Mathematics and the Physical World Quotes

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

“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

“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

“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

“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 study of molecular structure attempts to get at precisely the physical constituents of molecules.”
― 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

“displaced fifteen cubic inches of water, w2 pounds of gold would displace (w2/10) · 15 or (3/2)w2 cubic inches of water. Hence the crown should displace cubic inches of water. Archimedes measured the volume of water that the crown displaced and found it to be, let us say, twenty cubic inches. Hence he knew that (3) He also knew that (4) Archimedes now had two equations involving two unknowns and he proceeded to apply the machinery of algebra to find them. He multiplied both sides of equation (4) by 3 to obtain”
― Morris Kline, Mathematics and the Physical World

“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

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

“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

“Of course it must be noted that no decimal expression, no matter how many decimal places are used, will ever exactly equal an irrational number; a decimal is a fraction, and an irrational, we saw, cannot equal a fraction.”
― 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

“irrational number cannot equal a whole number or a fraction.”
― Morris Kline, Mathematics and the Physical World

“The number π, which is the ratio of the circumference of any circle to its diameter, is irrational, though this fact was not proved until 1768. Many other examples could be given.”
― Morris Kline, Mathematics and the Physical World

“The Pythagoreans were indeed baffled. Here was a totally new element in the universe that could not be described in terms of whole numbers. Their entire philosophy of nature, which was based on the principle that every phenomenon could be reduced to whole numbers, was threatened. They called this new number irrational, which term then meant unmentionable or unknowable though today it means a number not expressible as a ratio of whole numbers. There is a legend that the discovery of √2 was made by a member while the entire group of Pythagoreans was on a ship at sea. The member was thrown overboard and the rest of the group pledged to secrecy.”
― Morris Kline, Mathematics and the Physical World

“Thus, the machine’s memory device functions somewhat like the human memory. Since computing machines simulate the actions of nerves and memory, they may give us some clues to the functioning of the human brain and of nerve actions. Though these machines are in speed, accuracy, and endurance superior to the human brain, one should not infer, as many popular writers are now trying to suggest, that computers will ultimately replace brains. Machines do not think. They perform calculations. The machines, to use the words the Greeks used and which we mentioned at the beginning of this chapter, do logistica but not arithmetica. Nevertheless, we undoubtedly have in the machine a useful model for the study of some functions of the human brain.”
― 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

“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

“As an example, let us choose base six. To write the quantities from zero to five we would use the symbols 0, 1, 2, 3, 4, 5, as in base ten. The first essential difference comes up when we wish to denote six objects. Since six is to be the base we indicate this larger quantity by the symbols 10, the 1 denoting one times the base, just as in base ten the 1 in 10 denotes one times the base, or the quantity ten. Thus, the symbols 10 can mean different quantities, depending upon the base being employed. To write seven in base six we would write 11, because in base six these symbols mean 1.6+ 1, just as 11 in base ten means 1. 10 + 1. Similarly, to denote twenty in base six we write 32 because these symbols now mean 3 · 6 + 2. To indicate the quantity forty in base six we write 104, because these symbols mean 1 . 62 + 0 . 6 + 4, just as in base ten 104 means 1 · 102 + 0 · 10 + 4 or one hundred and four. It is clear that we can express quantity in base six. Moreover, we can perform the usual arithmetic operations in this base. We would, however, have to learn new addition and multiplication tables. For example, in base ten 4 + 5 = 9, but in base six 9 would be written 13.”
― 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

“The system of positional notation we use derives from the Hindus; however, the same scheme was used two milleniums earlier by the Babylonians, but to a more limited extent because they did not have a zero.”
― Morris Kline, Mathematics and the Physical World

“Because in our way of writing numbers the position of an integer determines the quantity it represents, the principle involved is called positional notation.”
― Morris Kline, Mathematics and the Physical World

“Another example may also help us to appreciate the abstractness of numbers. Mathematically, is equal to . But the corresponding physical fact may not be true.”
― Morris Kline, Mathematics and the Physical World

“Because the knowledge of counting, adding, subtracting, and the like is regarded as a preparation for “life” we are taught it mechanically from early childhood. The practice takes precedence over the principles. No doubt this introduction to life is not especially cheering.”
― 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

“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

“geometrical forms, such as triangles and circles, and the concepts of arithmetic, such as whole numbers and fractions, are abstractions of certain properties of physical objects.”
― Morris Kline, Mathematics and the Physical World

“Anyone who has played with integers, for example, has doubtless observed that the sum of the first two odd numbers, that is, 1 + 3, is the square of two or 4; the sum of the first three odd numbers, that is, 1 + 3 +5, is the square of three or 9. Similarly for the first four, five, and six odd numbers. Thus, simple calculation suggests a general statement, namely, that the sum of the first n odd numbers, where n is any integer, is the square of n. Of course this possible theorem is not proved by the above calculations. Nor could it ever be proved by such calculations, for mortal man could not make the infinite set of calculations that would be required to establish the conclusion for every n. The calculations do, however, give the mathematician something to work on.”
― Morris Kline, Mathematics and the Physical World

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