Mathematics for the Nonmathematician (Books on Mathematics)

by Morris Kline

Beyond improvement in the material life of man and beyond satisfaction of intellectual curiosity, the study of nature offers intangible values of another sort, especially the abolition of fear and terror and their replacement by a deep, quiet satisfaction in the ways of nature.

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. Moreover, the Greeks recognized that reason was the distinctive faculty which humans possessed. 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.”

It was by the application of reasoning to mathematics that the Greeks completely altered the nature of the subject. In fact, mathematics as we understand the term today is entirely a Greek gift,

With their gift of reason and with their explicit example of the power of reason, the Greeks founded Western civilization.

The Romans have also bequeathed gifts to Western civilization, but in the fields of mathematics and science their influence was negative rather than positive.

Since the Church did not favor Greek learning and since at any rate the illiterate Europeans had first to learn reading and writing, one is not surprised to find that mathematics and science were practically unknown in Europe until about 1100 A.D.

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.

Our principal intellectual doctrines and outlook were fashioned then, and we still live in the shadow of the Age of Reason.

The most profound in its intellectual significance was the creation of non-Euclidean geometry by Gauss. His discovery had both tantalizing and disturbing implications: tantalizing in that this new field contained entirely new geometries based on axioms which differ from Euclid’s, and disturbing in that it shattered man’s firmest conviction, namely that mathematics is a body of truths. With the truth of mathematics undermined, realms of philosophy, science, and even some religious beliefs went up in smoke.

The failure to penetrate social and biological problems by the deductive method, that is, the method of reasoning from axioms, caused social scientists to take over and develop further the mathematical theories of statistics and probability,

Today countries such as the United States, Russia, China, India, and Japan are also active. Though the last three of these did possess some native mathematics, it was limited and empirical as in Babylonia and Egypt.

Generations and even ages failed to note new ideas, despite the fact that all that was needed was not a technical achievement but merely a point of view.

“things are the shadows of ideas thrown on the screen of experience.”

“Mathematics considered as a science owes its origins to the idealistic needs of the Greek philosophers, and not as fable has it, to the practical demands of Egyptian economics.”

Thales might perhaps have lived in history as a leading businessman, but he is far better known as the father of Greek philosophy and mathematics. From his time onward, deductive proof became the standard in mathematics.

Plato stressed the unreliability of sensory perceptions. Empirical knowledge, as Plato put it, yields opinion only.

The formal proof, so to speak, merely sanctions the conquest already made by the intuition.

There is a well known “proof” that 2 = 1.

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. Stated otherwise, Euclidean geometry is the father of the science of logic.

It is because Euclidean geometry applies these principles of reasoning so clearly and so repeatedly that this subject is often taught as an approach to reasoning.

The crowning achievement of Hipparchus and Ptolemy was the creation of a new astronomical theory which described the paths of the heavenly bodies and enabled man to predict their positions.

One might say that, unlike the Pythagoreans, Plato did not wish to comprehend the physical world through mathematics but aimed at understanding the mathematical plan itself which observation of the physical world suggested very imperfectly.

There were no arts, no science, and no learning. The chief activities were eating, sleeping, carousing, and fighting other tribes. Since such activities are also characteristic of peoples we call civilized, we may say that to that extent the Germanic tribes were civilized.

The Renaissance world began to see man as the goal of God rather than God as the goal of man.

Sciences which arise in thought and end in thought do not give truths because no experience enters into these purely mental reflections, and without experience no thing is sure.

In his Treatise on Painting, a scientific treatise on painting and perspective, Leonardo gives his views. He opens with the statement, “Let no one who is not a mathematician read my works.”

The essential difference between the art of the Renaissance and that of the Middle Ages is the introduction of the third dimension, and Renaissance painting is characterized by the importance attached to realism, to the realistic rendering of space, distance, and forms, achieved by means of the mathematical system of perspective.

The Renaissance artist was a scientist, and painting was a science not merely in the sense that it had a highly technical and even mathematical content, but because it was inspired by the ultimate goal of science, understanding nature.

These conventions or agreements about points and a line at infinity obviate the necessity for making special statements when parallel lines happen to be involved in the theorems.

Famous are his epigrams, “The heart has its reasons which the reason knows nothing of” and “Nothing that has to do with faith can be the concern of reason.”

Ironically, Pascal, the defender of faith, helped immensely to found the ensuing Age of Reason.

The principle of duality is a remarkable property of projective geometry. It reveals the symmetry in the roles which point and line play in the structure of that geometry, and this symmetry in turn reveals that line and point are equally fundamental concepts.

logically projective geometry is the more fundamental and encompassing subject and that Euclidean geometry is in some sense a specialization.

algebra enables man to reason efficiently. It mechanizes thought, and hence produces almost automatically results that may otherwise be difficult to establish.

the rise of experimentation was not the reason that science suddenly blossomed in the seventeenth century. The value and import of seventeenth-century experimentation has been vastly overrated. Modern science owes its origins and present flourishing state to a new scientific method which was fashioned almost entirely by Galileo Galilei.

Galileo’s decision to aim for description was the most profound and the most fruitful thought that anyone has had about scientific methodology.

Randall says in his Making of the Modern Mind, “Science was born of a faith in the mathematical interpretation of Nature, held long before it had been empirically verified.”

The leading mathematicians of the seventeenth and eighteenth centuries were also the leading scientists. And the accomplishments of these two centuries were a triumph of mathematics and science conjoined.

even the best minds become absorbed in the problems of their times. Genius makes its contributions to the advancement of civilization, but the substance of its thoughts is determined by its age.

the way in which mathematics progresses. Ideas are first grasped intuitively and extensively explored before they become fully clarified and precisely formulated even in the minds of the best mathematicians. Gradually the ideas are refined and given the polish and rigor which one encounters in textbook presentations.

it is fortunate that mathematics and physics were so intimately related in the seventeenth and eighteenth centuries—so much so that they were hardly distinguishable—for the physical strength supported the weak logic of mathematics.

The creation of trigonometric functions was motivated by the study of vibratory or oscillatory motion.

The most significant revolutions in this world are not political. Political revolutions hardly change the daily life of man or, if they do, exert a short-term effect which may even be reversed by subsequent revolutions. The significant upheavals are caused by new ideas; these far more effectively, powerfully, and lastingly alter the lives of civilized human beings.

The two concepts which have most profoundly revolutionized our intellectual development since the nineteenth century are evolution and non-Euclidean geometry.

Insofar as the study of the physical world is concerned, mathematics has the same character as any of the sciences. It offers nothing but theories. And, as in science, new mathematical theories may replace older ones when experience or experiment shows that a new theory provides closer correspondence than an older one.

So far as the theories of mathematics are about reality, they are not certain; so far as they are certain, they are not about reality.

the creation of non-Euclidean geometry had the effect of divorcing mathematics from science.

the law of inertia in the world of ideas.

It would be more appropriate to say of man that he is surest of what he believes, than to claim that he believes what is sure.

Perhaps the most important difference between the deductive approach and statistical methods is that the latter tell us what happens to large groups and do not provide definite predictions about any one given case, whereas the former predicts precisely what must happen in individual instances.