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

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Henri PoincaréIt seems, therefore, that space cannot be constructed by considering sets of simultaneous sensations, which must be considered as sequences of sensations. Always go back to what I said before. Why do certain changes appear to us as changes of position and others as changes of state without a geometric character? For this we must first distinguish external changes that are involuntary and not accompanied by muscular sensations and internal changes that are the movements of our body and that we distinguish from others because they are voluntary and accompanied by muscle sensations . An external change can be corrected by internal loading, for example when we follow a moving object with the eye so as to always bring its image back to the same point of the retina. An external change susceptible of such a correction is a change of position; if it is not susceptible of it, it is a change of state.

Two external changes, which qualitatively are quite different, are regarded as corresponding to the same change of position if they can be corrected by the same internal change. In the same way, two internal changes can be constituted by sequences of muscular sensations which have nothing in common and yet correspond to the same change of position, if they can correct the same external change. This is what we express in ordinary language, saying that there are many ways to go from one point to another.

What matters then are the movements that must be made to reach a certain object, the consciousness of these movements being nothing else for us than all the muscular sensations that accompany them.

This posited, a certain object is in contact with one of my fingers, for example the index of the right hand; I feel a touch sensation T; at the same time I receive from this object the visual sensations V; the object moves away, the sensation T vanishes, the sensations V are replaced by the new visual sensations V’; this is an external change. I want to correct part of this external change by restoring the sensation T, that is to say, bring my index in contact with the object. For that I have to execute certain movements which are translated for me by a certain sequence of muscular sensations S; I know that, because many experiments, whether by myself or by my ancestors, have taught me that when the sensation T disappears, and that the visual sensations pass from V to V’, we could restore the sensation T by the movements corresponding to the sequence S. I also know that I could have obtained the same result by other movements translated for me, no longer by S, but by another sequence S’ or S”.

All these sequences of muscular sensations S, S’, S”, … have perhaps no common element, I bring them together because I know that both allow me to restore the sensation T whenever the sensations V have become V’. In our usual language, to us who already know geometry, we will say that the various sequences of movements that correspond to the series of muscular sensations S, S’, S”, have this in common that, in one as in the other, the initial position, as well as the final position of my index remains the same. Everything else can differ.

I am thus led not to distinguish these various suites S, S’, S”…, to look at them as a unique individual. I will not distinguish either the consequences of muscular sensations which differ too little. I will then have enough to build a physical continuum and I have, in fact, chosen the elements of this continuum which are suites of muscular sensations and I possess the “fundamental convention” which teaches me in which cases two of these elements must be regarded as identical and it is this continuum which has three dimensions.

But that’s not all, we have just defined a continuum which is a real space; it is the space considered as described by one of my fingers; but I have several fingers (and from the point of view that occupies me, all the points of my skin could serve as my fingers). Will my different fingers describe the same space? Yes, no doubt, but what does that mean? This implies a set of properties that it would not be easy to state in ordinary language, but that I can try to explain if I am allowed to use certain symbols. I consider two fingers that I will call α and β; the finger α will be, for example, the index of the right hand which we have used, to define the sequences S, S’, S”, …, we will write then:

S ≡ S’ (mod α)

and that will mean that if the movements corresponding to S restore the tactile sensation experienced by the finger α, it will be the same movements corresponding to S’ and conversely. I will write the same

S1 ≡ S1‘ (mod β)

to express that if the movements corresponding to S1 restore the tactile sensation experienced by the finger β, it will be the same movements corresponding to S1.

Having said that, I suppose that there are two particular suites of muscular sensations s and s1 which will be defined as follows: I suppose the finger β experiences a tactile sensation due to the contact of an object; make the movements corresponding to s, this sensation will disappear, but, finally, it will be the finger α which will experience a sensation of contact; I know from experience that this will happen whenever before these movements, the finger β felt a touch; or at least almost every time (I almost say, because it requires to succeed that the object has not moved in the meantime) in our ordinary language (which would be clearer for us, but I do not dare not use since I’m talking about beings who do not yet know the geometry), we would say that the movements corresponding to s have brought the finger α in the place primitively occupied by the finger β. For s1, it will be the opposite, the corresponding movements bring the finger β to the place primitively occupied by the finger α.
If these two suites s and s1 exist, the relation

S ≡ S’ (mod α)

will result in the relationship:

s + S + s1 ≡ s + S’+ s1 (mod β)

this is what we immediately convince ourselves if we remember the meaning of these symbols and we can easily deduce that the two spaces, generated by α and by β, are isomorphic and, in particular, that they have the same number of dimensions.

It would not be the same if the suites s and s1 did not exist. Suppose, indeed, that one can not find a sequence of movements such as a sensation of contact of the finger β with an object, it makes succeed a sensation of contact of the finger α with this same object, and this if not for sure, at least almost certainly, how then would we reason? We would say that the finger β feels the object without being at the same point of space, that it feels at a distance; otherwise, whenever the finger β would feel the object, it would be that it would be at the same point A of the space; then there should be a sequence of movements that would bring the finger α to point A; and since the object is at point A, the finger α should sense the object and this should always succeed. So if we suppose that there is no sequence of movements enjoying this property, we must admit that the finger β feels the contact at a distance, that is to say that the fact to be felt by this finger is not enough to determine the position of the object in space, that is to say finally, that the space must have more dimensions than the physical continuum generated by the finger β in the way we said.

I suppose, for example, that space has four dimensions, and I denote by x, y, z, t the four coordinates; I suppose that the finger β feels the contact of the object, whenever the 3 coordinates x, y, z are the same for the finger and the object, whatever may be the fourth coordinate; and on the other hand that the finger α feels the contact of the object, whenever the 3 coordinates x, y, t are the same for the object and this finger, whatever is the coordinate z. In these conditions, apply our rules for the construction of the physical continuum generated by β; we will find it only 3 dimensions, which will correspond to the three coordinates x, y, z, the coordinate t playing no role. Similarly, the physical continuum generated by α would have 3 dimensions corresponding to x, y and t. But we could not find a sequence of movements corresponding to a series of muscular sensations s, such that the sensation of contact for α succeeds, for sure, to the sensation of contact for β.

Let x1, y1, z1, t1 be the coordinates of the object, x0, y0, z0, t0 those of the finger β before the movement; x0‘, y0‘, z0‘, t0 those of the finger α after the move. We will express that the finger β feels the contact before the movement by writing:

(1) x0 = x1, y0 = y1, z0 = z1,

we will express that α feels the contact after the movement by writing:

(2) x0‘= x1, y0‘ = y1, t0‘ = t1.

For s to exist, we would have to choose x0, y0, z0, t0; x0‘, y0‘, z0‘, t0 so that relations (1) lead to relations (2) whatever moreover x1, y1, z1, t1. It is clear that this is impossible. It is precisely the impossibility of forming s which would reveal to us in such a case that space would have 4 dimensions and not 3 like the physical continuum generated by β.

And besides, we actually observe something analogous if we involve the sense of sight. Consider a point of the retina, we can make it play the same role as our fingers α or β. We can consider the sequence of movements necessary to bring back the image of an object at this point γ of the retina, or the corresponding sequence S of muscular sensations; we can use this sequence to define a physical continuum analogous to that generated by α or by β. This continuum will have only two dimensions. But we can not build a suite analogous to s, that is to say a sequence of movements that succeeds without fail, to the visual sensation felt at the point γ the tactile sensation felt on the finger α. In other words, it is not enough for us to find that the image of the object is in γ, so that we can determine the movements necessary to bring our finger in contact with this object; we are missing a datum which is the distance from the object. And that is why we say that the sight is exercised at a distance, and that space has three dimensions, one more than the continuum generated by γ.

We can see from this rapid exposition what are the experimental facts that led us to attribute three dimensions to space. In the presence of these facts, it was more convenient for us to attribute to him three than four or two; but this word of convenience is perhaps not strong enough here; a being who attributed two or four dimensions to space would have been in a world like ours, in a state of inferiority in the struggle for life; What is it to say? let me take my symbols and, for example, the congruences

S ≡ S’ (mod α)

whose meaning I explained above. To attribute two dimensions to space would be to admit such congruences which we ourselves do not admit; we would then be exposed to replacing the movements S which succeeds in the movements S’ which would not succeed. To attribute to it four dimensions would be, on the contrary, to reject congruences, which we ourselves admit; we would then deprive ourselves of the possibility of substituting for the movements S other movements S’ which would succeed just as well and which could present, in certain circumstances, particular advantages.

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