on a satisfactory deductive basis, and it passed from quasi-empirical art to a true science. Definitive works on perspective were written much later by eighteenth-century mathematicians Brook Taylor and J. H. Lambert.”

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Morris Kline

Источник: Mathematical Thought from Ancient to Modern Times (1972), p. 183

“One of the first algebraists to accept negative numbers was… placed negative numbers on a par with positive numbers and gave both roots of a quadratic equation, even when both were negative. Both Girard and Harriot used the minus sign for the operation of subtraction and for negative numbers.”

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Morris Kline

Источник: Mathematical Thought from Ancient to Modern Times (1972), pp. 252-253.

“Because they [the ancient Greeks] feared infinite processes they missed the limit process. In approximating a circle by a polygon they were content to make the difference smaller than any given quantity, but something positive was always left over. Thus the process remained clear to the intuition; the limit process, on the other hand, would have involved the infinitely small.”

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Morris Kline

Источник: Mathematical Thought from Ancient to Modern Times (1972), p. 176

“The chief innovator of symbolism in algebra was François Viète… an amateur in the sense that his professional life was devoted to the law… John Wallis… says that Viète, in denoting a class of numbers by a letter, followed the custom of lawyers who discussed legal cases by using arbitrary names [for the litigants]… and later the abbreviations… and still more briefly A, B, and C. Actually, letters had been used occasionally by the Greek Diophantus and by the Hindus. However, in these cases letters were confined to designating a fixed unknown number, powers of that number, and some operations. Viète recognized that a more extensive use of letters, and, in particular, the use of letters to denote classes of numbers, would permit the development of a new kind of mathematics; this he called logistica speciosa in distinction from logistica numerosa.”

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Morris Kline

...the growth of symbolism was slow. Even simple ideas take hold slowly. Only in the last few centuries has the use of symbolism become widespread and effective.
Источник: Mathematics and the Physical World (1959), p. 60

“Historically, it was Euclidean geometry that, developed to a large extent as a votive offering to the God of Reason, opened men's eyes to the possibility of design and to the possibility of uncovering it by the pursuit of mathematics.”

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Morris Kline

Источник: Mathematics and the Physical World (1959), p. 89

“Descartes… complained that Greek geometry was so much tied to figures "that is can exercise the understanding only on condition of greatly fatiguing the imagination." Descartes also deplored that the methods of Euclidean geometry were exceedingly diverse and specialized and did not allow for general applicability. Each theorem required a new kind of proof… What impressed Descartes especially was that algebra enables man to reason efficiently. It mechanizes thought, and hence produces almost automatically results that may otherwise be difficult to establish…. historically it was Descartes who clearly perceived and called attention to this feature. Whereas geometry contained the truth about the universe, algebra offered the science of method. It is… paradoxical that great thinkers should be enamored with ideas that mechanize thought. Of course, their goal is to get at more difficult problems, as indeed they do.”

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Morris Kline

Источник: Mathematics for the Nonmathematician (1967), pp. 255-256.

“Henri Poincaré thought the theory of infinite sets a grave malady and pathologic. "Later generations," he said in 1908, "will regard set theory as a disease from which one has recovered.”

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Morris Kline

[Morris Kline, Mathematics: The Loss of Certainty, http://books.google.com/books?id=RNwnUL33epsC&pg=PA203, 1982, Oxford University Press, 978-0-19-503085-3, 203]

“Though Wallis was advanced for his times and accepted negative numbers, he thought they were larger than infinity but not less than zero. In his Arithmetica Infinitorum (1655), he argued that since the ratio a/0, when a is positive, is infinite, then, when the denominator is changed to a negative number, as in a/b with b negative, the ratio must be greater than infinity.”

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Morris Kline

Источник: Mathematical Thought from Ancient to Modern Times (1972), p. 253.