It would be quite possible to build such a model if we had at our disposal all the necessary measurements of the Universe. But we do not have all the necessary measurements. Our measurements of the Universe are inadequate and incomplete. In a "geometrical" three-dimensional universe this is quite clear; the world cannot be fitted into the space of three co-ordinates. Too many things are left out things which cannot be measured. It is equally clear also in the "metageometrical" universe of four co-ordinates. The world with all its variety of phenomena does not fit into four-dimensional space, no matter how we take the fourth co-ordinate, whether as a quantity analogous to the first three or as an imaginary quantity determinable relatively to the ultimate physical velocity that has been found, i.e., the velocity of light.
The proof of the artificiality of the four-dimensional world in new physics lies first of all in the extreme complexity of its construction which requires a curved space. It is quite clear that this curvature of space indicates the presence in it of yet another dimension or dimensions.
The universe of four co-ordinates is as unsatisfactory as the universe of three co-ordinates. And to be more exact we can say that we do not possess all the measurements necessary for the construction of a model of the Universe because neither the three co-ordinates of old physics nor the four co-ordinates of new physics are sufficient for the description of all the variety of phenomena in the Universe; or, in other words, because we have not enough dimensions.
Let us imagine that somebody builds a model of a house having only the floor, one wall, and the roof. This will be a model corresponding to a three-dimensional model of the Universe. It will give a general impression of the house, but only on condition that both the model itself and the observer remain motionless. The slightest movement will destroy the whole illusion.
The four-dimensional model of the universe of new physics is the same model, only arranged so that it rotates, turning its front always to the observer. This can prolong the illusion for some time, but only on the condition of there being not more than one observer. Two people observing such a model from different sides will very soon see in what the trick consists.
Before attempting to make clear without any analogies what it actually means to say that the Universe does not fit into three-dimensional or four-dimensional space, and before attempting to discover what number of co-ordinates really determines the Universe, I must eliminate one of the most essential misunderstandings which exist with regard to its dimensions.
Mathematics detaches itself easily and simply from three-dimensional physics and Euclidean geometry because really it does not belong there at all.
It is quite wrong to think that all mathematical relations must have physical or geometrical meanings. On the contrary, only a very small and the most elementary part of mathematics has a permanent connection with geometry and with physics, and only very few geometrical and physical quantities can have permanent mathematical expression.
For us it is necessary to understand exactly that dimensions cannot be expressed mathematically and that consequently mathematics cannot serve as an instrument for the investigation of problems of space and time. Only measurements along previously agreed-upon co-ordinates can be expressed mathematically. It can, for instance, be said that the length of an object is 5 metres, the breadth 10 metres, and the height 15 metres. But the difference between the length, the breadth, and the height themselves cannot be expressed; mathematically they are equivalent. Mathematics does not feel dimensions as geometry and physics feel them. Mathematics cannot feel the difference between a point, a line, a surface, and a solid. The point, the line, the surface, and the solid can be expressed mathematically only by means of powers, that is to say, simply for the sake of designation: a, a line; a2, a surface; a3 a solid. But the fact is that the same designations would serve also for segments of a line of different lengths: a, 10 metres; a2, 100 metres; a3, 1000 metres.
The artificiality of designating dimensions by powers becomes perfectly clear if we reason in the following way.
We assume that a is a line, a2 is a square, a3 is a cube, a4 is a body of four dimensions; a5 and a6, as will be seen later, can be explained. But what will a25 mean, or a125, or a1000? Once we allow that dimensions correspond to powers, this will mean that powers actually express the dimensions. Consequently the number of dimensions must be the same as the number of powers. This would be an obvious absurdity, as the limitation of the Universe in relation to number of dimensions is quite obvious, and no one would seriously assert the possibility of an infinite, or even of a large, number of dimensions.
Having established this point, we may note once more, though it should be quite clear already, that three co-ordinates are not sufficient for the description of the Universe, for such a universe would contain no motion or, putting it differently, every observable motion would immediately destroy the Universe.
But motion by itself is a very complex phenomenon. At the very first approach to motion we meet with an interesting fact. Motion has in itself three clearly expressed dimensions: duration, velocity, and "direction". But this direction does not lie in Euclidean space, as it was taken by old physics; it is a direction from before to after, which for us never changes and never disappears.
Time is the measure of motion. If we represent time by a line, then the only line which will satisfy all the demands of time will be a spiral. A spiral is, so to speak, a "three-dimensional line", that is, a line which requires three co-ordinates for its construction and designation.
The three-dimensionality of time is completely analogous to the three-dimensionality of space. We do not measure space by cubes: we measure it linearly in different directions, and we do exactly the same with time, although in time we can measure only two co-ordinates out of three, namely the duration and the velocity; the direction of time is, for us, not a quantity but an absolute condition. Another difference is that in the case of space we realise that we are dealing with a three-dimensional continuum, whereas in the case of time we do not realise it. But, as has been said already, if we attempt to unite the three co-ordinates of time into one whole, we shall obtain a spiral.
This explains at once why the "fourth co-ordinate" is insufficient to describe time. Although it is admitted to be a curved line, its curvature remains undefined. Only three co-ordinates, or the "three-dimensional line", that is, the spiral, give an adequate description of time.
The three-dimensionality of time explains many phenomena which have hitherto remained incomprehensible, and makes unnecessary most of the elaborate hypotheses and suppositions which have been indispensable in the attempts to squeeze the Universe into the boundaries of the three- or four-dimensional continuum.
This also explains the failure of relativism to give a comprehensible form to its explanations. Excessive complexity in any construction is always the result of something having been omitted or wrongly taken at the beginning. The cause of the complexity in this case lies in the above-mentioned impossibility of squeezing the Universe into the boundaries of a three- or four-dimensional continuum. If we try to regard three-dimensional space as two-dimensional and to explain all physical phenomena as occurring on a surface, several further "principles of relativity" will be required.
The three dimensions of time can be regarded as the continuation of the dimensions of space, i.e. as the "fourth", the "fifth", and the "sixth" dimensions of space. A "six-dimensional" space is undoubtedly a Euclidean continuum, but of properties and forms totally incomprehensible to us. The six-dimensional form of a body is inconceivable for us, and if we were able to apprehend it with our senses we should undoubtedly see and feel it as three-dimensional. Three-dimensionality is the function of our senses. Time is the boundary of our senses. Six-dimensional space is reality, the world as it is. This reality we perceive only through the slit of our senses touch and vision and define as three-dimensional space, ascribing to it Euclidean properties. Every six-dimensional body becomes for us a three-dimensional body existing in time, and the properties of the fifth and the sixth dimensions remain for us imperceptible.
Six dimensions constitute a "period" beyond which there can be nothing except the repetition of the same period on a different scale. The period of dimensions is limited at one end by the point, and at the other end by the infinity of space multiplied by infinity of time, which in ancient symbolism was represented by two intersecting triangles, or a six-pointed star.
In spite of the fact that the counting of the dimensions in geometry begins with the line, actually, in the real physical sense, only the material point and the solid are objects which exist. Lines and surfaces are merely features and properties of a solid. They can also be regarded in another way: a line as the path of the motion of a point in space, and a surface as the path of the motion of a line along the direction perpendicular to it (or its rotation).
The same may be applied to the solid of time. In it only the point (the moment) and the solid are real. Moreover, the moment can change, that is, it can contract (disappear) and expand to become a solid, whereas the solid can contract and become a point, and so on.
The number of dimensions can be neither infinite nor very great; it cannot be more than six. The reason for this lies in the properties of the sixth dimension.
In order to understand this it is necessary to examine the content of the three dimensions of time taken in their "space" sense, that is, as the fourth, the fifth, and the sixth dimensions of space.
Let us take the line of time as we usually conceive it.
The line determined by the three points "before", "now", "after", is a line of the fourth dimension.
Let us now imagine several lines perpendicular to this line, before-now-after. These lines, each of which designates now for a given moment, will express the perpetual existence of past and possibility of future moments.
Each of these perpendicular lines is the perpetual now for some moment, and every moment has such a line of perpetual now.
This is the fifth dimension.
The fifth dimension forms a surface in relation to the line of time.
Everything we know, everything we recognise as existing, lies on the line of the fourth dimension; the line of the fourth dimension is the "historical time" of our section of existence. This is the only "time" we know, the only time we feel, the only time we recognise. But though we are not aware of it, sensations of the existence of other "times", both parallel and perpendicular, continually enter into our consciousness. These parallel "times" are completely analogous to our time and consist only of before-now-after, whereas the perpendicular "times" consist only of now, and are, as it were, cross-threads, the woof in a fabric, in their relation to the parallel lines of time which in this case represent the warp.
On the table before me are many different things. I may deal with these things in different ways. But I cannot, for instance, take from the table something that is not there. I cannot take from the table an orange that is not there, just as I cannot take from it the pyramid of Kheops or St Isaac's Cathedral. It looks as though there was actually no difference in this respect between an orange and a pyramid, and yet there is a difference. An orange could be on the table, but a pyramid could not be. However elementary all this is, it shows that there are different degrees of impossibility.
But at present we are concerned only with possibilities. As I have already mentioned, each moment contains a definite number of possibilities. I may actualise one of the existing possibilities, that is, I may do something. I may do nothing. But whatever I do, that is, whichever of the possibilities contained in the given moment is actualised, the actualisation of this possibility will determine the following moment of time, the following now. This second moment of time will again contain a certain number of possibilities, and the actualisation of one of these possibilities will determine the following moment of time, the following now, and so on.
Thus the line of direction of time can be defined as the line of the actualisation of one possibility out of the number of possibilities which were contained in the preceding point.
The line of this actualisation will be the line of the fourth dimension, the line of time. We visualise it as a straight line, but it would be more correct to think of it as a zigzag line.
For the modern mind eternity is an indefinite concept. In ordinary conversational language eternity is taken as a limitless extension of time. But religious or philosophical thought puts into the concept of eternity ideas which distinguish it from mere infinite extension homogeneous with finite extension. This is most clearly seen in Indian philosophy with its idea of Eternal Now as the state of Brahma.
In fact, the concept of eternity in relation to time is the same as the concept of a surface in relation to a line. A surface is a quantity incommensurable with a line. Infinity for a line need not necessarily be a line without end; it may be a surface, that is, an infinite number of finite lines.
Eternity can be an infinite number of finite "times".
It is difficult for us to think of "time" in the plural. Our thought is too much accustomed to the idea of one time, and though in theory the idea of plurality of "times" is already accepted by new physics, in practice we still think of time as one and the same, always and everywhere.
The sixth dimension will be the line of the actualisation of the possibilities which were contained in the preceding moment but were not actualised in "time". In every moment and at every point of the three-dimensional world there are a certain number of possibilities; in "time", that is, in the fourth dimension, one possibility is actualised every moment, and these actualised possibilities are laid out, one beside the other, in the fifth dimension. The line of time, repeated infinitely in eternity, leaves at every point unactualised possibilities. But these possibilities, which have not been actualised in one time, are actualised in the sixth dimension, which is an aggregate of "all times". The lines of the fifth dimension, which go perpendicular to the line of "time", form, as it were, a surface. The lines of the sixth dimension, which start from every point of "time" in all possible directions, form the solid or three-dimensional continuum of time, of which we know only one dimension. We are one-dimensional beings in relation to time. Because of this we do not see parallel time or parallel times; for the same reason we do not see the angles and turns of time, but see time as a straight line.
Until now we have taken all the lines of the fourth, the fifth, and the sixth dimensions as straight lines, as co-ordinates. But we must remember that these straight lines cannot be regarded as really existing. They are merely an imaginary system of co-ordinates for determining the spiral.
Generally speaking, it is impossible to establish and prove the real existence of straight lines beyond a certain definite scale and outside certain definite conditions. And even these "conditional straight lines" cease to be straight if we imagine them on a revolving body which possesses, besides, a whole series of other movements. This is quite clear as regards space lines: straight lines are nothing but imaginary co-ordinates which serve to measure the length, the breadth, and the depth of spirals. But time lines are geometrically in no way different from space lines. The only difference lies in the fact that in space we know three dimensions and are able to establish the spiral character of all cosmic movements, that is, movements which we take on a sufficiently large scale. But we dare not do this as regards "time". We try to lay out the whole space of time on one line of the great time which is general for everybody and everything. But this is an illusion; general time does not exist, and each separately existing body, each separately existing "system" (or what is accepted as such) has its own time. This is recognised by new physics. But what it means and what a separate existence means is not explained by new physics.
Separate time is always a completed circle. We can think of time as a straight line only on the great straight line of the great time. If the great time does not exist, every separate time can only be a circle, that is, a closed curve. But a circle or any closed curve requires two co-ordinates for its definition. The circle (circumference) is a two-dimensional figure. If the second dimension of time is eternity, this means that eternity enters into every circle of time and into every moment of the circle of time. Eternity is the curvature of time. Eternity is also movement, an eternal movement. And if we imagine time as a circle or as any other closed curve, eternity will signify eternal movement along this curve, eternal repetition, eternal recurrence.
The fifth dimension is movement in the circle, repetition, recurrence. The sixth dimension is the way out of the circle. If we imagine that one end of the curve rises from the surface, we visualise the third dimension of time the sixth dimension of space. The line of time becomes a spiral. But the spiral, of which I have spoken before, is only a very feeble approximation to the spiral of time, only its possible geometrical representation. The actual spiral of time is not analogous to any of the lines we know, for it branches off at every point. And as there can be many possibilities in every moment, so there can be many branches at every point. Our mind refuses not only to visualise, but even to think of, the resulting figure in curved lines, and we should lose the direction of our thought in this impasse if straight lines did not come to our aid.
A figure of three-dimensional time will appear to us in the form of a complicated structure consisting of radii diverging from every moment of time, each of them bearing within it its own time and throwing out new radii at every point. Taken together these radii will form the three-dimensional continuum of time.
We live and think and exist on one of these lines of time. But the second and third dimensions of time, that is, the surface on which this line lies and the solid in which this surface is included, enter every moment into our life and into our consciousness, and influence our "time". When we begin to feel the three dimensions of time we call them direction, duration, and velocity. But if we wish to understand the true interrelation of things even approximately, we must bear in mind the fact that direction, duration, and velocity are not real dimensions, but merely the reflections in our consciousness of the real dimensions.
In thinking of the time solid formed by the lines of all the possibilities included in each moment, we must remember that beyond these there can be nothing.
This is the point at which we can understand the limitedness of the infinite Universe.
The counting of dimensions in geometry begins with the line, the first dimension, and in a certain sense this is right. But both space and time have yet another, the zero dimension the point or the moment. And it must be understood that any space solid, up to the infinite sphere of old physics, is a point or a moment when taken in time.
The zero dimension, the first, the second, the third, the fourth, the fifth and the sixth dimensions form the period of dimensions. But a "figure" of the zero dimension, a point, is a solid of another scale. A figure of the first dimension, a line, is infinity in relation to a point. For itself a line is a solid, but a solid of another scale than a point. For a surface, that is, for a figure of two dimensions, a line is a point. A surface is three-dimensional for itself, whereas for a solid it becomes a point, and so on.
A line and a surface are for us only geometrical concepts, and it is at the first glance incomprehensible how they can be three-dimensional bodies for themselves. But it becomes more comprehensible if we begin with the solid which represents a real existent physical body. We know that a body is three-dimensional for itself as well as for other three-dimensional bodies of a scale near its own. It is also infinity for a surface, which is zero in relation to it, because no number of surfaces will make a solid. And the solid is also a point, a zero, a figure of the zero dimension for the fourth dimension first, because, however big it may be, a solid is a point, that is, a moment for time; and second, because no number of solids will make time. The whole of three-dimensional space is but a moment in time. It should be understood that "lines" and "surfaces" are only names which we give to dimensions which for us lie between the point and the solid. They have no real existence for us. Our Universe consists only of points and solids. A point is zero dimension, a solid is three dimensions. On another scale a solid must be taken as a time point, and on yet another scale again as a solid, but as a solid of three dimensions of time.
In such a simplified universe there would be no time and no motion. Time and motion are created precisely by these incompletely perceived solids, that is, by space and time lines and space and time surfaces. And the period of dimensions of the real Universe actually consists of seven "powers" of solids (a power is, of course, only a name in this case). (1) A point the hidden solid. (2) A line the solid of the second power. (3) A surface the solid of the third power. (4) A body or a solid the solid of the fourth power. (5) Time, or the existence of a body or a solid in time the solid of the fifth power. (6) Eternity, or the existence of time the solid of the sixth power. (7) That for which we have no name, the "six-pointed star", or the existence of eternity the solid of the seventh power.
Further it should be observed that dimensions are movable, i.e., any three consecutive dimensions form either "time" or "space", and the "period" can move upwards and downwards when one degree is added above and one is taken away from below, or when one degree is added below and one is taken away from above. Thus, if one dimension from "below" is added to the six dimensions we possess, then one dimension from "above" must disappear. The difficulty of understanding this eternally changing universe, which contracts and expands according to the size of the observer and the speed of his perception, is counterbalanced by the constancy of laws and relative positions in these changing conditions.
The "seventh dimension" is impossible, for it would be a line leading nowhere, running in a non-existent direction.
One could find many such examples. The whole of our life actually consists of phenomena of the "seventh dimension", that is, of phenomena of fictitious possibility, fictitious importance, and fictitious value. We live in the seventh dimension and cannot escape from it. And our model of the Universe can never be complete if we do not realise the place occupied in it by the "seventh dimension". But it is very difficult to realise this. We never even come near to understanding how many non-existent things play a rτle in our life, govern our fate and our actions. But again, as has been said before, even the non-existent and the impossible can be of different degrees and therefore it is perfectly justifiable to speak not of the seventh dimension, but generally of imaginary dimensions, the number of which is also imaginary.
In order to establish with complete exactitude the necessity for regarding the world as a world of six co-ordinates, it is necessary to examine the fundamental concepts of physics which have remained without definition, and see whether it is not possible to find definitions for them with the help of some of the principles we have established above.
We will deal with matter, space, motion, velocity, infinity, mass, light, etc.
In the usual views of both the old and the new physics, motion remains always the same. Distinction is made only between its properties duration, velocity, direction in space, discontinuity, continuity, periodicity, acceleration, retardation, and so on. The characteristics of these properties are attributed to motion itself, so that motion is divided into rectilinear, curvilinear, continuous, non-continuous, accelerated, retarded, etc. The principle of the relativity of motion led to the principle of the composition of velocities, and the working out of the principle of relativity led to the denial of the possibility of the composition of velocities when "terrestrial" velocities are compared with the velocity of light. This led to many other conclusions, suppositions, and hypotheses, but these do not interest us for the moment. One fact, however, must be established: namely, that the very concept "motion" is not defined. Equally, "velocity" is not defined. In regard to "light", opinions of physicists diverge.
For the present it is important for us to realise only that motion is always taken as a phenomenon of one kind. There are no attempts to establish different kinds of phenomena in motion itself. And this is especially strange because, for direct observation, there definitely exist four kinds of motion as four perfectly distinct phenomena.
In certain cases direct observation deceives us for instance, when it shows much non-existent motion. But phenomena themselves are one thing, and division of them is another. In this particular case direct observation brings us to real and unquestionable facts. One cannot reason about motion without having understood the division of motion into four kinds.
These four kinds of motion are as follows:
It is possible to determine exactly when we begin to see the point move and when we cease to see it move.
We see the movement of the point as movement if the point covers in 1/10 of a second one or two minutes of the arc of a circle, taking as the radius our distance from the wall. If the point moves more slowly, it will appear to us as motionless.
Let us suppose first that the point moves with the velocity of the hour-hand of a clock. Comparing its position with other, motionless, points, we first establish the fact of the movement of the point and, second, we determine the velocity of the movement; but we do not see the movement itself.
This will be the first kind of motion, invisible motion.
Further, if the point moves more quickly, covering two or more minutes of arc in 1/10 of a second, we see its motion as motion. This is the second kind of motion, visible motion. It can be very varied in its character and cover a large range of velocities, but when velocity is increased 4,000 to 5,000 times, and in certain cases less, it passes into the third kind of motion.
This means that if the point moves very fast, covering in 1/10 of a second the whole field of our vision, i.e. 160° or 9,600 minutes of arc, we shall see it not as a moving point but as a line.
This is the third kind of motion, with a visible trace, or motion in which the moving point is transformed into a line, motion with the apparent addition of one dimension.
Finally, if the point starts off at once with the velocity of, say, a rifle bullet, we shall not see it at all; but if the "point" possesses weight and mass, its motion may have many physical effects which we can observe and study. For instance, we can hear the motion, we can see other motions aroused by the invisible motion, and so on.
This is the fourth kind of motion, motion with an invisible but perceptible trace.
These four kinds of motion are absolutely real facts upon which depend the whole form, aspect, and correlation of phenomena in our Universe. This is so because the distinction of the four kinds of motion is not only subjective, i.e. they differ not only in our perception, but they differ physically in their results and in their action on other phenomena; and above all they are different in relation to one another, and this relation is permanent.
The ideas that have been set forth here may appear very naοve to a learned physicist. "What is the eye?" he would say. The eye has a strange capacity for "remembering" for about 1/10 of a second what it has seen; if the point moves sufficiently fast for the memory of each 1/10 of a second to merge with another memory, the result will be a line. There is no transformation of a point into a line here. It is all entirely subjective, that is, it all takes place only in us, only in our perceptions. In reality a moving point remains a moving point.
This is how the matter appears from a scientific point of view.
The objection is based on the supposition that we know that the observed phenomenon is produced by the motion of a point. But suppose we do not know? How can we ascertain it if we cannot come sufficiently near the line we observe, or arrest the motion, i.e., stop the supposed moving point?
Our eye sees a line; with a certain velocity of motion, a photographic camera will also "see" a line or a streak. The moving point is actually transformed into a line. We are quite wrong in not trusting our eye in this case. This is just a case in which our eye does not deceive us. The eye establishes an exact principle of division of velocities. The eye certainly establishes these divisions for itself, on its own level, on its own scale. And this scale may change. What will not change, for instance in connection with the distance, what will remain the same on any scale, is, first, the number of different kinds of motion there will always be four and next, the interrelation of the four velocities with their derivatives, i.e. with their results, or the interrelation of the four kinds of motion. This interrelation of the four kinds of motion creates the whole visible world. And the essence of this interrelation consists in the fact that one motion is not necessarily motion relatively to another motion, but only if the velocities which are compared do not differ greatly from one another.
Thus in the above example the visible motion of the point on the wall is motion in comparison both with invisible motion and with motion fast enough to form a line. But it will not be motion in relation to a flying bullet, for which it will be immobility, just as the line formed by a swiftly moving point will be a line and not motion for a slowly (invisibly) moving point. This can be formulated in the following way:
Dividing motion into four kinds according to the above principles, we observe that motion is motion (with increased or decreased velocity) only for kinds of motion that are near to one another, that is, within the limits of a definite correlation of velocities or, to put it more precisely, within the limits of a certain definite increase or decrease of velocity, which can probably be determined exactly. More remote kinds of motion, i.e. motions with very different velocities, for instance 4,000 or 5,000 times slower or faster than another, are for one another not motions of different velocity, but phenomena of a greater or lesser number of dimensions.
Motion is an apparent phenomenon dependent upon the extension of a body in the three dimensions of time. This means that every three-dimensional body possesses also three time-dimensions which we do not see as such and which we call the properties of motion or of existence. Our mind cannot embrace time-dimensions in their entirety; there exist no concepts which would express their essence in all their variety, for all existent "time concepts" express only one side, or only one dimension, each. Therefore the extension of three-dimensional bodies in the undefinable (for us) three dimensions of time appears to us as motion with all its properties.
We stand in exactly the same position in relation to dimensions of time as animals stand in relation to the third dimension of space.
I wrote in Tertium Organum about the perception of the third dimension by animals. All apparent movements are real for them. A house turns about when a horse runs past it, a tree jumps into the road. Even if an animal is motionless and only examines an equally motionless object, this object begins to manifest strange movements. The animal's own body, even in the state of rest, may manifest for it many strange movements which our bodies do not manifest for us.
Our relation to motion, and especially to velocity, is very similar to this. Velocity can be a property of space. The sensation of a velocity may be the sensation of the penetration into our consciousness of one of the dimensions of a higher space unknown to us.
Velocity can be regarded as an angle. And this at once explains all the properties of velocity and especially the fact that both great and small velocities cease to be velocities. An angle has naturally a limit both in one direction and in the other.
Let us again imagine a world of flat beings. Let us imagine these flat beings in the shape of squares with their organs of perception situated on one side of the square. Let is call this percipient side a.
Let us imagine that the "square" is turned with its percipient side towards two figures, let us say two "triangles" ABC and DEF, in the position shown in the diagram.
Of the triangle ABC it knows only the side AC, and this line is motionless for it. Of the triangle DEF it knows the lines DE and DF, which appear to it as one line, and these lines, which go out of the field of its vision, must undoubtedly differ from the line AC, must possess some property which the line AC does not possess. The "square" will call this property motion.
If the "square" happens to meet the triangle GHI, the lines GH and GI will also be "motion" for it, but a slower motion.
And if the "square" meets the triangle JKL, the lines JK and JL will be a swifter motion.
And finally, if the "square" meets lines almost perpendicular to its perception side, like the lines MN and MO, it will say that this is the limiting, maximal velocity and that there can be no higher velocity.
The idea of velocity as an angle makes not only clear but necessary the idea of a limiting velocity beyond which there can exist no other velocity, and also the idea of the impossibility of an infinite velocity, because an angle cannot be infinite and must have a limit which can always be established and measured.
So far, in all the above examples, velocity has been taken as uniform and unchangeable. But, on the basis of the same principle, it is easy to establish the meaning of acceleration, variable velocity, and so on.
Let us imagine that the receding line PQ is not a straight line but a line with an angle.
In examining such a line from the point P, the flat being will see this line as motion starting with one speed and then accelerating.
The line ST will appear as motion alternately accelerated and retarded.
And further, lines with angles, curves of different kinds, lines lying at various changing angles to the perception side, will represent different kinds of velocity: constant, variable, uniformly accelerated, uniformly retarded, periodically accelerated and retarded, and so on.
The essence of all that has been said is that a line receding at an angle will appear as motion only if it lies at certain definite degrees. A line lying at a very small angle to a motionless line which is parallel to the percipient side would appear motionless; at a greater angle it would appear as motion, and a line lying at an angle approaching the limit would appear something altogether different from motion. The "velocity" is only the property of certain definite angles, and as the angle does not depend on scale, it is quite possible that "velocity" is the only constant phenomenon in the universe.
This principle is in no way changed by the alteration of the angles on a spherical surface or, for instance, on the saddle-shaped surface used by Lobatchevsky, in comparison with the angles on a flat surface, because for every kind of surface the angles will remain unchangeable.
Now, starting from the above definitions of time, motion, and velocity, we shall pass to the definition of space, matter, mass, gravitation, infinity, commensurability and incommensurability, "negative quantity", etc.
Even the most complex mathematical and metageometrical views assert themselves each to the exclusion of all the others. "Spherical" space, "elliptical" space, space determined by the density of matter and by the laws of gravitation, "finite and yet limitless" space in each case this is the whole of space, and in each case the whole of space is uniform and homogeneous.
[In its essential features, this chapter was completed in 1912. The first part was written later, but in making a survey of the present state of physics I did not try to bring it fully up to date and to mention all the theories that had appeared by that time, because not one of them changed anything in my principal conclusions. The most complete exposition of views on space will be found by the reader in Prof. Eddington's book, Space, Time and Gravitation, particularly in the chapter Kinds of Space. At the beginning of this chapter Prof. Eddington quotes W K Clifford (1845-1879) who wrote in his book, Common Sense of the Exact Sciences:
The danger of asserting dogmatically that an axiom based on the experience of a limited region holds universally will now be to some extent apparent to the reader. It may lead us to entirely overlook, or when suggested at once reject, a possible explanation of phenomena. The hypothesis that space is not flat, and again that its geometrical character may change with the time, may or may not be destined to play a great part in the physics of the future; yet we cannot refuse to consider them as possible explanations of physical phenomena, because they may be opposed to the popular dogmatic belief in the universality of certain geometrical axioms a belief which has risen from centuries of indiscriminating worship of the genius of Euclid.
This may have a connection with the idea of the heterogeneity of space. PDO]
Of all the latest definitions of space the most interesting is the "mollusc" of Einstein. The "mollusc" anticipates many future discoveries. The "mollusc" is able to move by itself, to expand and to contract. The "mollusc" can be unequal to itself and heterogeneous with itself.
But still the "mollusc" is only an analogy, only a very timid example of the way in which space can and should be regarded. And behind this example, in order to make it possible, the whole arsenal of mathematics, metageometry, and new physics, with the "special" and "general" principles of relativity, is necessary.
In reality all this could be done much more simply if only the heterogeneity of space were understood.
Let us take space just as we took motion, from the point of view of direct observation,
(A) The space, occupied by the house in which I live, by the room in which I am now and by my body, is perceived by me as three-dimensional. Certainly this is not a pure "percept", for it has already passed through the prism of thinking, but as the three-dimensionality of the house, the room, and my body does not give rise to argument, it can be accepted.
(B) I look out of the window and see a portion of sky with several stars in it. The sky is two-dimensional for me. My mind knows that the sky possesses "depth". But my direct senses do not tell me so. On the contrary, they deny the truth of it.
(C) I am reflecting on the structure of matter and on a unit such as a molecule. One molecule has no dimensions for the direct senses but, by reasoning, I come to the conclusion that the space occupied by the molecule, consisting of atoms and electrons, must have six dimensions: three space dimensions and three time-dimensions; for otherwise, if the molecule did not possess the three time-dimensions, its three space-dimensions would be unable to produce any impression on my senses. A great quantity of molecules produces on me the impression of matter possessing mass only because of the six-dimensionality of the space occupied by every molecule.
Thus "space" is not homogeneous for me. The room is three-dimensional, the sky two-dimensional. The molecule has no dimension for direct perception; atoms and electrons have still less dimension, but owing to their six-dimensionality a multitude of molecules produces on me the impression of matter. If the molecules had no time-dimensions, matter would be emptiness for me.
What has been said above must leave several points requiring explanation. First, if the molecule has no dimension, how can atoms and electrons have still less? And second, how do time-dimensions affect our senses and why would not space-dimensions by themselves produce any effect on us?
In order to answer these questions, it is necessary to enlarge upon the above considerations.
Here again the objection may be raised, just as in the case of the four kinds of motion, that I take purely subjective sensations and attribute to them real meaning. And again, as in the case of the four kinds of motion, I may reply to this that what interests me is not sensations, but the interrelations of their causes. The causes are not subjective, but depend upon perfectly definite and perfectly objective conditions, namely, comparative magnitude and distance.
The house and the room are three-dimensional for me by virtue of their commensurability with my body. The "sky" is two-dimensional because it is remote. The "star" is a point because it is small as compared with the "sky". The "molecule" may be six-dimensional, but as a point, i.e. taken as a zero-dimensional body, it cannot produce any effect on my senses. These are all facts: there is nothing subjective in them.
But this is by no means all.
The dimensions of my space depend upon the size of my body. If the size of my body could change, the dimensions of the space around me would change also. "Dimension" corresponds to "size". If the dimensions of my world can change with a change in my size, then the size of my world can change.
But in what respect?
A right answer to this question will at once put us on the right road.
The smaller the "reference-body" or "reference-system", the smaller the world. Space is proportionate to the size of the reference-body, and all measurements of space are proportionate to the measurements of the reference-body. And yet it is the same space. Let us take an electron on the Sun in its relation to visible space and to the Earth. For the electron the whole of visible space will be (of course only approximately) a sphere of one kilometre in diameter; the distance from the Sun to the Earth will be a few centimetres, and the Earth itself will be almost a "material point". A ray of light from the Sun reaches the Earth (for the electron) instantaneously. This explains why we can never intercept a ray of light half-way.
If instead of an electron we take the Earth, then for the Earth distances will necessarily be much longer than they are for us. They will be longer by exactly as many times as the Earth is bigger than the human body. This is necessarily so if only because otherwise the Earth could not feel itself as the three-dimensional body we know it to be, but would be for itself some incomprehensible six-dimensional continuum. But such a self-feeling would contradict the rightly understood principle of the unity of laws. The reason is that if the Earth could be for itself a six-dimensional continuum, then we also should have to be for ourselves six-dimensional continua. And since we are for ourselves three-dimensional bodies, the Earth also must be for itself a three-dimensional body although at the same time it is not possible to assert with certainty that the Earth's conception of itself must necessarily coincide with our conception of it.
If we now try to imagine what the space occupied by terrestrial objects must be for the electron on the one hand and for the Earth on the other, we shall come to a very strange and, at first glance, paradoxical conclusion. Things which surround us tables, chairs, objects of daily use, other people, etc. cannot exist for the Earth, for they are too small for it. It is impossible to conceive of a chair in the planetary world. It is impossible to conceive of an individual man in relation to the Earth. An individual man cannot exist in relation to the Earth. The whole of humanity cannot exist by itself in relation to the Earth. It exists only together with all the vegetable and animal world and with all that has been made by the hand of man.
There can be no serious objection to this, because a particle of matter that is as small in relation to the human body as the human body or even all humanity is in relation to the Earth certainly cannot exist for us. And it is quite obvious that a chair cannot exist in the planetary world because it is too small. What is strange and what is paradoxical is the inference that a chair cannot exist for the electron or in the world of electrons also, and also because it is too small.
This seems an absurdity. "Logically" it ought to be that a chair cannot exist for the electron because a chair is too big compared with the electron. But it would be so only in a "logical", that is, in a three-dimensional, universe with a permanent space. The six-dimensional universe is illogical and the space in it can contract and expand on an incredibly large scale, preserving only one permanent property, namely angles. Therefore, the space existing for the electron which is proportionate to its size will be so small that a chair will occupy practically no room in this space.
The new model of the Universe establishes exactly the unity of space and time and the difference between them; it establishes also the principle that space can pass into time and time into space.
In old physics, space is always space and time is always time. In the new physics the two categories make one, space-time. In the new model of the Universe the phenomena of one category can pass into the phenomena of the other category, and vice versa.
When I write of space, space-concepts and space-dimensions, I mean space for us. For the electron, and most probably even for bodies much larger than the electron, our space is time.
The six-pointed star which represented the world in ancient symbolism is in reality the representation of space-time or the "period of dimensions", i.e. of the three space-dimensions and the three time-dimensions in their perfect union, where every point of space includes the whole of time and every moment of time includes the whole of space; when everything is everywhere and always.
But this state of six-dimensional space is incomprehensible and inaccessible for us, for our sense-organs and our minds enable us to establish a connection only with the material world, that is, with a world of certain definite limitations in relation to higher space.
Earlier in this chapter a definition by Prof. Chwolson was quoted:
In objectifying the cause of a sensation, that is, transferring this cause into a definite place in space, we conceive this place as containing something which we call matter or substance.
And further:
The use of the term "matter" was reserved exclusively for matter which is able to affect our organ of touch more or less directly.
Modern physics and chemistry have achieved much in the study of the structure and composition of matter, but they do not limit themselves by a definition of matter like that made by Prof. Chwolson. They apparently regard as matter everything that admits of objective study, everything that can be measured and weighed, even indirectly. In studying the structure and composition of matter, these sciences deal with divisions of matter which are so small that they can produce no effect on our organs of touch, but are nevertheless recognised as material.
In fact both the old view, which limited the concept of matter too closely, and the new view, which extends it too far, are incorrect.
In order to avoid contradictions, indefiniteness, and confusion of terms, it is necessary to establish the existence of several degrees of materiality.
It is necessary to keep these divisions in view, because without applying them it is impossible to find a way out of the chaos in which physical sciences find themselves.
What do these divisions signify from the standpoint of the above principles of "the new model of the Universe", and how can the degrees of materiality be defined?
Matter of the first kind is three-dimensional, i.e. any part of this matter and any "particle" can be measured in length, breadth, and height and exists in time, i.e. in the fourth dimension.
Matter of the second and third kinds, i.e., its components, molecules, atoms, and electrons have no space-dimensions in comparison with particles of matter of the first kind, and reach our consciousness only in large masses and only through their time-dimensions, the fourth, the fifth, and the sixth; in other words, they reach it only by virtue of their motion and the repetition of their motion.
Thus only the first degree of matter can be taken as existing in geometrical forms and in three-dimensional space. Atomic and electronic matter can with every right be regarded as matter belonging not to our, but to another, space for it requires six dimensions for its description. Its units molecules, atoms, and electrons if taken by themselves, can with every right be called immaterial.
"Materiality" is divided for us into three categories or three degrees.
The first kind of materiality is the state of matter of which our bodies consist. This matter and any part of it must possess (for us) three space-dimensions and one time-dimension; their fifth and sixth dimensions we cannot perceive.
In the materiality of the first kind there is (for us) more space than time.
The second and third kinds of materiality are the states of molecules, atoms, and electrons which (for our direct senses) have the zero dimension in space and reach our consciousness by virtue of their three dimensions of time.
In the materiality of the second and third kinds there is (for us) more time than space.
The world inside the molecules is completely analogous to the great space in which celestial bodies move. Electrons, atoms, molecules, planets, solar systems, agglomerations of stars all these are phenomena of the same order. Electrons move in their orbits in the atom just as planets move in the Solar System. Electrons are the same celestial bodies as planets; even their velocity is the same as the velocities of the planets. In the world of electrons and atoms it is possible to observe all the phenomena which are observed in the astronomical world. There are comets in this world which travel from one system to another, there are shooting stars, there are streams of meteorites. "As above so below". Science seems to have proved the old formula of the Hermetists. Unfortunately, however, it only seems so, for in actual fact the model of the Universe which science builds is too unstable and can fall to pieces at a single touch.
Indeed, what links together all these revolving particles or aggregations of matter? Why do not the planets of the Solar System fly apart in different directions? Why do they continue to revolve in their orbits round the central luminary? Why do electrons remain linked with one another, thus constituting an atom? Why do they not fly apart? Why does matter not resolve into nothing?
Science has always been confronted by these questions in one form or another, and even in our day it is unable to answer them without introducing two unknown quantities: "attraction" or "gravitation" and "aether".
"Attraction", says science, keeps the planets near the Sun and binds electrons into one whole; attraction, the mysterious force, the influence of a larger mass upon a smaller mass. This again produces a question: how can one mass influence another, even a smaller one, when it is at a great distance from it? If we imagine the Sun as a large apple, the Earth will be a poppy seed at a distance of twelve paces from this apple. How can the apple influence the poppy seed twelve paces from it? They must be linked in some way, for otherwise the influence of one body upon another remains totally incomprehensible and is in fact impossible.
Scientists have tried to find an answer to this problem by imagining a certain medium through which influence is transmitted and in which electrons and (possibly) also celestial bodies revolve.
All these hypotheses, and also the hypothesis of gravitation, are entirely unnecessary from the point of view of the new model of the Universe.
Atomic matter makes our consciousness aware of its existence through its motion. If the motion inside atoms were to stop, matter would turn into emptiness, into nothing. The effect of materiality, the impression of mass, is produced by the motion of the minutest particles, which demands time. If we take away time, if we imagine atoms without time, that is, if we imagine all the electrons constituting the atom as immovable, there will be no matter. Motionless small quantities are outside our scale of perception. We perceive not them, but their orbits, or the orbits of their orbits.
But in the case of celestial space we have learned sooner than we learned in the case of matter that what we see does not correspond to reality, though our science is still far from the right understanding of this reality.
Luminous points have turned into worlds moving in space. The universe of flying globes has come into being. But this picture is not the end of the possible understanding of celestial space.
If we represent schematically the interrelation of celestial bodies, we shall represent them as discs or points at a great distance from one another. But we know that they are not immovable, we know that they revolve round one another, and we know that they are not points. The Moon revolves around the Earth, the Earth revolves around the Sun, the Sun in turn revolves round some other luminary unknown to us or, at any rate, moves in a definite direction along a definite line. Consequently the Moon in revolving round the Earth at the same time revolves round the Sun and at the same time moves somewhere together with the Sun. And the Earth in revolving round the Sun at the same time revolves round an unknown centre.
If we wish to express graphically the paths of this motion, we shall represent the path of the Sun as a line, the path of the Earth as a spiral winding round this line, and the path of the Moon as a spiral winding round the spiral of the Earth. If we wish to represent the path of the whole Solar System, we shall have to represent the paths of all the planets and asteroids as spirals winding round the central line of the Sun, and the paths of the planets' satellites as spirals round the spirals of the planets. Such a drawing would be very difficult to make in fact with asteroids it would be impossible; and it would be still more difficult to construct an exact model from this drawing, especially if all the interrelations, distances, exact thickness of the spirals, etc., were to be strictly observed. But if we were to succeed in building such a model, it would be an exact model of a small particle of matter enlarged many times. The same model reduced a required number of times would appear to us as impenetrable matter, exactly identical with all the matter which surrounds us.
Matter or substances of which our bodies and all the objects surrounding us consist is built in exactly the same way as the Solar System; only we are incapable of perceiving electrons and atoms as immovable points but perceive them in the form of the complex and entangled traces of their movement which produce the effect of mass. If we were able to perceive the Solar System on a much smaller scale, it would produce on us the effect of matter. There would be no emptiness in the Solar System for us, just as there is no emptiness in the matter surrounding us.
The emptiness or fullness of space depends entirely upon the dimensions in which we perceive the matter or particles of matter contained in that space. The dimensions in which we perceive this matter depend upon the size of the particles of this matter in comparison with our body, upon the greater or lesser distance separating us from them, and upon our perception of their motion (which depends upon the velocity of their own motion and the rate of our perception), which creates the subjective aspect of the world.
All these conditions, taken together, determine the dimensions in which we perceive various agglomerations of matter.
A whole world, consisting of several suns, with their surrounding planets and satellites, rushing with terrific velocity through space, but separated from us by great distances, is perceived by us as an immovable point.
The almost immeasurably small electrons when moving are transformed into lines, and these lines intertwining among themselves create for us the impression of mass, i.e. of hard, impenetrable matter, of which the three-dimensional bodies surrounding us consist.
Matter is created by the fine web made by the traces of the motion of the smaller "material points".
The study of the principles of this motion is necessary for the understanding of the world, because it is only when we make these principles clear to ourselves that we shall have an exact conception of how the web created by the motion of the electrons is woven and thickened, and how the whole world of infinite variety of phenomena is constituted from this web.
To regard, for instance, the body of man as consisting of electrons or even of molecules is as wrong as it would be to regard the population of a large town or a company of soldiers or any gathering or people as consisting of cells. It is evident that the population of a large and even of a small town, or a company of soldiers, consists not of microscopic cells, but of individual men. Precisely in the same way the body of man consists of individual cells, or simply physically, of matter. Of course I have not in view a metaphor which would regard a gathering of people as an organism and individual people as cells of this organism.
A whole series of unnecessary hypotheses falls away as soon as we realise the general connectedness and cohesion which follow from the above definitions of matter and mass.
The first which falls away is the hypothesis of gravitation. Gravitation is necessary only in the "world of flying balls"; in the world of interconnected spirals it becomes unnecessary. Similarly there disappears the necessity of recognising a "medium" through which gravitation, or "action at a distance", is transmitted. Everything is connected. The world constitutes one whole.
The Sun, the Moon, the stars, which we see, are cross-sections of spirals which we do not see. These cross-sections do not fall out of the spirals because of the same principle by reason of which the cross-section of an apple cannot fall out of the apple.
But the apple falls to the ground as though aiming at the centre of the Earth in virtue of an entirely different principle, namely the "principle of symmetry". In the section on Symmetry in chapter II of this book, there is a description of that particular movement which I called movement from the centre and towards the centre along radii, and which, with its laws enumerated there, is the foundation and cause of the phenomena of symmetry.
The laws of symmetry, when they are established and elaborated, will occupy a very important place in the new model of the Universe. And it is quite possible that what is called the law of gravitation, in the sense of the formula for calculation, will prove to be a partial expression of the law of symmetry.
The definition of mass as the result of the motion of invisible points dispenses with any necessity for the hypothesis of aether. A ray of light has material structure, and so has electric current; but light and electricity are matter not formed into atoms, but remaining in the electronic state.
One of these undefined concepts is "infinity".
Infinity has a definite meaning only in mathematics. In geometry infinity needs to be defined, and still more does it need to be defined in physics. These definitions do not exist, nor have there even been attempts at such definition that are worthy of attention. "Infinity" is taken merely as something very big, bigger than anything else we can conceive, and at the same time as something completely homogeneous with the finite, yet incalculable. In other words, it is never said anywhere in a definite and exact form that the infinite is not homogeneous with the finite. I mean that it has not been established exactly what distinguishes the infinite from the finite either physically or geometrically.
In reality, both in the domain of geometry and in the domain of physics, infinity has a distinctive meaning which differs very greatly from the strictly mathematical meaning. The establishment of different meanings for infinity solves a number of otherwise insoluble problems and leads our thought out of a series of mazes and blind alleys created artificially or through misunderstanding.
First of all, an exact definition of infinity dispenses with the necessity for mixing up physics with geometry, which is the favourite idea of Einstein and the foundation of non-Euclidean geometry. I have pointed out earlier that the mixing up of physics and geometry, that is to say, the introduction of physics into geometry, or a physical revaluation of geometrical values, is unnecessary either for arguments concerning relativity or for anything else.
Physicists are quite right in feeling that geometry is not sufficient for them; in Euclidean space there is not enough room for them with their luggage. But the remarkable feature of the geometry of Euclid (and this is exactly why Euclidean geometry should be preserved intact) consists in the fact that it contains within itself an indication of the way out. There is no need to break up and destroy the geometry of Euclid. It can very well adapt itself to any kind of physical discovery and the key to this is infinity.
The difference between infinity in mathematics and infinity in geometry is quite clear at the first glance. Mathematics does not establish two infinities for one finite quantity. Geometry begins with this.
Let us take any finite line. What is infinity for this line? We have two answers: a line continued into infinity, or the square, of which the given line is a side. What is infinity for a square? An infinite plane, or the cube of which the given square constitutes a side. What is infinity for a cube? Infinite three-dimensional space, or a figure of four dimensions.
Thus the usual concept of an infinite line remains, but to it there is added another, the concept of infinity as a plane resulting from the motion of the line in a direction perpendicular to itself.
The infinite three-dimensional sphere remains; but a four-dimensional body constitutes infinity for a three-dimensional body.
Moreover, the problem becomes even simpler if we bear in mind that an "infinite" line, an "infinite" plane, and an "infinite" solid are pure abstractions; whereas a (finite) line in relation to a point, a square in relation to a line, and a cube in relation to a square, are real concrete facts.
So, remaining within the domain of facts, the principle of infinity in geometry can be formulated as follows: for every figure of a given number of dimensions, infinity is a figure of the given number of dimensions plus one.
At the same time, the figure of the lower number of dimensions is incommensurable with the figure of the higher number of dimensions. Incommensurability (in figures of different numbers of dimensions) creates infinity.
All this is very elementary. But if we firmly bear in mind the inferences to be drawn from these elementary propositions, they will enable us to free ourselves from the influence of the wrongly interpreted Aristotelian principle of the constancy of phenomena. The principle of Aristotle is true only within the limits of the finite, within the limits of commensurability. As soon as the infinite begins, we know nothing and have no right to assert anything in relation to the unity of phenomena and laws.
Continuing these arguments, we meet with another still more interesting fact, that is, that physical infinity differs from geometrical infinity as greatly as geometrical infinity differs from mathematical infinity. Or, to be more precise, physical infinity begins much sooner than geometrical infinity. And if mathematical infinity has only one meaning and geometrical infinity two meanings, physical infinity can have many meanings, that is, the mathematical meaning (incomputability), the geometrical meaning (the presence of an additional dimension or immeasurable extension), and purely physical meanings, that is, differences in function.
Infinity is created by incommensurability. But incommensurability can be arrived at in different ways. And in the physical world, incommensurability can be brought about because of the quantitative difference alone. As a rule, only quantities which are different qualitatively are regarded as incommensurable, and the qualitative difference is regarded as independent of the quantitative difference. But this is precisely where the chief mistake lies. Quantitative difference brings about qualitative difference.
In the mathematical world incommensurability is created by the incomputability of one of the quantities compared; in the geometrical world it is created either by the infinite extension of one of the quantities which are being compared or by the presence in it of a new dimension. In the physical world it is brought about simply by a difference in size which sometimes even permits of calculation.
All this means that infinity in geometry differs from infinity in mathematics in being relative. But geometrical infinity has no absolute meaning. A square is infinity for a line, but it is merely bigger than another smaller square or smaller than another bigger square.
In the physical world a large body is often incommensurable with a small one, and the small body bigger than the large one. A mountain is incommensurable with a mouse, and the mouse is bigger than the mountain.
The reason for this is that in the physical world things are determined by their functions (function in the sense of purpose). The function of every individual thing is possible only if the thing itself has a definite size. The reason why this has not been noticed and established long ago is to be found in a wrong understanding of the principle of Aristotle.
Physicists have often come upon manifestations of this law, namely, that the function of every individual thing is possible only if the thing itself has a definite size, but it has never arrested their attention and never led them to put together observations obtained in different domains. In the formulation of many physical laws we find qualifications that the particular law is true only of medium quantities, and that in the case of larger quantities or smaller quantities the law changes.
This law is still more clearly seen in the phenomena studied by biology and sociology.
The conclusion from what has been said can be formulated in the following way:
All that exists is what it is only within the limits of a certain and very small scale. On another scale it becomes something else. In other words, every thing and every event has a certain meaning only within the limits of a certain scale, when compared with things and events of proportions not very far removed from its own.
A chair cannot be a chair in the planetary world. Similarly, a chair cannot be a chair in the world of electrons. A chair has its meaning and its three dimensions only among objects created by the hand of man, serving the needs and requirements of man, and commensurable with man. On the planetary scale a chair cannot have individual existence because it cannot have any function. It is simply a small particle of matter inseparable from the matter surrounding it. As has been explained before, in the world of electrons also, a chair becomes too small for its function and therefore loses all its meaning and all its significance. A chair actually does not exist in comparison even with things which differ from it much less than planets or electrons. A chair in the midst of the ocean, or a chair in the midst of the Alpine ranges, would be a point having no dimension.
All this shows that incommensurability exists not only among things of different categories and denominations, and not only among things of a different number of dimensions, but also among things which merely differ considerably in size. A big object is incommensurable with a small object. A big object is often infinite in comparison with a small one.
Every separate thing and every separate phenomenon, in becoming bigger or smaller, ceases to be what it was and becomes something else something belonging to another category.
From the point of view of old physics, velocity, which was considered a generally understood phenomenon requiring no definition, always remained velocity; it could grow, increase, become an infinite velocity. It occurred to no one to doubt it. And having only accidentally stumbled upon the fact that the velocity of light is a limiting velocity, physicists were forced to admit that all was not well, and that the idea of velocity needed revision.
But physicists certainly could not surrender at once and admit that velocity can cease to be velocity and can become something else.
What did they actually stumble upon?
They stumbled upon an instance of infinity. The velocity of light is infinity as compared with all the velocities which can be observed or created experimentally. And, as such, it cannot be increased. In actual fact it ceases to be velocity and becomes an extension.
A ray of light possesses an additional dimension as compared with any object moving with "terrestrial velocities".
A line is infinity in relation to a point. The motion of the point does not alter this relation; a line will always remain a line.
The idea of limiting velocity presented itself when physicists hit upon a case of obvious infinity. But even apart from this, all the inconsistencies and contradictions in the old physics which were discovered and calculated by Prof. Einstein and supplied him with material for the building of his theories all these without exception result from the difference between the infinite and the finite. He himself often alludes to this.
Einstein's description of the example of "the behaviour of clocks and measuring rods on a rotating marble disc" suffers from one defect. Prof. Einstein forgot to say that the diameter of the "marble disc" to which are fastened the clocks which begin to go at different speeds with the movement of the disc, should be approximately equal to the distance from the Earth to Sirius; or else, the "clocks" must be the size of an atom (about five million of which can be put in the diameter of a full stop). With such a difference in size strange phenomena can actually occur, such as the unequal speed of the clocks or the change in the length of the measuring rods. But there could not be a "disc" with diameter equal to the distance from the Earth to Sirius, or clocks the size of an atom. Such clocks will cease to exist long before they change their speed, though this cannot be intelligible to modern physics which, as I pointed out before, cannot get free of the Aristotelian principle of the constancy of phenomena and therefore cannot notice that constancy is always destroyed by incommensurability. It can be assumed generally that, within the limits of terrestrial possibilities, the behaviour of both the clocks and the measuring-rods will be quite respectable, and for all practical purposes we can safely rely upon them. There is only one thing we must not do we must not set them any "problems with infinity".
After all, all the misunderstandings are caused by problems with infinity, chiefly because infinity is introduced on a level with finite quantities. The result will of course be different from what is expected. An unexpected result demands adaptation. The "special principle of relativity" and the "general principle of relativity" are very complicated and cumbersome adaptations for the explanation of the strange and unexpected results of "problems with infinity".
Prof. Einstein himself writes that proofs of his theories can be found either in astronomical phenomena or in the phenomena of electricity and light. In other words, he affirms by this that all problems requiring particular principles of relativity arise from problems with infinity or with incommensurability.
The special principle of relativity is based on the difficulty of establishing the simultaneity of two events separated by space, and above all on the impossibility of the composition of velocities in comparing terrestrial velocities with the velocity of light. This is precisely a case of the established heterogeneity of the finite and the infinite.
Of this heterogeneity I have spoken earlier; as regards the impossibility of establishing the simultaneity of two events, Prof. Einstein does not specify at what distance between two events the establishment of their simultaneity becomes impossible. If we insist upon an explanation we shall certainly receive the answer that the distance must be "very great". This "very great" distance again shows that Prof. Einstein presumes a problem with infinity.
Time really is different for different moving systems of bodies. But it is incommensurable (or it cannot be synchronised) only if the moving systems are separated by a large space which is actually infinity for them, or when they differ greatly in size or velocity, that is, when one of them is infinity in comparison with the other, or contains infinity.
To this may be added that not only time, but also space, is different for them, changing according to their size and velocity.
The general proposition is quite correct: "Every separately existing system has its own time".
But what does "separately existing" mean? And how can there be separate systems in a world of connected spirals? All that exists in the world constitutes one whole; there can be nothing separate.
The principle of the absence of separateness, of the impossibility of separateness, constitutes a very important part of certain philosophical teachings, for instance of Buddhism, where one of the first conditions for a right understanding of the world is considered to be the destruction of the "sense of separateness" in oneself.
From the point of view of the new model of the Universe, separateness exists but only relatively.
Let us imagine a system of cog-wheels, rotating with different velocities which depend on their size and the place occupied by each of them in the system. The system, for instance the mechanics of an ordinary watch, constitutes one whole, and from this point of view there can be nothing separate in it. From another point of view, each separate cog-wheel moves at its own velocity, i.e. it has a separate existence and its own time.
Let us try to begin the examination of these quantities in the same way as we began the examination of infinity and infinite quantities, that is, let us try to compare their meanings in mathematics, in geometry, and in physics.
Zero in mathematics has always one meaning. There is no reason to speak of zero quantities in mathematics.
Zero in mathematics and the point in geometry have approximately the same meaning, with the difference that the point in geometry indicates the place at which something begins or at which something ends, or at which something happens, for instance where two lines intersect one another; whereas in mathematics zero indicates the limit of certain possible operations. But in their essence there is no difference between zero and the point, because neither has independent existence.
The case is quite different in physics. The material point is a point only on the given scale. If the scale is changed, the point can prove to be a very complex and many-dimensional system of immense measurements.
Let us imagine a small map, on which even the bigger towns are points. Let us suppose that we have found the means to bring out the content of these points or to fill them with content. Then, what looked like a point will manifest a great many new properties and characteristics, and the extensions and measurements included in it. In the town will appear streets, parks, houses, people. How are the measurements of these streets, squares and people to be understood?
When the town was for us a point, they were smaller than a point. Is it not possible to call them negative dimensions?
The uninitiated, in the majority of cases, do not know that the concept "negative quantity" has no definition in mathematics. It has a certain meaning only in elementary arithmetic and in algebraical formulae, where it designates the operation to be performed rather than the difference in the properties of the quantities. In physics, "negative quantity" does not mean anything at all. Nevertheless we have already come upon negative quantities. It was when speaking of dimensions inside the atom that I had to point out that although the atom (or the molecule) had no dimensions for the direct senses, i.e. it is equal to zero, these dimensions or extensions inside the atom are still smaller, i.e. smaller than zero.
So we need no metaphors or analogies in order to speak about negative dimensions. These are the dimensions within what appears to be a material point. And this explains exactly why it is wrong to regard small particles of matter such as atoms or electrons as material. They are not material, because they are negative physically, i.e. smaller than physical zero.
Putting together all that has been set forth hitherto, we see that besides the period of six dimensions, we have imaginary dimensions, the seventh, the eighth, and so on, which proceed in non-existent directions and differ in the degrees of impossibility; and negative dimensions within the smallest particles representing for us material points.
I will complete the new model of the Universe by an analysis of a ray of light, but before beginning this analysis I must examine certain further properties of time taken as a three-dimensional continuum.
Until now I have taken time as the measure of motion. But motion in itself is the sensation of an incomplete perception of the space in question. For a dog, for a horse, for a cat, our third dimension is motion. For us motion begins in the fourth dimension and is a partial sensation of the fourth dimension. But as for animals the imaginary movements of objects which in reality constitute their third dimension merge into those movements which are movements for us, that is into the fourth dimension, so for us movements of the fourth dimension merge into movements of the fifth and sixth dimensions. Starting from this we must endeavour to establish something which will allow us to judge the properties of the fifth and sixth dimensions. Their relation to the fourth dimension must be analogous to the relation of the fourth dimension to the third, of the third to the second, and so on. This means that first of all the new, the higher, dimension must be incommensurable with the lower dimension and form infinity for it, seeming to repeat its characteristics an infinite number of times.
If we take "time" (that is extension from before to after) as the fourth dimension, what will be the fifth dimension in this case, that is, what forms infinity for time, what is incommensurable with time?
It is precisely phenomena of light that enable us to come into immediate contact with movements of the fifth and sixth dimensions.
The line of the fourth dimension is always and everywhere a closed curve, although on the scale of our three-dimensional perception we do not see either that this line is curved or that it is closed. This closed curve of the fourth dimension, or the circle of time, is the life or existence of every separate object, of every separate system, which is examined in time. But the circle of time does not break up or disappear; it continues to exist and, joining other, previously formed, circles, it passes into eternity. Eternity is the infinite repetition of the completed circle of life, an infinite repetition of existence. Eternity is incommensurable with time. Eternity is infinity for time.
Quanta of light are precisely such circles of eternity.
The third dimension of time (the sixth dimension of space) is the stretching out of these eternal circles into a spiral or a cylinder with a screw-thread in which each circle is locked in itself (and motion along it is eternal) and simultaneously passes into another circle which is also eternal, and so on.
This hollow cylinder with two kinds of thread would be a model of a ray of light a model of three-dimensional time.
The next question is: Where is the electron? What happens to the electron of the luminous molecule which sends out quanta of light? This is one of the most difficult questions for new physics.
From the point of view of the new model of the Universe the answer is clear and simple.
The electron is transformed into quanta; it becomes a ray of light. The point is transformed into a line, into a spiral, into a hollow cylinder.
As three-dimensional bodies, electrons do not exist for us. The fourth dimension of electrons, that is their existence (the completed circle), also has no measurement for us. It is too small, has too short duration, is shorter than our thought. We cannot know about them, i.e., we cannot perceive them in a direct way.
Only the fifth and sixth dimensions of electrons have certain measurements in our space-time. The fifth dimension constitutes the thickness of the ray and the sixth dimension its length.
Therefore in radiant energy, we deal not with electrons themselves but with their time dimensions, with the traces of their movement and existence, of which the primary web of any matter is woven.
Now if we accept the approximate description of the ray of light as a hollow cylinder consisting of quanta lying close to one another lengthwise along the ray, the picture becomes clearer.
First of all, the conflict between the theory of undulatory movements and the emission theory is settled, and it is settled in the sense that both theories prove to be equally true and equally necessary, though they refer to different phenomena or to different sides of the same kind of phenomenon.
Vibrations or undulatory movements, which were taken for the cause of light, are undulatory movements transmitted along already existing rays of light. What is called the "velocity of light" is probably the velocity of these vibrations passing along the ray. This explains why the calculations made on the basis of the theory of vibrations proved to be correct and made new discoveries possible. In itself a ray has no velocity; it is a line, a space concept, not a time concept.
No aether is necessary, for vibrations travel by light itself. At the same time light has "atomic structure", for a cross-section of a beam of light would show a network through the mesh of which the molecules of the gas it meets can easily slip.
In spite of the fact that scientists speak of the very accurate methods which they possess for counting electrons and measuring their velocities, it is permissible to have doubts whether they really mean electrons and not their extensions along the sixth dimension, the extensions which have already acquired space meaning for us.
The material structure of a ray of light explains also its possible deviations under the influence of forces acting upon it. But it is certain that these forces are not "attraction" in the Newtonian sense, although they may very possibly be magnetic attraction.
It was made clear before that the space of small units is proportionate to their size, and in exactly the same way their time is proportionate to their size. This means that their time, i.e. the time of their existence, is almost non-existent in comparison with our time.
Physics speaks of observing electrons and calculating their weight, velocity, etc. But an electron is for us only a phenomenon, and a phenomenon which is quicker than anything visible to our eyes; an atom as a whole is perhaps only a longer phenomenon, but longer on the same scale, just as there are various instantaneous speeds in a photographic camera. But both the atom and the electron are only time phenomena for us and, moreover, "instantaneous" phenomena; they are not bodies, not objects. Some scientists assert that they have succeeded in seeing molecules. But do they know how long by their clock a molecule can exist? During its very short existence a molecule of gas (which alone may be accessible to observation, if this be possible at all) travels through immense distances and will in no case appear either to our eye or to the photographic camera as a moving point. Seen as a line, it would inevitably intersect with other lines, so that it would be more difficult to trace a single molecule, even for the period of a small fraction of a second; and even if this became possible in some way, it would require such magnification as is actually impossible up to the present time.
All this must be kept in view in speaking, for instance, of phenomena of light. A great many misunderstandings fall away at once if we realise and carefully bear in mind the fact that an "electron" exists for an immeasurably small part of a second, which means that it can never under any condition be seen or measured by us, as we are.
It is impossible with existing scientific material to find firm ground for any theory as to the short existence of small units of matter. The material for such a theory is to be found in the idea of "different time in different cosmoses", which forms part of a special doctrine on the world, which will be the subject of another book.
1911-1929.