Frases de Morris Kline

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Morris Kline

Data de nascimento: 1. Maio 1908
Data de falecimento: 10. Junho 1992

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Morris Kline , foi professor de matemática e historiador de matemática norte-americano.

Citações Morris Kline

„The Greeks failed to comprehend the infinitely large, the infinitely small, and infinite processes.“

—  Morris Kline
Mathematical Thought from Ancient to Modern Times (1972), Context: The Greeks failed to comprehend the infinitely large, the infinitely small, and infinite processes. They "shrank before the silence of the infinite spaces." p. 57

„For these proofs Heron gave new proofs that avoided extending the lines, in order to meet the objection of anyone who would deny that the space was available for the extension.“

—  Morris Kline
Mathematical Thought from Ancient to Modern Times (1972), Context: Closely related to the problem of the parallel postulate is the problem of whether physical space is infinite. Euclid assumes in Postulate 2 that a straight-line segment can be extended as far as necessary; he uses this fact, but only to find a larger finite length—for example in Book I, Propositions 11, 16, and 20. For these proofs Heron gave new proofs that avoided extending the lines, in order to meet the objection of anyone who would deny that the space was available for the extension. p. 177

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„He had doubted the correctness of the law of refraction of light but when he found in 1661 that he could deduce it from his Principle, he not only resolved his doubts about the law but felt all the more certain that his Principle was correct.“

—  Morris Kline
Mathematical Thought from Ancient to Modern Times (1972), Context: Fermat knew that under reflection light takes the path requiring least time and, convinced that nature does indeed act simply and economically, affirmed in letters of 1657 and 1662 his Principle of Least Time, which states that light always takes the path requiring least time. He had doubted the correctness of the law of refraction of light but when he found in 1661 that he could deduce it from his Principle, he not only resolved his doubts about the law but felt all the more certain that his Principle was correct.... Huygens, who had at first objected to Fermat's Principle, showed that it does hold for the propagation of light in media with variable indices of refraction. Even Newton's first law of motion, which states that the straight line or shortest distance is the natural motion of a body, showed nature's desire to economize. These examples suggested that there might be a more general principle. The search for such a principle was undertaken by Maupertuis. p. 580

„Closely related to the problem of the parallel postulate is the problem of whether physical space is infinite. Euclid assumes“

—  Morris Kline
Mathematical Thought from Ancient to Modern Times (1972), Context: Closely related to the problem of the parallel postulate is the problem of whether physical space is infinite. Euclid assumes in Postulate 2 that a straight-line segment can be extended as far as necessary; he uses this fact, but only to find a larger finite length—for example in Book I, Propositions 11, 16, and 20. For these proofs Heron gave new proofs that avoided extending the lines, in order to meet the objection of anyone who would deny that the space was available for the extension. p. 177

„To avoid any assertion about the infinitude of the straight line, Euclid says a line segment“

—  Morris Kline
Mathematical Thought from Ancient to Modern Times (1972), Context: To avoid any assertion about the infinitude of the straight line, Euclid says a line segment (he uses the word "line" in this sense) can be extended as far as necessary. Unwillingness to involve the infinitely large is seen also in Euclid's statement of the parallel axiom. Instead of considering two lines that extend to infinity and giving a direct condition or assumption under which parallel lines might exist, his parallel axiom gives a condition under which two lines will meet at some finite point. p. 175

„The relationship of point to line“

—  Morris Kline
Mathematical Thought from Ancient to Modern Times (1972), Context: The relationship of point to line bothered the Greeks and led Aristotle to separate the two. Though he admits points are on lines, he says that a line is not made up of points and that the continuous cannot be made up of the discrete. This distinction contributed also to the presumed need for separating number from geometry, since to the Greeks numbers were discrete and geometry dealt with continuous magnitudes. p. 176

„The method has the form of the now-standard method of differential calculus, though it begs entirely the difficult theory of limits.“

—  Morris Kline
Mathematical Thought from Ancient to Modern Times (1972), Context: Fermat applied his method of tangents to many difficult problems. The method has the form of the now-standard method of differential calculus, though it begs entirely the difficult theory of limits. p. 346

„The attempt to avoid a direct affirmation about infinite parallel straight lines caused Euclid to phrase the parallel axiom in a rather complicated way. He realized that, so worded, this axiom lacked the self-sufficiency of the other nine axioms, and there is good reason to believe that he avoided using it until he had to. Many Greeks tried to find substitute axioms for the parallel axiom or to prove it on the basis of the other nine. …Simplicius“

—  Morris Kline
Mathematical Thought from Ancient to Modern Times (1972), Context: The attempt to avoid a direct affirmation about infinite parallel straight lines caused Euclid to phrase the parallel axiom in a rather complicated way. He realized that, so worded, this axiom lacked the self-sufficiency of the other nine axioms, and there is good reason to believe that he avoided using it until he had to. Many Greeks tried to find substitute axioms for the parallel axiom or to prove it on the basis of the other nine.... Simplicius cites others who worked on the problem and says further that people "in ancient times" objected to the use of the parallel postulate. p. 177

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„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.“

—  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]

„Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam egestas wisi a erat. Morbi imperdiet, mauris ac auctor dictum.“

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