An Introduction to Mathematics · Alfred North Whitehead
The world had to wait for eighteen hundred years till the Greek mathematical physicists found successors. In the sixteenth and seventeenth centuries of our era great Italians, in particular Leonardo da Vinci, the artist
(born 1452, died 1519), and Galileo (born 1564, died 1642), rediscovered the secret, known to Archimedes, of relating abstract mathematical ideas with the experimental investigation of natural phenomena. Meanwhile the slow advance of mathematics and the accumulation of accurate astronomical knowledge had placed natural philosophers in a much more advantageous position for research. Also the very egoistic self-assertion of that age, its greediness for personal experience, led its thinkers to want to see for themselves what happened; and the secret of the relation of mathematical theory and experiment in inductive reasoning was practically discovered. It was an act eminently characteristic of the age that Galileo, a philosopher, should have
dropped the weights from the leaning tower of Pisa. 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.
This matter of the dropping of weights from
the tower marks picturesquely an essential step in knowledge, no less a step than the first attainment of correct ideas on the science of dynamics, the basal science of the whole subject. The particular point in dispute was as to whether bodies of different weights would fall from the same height in the same time. According to a dictum of Aristotle, universally followed up to that epoch, the heavier weight would fall the quicker. Galileo affirmed that they would fall in the same time, and proved his point by dropping weights from the top of the leaning tower. The apparent exceptions to the rule all arise when, for some reason, such as extreme lightness or great speed, the air resistance is important. But neglecting the air the law is exact.
Galileo's successful experiment was not the
result of a mere lucky guess. It arose from his correct ideas in connection with inertia and mass. 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 more than a dry formula: it is also a pæan of triumph over defeated heretics. The point at issue can be understood by deleting from the law the phrase "or of uniform motion in a straight line." We there obtain what might be taken as 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. Thus, when 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.
The difference between the two points of view is well seen by reference to the theory of the motion of the planets. Copernicus, a
Pole, born at Thorn in West Prussia (born 1473, died 1543), showed how much simpler it was to conceive the planets, including the
earth as revolving round the sun in orbits which are nearly circular; and later, Kepler,
a German mathematician, in the year 1609 proved that, in fact, the orbits are practically ellipses, that is, a special sort of oval curves
which we will consider later in more detail. Immediately the question arose as to what are the forces which preserve the planets in this motion. According to the old false view,
held by Kepler, the actual velocity itself required
preservation by force. Thus he looked for tangential forces as in the accompanying figure (4). But according to the Newtonian law, apart from some force the planet would move for ever with its existing velocity in a straight line, and thus depart entirely from the sun. Newton, therefore, had to search
for a force which would bend the motion
round into its elliptical orbit. This he showed must be a force directed towards the sun as in the next figure (5). In fact, the force is the gravitational attraction of the sun acting according to the law of the inverse square of the distance, which has been stated above.
The science of mechanics rose among the
Greeks from a consideration of the theory of the mechanical advantage obtained by the use
of a lever, and also from a consideration of various problems connected with the weights of bodies. It was finally put on its true basis at the end of the sixteenth and during the seventeenth centuries, as the preceding account shows, partly with the view of explaining the theory of falling bodies, but chiefly in order to give a scientific theory of planetary motions. But since those days dynamics has taken upon itself a more ambitious task, and now claims to be the ultimate science of which the others are but branches. The claim amounts to this: namely, that the various qualities of things perceptible to the senses are merely our peculiar mode of appreciating changes in position on the part of things existing in space. For example, suppose we look at Westminster Abbey. It has been standing there, grey and immovable, for centuries past. But, according to modern scientific theory, that greyness, which so heightens our sense of the immobility of the building, is itself nothing but our way of appreciating the rapid motions of the ultimate molecules, which form the outer surface of the building and communicate vibrations to a substance called the ether. Again we lay our hands on its stones and note their cool, even temperature, so symbolic of the quiet repose of the building. But this feeling of temperature simply marks our sense of the transfer of heat from the
hand to the stone, or from the stone to the hand; and, according to modern science, heat is nothing but the agitation of the molecules of a body. Finally, the organ begins playing, and again sound is nothing but the result of motions of the air striking on the drum of the ear.
Thus the endeavour to give a dynamical explanation of phenomena is the attempt to explain them by statements of the general form, that such and such a substance or body was in this place and is now in that place. Thus we arrive at the great basal idea of modern science, that all our sensations are the result of comparisons of the changed configurations of things in space at various times. It follows therefore, that the laws of motion, that is, the laws of the changes of configurations of things, are the ultimate laws of physical science.
In the application of mathematics to the investigation of natural philosophy, science does systematically what ordinary thought does casually. When we talk of a chair, we usually mean something which we have been seeing or feeling in some way; though most of our language will presuppose that there is something which exists independently of our sight or feeling. Now in mathematical physics the opposite course is taken. The chair is conceived without any reference to
anyone in particular, or to any special modes of perception. The result is that the chair becomes in thought a set of molecules in space, or a group of electrons, a portion of the ether in motion, or however the current scientific ideas describe it. But the point is that science reduces the chair to things moving in space and influencing each other's motions. Then the various elements or factors which enter into a set of circumstances, as thus conceived, are merely the things, like lengths of lines, sizes of angles, areas, and volumes, by which the positions of bodies in space can be settled. Of course, in addition to these geometrical elements the fact of motion and change necessitates the introduction of the rates of changes of such elements, that is to say, velocities, angular velocities, accelerations, and suchlike things. Accordingly, mathematical physics deals with correlations between variable numbers which are supposed to represent the correlations which exist in nature between the measures of these geometrical elements and of their rates of change. But always the mathematical laws deal with variables, and it is only in the occasional testing of the laws by reference to experiments, or in the use of the laws for special predictions that definite numbers are substituted.
The interesting point about the world as
thus conceived in this abstract way throughout the study of mathematical physics, where only the positions and shapes of things are considered together with their changes, is that the events of such an abstract world are sufficient to "explain" our sensations. When we hear a sound, the molecules of the air have been agitated in a certain way: given the agitation, or air-waves as they are called, all normal people hear sound; and if there are no air-waves, there is no sound. And, similarly, a physical cause or origin, or parallel event (according as different people might like to phrase it) underlies our other sensations. Our very thoughts appear to correspond to conformations and motions of the brain; injure the brain and you injure the thoughts. Meanwhile the events of this physical universe succeed each other according to the mathematical laws which ignore all special sensations and thoughts and emotions.
Now, undoubtedly, this is the general aspect of the relation of the world of mathematical physics to our emotions, sensations, and thoughts; and a great deal of controversy has been occasioned by it and much ink spilled. We need only make one remark. The whole situation has arisen, as we have seen, from the endeavour to describe an external world "explanatory" of our various individual sensations and emotions, but a world
also, not essentially dependent upon any particular sensations or upon any particular individual. Is such a world merely but one huge fairy tale? But fairy tales are fantastic and arbitrary: if in truth there be such a world, it ought to submit itself to an exact description, which determines accurately its various parts and their mutual relations. Now, to a large degree, this scientific world does submit itself to this test and allow its events to be explored and predicted by the apparatus of abstract mathematical ideas. It certainly seems that here we have an inductive verification of our initial assumption. It must be admitted that no inductive proof is conclusive; but if the whole idea of a world which has existence independently of our particular perceptions of it be erroneous, it requires careful explanation why the attempt to characterise it, in terms of that mathematical remnant of our ideas which would apply to it, should issue in such a remarkable success.
It would take us too far afield to enter into
a detailed explanation of the other laws of motion. The remainder of this chapter must be devoted to the explanation of remarkable ideas which are fundamental, both to mathematical physics and to pure mathematics: these are the ideas of vector quantities and the parallelogram law for vector addition. We
have seen that the essence of motion is that a body was at A and is now at C. This transference from A to C requires two distinct elements to be settled before it is completely determined, namely its magnitude (i.e. the length AC ) and its direction. Now anything, like this transference, which is completely given by the determination of a magnitude
and a direction is called a vector. For example, a velocity requires for its definition the assignment of a magnitude and of a direction. It must be of so many miles per hour in such and such a direction. The existence and the independence of these two elements in the determination of a velocity are well illustrated by the action of the captain of a ship, who communicates with different subordinates respecting them: he tells the chief engineer the number of knots at which he is to steam, and the helmsman the compass
bearing of the course which he is to keep. Again the rate of change of velocity, that is velocity added per unit time, is also a vector quantity: it is called the acceleration. Similarly a force in the dynamical sense is another vector quantity. Indeed, the vector nature of forces follows at once according to dynamical principles from that of velocities and accelerations; but this is a point which we need not go into. It is sufficient here to say that a force acts on a body with a certain magnitude in a certain direction.
Now all vectors can be graphically represented by straight lines. All that has to be done is to arrange: (i) a scale according to which units of length correspond to units of magnitude of the vector--for example, one inch to a velocity of 10 miles per hour in the case of velocities, and one inch to a force of 10 tons weight in the case of forces--and (ii) a direction of the line on the diagram corresponding to the direction of the vector. Then a line drawn with the proper number of inches of length in the proper direction represents the required vector on the arbitrarily assigned scale of magnitude. This diagrammatic representation of vectors is of the first importance. By its aid we can enunciate the famous "parallelogram law" for the addition of vectors of the same kind but in different directions.
Consider the vector AC in figure 6 as representative
of the changed position of a body from A to C : we will call this the vector of transportation. It will be noted that, if the reduction of physical phenomena to mere changes in positions, as explained above, is correct, all other types of physical vectors are really reducible in some way or other to this single type. Now the final transportation from A to C is equally well effected by a transportation from A to B and a transportation from B to C , or, completing the parallelogram ABCD , by a transportation from A to D and a transportation from D to C. These transportations as thus successively applied are said to be added together. This is simply a definition of what we mean by the addition of transportations. Note further that, considering parallel lines as being lines drawn in the same direction, the transportations B to C and A to D may be conceived as the same transportation applied to bodies in the two initial positions B and A. With this conception we may talk of the transportation A to D as applied to a body in any position, for example at B. Thus we may say that the transportation A to C can be conceived as the sum of the two transportations A to B and A to D applied in any order. Here we have the parallelogram law for the addition of transportations: namely, if the transportations are A to B and A to D ,
complete the parallelogram ABCD , and then the sum of the two is the diagonal AC.
All this at first sight may seem to be very artificial. But it must be observed that nature itself presents us with the idea. For example, a steamer is moving in the direction AD (cf. fig. 6) and a man walks across its deck. If the steamer were still, in one minute he would arrive at B ; but during that minute his starting point A on the deck has moved to D , and his path on the deck has moved from AB to DC. So that, in fact, his transportation has been from A to C over the surface of the sea. It is, however, presented to us analysed into the sum of two transportations, namely, one from A to B relatively to the steamer, and one from A to D which is the transportation of the steamer.
By taking into account the element of time, namely one minute, this diagram of the man's transportation AC represents his velocity. For if AC represented so many feet of transportation, it now represents a transportation of so many feet per minute, that is to say, it represents the velocity of the man. Then AB and AD represent two velocities, namely, his velocity relatively to the steamer, and the velocity of the steamer, whose "sum" makes up his complete velocity. It is evident that diagrams and definitions concerning transportations
are turned into diagrams and definitions concerning velocities by conceiving the diagrams as representing transportations per unit time. Again, diagrams and definitions concerning velocities are turned into diagrams and definitions concerning accelerations
by conceiving the diagrams as representing velocities added per unit time.
Thus by the addition of vector velocities and of vector accelerations, we mean the addition according to the parallelogram law.
Also, according to the laws of motion a force is fully represented by the vector acceleration it produces in a body of given mass. Accordingly, forces will be said to be added when their joint effect is to be reckoned according to the parallelogram law.
Hence for the fundamental vectors of
science, namely transportations, velocities, and forces, the addition of any two of the same kind is the production of a "resultant" vector according to the rule of the parallelogram law.
By far the simplest type of parallelogram is a rectangle, and in pure mathematics it is
the relation of the single vector AC to the two component vectors, AB and AD , at right angles (cf. fig. 7), which is continually recurring. Let x , y , and r units represent the lengths of AB , AD , and AC , and let m units of angle represent the magnitude of the angle BAC. Then the relations between x , y , r , and m , in all their many aspects are the continually recurring topic of pure mathematics; and the results are of the type required for application to the fundamental vectors of mathematical physics. This diagram is the chief bridge over which the results of pure mathematics pass in order to obtain application to the facts of nature.
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End of part 4 of 19
Continue readingChapter V: The Symbolism of Mathematics
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