Changes of State and Changes of Position Icon

February 4, 2022

Law of Homogeneity

The first of these laws is the law of homogeneity. Suppose that by an external change we pass from the aggregate of impressions A to the ag- gregate B, and that then this change α is corrected by a correlative voluntary movement β, so that we are brought back to the aggregate A. Suppose now that another external change α 0 brings us again from the aggregate A to the aggregate B. Experiment then shows us that this change α 0 , like the change α, may be corrected by a volun- tary correlative movement β 0 , and that this movement β 0 corresponds to the same muscular sensations as the move- ment β which corrected α.

This fact is usually enunciated as follows:

Space is homogeneous and isotropic. We may also say that a movement which is once produced may be repeated a second and a third time, and so on, without any variation of its properties. In the first chapter, in which we discussed the nature of mathematical reasoning, we saw the importance that should be attached to the possibility of repeating the same operation indefinitely.

The virtue of mathematical reasoning is due to this repetition; by means of the law of homogeneity geometrical facts are apprehended. To be complete, to the law of homogeneity must be added a multitude of other laws, into the detailsspace and geometry.

of which I do not propose to enter, but which mathemati- cians sum up by saying that these displacements form a “group.”

The Non-Euclidean World

If geometrical space were a framework imposed on each of our representations considered individually, it would be impossible to represent to ourselves an image without this framework, and we should be quite unable to change our geometry.

But this is not the case; geometry is only the summary of the laws by which these images succeed each other.

There is nothing, therefore, to prevent us from imagining a series of representations, similar in every way to our ordinary representations, but succeeding one another ac- cording to laws which differ from those to which we are accustomed. We may thus conceive that beings whose education has taken place in a medium in which those laws would be so different, might have a very different geometry from ours.

Suppose, for example, a world enclosed in a large sphere and subject to the following laws:—The temper- ature is not uniform; it is greatest at the centre, and gradually decreases as we move towards the circumference of the sphere, where it is absolute zero. The law of this temperature is as follows:—If R be the radius of thescience and hypothesis

sphere, and r the distance of the point considered from the centre, the absolute temperature will be proportional to R 2 − r 2 . Further, I shall suppose that in this world all bodies have the same co-efficient of dilatation, so that the linear dilatation of any body is proportional to its absolute temperature. Finally, I shall assume that a body transported from one point to another of different temperature is instantaneously in thermal equilibrium with its new environment.

There is nothing in these hypotheses either contradictory or unimaginable. A moving object will become smaller and smaller as it approaches the circumference of the sphere. Let us observe, in the first place, that although from the point of view of our or- dinary geometry this world is finite, to its inhabitants it will appear infinite. As they approach the surface of the sphere they become colder, and at the same time smaller and smaller.

The steps they take are therefore also smaller and smaller, so that they can never reach the boundary of the sphere. If to us geometry is only the study of the laws according to which invariable solids move, to these imaginary beings it will be the study of the laws of motion of solids deformed by the differences of temperature alluded to.

No doubt, in our world, natural solids also experience variations of form and volume due to differences of tem- perature. But in laying the foundations of geometry we neglect these variations; for besides being but small they are irregular, and consequently appear to us to be accidental. In our hypothetical world this will no longer be the case, the variations will obey very simple and regular laws. On the other hand, the different solid parts of which the bodies of these inhabitants are composed will undergo the same variations of form and volume.

Let me make another hypothesis: suppose that light passes through media of different refractive indices, such that the index of refraction is inversely proportional to R 2 − r 2 . Under these conditions it is clear that the rays of light will no longer be rectilinear but circular. To justify what has been said, we have to prove that cer- tain changes in the position of external objects may be corrected by correlative movements of the beings which inhabit this imaginary world; and in such a way as to restore the primitive aggregate of the impressions expe- rienced by these sentient beings. Suppose, for example, that an object is displaced and deformed, not like an invariable solid, but like a solid subjected to unequal dilatations in exact conformity with the law of temper- ature assumed above. To use an abbreviation, we shallscience and hypothesis 78 call such a movement a non-Euclidean displacement. If a sentient being be in the neighbourhood of such a displacement of the object, his impressions will be mod- ified; but by moving in a suitable manner, he may re- construct them. For this purpose, all that is required is that the aggregate of the sentient being and the object, considered as forming a single body, shall experience one of those special displacements which I have just called non-Euclidean. This is possible if we suppose that the limbs of these beings dilate according to the same laws as the other bodies of the world they inhabit. Although from the point of view of our ordinary geometry there is a deformation of the bodies in this displacement, and although their different parts are no longer in the same relative position, nevertheless we shall see that the impressions of the sentient being remain the same as before; in fact, though the mutual distances of the different parts have varied, yet the parts which at first were in contact are still in contact. It follows that tactile impressions will be unchanged. On the other hand, from the hypothesis as to refraction and the cur- vature of the rays of light, visual impressions will also be unchanged. These imaginary beings will therefore be led to classify the phenomena they observe, and tospace and geometry.

distinguish among them the “changes of position,” which may be corrected by a voluntary correlative movement, just as we do.

If they construct a geometry, it will not be like ours, which is the study of the movements of our invariable solids; it will be the study of the changes of position which they will have thus distinguished, and will be “non- Euclidean displacements,” and this will be non-Euclidean geometry. So that beings like ourselves, educated in such a world, will not have the same geometry as ours. The World of Four Dimensions.—Just as we have pic- tured to ourselves a non-Euclidean world, so we may pic- ture a world of four dimensions. The sense of light, even with one eye, together with the muscular sensations relative to the movements of the eyeball, will suffice to enable us to conceive of space of three dimensions. The images of external objects are painted on the retina, which is a plane of two dimen- sions; these are perspectives. But as eye and objects are movable, we see in succession different perspectives of the same body taken from different points of view. We find at the same time that the transition from one per- spective to another is often accompanied by muscular sensations. If the transition from the perspective A toscience and hypothesis 80

the perspective B, and that of the perspective A 0 to the perspective B 0 are accompanied by the same muscular sensations, we connect them as we do other operations of the same nature. Then when we study the laws ac- cording to which these operations are combined, we see that they form a group, which has the same structure as that of the movements of invariable solids. Now, we have seen that it is from the properties of this group that we derive the idea of geometrical space and that of three dimensions. We thus understand how these per- spectives gave rise to the conception of three dimensions, although each perspective is of only two dimensions,— because they succeed each other according to certain laws. Well, in the same way that we draw the perspective of a three-dimensional figure on a plane, so we can draw that of a four-dimensional figure on a canvas of three (or two) dimensions. To a geometer this is but child’s play. We can even draw several perspectives of the same figure from several different points of view. We can eas- ily represent to ourselves these perspectives, since they are of only three dimensions. Imagine that the different perspectives of one and the same object to occur in suc- cession, and that the transition from one to the other is accompanied by muscular sensations. It is understoodspace and geometry.

that we shall consider two of these transitions as two operations of the same nature when they are associated with the same muscular sensations. There is nothing, then, to prevent us from imagining that these operations are combined according to any law we choose—for in- stance, by forming a group with the same structure as that of the movements of an invariable four-dimensional solid. In this there is nothing that we cannot represent to ourselves, and, moreover, these sensations are those which a being would experience who has a retina of two dimensions, and who may be displaced in space of four dimensions. In this sense we may say that we can repre- sent to ourselves the fourth dimension. Conclusions.—It is seen that experiment plays a con- siderable rôle in the genesis of geometry; but it would be a mistake to conclude from that that geometry is, even in part, an experimental science. If it were experimental, it would only be approximative and provisory. And what a rough approximation it would be! Geometry would be only the study of the movements of solid bodies; but, in reality, it is not concerned with natural solids: its object is certain ideal solids, absolutely invariable, which are but a greatly simplified and very remote image of them. The concept of these ideal bodies is entirely mental, andscience and hypothesis

experiment is but the opportunity which enables us to reach the idea. The object of geometry is the study of a particular “group”; but the general concept of group pre- exists in our minds, at least potentially. It is imposed on us not as a form of our sensitiveness, but as a form of our understanding; only, from among all possible groups, we must choose one that will be the standard, so to speak, to which we shall refer natural phenomena. Experiment guides us in this choice, which it does not impose on us. It tells us not what is the truest, but what is the most convenient geometry. It will be noticed that my description of these fantastic worlds has required no language other than that of ordinary geometry. Then, were we transported to those worlds, there would be no need to change that language. Beings educated there would no doubt find it more convenient to create a geometry different from ours, and better adapted to their impressions;

but as for us, in the presence of the same impressions, it is certain that we should not find it more convenient to make a change.