Morris Kline: Quotes about mathematics

Morris Kline was American mathematician. Explore interesting quotes on mathematics.

Morris Kline: 84 quotes0 likes

“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.”

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

“Laplace made many important discoveries in mathematical physics… Indeed, he was interested in anything that helped to interpret nature. He worked on hydrodynamics, the wave propagation of sound, and the tides. In the field of chemistry, his work on the liquid state of matter is classic. His studies of the tension in the surface layer of water, which accounts for the rise of liquids inside a capillary tube, and of the cohesive forces in liquids, are fundamental. Laplace and Lavoisier designed an ice calorimeter (1784) to measure heat and measured the specific heat of numerous substances; heat, to them, was still a special kind of matter. Most of Laplace's life was, however, devoted to celestial mechanics.”

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

“To the scientists of 1850, Hamilton's principle was the realization of a dream. …from the time of Galileo scientists had been striving to deduce as many phenomena of nature as possible from a few fundamental physical principles. …they made striking progress …But even before these successes were achieved Descartes had already expressed the hope and expectation that all the laws of science would be derivable from a single basic law of the universe. This hope became a driving force in the late eighteenth century after Maupertuis's and Euler's work showed that optics and mechanics could very likely be unified under one principle. Hamilton's achievement in encompassing the most developed and largest branches of physical science, mechanics, optics, electricity, and magnetism under one principle was therefore regarded as the pinnacle of mathematical physics.”

Source: Mathematical Thought from Ancient to Modern Times (1972), p. 441.

“Another feature of Alexandrian algebra is the absence of any explicit deductive structure. The various types of numbers… were not defined. Nor was there any axiomatic basis on which a deductive structure could be erected. The work of Heron, Nichomachus, and Diophantus, and of Archimedes as far as his arithmetic is concerned, reads like the procedural texts of the Egyptians and Babylonians… The deductive, orderly proof of Euclid and Apollonius, and of Archimedes' geometry is gone. The problems are inductive in spirit, in that they show methods for concrete problems that presumably apply to general classes whose extent is not specified. In view of the fact that as a consequence of the work of the classical Greeks mathematical results were supposed to be derived deductively from an explicit axiomatic basis, the emergence of an independent arithmetic and algebra with no logical structure of its own raised what became one of the great problems of the history of mathematics. This approach to arithmetic and algebra is the clearest indication of the Egyptian and Babylonian influences… Though the Alexandrian Greek algebraists did not seem to be concerned about this deficiency… it did trouble deeply the European mathematicians.”

Source: Mathematical Thought from Ancient to Modern Times (1972), p.144

“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, p.22.
Mathematics for the Nonmathematician (1967)

“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.”

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

“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.”

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

“The unnaturalness of mathematical symbolism is attested to by history. The algebra of the Egyptians, the Babylonians, the Greeks, the Hindus, and the Arabs was what is commonly called rhetorical algebra. …on the whole they used ordinary rhetoric to describe their mathematical work. Symbolism is a relatively modern invention of the sixteenth and seventeenth centuries…”

Source: Mathematics and the Physical World (1959), p. 59

“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.”

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

“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.”

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

“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.”

...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

“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.”

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