Mathematics for the Nonmathematician (Books on Mathematics)

by Morris Kline

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, and we should not be ashamed to point this out to students.

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.

Perhaps St. Augustine, with prophetic insight into the conflicts which were to arise later between the mathematically minded scientists of recent centuries and religious leaders, was seeking to discourage the further development of the subject. At any rate there is no question as to his attitude.

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.”

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

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

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

Mathematical formulation of physical data and mathematical methods of deriving new conclusions are today the substratum in all investigations of nature. The fact that mathematics is of central importance in the study of nature reveals almost immediately several values of this subject.

The deeper reason for the study of nature is to try to understand the ways of nature, that is, to satisfy sheer intellectual curiosity.

Indeed, to ask disinterested questions about nature is one of the distinguishing marks of mankind.

Food, shelter, sex, and television are enough to keep many happy. But others, aware of the pervasive natural mysteries, are more strongly obsessed to resolve them than any business man is to acquire wealth and power.

Those “seemingly unprofitable amusements of speculative brains” have freed us from serfdom, given us undreamed of powers, and, in fact, have replaced negative doctrines by positive mathematical laws which reveal a remarkable order and uniformity in nature.

Practical, scientific, philosophical, and artistic problems have caused men to investigate mathematics. But there is one other motive which is as strong as any of these — the search for beauty. Mathematics is an art, and as such affords the pleasures which all the arts afford.

In brief, we shall try to see that mathematics is an integral part of the modern world, one of the strongest forces shaping its thoughts and actions, and a body of living though inseparably connected with, dependent upon, and in turn valuable to all other branches of our culture.

An educated mind is, as it were, composed of all the minds of preceding ages. LE BOVIER DE FONTENELLE

Mathematics is also a cumulative development, that is, newer creations are built logically upon older ones, so that one must usually understand older results to master newer ones.

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 evident that equal numbers added to or subtracted from equal numbers give equal numbers. It was equally evident that two right angles are necessarily equal and that a circle can be drawn when center and radius are given. Hence they selected some of these obvious facts as a starting point and called them axioms.

Even today Euclid is the prime example of the power and accomplishments of reason. Hundreds of generations since Euclid’s days have learned from his geometry what reasoning is and what it can accomplish.

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 result of Copernicus’ thinking was a new system of astronomy in which the sun was immobile and the planets revolved around the sun. This heliocentric theory was considerably improved by Kepler.

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.

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.

A new science of motion was created by Galileo and Newton, and in the process two brand-new developments were added to mathematics. The first of these was the notion of a function, a relationship between variables best expressed for most purposes as a formula. The second, which rests on the notion of a function but represents the greatest advance in method and content since Euclid’s days, was the calculus.

With the aid of the calculus Newton was able to organize all data on earthly and heavenly motions into one system of mathematical mechanics which encompassed the motion of a ball falling to earth and the motion of the planets and stars.

Galileo’s and Newton’s plan of applying mathematics to sound physical principles not only succeeded in one major area but gave promise, in a rapidly accelerating scientific movement, of embracing all other physical phenomena.

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.

The lesson learned from the history of non-Euclidean geometry was that though mathematicians may start with axioms that seem to have little to do with the observable behavior of nature, the axioms and theorems may nevertheless prove applicable.

Hence mathematicians felt freer to give reign to their imaginations and to consider abstract concepts such as complex numbers, tensors, matrices, and n-dimensional spaces. This development was followed by an even greater advance in mathematics and, surprisingly, an increasing use of mathematics in the sciences.

It is rather a living plant that has flourished and languished with the rise and fall of civilizations.

While mathematicians produce formulas, no formula produces mathematicians.

The point in learning about these human variations, aside from satisfying our instinct to pry into other people’s lives, is that it explains to a large extent why the progress of the highly rational subject of mathematics has been highly irrational.

BELL, ERIC T.: Men of Mathematics, Simon and Schuster, New York, 1937.

Geometry will draw the soul toward truth and create the spirit of philosophy. PLATO

We must endeavor that those who are to be the principal men of our State go and learn arithmetic, not as amateurs, but they must carry on the study until they see the nature of numbers with the mind only; . . . arithmetic has a very great and elevating effect, compelling the soul to reason about abstract number, and rebelling against the introduction of visible or tangible objects into the argument.

Mathematics is in fact ideally suited to prepare the mind for higher forms of thought because on the one hand it pertains to the world of visible things and on the other hand it deals with abstract concepts.

Mathematics, then, is the best preparation for philosophy. For this reason Plato recommended that the future rulers, who were to be philosopher-kings, be trained for ten years, from age 20 to 30, in the study of the exact sciences, arithmetic, plane geometry, solid geometry, astronomy, and harmonics (music).

In making such idealizations, the mathematician deliberately distorts or approximates at least some features of the physical situation. Why does he do it? The reason usually is that he simplifies the problem and yet is quite sure that he has not introduced any gross errors.

All mathematical proofs must be deductive. Each proof is a chain of deductive arguments, each of which has its premises and conclusion.

In fact reasoning by analogy is a powerful method in science.

Work robbed man of time and energy for intellectual activities, the duties of citizenship and discussion.

To the Greeks the premises on which mathematics was to be built were self-evident truths, and they called these premises axioms.

We now know that axioms are suggested by experience and observation. Naturally, to be as certain as we can of these axioms we select those facts which seem clearest and most reliable in our experience. But we must recognize that there is no guarantee that we have selected truths about the world. Some mathematicians prefer to use the word assumptions instead of axioms to emphasize this point.

The most fertile source of mathematical ideas is nature herself. Mathematics is devoted to the study of the physical world, and simple experience or the more careful scrutiny of nature suggests idea after idea.

In the domains of algebra, calculus, and advanced analysis especially, the first-rate mathematician depends upon the kind of inspiration that we usually associate with the creation of music, literature, or art.

BELL, ERIC T.: The Development of Mathematics, 2nd ed., Chaps. 2 and 3, McGraw-Hill Book Co., N.Y., 1945.

WEDBERG, ANDERS: Plato’s Philosophy of Mathematics, Almqvist and Wiksell, Stockholm, 1955 (for students of philosophy).

A marvelous neutrality have these things Mathematical, and also a strange participation between things supernatural, immortal, intellectual, simple, and indivisible, and things natural, mortal, sensible, compounded and divisible. JOHN DEE (1527–1608)

The mathematicians’ refusal over centuries to grant irrationals the status of numbers illustrates one of the surprising features of the history of mathematics. New ideas are often as unacceptable in this field as they are in politics, religion, and economics.