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Henri Poincaré, Why space with three dimensions – Space and Nature

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But the question can be asked from a completely different point of view. We have so far placed ourselves in a purely subjective, purely psychological or, if you will, physiological point of view; we have only considered the relations of space with our senses. On the contrary, one could place oneself from the point of view of physics and wonder whether it would be possible to locate natural phenomena in a space other than ours and, for example, in a space with two or four dimensions. The laws revealed by physics are expressed by differential equations, and in these equations there are the three coordinates of certain material points. Is it impossible to express the same laws by other equations where, this time, there are other material points with four coordinates? Or would it be possible, but would the equations thus obtained be less simple? Or, finally, would they be just as simple and would we reject them simply because they shock our habits of mind?

What do we mean when we talk about expressing the same laws by other equations? Suppose two worlds M and M’; we can establish between the phenomena that occur or that could happen in these two worlds, a correspondence such that to any phenomenon Φ of the first corresponds a perfectly determined phenomenon Φ’ on the other, which is, so to speak, its image. So, if I suppose that the necessary effect of the phenomenon Φ, by virtue of the laws which govern the world M, is a certain phenomenon Φ1, and that the necessary effect of the phenomenon Φ’, image of Φ, by virtue of the laws which govern the world M’, is precisely the image Φ1, from the phenomenon Φ1, we can say that the two worlds obey the same laws. We do not care about the qualitative nature of the phenomena Φ and Φ’, we only need “parallelism” to be possible.

And, indeed, this qualitative nature of phenomena interests only our senses, and we have agreed to place ourselves in an extra-psychological point of view, to abstract, therefore, data of the senses and to consider only the mutual relations of phenomena. This is, indeed, what the physicist does when he substitutes, for example, the gases which experience reveals to us, and which gives us sensations of pressure and heat, with the gases of kinetic theory, where we see only material points in motion, or in the light of experience, and of the colored sensations it engenders, the vibrations of the ethereal medium.

It will suffice to consider a simple case, that of astronomical phenomena and the law of Newton. What we observe are not the coordinates of the stars, but only their distances; the natural expression of the laws of their movements, they are therefore differential equations between these distances and time. Now the distance of two points in space is a known and simple function of the coordinates of these two points. Let us transform our differential equations by substituting this function for each distance; we will then have these equations in their usual form, a form in which the coordinates of the stars appear.

But we could have replaced these distances with other functions, and we would have obtained other forms of these equations; all these forms would have been equally legitimate from the point of view which concerns us, since they would have respected the “parallelism” between the phenomena. Let us represent the stars as placed in four-dimensional space in such a way that the position of each of them is defined, no longer by three, but by four coordinates; let us then replace in our equations the quantity which we have hitherto considered as representing the distance of two stars by any function of the eight coordinates of these two stars; it is by no means necessary that this function be that which represents the distance of two points in ordinary four-dimensional space; it can be quite arbitrary since the “parallelism” will not be altered.

We will thus obtain a form of our equations where the coordinates of the stars will appear in the four-dimensional space; it will be a new expression of astronomical laws based on the hypothesis of a four-dimensional space and this expression will not be illegitimate since the condition of “parallelism” is respected. Only, it is clear that the equations thus obtained will be much less simple than our usual equations.

And it would undoubtedly be the same with the laws of physics. Is there a general reason for this to be so, as in all parts of Physics, it is the three-dimensional hypothesis which gives the equations their simplest form? Does this reason have any relation to that which was developed in the first part of this work and which compelled living beings to believe in the three dimensions or to act as if they believed it under pain of inferiority in the fight for life ?

Here, a short digression is necessary. Let’s go back, for a moment, to our old ordinary space. We say that it is relative and that means that the laws of Physics are the same in all the parts of this space, or in the mathematical language, that the differential equations which express these laws do not depend on the choice of the coordinate axes.

If we consider a perfectly isolated system, it makes no sense, we cannot observe the coordinates of the points of this system, but only their mutual distances, the observation will not be able to tell us if the properties of this system depend on the absolute position of the system in space, since this position is unobservable.

If the system is not isolated, it will not work either (if one wants to reason with rigor), since it will become impossible to express the laws which govern this system, without taking into account the action of the outer bodies. But there are systems almost isolated, surrounded by bodies close enough to be seen, too far away for their action to be sensible; this is what happens to our earthly world vis-a-vis the stars. We can then state the laws of this terrestrial world as if the stars did not exist, and yet bring this world back to a system of axes of coordinates perfectly defined and invariably related to these stars. So the experience shows us that the choice of these axes does not intervene, that the equations are not altered when making a change of axes. The set of possible axis changes form, as we know, a six-dimensional group.

Let us now renounce our ordinary space, replace our equations with others that will be equivalent, in the sense that they will respect the “parallelism” of phenomena. Whenever we are dealing with an almost isolated system, there will be an extremely general fact, a property of invariance which will subsist; there will be a group of transformations that will not alter the equations; these transformations will no longer have the meaning of a change of axes, their meaning may be any, but the group formed by these transformations must always remain isomorphic to the six-dimensional group of which we have just spoken; otherwise there would be no parallelism.

And this is because this group plays in any case an important role, because it is isomorphic to the group of axis changes in ordinary space, because it is so closely related to our three-dimensional space This is why our equations will take their simplest form when we highlight this group in the most natural way, that is, by introducing a three-dimensional space.

And since this group is isomorphic itself to that of the displacements of each of our members regarded as a solid body, as this property of solid bodies to move by obeying the laws of this group, is, in the last analysis, only a particular case of this property of invariance on which I just drew the attention, we see that there is no essential difference between the physical reason which leads us to attribute to the space three dimensions, and the psychological reasons developed in the first paragraphs of this chapter.

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