I have discussed this example in order to make you realise that in thinking of the possibilities of measurement in the s.p.a.ce-time manifold, we must not confine ourselves merely to those minor variations which might seem natural to human beings on the earth. Let us make therefore the general statement that four measurements, respectively of independent types (such as measurements of lengths in three directions and a time), can be found such that a definite event-particle is determined by them in its relations to other parts of the manifold.
If (p1, p2, p3, p4) be a set of measurements of this system, then the event-particle which is thus determined will be said to have p1, p2, p3, p4 as its co-ordinates in this system of measurement. Suppose that we name it the p-system of measurement. Then in the same p-system by properly varying (p1, p2, p3, p4) every event-particle that has been, or will be, or instantaneously is now, can be indicated. Furthermore, according to any system of measurement that is natural to us, three of the co-ordinates will be measurements of s.p.a.ce and one will be a measurement of time. Let us always take the last co-ordinate to represent the time-measurement. Then we should naturally say that (p1, p2, p3) determined a point in s.p.a.ce and that the event-particle happened at that point at the time p4. But we must not make the mistake of thinking that there is a s.p.a.ce in addition to the s.p.a.ce-time manifold.
That manifold is all that there is for the determination of the meaning of s.p.a.ce and time. We have got to determine the meaning of a s.p.a.ce-point in terms of the event-particles of the four-dimensional manifold. There is only one way to do this. Note that if we vary the time and take times with the same three s.p.a.ce co-ordinates, then the event-particles, thus indicated, are all at the same point. But seeing that there is nothing else except the event-particles, this can only mean that the point (p1, p2, p3) of the s.p.a.ce in the p-system is merely the collection of event-particles (p1, p2, p3, [p4]), where p4 is varied and (p1, p2, p3) is kept fixed. It is rather disconcerting to find that a point in s.p.a.ce is not a simple ent.i.ty; but it is a conclusion which follows immediately from the relative theory of s.p.a.ce.
Furthermore the inhabitant of Mars determines event-particles by another system of measurements. Call his system the q-system. According to him (q1, q2, q3, q4) determines an event-particle, and (q1, q2, q3) determines a point and q4 a time. But the collection of event-particles which he thinks of as a point is entirely different from any such collection which the man on earth thinks of as a point. Thus the q-s.p.a.ce for the man on Mars is quite different from the p-s.p.a.ce for the land-surveyor on earth.
So far in speaking of s.p.a.ce we have been talking of the timeless s.p.a.ce of physical science, namely, of our concept of eternal s.p.a.ce in which the world adventures. But the s.p.a.ce which we see as we look about is instantaneous s.p.a.ce. Thus if our natural perceptions are adjustable to the p-system of measurements we see instantaneously all the event-particles at some definite time p4, and observe a succession of such s.p.a.ces as time moves on. The timeless s.p.a.ce is achieved by stringing together all these instantaneous s.p.a.ces. The points of an instantaneous s.p.a.ce are event-particles, and the points of an eternal s.p.a.ce are strings of event-particles occurring in succession. But the man on Mars will never perceive the same instantaneous s.p.a.ces as the man on the earth. This system of instantaneous s.p.a.ces will cut across the earth-man"s system. For the earth-man there is one instantaneous s.p.a.ce which is the instantaneous present, there are the past s.p.a.ces and the future s.p.a.ces. But the present s.p.a.ce of the man on Mars cuts across the present s.p.a.ce of the man on the earth. So that of the event-particles which the earth-man thinks of as happening now in the present, the man on Mars thinks that some are already past and are ancient history, that others are in the future, and others are in the immediate present. This break-down in the neat conception of a past, a present, and a future is a serious paradox. I call two event-particles which on some or other system of measurement are in the same instantaneous s.p.a.ce "co-present"
event-particles. Then it is possible that A and B may be co-present, and that A and C may be co-present, but that B and C may not be co-present. For example, at some inconceivable distance from us there are events co-present with us now and also co-present with the birth of Queen Victoria. If A and B are co-present there will be some systems in which A precedes B and some in which B precedes A. Also there can be no velocity quick enough to carry a material particle from A to B or from B to A. These different measure-systems with their divergences of time-reckoning are puzzling, and to some extent affront our common sense. It is not the usual way in which we think of the Universe. We think of one necessary time-system and one necessary s.p.a.ce.
According to the new theory, there are an indefinite number of discordant time-series and an indefinite number of distinct s.p.a.ces. Any correlated pair, a time-system and a s.p.a.ce-system, will do in which to fit our description of the Universe. We find that under given conditions our measurements are necessarily made in some one pair which together form our natural measure-system. The difficulty as to discordant time-systems is partly solved by distinguishing between what I call the creative advance of nature, which is not properly serial at all, and any one time series. We habitually muddle together this creative advance, which we experience and know as the perpetual transition of nature into novelty, with the single-time series which we naturally employ for measurement. The various time-series each measure some aspect of the creative advance, and the whole bundle of them express all the properties of this advance which are measurable. The reason why we have not previously noted this difference of time-series is the very small difference of properties between any two such series. Any observable phenomena due to this cause depend on the square of the ratio of any velocity entering into the observation to the velocity of light. Now light takes about fifty minutes to get round the earth"s...o...b..t; and the earth takes rather more than 17,531 half-hours to do the same. Hence all the effects due to this motion are of the order of the ratio of one to the square of 10,000. Accordingly an earth-man and a sun-man have only neglected effects whose quant.i.tative magnitudes all contain the factor 1/108. Evidently such effects can only be noted by means of the most refined observations. They have been observed however. Suppose we compare two observations on the velocity of light made with the same apparatus as we turn it through a right angle. The velocity of the earth relatively to the sun is in one direction, the velocity of light relatively to the ether should be the same in all directions. Hence if s.p.a.ce when we take the ether as at rest means the same thing as s.p.a.ce when we take the earth as at rest, we ought to find that the velocity of light relatively to the earth varies according to the direction from which it comes.
These observations on earth const.i.tute the basic principle of the famous experiments designed to detect the motion of the earth through the ether. You all know that, quite unexpectedly, they gave a null result.
This is completely explained by the fact that, the s.p.a.ce-system and the time-system which we are using are in certain minute ways different from the s.p.a.ce and the time relatively to the sun or relatively to any other body with respect to which it is moving.
All this discussion as to the nature of time and s.p.a.ce has lifted above our horizon a great difficulty which affects the formulation of all the ultimate laws of physics--for example, the laws of the electromagnetic field, and the law of gravitation. Let us take the law of gravitation as an example. Its formulation is as follows: Two material bodies attract each other with a force proportional to the product of their ma.s.ses and inversely proportional to the square of their distances.
In this statement the bodies are supposed to be small enough to be treated as material particles in relation to their distances; and we need not bother further about that minor point. The difficulty to which I want to draw your attention is this: In the formulation of the law one definite time and one definite s.p.a.ce are presupposed. The two ma.s.ses are a.s.sumed to be in simultaneous positions.
But what is simultaneous in one time-system may not be simultaneous in another time-system. So according to our new views the law is in this respect not formulated so as to have any exact meaning. Furthermore an a.n.a.logous difficulty arises over the question of distance. The distance between two instantaneous positions, _i.e._ between two event-particles, is different in different s.p.a.ce-systems. What s.p.a.ce is to be chosen?
Thus again the law lacks precise formulation, if relativity is accepted.
Our problem is to seek a fresh interpretation of the law of gravity in which these difficulties are evaded. In the first place we must avoid the abstractions of s.p.a.ce and time in the formulation of our fundamental ideas and must recur to the ultimate facts of nature, namely to events.
Also in order to find the ideal simplicity of expressions of the relations between events, we restrict ourselves to event-particles. Thus the life of a material particle is its adventure amid a track of event-particles strung out as a continuous series or path in the four-dimensional s.p.a.ce-time manifold. These event-particles are the various situations of the material particle. We usually express this fact by adopting our natural s.p.a.ce-time system and by talking of the path in s.p.a.ce of the material particle as it exists at successive instants of time.
We have to ask ourselves what are the laws of nature which lead the material particle to adopt just this path among event-particles and no other. Think of the path as a whole. What characteristic has that path got which would not be shared by any other slightly varied path? We are asking for more than a law of gravity. We want laws of motion and a general idea of the way to formulate the effects of physical forces.
In order to answer our question we put the idea of the attracting ma.s.ses in the background and concentrate attention on the field of activity of the events in the neighbourhood of the path. In so doing we are acting in conformity with the whole trend of scientific thought during the last hundred years, which has more and more concentrated attention on the field of force as the immediate agent in directing motion, to the exclusion of the consideration of the immediate mutual influence between two distant bodies. We have got to find the way of expressing the field of activity of events in the neighbourhood of some definite event-particle E of the four-dimensional manifold. I bring in a fundamental physical idea which I call the "impetus" to express this physical field. The event-particle E is related to any neighbouring event-particle P by an element of impetus. The a.s.semblage of all the elements of impetus relating E to the a.s.semblage of event-particles in the neighbourhood of E expresses the character of the field of activity in the neighbourhood of E. Where I differ from Einstein is that he conceives this quant.i.ty which I call the impetus as merely expressing the characters of the s.p.a.ce and time to be adopted and thus ends by talking of the gravitational field expressing a curvature in the s.p.a.ce-time manifold. I cannot attach any clear conception to his interpretation of s.p.a.ce and time. My formulae differ slightly from his, though they agree in those instances where his results have been verified. I need hardly say that in this particular of the formulation of the law of gravitation I have drawn on the general method of procedure which const.i.tutes his great discovery.
Einstein showed how to express the characters of the a.s.semblage of elements of impetus of the field surrounding an event-particle E in terms of ten quant.i.ties which I will call J_{11}, J_{12} (=J_{21}), J_{22}, J_{23}(=J_{32}), etc. It will be noted that there are four spatio-temporal measurements relating E to its neighbour P, and that there are ten pairs of such measurements if we are allowed to take any one measurement twice over to make one such pair. The ten J"s depend merely on the position of E in the four-dimensional manifold, and the element of impetus between E and P can be expressed in terms of the ten J"s and the ten pairs of the four spatio-temporal measurements relating E and P. The numerical values of the J"s will depend on the system of measurement adopted, but are so adjusted to each particular system that the same value is obtained for the element of impetus between E and P, whatever be the system of measurement adopted. This fact is expressed by saying that the ten J"s form a "tensor." It is not going too far to say that the announcement that physicists would have in future to study the theory of tensors created a veritable panic among them when the verification of Einstein"s predictions was first announced.
The ten J"s at any event-particle E can be expressed in terms of two functions which I call the potential and the "a.s.sociate-potential" at E. The potential is practically what is meant by the ordinary gravitation potential, when we express ourselves in terms of the Euclidean s.p.a.ce in reference to which the attracting ma.s.s is at rest.
The a.s.sociate-potential is defined by the modification of subst.i.tuting the direct distance for the inverse distance in the definition of the potential, and its calculation can easily be made to depend on that of the old-fashioned potential. Thus the calculation of the J"s--the coefficients of impetus, as I will call them--does not involve anything very revolutionary in the mathematical knowledge of physicists. We now return to the path of the attracted particle. We add up all the elements of impetus in the whole path, and obtain thereby what I call the "integral impetus." The characteristic of the actual path as compared with neighbouring alternative paths is that in the actual paths the integral impetus would neither gain nor lose, if the particle wobbled out of it into a small extremely near alternative path. Mathematicians would express this by saying, that the integral impetus is stationary for an infinitesimal displacement. In this statement of the law of motion I have neglected the existence of other forces. But that would lead me too far afield.
The electromagnetic theory has to be modified to allow for the presence of a gravitational field. Thus Einstein"s investigations lead to the first discovery of any relation between gravity and other physical phenomena. In the form in which I have put this modification, we deduce Einstein"s fundamental principle, as to the motion of light along its rays, as a first approximation which is absolutely true for infinitely short waves. Einstein"s principle, thus partially verified, stated in my language is that a ray of light always follows a path such that the integral impetus along it is zero. This involves that every element of impetus along it is zero.
In conclusion, I must apologise. In the first place I have considerably toned down the various exciting peculiarities of the original theory and have reduced it to a greater conformity with the older physics. I do not allow that physical phenomena are due to oddities of s.p.a.ce. Also I have added to the dullness of the lecture by my respect for the audience. You would have enjoyed a more popular lecture with ill.u.s.trations of delightful paradoxes. But I know also that you are serious students who are here because you really want to know how the new theories may affect your scientific researches.
CHAPTER IX
THE ULTIMATE PHYSICAL CONCEPTS
The second chapter of this book lays down the first principle to be guarded in framing our physical concept. We must avoid vicious bifurcation. Nature is nothing else than the deliverance of sense-awareness. We have no principles whatever to tell us what could stimulate mind towards sense-awareness. Our sole task is to exhibit in one system the characters and inter-relations of all that is observed.
Our att.i.tude towards nature is purely "behaviouristic," so far as concerns the formulation of physical concepts.
Our knowledge of nature is an experience of activity (or pa.s.sage). The things previously observed are active ent.i.ties, the "events." They are chunks in the life of nature. These events have to each other relations which in our knowledge differentiate themselves into s.p.a.ce-relations and time-relations. But this differentiation between s.p.a.ce and time, though inherent in nature, is comparatively superficial; and s.p.a.ce and time are each partial expressions of one fundamental relation between events which is neither spatial nor temporal. This relation I call "extension."
The relation of "extending over" is the relation of "including," either in a spatial or in a temporal sense, or in both. But the mere "inclusion" is more fundamental than either alternative and does not require any spatio-temporal differentiation. In respect to extension two events are mutually related so that either (i) one includes the other, or (ii) one overlaps the other without complete inclusion, or (iii) they are entirely separate. But great care is required in the definition of spatial and temporal elements from this basis in order to avoid tacit limitations really depending on undefined relations and properties.
Such fallacies can be avoided by taking account of two elements in our experience, namely, (i) our observational "present," and (ii) our "percipient event."
Our observational "present" is what I call a "duration." It is the whole of nature apprehended in our immediate observation. It has therefore the nature of an event, but possesses a peculiar completeness which marks out such durations as a special type of events inherent in nature. A duration is not instantaneous. It is all that there is of nature with certain temporal limitations. In contradistinction to other events a duration will be called infinite and the other events are finite[10]. In our knowledge of a duration we distinguish (i) certain included events which are particularly discriminated as to their peculiar individualities, and (ii) the remaining included events which are only known as necessarily in being by reason of their relations to the discriminated events and to the whole duration. The duration as a whole is signified[11] by that quality of relatedness (in respect to extension) possessed by the part which is immediately under observation; namely, by the fact that there is essentially a beyond to whatever is observed. I mean by this that every event is known as being related to other events which it does not include. This fact, that every event is known as possessing the quality of exclusion, shows that exclusion is as positive a relation as inclusion. There are of course no merely negative relations in nature, and exclusion is not the mere negative of inclusion, though the two relations are contraries. Both relations are concerned solely with events, and exclusion is capable of logical definition in terms of inclusion.
[10] Cf. note on "significance," pp. 197, 198.
[11] Cf. Ch. III, pp. 51 et seq.
Perhaps the most obvious exhibition of significance is to be found in our knowledge of the geometrical character of events inside an opaque material object. For example we know that an opaque sphere has a centre.
This knowledge has nothing to do with the material; the sphere may be a solid uniform billiard ball or a hollow lawn-tennis ball. Such knowledge is essentially the product of significance, since the general character of the external discriminated events has informed us that there are events within the sphere and has also informed us of their geometrical structure.
Some criticisms on "The Principles of Natural Knowledge" show that difficulty has been found in apprehending durations as real stratifications of nature. I think that this hesitation arises from the unconscious influence of the vicious principle of bifurcation, so deeply embedded in modern philosophical thought. We observe nature as extended in an immediate present which is simultaneous but not instantaneous, and therefore the whole which is immediately discerned or signified as an inter-related system forms a stratification of nature which is a physical fact. This conclusion immediately follows unless we admit bifurcation in the form of the principle of psychic additions, here rejected.
Our "percipient event" is that event included in our observational present which we distinguish as being in some peculiar way our standpoint for perception. It is roughly speaking that event which is our bodily life within the present duration. The theory of perception as evolved by medical psychology is based on significance. The distant situation of a perceived object is merely known to us as signified by our bodily state, _i.e._ by our percipient event. In fact perception requires sense-awareness of the significations of our percipient event together with sense-awareness of a peculiar relation (situation) between certain objects and the events thus signified. Our percipient event is saved by being the whole of nature by this fact of its significations.
This is the meaning of calling the percipient event our standpoint for perception. The course of a ray of light is only derivatively connected with perception. What we do perceive are objects as related to events signified by the bodily states excited by the ray. These signified events (as is the case of images seen behind a mirror) may have very little to do with the actual course of the ray. In the course of evolution those animals have survived whose sense-awareness is concentrated on those significations of their bodily states which are on the average important for their welfare. The whole world of events is signified, but there are some which exact the death penalty for inattention.
The percipient event is always here and now in the a.s.sociated present duration. It has, what may be called, an absolute position in that duration. Thus one definite duration is a.s.sociated with a definite percipient event, and we are thus aware of a peculiar relation which finite events can bear to durations. I call this relation "cogredience."
The notion of rest is derivative from that of cogredience, and the notion of motion is derivative from that of inclusion within a duration without cogredience with it. In fact motion is a relation (of varying character) between an observed event and an observed duration, and cogredience is the most simple character or subspecies of motion. To sum up, a duration and a percipient event are essentially involved in the general character of each observation of nature, and the percipient event is cogredient with the duration.
Our knowledge of the peculiar characters of different events depends upon our power of comparison. I call the exercise of this factor in our knowledge "recognition," and the requisite sense-awareness of the comparable characters I call "sense-recognition." Recognition and abstraction essentially involve each other. Each of them exhibits an ent.i.ty for knowledge which is less than the concrete fact, but is a real factor in that fact. The most concrete fact capable of separate discrimination is the event. We cannot abstract without recognition, and we cannot recognise without abstraction. Perception involves apprehension of the event and recognition of the factors of its character.
The things recognised are what I call "objects." In this general sense of the term the relation of extension is itself an object. In practice however I restrict the term to those objects which can in some sense or other be said to have a situation in an event; namely, in the phrase "There it is again" I restrict the "there" to be the indication of a special event which is the situation of the object. Even so, there are different types of objects, and statements which are true of objects of one type are not in general true of objects of other types. The objects with which we are here concerned in the formulation of physical laws are material objects, such as bits of matter, molecules and electrons. An object of one of these types has relations to events other than those belonging to the stream of its situations. The fact of its situations within this stream has impressed on all other events certain modifications of their characters. In truth the object in its completeness may be conceived as a specific set of correlated modifications of the characters of all events, with the property that these modifications attain to a certain focal property for those events which belong to the stream of its situations. The total a.s.semblage of the modifications of the characters of events due to the existence of an object in a stream of situations is what I call the "physical field" due to the object. But the object cannot really be separated from its field.
The object is in fact nothing else than the systematically adjusted set of modifications of the field. The conventional limitation of the object to the focal stream of events in which it is said to be "situated" is convenient for some purposes, but it obscures the ultimate fact of nature. From this point of view the ant.i.thesis between action at a distance and action by transmission is meaningless. The doctrine of this paragraph is nothing else than another way of expressing the unresolvable multiple relation of an object to events.
A complete time-system is formed by any one family of parallel durations. Two durations are parallel if either (i) one includes the other, or (ii) they overlap so as to include a third duration common to both, or (iii) are entirely separate. The excluded case is that of two durations overlapping so as to include in common an aggregate of finite events but including in common no other complete duration. The recognition of the fact of an indefinite number of families of parallel durations is what differentiates the concept of nature here put forward from the older orthodox concept of the essentially unique time-systems.
Its divergence from Einstein"s concept of nature will be briefly indicated later.
The instantaneous s.p.a.ces of a given time-system are the ideal (non-existent) durations of zero temporal thickness indicated by routes of approximation along series formed by durations of the a.s.sociated family. Each such instantaneous s.p.a.ce represents the ideal of nature at an instant and is also a moment of time. Each time-system thus possesses an aggregate of moments belonging to it alone. Each event-particle lies in one and only one moment of a given time-system. An event-particle has three characters[12]: (i) its extrinsic character which is its character as a definite route of convergence among events, (ii) its intrinsic character which is the peculiar quality of nature in its neighbourhood, namely, the character of the physical field in the neighbourhood, and (iii) its position.
[12] Cf. pp. 82 et seq.
The position of an event-particle arises from the aggregate of moments (no two of the same family) in which it lies. We fix our attention on one of these moments which is approximated to by the short duration of our immediate experience, and we express position as the position in this moment. But the event-particle receives its position in moment M in virtue of the whole aggregate of other moments M{"}, M{""}, etc., in which it also lies. The differentiation of M into a geometry of event-particles (instantaneous points) expresses the differentiation of M by its intersections with moments of alien time-systems. In this way planes and straight lines and event-particles themselves find their being. Also the parallelism of planes and straight lines arises from the parallelism of the moments of one and the same time-system intersecting M. Similarly the order of parallel planes and of event-particles on straight lines arises from the time-order of these intersecting moments.
The explanation is not given here[13]. It is sufficient now merely to mention the sources from which the whole of geometry receives its physical explanation.
[13] Cf. _Principles of Natural Knowledge_, and previous chapters of the present work.
The correlation of the various momentary s.p.a.ces of one time-system is achieved by the relation of cogredience. Evidently motion in an instantaneous s.p.a.ce is unmeaning. Motion expresses a comparison between position in one instantaneous s.p.a.ce with positions in other instantaneous s.p.a.ces of the same time-system. Cogredience yields the simplest outcome of such comparison, namely, rest.
Motion and rest are immediately observed facts. They are relative in the sense that they depend on the time-system which is fundamental for the observation. A string of event-particles whose successive occupation means rest in the given time-system forms a timeless point in the timeless s.p.a.ce of that time-system. In this way each time-system possesses its own permanent timeless s.p.a.ce peculiar to it alone, and each such s.p.a.ce is composed of timeless points which belong to that time-system and to no other. The paradoxes of relativity arise from neglecting the fact that different a.s.sumptions as to rest involve the expression of the facts of physical science in terms of radically different s.p.a.ces and times, in which points and moments have different meanings.
The source of order has already been indicated and that of congruence is now found. It depends on motion. From cogredience, perpendicularity arises; and from perpendicularity in conjunction with the reciprocal symmetry between the relations of any two time-systems congruence both in time and s.p.a.ce is completely defined (cf. _loc. cit._).
The resulting formulae are those for the electromagnetic theory of relativity, or, as it is now termed, the restricted theory. But there is this vital difference: the critical velocity c which occurs in these formulae has now no connexion whatever with light or with any other fact of the physical field (in distinction from the extensional structure of events). It simply marks the fact that our congruence determination embraces both times and s.p.a.ces in one universal system, and therefore if two arbitrary units are chosen, one for all s.p.a.ces and one for all times, their ratio will be a velocity which is a fundamental property of nature expressing the fact that times and s.p.a.ces are really comparable.
The physical properties of nature are expressed in terms of material objects (electrons, etc.). The physical character of an event arises from the fact that it belongs to the field of the whole complex of such objects. From another point of view we can say that these objects are nothing else than our way of expressing the mutual correlation of the physical characters of events.
The spatio-temporal measurableness of nature arises from (i) the relation of extension between events, and (ii) the stratified character of nature arising from each of the alternative time-systems, and (iii) rest and motion, as exhibited in the relations of finite events to time-systems. None of these sources of measurement depend on the physical characters of finite events as exhibited by the situated objects. They are completely signified for events whose physical characters are unknown. Thus the spatio-temporal measurements are independent of the objectival physical characters. Furthermore the character of our knowledge of a whole duration, which is essentially derived from the significance of the part within the immediate field of discrimination, constructs it for us as a uniform whole independent, so far as its extension is concerned, of the un.o.bserved characters of remote events. Namely, there is a definite whole of nature, simultaneously now present, whatever may be the character of its remote events. This consideration reinforces the previous conclusion. This conclusion leads to the a.s.sertion of the essential uniformity of the momentary s.p.a.ces of the various time-systems, and thence to the uniformity of the timeless s.p.a.ces of which there is one to each time-system.
The a.n.a.lysis of the general character of observed nature set forth above affords explanations of various fundamental observational facts: (a) It explains the differentiation of the one quality of extension into time and s.p.a.ce. () It gives a meaning to the observed facts of geometrical and temporal position, of geometrical and temporal order, and of geometrical straightness and planeness. (?) It selects one definite system of congruence embracing both s.p.a.ce and time, and thus explains the concordance as to measurement which is in practice attained. (d) It explains (consistently with the theory of relativity) the observed phenomena of rotation, _e.g._ Foucault"s pendulum, the equatorial bulge of the earth, the fixed senses of rotation of cyclones and anticyclones, and the gyro-compa.s.s. It does this by its admission of definite stratifications of nature which are disclosed by the very character of our knowledge of it. (e) Its explanations of motion are more fundamental than those expressed in (d); for it explains what is meant by motion itself. The observed motion of an extended object is the relation of its various situations to the stratification of nature expressed by the time-system fundamental to the observation. This motion expresses a real relation of the object to the rest of nature. The quant.i.tative expression of this relation will vary according to the time-system selected for its expression.