Morris Kline > Quotes

“Logic is the art of going wrong with confidence.”
― Morris Kline

“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 sense tried by the present-day kaleidoscope of events.”
― Morris Kline

“All mathematical proofs must be deductive. Each proof is a chain of deductive arguments, each of which has its premises and conclusion.”
― Morris Kline, Mathematics for the Nonmathematician

“A good expository paper will benefit far more people than most research papers. A good text is worth a thousand of the usual trifles that appear in research journals.”
― Morris Kline, Why the Professor Can't Teach: Mathematics and the Dilemma of American Undergraduate Education

“The feeling that one must be an authority in a subject to say anything about it is unfounded. We are all laymen outside the field of our own specialty,”
― Morris Kline, Mathematics for the Nonmathematician

“Because we are forced to learn about numbers and operations with numbers while we are still too young to appreciate them—a preparation for life which hardly excites our interest in the future—we grow up believing that numbers are drab and uninteresting. But the number system warrants attention not only as the basis of mathematics, but because it contains weighty and beautiful ideas which lend themselves to powerful applications.”
― Morris Kline, Mathematics for the Nonmathematician

“Man knew how to feed, clothe, and house himself millenniums before mathematics existed.”
― Morris Kline, Mathematics for the Nonmathematician

“Base two especially impressed the seventeenth-century religious philosopher and mathematician Gottfried Wilhelm Leibniz. He observed that in this base all numbers were written in terms of the symbols 0 and 1 only. Thus eleven, which equals 1 · 23 + 0 · 22 + 1 · 2 + 1, would be written 1011 in base two. Leibniz saw in this binary arithmetic the image and proof of creation. Unity was God and zero was the void. God drew all objects from the void just as the unity applied to the zero creates all numbers. This conception, over which the reader would do well not to ponder too long, delighted Leibniz so much that he sent it to Grimaldi, the Jesuit president of the Chinese tribunal for mathematics, to be used as an argument for the conversion of the Chinese emperor to Christianity.”
― Morris Kline, Mathematics and the Physical World

“The supreme contribution of the Greeks was to call attention to, employ, and emphasize the power of human reason. This recognition of the power of reasoning is the greatest single discovery made by man.”
― Morris Kline, Mathematics for the Nonmathematician

“Even the greatest Greek algebraist, Diophantus, who lived during the latter part of the Alexandrian Greek civilization (around A.D. 250), rejected irrationals as numbers.”
― Morris Kline, Mathematics and the Physical World

“Aristotle says, “Now what is characteristic of any nature is that which is best for it and gives most joy. Such to man is the life according to reason, since it is that which makes him man.”
― Morris Kline, Mathematics for the Nonmathematician

“While 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.”
― Morris Kline, Mathematics and the Physical World

“Geometry . . . is the science that it hath pleased God hitherto to bestow on mankind. —THOMAS HOBBES”
― Morris Kline, Mathematics and the Physical World

“predecessor of Isaac Newton at Cambridge University, maintained that irrational numbers have no meaning independent of geometric lengths.”
― Morris Kline, Mathematics and the Physical World

“The study of molecular structure attempts to get at precisely the physical constituents of molecules.”
― Morris Kline, Mathematics and the Physical World

“As far back as about the year 400 A.D., St. Augustine, Bishop of Hippo in Africa and one of the great fathers of Christianity, had this to say: The good Christian should beware of mathematicians and all those who make empty prophecies. The danger already exists that the mathematicians have made a covenant with the devil to darken the spirit and to confine man in the bonds of Hell.”
― Morris Kline, Mathematics for the Nonmathematician

“The mathematics and science that developed in Europe after the Renaissance became much more dependent upon quantitative results and hence upon the use of all types of numbers.”
― Morris Kline, Mathematics and the Physical World

“The second basic function of algebra is to convert expressions into more useful ones. Gauss’s”
― Morris Kline, Mathematics and the Physical World

“Negative numbers, equations involving unknowns, formulas, derivatives, integrals, and other concepts we shall encounter are abstractions built upon abstractions.”
― Morris Kline, Mathematics and the Physical World

“Can we combine √2 + √3 into the simpler √5? Let us test this operation on whole numbers expressed as roots. Certainly √9 + √16 does not equal , that is, 3 + 4 does not equal 5. Hence we should not assert the analogous relation for irrational numbers.”
― Morris Kline, Mathematics and the Physical World

“It would seem that to depart from the real world by concentrating on just a few abstract properties of physical objects, such as the straightness of some physical lengths, would rob mathematics of effectiveness. Yet part of the secret of mathematical power lies in its use of abstract concepts. By this means we free our minds from burdensome and irrelevant detail and are thereby able to accomplish more. For example, if one should study fruits and attempt to encompass in one theory color, shape, structure, nature of skin, relative hardness, nature of pulp, and other properties he might get nowhere because he had tackled too big a problem.”
― Morris Kline, Mathematics and the Physical World

“If hydrogen, the element of lowest atomic weight, should be the building block (aside from other particles of negligible weight) then something must be lost in the process of fusing hydrogen atoms to form elements of larger atomic weight. This loss of mass, which has been known to be equivalent to energy since Einstein’s work of 1905, is called the binding energy. The idea occurred that it might perhaps be utilized. This is now done in the fusion process that takes place when a hydrogen bomb explodes and the extra mass is converted to radiated energy.”
― Morris Kline, Mathematics and the Physical World

“seemed almost certain to the mathematicians that since the general first, second, third, and fourth degree equations can be solved by means of the usual algebraic operations such as addition, subtraction, and roots, then the general fifth degree equation and still higher degree equations could also be solved. For three hundred years this problem was a classic one. Hundreds of mature and expert mathematicians sought the solution, but a little boy found the full answer. The Frenchman Évariste Galois (1811— 1832), who refused to conform to school examinations but worked brilliantly and furiously on his own, showed that general equations of degree higher than the fourth cannot be solved by algebraic operations. To establish this result Galois created the theory of groups, a subject that is now at the base of modern abstract algebra and that transformed algebra from a a series of elementary techniques to a broad, abstract, and basic branch of mathematics.”
― Morris Kline, Mathematics and the Physical World

“Goldbach’s hypothesis”
― Morris Kline, Mathematics and the Physical World

“the nerve cells respond to electrical impulses much as an electron tube does.”
― Morris Kline, Mathematics and the Physical World

“Is then mathematics a collection of diamonds hidden in the depths of the universe and gradually unearthed, or is it a collection of synthetic stones manufactured by man, yet so brilliant nevertheless that they bedazzle those mathematicians who are already partially blinded by pride in their own creations?”
― Morris Kline

“But Archimedes wished to determine how much silver and how much gold were in the crown, and here he found that algebra would help him. He supposed that the crown, which, let us say, weighed ten pounds, was made up of w1 pounds of silver and w2 pounds of gold. He found that ten pounds of pure silver displaced thirty cubic inches of water. Hence, w1 pounds of silver would displace (w1/10) · 30 or 3 w1 cubic inches of water. Since ten pounds of pure gold”
― Morris Kline, Mathematics and the Physical World

“Just what did the Greeks seek in probing nature? They sought no material gain and no power over nature; they sought merely to satisfy their minds. Because they enjoyed reasoning and because nature presented the most imposing challenge to their understanding, the Greeks undertook the purely intellectual study of nature. Thus the Greeks are the founders of science in the true sense.”
― Morris Kline, Mathematics for the Nonmathematician

“Though Leibniz may have had special reasons for considering base two, neither this base nor any other had until recently been seriously considered as a substitute for base ten. In fact, aside from incidental uses of other bases in higher mathematics to facilitate an occasional proof, the subject of bases other than ten was regarded until recently as an intellectual amusement.”
― Morris Kline, Mathematics and the Physical World

“The danger already exists that the mathematicians have made a covenant with the devil to darken the spirit and to confine man in the bonds of Hell.”
― Morris Kline, Mathematics for the Nonmathematician