Morris Kline: News

“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 goal of deriving all the phenomena of nature from a few basic physical laws and the axioms of mathematics had been set by Galileo…
In studying curvilinear motions on the earth Galileo had found the parabola to be the basic curve. In the heavens… Kepler… had found the ellipse to be the basic curve. Why this difference? …since parabola and ellipse are both conic sections there was the provocative suggestion that perhaps some physical law unified these related paths of motion. …
It has often happened in the history of mathematics and science that major problems remained outstanding… great minds… succeeded only in revealing the true difficulties… and in generating an atmosphere of dispair… Then a genius appeared… with ideas that seemed remarkably simple once propounded, clarified the entire situation, dispelled the confusion, restored order, and produced a new synthesis that embraced far more even than the phenomena under consideration. The genius who… picked up the torch of science dropped by Galileo, was Isaac Newton.”

Morris Kline

Source: Mathematics and the Physical World (1959), p. 225

Law , History , Mathematics , Idea

“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

“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

“Laplace created a number of new mathematical methods that were subsequently expanded into branches of mathematics, but he never cared for mathematics except as it helped him to study nature.”

Morris Kline

Source: Mathematical Thought from Ancient to Modern Times (1972), p. 495

Mathematics , Nature , Study , Help

“Over and above the specific theorems created by men such as Desargues, Pascal and La Hire, several new ideas and outlooks were beginning to appear. The first is the idea of continuous change of a mathematical entity from one state to another… [i. e., of a] a geometrical figure. It was Kepler, in his Astronomiae Optica of 1604, who first seemed to grasp the fact that parabola, ellipse, hyperbola, circle, and the degenerate conic consisting of a pair of lines are continuously derivable from each other…. The notion of a continuous change in a figure was also employed by Pascal. He allowed two consecutive vertices of his hexagon to approach each other so that the figure became a pentagon. In the same manner he passed from pentagons to quadrilaterals.
The second idea to emerge from the work of the projective geometers is that of transformation and invariance. To project a figure from some point and then take a section of that projection is to transform the figure to a new one. The properties… of interest are those that remain invariant under transformation. Other geometers of the seventeenth century, for example, Gregory of St. Vincent… and Newton, introduced transformations other than projection and section.”

Morris Kline

Source: Mathematical Thought from Ancient to Modern Times (1972), pp. 298-299

Men , Mathematics , Idea , Appearance

“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 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

“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