THE next step in non-Euclidean Geometry was its extension to a space that had nothing in common with Euclidean space, i.e., to a space of more than the three accepted dimensions. This jump was made by Minkowski, a disciple of Riemann, and in it he was followed by Einstein and his fellow Relativists. This was a still more exaggerated case of attributing analytical formulae to physical conditions, making the latter conform to the former instead of contrariwise.

The new Geometry was constructed on the assumption that space and time were not separate entities, not essentially distinct, but merely special aspects of a more fundamental world element. Space, therefore, no longer really exists, nor does time exist, but a new element called space-time is the only reality. This space-time constitutes a four-dimensional continuum, which replaces the three-dimensional continuum of ordinary geometry, which is the geometry of solids as we know them. In this continuum time is taken as the fourth dimension. The Geometry of this four-dimensional continuum is the Geometry introduced by Minkowski, and adopted by Einstein as the mathematical basis of his system of Relativity. This four-dimensional continuum constitutes a kind of “hyperspace” with four coordinates instead of three, time constituting the fourth.

The idea of using time as a function of the position of a point in space is not exactly new in mechanics. Time has long been used as a vector in the geometrical representation of motion and position. A four-coordinate analysis for mechanical problems was then by no means a fresh discovery, for the idea had been foreshadowed by mathematicians and physicists; it had been a topic of abstract discussion for perhaps a century before Minkowski. Thus Lagrange expresses the idea: “The position of a point in space depends on three rectangular coordinates; these coordinates in the problems of mechanics are conceived as being functions of t. Thus we may regard mechanics as a geometry of four dimensions.”¹ Lagrange still separates mechanics from geometry pure; it remained for Minkowski and Einstein to confuse them, and in this rests their discovery.

The existence of such an element is primarily a question of metaphysics and criteriology, since it brings with it a new and extraordinary conception of space and time, and sets aside and completely contradicts all our sense perceptions and intuitions concerning them. In the new Geometry our old conceptions must all go by the board, and are to be replaced by new and inconceivable ideas. It is not then primarily a question of geometry at all, since it goes much deeper; but because there are geometric implications in it, we must see how far these agree with, or contradict what is otherwise accepted in the science.

In the first place, the thing itself is not conceivable, and next, there is no argument for such a conception. A fourth dimension is not conceivable, and again time is not conceivable as a dimension, either fourth or any other. We cannot represent to ourselves a four-dimensional thing; and by representation we do not mean merely to visualize or imagine, but we are not even able to conceive it intellectually. “Hyperspace” or a “supersolid” is something inconceivable geometrically both in itself and in its geometrical elements.

We represent a solid or physical body as something consisting of three dimensions, and its position can be geometrically represented by three coordinates, or lines at right angles to one another. Anything beyond that is inconceivable. We can not only not conceive what a “supersolid” of four dimensions is, but even see in what way it can be composed of elements. We cannot represent the fourth dimension geometrically. The other three dimensions can be represented by lines at right angles to one another, but when this has been done, there is no room for a fourth line at right angles to the other three giving us a new direction. There is no new direction conceivable, and no room for the necessary right angle. Yet, if there is to be a fourth dimension, it must have a definite coordinate direction, a fact to which Minkowski particularly called attention.

If we apply any of the conditions that govern the relations of one dimension to another, we cannot fit them to a fourth dimension. For instance, a point in motion produces a line; a line in motion at right angles to itself produces a plane; a plane in motion at right angles to itself produces a solid. But a solid in any motion whatsoever can produce nothing but a solid, and never a “supersolid.” The solid can never move at right angles to itself, but only to one of its surface planes, and therefore it will only produce an increase of solid magnitude, and not a “supersolid.” In other words, a new motion at right angles to each of the three primary directions is inconceivable, and hence the “supersolid” is inconceivable. It is then not merely inconceivable in itself, but the element on which it depends, motion in the fourth direction, is not only inconceivable, but its contradictory is imposed on us by the very necessity of our understanding, i.e., that any motion of a solid can only be in relation to its planes, and not to itself as a solid.

Another case; if we take one dimension, or a line, it is possible to start from any point outside of it in the second dimension and reach any point in the line without passing through any other point in the line. Similarly, in respect to two dimensions, we can take any point in the third dimension, and from it reach any point in the plane of two dimensions without passing through any other point in the plane. Following the analogy, if there be a fourth dimension, we should be able to take any point in the fourth dimension and move from it to any point in three dimensions without passing through any other point of the three-dimensional continuum. A four-dimensional being should be able to step into the interior of a completely closed room, and he should be able to see into a solid as we now see into a square. But this is physically impossible. It can neither be represented nor thought. Thus we cannot get to the center of a solid without passing through all the points between the center and the surface. So again the fourth dimension is inconceivable according to our mode of conception.

Furthermore, we cannot create the higher dimensions by adding the lower together. An addition of points will not give us a line; an addition of lines will not give us a plane; an addition of planes will not give us a solid. An addition of solids will not give us a “supersolid.” All that these additions do, is to give us a multiplicity or a multitude of the order in which the addition is constructed. It is for this reason that motion is necessary to generate a higher dimensionality out of a lower. Now there is no possible or conceivable way to create a “supersolid” out of a solid by motion, unless we can point out a new direction not included in the directions given in the three dimensions. This is a thing that can neither be conceived nor be geometrically constructed. Yet we are told by Riemann that this thing is easy.

Again two lines cut in a point; two planes cut in a line; but two solids coincide in volume when they cut, but meet in a plane. Two solids then cut in a solid, and not two “supersolids.” If there were such things as “supersolids” conceivable, then these “supersolids” would have to cut in a “supersolid,” which is unthinkable. It is likewise unthinkable that they should meet in a solid just as two solids meet in a plane; for we can neither imagine nor think nor describe the solids and their relation to one another in which two such “supersolids” would meet. The fact that lines cut in a point, and planes in a line, but that nothing can cut in a plane, shows that our conception of volume or solid is final, and is the only real spatial entity, and that it cannot be part or element of any other, since itself embraces the total three elements. If the three dimensions were but elements of a fourth, then this should not hold. The four-dimensional continuum is therefore inconceivable.

Furthermore, we can project a two-dimensional figure on a line, a three-dimensional figure, or a volume, on a surface; but we can by no geometric means available to the human intellect project a four-dimensional figure on a three-dimensional one. Relativists go to a great deal of pains to try to explain their four-dimensional Geometry by means of projective geometry, but all their illustrations are drawn from solid figures projected on plane ones, and since there is no parity or analogy, the whole illustration falls without any possible deduction. Not only then is the four-dimensional continuum inconceivable in itself, but it is entirely impossible to conceive of any method whereby such a continuum can be projected either in a three-dimensional or two-dimensional continuum.

Finally, from the geometric viewpoint, four-dimensional space cannot be spherical, as is required by Riemann’s Geometry, nor can it be represented spherically. A sphere, as we understand it, is three-dimensional. What then can be a four-dimensional sphere? Sphere simply means a three-dimensional volume limited by a spherical surface. In what sense can a spherical surface limit four-dimensionality. A spherical surface is three-dimensional. How can a superficies of three dimensions enclose a volume of four dimensions? A four-dimension spherical surface is inconceivable. A four-dimensional sphere is then also inconceivable. For if space is limited and curved, the surface of total space must be spherical; then within such a spherical surface there are only three dimensions. A spherical surface surrounding four dimensions is an absurdity.

There can then be no question of a four-dimensional geometry in the ordinary understanding of the term. There is no possibility either of constructing figures, or demonstrating anything whatever with regard to it, since we cannot even conceive the elements geometrically. To state then that there is a four-dimensional geometry is simply playing with words and throwing dust into the eyes of others by using terms that have no meaning. Geometry, as we understand it, represents its elements by lines and directions, but we cannot represent a four-dimensional geometry by these means, even projectively, and therefore there is no such thing as a four-dimensional geometry.

Not only is a four-dimensional continuum geometrically inconceivable, but time especially is inconceivable as one of the dimensions. Time does not fit into our conception of dimensional space as one of the corresponding dimensions; it is rather a separate continuity. Every continuum must be measured by some standard outside of itself. In this it differs from discrete quantity which has the reasons of its measure in itself in the common nature of the individual. Since there is no individual distinction in the continuum, the measure must be applied from the outside, and is arbitrarily or conventionally set. Thus, we can use a yardstick to measure the spatial dimension known as distance, but we can just as well use the meter or any other conventional measure. The precise individual unit of measurement is not naturally determined; we can conventionally take any unit we wish for our own convenience. But the nature of the unit must be of the common nature of the whole continuum. Just as in discrete quantity what enables us to measure the multitude is the common nature that underlies all individual units, so in the continuum we are able to apply a unit only when that unit possesses the same nature as the whole continuum. The only difference, as far as measurement is concerned, between discrete and continuous quantity is that in the former the unit is determined by nature, in the latter by convention, but in all cases the unit can be applied to measure the whole only under the condition that it is of a common nature with the whole.

A measure of space must therefore always be some unit of distance. We cannot measure extensive quantity by any other unit than a unit of extensive quantity or distance. This is a point, and a very simple point at that, that Riemann and all his successors, the Relativist mathematicians, have entirely missed. There is no common measure of time and distance. Distance is measured relatively to some position; time is measured according to the motion of some accepted aspect of change. The measure of distance is something static; that of time something changeable and changing. Time itself is the measure of motion; motion is our means of measuring time. In other words, it is through motion that we perceive time.

We cannot have a continuum made up of two different, disparate, and incommensurable units of measurement. We cannot have a mile of time, or a foot or a yard or a meter of time, to make a new dimension with the distance measure of the other three dimensions.² There is no possible common measure of the three dimensions of space and the fourth dimension of time. There is then no possible fourth coordinate in relation to the other three coordinates. Geometers of four dimensions leap this difficulty in their analytical treatment of the subject by taking all the dimensions as infinitesimal, therefore not only begging the question by assuming that the measure of time is the same as the measure of distance, not only in kind, but actually in unit of measurement. This is certainly not brilliant mathematical, physical, or metaphysical reasoning. It is only a blundering, myopic intellect which cannot perceive the ground on which it is trying to walk, that could ever do this; certainly not a genius of discovery. Here is how it is done.

Every point in space can be defined by three numbers or measures which are the coordinates of that point, and distinguish it from all other points. Where the equations are differential, they contain only the distance ds, which may then be considered a line element, and the coordinates in the other member of the equation are also infinitesimals. Thus in Cartesian coordinates the line element assumes the form of the equation

If x, y and z were equal, then we should reduce the equation to a differential of   \/^— 3x².

Now Riemann’s equation for the line element takes this other form in the differential of x

All that was required by Minkowski for the application of his four-dimensional Geometry was the change

where the time element was put on the same basis as the distance element. This is equivalent to taking a four-dimensional quantity of 2 ft. by 2 ft. by 2 ft. space by 2 ft. of time element. The fact that the equation is differential and the elements infinitesimal does not make any difference, since the differential equation must be based on the actual value of the variables in an equation of real values.

The whole concept of a space-time continuum shows an entire lack of an intelligent idea of time. Time for Relativists and four-dimensional Geometers loses all its physical and metaphysical meaning, and becomes no more than the symbol t, a unit to be handled mathematically. As a matter of fact time is not a function of three dimensions even mathematically, but goes with one dimension as its function. Motion may take place in any or all of the three dimensions; it may be in one dimension as motion in a straight line; it may take place in two directions or dimensions as curvilinear motion in a plane; or it may take place in three dimensions as helical motion. But motion itself is always geometrically conceived and represented as a line, or as one direction, and a thing in motion as a point or position which indeed may change its position in space, but its direction ds is always considered as one. Time is, therefore, not a geometrical element at all, but a mechanical one, and is the function of only one dimension at the same time.

This is the reason, as we have seen above, that the motion of the point generates the line, the line the plane, the plane the solid, but the solid not a “supersolid,” but only increase of volume. All these motions are in a straight line and in one direction; the point, the plane, and the solid move in the very same way with the same kind of motion, and time is the function of but one dimension, that denoted by the vector of motion and velocity. There can be no motion then to create a “supersolid.” Time cannot be a fourth dimension.

We have seen that the higher dimension cannot be generated from a lower by means of addition, but can only be realized in this way through motion. The first dimension is created by the motion of a point; the second by motion of a line; the third by the motion of a plane. It is clear then that time cannot be considered as a fourth dimension in the sense that the motion of a solid of three dimensions through time the fourth dimension will give a “supersolid,” or a continuum of four dimensions. For all the motions creating all the dimensions from the single dimension to three dimensions are motions in time. The motion of the point to produce the line is a motion in time; the motion of the line to produce the plane is in time; the motion of the plane to produce the volume is in time. Time is not, therefore, a fourth direction through which a solid may move to create a four-dimensional continuum. For, if it were, the motion of a point would already give two dimensions, its direction in space and its direction in time.

Time has no relation at all to distance as distance, or as a dimension. What time really is, is this; it is a function of motion in distance or space, such that time can be taken as the external measure or unit of motion in distance, and vice versa. It has no connection with distance except through motion. If there were no movement, time would not enter into relation with distance at all, or with dimensionality. Time is, then, both as a unit of measurement and as a physical thing, not only entirely separate from dimension, but its concept is entirely different, and is irreducible to any common denominator with the dimensions, to say nothing of its being part and parcel of them. Time is, therefore, a mechanical and not a geometric element.

If time is not considered thus, and if it is not treated solely as a function of motion, all mechanics is immediately destroyed, and this is exactly what the Relativists do in favor of their newfangled Geometry. If in actual physics motion is simply a mere curve in the space-time continuum, there will be no means of measuring the velocity of this motion, and there could be no such motion, since it would become a static curve, the same as any curve drawn in two or three-dimensional space. The time element would not be external to the dimension for a measure of motion. There would be no truth in a velocity of so many feet a minute, for there could be no longer any minutes, since the time element would have to have the same measure as the space element, and not a disparate one. But, of course, the whole conception is inconceivable and fundamentally absurd. Metaphysically the absurdities are still greater. For instance, if four-dimensionality is a reality, then there is no change or motion whatsoever, and just as we are now extended by our physical bulk in three dimensions, so according to the four-dimensional theory we would be spread out at the same time from birth to death over four dimensions, a kind of monster everywhere changeless and asleep, except that small particle of the mass represented by the now of time and our physical bulk which is enlivened by consciousness. The only change possible would be that in consciousness itself as it becomes cognizant of some separate small part of the whole. A strange monster indeed. But this concerns the metaphysics and psychology of four-dimensionality, a point that we shall
examine again.

Now as to the arguments by which this preposterous conception is supported. Relativists and neo-Geometers have greatly exerted themselves to make this four dimensionality intelligible. G. T. Fechner tries to arrive at a notion of the fourth dimension in this way. If we imagine the picture of a man as reflected in a camera obscura to be a living intelligent being, it would live and move in a plane only, as being a planar or two-dimensional being. If now we picture to ourselves a continual displacement of this hypothetical being through the third dimension, “the latter would enter its experience as a succession of different two-dimensional states.” That is equivalent to saying that the third dimension would then actually exist for this being but disintegrated into a time succession. Then Fechner adds: “There is a time in which all changes, ourselves included. What is the reason of this? The movement of our three-dimensional space through the fourth dimension.” This argument seems to have pleased all the neo-Geometers and Relativists, for among others it has been brought forward by Riemann, A. F. Lange, Helmholtz, Einstein, and Poincaré. But it is false and misleading, and no argument at all, as we shall see. In the first place, there is the impossibility of such a being; in the second place, it is misleading, because analogy is not argument, or only rhetorical argument whose aim is persuasion, not truth; finally, even the analogy is lacking, and the parallel fails.

In the first place, the conception of a two-dimensional being is false. All admit that it is fictitious and not physically possible; under present conditions of existence no planar being is physically possible. But this is not the point of our objection. We are not precisely objecting to the physical possibility of the planar being, for it might be legitimate to imagine the unreal to illustrate the real. What we object to is this: it is the supposition in back of the imaginary illustration, that the higher dimensions are made out of the lower. First comes one dimension, then two dimensions, then three, and so on.

This may be true in a way, i.e., mathematically and logically, but it is not true physically and metaphysically. As far as reality is concerned, the first dimensionality we have is that of volume, or three dimensions. It is only by mental abstraction that we obtain two dimensions and one dimension. Really, then, one dimension and two dimensions are obtained by analysis of volume, or three dimensions, and not contrariwise, volume by synthesis of three dimensions. We cannot argue from a lower dimension to higher dimensions, except in so far as the facts of the lower taken from the higher allow. The planar being is then not merely an actual physical impossibility, but a pure logical creation, an ens rationis, which cannot be used to illustrate reality. It is then not a matter of actual lack of physical existence as such, or even of possible existence that we object to, but to the argument founded on such a conception.

Einstein falls into the same blunder in reasoning. Thus he says: “In virtue of the latter property we speak of a ‘continuum,’ and owing to the fact that there are three coordinates we speak of it as being ‘three-dimensional.’ ”³ Here we have in a nutshell the fallacy that is at the basis of the whole four-dimension theory. A continuum is three-dimensional because it can be represented by three coordinates, instead of the other way around, that it can be represented by three coordinates because it is three-dimensional. Einstein not only puts the cart before the horse, but endeavors to make the cart pull the horse into four-dimensionality. Here is the argument; it follows what was quoted above. “Similarly, the world of phys phenomena which was briefly called ‘world’ by Minkowski, is naturally four-dimensional in the space-time sense. For it is composed of individual events, each of which is described by four numbers, namely, three space ordinates, x, y, z, and a time coordinate, the time-value t.” We have here a two-fold fallacy: first, the assumption to which we called attention above, that dimensionality depends on the coordinates instead of the verse; and second, that any four determining numb are four coordinates, which is a complete begging of the question.

The sophism that is basic in the whole system of Relativity and four-dimensionality is right here. In the 1 ginning of his work on Relativity, Einstein tells us: “In order to have a complete (italics his) description of 1 motion, we must specify how the body alters its position with time; i.e., for every point on the trajectory it ml be stated at what time the body is situated there.”⁴ This is just the point at issue. A complete knowledge of a motion requires two things, a knowledge of its relative positions, i.e., of the length and direction of trajectory, and next, a knowledge of its relative rate change, or its velocity. One of these elements is static the other kinetic; one belongs to geometry, the other mechanics. For a static or geometric knowledge, even complete, we do not require the time; we only need time to mark the rate of change of something changing. The two elements of knowledge are in different categories, and hence in different sciences, and they are so kept classical mechanics. The one can be geometrically represented by coordinates, not the other. Relativists class them together as in the same category, and represent them all by coordinates, and thus, if they are logical, destroy all change and all motion, and reduce all to a static four-dimensionality.

Moreover, there is no argument to be drawn from such an analogy as that given above; even if allowed, it would be naught but illustration. Let us grant the possibility of the impossible, a two-dimensional material being, and let us further grant the possibility of an analogy between such a being and real-dimensional beings. Let us then see what there is in the analogy. If this planar being were to move in three-dimensional space, would his motion be for him a mere succession of two-dimensional states? Decidedly not, if the being were an intelligent being. Fechner supposes his planar being to live and move in a planar field. This intelligent being would already know two dimensions and motion in time in these two dimensions. Now the supposition is that he moves in a third dimension. He would either be conscious of such motion or he would not. If not conscious, then of course the motion does not exist for him. If he is conscious of succession, he can only be conscious of dimensional movement, by being able to construct the distance through which his motion has made him pass. Since by hypothesis he is intelligent, he must be supposed to know plane geometry already, and among other things how to construct right angles out of lines perpendicular to one another. It would, therefore, be a simple matter for him to make a new right angle, by taking the new line of motion, which is perpendicular to his own being, and constructing it with any line in his own being to have a right angle. Either he should be able to construct the line of such motion. or he could not be conscious of it. He would then have a new plane geometry in a new direction. If, therefore, he did not conclude to the possibility of the existence of a three-dimensional continuum, the fault would be in his intelligence and not in the data supplied him. If he exists unconsciously in this dimension, there would still be no analogy, for he would not even be conscious of time in this dimension, nor would he be conscious of the change for which time is the measure.

There is then in no case a parallel between this little two-dimensional being consciously or unconsciously existing in the third dimension and ourselves, three-dimensional beings moving through time in the fourth. If it is not conscious of the third dimension then the analogy would lead us to conclude that we are equivalently unconscious of the fourth; then all the metageometers around the world and back again could not make it. a reality, since there would be no conscious basis for it. For the little being could have no time consciousness without space consciousness. If he is supposed conscious in any way of the movement, then the analogy would call for our being equivalently conscious of movement in the fourth dimension, and it should be no more a mystery to us than the three dimensions would be to the two-dimensional being. We are perfectly conscious of time, and movement in time, but of no movement in a dimension

The analogy is, therefore, entirely illusory, and furthermore is certainly not a sign of penetration and clearness even in marshalling ideas. If the instance of the planar being means anything, it means that the planar being is as much in four dimensions as we are supposed to be ourselves. For it really is made to move in three dimensions as we do, and its motion takes place in time, the same as ours. If then we exist in four-dimensional space, so does this fanciful planar being. But it would balk, the same as any intelligent three-dimensional being, at having such a four-dimensional continuum created for him by people who are unable to create a better analogy than the one given above, or not having been able to do better, did not have at least the good sense to keep it under cover, and not spread it out on paper for everyone to read. Such is about the nature of the only argument ever brought forward for four-dimensional space, and it is nothing but an abortive attempt to make it conceivable, not to prove it.

Not only is the idea of four-dimensionality inconceivable, but it is geometrically unrepresentable. Any treatment of more than three dimensions must be purely analytic, and the formal reasons for more than three coordinates can only be algebraic or analytic. Now analysis does not impose itself on geometry, but conversely. Analytic representation of geometric problems rests entirely on a convention, and hence analysis can never give any more than we actually put into it. It depends entirely on the data of the actual problem, which in turn depend on reality and experience. Analysis is, therefore, limited by the conditions that are naturally imposed upon it as far as geometry is concerned, which are those of experience, and beyond these limits analysis cannot be converted into the geometric relations of reality.

For instance: we cannot extend to four dimensions the data gotten from experience of three dimensions. Abstraction from reality stops there; it cannot get beyond three. There is then no geometric conception of four dimensions. If, however, we extend by analysis the terminology of dimensions as used in geometry to analysis where a certain quantity depends upon four separate quantities for its value, these latter quantities are no longer dimensions, but only parameters, which is quite a different conception from that of dimension. These parameters may be made in any number that the conditions of the analytic problems allow, but as far as geometry is concerned, they cannot represent or be made to represent anything more than three dimensions. It is also an equal blunder to confound parameters with coordinates, for a coordinate is more than a parameter, since it is directional.

Geometrical relations can be made analytical only within certain limits, and analytical relations can be reconverted into geometric only under the same limits. Analysis is only numerical mathematics which deals with purely logical relations. The relations of geometry are real relations, due to the fact that geometry deals with things that have other than a logical existence. Its mathematics must therefore be founded on objective conditions and not on pure logical relations.

Let us take, for instance, analytic geometry. Analytic geometry considers the position and relation of the elements of a geometric figure as vectors of direction, and by analyzing these into the terms of their coordinates is able to reduce the relations of distance and the relations of form to algebraic equations. But from this it does not follow that all possible algebraic equations can be construed as geometric forms and figures. This would be a pure fallacy, and it is this fallacious reasoning that is implicitly made by all those who use analytical methods to solve problems by more than the three coordinates of Euclidean or natural space.