Mathematics for the Nonmathematician (Books on Mathematics)

by Morris Kline

Just as we gradually add to our knowledge of the varieties of human beings and animals which exist in our physical world, so must we broaden our knowledge of the varieties of numbers and with true liberality accept these strangers on the same basis as the already familiar numbers.

Bad analogy

Though machines are, in speed, accuracy, and endurance, superior to the human brain, one should not infer, as many popular writers are now suggesting, that machines will ultimately replace brains. Machines do not think.

They perform the calculations which they are directed to perform by people who have the brains to know what calculations are wanted. Nevertheless, we undoubtedly have in the machine a useful model for the study of some functions of the human brain and nerves.

On Games of Chance,

ORE, OYSTEIN: Cardano, The Gambling Scholar, Chaps. 1 through 5, Princeton University Press, Princeton, 1953.

AXIOM 6. Things equal to the same thing are equal to each other.

AXIOM 8. If equals be subtracted from equals, the remainders are equal.

Many of these theorems are indeed simple to prove and obviously true of the geometrical figures involved. But Euclid’s purpose in proving them was to play safe.

The mathematician might then consider the question, Which pentagon of all pentagons with the same perimeter has maximum area?

And now the conjecture seems reasonable that of two regular polygons with the same perimeter, the one with more sides will have more area. This is so. Where does this result lead? One can form regular polygons of more and more sides, which all have the same perimeter. As the number of sides increases, the area increases. But as the number of sides increases, the regular polygon approaches the circle in shape. Hence the circle should have more area than any regular polygon of the same perimeter.

Nature not only sets problems for us, but often solves them too, if we are but keen enough in our observations.

Each science must start with fundamental principles relevant to its field and proceed by deductive demonstrations of new truths.

By teaching mankind the principles of correct reasoning, Euclidean geometry has influenced thought even in fields where extensive deductive systems could not be or have not thus far been erected.

BOYS, C. VERNON: Soap Bubbles, Dover Publications, Inc., New York, 1959.

COURANT, R. and H. ROBBINS: What is Mathematics?, pp. 329–338, pp. 346–361, Oxford University Press, New York, 1941.

KLINE, MORRIS: Mathematics: A Cultural Approach, Sections 6–8, Addison-Wesley Publishing Co., Reading, Mass., 1962.

KLINE, MORRIS: Mathematics and the Physical World, Chaps. 6 and 17, T. Y. Crowell Co., New York, 1959. Also in paperback, Doubleday and Co., N.Y., 1963.

SAWYER, W. W.: Mathematician’s Delight, Chaps. 2 and 3, Penguin Books, Harmon...

TAYLOR, LLOYD WM.: Physics, The Pioneer Science, Chaps. 29–32, Dover Publications, Inc., New York, 1959.

Archimedes, the greatest intellect of the Alexandrian world, constructed a planetarium which reproduced the motions of the heavenly bodies and designed a pump for raising water from a river to land.

Before we venture onto vast stretches of the earth’s surface or into the heavens, let us see what we can do with trigonometric ratios in rather simple, homely situations.

The classical Greeks had indeed speculated about these sizes and distances, but since they relied far more upon aesthetically pleasing principles than upon keen observation, measurement of angles, and numerical calculation, their conclusions were often absurd.

We shall consider first how one can find the distance to the moon.

Well, no decently behaving light ray would wish to disobey the mathematical law of refraction.

It would appear from Fermat’s analysis that nature is wise. It knows mathematics and employs it in the interest of economy.

The power of mathematics to describe and analyze nature’s ways was advanced well beyond the stage at which Euclid and Apollonius had left it.

TAYLOR, LLOYD W.: Physics, The Pioneer Science, pp. 442–470, Dover Publications, Inc., New York, 1959.

THE MATHEMATICAL ORDER OF NATURE

Great men! elevated above the common standard of human nature, by discovering the laws which celestial occurrences obey, and by freeing the wretched mind of man from the fears which the eclipses inspired.

THE GREEK CONCEPT OF NATURE

Possessed with insatiable curiosity and courage, they asked and answered the questions which occur to many, are tackled by few, and are resolved only by individuals of the highest intellectual caliber.

They affirmed that nature is rationally and indeed mathematically designed.

The Greeks were the first people with the audacity to conceive of such law and order in the welter of phenomena and the first with the genius to uncover a pattern to which nature conforms. They dared to ask for and they found a design underlying the greatest spectacle man beholds, the motion of the brilliant sun, the changing shapes of the many-hued moon, the piercing shafts of the planets, the broad panorama of lights from the canopy of stars, and the seemingly miraculous eclipses of the sun and moon.

To appreciate the originality and boldness of the steps which the Greeks took in this direction, one must compare their attitude with what preceded.

The visible figures in the heavens are far inferior to the true objects, namely those objects that are to be apprehended by reason and mental conceptions.

The theory of Hipparchus and Ptolemy is the final Greek answer to Plato’s problem of rationalizing the appearances in the heavens and is the first really great scientific synthesis.

THE EVIDENCE FOR THE MATHEMATICAL DESIGN OF NATURE

Those in which lightness dominated (for example, fire) always sought to rise. Those in which heaviness dominated (for example, metals) sought to fall. Every object had a natural place and, when not hindered, sought it. Thus the natural place of light objects was a region near the moon, whereas heavy objects tended to congregate at the center of the universe which was, of course, the center of the earth.

From these scientific investigations one major fact stood forth: the universe is mathematically designed. Mathematics is immanent in nature; it is the truth about its structure, or, as Plato would have it, the reality of the physical world.

Moreover, human reason could penetrate the divine plan and reveal the mathematical structure of nature. Almost all of the mathematical and scientific research which has taken place since Greek times has been inspired by the conviction that there is law and order in the universe, and that mathematics is the key to this order. The Greek miracle has not been rivaled, not even by our modern civilization. A relative handful of people produced in a few hundred years supreme

DAMPIER-WHETHAM, WM. C.D.: A History of Science, Chap. 1, Cambridge University Press, Cambridge, 1929.

KUHN, THOMAS S.: The Copernican Revolution, Chaps. 1 through 3, Harvard University Press, Cambridge, 1957.

There were no arts, no science, and no learning. The chief activities were eating, sleeping, carousing, and fighting other tribes.

Man was the center of the universe not merely geographically but also in terms of the ultimate purposes served by nature.

MATHEMATICS IN THE MEDIEVAL PERIOD

Europe revolted against scholastic domination of thought, rigid authority, and restrictions on the physical life. It revolted against the Scriptures as the source of all knowledge and the authority for all assertions. It revolted against enforced conformity to the established canons of conduct.

A leading figure in the revolt from the old modes of thought is Leonardo da Vinci (1452–1519). Because he saw how most scholars accepted as authoritative all that they read, he distrusted the men who took their learning only from books and professed their knowledge so dogmatically. He describes them as puffed up and pompous, strutting about, and adorned only by the labors of others whom they merely repeated. These were only the reciters and trumpeters of other people’s learning.

Leonardo determined to learn for himself and made exhaustive studies of plants, animals, the human body, light, the principles of mechanical devices, rocks, the fli...

Many European thinkers finally broke away from the endless rationalizing on the basis of dogmatic principles which were vague in meaning and unrelated to experience, and chose human inquiry rather than divine authority.

Leonardo is a representative figure in this shift to nature as the prime focus. He almost boasts that he is not a man of letters and that he chose to learn from experience. His observations and inventions recorded in his notebooks give evidence of his extensive and detailed physical studies.