Morris Kline Quotes (42 quotes) | Quotes of famous people

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

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

Morris Kline

Source: Mathematical Thought from Ancient to Modern Times (1972), p. 346
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.

“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

Source: Mathematical Thought from Ancient to Modern Times (1972), p. 177
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.

“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

Source: Mathematical Thought from Ancient to Modern Times (1972), p. 177
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.

Way

“The relationship of point to line”

Morris Kline

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

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

Morris Kline

Source: Mathematical Thought from Ancient to Modern Times (1972), p. 175
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.

Morris Kline

Problem

“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

Source: Mathematical Thought from Ancient to Modern Times (1972), p. 580
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.

Law , Light

“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.”

Morris Kline

Source: Mathematical Thought from Ancient to Modern Times (1972), p. 143.

Time

“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.”

Morris Kline

Morris Kline, p.22.
Mathematics for the Nonmathematician (1967)

History , Mathematics , Idea

“In the field of non-Euclidean geometry, Riemann… began by calling attention to a distinction that seems obvious once it is pointed out: the distinction between an unbounded straight line and an infinite line. The distinction between unboundedness and infiniteness is readily illustrated. A circle is an unbounded figure in that it never comes to an end, and yet it is of finite length. On the other hand, the usual Euclidean concept of a straight line is also unbounded in that it never reaches an end but is of infinite length.
…he proposed to replace the infiniteness of the Euclidean straight line by the condition that it is merely unbounded. He also proposed to adopt a new parallel axiom… In brief, there are no parallel lines. This … had been tried… in conjunction with the infiniteness of the straight line and had led to contradictions. However… Riemann found that he could construct another consistent non-Euclidean geometry.”

Morris Kline

Source: Mathematical Thought from Ancient to Modern Times (1972), p. 454

“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.”

Morris Kline

Source: Mathematical Thought from Ancient to Modern Times (1972), p. 192.

“The concept of the infinitely small is involved in the relation of points to a line or the relation of the discrete to the continuous, and Zeno's paradoxes may have caused the Greeks to shy away from this subject.”

Morris Kline

Source: Mathematical Thought from Ancient to Modern Times (1972), p. 175

“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.”

Morris Kline

Source: Mathematical Thought from Ancient to Modern Times (1972), p. 175

Nature

“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.”

Morris Kline

Source: Mathematical Thought from Ancient to Modern Times (1972), p. 252.

Parting , Falseness

“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.”

Morris Kline

Source: Mathematics and the Physical World (1959), pp. 49-50.

Help

“By 1700 all of the familiar members of the [number] system… were known. However, opposition to the newer types of numbers was expressed throughout the century. Typical are the objections of… Baron Francis Masères… in 1759 his Dissertation on the Use of the Negative Sign in Algebra… shows how to avoid negative numbers… and especially negative roots, by carefully segregating the types of quadratic equations so that those with negative roots are considered separately; and… the negative roots are to be rejected.”

Morris Kline

Source: Mathematical Thought from Ancient to Modern Times (1972), p. 592.

“The Hindus introduced negative numbers… The first known use is Brahmagupta about 628; he also states the rules for the four operations with negative numbers. Bhāskara points out that the square root of a positive number is twofold, positive and negative. He brings up the matter of the square root of a negative number but says that there is no square root because a negative number is not a square. No definitions, axioms, or theorems are given.
The Hindus did not unreservedly accept negative numbers. Even Bhāskara, while giving 50 and -5 as two solutions of a problem, says, "The second value is in this case not to be taken, for it is inadequate; people do not approve of negative solutions."”

Morris Kline

However, negative numbers gained acceptance slowly.
Source: Mathematical Thought from Ancient to Modern Times (1972), p. 185.

People , Problem

“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.”

Morris Kline

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

Books , Reading

“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.”

Morris Kline

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

God , Law , Behavior , Mathematics

“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.”

Morris Kline

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

Law , Search , Nature , Knowledge

“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.”

Morris Kline

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

Falseness

“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.”

Morris Kline

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

School , Art , Mathematics , Science

“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.”

Morris Kline

Source: 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.”

Morris Kline

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

Fear

“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.”

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.
Source: Mathematics and the Physical World (1959), p. 60

Life , Law , Mathematics , Idea

“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.”

Morris Kline

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

Men , God , Mathematics

“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.”

Morris Kline

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

Truth , Exercises , Imagination , Idea

“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]

Disease

“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.”

Morris Kline

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

Change , Time