Mathematics for the Nonmathematician

A HISTORICAL ORIENTATION

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

LE BOVIER DE FONTENELLE

Our first objective will be to gain some historical perspective on the subject of mathematics. Although the logical development of mathematics is not markedly different from the historical, there are nevertheless many features of mathematics which are revealed by a glimpse of its history rather than by an examination of concepts, theorems, and proofs. Thus we may learn what the subject now comprises, how the various branches arose, and how the character of the mathematical contributions made by various civilizations was conditioned by these civilizations. This historical survey may also help us to gain some provisional understanding of the nature, extent, and uses of mathematics. Finally, a preview may help us to keep our bearings. In studying a vast subject, one is always faced with the danger of getting lost in details. This is especially true in mathematics, where one must often spend hours and even days in seeking to understand some new concepts or proofs.

2–2 MATHEMATICS IN EARLY CIVILIZATIONS

Aside possibly from astronomy, mathematics is the oldest and most continuously pursued branch of human thought. Moreover, unlike science, philosophy, and social thought, very little of the mathematics that has ever been created has been discarded. 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. These facts recommend that we go back to the very origins of mathematics.

As we examine the early civilizations, one remarkable fact emerges immediately. Though there have been hundreds of civilizations, many with great art, literature, philosophy, religion, and social institutions, very few possessed any mathematics worth talking about. Most of these civilizations hardly got past the stage of being able to count to five or ten.

In some of these early civilizations a few steps in mathematics were taken. In prehistoric times, which means roughly before 4000 B.C., several civilizations at least learned to think about numbers as abstract concepts. That is, they recognized that three sheep and three arrows have something in common, a quantity called three, which can be thought about independently of any physical objects. Each of us in his own schooling goes through this same process of divorcing numbers from physical objects. The appreciation of “number” as an abstract idea is a great, and perhaps the first, step in the founding of mathematics.

Another step was the introduction of arithmetical operations. It is quite an idea to add the numbers representing two collections of objects in order to arrive at the total instead of counting the objects in the combined collections. Similar remarks apply to subtraction, multiplication, and division. The early methods of carrying out these operations were crude and complicated compared with ours, but the ideas and the applications were there.

Only a few ancient civilizations, Egypt, Babylonia, India, and China, possessed what may be called the rudiments of mathematics. The history of mathematics, and indeed the history of Western civilization, begins with what occurred in the first two of these civilizations. The role of India will emerge later, whereas that of China may be ignored because it was not extensive and moreover had no influence on the subsequent development of mathematics.

Our knowledge of the Egyptian and Babylonian civilizations goes back to about 4000 B.C. The Egyptians occupied approximately the same region that now constitutes modern Egypt and had a continuous, stable civilization from ancient times until about 300 B.C. The term “Babylonian” includes a succession of civilizations which occupied the region of modern Iraq. Both of these peoples possessed whole numbers and fractions, a fair amount of arithmetic, some algebra, and a number of simple rules for finding the areas and volumes of geometrical figures. These rules were but the incidental accumulations of experience, much as people learned through experience what foods to eat. Many of the rules were in fact incorrect but good enough for the simple applications made then. For example, the Egyptian rule for finding the area of a circle amounts to using 3.16 times the square of the radius; that is, their value of π was 3.16. This value, though not accurate, was even better than the several values the Babylonians used, one of these being 3, the value found in the Bible.

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 gift of the Nile. This bit of history is a partial truth. The redetermination of boundaries was undoubtedly an application, but geometry existed in Egypt long before the date of 1400 B.C. mentioned by Herodotus for its origin. Herodotus would have been more accurate to say that Egypt is a gift of the Nile, for it is true today as it was then that the only fertile land in Egypt is that along the Nile; and this because the river deposits good soil on the land as it overflows.

Applications of geometry, simple and crude as they were, did play a large role in Egypt and Babylonia. Both peoples were great builders. The Egyptian temples, such as those at Karnak and Luxor, and the pyramids still appear to be admirable engineering achievements even in this age of skyscrapers. The Babylonian temples, called ziggurats, also were remarkable pyramidal structures. The Babylonians were, moreover, highly skilled irrigation engineers, who built a system of canals to feed their hot dry lands from the Tigris and Euphrates rivers.

Perhaps a word of caution is necessary with respect to the pyramids. Because these are impressive structures, some writers on Egyptian civilization have jumped to the conclusion that the mathematics used in the building of pyramids must also have been impressive. These writers point out that the horizontal dimensions of any one pyramid are exactly of the same length, the sloping sides all make the same angle with the ground, and the right angles are right. However, not mathematics but care and patience were required to obtain such results. A cabinetmaker need not be a mathematician.

Mathematics in Egypt and Babylonia was also applied to astronomy. Of course, astronomy was pursued in these ancient civilizations for calendar reckoning and, to some extent, for navigation. The motions of the heavenly bodies give us our fundamental standard of time, and their positions at given times enable ships to determine their location and caravans to find their bearings in the deserts. Calendar reckoning is not only a common daily and commercial need, but it fixes religious holidays and planting times. In Egypt it was also needed to predict the flood of the Nile, so that farmers could move property and cattle away beforehand.

It is worthy of note that by observing the motion of the sun, the Egyptians managed to ascertain that the year contains 365 days. There is a conjecture that the priests of Egypt knew that 365 was a more accurate figure but kept the knowledge secret. The Egyptian calendar was taken over much later by the Romans and then passed on to Europe. The Babylonians, by contrast, developed a lunar calendar. Since the duration of the month as measured from new moon to new moon varies from 29 to 30 days, the twelve-month year adopted by the Babylonians did not coincide with the year of the seasons. Hence the Babylonians added extra months, up to a total of seven, in every 19-year cycle. This scheme was also adopted by the Hebrews.

Astronomy served not only the purposes just described, but from ancient times until recently it also served astrology. In ancient Babylonia and Egypt the belief was widespread that the moon, the planets, and the stars directly influenced and even controlled affairs of the state. This doctrine was gradually extended and later included the belief that the health and welfare of the individual were also subject to the will of the heavenly bodies. Hence it seemed reasonable that by studying the motions and relative positions of these bodies man could determine their influences and even predict his future.

When one compares Egyptian and Babylonian accomplishments in mathematics with those of earlier and contemporary civilizations, one can indeed find reason to praise their achievements. But judged by other standards, Egyptian and Babylonian contributions to mathematics were almost insignificant, although these same civilizations reached relatively high levels in religion, art, architecture, metallurgy, chemistry, and astronomy. 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 zig-gurats. 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 results, was not even envisioned. Experience recommended their procedures and rules, and with this support they were content. Egyptian and Babylonian mathematics is best described as empirical and hardly deserves the appellation mathematics in view of what, since Greek times, we regard as the chief features of the subject. Some flesh and bones of concrete mathematics were there, but the spirit of mathematics was lacking.

The lack of interest in theoretical or systematic knowledge is evident in all activities of these two civilizations. The Egyptians and Babylonians must have noted the paths of the stars, planets, and moon for thousands of years. Their calendars, as well as tables which are extant, testify to the scope and accuracy of these observations. But no Egyptian or Babylonian strove, so far as we know, to encompass all these observations in one major plan or theory of heavenly motions. Nor does one find any other scientific theory or connected body of knowledge.

2–3 THE CLASSICAL GREEK PERIOD

We have seen so far that mathematics, initiated in prehistoric times, struggled for existence for thousands of years. It finally obtained a firm grip on life in the highly congenial atmosphere of Greece. This land was invaded about 1000 B.C. by people whose origins are not known. By about 600 B.C. these people occupied not only Greece proper but many cities in Asia Minor on the Mediterranean coast, islands such as Crete, Rhodes, and Samos, and cities in southern Italy and Sicily. Though all of these areas bred famous men, the chief cultural center during the classical period, which lasted from about 600 B.C. to 300 B.C., was Athens.

Greek culture was not entirely indigenous. The Greeks themselves acknowledge their indebtedness to the Babylonians and especially to the Egyptians. Many Greeks traveled in Egypt and in Asia Minor. Some went there to study. Nevertheless, what the Greeks created differs as much from what they took over from the Egyptians and Babylonians as gold differs from tin. Plato was too modest in his description of the Greek contribution when he said, “Whatever we Greeks receive we improve and perfect.” The Greeks not only made finished products out of the raw materials imported from Egypt and Babylonia, but they created totally new branches of culture. Philosophy, pure and applied sciences, political thought and institutions, historical writings, almost all our literary forms (except fictional prose), and new ideals such as the freedom of the individual are wholly Greek contributions.

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, though in this case we need not heed Virgil’s injunction to fear such benefactions. But how did the Greeks plan to employ reason in mathematics? Whereas the Egyptians and Babylonians were content to pick up scraps of useful information through experience or trial and error, 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 decided to concentrate on these important concepts. Moreover, they were explicit about their intention to treat general abstract concepts rather than particular physical realizations. Thus they would consider the ideal circle rather than the boundary of a field or the shape of a wheel. They then observed that certain facts about these concepts are both obvious and basic. 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. Their next idea was to apply reasoning, with these facts as premises, and to use only the most reliable methods of reasoning man possesses. If the reasoning were successful, it would produce new knowledge. Also, since they were to reason about general concepts, their conclusions would apply to all objects of which the concepts were representative. Thus if they could prove that the area of a circle is π times the square of the radius, this fact would apply to the area of a circular field, the floor area of a circular temple, and the cross section of a circular tree trunk. Such reasoning about general concepts might not only produce knowledge of hundreds of physical situations in one proof, but there was always the chance that reasoning would produce knowledge which experience might never suggest. All these advantages the Greeks expected to derive from reasoning about general concepts on the basis of evident reliable facts. A neat plan, indeed!

It is perhaps already clear that the Greeks possessed a mentality totally different from that of the Egyptians and Babylonians. They reveal this also in the plans they had for the use of mathematics. The application of arithmetic and algebra to the computation of interest, taxes, or commercial transactions, and of geometry to the computation of the volumes of granaries was as far from their minds as the most distant star. As a matter of fact, their thoughts were on the distant stars. 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.

Just what did the Greeks seek in probing nature? They sought no material gain and no power over nature; they sought merely to satisfy their minds. Because they enjoyed reasoning and because nature presented the most imposing challenge to their understanding, the Greeks undertook the purely intellectual study of nature. Thus the Greeks are the founders of science in the true sense.

The Greek conception of nature was perhaps even bolder than their conception of mathematics. Whereas earlier and later civilizations viewed nature as capricious, arbitrary, and terrifying, and succumbed to the belief that magic and rituals would propitiate mysterious and feared forces, the Greeks dared to look nature in the face. 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 Greek mind rejected traditional doctrines, supernatural causes, superstitions, dogma, authority, and other such trammels on thought and undertook to throw the light of reason on the processes of nature. In seeking to banish the mystery and seeming arbitrariness of nature and in abolishing belief in dreaded forces, the Greeks were pioneers.

For reasons which will become clearer in a later chapter, the Greeks favored geometry. By 300 B.C., Thales, Pythagoras and his followers, Plato’s disciples, notably Eudoxus, and hundreds of other famous men had built up an enormous logical structure, most of which Euclid embodied in his Elements. This is, of course, the geometry we still study in high school. Though they made some contributions to the study of the properties of numbers and to the solution of equations, almost all of their work was in geometric form, and so there was no improvement over the Egyptians and Babylonians in the representation of, and calculation with, numbers or in the symbolism and techniques of algebra. For these contributions the world had to wait many more centuries. But the vast development in geometry exerted an enormous influence in succeeding civilizations and supplied the inspiration for mathematical activity in civilizations which might otherwise never have acquired even the very concept of mathematics.

The Greek accomplishments in mathematics had, in addition, the broader significance of supplying the first impressive evidence of the power of human reason to deduce new truths. In every culture influenced by the Greeks, this example inspired men to apply reason to philosophy, economics, political theory, art, and religion. 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. Modern man as well as the ancient Greeks learned from the Euclidean document how exact reasoning should proceed, how to acquire facility in it, and how to distinguish correct from false reasoning. Although many people depreciate this value of mathematics, it is interesting nevertheless that when these people seek to offer an excellent example of reasoning, they inevitably turn to mathematics.

This brief discussion of Euclidean geometry may show that the subject is far from being a relic of the dead past. It remains important as a stepping-stone in mathematics proper and as a paradigm of reasoning. With their gift of reason and with their explicit example of the power of reason, the Greeks founded Western civilization.

2–4 THE ALEXANDRIAN GREEK PERIOD

The intellectual life of Greece was altered considerably when Alexander the Great conquered Greece, Egypt, and the Near East. Alexander decided to build a new capital for his vast empire and founded the city in Egypt named after him. The center of the new Greek world became Alexandria instead of Athens. Moreover, Alexander made deliberate efforts to fuse Greek and Near Eastern cultures. Consequently, the civilization centered at Alexandria, though predominantly Greek, was strongly influenced by Egyptian and Babylonian contributions. This Alexandrian Greek civilization lasted from about 300 B.C. to 600 A.D.

The mixture of the theoretical interests of the Greeks and the practical outlook of the Babylonians and Egyptians is clearly evident in the mathematical and scientific work of the Alexandrian Greeks. The purely geometric investigations of the classical Greeks were continued, and two of the most famous Greek mathematicians, Apollonius and Archimedes, pursued their studies during the Alexandrian period. In fact, Euclid also lived in Alexandria, but his writings reflect the achievements of the classical period. For practical applications, which usually require quantitative results, the Alexandrians revived the crude arithmetic and algebra of Egypt and Babylonia and used these empirically founded tools and procedures, along with results derived from exact geometrical studies. There was some progress in algebra, but what was newly created by men such as Nichomachus and Diophantus was still short of even the elementary methods we learn in high school.

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.

During the centuries in which the Alexandrian civilization flourished, the Romans grew strong, and by the end of the third century B.C. they were a world power. After conquering Italy, the Romans conquered the Greek mainland and a number of Greek cities scattered about the Mediterranean area. Among these was the famous city of Syracuse in Sicily, where Archimedes spent most of his life, and where he was killed at the age of 75 by a Roman soldier. According to the account given by the noted historian Plutarch, the soldier shouted to Archimedes to surrender, but the latter was so absorbed in studying a mathematical problem that he did not hear the order, whereupon the soldier killed him.

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. 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. As the great philosopher Alfred North Whitehead remarked, “No Roman ever lost his life because he was absorbed in the contemplation of a mathematical diagram.”

Indirectly as well as directly, the Romans brought about the destruction of the Greek civilization at Alexandria, directly by conquering Egypt and indirectly by seeking to suppress Christianity. The adherents to this new religious movement, though persecuted cruelly by the Romans, increased in number while the Roman Empire grew weaker. In 313 A.D. Rome legalized Christianity and, under the Emperor Theodosius (379–395), adopted it as the official religion of the empire. But even before this time, and certainly after it, the Christians began to attack the cultures and civilizations which had opposed them. By pillage and the burning of books, they destroyed all they could reach of ancient learning. Naturally the Greek culture suffered, and many works wiped out in these holocausts are now lost to us forever.

The final destruction of Alexandria in 640 A.D. was the deed of the Arabs. The books of the Greeks were closed, never to be reopened in this region.

2–5 THE HINDUS AND ARABS

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. They conquered all the land around the Mediterranean Sea. In the Near East they took over Persia and penetrated as far as India. In southern Europe they occupied Spain, southern France, where they were stopped by Charles Martel, southern Italy and Sicily. Only the Byzantine or Eastern Roman Empire was not subdued and remained an isolated center of Greek and Roman learning. In rather surprisingly quick time as the history of nations goes, the Arabs settled down and built a civilization and culture which maintained a high level from about 800 to 1200 A.D. Their chief centers were Bagdad in what is now Iraq, and Cordova in Spain. Realizing that the Greeks had created wonderful works in many fields, the Arabs proceeded to gather up and study what they could still find in the lands they controlled. They translated the works of Aristotle, Euclid, Apollonius, Archimedes, and Ptolemy into Arabic. In fact, Ptolemy’s chief work, whose title in Greek meant “Mathematical Collection,” was called the Almagest (The Greatest Work) by the Arabs and is still known by this name. Incidentally, other Arabic words which are now common mathematical terms are algebra, taken from the title of a book written by Al-Khowarizmi, a ninth-century Arabian mathematician, and algorithm, now meaning a process of calculation, which is a corruption of the man’s name.

Though they showed keen interest in mathematics, optics, astronomy, and medicine, the Arabs contributed little that was original. It is also peculiar that, although they had at least some of the Greek works and could therefore see what mathematics meant, their own contributions, largely in arithmetic and algebra, followed the empirical, concrete approach of the Egyptians and Babylonians. They could on the one hand appreciate and critically review the precise, exact, and abstract mathematics of the Greeks while, on the other, offer methods of solving equations which, though they worked, had no reasoning to support them. During all the centuries in which Greek works were in their possession, the Arabs manfully resisted the lures of exact reasoning in their own contributions.

We are indebted to the Arabs not only for their resuscitation of the Greek works but for picking up some simple but useful ideas from India, their neighbor on the East. The Indians, too, had built up some elementary mathematics comparable in extent and spirit with the Egyptian and Babylonian developments. However, after about 200 A.D., mathematical activity in India became more appreciable, probably as a result of contacts with the Alexandrian Greek civilization. The Hindus made a few contributions of their own, such as the use of special number symbols from 1 to 9, the introduction of 0, and the use of positional notation with base ten, that is, our modern method of writing numbers. They also created negative numbers. These ideas were taken over by the Arabs and incorporated in their mathematical works.

Because of internal dissension the Arab Empire split into two independent parts. The Crusades launched by the Europeans and the inroads made by the Turks further weakened the Arabs, and their empire and culture disintegrated.

2–6 EARLY AND MEDIEVAL EUROPE

Thus far Europe proper has played no role in the history of mathematics. The reason is simple. The Germanic tribes who occupied central Europe and the Gauls of western Europe were barbarians. Among primitive civilizations, theirs were primitive indeed. They had no learning, no art, no science, not even a system of writing.

The barbarians were gradually civilized. While the Romans were still successful in holding the regions now called France, England, southern Germany, and the Balkans, the barbarians were in contact with, and to some extent influenced by, the Romans. When the Roman Empire collapsed, the Church, already a strong organization, took on the task of civilizing and converting the barbarians. 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.

2–7 THE RENAISSANCE

Insofar as the history of mathematics is concerned, the Arabs served as the agents of destiny. Trade with the Arabs and such invasions of the Arab lands as the Crusades acquainted the Europeans, who hitherto possessed only fragments of the Greek works, with the vast stores of Greek learning possessed by the Arabs. The ideas in these works excited the Europeans, and scholars set about acquiring them and translating them into Latin. Through another accident of history another group of Greek works came to Europe. We have already noted that the Eastern Roman or Byzantine Empire had survived the Germanic and the Arab aggrandizements. But in the fifteenth century the Turks captured the Eastern Roman Empire, and Greek scholars carrying precious manuscripts fled the region and went to Europe.

We shall leave for a later chapter a fuller account of how the European world was aroused by the renaissance of the novel and weighty Greek ideas, and of the challenge these ideas posed to the European beliefs and way of life.* From the Greeks the Europeans acquired arithmetic, a crude algebra, the vast development of Euclidean geometry, and the trigonometry of Hip-parchus and Ptolemy. Of course, Greek science and philosophy also became known in Europe.

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. They applied Euclidean geometry to create a new system of perspective which permitted them to paint realistically. Specifically, the artists created a new style of painting which enabled them to present on canvas, scenes making the same impression on the eye as the actual scenes themselves. From the work of the artists, the mathematicians derived ideas and problems that led to a new branch of mathematics, projective geometry.

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. 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. Its effects on religion, philosophy, science, and on man’s estimations of his own importance were profound. The heliocentric theory also raised scientific and mathematical problems which were a direct incentive to new mathematical developments.

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. The compass became known to the Europeans and made possible long-range navigation, which the merchants sponsored for the purpose of finding new sources of raw materials and better trade routes. One result was the discovery of America and the consequent influx of new ideas into Europe. The invention of printing and of paper made of rags afforded books in large quantities and at cheap prices, so that learning spread far more than it ever had in any earlier civilizations. The Protestant Revolution stirred debate and doubts concerning doctrines that had been unchallenged for 1500 years. The rise of a merchant class and of free men engaged in labor in their own behalf stimulated an interest in materials, methods of production, and new commodities. All of these needs and influences challenged the Europeans to build a new culture.

2–8 DEVELOPMENTS FROM 1550 TO 1800

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. This is the period in which algebra was built as a branch of mathematics and in which much of the algebra we learn in high school was created. Almost all the great mathematicians of the sixteenth and seventeenth centuries, Cardan, Tartaglia, Vieta, Descartes, Fermat, and Newton, men we shall get to know better later, contributed to the subject. 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.

The next development of consequence, coordinate geometry, came from two men, both interested in method. One was René Descartes. Descartes is perhaps even more famous as a philosopher than as a mathematician, though he was one of the major contributors to our subject. As a youth Descartes was already troubled by the intellectual turmoil of his age. 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. He found the clue to this method in mathematics, and with it constructed the first great modern philosophical system. 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. He found the answer in the use of algebra. Pierre de Fermat’s interest in method was confined to mathematics proper, but he too appreciated the need for more effective ways of working with curves and also arrived at the idea of applying algebra. In this development of coordinate geometry we have one of the remarkable examples of how the times influence the direction of men’s thoughts.

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. These activities and interests created problems of materials, methods of production, quality of the product, and utilization of devices to replace or increase the effectiveness of manpower. The people involved, like the artists, had become aware of Greek mathematics and science and sensed that it could be helpful. And so they too sought to employ this knowledge in their own behalf. Thereby arose a new motive for the study of mathematics and science. 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. Hence mathematicians and scientists turned earnestly to an enlarged program in which both understanding and mastery of nature were to be sought.

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. The plan, perhaps oversimply stated, was that experience and experiment were to supply basic mathematical principles and mathematics was to be applied to these principles to deduce new truths, just as new truths are deduced from the axioms of geometry.

The most pressing scientific problem of the seventeenth century was the study of motion. On the practical side, investigations of the motion of projectiles, of the motion of the moon and planets to aid navigation, and of the motion of light to improve the design of the newly discovered telescope and microscope, were the primary interests. On the theoretical side, the new heliocentric astronomy invalidated the older, Aristotelian laws of motion and called for totally new principles. It was one thing to explain why a ball fell to earth on the assumption that the earth was immobile and the center of the universe, and another to explain this phenomenon in the light of the fact that the earth was rotating and revolving around the sun. 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. The subject matter of mathematics and the power of mathematics expanded so greatly that at the end of the seventeenth century Leibniz could say,

Taking mathematics from the beginning of the world to the time when Newton lived, what he had done was much the better half.

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. This great creation produced universal laws which not only united heaven and earth but revealed a design in the universe far more impressive than man had ever conceived. 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.

We learn in history that the end of the seventeenth century and the eighteenth century were marked by a new intellectual attitude briefly described as the Age of Reason. We are rarely told that this age was inspired by the successes which mathematics, to be sure in conjunction with science, had achieved in organizing man’s knowledge. Infused with the conviction that reason, personified by mathematics, would not only conquer the physical world but could solve all of man’s problems and should therefore be employed in every intellectual and artistic enterprise, the great minds of the age undertook a sweeping reorganization of philosophy, religion, ethics, literature, and aesthetics. The beginnings of new sciences such as psychology, economics, and politics were made during these rational investigations. Our principal intellectual doctrines and outlook were fashioned then, and we still live in the shadow of the Age of Reason.

While these major branches of our culture were being transformed, eighteenth-century scientists continued to win victories over nature. The calculus was soon extended to a new branch of mathematics called differential equations, and this new tool enabled scientists to tackle more complex problems in astronomy, in the study of the action of forces causing motions, in sound, especially musical sounds, in light, in heat (especially as applied to the development of the steam engine), in the strength of materials, and in the flow of liquids and gases. Other branches, which can be merely mentioned, such as infinite series, the calculus of variations, and differential geometry, added to the extent and power of mathematics. The great names of the Bernoullis, Euler, Lagrange, Laplace, d’Alembert, and Legendre, belong to this period.

2–9 DEVELOPMENTS FROM 1800 TO THE PRESENT

During the nineteenth century, developments in mathematics came at an ever increasing rate. Algebra, geometry and analysis, the last comprising those subjects which stem from calculus, all acquired new branches. The great mathematicians of the century were so numerous that it is impractical to list them. We shall encounter some of the greatest of these, Karl Friedrich Gauss and Bernhard Riemann, in our work. We might mention also Henri Poincaré and David Hilbert, whose work extended into the twentieth century.

Undoubtedly the primary cause of this expansion in mathematics was the expansion in science. The progress made in the seventeenth and eighteenth centuries had sufficiently illustrated the effectiveness of science in penetrating the mysteries of the physical world and in giving man control over nature, to cause an all the more vigorous pursuit of science in the nineteenth century. In that century also, science became far more intimately linked with engineering and technology than ever before. Mathematicians, working closely with the scientists as they had since the seventeenth century, were presented with thousands of significant physical problems and responded to these challenges.

Perhaps the major scientific development of the century, which is typical in its stimulation of mathematical activity, was the study of electricity and magnetism. While still in its infancy this science yielded the electric motor, the electric generator, and telegraphy. Basic physical principles were soon expressed mathematically, and it became possible to apply mathematical techniques to these principles, to deduce new information just as Galileo and Newton had done with the principles of motion. In the course of such mathematical investigations, James Clerk Maxwell discovered electromagnetic waves of which the best known representatives are radio waves. A new world of phenomena was thus uncovered, all embraced in one mathematical system. Practical applications, with radio and television as most familiar examples, soon followed.

Remarkable and revolutionary developments of another kind also took place in the nineteenth century, and these resulted from a re-examination of elementary mathematics. 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. So shocking were the implications that even mathematicians refused to take non-Euclidean geometry seriously until the theory of relativity forced them to face the full significance of the creation.

For reasons which we trust will become clearer further on, the devastation caused by non-Euclidean geometry did not shatter mathematics but released it from bondage to the physical world. 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.

Even before the nineteenth century, the rationalistic spirit engendered by the success of mathematics in the study of nature penetrated to the social scientists. They began to emulate the physical scientists, that is, to search for the basic truths in their fields and to attempt reorganization of their subjects on the mathematical pattern. But these attempts to deduce the laws of man and society and to erect sciences of biology, economics, and politics did not succeed, although they did have some indirect beneficial effects.

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, which had already been initiated by mathematicians for various purposes ranging from problems of gambling to the theory of heat and astronomy. These techniques have been remarkably successful and have given some scientific methodology to what were largely speculative domains.

This brief sketch of the mathematics which will fall within our purview may make it clear that mathematics is not a closed book written in Greek times. It is rather a living plant that has flourished and languished with the rise and fall of civilizations. Since about 1600 it has been a continuing development which has become steadily vaster, richer, and more profound. The character of mathematics has been aptly, if somewhat floridly, described by the nineteenth-century English mathematician James Joseph Sylvester.

Mathematics is not a book confined within a cover and bound between brazen clasps, whose contents it needs only patience to ransack; it is not a mine, whose treasures may take long to reduce into possession, but which fill only a limited number of veins and lodes; it is not a soil, whose fertility can be exhausted by the yield of successive harvests; it is not a continent or an ocean, whose area can be mapped out and its contour defined; it is as limitless as the space which it finds too narrow for its aspirations; its possibilities are as infinite as the worlds which are forever crowding in and multiplying upon the astronomer’s gaze; it is incapable of being restricted within assigned boundaries or being reduced to definitions of permanent validity as the consciousness, the life, which seems to slumber in each monad, in every atom of matter, in each leaf and bud and cell and is forever ready to burst forth into new forms of vegetable and animal existence.

Our sketch of the development of mathematics has attempted to indicate the major eras and civilizations in which the subject has flourished, the variety of interests which induced people to pursue mathematics, and the branches of mathematics that have been created. Of course, we intend to investigate more carefully and more fully what these creations are and what values they have furnished to mankind. One fact of history may be noted by way of summary here. Mathematics as a body of reasoning from axioms stems from one source, the classical Greeks. All other civilizations which have pursued or are pursuing mathematics acquired this concept of mathematics from the Greeks. The Arab and Western European were the next civilizations to take over and expand on the Greek foundation. 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. Modern mathematical activity in these five countries and wherever else it is now taking hold was inspired by Western European thought and actually learned by men who studied in Europe and returned to build centers of teaching in their own countries.

2–10 THE HUMAN ASPECT OF MATHEMATICS

One final point about mathematics is implicit in what we have said. We have spoken of problems which gave rise to mathematics, of cultures which emphasized some directions of thinking as opposed to others, and of branches of mathematics, as though all these forces and activities were as impersonal as the force of gravitation. But ideas and thinking are conveyed by people. Mathematics is a human creation. Although most Greeks did believe that mathematics existed independently of human beings as the planets and mountains seem to, and that all that human beings do is discover more and more of the structure, the prevalent belief today is that mathematics is entirely a human product. The concepts, the axioms, and the theorems established are all created by human beings in man’s attempt to understand his environment, to give play to his artistic instincts, and to engage in absorbing intellectual activity.

The lives and activities of the men themselves are also fascinating. While mathematicians produce formulas, no formula produces mathematicians. They have come from all levels of society. The special talent, if there is such, which makes mathematicians has been found in Casanovas and ascetics, among business men and philosophers, among atheists and the profoundly religious, among the retiring and the worldly. Some, like Blaise Pascal and Gauss, were precocious; Évariste Galois was dead at 21, and Niels Hendrik Abel at 27. Others, like Karl Weierstrass and Henri Poincaré, matured more normally and were productive throughout their lives. Many were modest; others extremely egotistical and vain beyond toleration. One finds scoundrels, such as Cardan, and models of rectitude. Some were generous in their recognition of other great minds; others were resentful and jealous and even stole ideas to boost their own reputations. Disputes about priority of discovery abound.

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. Of course, the major historical forces, which we sketched above, limit the actions and influence the outlook of individuals, but we also find in the history of mathematics all the vagaries which he have learned to associate with human beings. Leading mathematicians have failed to recognize bright ideas suggested by younger men, and the authors died neglected. Big men and little men made unsuccessful attempts to solve problems which their successors solved with ease. On the other hand, some supposed proofs offered even by masters were later found to be false. 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. The examples of the blindness of human beings to ideas which later seem simple and obvious furnish fascinating insight into the working of the human mind.

Recognition of the human element in mathematics explains in large measure the differences in the mathematics produced by different civilizations and the sudden spurts made in new directions by virtue of insights supplied by genius. Though no subject has profited as much as mathematics has by the cumulative effect of thousands of workers and results, in no subject is the role of great minds more readily discernible.

EXERCISES

1. Name a few civilizations which contributed to mathematics.

2. What basis did the Egyptians and Babylonians have for believing in their mathematical methods and formulas?

3. Compare Greek and pre-Greek understanding of the concepts of mathematics.

4. What was the Greek plan for establishing mathematical conclusions?

5. What was the chief contribution of the Arabs to the development of mathematics?

6. In what sense is mathematics a creation of the Greeks rather than of the Egyptians and Babylonians?

7. Criticize the statement “Mathematics was created by the Greeks and very little was added since their time.”

Topics for Further Investigation

To write on the following topics use the books listed under Recommended Reading.

1. The mathematical contributions of the Egyptians or Babylonians.

2. The mathematical contributions of the Greeks.

CHAPTER 2