Mathematics for the Nonmathematician (Books on Mathematics)

by Morris Kline

He says, “If you do not rest on the good foundation of nature, you will labor with little honor and less profit.” 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.

The new impulse to study nature and the decision to apply reason instead of relying upon authority were forces which might in themselves have led to mathematical activity.

“No human inquiry can be called true science unless it proceeds through mathematical demonstrations.” He

Galileo speaks of mathematics as the language in which God wrote the great book—the universe—and unless one knows this language, it is impossible to comprehend a single word.

Mathematics remained the one accepted body of truths amid crumbling philosophical systems, disputed theological beliefs, and changing ethical values. Mathematical knowledge was certain knowledge and offered a secure foothold in a morass. The search for truth was redirected toward mathematics.

However, in the sixteenth century the new goal in the intellectual world became to study nature through mathematics and indeed to uncover the mathematical design of nature.

CARDAN, JEROME: The Book of My Life, E. P. Dutton and Co., New York, 1930.

CROMBIE, A. C: Augustine to Galileo, Chaps. 1 to 5, Falcon Press, London, 1952.

CROMBIE, A. C.: Robert Grosseteste and the Origins of Experimental Science, Oxford University Press, London, 1953.

Mighty is geometry; joined with art, resistless. EURIPIDES

Greek conviction that the essence of nature’s behavior should be sought in mathematical laws, bore fruit first in the field of art rather than science.

Thus Leonardo da Vinci, in offering his services to Lodovico Sforza, ruler of Milan, promises to serve as engineer, constructor of military works, and designer of war machines, as well as architect, sculptor, and painter. The artist was even expected to predict the motion of cannon balls, by no means a simple problem for the mathematics of those times. In view of these manifold activities the painter necessarily had to be something of a scientist.

The Renaissance painters went so far in assimilating this knowledge and in applying mathematics to painting that they produced the first really new mathematics in Europe. In the fifteenth century they were the most accomplished and also the most original mathematicians.

This point of view is interesting

Nature was to be the authority for what appeared on canvas, and painting was to be the science of reproducing nature accurately. The objective of painting, says Leonardo da Vinci, is to reproduce nature and the merit of a painting lies in the exactness of the reproduction.

This highly intellectual painter with a passion for geometry planned all his works mathematically to the last detail. Each scene to be painted was a mathematical problem.

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.

Cool observation

The artist of that period regarded himself as the servant of science. These men who explored and represented nature with methods peculiar to their art were motivated precisely by the spirit and objectives of the scientists who studied astronomy, light, motion, and other phenomena.

Thus this ratio of ratios, or cross ratio as it is called, is a projective invariant. This is a very surprising fact. It does not matter where the points A, B, C, and D lie on the line l or which points are labeled A, B, C, or D. The cross ratio of the lengths they determine and the cross ratio of the corresponding lengths in the section will be the same.

“to free geometry from the hieroglyphics of analysis.”

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

In order to seek truth it is necessary once in the course of our life to doubt as far as possible all things. DESCARTES

At the end of his course of study he concluded that all his education had advanced him only to the point of discovering man’s ignorance.

Descartes is very entertaining

Descartes decided to learn some things that were not in books.

Convinced that the knowledge he had acquired in school was either unreliable or worthless, Descartes swept away all opinions, prejudices, dogmas, pronouncements of authorities, and, so he believed, preconceived notions.

He would begin, in other words, with unquestionable, self-evident truths. The next principle of his method was to break down larger problems into smaller ones; he would proceed from the simple to the complex.

After much critical reflection he decided that he was sure of the following truths: (a) I think, therefore I am. (b) Each phenomenon must have a cause, (c) An effect cannot be greater than the cause. (d) The mind has within it the ideas of perfection, space, time, and motion.

Descartes also deplored that the methods of Euclidean geometry were exceedingly diverse and specialized and did not allow for general applicability.

I have resolved to quit only abstract geometry, that is to say, the consideration of questions which serve only to exercise the mind, and this, in order to study another kind of geometry, which has for its object the explanation of the phenomena of nature.

Goal: read everything Descartes wrote

Whereas geometry contained the truth about the universe, algebra offered the science of method. It is, incidentally, somewhat paradoxical that great thinkers should be enamored of ideas which mechanize thought. Of course, their goal is to get at more difficult problems, as indeed they do.

The great merit of the idea conceived by Fermat and Descartes is that it permits us to represent a curve algebraically and, as we shall see later, learn much about the curve by working with the equation. But another value of their idea, hardly secondary in importance, is that any equation in x and y determines a curve.

Four-dimensional geometry is entirely a creation of the mind; it is a geometry without pictures.

This brief presentation of four-dimensional geometry may give some further indication of the direction in which scientific thought has been moving with the aid of mathematics. Copernicus asked the world to accept a theory of planetary motions which violated some sense impressions for the sake of a better mathematical account. The utilization of a four-dimensional geometry which has no sensuous or visual content means complete reliance upon the mind.

In other words, Descartes and Fermat made possible the algebraic representation and the study by algebraic means of the various objects and paths of interest to scientists. In addition, algebra supplies quantitative knowledge. This method of working with curves and surfaces is so basic in science that Descartes and Fermat may very well be called the founders of mathematical physics.

Part of Descartes’ greatness and perhaps the largest part of his contribution was his vision of what his method accomplished; he said he had “reduced physics to mathematics.”

SCOTT, J. F.: The Scientific Work of René Descartes, Taylor and Francis, Ltd., London, 1952.

THE SIMPLEST FORMULAS IN ACTION

When you can measure what you are talking about and express it in numbers, you know something about it. LORD KELVIN

Bacon criticizes the Greeks. He says that the interrogation of nature should be pursued not to delight scholars but to serve man. It is to relieve suffering, to better the mode of life, and to increase happiness. Let us put nature to use.

Modern science owes its origins and present flourishing state to a new scientific method which was fashioned almost entirely by Galileo Galilei. Galileo’s method is doubly important to us because, as we shall see, it assigned a major role to mathematics.

THE SCIENTIFIC METHOD OF GALILEO

At the age of 23 when his application for a teaching position at the University of Bologna was rejected because he did not seem worthy of an appointment, he accepted a professorship of mathematics at Pisa.

Galileo was made to feel uncomfortable, and left in 1592 to accept the position of professor of mathematics at the University of Padua.

To the medievalists who kept repeating Aristotle and debating the meaning of his works, Galileo addressed the criticism that knowledge comes from observation and not from books.

He tried to discard the relatively unimportant and nonessential, and here he showed genius, for, as any card player knows, to recognize what to discard is wisdom.

Galileo was fully conscious of what he had accomplished. He says toward the end of his Two New Sciences, “So that we may say the door is now opened, for the first time, to a new method fraught with numerous and wonderful results which in future years will command the attention of other minds.”

But others were also aware of Galileo’s greatness. The seventeenth-century philosopher Thomas Hobbes said of Galileo, “He has been the first to open to us the door to the whole realm of Physics.

The scholars who fashioned modern science, Descartes, Galileo, and Newton, approached the study of nature as mathematicians.

To express the physical principles in the manner he regarded as significant, Galileo introduced a new mathematical concept, the extremely important concept of a function.

Galileo not only formulated the general program for science, but he put it into effect. And here, too, he showed immense wisdom.

As we noted earlier in this chapter, Galileo thought as a mathematician, and he began his work by idealizing the problem he set out to solve.