But a little investigation of the ways of our society does show that some reasoning can be useful. The doctor who seeks to diagnose an illness reasons from symptoms to the cause. The lawyer who wishes to convince a jury often resorts to reason. The engineer reasons continually in order to design or produce a new device. The scientist observes or experiments and draws conclusions from the evidence he obtains. All of these people utilize and even depend upon reasoning. The purpose and value of all such reasoning is the derivation of knowledge that is not otherwise obtainable or is obtainable by other means but only with considerably greater expense and effort.

Whereas in the professions of medicine, law, engineering, and science reasoning is employed somewhat, in mathematics reasoning is the sum and substance. Since the case for mathematics rests entirely upon what can be achieved by reasoning, perhaps we should convince ourselves at the very outset that mathematical reasoning can yield desirable knowledge. The examples we shall consider at the moment are not the most impressive but may furnish grounds for the further exploration of the subject.

Let us be practical and consider first a matter of dollars and cents. Suppose that a young man has a choice between two jobs. Each offers a starting salary of $1800 per year, but the first one would lead to an annual raise of $200 whereas the second would lead to a semiannual raise of $50. Which job is preferable? One would think that the answer is obvious. A raise of $200 per year seems better than one that apparently would amount to only $100 per year. But let us do a little arithmetic and put down what each job offers during successive six-month periods.

The first job will pay

The second job, which bears a semiannual increase of $50, will pay

It is clear from a comparison of these two sets of salaries that the second job brings a better return during the second half of each year and does as well as the first job during the first half. The second job is the better one. With the arithmetic before us it is possible to see more readily why the second job is better. The semiannual increase of $50 means that the salary will be higher at the rate of $50 for six months or at the rate of $100 for the year because the recipient will get $50 more for each of the six-month periods. Hence two such increases per year amount to an increase at the rate of $200 per year. Thus far the two jobs seem to be equally good. But on the second job the increases start after the first six months, whereas on the first job they do not start until one year has elapsed. Hence the second job will pay more during the latter six months of each year.

Let us consider another simple problem. Suppose that a merchant sells his apples at two for 5 cents and his oranges at three for 5 cents. Being somewhat annoyed with having to do considerable arithmetic on each sale, the merchant decides to commingle apples and oranges and to sell any five pieces of fruit for 10 cents. This move seems reasonable because if he sells two apples and three oranges he sells five pieces of fruit and receives 10 cents. But now he can charge 2 cents apiece and his arithmetic on each sale is simple.

The dealer is cheating himself. Just to check quickly we shall assume that he has one dozen apples and one dozen oranges for sale. If he sells apples normally at two for 5 cents he receives 30 cents for the dozen apples. If he sells oranges at three for 5 cents he receives 20 cents for the dozen oranges. His total receipts are then 50 cents. However, if he sells the twenty-four pieces at five for 10 cents he will receive 2 cents per article or 48 cents.

The loss is due to poor reasoning on the part of the dealer. He assumed that the average price of the apples and oranges should be 2 cents each. But the average price per apple is 2½ cents and the average price per orange is 1

cents and the average price of two such items is

The foregoing examples involved only simple arithmetic and since we all do such arithmetic rather mechanically we may hardly feel that we have reasoned. We shall therefore undertake a slightly more advanced problem. A farmer interested in land on which to grow crops is offered two parcels. Both are right triangles, one having the dimensions 30, 40, and 50 feet and the other, the dimensions 90, 120, and 150 feet (fig. 1). The dimensions of the second triangle, it will be noted, are three times that of the first. However the price of the second parcel is five times the price of the first.

Since the farmer is naturally interested in buying acreage as cheaply as possible, he must now decide which is the better buy.

Even this last example of the use of mathematical reasoning may not be too impressive to some people. They may claim that the proper visualization of the areas would have sufficed to determine the correct conclusion. Moreover, with continued experience we all develop what we commonly call intuition, a combination of experience, sense impressions, speculation, and crude guessing. Since this faculty helps us to arrive at some conclusions without mathematical reasoning, we had better satisfy ourselves that it is subject to severe limitations.

Let us consider two related problems. First, we have a garden, circular in shape and with a radius of 10 feet. We wish to protect the garden by a fence that is to be at each point one foot away from the boundary of the garden (fig. 2). How much longer is the fence than the circumference of the garden itself? The answer is readily obtained. The circumference of the garden is given by a formula of geometry; this says that the circumference is 2π times the radius, π being the symbol for a number which is approximately

. Hence the circumference of the garden is 2π · 10 (the dot standing for “multiplied by”). The condition that the fence be one foot beyond the garden means that the radius of the circular fence is to be 11 feet. Hence the length of the fence is 2π · 11. The difference in these two circumferences is 22π—20π or 2π. Hence the fence should be 2π feet longer than the circumference of the garden.

We now consider the related problem. Suppose we were to build a roadway around the earth—a trivial task for modern engineers—and the height of the roadway were to be one foot above the surface of the earth all the way round (fig. 2). How much longer than the circumference of the earth would the roadway be? Before calculating this quantity let us use our intuition to at least estimate it. The radius of the earth is about 4000 miles or 21,120,000 feet. Since this radius is roughly 2,000,000 times that of the garden we considered, one might expect that the additional length of the roadway should be about 2,000,000 times the additional length of fence required to enclose the garden. The latter quantity was just 2π feet. Hence an intuitive argument for the additional length of roadway would seemingly lead to the figure of 2,000,000 · 2π feet. Whether or not the reader would agree to this argument he would almost certainly estimate that the length of the roadway would be very much greater than the circumference of the earth.

A little mathematics tells the story. To avoid calculation with large numbers let us denote the radius of the earth by r. The circumference of the earth is then 2πr. The circumference or length of the roadway is 2π(r + 1). But the latter is 2πr + 2π. Hence the difference between the length of the roadway and the circumference of the earth is just 2π feet, precisely the same figure that we obtained for the difference between the length of fence and the circumference of the garden, despite the fact that the roadway encircles an enormous earth whereas the fence encircles a small garden. In fact the mathematics tells us even more. Regardless of what the value of r is, the difference, 2π(r + 1)—2πr, is always 2π, and this means that the circumference of the outer circle, if it is at each point one foot away from the inner circle, will always be just 2π feet longer than the circumference of the inner circle.

There are many problems in which intuition can be only of incidental use and wherein mathematical reasoning must carry the entire burden. One of the simplest and yet most impressive examples of what such reasoning can achieve is Eratosthenes’ calculation of the circumference of the earth. Eratosthenes (275—194 B.C.) was a famous scholar, poet, historian, astronomer, geographer, and mathematician who lived during the latter period of the ancient Greek civilization when the center of that culture was in Alexandria. In common with most learned Greeks Eratosthenes knew that the earth is spherical; he set out to determine its circumference.

Because he was learned in geography he knew that the city of Alexandria was due north of the city of Syene (fig. 3) and that the measured distance along the surface of the earth from one city to the other was 500 miles. At the summer solstice the noon sun shone directly down into a well at Syene. This means, as Eratosthenes appreciated, that the sun was directly overhead at that time; that is, the direction of the sun was OBS’. At Alexandria at the same instant the direction of the sun was AS, whereas the overhead direction is OAD. Now the sun is so far away that the direction AS is the same as BS’, or, mathematically speaking, AS and BS’ are parallel lines. Hence it follows from a theorem of geometry, which we may accept for the present, that angles DAS and AOB are equal. Eratosthenes measured angle DAS and found it to be 7½°. This, then, is the size of angle AOB. But this angle is 7 ½/360 or

of the entire angle at O. It follows that arc AB is

of the entire circumference and since AB is 500 miles, the entire circumference is 48 · 0 or 24,000 miles. Curiously enough, this figure was reduced by later geographers, who used poorer methods, to about 17,000 miles. This figure is the one that reached Columbus; had he known the truer figure of Eratosthenes he might never have undertaken to sail to India because he might have been daunted by the distance to be covered.

It may be clear from even these few simple examples that mathematical reasoning can produce knowledge which guesswork, intuition, and experience cannot produce or can produce only inaccurately. But why should mathematics be studied so extensively and why should it be of interest to all educated people? The primary motivation for the development of mathematics proper and the primary reason for the great importance of the subject is its value in the study of nature. Mathematical concepts and mathematical methods of obtaining knowledge have been most effective in representing and investigating the motions of the heavenly bodies and the motions of objects on and near the surface of the earth, the phenomena of sound, light, heat, electricity, and electromagnetic waves, the structure of matter, the chemical reactions of various substances, the structure of the eye, ear, and other organs of the human body, and dozens of other major scientific phenomena.

But if the main motivation for the development and study of mathematical ideas is the study of nature, why should one study nature? If, for example, one works all his life to earn money, then it is reasonable to ask, what purposes does money serve? Of course the most obvious reasons for the study of nature are the practical advantages that ensue. The farmer studies soil, seeds, methods of cultivation, and the weather to produce good crops. The hunter studies the habits of animals in order to track them down. The sailor studies the currents of large bodies of water, the tides, the weather, and the stars to aid him in navigation. A metallurgist studies minerals in order to produce better iron or steel. Such studies of nature, found even in primitive societies, are obviously useful. In our own civilization the practical advantages derived from the study of nature have reached staggering proportions. We have but to think of our trains, airplanes, and ships, our movies, radio, television, and telephone, our mass production of highly useful equipment for the home, the services performed by electricity and modern medical treatments to appreciate that man has profited by the study of nature.

While the material improvements which have resulted from the study of nature appeal most strongly to people, it is not true that these are really the most compelling reasons for that study. Nor, as a matter of fact, have the practical gains generally been obtained from investigations so motivated. The deeper reason for the study of nature is to satisfy man’s intellectual curiosity. Even in the most primitive civilizations, and certainly in ours, there have been people who have sought to answer such questions as, How did the universe come about? Why was man born and what will happen to him after death? What causes and what is light? Is there plan and organization in the ...