An Introduction to Mathematics

"The study of mathematics is apt to commence in disappointment. ...[L]ike the ghost of Hamlet's father, this great science eludes the efforts of our mental weapons to grasp it—'"Tis here, 'tis there, 'tis gone"...—and too noble for our gross methods."

""A show of violence,"...may surely be "offered" to the trivial results which occupy the pages of some elementary mathematical treatises."

"[I]ts fundamental ideas are not explained to the student disentangled from the technical procedure which has been invented to facilitate their exact presentation in particular instances. ...[T]he unfortunate learner finds himself struggling to acquire a knowledge of a mass of details which are not illuminated by any general conception."

"[I]t is... an error to confine attention to technical processes, excluding consideration of general ideas. Here lies the road to pedantry."

"The object of the following chapters is not to teach mathematics, but to enable students from the very beginning... to know what the science is about, and why it is necessarily the foundation of exact thought as applied to natural phenomena."

"Arithmetic... will be a good subject to consider in order to discover, if possible, the most obvious characteristic of the science."

"[A]rithmetic... applies to everything... of all things it is true that two and two make four. Thus... mathematics... deals with properties and ideas which are applicable to things just because they are things, and apart from... feelings, or emotions, or sensations... This is what is meant by calling mathematics an abstract science."

"Swift, in his description of Gulliver's voyage to ... describes the mathematicians of that country as silly and useless dreamers, whose attention has to be awakened by flappers. ...Swift ...lived at a time peculiarly unsuited for gibes at contemporary mathematicians. Newton's Principia had just been written, one of the great forces which have transformed the modern world. Swift might just as well have laughed at an earthquake."

"The progress of science consists in observing... interconnections and in showing with a patient ingenuity that the events of this evershifting world are but examples of a few general connections or relations called laws."

"To see what is general in what is particular and what is permanent in what is transitory is the aim of scientific thought."

"In the eye of science, the fall of an apple, the motion of a planet round a sun, and the clinging of the atmosphere to the earth are all seen as examples of the law of gravity. This possibility of disentangling the most complex evanescent circumstances into various examples of permanent laws is the controlling idea of modern thought."

"[W]e ascribe the origin of... sensations to relations between the things which form the external world."

"[W]e... endeavour to imagine the world as one connected set of things which underlies all the perceptions of all people."

"[W]e hear and we touch the same world as we see. ...[W]e want to describe the connections between these external things in some way which does not depend on any particular sensations, nor even on all the sensations of any particular person."

"The laws... are to be described, if possible, in a neutral universal fashion, the same for blind men as for deaf men, and the same for beings with faculties beyond our ken as for normal human beings."

"But when we have put aside our immediate sensations, the most serviceable part—from its clearness, definiteness, and universality—of what is left is composed of our general ideas of the abstract formal properties of things... the abstract mathematical ideas..."

"[S]tep by step, and not realizing the full meaning of the process, mankind has been led to search for a mathematical description of the properties of the universe, because in this way only can a general idea of the course of events be formed, freed from reference to particular persons or to particular types of sensation."

"[S]cience seeks to describe an apple in terms of the positions and motions of molecules, a description which ignores me and you and him, and also ignores sight and touch and taste and smell."

"[M]athematical ideas, because they are abstract, supply just what is wanted [needed] for a scientific description of the course of events. This point has usually been misunderstood, from being thought of in too narrow a way. Pythagoras had a glimpse of it when he proclaimed that number was the source of all things. In modern times the belief that the ultimate explanation of all things was to be found in Newtonian mechanics was an adumbration of the truth that all science as it grows towards perfection becomes mathematical in its ideas."

"Mathematics as a science commenced when first someone... proved propositions about any things or about some things, without specification of definite particular things. These propositions were first enunciated by the Greeks for geometry; and, accordingly, geometry was the great Greek mathematical science."

"The ideas of any and of some are introduced into algebra by the use of letters, instead of the definite numbers of arithmetic."

"[I]n the place of saying that 3 > 2, we generalize and say that if x be any number there exists some number (or numbers) y such that y > x. ...[T]his latter assumption... is of vital importance, both to philosophy and to mathematics; for by it the notion of infinity is introduced."

"After the rise of algebra the differential calculus was invented by Newton and Leibniz... a pause in the progress of the philosophy of mathematical thought occurred so far as these notions are concerned; and it was not till within the last few years that it has been realized how fundamental any and some are to the very nature of mathematics..."

"One of the causes of the apparent triviality of much of elementary algebra is the preoccupation of the text-books with the solution of equations."

"[T]he majority of interesting formulae, especially when the idea of some is present, involve more than one variable. For example, the consideration of the pairs of numbers x and y (fractional or integral) which satisfy x + y = 1 involves the idea of two correlated variables, x and y. When two variables are present the same two main types of statement occur. For example, (1) for any pair of numbers, x and y, x+ y=y + x, and (2) for some pairs of numbers, x and y, x + y = 1."

"[T]here 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... Accordingly, however accurately we have calculated that some event must occur, the doubt always remains—Is the law true? ...[W]e have no faculty capable of observation with ideal precision, so... 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... the force on either body... is proportional to \frac{mM}{d^2}; thus this force can be written as equal to \frac{kmM}{d^2} where k is a definite number depending on the absolute magnitude of this attraction and... the scale by which we... measure forces. ...[W]e have now got our formula for the force of attraction... F = k\frac{mM}{d^2}, giving the correlation between the variables F, m, M, and d."

"[I]t 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... had to be generated; otherwise Newton could never have thought of a formula representing the force between any two masses at any distance."

"[W]hat are the meanings of the terms employed, Force, Mass, Distance? Take the easiest of these terms, Distance. ...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... 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... 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 realizing 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 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. ...The Greeks knew that (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 was invented, by which these effects could be intensified. Some organized knowledge was being obtained; but still no relevent 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 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 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 [these] phenomena were studied by men who were dominated by abstract theoretic interests."

"The discovery of the is due to... Galvani in 1780, and Volta in 1792. This... 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 inter-connected 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... 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, Oersted... discovered that electric currents exert a force on magnets, and almost immediately afterwards the mathematical law of the force was correctly formulated by Ampere... who also proved that two electric currents exerted forces on each other."

"The momentous laws of induction between currents and between currents and magnets were discovered by Michael Faraday in 1831. ...Faraday's child... 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."

"Herz... in 1888, following on Maxwell's ideas, succeeded in producing electric vibrations by direct electrical methods. His experiments are the basis of our ."

"[B]y 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."

"[W]e can generalize our point of view still further, and direct our attention to the growth [in the barest outlines] of mathematical physics considered as one great chapter of scientific thought. ...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. ...[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."

"[L]and-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 and optics. Indeed, Archimedes, who combined a genius for mathematics with physical insight, must rank with Newton, who lived nearly two thousand years later, as one of the founders of mathematical physics."

"In these days an infinite number of chemical tests would be available. But then Archimedes had to think... 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! ...) This day... ought to be celebrated as the birthday of mathematical physics; the science [that later] came of age when Newton sat in his orchard. Archimedes... had 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. ...Hence if W lb. be the [known] weight of the crown, as weighed in air, and w lb. be the [unknown] weight of the water which it displaces when completely immersed, W - w [from which (knowing W) the weight w of the equal volume of water can be derived,] would be the extra upward force necessary to sustain the crown as it hung in the water. [Alternatively, the weight of water, equaling the volume of the crown, and overflowing a tub, could be weighed directly.] Now, this upward force can easily be obtained by weighing the body as it hangs in the water [Fig. 3]...But \frac{w}{W} ...is the same for any lump of metal of the same material: it is now called the ... Archimedes had only to take a lump of indisputably pure gold and find its specific gravity by the same process. ...[N]ot only' is it the first precise example of the application of mathematical ideas to physics, but also... a perfect and simple example of what must be the method and spirit of the science for all time. The discovery of the theory of specific gravity marks a genius of the first rank."

"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 [as did Archimedes] because he was absorbed in the contemplation of a mathematical diagram."

"The world had to wait for eighteen hundred years till the Greek mathematical physicists found successors. In the sixteenth and seventeenth centuries... Leonardo da Vinci... and Galileo... rediscovered the secret, known to Archimedes, of relating abstract mathematical ideas with the experimental investigation of natural phenomena."

"[T]he slow advance of mathematics and the accumulation of accurate astronomical knowledge had placed natural philosophers in a much more advantageous position for research. ...[T]he very egoistic self-assertion of that age, its greediness for personal experience, led its thinkers to want to see for themselves... and the secret of the relation of mathematical theory and experiment in inductive reasoning was practically discovered."

"There are always men of thought and men of action; mathematical physics is the product of an age which combined in the same men impulses to thought with impulses to action."

"The first law of motion, as following Newton we now enunciate it, is—Every body continues in its state of rest or of uniform motion in a straight line, except so far as it is compelled by impressed force to change that state. This law... is... a paean of triumph over defeated heretics. The point at issue... the Aristotelian opposition formula: "Every body continues in its state of rest except so far as it is compelled by impressed force to change that state." In this last false formula it is asserted that, apart from force, a body continues in a state of rest; and accordingly that, if a body is moving, a force is required to sustain the motion; so that when the force ceases, the motion ceases."

"The true Newtonian law takes diametrically the opposite point of view. The state of a body unacted on by force is that of uniform motion in a straight line, and no external force or influence is to be looked for as the cause, or, if you like to put it so, as the invariable accompaniment of this uniform rectilinear motion. Rest is merely a particular case of such motion, merely when the velocity is and remains zero."

"[W]hen a body is moving, we do not seek for any external influence except to explain changes in the rate of the velocity or changes in its direction. So long as the body is moving at the same rate and in the same direction there is no need to invoke the aid of any forces."