Mathematics for the Nonmathematician (our commented edition)


In mathematics I can report no deficience, except it be that men do not sufficiently understand the excellent use of the Pure Mathematics....


One can wisely doubt whether the study of mathematics is worth while and can find good authority to support him. 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.

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.
            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."
            Even the distinguished seventeenth-century contributor to mathematics, Blaise Pascal, decided after studying mankind that the pure sciences were not suited to it. 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. Many other great men, for example, the poet Johann Wolfgang Goethe and the historian Edward Gibbon, have felt likewise and have not hesitated to express themselves. And so the student who dislikes the subject can claim to be in good, if not living, company.
            In view of the support he can muster from authorities, the student may well inquire why he is asked to learn mathematics. ls 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 needs mathematics so much? Perhaps the subject got into the curriculum by mistake, and no one has taken the trouble to throw it out. Certainly the student is justified in asking his teacher the very question which Mephistopheles put to Faust:

Is it right, I ask, is it even prudence,
To bare thyself and bore the students?

            Perhaps we should begin our answers to these questions by pointing out that the men we cited as disliking or disapproving of mathematics were really exceptional. In the great periods of culture which preceded the present one, almost all educated people valued mathematics. The Greeks, who created the modern concept of mathematics, spoke unequivocally for its importance. During the Middle Ages and in the Renaissance, mathematics was never challenged as one of the most important studies. The seventeenth century was aglow not only with mathematical activity but with popular interest in the subject. We have the instance of Samuel Pepys, so much attracted by the rapidly expanding influence of mathematics that at the age of thirty he could no longer tolerate his own ignorance and begged to learn the subject. He began, incidentally, with the multiplication table, which he subsequently taught to his wife. In 1681 Pepys was elected president of the Royal Society, a post later held by Isaac Newton.
            In perusing eighteenth-century literature, one is struck by the fact that the journals which were on the level of our Harper's and the Atlantic Monthly contained mathematical articles side by side with literary articles. The educated man and woman of the eighteenth century knew the mathematics of their day, felt obliged to be an courant with all important scientific developments, and read articles on them much as modern man reads articles on politics. These people were as much at home with Newton's mathematics and physics as with Pope's poetry.
            The vastly increased importance of mathematics in our time makes it all the more imperative that the modern person know something of the nature and role of mathematics. It is true that the role of mathematics in our civilization is not always obvious, and the deeper and more complex modern applications are not readily comprehended even by specialists. But the essential nature and accomplishments of the subject can still be understood.
            Perhaps we can see more easily why one should study mathematics if we take a moment to consider what mathematics is. Unfortunately the answer cannot be given in a single sentence or a single chapter. The subject has many facets or, some might say, is Hydra-headed. One can look at mathematics as a language, as a particular kind of logical structure, as a body of knowledge about number and space, as a series of methods for deriving conclusions, as the essence of our knowledge of the physical world, or merely as an amusing intellectual activity. Each of these features would in itself be difficult to describe accurately in a brief space.
            Because it is impossible to give a concise and readily understandable definition of mathematics, some writers have suggested, rather evasively, that mathematics is what mathematicians do. But mathematicians are human beings,and most of the things they do are uninteresting and some, embarrassing to relate. The only merit in this proposed definition of mathematics is that it points up the fact that mathematics is a human creation.
            A variation on the above definition which promises more help in understanding the nature, content, and values of mathematics, is that mathematics is what mathematics does. If we examine mathematics from the standpoint of what it is intended to and does accomplish, we shall undoubtedly gain a truer and clearer picture of the subject.
            Mathematics is concerned primarily with what can be accomplished by reasoning. And here we face the first hurdle. Why should one reason? It isnot a natural activity for the human animal. It is clear that one does not need reasoning to learn how to eat or to discover what foods maintain life. Manknew how to feed, clothe, and house himself millenniums before mathematics existed. Getting along with the opposite sex is an art rather than a sciencemastered by reasoning. One can engage in a multitude of occupations and even climb high in the business and industrial world without much use ofreasoning and certainly without mathematics. One's social position is hardly elevated by a display of his knowledge of trigonometry. In fact, civilizationsin which reasoning and mathematics played no role have endured and even flourished. If one were willing to reason, he could readily supply evidence toprove that reasoning is a dispensable activity. Those who are opposed to reasoning will readily point out other methodsof obtaining knowledge. Most people are in fact convinced that their senses are really more than adequate. The very common assertion "seeing is believ-ing" expresses the common reliance upon the senses. But everyone should recognize that the senses are limited and often fallible and, even where ac-curate, must be interpreted. Let us consider, as an example, the sense of sight. L

4 wmr MATHENIATICSPHow big is the sun? Our eyes tell us that it is about as large as a rubber ball. This then is what we should believe. On the other hand, we do not see theair around us, nor for that matter can we feel, touch, smell, or taste it. Hence we should not believe in the existence of air.To consider a somewhat more complicated situation, suppose a teacher should hold up a fountain pen and ask, What is it? A student coming fromsome primitive society might call it a shiny stick, and indeed this is what the eyes see. Those who call it a fountain pen are really calling upon educationand experience stored in their minds. 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." -Every day we see the sun where it is not. For about five minutes before what we call sunset, the sun is actually below the geometrical horizon andshould therefore be invisible. But the rays of light from the sun curve toward us as they travel in the eartlfr atmosphere, and the observer at P (Fig. 1-1)not only "sees" the sun but thinks the light is coming from the direction O'P. Hence he believes the sun is in that direction.Apparent position U I _ K, E .,,, M 0/ `c ~ 3;Y .v 4# `i* " ?lA*' V *ii`i `i `E~: `ii:l":"`:` ` il . ' EA* >>*. , * >> E,;_.I `,_True position . `%" 4.;;iZ--*`;Y'#IE*Z`<.--;fiQ"`_$` ` i o " " ` ,;.Iffi` lX _,.=J*:'; i i i -Fig. 1-1. Deviation of a ray by the earth's atmosphere. The senses are obviously helpless in obtaining some kinds of knowledge,such as the distance to the sun, the size of the earth, the speed of a bullet (unless one wishes to feel its velocity), the temperature of the sun, the predic-tion of eclipses, and dozens of other facts. If the senses are inadequate, what about experimentation or, in simple cases,measurement? One can and in fact does learn a great deal by such means. But suppose one wants to find a very simple quantity, the area of a rectangle. Toobtain it by measurement, one could lay off unit squares to cover the area and then count the number of squares. It is at least a little simpler to measure thelengths of the sides and then use a formula obtained by reasoning, namely, that the area is the product of length and width. In the only slightly more com-plicated problem of determining how high a projectile will go, we should certainly not consider traveling with the projectile.As to experimentation, let us consider a relatively simple problem of modern technology. One wishes to build a bridge across a river. How long and

wi-rv MArm;MArrcs? 5how thick should the many beams be? What shape should the bridge take? lf it is to be supported by cables, how long and how thick should these be? Ofcourse one could arbitrarily choose a number of lengths and thicknesses for the beams and cables and build the bridge. In this event, it would only be fair thatthe experimenter be the first to cross this bridge. It may be clear from this brief discussion that the senses, measurement,and experimentation, to consider three alternative ways of acquiring knowledge, are by no means adequate in a variety of situations. Reasoning is essential.The lawyer, the doctor, the scientist, and the engineer employ reasoning daily to derive knowledge that would otherwise not be obtainable or perhaps obtain-able only at great expense and effort. Mathematics more than any other human endeavor relies upon reasoning to produce knowledge. `One may be willing to accept the fact that mathematical reasoning is an effective procedure. But just what does mathematics seek to accomplish withits reasoning? The primary objective of all mathematical work is to help man study nature, and in this endeavor mathematics cooperates with science. Itmay seem, then, that mathematics is merely a useful tool and that the real pursuit is science. We shall not attempt at this stage to separate the roles ofmathematics and science and to evaluate the relative merits of their contributions. We shall simply state that their methods are different and that mathe-matics is at least an equal partner with science. We shall see later how observations of nature are framed in statementscalled axioms. Mathematics then discloses by reasoning secrets which nature may never have intended to reveal. The determination of the pattern of mo-tion of celestial bodies, the discovery and control of radio waves, the understanding of molecular, atomic, and nuclear structures, and the creation ofartificial satellites are a few basically mathematical achievements. Mathematical formulation of physical data and mathematical methods of deriving new con-clusions are today the substratum in all investigations of nature. The fact that mathematics is of central importance in the study of naturereveals almost immediately several values of this subject. The first is the practical value. The construction of bridges and skyscrapers, the harnessing of thepower of water, coal, electricity, and the atom, the effective employment of light, sound, and radio in illumination, communication, navigation, and evenentertainment, and the advantageous employment of chemical knowledge in the design of materials, in the production of useful forms of oil, and in medi-cine are but a few of the practical achievements already attained. And the future promises to dwarf the past.However, material progress is not the most compelling reason for the study of nature, nor have practical results usually come about from investigations sodirected. In fact, to overemphasize practical values is to lose sight of the greater significance of human thought. The deeper reason for the study ofnature is to try to understand the ways of nature, that is, to satisfy sheer in6 WHY Marummrics?tellectual curiosity. Indeed, to ask disinterested questions about nature is one of the distinguishing marks of mankind. In all civilizations some people at leasthave tried to answer such questions as: How did the universe come about? How old is the universe and the earth in particular? I-Iow large are the sunand the earth? Is man an accident or part of a larger design? Will the solar system continue to function or will the earth some day fall into the sun? Whatis light? Of course, not all people are interested in such questions. Food, shelter, sex, and television are enough to keep many happy. But others, awareof the pervasive natural mysteries, are more strongly obsessed to resolve them than any business man is to acquire wealth and power.Beyond improvement in the material life of man and beyond satisfaction of intellectual curiosity, the study of nature offers intangible values of anothersort, especially the abolition of fear and terror and their replacement by a deep, quiet satisfaction in the ways of nature. To the uneducated and to thoseuninitiated in the world of science, many manifestations of nature have appeared to be agents of destruction sent by angry gods. Some of the beliefs inancient and even medieval Europe may be of special interest in view of what happened later. The sun was the center of all life. As winter neared and thedays became shorter, the people believed that a battle between the gods of light and darkness was taking place. Thus the god Wodan was supposed to beriding through heaven on a white horse followed by demons, all of whom sought every opportunity to harm people. VVhen, however, the days began tolengthen and the sun began to show itself higher in the sky each day, the people believed that the gods of light had won. They ceased all work and cele-brated this victory. Sacrifices were oifered to the benign gods. Symbols of fertility such as fruit and nuts, whose growth is, of course, aided by the sun,were placed on the altars. To symbolize further the desire for light and the joy in light, a huge log was placed in the fire to burn for twelve days, andcandles were lit to heighten the brightness. The beliefs and superstitions which have been attached to events we takein stride are incredible to modern man. An eclipse of the sun, a threat to the continuance of the light and heat which causes crops to grow, meant that theheavenly body was being swallowed up by a dragon. Many Hindu people believe today that a demon residing in the sky attacks the sun once in a whileand 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 clearthat these rituals were the effective agent and so had to be pursued on every such occasion. In addition, special magic potions drunk during eclipses insuredhealth, happiness, and wisdom. To primitive peoples of the past, thunder, lightning, and storms werepunishments visited by the gods on people who had apparently sinned in some way. The stories in the Old Testament of the flood and of the destruction ofSodom and Gomorrah by fire and brimstone are examples of such acts of

WHY MAT1-1EMAr1cs? 7wrath by the God of the Hebrews. Hence there was continual concern and even dread about what the gods might have in mind for helpless humans. Theonly recourse was to propitiate the divine powers, so that they would bring good fortune instead of evil.Fears, dread, and superstitions have been eliminated, at least in our Western civilization, by just those intellectually curious people who have studied na-ture's mighty displays. 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. Man has emerged as theproud possessor of knowledge which has enabled him to view nature calmly and objectively. An eclipse of the sun occurring on schedule is no longer anoccasion for trembling but for quiet satisfaction that we know nature's ways. We breathe freely, knowing that nature will not be willful or capricious.Indeed, man has been remarkably successful in his study of nature. History is said to repeat itself, but, in general, the circumstances of the supposed repe-tition are not the same as those of the earlier occurrence. As a consequence, the history of man has not been too effective a guide for the future. Nature iskinder. When nature repeats herself, and she does so constantly, the repetitions are exact facsimiles of previous events, and therefore man can anticipate na-ture's behavior and be prepared for what will take place. We have learned to recognize the patterns of nature and we can speak today of the uniformity ofnature and delight in the regularity of her behavior. The successes of mathematics in the study of inanimate nature have in-spired in recent times the mathematical study of human nature. Mathematics has not only contributed to the very practical institutions such as banking,insurance, pension systems, and the like, but it has also supplied some substance, spirit, and methodology to the infant sciences of economics, politics, and so-ciology. Number, quantitative studies, and precise reasoning have replaced vague, subjective, and ineffectual speculations and have already given evidenceof greater values to come. As man turns to thoughts about himself and his fellow man, other questionsoccur to him which are as fundamental as any he can ask. Why is man born? What purposes does he serve or should he serve? What future awaits him?The knowledge acquired about our physical universe has profound implications for the origin and role of man. Moreover, as mathematics and science haveamassed increasing knowledge and power, they have gradually encompassed the biological and psychological sciences, which in turn have shed further lighton man's physical and mental life. Thus it has come about that mathematics and science have profoundly affected philosophy and religion.Perhaps the most profound questions in the realm of philosophy are, What is truth and how does man acquire it? Though we have no final answer tothese questions, the contribution of mathematics toward this end is paramount.

8 WHY MATHEMATICS?For two millenniums mathematics was the prime example of truths man had unearthed. Hence all investigations of the problem of acquiring truths neces-sarily reckoned with mathematics. Though some startling developments in the nineteenth century altered completely our understanding of the nature ofmathematics, the effectiveness of the subject, especially in representing and analyzing natural phenomena, has still kept mathematics the focal point of allinvestigations into the nature of knowledge. Not the least significant aspect of this value of mathematics has been the insight it has given us into the ways andpowers of the human mind. Mathematics is the supreme and most remarkable example of the mind's power to cope with problems, and as such it is worthyof study. Among the values which mathematics offers are im services to the arts.Most people are inclined to believe that the arts are independent of mathematics, but we shall see that mathematics has fashioned major styles of paintingand architecture, and the service mathematics renders to music has not only enabled man to understand it, but has spread its enjoyment to all comers ofour globe. Practical, scientific, philosophical, and artistic problems have caused mento 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 suchaffords the pleasures which all the arts afford. This last statement may come as a shock to people who are used to the conventional concept of the true artsand mentally contrast these with mathematics to the detriment of the latter. But the average person has not thought through what the arts really are andwhat they offer. All that many people actually see in painting, for example, are familiar scenes and perhaps bright colors. These qualities, however, are notthe ones which make painting an art. The real values must be learned, and a genuine appreciation of art calls for much study.Nevertheless, we shall not insist on the aesthetic values of mathematics. It may be fairer to rest on the position that just as there are tone-deaf and color-blind people, so may there be some who temperamentally are intolerant of cold argumentation and the seemingly overfine distinctions of mathematics.To many people, mathematics offers intellectual challenges, and it is well known that such challenges do engross humans. Games such as bridge, cross-word puzzles, and magic squares are popular. Perhaps the best evidence is the attraction of puzzles such as the following: A wolf, a goat, and cabbage areto 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 wolfdoes 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 theycross so that no woman is in the company of a man unless her husband is also present? Such puzzles go back to Greek and Roman times. The mathematicianTartaglia, who lived in the sixteenth century, tells us that they were afterdinner amusements.

WHY MATHEMATICS? 9People do respond to intellectual challenges, and once one gets a slight start in mathematics, he encounters these in abundance. In view of the addi-tional values to be derived from the subject, one would expect people to spend time on mathematical problems as opposed to the more superficial, and in someinstances cheap, games which lack depth, beauty, and importance. The tantalizing and compelling pursuit of mathematical problems offers mental absorp-tion, peace of mind amid endless challenges, repose in activity, battle without conflict, and the beauty which the ageless mountains present to senses tried bythe kaleidoscopic rush of events. The appeal offered by the detachment and objectivity of mathematical reasoning is superbly described by Bertrand Rus-sell. Remote from human passions, remote even from the pitiful facts of nature, thegenerations 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 canescape from the dreary exile of the actual world. The creation and contemplation of mathematics offer such values.Despite all these arguments for the study of mathematics, the reader may have justifiable doubts. The idea that thinking about numbers and figures leadsto deep and powerful conclusions which influence almost all other branches of thought may seem incredible. The study of numbers and geometrical figuresmay not seem a sufficiently attractive and promising enterprise. Not even the founders of mathematics envisioned the potentialities of the subject.So we start with some doubts about the worth of our enterprise. We could encourage the reader with the hackneyed maxim, nothing ventured,nothing gained. We could call to his attention the daily testimony to the power of mathematics offered by almost every newspaper and journal. Butsuch appeals are hardly inspiring. Let us proceed on the very weak basis that perhaps those more experienced in what the world has to offer may also havethe wisdom to recommend worth-while studies. Hence, despite St. Augustine, the reader is invited to tempt hell and dam-nation by engaging in a study of the subject. Certainly he can be assured that the subject is within his grasp and that no special gifts or qualities of mind areneeded to learn mathematics. It is even debatable whether the creation of mathematics requires special talents as does the creation of music or greatpaintings, but certainly the appreciation of what others have done does not demand a "mathematical mind" any more than the appreciation of art requiresan "artistic mind." Moreover, since we shall not draw upon any previously acquired knowledge, even this potential source of trouble will not arise.Let us review our objectives. We should like to understand what mathematics is, how it functions, what it accomplishes for the world, and what ithas to offer in itself. We hope to see that mathematics has content which serves the physical and social scientist, the philosopher, logician, and the artist;content which influences the doctrines of the statesman and the theologian;

10 WHY MATHEMATICS?content which satisfies the curiosity of the man vt ho surveys the heavens and the man who muses on the sweetness of musical sounds; and content which hasundeniably, if sometimes imperceptibly, shaped the course of modern history. In brief, we shall try to see that mathematics is an integral part of the modernworld, one of the strongest forces shaping its thoughts and actions, and a body of living though inseparably connected with, dependent upon, and in turnvaluable to all other branches of our culmre. Perhaps we shall also see how by suffusing and inHuencing all thought it has set the intellectual temper of ourtimes. EXERCISES `1. A wolf, a goat, and a cabbage are to be rowed across a river in a boat holding only one of these three objects besides the oarsman. How should he carry themacross so that the goat should not eat the cabbage or the wolf devour the goat? 2. Another hoary teaser is the following: A man goes to a tub of water with twojars, one holding 3 pt and the other 5 pt. How can he bring back exactly 4 pt? 3. Two husbands and two wives have to cross a river in a boat which can holdonly two people. How can they cross so that no woman is in the company of a man unless her husband is also present?Recommended Reading Russau., BERTRAND: "The Study of Mathematics," an essay in the collection entitledMysticism and Logic, Longmans, Green and Co., New York, 1925. W1-nramaao, ALP`RED Noirrnz "The Mathematical Curriculum," an essay in the col-lection entitled Tbe Ainzr of Education, The New American Library, New York, 1949.Wuiraumn, ALFRED Norm; Science and the Modern World, Chaps. 2 and 3, Cambridge University Press, Cambridge, 1926.

CHAPTER 2A HISTORICAL ORIENTATION An educated mind is, as it were, composed 0f all the minds 0f precedingage;. LE Bovieiz DE FONTENELLE2-1 INTRODUCTION Our first objective will be to gain some historical perspective on the subject ofmathematics. Although the logical development of mathematics is not markedly different from the historical, there are nevertheless many features ofmathematics which are revealed by a glimpse of its history rather than by an examination of concepts, theorems, and proofs. Thus we may learn what thesubject now comprises, how the various branches arose, and how the character of the mathematical contributions made by various civilizations was condi-tioned 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 isespecially 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 contin-uously pursued branch of human thought. Moreover, unlike science, philosophy, and social thought, very little of the mathematics that has ever beencreated has been discarded. Mathematics is also a cumulative development, that is, newer creations are built logically upon older ones, so that one mustusually 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 withgreat art, literature, philosophy, religion, and social institutions, very few possessed any mathematics worth talking about. Most of these civilizationshardly got past the stage of being able to count to Hve or ten. 11

12 A HISTORICAL oR1ENrAr1oNIn some of these early civilizations a few steps in mathematics were taken. In prehistoric times, which means roughly before 4000 B.c., several civilizationsat least learned to think about numbers as abstract concepts. That is, they recognized that three sheep and three arrows have something in common, aquantity called three, which can be thought about independently of any physical objects. Each of us in his own schooling goes through this same processof divorcing numbers from physical objects. The appreciation of "number'" as an abstract idea is a great, and perhaps the first, step in the founding ofmathematics. Another step was the introduction of arithmetical operations. It is quitean 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 collec-tions. Similar remarks apply to subtraction, multiplication, and division. The early methods of carrying out these operations were crude and complicatedcompared with ours, but the ideas and the applications were there. Only a few ancient civilizations, Egypt, Babylonia, India, and China, pos-sessed what may be called the rudiments of mathematics. The history of mathematics, and indeed the history of Western civilization, begins with whatoccurred 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 andmoreover had no influence on the subsequent development of mathematics. Our knowledge of the Egyptian and Babylonian civilizations goes back toabout 4000 B.c. The Egyptians occupied approximately the same region that now constitutes modern Egypt and had a continuous, stable civilization fromancient times until about 300 B.c. The term "Babylonian" includes a succession of civilizations which occupied the region of modern Iraq. Both of thesepeoples possessed whole numbers and fractions, a fair amount of arithmetic, some algebra, and a number of simple rules for finding the areas and volumesof 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 areaof a circle amounts to using 3.16 times the square of the radius; that is, their value of ar was 3.16. This value, though not accurate, was even better than theseveral 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 tabletsof the Babylonians, both civilizations used arithmetic and algebra largely in commerce and state administration, to calculate simple and compound intereston 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 grainwould 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

MATHEMATICS IN EARLY CIVILIZATIONS 13of land, the volumes of structures, and the quantity of bricks or stones needed to erect a temple or pyramid. The ancient Greek historian Herodotus saysthat 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 applica-tion, but geometry existed in Egypt long before the date of 1400 mc. mentioned by Herodotus for its origin. Herodotus would have been more accurateto 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 riverdeposits good soil on the land as it overflows. Applications of geometry, simple and crude as they were, did play a largerole 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 beadmirable engineering achievements even in this age of skyscrapers. The Babylonian temples, called ziggurats, also were remarkable pyramidal struc-tures. The Babylonians were, moreover, highly skilled irrigation engineers, who built a system of canals to feed their hot dry lands from the Tigris andEuphrates rivers. Perhaps a word of caution is necessary with respect to the pyramids. Be-cause these are impressive structures, some writers on Egyptian civilization have jumped to the conclusion that the mathematics used in the building ofpyramids must also have been impressive. These writers point out that the horizontal dimensions of any one pyramid are exactly of the same length, thesloping 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 ob-tain such results. A cabinetmaker need not be a mathematician. Mathematics in Egypt and Babylonia was also applied to astronomy. Ofcourse, astronomy was pursued in these ancient civilizations for calendar reckoning and, to some extent, for navigation. The motions of the heavenly bodiesgive us our fundamental standard of time, and their positions at given times enable ships to determine their location and caravans to find their bearings inthe deserts. Calendar reckoning is not only a common daily and commercial need, but it fixes religious holidays and planting times. In Egypt it was alsoneeded 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 conjecturethat the priests of Egypt knew that 365j; was a more accurate figure but kept the knowledge secret. The Egyptian calendar was taken over much later bythe Romans and then passed on to Europe. The Babylonians, by contrast, developed a lunar calendar. Since the duration of the month as measured fromnew 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.

14 A msromcm. oiuENrAT1oNHence 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 Egyptthe belief was widespread that the moon, the planets, and the St81`S directly influenced and even controlled affairs of the state. This doctrine was graduallyextended 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 seemedreasonable 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 indeedfind reason to praise their achievements. But judged by other standards, Egyptian and Babylonian contributions to mathematics were almost insignifi-cant, although these same civilizations reached relatively high levels in religion, art, architecture, metallurgy, chemistry, and astronomy. Compared with theaccomplishments of their immediate successors, the Greeks, the mathematics of the Egyptians and Babylonians is the scrawling of children just learning howto 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 thesubject. Moreover, the very essence of mathematics, namely, reasoning to establish the validity of methods and results, was not even envisioned. Experi-ence recommended their procedures and rules, and with this support they were content. Egyptian and Babylonian mathematics is best described as empiricaland 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 ofconcrete mathematics were there, but the spirit of mathematics was lacking. The lack of interest in theoretical or systematic knowledge is evident inall 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 fatas we know, to encompass all these observations in one major plan or theory of heavenly motions. Nor does one find any other scientinc theory or connectedbody of knowledge. 2-3 THE CLASSICAL GREEK PERIODWe 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 inthe highly congenial atmosphere of Greece. This land was invaded about e'

THE CLASSICAL GREEK PERIOD 151000 E.c. by people whose origins are not known. By about 600 n.c. these people occupied not only Greece proper but many cities in Asia Minor on theMediterranean coast, islands such as Crete, Rhodes, and Samos, and cities in southern Italy and Sicily. Though all of these areas bred famous men, the chiefcultural 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 ac* knowledge their indebtedness to the Babylonians and especially to the Egyp-tians. Many Greeks traveled in Egypt and in Asia Minor. Some went there to study. Nevertheless, what the Greeks created differs as much from whatthey took over from the Egyptians and Babylonians as gold differs from tin. Plato was too modest in his description of the Greek contribution when hesaid, "Whatever we Greeks receive we improve and perfect." The Greeks not only made finished products out of the raw materials imported from Egyptand 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 ofreasoning 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 itis that which makes him man." It was by the application of reasoning to mathematics that the Greekscompletely altered the nature of the subject. In fact, mathematics as we understand the term today is entirely a Greek gift, though in this case weneed not heed Virgil's injunction to fear such benefactions. But how did the Greeks plan to employ reason in mathematics? Whereas the Egyptians andBabylonians were content to pick up scraps of useful information through experience or trial and error, the Greeks abandoned empiricism and undertooka systematic, rational attack on the whole subject. First of all, the Greeks saw clearly that numbers and geometric forms occur everywhere in the heavensand on earth. Hence they decided to concentrate on these important concepts. Moreover, they were explicit about their intention to treat general abstractconcepts rather than particular physical realizations. Thus they would consider the ideal circle rather than the boundary of a Held 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 equalnumbers 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 aregiven. Hence they selected some of these obvious facts as a starting point and ;

16 A H1sToR1cAL ORIENTATIONcalled 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 pos-sesses. If the reasoning were successful, it would produce new knowledge. Also, since they were to reason about general concepts, their conclusionswould apply to all objects of which the concepts were representative. Thus if they could prove that the area of a circle is rr 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 aboutgeneral concepts might not only produce knowledge of hundreds of physical situations in one proof, but there was always the chance that reasoning wouldproduce knowledge which experience might never suggest. All these advantages the Greeks expected to derive from reasoning about general concepts onthe basis of evident reliable facts. A neat plan, indeed! It is perhaps already clear that the Greeks possessed a mentality totallydifferent from that of the Egyptians and Babylonians. They reveal this also in the plans they had for the use of mathematics. The application of arithmeticand algebra to the computation of interest, taxes, or commercial transactions, and of geometry to the computation of the volumes of granaries was as farfrom 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 manyrespects, 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, theheavenly bodies attracted them most. Thus, though the Greeks also studied light, sound, and the motions of bodies on the earth, astronomy was their chiefscientific interest. just what did the Greeks seek in probing nature? They sought no mate-rial gain and no power over nature, they sought merely to satisfy their minds. Because they enjoyed reasoning and because nature presented the most im-posing challenge to their understanding, the Greeks undertook the purely intellectuai study of nature. Thus the Greeks are the founders of science in thetrue sense. The Greek conception of nature was perhaps even bolder than their con-ception of mathematics. VVhereas earlier and later civilizations viewed nature as capricious, arbitrary, and terrifying, and succumbed to the belief that magicand rituals would propitiate mysterious and feared forces, the Greeks dared to look nature in the face. They dared to afiirm that nature was rationally andindeed mathematically designed, and that man"s reason, chiefly through the aid of mathematics, would fathom that design. The Greek mind rejectedtraditional doctrines, supernatural causes, superstitious, dogma, authority, and other such trammels on thought and undertook to throw the light of reasonon the processes of nature. In seeking to banish the mystery and seeming arbitrariness of nature and in abolishing belief in dreaded forces, the Greekswere pioneers.

THF ALr;xANmuAN GREEK PERIOD 17For reasons which will become clearer in a later chapter, the Greeks favored geometry. By 300 12.c., Thales, Pythagoras and his followers, Plato'sdisciples, 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 thesolution of equations, almost all of their work was in geometric form, and so there was no improvement over the Egyptians and Babylonians in the repre-sentation 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 incivilizations 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 humanreason to deduce new truths. In every culture influenced by the Greeks, this example inspired men to apply reason to philosophy, economics, political the-ory, art, and religion. Even today Euclid is the prime example of the power and accomplishments of reason. Hundreds of generations since Euelid's dayshave learned from his geometry what reasoning is and what it can accomplish. Modern man as well as the ancient Greeks learned from the Euclidean docu-ment how exact reasoning should proceed, how to acquire facility in it, and how to distinguish correct from false reasoning. Although many people de-preciate this value of mathematics, it is interesting nevertheless that when these people seek to offer an excellent example of reasoning, they inevitably turn tomathematics. This brief discussion of Euclidean geometry may show that the subject isfar from being a relic of the dead past. lt remains important as a stepping stone in mathematics proper and as a paradigm of reasoning. With their giftof 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 theGreat conquered Greece, Egypt, and the Near East. Alexander decided to build a new capital for his vast empire and founded the city in Egypt namedafter him. The center of the new Greek world became Alexandria instead of Athens. Moreover, Alexander made deliberate efforts to fuse Greek and NearEastern cultures. Consequently, the civilization centered at Alexandria, though predominantly Greek, was strongly infiuenced by Egyptian and Babyloniancontributions. This Alexandrian Greek civilization lasted from about 300 ac. to 600 A.u.L

18 A HIS'l`()Rl(IAl. o1ui;N'rA'rioNThe mixture of the theoretical interests of the Greeks and the practical outlook of the Babylonians and Egyptians is clearly evident in the mathematicaland scientific work of the Alexandrian Greeks. The purely geometric investigations of the classical Greeks were continued, and two of the most famousGreek mathematicians, Apollonius and Archimedes, pursued their studies during the Alexandrian period. In fact, Euclid also lived in Alexandria, but hiswritings reflect the achievements of the classical period. For practical applications, which usually require quantitative results, the Alexandrians revived thecrude arithmetic and algebra of Egypt and Babylonia and used these empirically founded tools and procedures, along with results derived from exactgeometrical studies. There was some progress in algebra, but what was newly created by men such as Nichomachus and Diophantus was still short of eventhe elementary methods we learn in high school. The attempt to be quantitative, coupled with the classical Greek love forthe mathematical study of nature, stimulated two of the most famous astronomers of all time, Hipparchus and Ptolemy, to calculate the sizes and distancesof the heavenly bodies and to build a sound and, for those times, accurate astronomical theory, which is still known as Ptolemaic theory. Hipparchusand 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 aworld 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 Romansoldier. According to the account given by the noted historian Plutarch, the soldier shouted to Archimedes to surrender, but the latter was so absorbed instudying 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 \Vestern civilization, but in the Helds of mathe-matics and science their influence was negative rather than positive. The Romans were a practical people and even boasted of their practicality. Theysought wealth and world power and were willing to undertake great engineering enterprises, such as the building of roads and viaducts, which might helpthem 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 \Vhitehead remarked, "No Roman ever lost his life because he was absorbed in the contemplation of a mathe-matical diagram." Indirectly as well as directly, the Romans brought about the destructionof the Greek civilization at Alexandria, directly by conquering Egypt and

THE 1-iiN1>Us AND ARABs 19indirectly by seeking to suppress Christianity. The adherents to this new religious movement, though persecuted cruelly by the Romans, increased innumber while the Roman Empire grew weaker. In 313 A.D. Rome legalized Christianity and, under the Emperor Theodosius (379-395), adopted it as theofficial religion of the empire. But even before this time, and certainly after it, the Christians began to attack the cultures and civilizations which had opposedthem. By pillage and the burning of books, they destroyed all they could reach of ancient learning. Naturally the Greek culture suffered, and manyworks wiped out in these holocausts are now lost to us forever. The final destruction of Alexandria in 640 A.u. 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 ARABSThe Arabs, who suddenly appeared on the scene of history in the role of destroyers, had been a nomadic people. They were unified under the leader-ship of the prophet Mohammed and began an attempt to convert the world to Mohammedanism, using the sword as their most decisive argument. Theyconquered 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 oe-cupied Spain, southern France, where they were stopped by Charles Martel, southern Italy and Sicily. ()nly the Byzantine or Eastern Roman Empire wasnot subdued and remained an isolated center of Greek and Roman learning. In rather surprisingly quick time as the history of nations goes, the Arabssettled 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 nowIraq, and Cordova in Spain. Realizing that the Greeks had created wonderful works in many fields, the Arabs proceeded to gather up and study what theycould 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 Alzmzgest ( The Greatest \/Vork) by the Arabs and is still knownby 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. lt 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 andalgebra, followed the empirical, concrete approach of the Egyptians and Babylonians. They could on the one hand appreciate and critically review theprecise, exact, and abstract mathematics of the Greeks while, on the other, offer

20 A His*1`oi<1cAi. ORIENTATIONmethods of solving equations which, though they worked, had no reasoning to support them. During all the centuries in which Greek works were in theirpossession, the Arabs manfully resisted the lures of exact reasoning in their own contributions.\Ve 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 neigh-bor on the East. The Indians, too, had built up some elementary mathematics comparable in extent and spirit with the Egyptian and Babylonian develop-ments. However, after about 200 Aa)., mathematical activity in India became more appreciable, probably as a result of contacts with the Alexandrian Greekcivilization. The Hindus made a few contributions of their own, such as the use of special number symbols from I to 9, the introduction of 0, and the useof positional notation with base ten, that is, our modern method of writing numbers. They also created negative numbers. These ideas were taken overby the Arabs and incorporated in their mathematical works. Because of internal dissension the Arab Empire split into two independentparts. 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. Thereason is simple. The Germanic tribes who occupied central Europe and the Gauls of western Europe were barbarians. Among primitive civilizations, theirswere primitive indeed. They had no learning, no art, no science, not even a system of writing.The barbarians were gradually civilized. VVhile the Romans were still successful in holding the regions now called France, England, southern Ger-many, and the Balkans, the barbarians were in contact with, and to some extent influenced by. the Romans, \Vhen 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 anyrate the illiterate Europeans had first to learn reading and writing, one is not surprised to find that mathematics and science were practically unknown inEurope until about l 100 Ap. 2-7 THE RENAISSANCEinsofar 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 landsas the Crusades acquainted the Europeans, who hitherto possessed only fragments of the Greek works, with the vast stores of Greek learning possessed bythe Arabs. The ideas in these works excited the Europeans, and scholars set about acquiring them and translating them into Latin. Through another accirms RENAISSANCE 21dent of history another group of Greek works came to Europe. We have already noted that the Eastern Roman or Byzantine Empire had survived theGermanic and the Arab aggrandizements. But in the fifteenth century the Turks captured the Eastern Roman Empire, and Greek scholars carryingprecious manuscripts fled the region and went to Europe. We shall leave for a later chapter a fuller account of how the Europeanworld 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 oflife.* 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. lmbued with the Greek doctrines that man must studyhimself and the real world, the artists began to paint reality as they actually perceived it instead of interpreting religious themes in symbolic styles. Theyapplied Euclidean geometry to create a new system of perspective which permitted them to paint realistically. Specifically, the artists created a newstyle of painting which enabled them to present on canvas, scenes making the same impression on the eye as the actual scenes themselves. From the workof 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 Greekdoctrine that the world is mathematically designed, Nicolaus Copernicus sought to show that God had done a better job than Hipparchus and Ptolemyhad described. The result of Copernicus' thinking was a new system of astronomy in which the sun was immobile and the planets revolved aroundthe sun. This heliocentric theory was considerably improved by Kepler. Its effects on religion, philosophy, science, and on man"s estimations of his ownimportance were profound. The heliocentric theory also raised scientific and mathematical problems which were a direct incentive to new mathematicaldevelopments. just how much mathematical activity the revival of Greek works mighthave stimulated cannot be determined, for simultaneously with the translation and absorption of these works, a number of other revolutionary developmentsaltered the social, economic, religious, and intellectual life of Europe. The introduction of gunpowder was followed by the use of muskets and latercannons. These inventions revolutionized methods of warfare and gave the newly emerging social class of free common men an important role in thatdomain. The compass became known to the Europeans and made possible long--range navigation, which the merchants sponsored for the purpose of* See Chapter 9. --

22 A HISTORICAL ORIENTATIONfinding 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 hadin any earlier civilizations. The Protestant Revolution stirred debate and doubts concerning doctrines that had been unchallenged for 1500 years. The rise of amerchant class and of free men engaged in labor in their own behalf stimulated an interest in materials, methods of production, and new commodities. All ofthese 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 becamecenters of attention. A remarkable improvement in these mathematical fields followed. This is the period in which algebra was built as a branch of mathe-matics 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 torepresent a class of numbers, a device which gives algebra its generality and power, was introduced by Vieta. In this same period, logarithms were createdto facilitate the calculations of astronomers. The history of arithmetic and algebra illustrates one of the striking and curious features of the history ofmathematics. 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 Rene' Descartes. Descartes isperhaps 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 wasalready troubled by the intellectual turmoil of his age. He found no certainty in any of the knowledge taught him, and he therefore concentrated for yearson 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 modernphilosophical system. Because the scientific problems of his time involved work with curves, the paths of ships at sea, of the planets, of objects in motionnear 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 curvesand also arrived at the idea of applying algebra. In this development of coordinate geometry we have one of the remarkable examples of how thetimes influence the direction of men's thoughts.

mtvaLo1>MEN*rs mom 1550 TO 1800 23We 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 ofmaterials, methods of production, quality of the product, and utilization of devices to replace or increase the effectiveness of manpower. The peopleinvolved, like the artists, had become aware of Greek mathematics and science and sensed that it could be helpful. And so they too sought to employthis knowledge in their own behalf. Thereby arose a new motive for the study of mathematics and science. Whereas the Greeks had been content tostudy nature merely to satisfy their own curiosity and to organize their conclusions in patterns pleasing to the mind, the new goal, effectively pro-claimed by Descartes and Francis Bacon, was to make nature serve man. Hence mathematicians and scientists turned earnestly to an enlarged programin which both understanding and mastery of nature were to be sought. However, Bacon had cautioned that nature can be commanded only whenone learns to obey her. One must have facts of nature on which to base reasoning about nature. Hence mathematicians and scientists sought to acquirefacts from the experience of artists, technicians, artisans, and engineers. The alliance of mathematics and experience was gradually transformed into analliance of mathematics and experimentation, and a new method for the pursuit of the truths of nature, first clearly perceived and formulated by GalileoGalilei (1564-1642) and Newton, was gradually evolved. The plan, perhaps oversimply stated, was that experience and experiment were to supply basicmathematical principles and mathematics was to be applied to these principles to deduce new truths, just as new truths are deduced from the axioms ofgeometry. The most pressing scientific problem of the seventeenth century was thestudy of motion. On the practical side, investigations of the motion of projectiles, of the motion of the moon and planets to aid navigation, and ofthe motion of light to improve the design of the newly discovered telescope and microscope, were the primary interests. On the theoretical side, the newheliocentric astronomy invalidated the older, Aristotelian laws of motion and called for totally new principles. It was one thing to explain why a ball fellto 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 thatthe 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-newdevelopments were added to mathematics. The first of these was the notion of a function, a relationship between variables best expressed for most purposesas 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 thecalculus. The subject matter of mathematics and the power of mathematics

24 A msromcat omenrarionexpanded so greatly that at the end of the seventeenth century Leibniz could say,Taking mathematics from z/ae beginning of the world to the time when Newton lived, what he had done was muck 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 mechanicswhich encompassed the motion of a ball falling to earth and the motion of the planets and stars. This great creation produced universal laws which notonly 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 ofapplying mathematics to sound physical principles not only succeeded in one major area but gave promise, in a rapidly accelerating scientific movement, ofembracing all other physical phenomena. We learn in history that the end of the seventeenth century and theeighteenth century were marked by a new intellectual attitude briehy described as the Age of Reason. We are rarely told that this age was inspired by thesuccesses which mathematics, to be sure in conjunction with science, had achieved in organizing man's knowledge. infused with the conviction thatreason, personified by mathematics, would not only conquer the physical world but could solve all of man's problems and should therefore be employedin 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, economies, and politics were made during these rational investigations. Our principalintellectual 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. Thecalculus was soon extended to a new branch of mathematics called differential equations, and this new tool enabled scientists to tackle more complex problemsin 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 develop-ment of the steam engine), in the strength of materials, and in the How of liquids and gases. Other branches, which can be merely mentioned, such asinhnite 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 PRESENTDuring the nineteenth century, developments in mathematics came at an ever increasing rate. Algebra, geometry and analysis, the last comprising those

1>Evi:i.oi#s11zNrs mom 1800 T0 ini; viuzsrxr 25subjects which stem from calculus, all acquired new branches. The great mathematicians of the century were so numerous that it is impractical to listthem. \Ve shall encounter some of the greatest of these, Karl Friedrich Gauss and Bernhard Riemann, in our work. \Ve might mention also Henri Poincareand David Hilbert, whose work extended into the twentieth century. Undoubtedly the primary cause of this expansion in mathematics was theexpansion in science. The progress made in the seventeenth and eighteenth centuries had sufiiciently illustrated the effectiveness of science in penetratingthe 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.ln that century also, science became far more intimately linked with engineering and technology than ever before. Mathematicians, working closelywith the scientists as they had since the seventeenth century, were presented with thousands of significant physical problems and responded to thesechallenges. Perhaps the major scientific development of the century, which is typicalin its stimulation of mathematical activity, was the study of electricity and magnetism. VVhile 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 tech-niques to these principles, to deduce new information just as Galileo and Newton had done with the principles of motion. In the course of suchmathematical investigations, _]ames Clerk Maxwell discovered electromagnetic waves of which the best known representatives are radio waves. A new worldof 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 tookplace in the nineteenth century, and these resulted from a re-examination of elementary mathematics. The most profound in its intellectual significancewas the creation of non-Euclidean geometry by Gauss. His discovery had both tantalizing and disturbing implications; tantalizing in that this new fieldcontained entirely new geometries based on axioms which differ from Euclid's, and disturbing in that it shattered man's iirmest conviction, namely thatmathematics is a body of truths. With the truth of mathematics undermined, realms of philosophy, science, and even some religious beliefs went up insmoke. So shocking were the implications that even mathematicians refused to take non--Euclidean geometry seriously until the theory of relativity forcedthem to face the full significance of the creation. For reasons which we trust will become clearer further on, the devastationcaused by non--Euclidean geometry did not shatter mathematics but released it from bondage to the physical world. The lesson learned from the historyof non--Euclidean geometry was that though mathematicians may start with _

26 A HISTORICAL omicxrariotcaxioms that seem to have little to do with the observable behavior of nature, the axioms and theorems may nevertheless prove applicable. Hence mathe-maticians felt freer to give reign to their imaginations and to consider abstract concepts such as complex numbers, tensors, matrices, and n-dimensionalspaces. 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 socialscientists. They began to emulate the physical scientists, that is, to search for the basic truths in their fields and to attempt reorganization of theirsubjects on the mathematical pattern. But these attempts to deduce the laws of man and society and to erect sciences of biology, economics, and politicsdid not succeed, although they did have some indirect beneficial effects. The failure to penetrate social and biological problems by the deductivemethod, that is, the method of reasoning from axioms, caused social scientists to take over and develop further the mathematical theories of statistics andprobability, which had already been initiated by mathematicians for various purposes ranging from problems of gambling to the theory of heat andastronomy. 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 Greektimes. lt is rather a living plant that has flourished and languished with the rise and fall of civilizations. Since about 1600 it has been a continuingdevelopment which has become steadily vaster, richer, and more profound. The character of mathematics has been aptly, if somewhat floridly, describedby the nineteenth--century English mathematician james joseph Sylvester. Mathematics is not a book confined within a cover and bound between brazenclasps, whose contents it needs only patience to ransack; it is not a mine, whose treasures may take long to reduce into possession, but which jill only a limitednumber 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 bemapped 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 whichare forever crowding in and multiplying upon the astronomer's gaze; it is incapable of being restricted within assigned boundaries or being reduced to de/initions ofpermanent 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 readyto burst forth into new forms of vegetable and animal existence. Our sketch of the development of mathematics has attempted to indicatethe major eras and civilizations in which the subject has flourished, the variety of interests which induced people to pursue mathematics, and thebranches of mathematics that have been created. Of course, we intend to

THE HUAIAN .--xs1#nc*r or I\1ATHl-ZX\lATI(ZS 27investigate more carefully and more fully what these creations are and what values they have furnished to mankind. One fact of history may be notedby way of summary here. Mathematics as a body of reasoning from axioms stems from one source, the classical Greeks. All other civilizations whichhave pursued or are pursuing mathematics acquired this concept of mathematics from the Greeks. The Arab and VVestern European were the nextcivilizations to take over and expand on the Greek foundation. Today countries such as the United States, Russia, China, India, and japan arealso active. Though the last three of these did possess some native mathematics, it was limited and empirical as in Babylonia and Egypt. Modernmathematical activity in these five countries and wherever else it is now taking hold was inspired by Western European thought and actually learnedby 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. \Ve havespoken of problems which gave rise to mathematics, of cultures which emphasized some directions of thinking as opposed to others, and of branches ofmathematics, 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 moun-tains 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 humanproduct. The concepts, the axioms, and the theorems established arc all created by human beings in man's attempt to understand his environment, togive play to his artistic instincts, and to engage in absorbing intellectual activity.The lives and activities of the men themselves are also fascinating. VVhile mathematicians produce formulas, no formula produces mathematicians. Theyhave come from all levels of society, The special talent, if there is such, which makes mathematicians has been found in Casanovas and ascetics, amongbusiness men and philosophers, among atheists and the profoundly religious, among the retiring and the worldly. Some, like Blaise Pascal and Gauss, wereprecocious, Evariste Galois was dead at 21, and Niels Hendrik Abel at 27. Others, like Karl Weierstrass and Henri Poincare', matured more normally andwere productive throughout their lives. Many were modest; others extremely egotistical and vain beyond toleration. One finds scoundrels, such as Cardan,and models of rectitude. Sonic were generous in their recognition of other great minds; others were resentful and iealous and even stole ideas to boosttheir own reputations. l)isputes about priority of discovery abound. >

28 A HISTORICAL ()RlEN'l`A'l`l0NThe 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 extentwhy 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 associatewith human beings. Leading mathematicians have failed to recognize bright ideas suggested by younger men, and the authors died neglected. Big men andlittle men made unsuccessful attempts to solve problems which their successors solved with ease. On the other hand, some supposed proofs offered even bymasters 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 achieve-ment but merely a point of view. The examples of the blindness of human beings to ideas which later seem simple and obvious furnish fascinating insightinto the working of the human mind. Recognition of the human element in mathematics explains in large measurethe 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 greatminds more readily discernible. EXERCISES1. Name a few civilizations vn hich contributed to mathematics. 2. VVhat basis did the Egyptians and Babylonians have for believing in their mathe-matical methods and formulas? 3. Compare Greek and pre--Greek understanding of the concepts of mathematics.4. VVhat was the Greek plan for establishing mathematical conclusions? 5. VVhat was the chief contribution of the Arabs to the development of mathe-matics? 6. ln what sense is mathematics a creation of the Greeks rather than of the Egyp-tians and Babylonians? 7. Criticize the statement "Mathematics was created by the Greeks and very littlewas added since their time." Topics for Further InvestigationTo 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.

THE HUMAN ASPECT or NIATI-IE1V1A'[`ICS 29Recommended Reading BALL, W. W. R0UsE: A Short Account of the History of Mathematics, Dover Pub-lications, Inc., New York, 1960. BELL, Eiuc T.: Men of Mathematics, Simon and Schuster, New York, 1937.Ci-m.#E, V. GoE.#oN; Man Makes Himself, The New American Library, New York, 1951.EVES, HOWARD: An Introduction to the History of Mathematics, Rev. ed., I-Iolt, Rinehart and Winston, Inc., New York, 1964.NEUGEBAUER, Orroz The Exact Sciences in Antiquity, Princeton University Press, Princeton, 1952.Scorr, j. F.: A History of Mathematics, Taylor and Francis, Ltd., London, 1958. Smrrn, DAVID EUGENE; History of Mathematics, Vol. I, Dover Publications, Inc.,New York, 1958. Srnuix, Dmx ].; A Concise History of Mathematics, Dover Publications, Inc., NewYork, 1948.

lCHAPTER 3 LOGIC AND MATHEMATICSGeometry will draw the soul toward truth and create the spirit of philosophy.PLATO 3-1 INTRODUCTIONMathematics has its own ways of establishing knowledge, and the understanding of mathematics is considerably promoted if one learns first just what thoseways are. In this chapter we shall study the concepts which mathematics treats; the method, called deductive proof, by which mathematics establishes its con-clusions; and the principles or axioms on which mathematics rests. Study of the contents and logical structure of mathematics leaves untouched the subjectof how the mathematician knows what conclusions to establish and how to prove them. We shall therefore present a brief and preliminary discussion ofthe creation of mathematics. This topic will recur as we examine the subject matter itself in subsequent chapters.Since mathematics, as we conceive the subject today, was fashioned by the Greeks, we shall also attempt to see what features of Greek thought andculture caused these people to remodel what the Egyptians and Babylonians had pursued for several thousand years.3-2 THE CONCEPTS OF MATHEMATICS The first major step which the Greeks made was to insist that mathematicsmust deal with abstract concepts. Let us see just what this means. VI/'hen we first learn about numbers we are taught to think about collections of particularobjects such as two apples, three men, and so on. Gradually and rather subconsciously we begin to think about the numbers 2, 3, and other wholenumbers without having to associate them with physical objects. \Ve soon reach the more advanced stage of adding, subtracting, and performing otheroperations with numbers without having to handle collections of objects in order to understand these operations or to see that the results agree withexperience. Thus we soon become convinced that 4 times 5 must be 20, whether these numbers represent quantities of apples, horses, or even purely30

VTHE CONCEPTS or MAT!-1iaMArics 31 imaginary objects. By this time we are really dealing with concepts or ideas,for the whole numbers do not exist in nature. Any whole number is rather an abstraction of a property which is common to many different collections orsets of objects. The whole numbers then are ideas, and the same is true of fractions suchas g, Q, and so on. In the latter case, too, the formulation of the physical relationship of a part of an object to the whole, whether it refers to pies,bushels of wheat, or to a smaller monetary value in relation to a larger one, again leads to an abstraction. Mathematicians formulate operations withfractions, that is, combining parts of an object, taking one part away from the other, or taking a part of a part, in such a way that the result of anyoperation on abstract fractions agrees with the corresponding physical occurrence. Thus the mathematical process of, say adding $ and {$, which yields $,expresses the addition of $ of a pie and g of a pie, and the result tells us how many parts of a pie one would actually have.Whole numbers, fractions, and the various operations with whole numbers and fractions are abstractions. Although this fact is rather easy to understand,we tend to lose sight of it and cause ourselves unnecessary confusion. Let us consider an example. A man goes into a shoe store and buys 3 pairs of shoesat 10 dollars per pair. The storekeeper reasons that 3 pairs times 10 dollars is 30 dollars and asks for 30 dollars in return for the 3 pairs of shoes. lf thisreasoning is correct, then it is equally correct for the customer to argue that 3 pairs times 10 dollars is 30 pairs of shoes and to walk out with 30 pairs ofshoes without handing the storekeeper one cent. The customer may end up in jail, but he may console himself while he languishes there that his reasoningis as sound as the storekeeper"s. The source of the difficulty is, of course, that one cannot multiply shoesby dollars. One can multiply the number 3 by the number l0 and obtain the number 30. The practical and no doubt obligatory physical interpretation ofthe answer in the above situation is that one must pay 30 dollars rather than walk out with 30 pairs of shoes. We see, therefore, that one must distinguishbetween the purely mathematical operation of multiplying 3 by 10 and the physical objects with which these numbers may be associated.The same point is involved in a slightly different situation. Mathematically

$is equal to g. But the corresponding physical fact may not be true. One maybe willing to accept 4 half--pies instead of 2 whole pies, but no woman would accept 4 half-dresses in place of 2 dresses or 4 half--shoes in place of 1 pair ofwhole shoes. The Egyptians and Babylonians did reach the stage of working with purenumbers dissociated from physical objects. But like young children of our civilization, they hardly recognized that they were dealing with abstractentities. By contrast, the Greeks not only recognized numbers as ideas but emphasized that this is the way we must regard them. The Greek philosopherK

32 Locic AND MATHEAIATICSPlato, who lived from 428 to 348 B.c. and whose ideas are representative of the classical Greek period, says in his famous work, the Republic,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 untilthey 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.The Greeks not only emphasized the distinction between pure numbers and the physical applications of such numbers, but they preferred the formerto the latter. The study of the properties of pure numbers, which they called arithmetica, was esteemed as a worthy activity of the mind, whereas the useof numbers in practical applications, logistica, was deprecated as a mere skill. Geometrical thinking prior to the classical Greek period was even lessadvanced than thinking about numbers. To the Egyptians and Babylonians the words "straight linc" meant no more than a stretched rope or a line tracedin sand, and a rectangle was a piece of land of a particular shape. The Greeks began the practice of treating point, line, triangle, and other geometricalnotions as concepts. They did of course appreciate that these mental notions are suggested by physical objects, but they stressed that the concepts differfrom the physical examples as sharply as the concept of time differs from the passage of the sun across the sky. The stretched string is a physical objectillustrating the concept of line, but the mathematical line has no thickness, no color, no molecular structure, and no tension.The Greeks were explicit in asserting that geometry deals with abstractions. Speaking of mathematicians, Plato says,And do you not know also that although they make use of the visible forms and reason about them, they are thinking 7lOt of these, but of the ideals which theyresemble; 7IOt of the figures which they draw, but of the absolute square and the absolute diameter . . . they are really seeking to behold the things themselves,which can be seen only with the eye of the mind? On the basis of elementary abstractions, mathematics creates others whichare even more remote from anything real. Negative numbers, equations involving unknowns, formulas, and other concepts we shall encounter areabstractions built upon abstractions. Fortunately, every abstraction is ultimately derived from, and therefore understandable in terms of, intuitively meaningfulobjects or phenomena. The mind does play its part in the creation of mathematical concepts, but the mind does not function independently of the outsideworld. Indeed the mathematician who treats concepts that have no physically real or intuitive origins is almost surely talking nonsense. The intimate connec-tion between mathematics and objects and events in the physical world is

Tru; coxcisvrs or A1.\'l`Hl#.\l.\ll(Ih 33reassuring, for it means that we can not only hope to understand the mathematics proper, but also expect physically meaningful and valuable conclusions.The use of abstractions is not peculiar to mathematics. The concepts of force, mass, and energy, which are studied in physics, are abstractions fromreal phenomena. The concept of wealth, an abstraction from material possessions such as land, buildings, and jewelry, is studied in economics. Theconcepts of liberty, justice, and democracy are familiar in political science. Indeed, with respect to the use of abstract concepts, the distinction betweenmathematics on the one hand and the physical and social sciences on the other is not a sharp one. ln fact, the influence of mathematics and mathematicalways of thinking on the physical sciences especially has led to ever increasing use of abstract concepts including some, as wc shall see, which may have nodirect real counterpart at all, any more than a mathematical formula has a direct real counterpart.The very fact that other studies also engage in abstractions raises an important question. Mathematics is confined to some abstractions, numbersand geometrical forms, and to concepts built upon these basic ones. Abstractions such as mass, force, and energy belong to physics, and still otherabstractions belong to other subjects. \Vhy doesn`t mathematics also treat forces, wealth, and justice? Certainly these concepts are also worthy of study.Did the mathematicians make an agreement with physicists, economists, and others to divide the concepts among themselves? The restriction of mathe-matics to numbers and geometrical forms is partly a historical accident and partly a deliberate decision made by the Greeks. Numbers and geometricalforms had already been introduced by the Egyptians and Babylonians, and their utility in daily life was established. Since the Greeks learned the rudi-ments of mathematics from these civilizations, the sheer weight of tradition might have caused them to continue the practice of regarding mathematics asthe study of numbers and geometrical figures. But people as original and bold in thought as the Greeks would not have been bound merely by tradition, hadthey not found in numbers and geometrical forms sharp and clear notions which appealed to their delight in the processes of exact thinking. However,an even more compelling reason was their belief that numerical and geometrical properties and relationships were basic, that they underlay the phenomena ofthe physical world and the design of the entire universe. Hence to understand the world one should seek this mathematical essence. The brilliance anddepth of their conception of the universe will be revealed more and more as we proceed.When one compares the pre-Greek and Greek understanding of the concepts of mathematics and notes the sharp transition from the concrete to theabstract, another question presents itself. The Greeks eliminated the physical substance and retained only the idea. \Vhy did they do it? Surely it is morediflicult to think about abstractions than about concrete things. Also it would

34 Locic AND IV[ATHEl\IA'1`ICSseem that an attempt to study nature by concentrating on just a few aspects of physical objects rather than on the objects themselves would fall far short ofeffectiveness. Insofar as the emphasis on abstractions is concerned, the Greeks saw atonce what any thinking people would see sooner or later. One advantage of treating abstractions is the gain in generality. When a child learns that5 + 5= 10, he acquires in one swoop a fact which applies to hundreds of situations. Likewise a theorem proved about the abstract triangle applies to atriangular piece of land, a musical percussion instrument, and a triangle determined by three heavenly bodies at any instant of time. lt has been saidthat the process of abstraction amounts to giving the same name to different things, but this very recognition that different objects possess the commonproperty named in the abstraction carries with it the implication that anything true of the abstraction will apply to the several objects. Part of the secret ofthe power of mathematics is that it deals with abstractions. Another advantage of abstraction was also clear to the Greeks. Abstract-ing from a physical situation just those properties which are to be studied frees the mind from burdensome and irrelevant details and enables one to concen-trate on the features of interest. When one wishes to determine the area of a piece of land, only shape and size are relevant, and it is desirable to think onlyabout these and not about the fertility of the soil. The emphasis on mathematical abstractions by the classical Greeks waspart and parcel of their outlook on the entire universe. They were concerned with truths, and leading philosophical schools, notably the Pythagorcans andthe Platonists, maintained that truths could be established only about abstractions. Let us follow their argument. The physical world presents variousobjects to the senses. But the impressions received by the senses are inexact, transitory, and constantly changing, indeed, the senses may be even deceived,as by mirages. However, truth, by its very meaning, must consist of permanent, unchanging, definite entities and relationships. Fortunately, the intelli-gence of man excited to reflection by the impressions of sensible objects may rise to higher conceptions of the realities faintly exhibited to the senses, and soman may rise to the contemplation of ideas. These are eternal realities and the true goal of thought, whereas mere "things are the shadows of ideas thrown onthe screen of experience." Thus Plato would that there is nothing real in a horse, a house, or abeautiful woman. The reality is in the universal type or idea of a horse, a home, or a woman. The ideas, among which Plato emphasized Beauty, justice,Intelligence, Goodness, Perfection, and the State, are independent of the superficial appearances of things, of the fiux of life, and of the biases and warpeddesires of man; they are in fact constant and invariable, and knowledge concerning them is firm and indestructible. Real and eternal knowledge concernsthese ideas, rather than sensuous objects. This distinction between the intelligible world and the world revealed by the senses is all-important in Plato.

THIZ (LON(ZliP'1`S OF A1A'1`HENIA'l`lCSj Y AFig. 3-1. Polyclitusz Spear-bearer (Daryphorus). National Museum,Naples. To put Plato`s doctrine in ever) day language, fundamental knowledgedoes not concern itself with what john ate. Xlary heard, or William felt. Knowledge must rise above individuals and particular objects and tell us aboutbroad classes of objects and about man as a whole. True knowledge must therefore of necessity concern abstractions. Plato admits that physical or sen-sible objects suggest the ideas just as diagrams of geometry suggest abstract geometrical concepts. Hence there is a point to studying physical objects, butone must not lose himself in trivial and confusing minutiae. .Tl1C 2llW$l'I`HCfl(>l1< (lf IU:lCl\CIU1lKlCS p()$.%CsA*C(l 11 5pCClzll llI]p()l'I1lDCC f()l` fl\A*Greeks. The philosophers pointed out that, to pass from a knowledge of the world of matter to the world of ideas, man must train his mind to grasp theL

:6 rorar; isn umn yrancs {y*#'i->>5s&>>&(C)~= Q. ~s / Ee i `\*/ Y Q_, FIG. 3-2. .##` asf Bustof Caesar. Vatican.ideas. These highest realities blind the person who is not prepared to contemplate them, Ile is, to use Plato's famous simile, like one who lives continuouslyin the deep shadows of a cave and is suddenly brought out into the sunlight. The study of mathematics helps make the transition from darkness to light.Nlathematics 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 andon the other hand it deals with abstract concepts. Hence through the study of mathematics man learns to pass from concrete figures to abstract forms; more-over. this study purifies the mind by drawing it away from the contemplation of the sensible and perishable and leading it to the eternal ideas. These latterabstractions are on the same mental level as the concepts of mathematics. Thus, Socrates says, "Thc understanding of mathematics is necessary for asound grasp of ethics." To sum up Plato"s position we may say that while a little knowledge ofgeometry and calculation suffices for practical needs, the higher and more advanced portions tend to lift the mind above mundane considerations andenable it to apprehend the final aim of philosophy, the idea of the Good. Mathematics, then, is the best preparation for philosophy. For this reasonPlato 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 harmonies (music). The oft-repeated inscription over the doors of Plato`s Academy, stating thatno one ignorant of mathematics should enter, fully expresses the importance

rua concaars or Maramomcs 37g 4 gw e , 'i * Y r * _ iaa.;VV, V >>w>>;;a~a~--saaa w;;Q il *3* Q} V 5**4 M 5 7 ` . la. e .4. P el ~ , t s4EA >> 3 ` `l ` Ml *'G* -4-`YY `. .l';`l * """ -- ` if V * Y %`t ~;,,;`$o$?*; L-? r 'F gr j ~ 'T E! * * 4---. TJ _ -?**"* ` . * l ..,.,:.,*Las..a_">>.:*:>>-242 _ V __ g*<-rT*;-##*u#~--.**>>. E` M ` -..& 2--;""":'..L':.Q"i4#+**~>>.'D" iw- r * F"` -. ~.'** Q nFig. 3-3. Parthenon, Athens. he attached to the subject, although modern critics of Plato read into thesewords his admission that one would not be able to learn it after entering. This value of mathematical training led one historian to remark, "l\/lathematicsconsidered 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 Egyptiancconomics." The preference of the Greeks for abstractions is equally evident in the artof the great sculptors, Polyclitus, Praxiteles, and Phidias. One has only to glance at the face in Fig. 3-I to observe that Greek sculpture of the classicalperiod dwelt not on particular men and women but on types, ideal types. ldealization extended to standardization of the ratios of the parts of the bodyto each other. Polyclitus believed, in fact, that there were ideal numerical ratios which fix the proportions of the human body. Perfect art must follow theseideal proportions. He wrote a book, The Canon, on the subject and constructed the "Spear-bearer" to illustrate the thesis. These abstract types con-trast sharply with what is found in numerous busts and statues of private individuals and military and political leaders made by Romans (Fig. 3-2).Greek architecture also reveals the emphasis on ideal forms. The simple and austere buildings were always rectangular in shape; even the ratios ofthe dimensions employed were fixed. The Parthenon at Athens (Fig. 3-3) is an example of the style and proportions found in almost all Greek temples.

38 Looic AND AIATHEAIATICSEXERCISES 1. Suppose 5 r 4 each. To ansis 20 men. On the other i . number of trucks is 20 trucks. 4 men times 5 true yields 0 truck;.How do you know that the answer is 20 men in one case and 20 trucks intl: other?2. If the product of 25c and 25c is obtained by multiplying 0.25 by 0.25 the result is 0.0625 or 6&c. Does it pay to multiply money?3. Can you suggest some abstract political or ethical concepts? 4. Suppose 30 books are to be distributed among 5 people. Since 30 books dividedby 5 people yields 6 books, each person gets 6 books. Criticize the reasoning. e 5. A store advertises that it will give a credit of $1 for each purchase amounting tot$1. A man who spends $6 reasons that he should receive a credit of $6 times $1,i or $6. But $6 is 600c and $1 is 100c. Hence 600c times 100c is 60,000c, or $600.It would seem that it is more profitable to operate with the almost worthless cent than with dollars! What is wrong?6. What does the statement that mathematics deals with abstractions mean? 7. Why did the Greeks make mathematics abstract?3-3 IDEALIZATION The geometrical notions of mathematics are abstract in the sense that shapesare mental concepts which actual physical objects merely approximate. The sides of a rectangular piece of land may not be exactly straight nor would eachangle be exactly 90o. Hence, in adopting such abstract concepts, mathematics does idealize. But in studying the physical world, mathematics also idealizesin another sense which is equally important. Very often mathematicians undertake to study an object which is not a sphere and yet choose to regard it assuch. For example, the earth is not a sphere but a spheroid, that is, a sphere flattened at the top and bottom. Yet in many physical problems which aretreated mathematically the earth is represented as a perfect sphere. In problems of astronomy a large mass such as the earth or the sun is often regarded asconcentrated at one point. ` In making such idealizations, the mathematician deliberately distorts orapproximates at least some features of the physical situation. Why does he 1 do it? The reason usually is that he simplifies the problem and yet is quitesure that he has not introduced any gross errors. If one is to investigate, for example, the motion of a shell which travels ten miles, the difference betweenthe assumed spherical shape of the earth and the true spheroidal shape does not matter. In fact, in the study of any motion which takes place over a limitedregion, say one mile, it may be suf`Hcient to treat the earth as a flat surface. On the other hand, if one were to draw a very accurate map of the earth, he would

Meruons or Rmsomxo 39take into account that the shape is spheroidal. As another example, to find the distance to the moon, it is good enough to assume that the moon is a point inspace. However, to find the size of the moon, it is clearly pointless to regard the moon as a point.The question does arise, how does the mathematician know when idealization is justified? There is no simple answer to this question. If he has to solvea series of like problems, he may solve one using the correct figure, and another, using a simplified figure, and compare results. If the difference doesnot matter for his purposes, he may then retain the simpler figure for the remaining problems. Sometimes he can estimate the error introduced by usingthe simpler figure and may find that this error is too small to matter. Or the mathematician may make the idealization and use the result because it is thebest he can do. Then he must accept experience as his guide in deciding whether the result is good enough.To idealize by deliberately introducing a simplification is to lie a little, but the lie is a white one. Using idealizations to study the physical world doesimpose a limitation on what mathematics accomplishes, but we shall find that even where idealizations are employed, the knowledge gained is of immensevalue. EXERCISESl. Distinguish between abstraction and idealization. 2. Is it correct to assume that the lines of sight to the sun from two places A and Bon the earth's surface are parallel? 3. Suppose you wished to measure the height of a fiagpole. Would it be wise toregard the flagpole as a line segment? 3-4 METHODS OF REASONINGThere are many ways, more or less reliable, of obtaining knowledge. One can resort to authority as one often does in obtaining historical knowledge. Onemay accept revelation as many religious people do. And one may rely upon experience. The foods we eat are chosen on the basis of experience. No onedetermined in advance by careful chemical analysis that bread is a healthful food.We may pass over with a mere mention such sources of knowledge as authority and revelation, for these sources cannot be helpful in buildingmathematics or in acquiring knowledge of the physical world. It is true that in the medieval period of Western European culture men did contend thatall desirable knowledge of nature was revealed in the Bible. However in no significant period of scientinc thought has this view played any role. Experi-ence, 0n the other hand, is a useful source of knowledge. But there are --

40 LOGIC AND AIATHEMATICSdifficulties in employing this method. \Ve should not wish to build a fiftystory building in order to decide whether a steel beam of specified dimensionsis strong enough to be used in the foundation. Moreover, even if one should happen to choose workable dimensions, the choice may be wasteful of ma-terials. Of course, experience is of no use in determining the size of the earth or the distance to the moon.Closely related to experience is the method of experiment which amounts l 'to setting up and going through a series of purposive, systematic experiences.It is true that experimentation fundamentally is experience, but it is usually accompanied by careful planning which eliminates extraneous factors, andthe experience is repeated enough times to yield reliable information. I-lowever, experimentation is subject to much the same limitations as experience.Are authority, revelation, experience, and even experimentation the only methods of obtaining knowledge? The answer is no. The major method isreasoning, and within the domain of reasoning there are several forms. One can reason by analogy. A boy who is considering a college career may notethat his friend went to college and handled it successfully. He argues that since he is very much like his friend in physical and mental qualities, he tooshould succeed in college work. The method of reasoning just illustrated is to find a similar situation or circumstance and to argue that what was true forthe similar case should be true of the one in question. Of course, one must be able to find a similar situation and one must take the chance that the differencesdo not matter. Another common method of reasoning is induction. People use thismethod of reasoning every day. Because a person may have had unfortunate experiences in dealing with a few department stores, he concludes that alldepartment stores are bad to deal with. Or, for example, experimentation would show that iron, copper, brass, oil, and other substances expand whenheated, and one consequently concludes that all substances expand when heated. Inductive reasoning is in fact the common method used in experimen-tation. An experiment is generally performed many times, and if the same result is obtained each time, the experimenter concludes that the result willalways follow. The essence of induction is that one observes repeated occurrences of the same pbewomenoiz and concludes that the phenomenon willalways occur. Conclusions obtained by induction seem well warranted by the evidence, especially when the number of instances observed is large. Thus thesun is observed so often to rise in the morning that one is sure it has risen even on those mornings when it is hidden by clouds.There is still a third method of reasoning, called deduction. Let us consider some examples. If we accept as basic facts that honest people return foundmoney and that john is honest, we may conclude unquestionably that john will return n1oney that he finds. Likewise, if we start with the facts that nomathematician is a fool and that john is a mathematician, then we may con7 Txtmnons or Rmsoxnru 41 clude with certainty that john is not a fool. ln deductive reasoning we startwith certain statements, called premises, and assert a conclusion which is a neceymry or inescapable consequence of the premises.All three methods of reasoning, analogy, induction, and deduction, and other methods we could describe, are commonly employed. There is oneessential difference, however, between deduction on the one hand and all other methods of reasoning on the other. \Vhereas the conclusion drawn by analogyor induction has only a probability of being correct, the conclusion drawn by deduction necessarily holds. Thus one might argue that because lions aresimilar to cows and cows eat grass, lions also eat grass. This argument by analogy leads to a false conclusion. The same is true for induction: althoughexperiment may indeed show that two dozen different substances expand when heated, it does not necessarily follow that all substances do. Thus water, forexample, when heated from Oo to -l*o centigrade* does not expand; it contracts. Since deductive reasoning has the outstanding advantage of yielding anindubitable conclusion, it would seem obvious that one should always use this method in preference to the others. But the situation is not that simple.For one thing analogy and induction are often easier to employ. Tn the case of analogy, a similar situation may be readily available. ln the case of induction,experience often supplies the facts with no effort at all. The fact that the sun rises every morning is noticed by all of us almost automatically. Furthermore,deductive reasoning calls for premises which it may be impossible to obtain despite all efforts. Fortunately we can use deductive reasoning in a variety ofsituations. For example, we can use it to find the distance to the moon. In this instance, both analogy and induction are powerless, whereas, as we shallsee later, deduction will obtain the result quickly. It is also apparent that where deduction can replace induction based on expensive experimentation,deduction is preferred. Because we shall be concerned primarily with deductive reasoning, let usbecome a little more familiar with it. \Ve have given several examples of deductive reasoning and have asserted that the conclusions are inescapableconsequences of the premises. Let us consider, however, the following example. We shall accept as premises thatAll good cars are expensive and.--\ll Locomobiles are expensive. We might conclude thatAll Locomobiles are good cars. ` ln scientific texts, "cclsius" is considered to be the more precise term.L.

42 LOGIC AND ]\lATHhjA[A'l`lCS],;\]#_mi\,(_ l.t*arncd aiijwit #**`**1>l*`Loeonroluiles lm'foS"`"`S 9 @ (R)Fig. 3-4 Fig. 3-5 The reasoning here is intended as deductive; that is, the presumption in draw-ing this conclusion is that it is an inevitable consequence of the premises. Unfortunately, the reasoning is not correct. How can we see that it is notcorrect? A good way of picturing deductive arguments which enables us to see whether or not they are correct is called the circle test.We note that the Hrst premise deals with cars and expensive objects. Let us think of all the expensive objects in this world as represented by the pointsof a circle, the largest circle in Fig. 3-4. The statement that all good cars are expensive means that all good cars are a part of the collection of expensiveobjects. Hence we draw another circle within the circle of expensive objects, and the points of this smaller circle represent all the good cars. The secondpremise says that all Locomobiles are expensive. Hence if we represent all ~ Locomobiles by the points of a circle, this circle, too, must be drawn within thecircle of expensive objects. However we do not know, on the basis of the two premises, where to place the circle representing all Locomobiles. It can, as faras we know, fall in the position shown in the figure. Then we cannot conclude that all Loeomobiles are good cars, because if that conclusion were inevitable,the circle representing Locomobiles must fall inside the circle representing good cars.Many people do conclude from the above premises that all Locomobiles are good cars and the reason that they err is that they confuse the premise "Allgood cars are expensive" with the statement that ".--\ll expensive cars are good." \>>Vere the latter statement our first premise then the deductive argument wouldbe valid or correct. Let us consider another example. Suppose we take as our premises thatAll professors are learned people andSONIC p1'UfC$$()I'5 1lI`A* iHfA*lllgCIlI pC()plC. May we necessarily conclude thatSome intelligent people are learned?

Mtrnoos or Rmsoxrrsc 43It may or may not be obvious that this conclusion is correct. Let us use the circle test. We draw a circle representing the class of learned people (Fig 3-5).Since the first premise tells us that all professors are learned people, the circle representing the class of professors must fall within the circle represent-ing learned people. The second premise introduces the class of intelligent people, and we now have to determine where to draw that circle. This classmust include some professors. Hence the circle must intersect the circle of professors. Since the latter is inside the circle of learned people, some intelli-gent people must fall within the class of learned people. These examples of deductive reasoning may make another point clear. Indetermining whether a given argument is correct or valid, we must rely only upon the facts given in the premises. \Ve may not use information which isnot explicitly there. For example, we may believe that learned people are intelligent because to acquire learning they must possess intelligence. Butthis belief or fact, if it is a fact, cannot enter into the argument. Nothing that one may happen to know or believe about learned or intelligent people is tobe used unless explicitly stated in the premises. In fact, as far as the validity of the argument is concerned, we might just as well have considered thepremises All x's are y's,Some x's are 2's, and the conclusion, then, isSome z's are y's. Here we have used x for professor, y for learned person, and 2 for intelligentperson. The use of x, y, and 2 does make the argument more abstract and more difficult to retain in the mind, but it emphasizes that we must look onlyat the information in the premises and avoids bringing in extraneous information about professors, learned people, and intelligent people. When we writethe argument in this more abstract form, we also see more clearly that what determines the validity of the argument is the form of the premises rather thanthe meaning of x, y, and 2. A great deal of deductive reasoning falls into the patterns we have beenillustrating. There are, however, variations that should be noted. lt is quite customary, especially in the geometry we learn in high school, to statetheorems in what is called the "if . . . then"` form. Thus one might say, if a triangle is isosceles, then its base angles are equal. One could as well say, allisosceles triangles have equal base angles, or, the base angles of an isosceles triangle are equal. All three versions say the same thing.Connected with the "if . . . then'" form of a premise is a related statement which is often misunderstood. The statement "if a man is a professor, he islearned" offers no difficulty. As noted in the preceding paragraph, it is equivalent to "all professors are learned."' However the statement "only if aL

44 LOGIC .\\n \IA'IH1*ZAlA'l`I(ZSman is a professor, is he learned" has quite a different meaning. It means that to be learned one must be a professor or that if a man is learned, he must bea professor. Thus the addition of the word only has the significance of interchanging the "if" clause and the "then" clause.\Ve shall encounter numerous instances of deductive reasoning in our work. The subiect of deductive reasoning is customarily studied in logic, adiscipline which treats more thoroughly the valid forms of reasoning. However, xve shall not need to depend upon formal training in logic. ln IHOSI cases,common experience will enable us to ascertain whether the reasoning is or is not valid. \\i'hen in doubt, we can use the circle test. Moreover, mathematicsitself is the superb field from which to learn reasoning and is the best exercise in logic. The laws of logic were in fact formulated by the Greeks on the 'basis of their experiences with mathematical arguments. EXERCISES1. A coin is tossed ten times and each time it falls heads. \Vhat conclusion does inductive reasoning warrant?2. Characterize deductive reasoning. $. VVhat superior features does deductive reasoning possess compared with induc-tion and analogy? 4. Can you prove deductivcly that George \Vashington was the best president ofthe United States? 5. (Zan one always apply deductive reasoning to prove a desired statement?6. Can you prove deductively that monogamy is the best system of marriage? 7. Are the follow ing purportedly deductive arguments valid?a) All good cars are expensive. A Daffy is an expensive car. Therefore a Daffy is a good car.b) All New Yorkers are good citizens. All good citizens give to charity. Therefore all New Yorkers give to charity.c) All college students are clever. All young boys are clever. Therefore all young boys are college students.d) The same premises as in (c), but the conclusion: All college students are young boys.e) lt rains every Monday and it is raining today; hence today must be Monday. f) No decent people curse; Americans are decent; therefore Americans donot curse. g) No decent people curse; Americans curse; therefore some Americans arenot decent. Ii) No decent people curse; some Americans are not decent; therefore someAmericans curse. i) No undergraduates have a bachelor--of--arts degree; no freshmen have abachelor--of-arts degree. Therefore all freshmen are undergraduates.

MATi~1EA1ATicA1. vnoor 458. lf someone gave you a valid deductive argument but the conclusion was not true, where would you look for the difficulty?9. Distinguish between the validity of a deductive argument and the truth of the conclusion.3-5 MATHEMATICAL PROOF We have seen so far in our discussion of reasoning that there are severalmethods of reasoning and that all are useful. These methods can be applied to mathematical problems. Let us suppose that one wished to determine the sumof the angles of a triangle. He could draw on paper many different triangles or construct some out of wood or metal and measure the angles. In eachease he would find that the sum is as close to 180o as the eye and hand can determine. By inductive reasoning he could conclude that the sum of theangles in every triangle is 180o. As a matter of fact, the Babylonians and Egyptians did in effect use inductive reasoning to establish their mathematicalresults. They must have determined by measurement that the area of a triangle is one-half the base times the altitude and, having used this formula repeatedlyand having obtained reliable results, they concluded that the formula is correct. The mld-points of parallelchords lie on a straight la) u>> une.To see that reasoning by analogy can be used in mathematics, let us note first that the centers of a set of parallel chords of a circle lie on a straight line(Fig. 3-6a). In fact this line is a diameter of the circle. Now an ellipse (Fig. 3-6b) is very much like a circle. Hence one might conclude that thecenters of a set of parallel chords of an ellipse also lie on a straight line. Deduction is certainly applicable in mathematics. The proofs which onelearns in Euclidean geometry are deductive. As another illustration we might consider the following algebraic argument. Suppose one wishes to solve theequation x -- 3: 7. One knows that equals added to equals give equals. If we added 3 to both sides of the preceding equation, we would be adding equals toan equality. Hence the addition of 3 to both sides is justified. \Vhen this is done, the result is x= I0, and the equation is solved.Thus all three methods are applicable. There is a lot to be said for the use of induction and analogy. The inductive argument for the sum of the anglesof a triangle can be carried out in a matter of minutes. The argument by analogy given above is also readily made. On the other hand, finding deduc46 Looic AND :\rATui:xiA`r1cs * itive proofs for these same conclusions might take weeks or might never be accomplished by the average person. As a matter of fact, we shall soon en- `counter some examples of conjectures for which the inductive evidence is , overwhelming but for which no deductive proof has been thus far obtainedeven by the best mathematicians. l Despite the usefulness and advantages of induction and analogy, mathe- `matics does not rely upon these methods to establish its conclusions. All . uuztheumticul proofs must be deductive. Each proof is a chain of deductivearguments, each of which has its premises and conclusion. Before examining the reasons for this restriction to deductive proof, we *might contrast the method of mathematics with those of the physical and . social sciences. The scientist feels free to draw conclusions by any method ofreasoning and, for that matter, on the basis of observation, experimentation, i and experience. He may reason by analogy as, for example, when he reasons oabout sound waves by observing water waves or when he reasons about a possible cure for a disease afiecting human beings by testing the cure on tanimals. In fact reasoning by analogy is a powerful method in science. The scientist may also reason inductively: if he observes many times that hydrogenand oxygen combine to form water, he will conclude that this combination will always form W21tA*1`. At some stages of his work the scientist may also `reason deductively and, in fact, even employ the concepts and methods of mathematics proper. _To contrast further the method of mathematics with that of the scientist i --and perhaps to illustrate just how stubborn the mathematician can be--wemight consider a rather famous example. Mathematicians are concerned with whole numbers, or integers, and among these they distinguish the primenumbers. A prime is a number which has no integral divisors other than itself , and 1. Thus 11 is a prime number, whereas 12 is not because it is divisible by l,2 for example. Now by actual trial one finds that each of the first few even numbers can be expressed as the sum of two prime numbers. For example, .2=l+1;4:2+2;6=3+3;8:3--l-5;10:3+7; .... Ifoneinvesti-- . gates larger and larger even numbers, one finds without exception that everyeven number can be expressed as the sum of two primes. Hence by inductive ` reasoning one could conclude that every even number is the sum of two prime "numbers. But the mathematician does not accept this conclusion as a theorem ofmathematics because it has not been proved deductively from acceptable i premises. The conjecture that every even number is the sum of two primes, iknown as Goldbaclfs hypothesis because it was first suggested by the eighteenth--century mathematician Christian Goldbach, is an unsolved problemof mathematics. The mathematician will insist on a deductive proof even if it takes thousands of years, as it literally has in some instances, to find one.However a scientist would not hesitate to use this inductively well supported conclusion.

MATHEAIATICAL vnoor 47Of course, the scientist should not be surprised to find that some of his conclusions are false because, as we have seen, induction and analogy do notlead to sure conclusions. But it does seem as though the scientist"s procedure is wiser since he can take advantage of any method of reasoning which willhelp him advance his knowledge. The mathematician by comparison appears to be narrow-minded or shortsighted. He achieves a reputation for certainty,but at the price of limiting his results to those which can be established deductively. How wise the mathematician may be in his insistence on deduc-tive proof we shall learn as we proceed. The decision to confine mathematical proof to deductive reasoning wasmade by the Greeks of the classical period. And they not only rejected all other methods of proof in mathematics, but they also discarded all theknowledge which the Egyptians and Babylonians had acquired over a period of four thousand years because it had only an empirical justification. Whydid the Greeks do it? The intellectuals of the classical Greek period were largely absorbed inphilosophy and these same men, because they possessed intellectual interests, were the very ones who developed mathematics as a system of thought. Thelonians, the Pythagoreans, the Sophists, the Platonists, and the Aristotelians were the leading philosophers who gave mathematics its definitive form. Thecredit for initiating this step probably belongs to one school of Greek philosopher-mathematicians, known as the Ionian school. However, if creditcan be assigned to any one person, it belongs to Thales, who lived about 600 B.c. Though a native of Miletus, a Greek city in Asia Minor, Thales spentmany years in Egypt as a merchant. There he learned what the Egyptians had to offer in mathematics and science, but apparently he was not satisfied, for hewould accept no results that could not be established by deductive reasoning from clearly acceptable axioms. In his wisdom Thales perceived what weshall perceive as we follow the story of mathematics, that the obvious is far more suspect than the abstruse.Thales probably supplied the proof of many geometrical theorems. He acquired great fame as an astronomer and is believed to have predicted aneclipse of the sun in 585 B.c. A philosopher--astronomer--mathematician might readily be accused of being an impractical stargazer, but Aristotle tells usotherwise. In a year when olives promised to be plentiful, Thales shrewdly cornered all the oil presses to be found in Miletus and in Chios. When theolives were ripe for pressing, Thales was in a position to rent out the presses at his own price. Thales might perhaps have lived in history as a leadingbusinessman, but he is far better known as the father of Greek philosophy and mathematics. From his time onward, deductive proof became the standard inmathematics. It is to be expected that philosophers would favor deductive reasoning.Whereas scientists select particular phenomena for observation and experimentation and then draw conclusions by induction or analogy, philosophers are

48 Locic Aww Marumxarrcsconcerned with broad knowledge about man and the physical world. To establish universal truths, such as that man is basically good, that the world is --designed, or that man's life has purpose, deductive reasoning from acceptable principles is far more feasible than induction or analogy. As Plato put it in hisRep!/blir, "If persons cannot give or receive a reason, they cannot attain that knowledge which, as we have said, man ought to have." QThere is another reason that philosophers favor deductive reasoning. These men seek truths, the eternal verities. We have seen that of all the methods of _reasoning only deductive reasoning grants sure and exact conclusions. Hence E this is the method which philosophers would almost necessarily adopt. Not 3only do induction and analogy fail to yield absolutely unquestionable conclu- ` sions, but many Greek philosophers would not have accepted as facts the datawith which these methods operate, because these are acquired by the senses. Plato stressed the unreliability of sensory perceptions. Empirical knowledge,as Plato put it, yields opinion only. The Greek preference for deduction had a sociological basis. Contrary toour own society wherein bankers and industrialists are respected most, in classical Greek society, the philosophers, mathematicians, and artists were theleading citizens. The upper class regarded earning a living as an unfortunate necessity. VVorlY robbed man of time and energy for intellectual activities,the duties of citizenship and discussion. These Greeks did not hesitate to express their disdain for work and business. The Pythagoreans, who, as weshall see, delighted in the properties of numbers and applied numbers to the study of nature, derided the use of numbers in commerce. They boasted thatthey sought knowledge rather than wealth. Plato, too, maintained that knowledge rather than trade was the goal in studying arithmetic. Freemen, hedeclared, who allowed themselves to become preoccupied with business should be punished, and a civilization which is concerned mainly with the materialwants of man is no more than a "city of happy pigs." Xenophon, the famous Greek general and historian, says, "\Vhat are called the mechanical arts carrya social stigma and arc rightly dishonored in our cities." Aristotle wanted an ideal society in which citizens would not have to practice any mechanical arts.Among the Boeotians, one of the independent tribes of Ancient Greece, those who defiled themselves, with commerce were by law excluded from statepositions for ten years. \Vho did the daily work of providing food, shelter, clothing, and the othernecessities of life? Slaves and free men ineligible for citizenship ran the businesses and the households, did unskilled and technical work, managed the in-dustries, and carried on the professions such as medicine. They produced even the articles of refinement and luxury.In view of this attitude of the Greek upper class towards commerce and trade, it is not hard to understand the classical Greek's preference for deduction.People who do not "live" in the workaday world can learn little from experiMA`rHrtMA`1`1cA1. vkoor 49{ ence, and people who will not observe and use their hands to experiment willnot have the facts on which to base reasoning by analogy or induction. In fact the institution of slavery in classical Greek society fostered a divorce oftheory from practice and favored the development of speculative and deductive science and mathematics at the expense of experimentation and practicalapplications. Over and above the various cultural forces which inclined the Greekstoward deduction were a farsightedness and a wisdom which mark true genius. The Greeks were the first to recognize the power of reason. The mind was afaculty not only additional to the senses but more powerful than the senses. The mind can survey all the whole numbers, but the senses are limited toperceiving only a few at a time. The mind can encompass the earth and the heavens; the sense of sight is confined to a small angle of vision. Indeed themind can predict future events which the senses of contemporaries will not live to perceive. This mental faculty could be exploited. The Greeks sawclearly that if man could obtain some truths, he could establish others entirely by reasoning, and these new truths, together with the original ones, enabledman to establish still other truths. Indeed the possibilities would multiply at an enormous rate. Here was a means of acquiring knowledge which had beeneither overlooked or neglected. This was indeed the plan which the Greeks projected for mathematics.By starting with some truths about numbers and geometrical figures they could deduce others. A chain of deductions might lead to a significant newfact which would be labeled a theorem to call attention to its importance. Each theorem added to the stock of truths that could serve as premises fornew deductive arguments, and so one could build an immense body of knowledge about the basic concepts.Although the Greeks may have been guilty of overemphasizing the power of the mind unaided by experience and observation to obtain truths, there isno doubt that in insisting on deductive proof as the sole method, they rose above the practical level of carpenters, surveyors, farmers, and navigators. Atthe same time they elevated the subject of mathematics to a system of thought. Moreover the preference for reason which they exhibited gave this faculty thehigh prestige which it now enjoys and permitted it to exercise its true powers. When we have surveyed some of the creations of the mind that succeedingcivilizations building on the Greek plan contributed, we shall appreciate the true depth of the Greek vision.EXERCISES 1. Compare Greek and pre-Greek standards of proof in mathematics. Reread therelevant parts of Chapter 2. 2. Distinguish science and mathematics with respect to ways of establishing con-clusions. L J

V50 Looic AND Marummrics 3. Explain the $t3tCI11A*I1( that the Greeks converted mathematics from an empiricalscience to a deductive system. 4. Are the following deductive arguments valid?a) All even numbers are divisible by 4. Ten is an even number. Hence 10 is l divisible by 4.b) Equals divided by equals give equals. Dividing both sides of 3x:6 by 3 is l dividing equals by equals. Hence x : 2.5. Does it follow from the fact that the square of any odd number is odd that the square of any even number is even? l6. Criticize the argument; i The square of every even number is even because 22:4, 42: 16, 62: 36, andit is obvious that the square of any larger even number also is even. i 7. lf we accept the premises that the square of any odd number is odd and the isquare of any even number is even, does it follow deductively that if the square g of a number is even, the number must be even? l8. \Vhy did the Greeks insist on deductive proof in mathematics? 9. Let us take for granted that if a triangle has two equal sides, the opposite langles are equal and that we have a triangle in which all three sides are equal. l Prove deductively that all three angles are equal in the triangle under considera-- ltion. You may also use the premise that things equal to the same thing are equal to one another. 110. How did the Greeks propose to obtain new truths from known ones? l 24 3-6 AXIOMS AND DEFINITIONSFrom our discussion of deductive reasoning we know that to apply such reasoning we must have premises. Hence the question arises, what premisesdoes the mathematician use? Since the mathematician reasons about numbers { and geometrical figures, he must of course have facts about these concepts.These cannot be obtained deductively because then there would have to be prior premises, and if one continued this process backward, there would be nostarting point. The Greeks readily found premises. It seemed indisputable, for ' example, that two points determine one and only one line and that equalsadded to equals give equals. To the Greeks the premises on which mathematics was to be built wereself--evident truths, and they called these premises axioms. Socrates and Plato believed, as did many later philosophers, that these truths were already in ourminds at birth and that we had but to recall them. And since the Greeks believed that axioms were truths and since deductive reasoning yielded unques-tionable conclusions, they also believed that theorems were truths. This view is no longer held, and we shall see later in this book why mathematiciansabandoned it. We now know that axioms are suggested by experience and observation. Naturally, to be as certain as we can of these axioms we selectthose facts which seem clearest and most reliable in our experience. But we

THE CREATION or MATHEl\1ATICS 51must recognize that there is no guarantee that we have selected truths about the world. Some mathematicians prefer to use the word assumptions insteadof axioms to emphasize this point. The mathematician also takes care to state his axioms at the outset and tobe sure as he performs his reasoning that no assumptions or facts are used which were not so stated. There is an interesting story told by former Presi-dent Charles W. Eliot of Harvard which illustrates the likelihood of introducing unwarranted premises. He entered a crowded restaurant and handedhis hat to the doorman. When he came out, the doorman at once picked Eliot's hat out of hundreds on the racks and gave it to him. He was amazed that thedoorman could remember so well and asked him, "How did you know that was my hat?" "I didn't," replied the doorman. "Why, then, did you hand it tome?" The doorman's reply was, "Because you handed it to me, sir." Undoubtedly no harm would have been done if the doorman had assumedthat the hat he returned to President Eliot belonged to the man. But the mathematician interested in obtaining conclusions about the physical worldmight be wasting his time if he unwittingly introduced an assumption that he had no right to makeThere is one other element in the logical structure of mathematics about which we shall say a few words now and return to in a later chapter (Chap-ter 20). Like other studies mathematics uses definitions. Whenever we have occasion to use a concept whose description requires a lengthy statement, weintroduce a single word or phrase to replace that lengthy statement. For example, we may wish to talk about the figure which consists of three dis-tinct points which do not lie on the same straight line and of the line segments joining these points. It is convenient to introduce the word triangle to repre-sent this long description. Likewise the word circle represents the set of all points which are at a fixed distance from a definite point. The definite pointis called the center, and the fixed distance is called the radius. Definitions promote brevity.EXERCISES 1. What belief did the Greeks hold about the axioms of mathematics?2. Summarize the changes which the Greeks made in the nature of mathematics. 3. ls it fair to say that mathematics is the child of philosophy?3-7 THE CREATION OF MATHEMATICS Because mathematical proof is strictly deductive and merely reasonable orappealing arguments may not be used to establish a conclusion, mathematics has been described as a deductive science, or as the science which derivesnecessary conclusions, that is, conclusions which necessarily or inevitably L

52 Looic AND MATHEMATICSfollow from the axioms. This description of mathematics is incomplete. Mathematicians must also discover what to prove and how to go about estab-lishing proofs. These processes are also part of mathematics and they are not deductive.How does the mathematician discover what to prove and the deductive arguments that lead to the conclusions? The most fertile source of mathemat-ical ideas is nature herself. Mathematics is devoted to the study of the physical world, and simple experience or the more careful scrutiny of nature suggestsidea after idea. Let us consider here a few simple examples. Once mathematicians had decided to devote themselves to geometric forms, it was onlynatural that such questions should arise as, what are the area, perimeter, and sum of the angles of common figures? Moreover, it is even possible to seehow the precise statement of the theorem to be proved would follow from direct experience with physical objects. The mathematician might measurethe sum of the angles of various triangles and find that these measurements all yield results close to l80o. Hence the suggestion that the sum of the anglesin every triangle is l80o occurs as a possible theorem. To decide the question, which has more area, a polygon or a circle having the same perimeter, onemight cut out cardboard figures and weigh them. The relative weights would suggest the statement of the theorem to be proved.After some theorems have been suggested by direct physical problems, others are readily conceived by generalizing or varying the conditions. Thusknowing the problem of determining the sum of the angles of a triangle, one might ask, What is the sum of the angles of a quadrilateral, a pentagon, and soforth? That is, once the mathematician begins an investigation which is suggested by a physical problem, he can easily find new problems which go be-yond the original one. In the domains of arithmetic and algebra direct calculation with numbers,which is analogous to measurement in geometry, will suggest possible theorems. Anyone who has played with integers, for example, has doubtless observed thefollowing facts: l = 1,1 + 2 = 4 : 22, 1 + 1 + s = 9 Z sz,1+s+s+7=16:42, We note that each number on the right is the square of the number of oddnumbers appearing on the left; thus in the fourth line, there are four numbers on the left side, and the right side is 42. The general result which these cal-culations suggest is that if the first Tl odd numbers were on the left side, then the sum would be 112. Of course, this possible theorem is not proved by the

THE CREATION or Z\1ATHEI\IA'l`I(jS 53above calculations. Nor could it ever be proved by such calculations, for no mortal man could make the infinite set of computations required to establishthe conclusion for every 11. The calculations do, however, give the mathematician something to work on.These simple illustrations of how observation, measurement, and calculation suggest possible theorems are not too striking or very profound. Weshall see in the course of later work how physical problems suggest more significant mathematical theorems. However, experience, measurement, cal-culation, and generalization do not include the most fertile source of possible theorems. And it is especially true in seeking methods of proof that morethan routine techniques must be utilized. In both endeavors the most important source is the creative act of the human mind. ~___ -1 _____, 4 LB (` Fig. 3-7 Let us consider the matter of proof. Suppose one has discovered bymeasurements that the sum of the angles of various triangles is l80o. One must now prove this result deductively. No obvious method will do the job.Some new idea is required, and the reader who remembers his elementary geometry will recall that the proof is usually made by drawing a line throughone vertex (A in Fig. 3-7) and parallel to the opposite side. It then turns out as a consequence of a previously established theorem on parallel lines that theangles 1 and 2 are equal, as are the angles 3 and 4. However the angles 1, 3, and the angle A of the triangle itself do add up to l80o, and so the same is truefor the angles of the triangle. This method of proof is not routine. The idea of drawing the line through A must be supplied by the mind. Some methodsof proof seem so devious and artificial that they have provoked critical comments. The philosopher Arthur Schopenhauer called Euclid's proof of thePythagorean theorem "a mouse-trap proof" and "a proof walking on stilts, nay, a mean, underhand proof."The above example has been offered to emphasize the fact that ingenious mathematical work must be done in finding methods of proofs even after thequestion of what to prove is disposed of. In the search for a method of proof, as in finding what to prove, the mathematician must use audacious imagination,insight, and creative ability. His mind must see possible lines of attack where others would not. ln the domains of algebra, calculus, and advanced analysisespecially, the first-rate mathematician depends upon the kind of inspiration that we usually associate with the creation of music, literature, or art. Thecomposer feels that he has a theme which when properly developed will pro`

54 LOGIC AND Marr-mmarrcsduce true music. Experience and a knowledge of music aid him in arriving at this conviction. Similarly, the mathematician surmises that he has a conclusionwhich will follow from the axioms of mathematics. Experience and knowledge may guide his thoughts into the proper channels. Modifications of one sort oranother may be required before a correct proof and a satisfactory statement of the new theorem are achieved. But essentially both mathematician and com-poser are moved by an aHiatus which enables them to see the final edifice before a single stone is laid.We do not know just what mental processes may lead to correct insight any more than we know how it was possible for Keats to write fine poetryor why Rembrandt was able to turn out fine paintings. One might say of mathematical creation what P. W. Bridgman, the noted physicist, has said ofscientific method, that it consists of "doing one's damnedest with one's mind, no holds barred." There is no logic or infallible guide which tells the mindhow to think. The very fact that many great mathematicians have tackled a problem and failed and that another comes along and solves it shows that themind has something to contribute. The preceding discussion of the creation of mathematics should correctseveral mistaken popular impressions. When creating a mathematical proof, the mind does not see the cold, ordered arguments which one reads in texts,but rather it perceives an idea or a scheme which when properly formulated constitutes the deductive proof. The formal proof, so to speak, merely sanc-tions the conquest already made by the intuition. Secondly, the deductive proof is not the preferable form by which to grasp the idea or method em-ployed. In fact the deductive argument often conceals the idea because the logical form is not perspicuous to the intuition. At the very least the detailsof the arguments obscure the main threads. The value of the deductive organization of the proof is that it enables the creator and the reader to test thearguments by the standards of exact reasoning. Thirdly, there is the prevalent but mistaken notion that scientists and mathematicians must keep their mindsopen and unbiased in pursuing an investigation. They are not supposed to prejudge the conclusion. Actually the mathematician must first decide whatto prove, and this conclusion not only does but must precede the search for the proof, or else he would not know where to head. This is not to say thatthe mathematician may not sometimes make a false conjecture. If he does, his search for a proof will fail or in the course of the search he will realize thathe cannot prove what he is after, and he will correct his conjecture. But in any case he knows what he is trying to prove.EXERCISES 1. Consider the parallelogram ABCD (Fig. 3-8). By definition, the opposite sidesare parallel. Now introduce the diagonal BD. Does observation suggest a possible theorem relating the triangles ABD and BDC?

rm; camrion or Mari-nsmarics 55. E "

* B AF HC D ' D G (Fig. 3-8 Flg. 3-9 2. Consider any quadrilateral ABCD (Fig. 3-9) and the figure formed by `joiningthe mid-points E, F, G, H of the sides of the quadrilateral. Does observation or intuition suggest any significant fact about the quadrilateral EFGH?3. The formula 712 --- 71 + 41 is supposed to yield primes for various values of 77. Thus when 72 : 1,12 -- 1 + 41 = 41, and this is a prime. When 77 : 2,22 -- 2 + 41 : 43, and this is a prime. Test the formula for 71; 3 and 71:4. Are the resultingvalues of the formula primes? I-Iave you proved, then, that the formula always yields primes?4. Can you specify conditions under which two quadrilaterals will be congruent, that is, have the same size and shape?5. The following lines show some calculations with the sum of the cubes of whole numbers:13 Z l, l3+23:1+8:9:32:(1+2)2,13+23+23:1-1-8+27:s6:62=(1+2+s)2 13+23-}-33+43:1+8-1-27+64=100=102=<1+2+s+4>2- What generalization do these few calculations suggest?REVIEW EXERCISES 1. What basis did the Egyptians and Babylonians have for believing in the cor-rectness of their mathematical conclusions? 2. Compare Greek and pre-Greek understanding of the concepts of mathematics.3. What was the Greek plan for establishing mathematical conclusions? _.

56 LOGIC AND AIATHENIATICS4. In what sense is mathematics a creation of the Greeks rather than of the Egyptians and the Babylonians?5. Suppose we accept the premises that all professors are intelligent people and all professors are learned people. Which of the following conclusions isvalidly deduced? a) Some intelligent people are learned.b) Some learned people are intelligent. c) All intelligent people are learned.d) All learned people are intelligent. 6. Suppose we accept the premises that all college students are wise, and no pro-fessors are college students. Which of the following conclusions is validly deduced?a) No professors are wise. b) Some professors are wise.c) All professors are wise. 7. ls the following argument valid?All parallelograms are quadrilaterals, and figure ABCD is a quadrilateral. Hence figure ABCD is a parallelogram.8. What conclusion can you deduce from the premises, Every successful student must work hard,and john does not work hard?9. Smith says, If it rains I go to the movies.If Smith went to the movies, what can you conclude deductively? 10. Smith says,I go to the movies only if it rains. If Smith went to the movies, what can you conclude deductively?Topics for Further Investigation To pursue any of these topics use the books listed below under RecommendedReading. 1. The life and work of the Pythagoreans2. The life and work of Euclid Recommended ReadingBELL, Emc T.; The Development of Mathematics, 2nd ed., Chaps. 2 and 3, McGrawHill Book C0., N.Y., 1945.BELL, Emc T.: M en of Mathematics, Simon and Schuster, New York, 1937.

THE CREATION OF 1\1ATHE1\4ATICS 57Cttxccrr, MA1zsuALL; Greek Science in Antiquity, Chap. 2, Abelard-Schuman, Inc., New York, 1955.Conan, Momus R. and E. Ntxcm.: An Introduction to Logic and Scientific Method, Chaps. 1 through 5, Harcourt Brace and Co., New York, 1934.COOLIDGE, ]. L.; The Mathematics of Great Amateurs, Chap. 1, Dover Publications, Inc., New York, 1963.HAM1LToN, EDITH; The Greele Way to Western Civilization, Chaps. 1 through 3, The New American Library, New York, 1948.jams, Sm JAMES; The Growth of Physical Science, 2nd ed., Chap. 2, Cambridge University Press, Cambridge, 1951.NEUGEBAUER, OTTo; The Exact Sciences in Antiquity, Princeton University Press, Princeton, 1952. .Szvnm, DAv1# Eucmmz History of Mathematics, Vol. I., Dover Publications, Inc., New York, 1958.STRUTK, Dum ].: A Concise History of Mathematics, Dover Publications, Inc., New York, 1948.TAYLoR, HENRY OsB0RN; Ancient Ideals, 2nd ed., Vol. I, Chaps. 7 through 13, The Macmillan Co., New York, 1913.Wcnsmzc, Awnxns; Plat0's Philosophy of Mathematics, Almqvist and Wiksell, Stockholm, 1955 (for students of philosophy).l