I have been speaking of the timeless s.p.a.ces which are a.s.sociated with time-systems. These are the s.p.a.ces of physical science and of any concept of s.p.a.ce as eternal and unchanging. But what we actually perceive is an approximation to the instantaneous s.p.a.ce indicated by event-particles which lie within some moment of the time-system a.s.sociated with our awareness. The points of such an instantaneous s.p.a.ce are event-particles and the straight lines are rects. Let the time-system be named a, and let the moment of time-system a to which our quick perception of nature approximates be called M. Any straight line r in s.p.a.ce a is a locus of points and each point is a point-track which is a locus of event-particles. Thus in the four-dimensional geometry of all event-particles there is a two-dimensional locus which is the locus of all event-particles on points lying on the straight line r. I will call this locus of event-particles the matrix of the straight line r. A matrix intersects any moment in a rect. Thus the matrix of r intersects the moment M in a rect ?. Thus ? is the instantaneous rect in M which occupies at the moment M the straight line r in the s.p.a.ce of a.
Accordingly when one sees instantaneously a moving being and its path ahead of it, what one really sees is the being at some event-particle A lying in the rect ? which is the apparent path on the a.s.sumption of uniform motion. But the actual rect ? which is a locus of event-particles is never traversed by the being. These event-particles are the instantaneous facts which pa.s.s with the instantaneous moment.
What is really traversed are other event-particles which at succeeding instants occupy the same points of s.p.a.ce a as those occupied by the event-particles of the rect ?. For example, we see a stretch of road and a lorry moving along it. The instantaneously seen road is a portion of the rect ?--of course only an approximation to it. The lorry is the moving object. But the road as seen is never traversed. It is thought of as being traversed because the intrinsic characters of the later events are in general so similar to those of the instantaneous road that we do not trouble to discriminate. But suppose a land mine under the road has been exploded before the lorry gets there. Then it is fairly obvious that the lorry does not traverse what we saw at first. Suppose the lorry is at rest in s.p.a.ce . Then the straight line r of s.p.a.ce a is in the direction of in s.p.a.ce a, and the rect ? is the representative in the moment M of the line r of s.p.a.ce a. The direction of ? in the instantaneous s.p.a.ce of the moment M is the direction of in M, where M is a moment of time-system a. Again the matrix of the line r of s.p.a.ce a will also be the matrix of some line s of s.p.a.ce which will be in the direction of a in s.p.a.ce . Thus if the lorry halts at some point P of s.p.a.ce a which lies on the line r, it is now moving along the line s of s.p.a.ce . This is the theory of relative motion; the common matrix is the bond which connects the motion of in s.p.a.ce a with the motions of a in s.p.a.ce .
Motion is essentially a relation between some object of nature and the one timeless s.p.a.ce of a time-system. An instantaneous s.p.a.ce is static, being related to the static nature at an instant. In perception when we see things moving in an approximation to an instantaneous s.p.a.ce, the future lines of motion as immediately perceived are rects which are never traversed. These approximate rects are composed of small events, namely approximate routes and event-particles, which are pa.s.sed away before the moving objects reach them. a.s.suming that our forecasts of rectilinear motion are correct, these rects occupy the straight lines in timeless s.p.a.ce which are traversed. Thus the rects are symbols in immediate sense-awareness of a future which can only be expressed in terms of timeless s.p.a.ce.
We are now in a position to explore the fundamental character of perpendicularity. Consider the two time-systems a and , each with its own timeless s.p.a.ce and its own family of instantaneous moments with their instantaneous s.p.a.ces. Let M and N be respectively a moment of a and a moment of . In M there is the direction of and in N there is the direction of a. But M and N, being moments of different time-systems, intersect in a level. Call this level ?. Then ? is an instantaneous plane in the instantaneous s.p.a.ce of M and also in the instantaneous s.p.a.ce of N. It is the locus of all the event-particles which lie both in M and in N.
In the instantaneous s.p.a.ce of M the level ? is perpendicular to the direction of in M, and in the instantaneous s.p.a.ce of N the level ?
is perpendicular to the direction of a in N. This is the fundamental property which forms the definition of perpendicularity. The symmetry of perpendicularity is a particular instance of the symmetry of the mutual relations between two time-systems. We shall find in the next lecture that it is from this symmetry that the theory of congruence is deduced.
The theory of perpendicularity in the timeless s.p.a.ce of any time-system a follows immediately from this theory of perpendicularity in each of its instantaneous s.p.a.ces. Let ? be any rect in the moment M of a and let ? be a level in M which is perpendicular to ?. The locus of those points of the s.p.a.ce of a which intersect M in event-particles on ? is the straight line r of s.p.a.ce a, and the locus of those points of the s.p.a.ce of a which intersect M in event-particles on ? is the plane l of s.p.a.ce a. Then the plane l is perpendicular to the line r.
In this way we have pointed out unique and definite properties in nature which correspond to perpendicularity. We shall find that this discovery of definite unique properties defining perpendicularity is of critical importance in the theory of congruence which is the topic for the next lecture.
I regret that it has been necessary for me in this lecture to administer such a large dose of four-dimensional geometry. I do not apologise, because I am really not responsible for the fact that nature in its most fundamental aspect is four-dimensional. Things are what they are; and it is useless to disguise the fact that "what things are" is often very difficult for our intellects to follow. It is a mere evasion of the ultimate problems to shirk such obstacles.
CHAPTER VI
CONGRUENCE
The aim of this lecture is to establish a theory of congruence. You must understand at once that congruence is a controversial question. It is the theory of measurement in s.p.a.ce and in time. The question seems simple. In fact it is simple enough for a standard procedure to have been settled by act of parliament; and devotion to metaphysical subtleties is almost the only crime which has never been imputed to any English parliament. But the procedure is one thing and its meaning is another.
First let us fix attention on the purely mathematical question. When the segment between two points A and B is congruent to that between the two points C and D, the quant.i.tative measurements of the two segments are equal. The equality of the numerical measures and the congruence of the two segments are not always clearly discriminated, and are lumped together under the term equality. But the procedure of measurement presupposes congruence. For example, a yard measure is applied successively to measure two distances between two pairs of points on the floor of a room. It is of the essence of the procedure of measurement that the yard measure remains unaltered as it is transferred from one position to another. Some objects can palpably alter as they move--for example, an elastic thread; but a yard measure does not alter if made of the proper material. What is this but a judgment of congruence applied to the train of successive positions of the yard measure? We know that it does not alter because we judge it to be congruent to itself in various positions. In the case of the thread we can observe the loss of self-congruence. Thus immediate judgments of congruence are presupposed in measurement, and the process of measurement is merely a procedure to extend the recognition of congruence to cases where these immediate judgments are not available.
Thus we cannot define congruence by measurement.
In modern expositions of the axioms of geometry certain conditions are laid down which the relation of congruence between segments is to satisfy. It is supposed that we have a complete theory of points, straight lines, planes, and the order of points on planes--in fact, a complete theory of non-metrical geometry. We then enquire about congruence and lay down the set of conditions--or axioms as they are called--which this relation satisfies. It has then been proved that there are alternative relations which satisfy these conditions equally well and that there is nothing intrinsic in the theory of s.p.a.ce to lead us to adopt any one of these relations in preference to any other as the relation of congruence which we adopt. In other words there are alternative metrical geometries which all exist by an equal right so far as the intrinsic theory of s.p.a.ce is concerned.
Poincare, the great French mathematician, held that our actual choice among these geometries is guided purely by convention, and that the effect of a change of choice would be simply to alter our expression of the physical laws of nature. By "convention" I understand Poincare to mean that there is nothing inherent in nature itself giving any peculiar _role_ to one of these congruence relations, and that the choice of one particular relation is guided by the volitions of the mind at the other end of the sense-awareness. The principle of guidance is intellectual convenience and not natural fact.
This position has been misunderstood by many of Poincare"s expositors.
They have muddled it up with another question, namely that owing to the inexact.i.tude of observation it is impossible to make an exact statement in the comparison of measures. It follows that a certain subset of closely allied congruence relations can be a.s.signed of which each member equally well agrees with that statement of observed congruence when the statement is properly qualified with its limits of error.
This is an entirely different question and it presupposes a rejection of Poincare"s position. The absolute indetermination of nature in respect of all the relations of congruence is replaced by the indetermination of observation with respect to a small subgroup of these relations.
Poincare"s position is a strong one. He in effect challenges anyone to point out any factor in nature which gives a preeminent status to the congruence relation which mankind has actually adopted. But undeniably the position is very paradoxical. Bertrand Russell had a controversy with him on this question, and pointed out that on Poincare"s principles there was nothing in nature to determine whether the earth is larger or smaller than some a.s.signed billiard ball. Poincare replied that the attempt to find reasons in nature for the selection of a definite congruence relation in s.p.a.ce is like trying to determine the position of a ship in the ocean by counting the crew and observing the colour of the captain"s eyes.
In my opinion both disputants were right, a.s.suming the grounds on which the discussion was based. Russell in effect pointed out that apart from minor inexact.i.tudes a determinate congruence relation is among the factors in nature which our sense-awareness posits for us. Poincare asks for information as to the factor in nature which might lead any particular congruence relation to play a preeminent _role_ among the factors posited in sense-awareness. I cannot see the answer to either of these contentions provided that you admit the materialistic theory of nature. With this theory nature at an instant in s.p.a.ce is an independent fact. Thus we have to look for our preeminent congruence relation amid nature in instantaneous s.p.a.ce; and Poincare is undoubtedly right in saying that nature on this hypothesis gives us no help in finding it.
On the other hand Russell is in an equally strong position when he a.s.serts that, as a fact of observation, we do find it, and what is more agree in finding the same congruence relation. On this basis it is one of the most extraordinary facts of human experience that all mankind without any a.s.signable reason should agree in fixing attention on just one congruence relation amid the indefinite number of indistinguishable compet.i.tors for notice. One would have expected disagreement on this fundamental choice to have divided nations and to have rent families.
But the difficulty was not even discovered till the close of the nineteenth century by a few mathematical philosophers and philosophic mathematicians. The case is not like that of our agreement on some fundamental fact of nature such as the three dimensions of s.p.a.ce. If s.p.a.ce has only three dimensions we should expect all mankind to be aware of the fact, as they are aware of it. But in the case of congruence, mankind agree in an arbitrary interpretation of sense-awareness when there is nothing in nature to guide it.
I look on it as no slight recommendation of the theory of nature which I am expounding to you that it gives a solution of this difficulty by pointing out the factor in nature which issues in the preeminence of one congruence relation over the indefinite herd of other such relations.
The reason for this result is that nature is no longer confined within s.p.a.ce at an instant. s.p.a.ce and time are now interconnected; and this peculiar factor of time which is so immediately distinguished among the deliverances of our sense-awareness, relates itself to one particular congruence relation in s.p.a.ce.
Congruence is a particular example of the fundamental fact of recognition. In perception we recognise. This recognition does not merely concern the comparison of a factor of nature posited by memory with a factor posited by immediate sense-awareness. Recognition takes place within the present without any intervention of pure memory. For the present fact is a duration with its antecedent and consequent durations which are parts of itself. The discrimination in sense-awareness of a finite event with its quality of pa.s.sage is also accompanied by the discrimination of other factors of nature which do not share in the pa.s.sage of events. Whatever pa.s.ses is an event. But we find ent.i.ties in nature which do not pa.s.s; namely we recognise samenesses in nature. Recognition is not primarily an intellectual act of comparison; it is in its essence merely sense-awareness in its capacity of positing before us factors in nature which do not pa.s.s. For example, green is perceived as situated in a certain finite event within the present duration. This green preserves its self-ident.i.ty throughout, whereas the event pa.s.ses and thereby obtains the property of breaking into parts. The green patch has parts. But in talking of the green patch we are speaking of the event in its sole capacity of being for us the situation of green. The green itself is numerically one self-identical ent.i.ty, without parts because it is without pa.s.sage.
Factors in nature which are without pa.s.sage will be called objects.
There are radically different kinds of objects which will be considered in the succeeding lecture.
Recognition is reflected into the intellect as comparison. The recognised objects of one event are compared with the recognised objects of another event. The comparison may be between two events in the present, or it may be between two events of which one is posited by memory-awareness and the other by immediate sense-awareness. But it is not the events which are compared. For each event is essentially unique and incomparable. What are compared are the objects and relations of objects situated in events. The event considered as a relation between objects has lost its pa.s.sage and in this aspect is itself an object.
This object is not the event but only an intellectual abstraction. The same object can be situated in many events; and in this sense even the whole event, viewed as an object, can recur, though not the very event itself with its pa.s.sage and its relations to other events.
Objects which are not posited by sense-awareness may be known to the intellect. For example, relations between objects and relations between relations may be factors in nature not disclosed in sense-awareness but known by logical inference as necessarily in being. Thus objects for our knowledge may be merely logical abstractions. For example, a complete event is never disclosed in sense-awareness, and thus the object which is the sum total of objects situated in an event as thus inter-related is a mere abstract concept. Again a right-angle is a perceived object which can be situated in many events; but, though rectangularity is posited by sense-awareness, the majority of geometrical relations are not so posited. Also rectangularity is in fact often not perceived when it can be proved to have been there for perception. Thus an object is often known merely as an abstract relation not directly posited in sense-awareness although it is there in nature.
The ident.i.ty of quality between congruent segments is generally of this character. In certain special cases this ident.i.ty of quality can be directly perceived. But in general it is inferred by a process of measurement depending on our direct sense-awareness of selected cases and a logical inference from the transitive character of congruence.
Congruence depends on motion, and thereby is generated the connexion between spatial congruence and temporal congruence. Motion along a straight line has a symmetry round that line. This symmetry is expressed by the symmetrical geometrical relations of the line to the family of planes normal to it.
Also another symmetry in the theory of motion arises from the fact that rest in the points of corresponds to uniform motion along a definite family of parallel straight lines in the s.p.a.ce of a. We must note the three characteristics, (i) of the uniformity of the motion corresponding to any point of along its correlated straight line in a, and (ii) of the equality in magnitude of the velocities along the various lines of a correlated to rest in the various points of , and (iii) of the parallelism of the lines of this family.
We are now in possession of a theory of parallels and a theory of perpendiculars and a theory of motion, and from these theories the theory of congruence can be constructed. It will be remembered that a family of parallel levels in any moment is the family of levels in which that moment is intersected by the family of moments of some other time-system. Also a family of parallel moments is the family of moments of some one time-system. Thus we can enlarge our concept of a family of parallel levels so as to include levels in different moments of one time-system. With this enlarged concept we say that a complete family of parallel levels in a time-system a is the complete family of levels in which the moments of a intersect the moments of . This complete family of parallel levels is also evidently a family lying in the moments of the time-system . By introducing a third time-system ?, parallel rects are obtained. Also all the points of any one time-system form a family of parallel point-tracks. Thus there are three types of parallelograms in the four-dimensional manifold of event-particles.
In parallelograms of the first type the two pairs of parallel sides are both of them pairs of rects. In parallelograms of the second type one pair of parallel sides is a pair of rects and the other pair is a pair of point-tracks. In parallelograms of the third type the two pairs of parallel sides are both of them pairs of point-tracks.
The first axiom of congruence is that the opposite sides of any parallelogram are congruent. This axiom enables us to compare the lengths of any two segments either respectively on parallel rects or on the same rect. Also it enables us to compare the lengths of any two segments either respectively on parallel point-tracks or on the same point-track. It follows from this axiom that two objects at rest in any two points of a time-system are moving with equal velocities in any other time-system a along parallel lines. Thus we can speak of the velocity in a due to the time-system without specifying any particular point in . The axiom also enables us to measure time in any time-system; but does not enable us to compare times in different time-systems.
The second axiom of congruence concerns parallelograms on congruent bases and between the same parallels, which have also their other pairs of sides parallel. The axiom a.s.serts that the rect joining the two event-particles of intersection of the diagonals is parallel to the rect on which the bases lie. By the aid of this axiom it easily follows that the diagonals of a parallelogram bisect each other.
Congruence is extended in any s.p.a.ce beyond parallel rects to all rects by two axioms depending on perpendicularity. The first of these axioms, which is the third axiom of congruence, is that if ABC is a triangle of rects in any moment and D is the middle event-particle of the base BC, then the level through D perpendicular to BC contains A when and only when AB is congruent to AC. This axiom evidently expresses the symmetry of perpendicularity, and is the essence of the famous pons asinorum expressed as an axiom.
The second axiom depending on perpendicularity, and the fourth axiom of congruence, is that if r and A be a rect and an event-particle in the same moment and AB and AC be a pair of rectangular rects intersecting r in B and C, and AD and AE be another pair of rectangular rects intersecting r in D and E, then either D or E lies in the segment BC and the other one of the two does not lie in this segment. Also as a particular case of this axiom, if AB be perpendicular to r and in consequence AC be parallel to r, then D and E lie on opposite sides of B respectively. By the aid of these two axioms the theory of congruence can be extended so as to compare lengths of segments on any two rects.
Accordingly Euclidean metrical geometry in s.p.a.ce is completely established and lengths in the s.p.a.ces of different time-systems are comparable as the result of definite properties of nature which indicate just that particular method of comparison.
The comparison of time-measurements in diverse time-systems requires two other axioms. The first of these axioms, forming the fifth axiom of congruence, will be called the axiom of "kinetic symmetry." It expresses the symmetry of the quant.i.tative relations between two time-systems when the times and lengths in the two systems are measured in congruent units.
The axiom can be explained as follows: Let a and be the names of two time-systems. The directions of motion in the s.p.a.ce of a due to rest in a point of is called the "-direction in a" and the direction of motion in the s.p.a.ce of due to rest in a point of a is called the "a-direction in ." Consider a motion in the s.p.a.ce of a consisting of a certain velocity in the -direction of a and a certain velocity at right-angles to it. This motion represents rest in the s.p.a.ce of another time-system--call it p. Rest in p will also be represented in the s.p.a.ce of by a certain velocity in the a-direction in and a certain velocity at right-angles to this a-direction. Thus a certain motion in the s.p.a.ce of a is correlated to a certain motion in the s.p.a.ce of , as both representing the same fact which can also be represented by rest in p. Now another time-system, which I will name s, can be found which is such that rest in its s.p.a.ce is represented by the same magnitudes of velocities along and perpendicular to the a-direction in as those velocities in a, along and perpendicular to the -direction, which represent rest in p. The required axiom of kinetic symmetry is that rest in s will be represented in a by the same velocities along and perpendicular to the -direction in a as those velocities in along and perpendicular to the a-direction which represent rest in p.
A particular case of this axiom is that relative velocities are equal and opposite. Namely rest in a is represented in by a velocity along the a-direction which is equal to the velocity along the -direction in a which represents rest in .
Finally the sixth axiom of congruence is that the relation of congruence is transitive. So far as this axiom applies to s.p.a.ce, it is superfluous.
For the property follows from our previous axioms. It is however necessary for time as a supplement to the axiom of kinetic symmetry. The meaning of the axiom is that if the time-unit of system a is congruent to the time-unit of system , and the time-unit of system is congruent to the time-unit of system ?, then the time-units of a and ? are also congruent.
By means of these axioms formulae for the transformation of measurements made in one time-system to measurements of the same facts of nature made in another time-system can be deduced. These formulae will be found to involve one arbitrary constant which I will call k.
It is of the dimensions of the square of a velocity. Accordingly four cases arise. In the first case k is zero. This case produces nonsensical results in opposition to the elementary deliverances of experience. We put this case aside.
In the second case k is infinite. This case yields the ordinary formulae for transformation in relative motion, namely those formulae which are to be found in every elementary book on dynamics.
In the third case, k is negative. Let us call it -c, where c will be of the dimensions of a velocity. This case yields the formulae of transformation which Larmor discovered for the transformation of Maxwell"s equations of the electromagnetic field. These formulae were extended by H. A. Lorentz, and used by Einstein and Minkowski as the basis of their novel theory of relativity. I am not now speaking of Einstein"s more recent theory of general relativity by which he deduces his modification of the law of gravitation. If this be the case which applies to nature, then c must be a close approximation to the velocity of light _in vacuo_. Perhaps it is this actual velocity. In this connexion "_in vacuo_" must not mean an absence of events, namely the absence of the all-pervading ether of events. It must mean the absence of certain types of objects.
In the fourth case, k is positive. Let us call it h, where h will be of the dimensions of a velocity. This gives a perfectly possible type of transformation formulae, but not one which explains any facts of experience. It has also another disadvantage. With the a.s.sumption of this fourth case the distinction between s.p.a.ce and time becomes unduly blurred. The whole object of these lectures has been to enforce the doctrine that s.p.a.ce and time spring from a common root, and that the ultimate fact of experience is a s.p.a.ce-time fact. But after all mankind does distinguish very sharply between s.p.a.ce and time, and it is owing to this sharpness of distinction that the doctrine of these lectures is somewhat of a paradox. Now in the third a.s.sumption this sharpness of distinction is adequately preserved. There is a fundamental distinction between the metrical properties of point-tracks and rects. But in the fourth a.s.sumption this fundamental distinction vanishes.
Neither the third nor the fourth a.s.sumption can agree with experience unless we a.s.sume that the velocity c of the third a.s.sumption, and the velocity h of the fourth a.s.sumption, are extremely large compared to the velocities of ordinary experience. If this be the case the formulae of both a.s.sumptions will obviously reduce to a close approximation to the formulae of the second a.s.sumption which are the ordinary formulae of dynamical textbooks. For the sake of a name, I will call these textbook formulae the "orthodox" formulae.