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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.
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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.
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Aristotle had considered the question of whether space is infinite and gave six nonmathematical arguments to prove that it is finite; he foresaw that this question would be troublesome.
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In arithmetic the Arabs took one step backward. Though they were familiar with negative numbers and the rules for operating with them through the work of the Hindus, they rejected negative numbers.
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The Greeks failed to comprehend the infinitely large, the infinitely small, and infinite processes. They "shrank before the silence of the infinite spaces."
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Bertrand Russell... wrote in My Philosophical Development, "Those who taught me the infinitesimal calculus did not know the valid proofs of its fundamental theorems and tried to persuade me to accept the official sophistries as an act of faith."
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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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While the mathematicians were still looking askance at the Greek gift of the irrational number, the Hindus of India were preparing another brain-teaser, the negative number, which they introduced about A.D. 700. The Hindus saw that when the usual, positive numbers were used to represent assets, it was helpful to have other number represent debts.
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Indeed it is paradoxical that abstractions so remote from reality should achieve so much. Artificial the mathematical account may be, a fairy tale perhaps, but one with a moral.
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The use of canon raised numerous questions concerning the paths of projectiles....One might determine... what type of curve a projectile follows and.... prove some geometrical facts about this curve, but geometry could never answer such questions as how high the projectile would go or how far from the starting point it would land. The seventeenth century sought the quantitative or numerical information needed for practical applications, and such information is provided by algebra.
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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.
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The history of arithmetic and algebra illustrates one of the striking and curious features of the history of mathematics. Ideas that seem remarkably simple once explained were thousands of years in the making.
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The Hindus saw clearly that if the arithmetic operations... were properly defined for negative numbers, these numbers could be employed to as good advantage as people had previously derived from positive numbers....To people to whom the word number had always meant positive whole numbers and positive fractions, the very idea that there could be other numbers came hard. For many centuries negative numbers were either rejected or treated as second-class citizens.
What was especially difficult for mathematicians to swallow was that negative numbers could be acceptable roots of equations.
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If potential application is the goal, then as the great physical chemist Josiah Willard Gibbs remarked, the pure mathematician can do what he pleases, but the applied mathematician must be at least partially sane.
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Mathematics is a model of exact reasoning, an absorbing challenge to the mind, an esthetic experience for creators and some students, a nightmarish experience to other students, and an outlet for the egotistic display of mental power.
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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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Aristotle says the infinite is imperfect, unfinished, and therefore unthinkable; it is formless and confused. Only as objects are delimited and distinct do they have a nature.
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As for negative numbers... most mathematicians of the sixteenth and seventeenth centuries did not accept them... In the fifteenth century Nicolas Chuquet and, in the sixteenth, Stifel both spoke of negative numbers as absurd numbers....Descartes accepted them, in part....he had shown that, given an equation, one can obtain another whose roots are larger than the original one by any given quantity. Thus an equation with negative roots could be transformed into one with positive roots. Since we can turn false roots into real roots, Descartes was willing to accept negative numbers. Pascal regarded the subtraction of 4 from zero as utter nonsense.
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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.
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The conquest of new domains of mathematics proceeds somewhat as do military conquests. Bold dashes into enemy territory capture strongholds. These incursions must then be followed up and supported by broader, more thorough and more cautious operations to secure what has been only tentatively and insecurely grasped.
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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.
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The tantalizing and compelling pursuit of mathematical problems offers mental absorption, peace of mind amid endless challenges, repose in activity, battle without conflict, refuge from the goading urgency of contingent happenings, and the sort of beauty changeless mountains present to senses tried by the present day kaleidoscope of events.
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When an equation...clearly leads to two negative or imaginary roots, [Diophantus] retraces his steps and shows by how by altering the equation, he can get a new one that has rational roots....Diophantus is a pure algebraist; and since algebra in his time did not recognize irrational, negative, and complex numbers, he rejected equations with such solutions.
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Mathematics is a body of knowledge, but it contains no truths.
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Theoretical Science is a game of mathematical make-believe.
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Morris Kline
Born:
May 1, 1908
Died:
June 10, 1992
(aged 84)
Bio:
Morris Kline was a Professor of Mathematics, a writer on the history, philosophy, and teaching of mathematics, and also a popularizer of mathematical subjects.
Known for:
Mathematics: The Loss of Certainty (1980)
Mathematics for the Nonmathematician (1967)
Mathematics in Western culture (1953)
Calculus: An Intuitive and Physical Approach (1967)
Morris Kline on Wikipedia
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