Mathematics for the Nonmathematician (Books on Mathematics)

by Morris Kline

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.

At about the same time that St. Augustine lived, the Roman jurists ruled, under the Code of Mathematicians and Evil-Doers, that “to learn the art of geometry and to take part in public exercises, an art as damnable as mathematics, are forbidden.”

In a letter to Fermat written on August 10, 1660, Pascal says: “To speak freely of mathematics, I find it the highest exercise of the spirit; but at the same time I know that it is so useless that I make little distinction between a man who is only a mathematician and a common artisan. Also, I call it the most beautiful profession in the world; but it is only a profession; and I have often said that it is good to make the attempt [to study mathematics], but not to use our forces: so that I would not take two steps for mathematics, and I am confident that you are strongly of my opinion.” Pascal’s famous injunction was, “Humble thyself, impotent reason.”

The philosopher Arthur Schopenhauer, who despised mathematics, said many nasty things about the subject, among others that the lowest activity of the spirit is arithmetic, as is shown by the fact that it can be performed by a machine.

Is it because Plato, some 2300 years ago, advocated mathematics to train the mind for philosophy?

Is it because the Church in medieval times taught mathematics as a preparation for theological reasoning?

Or is it because the commercial, industrial, and scientific life of the Western world ...

Mathematics is concerned primarily with what can be accomplished by reasoning.

Getting along with the opposite sex is an art rather than a science mastered by reasoning. One can engage in a multitude of occupations and even climb high in the business and industrial world without much use of reasoning and certainly without mathematics.

Likewise, when we look at a tall building from a distance, it is experience which tells us that the building is tall. Hence the old saying that “we are prone to see what lies behind our eyes, rather than what appears before them.”

Mathematics more than any other human endeavor relies upon reasoning to produce knowledge.

The primary objective of all mathematical work is to help man study nature, and in this endeavor mathematics cooperates with science.

We shall see later how observations of nature are framed in statements called axioms.

The determination of the pattern of motion of celestial bodies, the discovery and control of radio waves, the understanding of molecular, atomic, and nuclear structures, and the creation of artificial satellites are a few basically mathematical achievements.

of deriving new conclusions are today the substratum in all inves...

Of course, not all people are interested in such questions. Food, shelter, sex, and television are enough to keep many happy.

dragon. Many Hindu people believe today that a demon residing in the sky attacks the sun once in a while and that this is what causes the eclipse. Of course, when prayers, sacrifices, and ceremonies were followed by the victory of the sun or moon, it was clear that these rituals were the effective agent and so had to be pursued on every such occasion.

A wolf, a goat, and cabbage are to be transported across a river by a man in a boat which can hold only one of these in addition to the man. How can he take them across so that the wolf does not eat the goat or the goat the cabbage? Two husbands and two wives have to cross a river in a boat which can hold only two people. How can they cross so that no woman is in the company of a man unless her husband is also present?

The tantalizing and compelling pursuit of mathematical problems offers mental absorption, peace of mind amid endless challenges, repose in activity, battle without conflict, and the beauty which the ageless mountains present to senses tried by the kaleidoscopic rush of events.

Remote from human passions, remote even from the pitiful facts of nature, the generations have gradually created an ordered cosmos, where pure thought can dwell as in its natural home and where one, at least, of our nobler impulses can escape from the dreary exile of the actual world.

RUSSELL, BERTRAND: “The Study of Mathematics,” an essay in the collection entitled Mysticism and Logic, Longmans, Green and Co., New York, 1925. WHITEHEAD, ALFRED NORTH: “The Mathematical Curriculum,” an essay in the collection entitled The Aims of Education, The New American Library, New York, 1949. WHITEHEAD, ALFRED NORTH: Science and the Modern World, Chaps. 2 and 3, Cambridge University Press, Cambridge, 1926.

What did these early civilizations do with their mathematics? If we may judge from problems found in ancient Egyptian papyri and in the clay tablets of the Babylonians, both civilizations used arithmetic and algebra largely in commerce and state administration, to calculate simple and compound interest on loans and mortgages, to apportion profits of business to the owners, to buy and sell merchandise, to fix taxes, and to calculate how many bushels of grain would make a quantity of beer of a specified alcoholic content.

Geometrical rules were applied to calculate the areas of fields, the estimated yield of pieces of land, the volumes of structures, and the quantity of bricks or stones needed to erect a temple or pyramid. The ancient Greek historian Herodotus says that because the annual overflow of the Nile wiped out the boundaries of the farmers’ lands, geometry was needed to redetermine the boundaries.

In fact, Herodotus speaks of geometry as the g...

It is worthy of note that by observing the motion of the sun, the Egyptians managed to ascertain that the year contains 365 days.

Compared with the accomplishments of their immediate successors, the Greeks, the mathematics of the Egyptians and Babylonians is the scrawling of children just learning how to write as opposed to great literature. They barely recognized mathematics as a distinct subject.

It was a tool in agriculture, commerce, and engineering, no more important than the other tools they used to build pyramids and ziggurats. Over a period of 4000 years hardly any progress was made in the subject.

Moreover, the very essence of mathematics, namely, reasoning to establish the validity of methods and res...

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.

“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.”

the Greeks abandoned empiricism and undertook a systematic, rational attack on the whole subject. First of all, the Greeks saw clearly that numbers and geometric forms occur everywhere in the heavens and on earth.

Hence they selected some of these obvious facts as a starting point and called them axioms.

The Greeks found mathematics valuable in many respects, as we shall learn later, but they saw its main value in the aid it rendered to the study of nature; and of all the phenomena of nature, the heavenly bodies attracted them most. Thus, though the Greeks also studied light, sound, and the motions of bodies on the earth, astronomy was their chief scientific interest.

They sought no material gain and no power over nature; they sought merely to satisfy their minds.

They dared to affirm that nature was rationally and indeed mathematically designed, and that man’s reason, chiefly through the aid of mathematics, would fathom that design.

The attempt to be quantitative, coupled with the classical Greek love for the mathematical study of nature, stimulated two of the most famous astronomers of all time, Hipparchus and Ptolemy, to calculate the sizes and distances of the heavenly bodies and to build a sound and, for those times, accurate astronomical theory, which is still known as Ptolemaic theory. Hipparchus and Ptolemy also created the chief tool they needed for this purpose, the mathematical subject known as trigonometry.

The contrast between Greek and Roman cultures is striking. The Romans have also bequeathed gifts to Western civilization, but in the fields of mathematics and science their influence was negative rather than positive. The Romans were a practical people and even boasted of their practicality.

Greeks: academics, Romans: lean, y combinator

They sought wealth and world power and were willing to undertake great engineering enterprises, such as the building of roads and viaducts, which might help them to expand, control, and administer their empire, but they would spend no time or effort on theoretical studies which might further these activities.

The Arabs, who suddenly appeared on the scene of history in the role of destroyers, had been a nomadic people. They were unified under the leadership of the prophet Mohammed and began an attempt to convert the world to Mohammedanism, using the sword as their most decisive argument.

The first major European development in mathematics occurred in the work of the artists. Imbued with the Greek doctrines that man must study himself and the real world, the artists began to paint reality as they actually perceived it instead of interpreting religious themes in symbolic styles.

Wish this were true of deep learning. Perhaps if society goes to shit and all this knowledge is lost, it'll be picked up again by artists

Stimulated by Greek astronomical ideas, supplied with data and the astronomical theory of Hipparchus and Ptolemy, and steeped in the Greek doctrine that the world is mathematically designed, Nicolaus Copernicus sought to show that God had done a better job than Hipparchus and Ptolemy had described.

Just how much mathematical activity the revival of Greek works might have stimulated cannot be determined, for simultaneously with the translation and absorption of these works, a number of other revolutionary developments altered the social, economic, religious, and intellectual life of Europe. The introduction of gunpowder was followed by the use of muskets and later cannons. These inventions revolutionized methods of warfare and gave the newly emerging social class of free common men an important role in that domain.

Since many of the problems raised by the motion of cannon balls, navigation, and industry called for quantitative knowledge, arithmetic and algebra became centers of attention. A remarkable improvement in these mathematical fields followed.

In particular, the use of letters to represent a class of numbers, a device which gives algebra its generality and power, was introduced by Vieta.

In this same period, logarithms were created to facilitate the calculations of astronomers. 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.

He found no certainty in any of the knowledge taught him, and he therefore concentrated for years on finding the method by which man can arrive at truths.

Because the scientific problems of his time involved work with curves, the paths of ships at sea, of the planets, of objects in motion near the earth, of light, and of projectiles, Descartes sought a better method of proving theorems about curves.

We have already noted that a new society was developing in Europe. Among its features were expanded commerce, manufacturing, mining, large-scale agriculture, and a new social class—free men working as laborers or as independent artisans.

Whereas the Greeks had been content to study nature merely to satisfy their own curiosity and to organize their conclusions in patterns pleasing to the mind, the new goal, effectively proclaimed by Descartes and Francis Bacon, was to make nature serve man.

However, Bacon had cautioned that nature can be commanded only when one learns to obey her. One must have facts of nature on which to base reasoning about nature. Hence mathematicians and scientists sought to acquire facts from the experience of artists, technicians, artisans, and engineers. The alliance of mathematics and experience was gradually transformed into an alliance of mathematics and experimentation, and a new method for the pursuit of the truths of nature, first clearly perceived and formulated by Galileo Galilei (1564–1642) and Newton, was gradually evolved.

This approach of obedience and the mastery is something I identify with. It seems to be a the root of the mlux approach