An Introduction to Mathematics · Alfred North Whitehead
The way in which the idea of variables satisfying a relation occurs in the applications of mathematics is worth thought, and by devoting some time to it we shall clear up our thoughts on the whole subject.
Let us start with the simplest of examples:--Suppose that building costs 1 s. per cubic foot and that 20 s. make £ 1. Then in all the complex circumstances which attend the building of a new house, amid all the various sensations and emotions of the owner, the architect, the builder, the workmen, and the onlookers as the house has grown to completion, this fixed correlation is by the law assumed to hold between the cubic content and the cost to the owner, namely that if x be the number of cubic feet, and £ y the cost, then 20y = x. This correlation of x and y is assumed to be true for the building of any house by any owner. Also, the volume of the house and the cost are not supposed to have been perceived or apprehended by any particular sensation or faculty, or by any
particular man. They are stated in an abstract general way, with complete indifference to the owner's state of mind when he has to pay the bill.
Now think a bit further as to what all this means. The building of a house is a complicated set of circumstances. It is impossible to begin to apply the law, or to test it, unless amid the general course of events it is possible to recognize a definite set of occurrences as forming a particular instance of the building of a house. In short, we must know a house when we see it, and must recognize the events which belong to its building. Then amidst these events, thus isolated in idea from the rest of nature, the two elements of the cost and cubic content must be determinable; and when they are both determined, if the law be true, they satisfy the general formula 20y = x. But is the law true? Anyone who has had much to do with building will know that we have here put the cost rather high. It is only for an expensive type of house that it will work out at this price. This brings out another point which must be made clear. While we are making mathematical calculations connected with the formula 20y = x , it is indifferent to us whether the law be true or
false. In fact, the very meanings assigned to x and y , as being a number of cubic feet and a number of pounds sterling, are indifferent. During the mathematical investigation we are, in fact, merely considering the properties of this correlation between a pair of variable numbers x and y. Our results will apply equally well, if we interpret y to mean a number of fishermen and x the number of fish caught, so that the assumed law is that on the average each fisherman catches twenty fish. The mathematical certainty of the investigation only attaches to the results considered as giving properties of the correlation 20y = x between the variable pair of numbers x and y. There is no mathematical certainty whatever about the cost of the actual building of any house. The law is not quite true and the result it gives will not be quite accurate. In fact, it may well be hopelessly wrong.
Now all this no doubt seems very obvious. But in truth with more complicated instances there is no more common error than to assume that, because prolonged and accurate mathematical calculations have been made, the application of the result to some fact of nature is absolutely certain. The conclusion of no argument can be more certain than the assumptions from which it starts. All mathematical calculations about the course of
nature must start from some assumed law of nature, such, for instance, as the assumed law of the cost of building stated above. Accordingly, however accurately we have calculated that some event must occur, the doubt always remains--Is the law true? If the law states a precise result, almost certainly it is not precisely accurate; and thus even at the best the result, precisely as calculated, is not likely to occur. But then we have no faculty capable of observation with ideal precision, so, after all, our inaccurate laws may be good enough.
We will now turn to an actual case, that of Newton and the Law of Gravity. This law states that any two bodies attract one another with a force proportional to the product of their masses, and inversely proportional to the square of the distance between them. Thus if m and M are the masses of the two bodies, reckoned in lbs. say, and d miles is the distance between them, the force on either body, due to the attraction of the other and directed towards it, is proportional to mM/d^2 ; thus this force can be written as equal to kmM/d^2 , where k is a definite number depending on the absolute magnitude of this attraction and also on the scale by which we choose to measure forces. It is easy to see that, if we
wish to reckon in terms of forces such as the weight of a mass of 1 lb., the number which k represents must be extremely small; for when m and M and d are each put equal to 1 , kmM/d^2 becomes the gravitational attraction of two equal masses of 1 lb. at the distance of one mile, and this is quite inappreciable.
However, we have now got our formula for the force of attraction. If we call this force F , it is F = kmM/d^2 , giving the correlation between the variables F , m , M , and d. We all know the story of how it was found out. Newton, it states, was sitting in an orchard and watched the fall of an apple, and then the law of universal gravitation burst upon
his mind. It may be that the final formulation of the law occurred to him in an orchard, as well as elsewhere--and he must have been somewhere. But for our purposes it is more instructive to dwell upon the vast amount of preparatory thought, the product of many minds and many centuries, which was necessary before this exact law could be formulated. In the first place, the mathematical habit of mind and the mathematical procedure explained in the previous two chapters had to be generated; otherwise Newton could never have thought of a formula representing the force between any two masses
at any distance. Again, what are the meanings
of the terms employed, Force, Mass, Distance?
Take the easiest of these terms, Distance. It seems very obvious to us to conceive all material things as forming a definite geometrical whole, such that the distances of the various parts are measurable in terms of some unit length, such as a mile or a yard. This is almost the first aspect of a material structure which occurs to us. It is the gradual outcome of the study of geometry and of the theory of measurement. Even now, in certain cases, other modes of thought are convenient. In a mountainous country distances are often reckoned in hours. But leaving distance, the other terms, Force and Mass, are much more obscure. The exact comprehension of the ideas which Newton
meant to convey by these words was of slow growth, and, indeed, Newton himself was the first man who had thoroughly mastered the true general principles of Dynamics.
Throughout the middle ages, under the influence of Aristotle, the science was entirely
misconceived. Newton had the advantage of coming after a series of great men, notably Galileo, in Italy, who in the previous two
centuries had reconstructed the science and had invented the right way of thinking about it. He completed their work. Then, finally, having the ideas of force, mass, and distance,
clear and distinct in his mind, and realising their importance and their relevance to the fall of an apple and the motions of the planets, he hit upon the law of gravitation and proved it to be the formula always satisfied in these various motions.
The vital point in the application of mathematical formulæ is to have clear ideas and a correct estimate of their relevance to the phenomena under observation. No less than ourselves, our remote ancestors were impressed with the importance of natural phenomena and with the desirability of taking energetic measures to regulate the sequence of events. Under the influence of irrelevant ideas they executed elaborate religious ceremonies to aid the birth of the new moon, and performed sacrifices to save the sun during the crisis of an eclipse. There is no reason to believe that they were more stupid than we are. But at that epoch there had not been opportunity for the slow accumulation of clear and relevant ideas.
The sort of way in which physical sciences
grow into a form capable of treatment by mathematical methods is illustrated by the history of the gradual growth of the science of electromagnetism. Thunderstorms are events on a grand scale, arousing terror in men and even animals. From the earliest times they must have been objects of wild
and fantastic hypotheses, though it may be doubted whether our modern scientific discoveries in connection with electricity are not more astonishing than any of the magical explanations of savages. The Greeks knew that amber (Greek, electron) when rubbed would attract light and dry bodies. In 1600 A.D., Dr. Gilbert, of Colchester, published
the first work on the subject in which any scientific method is followed. He made a list of substances possessing properties similar to those of amber; he must also have the credit of connecting, however vaguely, electric and magnetic phenomena. At the end of the seventeenth and throughout the eighteenth century knowledge advanced. Electrical machines were made, sparks were obtained from them; and the Leyden Jar was invented, by which these effects could be intensified. Some organised knowledge was being obtained; but still no relevant mathematical ideas had been found out. Franklin,
in the year 1752, sent a kite into the clouds and proved that thunderstorms were electrical.
Meanwhile from the earliest epoch (2634 B.C.) the Chinese had utilized the characteristic property of the compass needle, but do not seem to have connected it with any theoretical ideas. The really profound changes in human life all have their ultimate origin in knowledge
pursued for its own sake. The use of the compass was not introduced into Europe till the end of the twelfth century A.D., more than 3000 years after its first use in China. The importance which the science of electromagnetism has since assumed in every department of human life is not due to the superior practical bias of Europeans, but to the fact that in the West electrical and magnetic phenomena were studied by men who were dominated by abstract theoretic interests.
The discovery of the electric current is due
to two Italians, Galvani in 1780, and Volta
in 1792. This great invention opened a new series of phenomena for investigation. The scientific world had now three separate, though allied, groups of occurrences on hand--the effects of "statical" electricity arising from frictional electrical machines, the magnetic phenomena, and the effects due to electric currents. From the end of the eighteenth century onwards, these three lines of investigation were quickly interconnected and the modern science of electromagnetism was constructed, which now threatens to transform human life.
Mathematical ideas now appear. During the decade 1780 to 1789, Coulomb, a Frenchman,
proved that magnetic poles attract or repel each other, in proportion to the inverse square of their distances, and also that the
same law holds for electric charges--laws curiously analogous to that of gravitation. In 1820, Öersted, a Dane, discovered that
electric currents exert a force on magnets, and almost immediately afterwards the mathematical law of the force was correctly formulated by Ampère, a Frenchman, who
also proved that two electric currents exerted forces on each other. "The experimental investigation by which Ampère established the law of the mechanical action between electric currents is one of the most brilliant achievements in science. The whole, theory and experiment, seems as if it had leaped, full-grown and full armed, from the brain of the 'Newton of Electricity.' It is perfect
in form, and unassailable in accuracy, and it is summed up in a formula from which all the phenomena may be deduced, and which must always remain the cardinal formula of electro-dynamics." (Electricity and Magnetism, Clerk Maxwell, Vol. II.,
ch. iii.)
The momentous laws of induction between currents and between currents and magnets were discovered by Michael Faraday in 1831--82.
Faraday was asked: "What is the use of this discovery?" He answered: "What is the use of a child--it grows to be a man." Faraday's child has grown to be a man and is now the basis of all the modern applications
of electricity. Faraday also reorganized the whole theoretical conception of the science. His ideas, which had not been fully understood by the scientific world, were extended and put into a directly mathematical form by Clerk Maxwell in 1873. As a result of his
mathematical investigations, Maxwell recognized that, under certain conditions, electrical vibrations ought to be propagated. He at once suggested that the vibrations which form light are electrical. This suggestion has
since been verified, so that now the whole theory of light is nothing but a branch of the
great science of electricity. Also Herz, a
German, in 1888, following on Maxwell's ideas, succeeded in producing electric vibrations by direct electrical methods. His experiments are the basis of our wireless telegraphy.
In more recent years even more fundamental discoveries have been made, and the science continues to grow in theoretic importance and in practical interest. This rapid sketch of its progress illustrates how, by the gradual introduction of the relevant theoretic ideas, suggested by experiment and themselves suggesting fresh experiments, a whole mass of isolated and even trivial phenomena are welded together into one coherent science, in which the results of abstract mathematical deductions, starting from a few simple assumed
laws, supply the explanation to the complex tangle of the course of events.
Finally, passing beyond the particular sciences of electromagnetism and light, we can generalize our point of view still further, and direct our attention to the growth of mathematical physics considered as one great chapter of scientific thought. In the first place, what in the barest outlines is the story of its growth?
It did not begin as one science, or as the product of one band of men. The Chaldean shepherds watched the skies, the agents of Government in Mesopotamia and Egypt measured the land, priests and philosophers brooded on the general nature of all things. The vast mass of the operations of nature appeared due to mysterious unfathomable forces. "The wind bloweth where it listeth" expresses accurately the blank ignorance then existing of any stable rules followed in detail by the succession of phenomena. In broad outline, then as now, a regularity of events was patent. But no minute tracing of their interconnection was possible, and there was no knowledge how even to set about to construct such a science.
Detached speculations, a few happy or unhappy shots at the nature of things, formed the utmost which could be produced.
Meanwhile land-surveys had produced geometry,
and the observations of the heavens disclosed the exact regularity of the solar system. Some of the later Greeks, such as Archimedes, had just views on the elementary
phenomena of hydrostatics and optics. Indeed, Archimedes, who combined a genius for mathematics with a physical insight, must rank with Newton, who lived nearly two
thousand years later, as one of the founders of mathematical physics. He lived at Syracuse, the great Greek city of Sicily. When the Romans besieged the town (in 212 to 210 B.C.), he is said to have burned their ships by concentrating on them, by means of mirrors, the sun's rays. The story is highly improbable, but is good evidence of the reputation which he had gained among his contemporaries for his knowledge of optics. At the end of this siege he was killed. According to one account given by Plutarch, in his life of
Marcellus, he was found by a Roman soldier
absorbed in the study of a geometrical diagram which he had traced on the sandy floor of his room. He did not immediately obey the orders of his captor, and so was killed. For the credit of the Roman generals it must be said that the soldiers had orders to spare him. The internal evidence for the other famous story of him is very strong; for the discovery attributed to him is one eminently worthy of his genius for mathematical and physical research.
Luckily, it is simple enough to be explained here in detail. It is one of the best easy examples of the method of application of mathematical ideas to physics.
Hiero, King of Syracuse, had sent a quantity
of gold to some goldsmith to form the material of a crown. He suspected that the craftsmen had abstracted some of the gold and had supplied its place by alloying the remainder with some baser metal. Hiero sent the crown to Archimedes and asked him to test it. In these days an indefinite number of chemical tests would be available. But then Archimedes had to think out the matter afresh. The solution flashed upon him as he lay in his bath. He jumped up and ran through the streets to the palace, shouting Eureka! Eureka! (I have found it, I have found it). This day, if we knew which it was, ought to be celebrated as the birthday of mathematical physics; the science came of age when Newton sat in his
orchard. Archimedes had in truth made a great discovery. He saw that a body when immersed in water is pressed upwards by the surrounding water with a resultant force equal to the weight of the water it displaces. This law can be proved theoretically from the mathematical principles of hydrostatics and can also be verified experimentally. Hence, if W lb. be the weight of the crown, as weighed
in air, and w lb. be the weight of the water which it displaces when completely immersed, W - w would be the extra upward force necessary to sustain the crown as it hung in water.
Now, this upward force can easily be ascertained by weighing the body as it hangs in water, as shown in the annexed figure. If
the weights in the right-hand scale come to F lb., then the apparent weight of the crown in water is F lb.; and we thus have F = W - w and thus w = W - F, and W/w = W/W - F (A) where W and F are determined by the easy, and fairly precise, operation of weighing.
Hence, by equation (A), W/w is known. But W/w is the ratio of the weight of the crown to the weight of an equal volume of water. This ratio is the same for any lump of metal of the same material: it is now called the specific gravity of the material, and depends only on the intrinsic nature of the substance and not on its shape or quantity. Thus to test if the crown were of gold, Archimedes had only to take a lump of indisputably pure gold and find its specific gravity by the same process. If the two specific gravities agreed, the crown was pure; if they disagreed, it was debased.
This argument has been given at length, because not only is it the first precise example of the application of mathematical ideas to physics, but also because it is a perfect and simple example of what must be the method and spirit of the science for all time.
The death of Archimedes by the hands of a Roman soldier is symbolical of a world-change of the first magnitude: the theoretical Greeks, with their love of abstract science, were superseded in the leadership of the European world by the practical Romans. Lord Beaconsfield,
in one of his novels, has defined a practical man as a man who practises the errors of his forefathers. The Romans were a great race, but they were cursed with the sterility
which waits upon practicality. They did not improve upon the knowledge of their forefathers, and all their advances were confined to the minor technical details of engineering. They were not dreamers enough to arrive at new points of view, which could give a more fundamental control over the forces of nature. No Roman lost his life because he was absorbed in the contemplation of a mathematical diagram.
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End of part 3 of 19
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