2013 June 7
PART 3: Poincaré's words are in boldface.
The space so created is only a little space extending no farther than my arm can reach ; the intervention of the memory is necessary to push back its limits. There are points which will remain out of my reach, whatever effort I make to stretch forth my hand ; if I were fastened to the ground like a hydropolyp, for instance, which can only extend its tentacles, all these points would be outside of space, since the sensations we could experience from the action of bodies there situated, would be associated with the idea of no movement allowing us to reach them, of no appropriate parry. These sensations would not seem to us to have any spatial character, and we should not seek to localize them.
But we are not fixed to the ground like some of the lower animals ; we can, if the enemy be too far away, advance toward him first and extend a hand when we are sufficiently near. This is still a parry, but a parry at long range.
On the other hand, it is a complex parry, and into the representation we make of it enters the representation of the muscular sensations caused by the movements of the legs, that of the muscular sensations caused by the final movement of the arm, that of the sensations of the semicircular canals, etc. We must besides represent to ourselves, not a complex of simultaneous sensations, but a complex of successive sensations, following each other in a determinate order, and this is why I have just said that the intervention of memory was necessary.
Notice moreover that to reach the same point I may approach nearer the mark to be attained, so as to have to stretch my arm less. What more? It is not one, it is a thousand parries I can oppose to the same danger. All these parries are made of sensations which may have nothing in common and yet we regard them as defining the same point of space, since they may respond to the same danger and are all associated with the notion of this danger. The potentiality of warding off the same stroke makes the unity of these different parries, just as the possibility of being parried in the same way makes the unity of such different kinds of strokes which may menace us from the same point of space. It is this double unity which makes the individuality of each point of space, and, in the notion of point, there is nothing else.
The space before considered, which might be called restricted space, was referred to coordinate axes attached to my body; these axes were fixed, since my body did not move and only my members changed their position.
What are the axes to which we naturally refer extended space, that is to say, the new space just defined? We define a point by the sequence of movements to be made to reach it, starting from a certain initial position of the body. The axes are therefore fixed to this initial position of the body.
But the position I call initial may be arbitrarily chosen among all the positions my body has successively occupied ; if the more or less unconscious memory of these successive positions is necessary for the genesis of the notion of space, this memory may go back more or less remotely into the past. Thence results in the definition itself of space a certain indeterminateness, and it is precisely this indetermination which constitutes its relativity.
There is no absolute space; there is only space relative to a certain initial position of the body. For a conscious being fixed to the ground like some of the lower animals, and consequently knowing only restricted space, space would still be relative (since it would have reference to his body), but this being would not be conscious of this relativity, because the axes of reference for this restricted space would be unchanging. Doubtless the rock to which this being would be fettered would not be motionless, since it would be carried along in the movement of our planet; for us consequently these axes would change at each instant; but for him they would be changeless.
We have the faculty of referring our extended space now to the position A of our body, considered as initial, again to the position B, which it had some moments afterward, and which we are free to regard in its turn as initial ; we make therefore at each instant unconscious transformations of coordinates. This faculty would be lacking in our imaginary being, and from not having traveled, he would think space absolute. At every instant, his system of axes would be imposed upon him; this system would have to change greatly in reality, but for him it would be always the same, since it would be always the only system. Quite otherwise is it with us, who at each instant have many systems among which we may choose at will, on condition of going back by memory more or less far into the past.
This is not all. Restricted space would not be homogeneous ; the different points of this space could not be regarded as equivalent, since some could be reached only at the cost of the greatest efforts, while others could be easily attained. On the contrary, our extended space seems to us homogeneous, and we say all points are equivalent.
What does this mean? If we start from a certain place A, we can, from this position, make certain move- ments M, characterized by a certain complex of muscular sensations. But, starting from another position B, we make movements M' characterized by the same muscular sensations. Let a, then, be the situation of a certain point of the body, the end of the index finger of the right hand for example, in the initial position A, and b the situation of this same index when, starting from this position A, we have made the motions M. Afterwards, let a' be the situa- tion of this index in the position B, and b' its situation when, starting from the position B, we have made the motions M'.
I am accustomed to say that the points of space a and b are related to each other just as are the points a' and b', and this simply means that the two series of movements M and M' are accompanied by the same muscular sensations. And as I am conscious that, in passing from the position A to the position B, my body has remained capable of the same movements, I know there is a point of space related to the point a' just as any point b is to the point a, so that the two points a and a' are equivalent. This is what is called the homogeneity of space. And at the same time this is why space is relative, since its properties remain the same whether it be referred to the axes A or to the axes B, so that the relativity of space and its homogeneity are one sole and same thing.
Now if I wish to pass to the great space, which no longer serves only for me, but where I may lodge the universe, I get there by an act of imagination. I imagine how a giant would feel who could reach the planets in a few steps; or, if you choose, what I myself should feel in presence of a miniature world where these planets were replaced by little balls , while on one of these little balls moved a liliputian whom I should call myself. But this act of imagination would be impossible for me, had I not previously constructed my restricted space and my extended space for my own use.
For a mobile organism like Homo sapiens, the natural coordinates associated with one's body are not fixed, as they would be for a plant or a coral. Rather, each complicated motion produces a new configuration of the body, and thus (for Poincaré) a new coordinate system. In the Twentieth Century, a major and perplexing field of engineering -- the control of robotic arms -- would use mainly trial-and-error or brute-force computational methods to study such complex forms of motion. Apart from the work of Stanford's Thomas Kane, little was done to address the problem analytically until the dawn of the Twenty-first Century, when it was realised (by Jean-Claude Latombe, also of Stanford, among others) that the Nineteenth Century mathematics of quaternions was well suited to the job.
Poincaré however is here as usual looking at things from a higher standpoint. Rather than discuss the complicated geometrical problem of describing particular body-configurations and how one may be transformed into another, he views them all as trajectories in an abstract space, trajectories which cross at points in "real" physical space. While a sessile organism would find use for an "æther" in the form of the rock to which it attached itself, a mobile organism has no fixed reference-frame except its own constantly-changing body.
THE RELATIVITY OF SPACE, by Henri Poincaré