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Henri Poincaré, Space and time (1)

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Newton's reflector, the first reflecting telescope

One of the reasons that led me to return to one of the questions I have most often dealt with, is the recent revolution in our ideas on mechanics. Will the principle of relativity, as conceived by Lorentz, not impose on us an entirely new conception of space and time and thereby force us to abandon conclusions that might seem to be acquired? Have we not said that geometry was built by the mind on the occasion of the experiment, without doubt, but without being imposed on us by experience, so that, once constituted, it is immune to any revision, she is out of reach of new assaults of experience? and yet the experiments on which the new mechanics is based do not seem to have shaken it? To see what must be thought of it, I must briefly recall some of the basic ideas that I have sought to highlight in my earlier writings.

I will first dismiss the idea of ​​an alleged sense of space that would make us locate our sensations in a ready-made space, the notion of which would preexist any experience, and which before any experiment would have all the properties of geometer’s space. What is this so-called sense of space? When we want to know if an animal has it, what experience do we have to do? We place in his neighborhood objects that he covets, and we look to see if he can do without trial the movements that allow him to reach them. And how do we see that other men are endowed with this precious sense of space? it is because they, too, are able to contract their muscles in order to reach the objects whose presence is revealed to them by certain sensations. What more is there when we see the sense of space in our own consciousness? Here again, in the presence of various sensations, we know that we could make movements that would allow us to reach the objects we see as the cause of these sensations, and thereby to act on these sensations, to make them disappear or make them more intense; the only difference is that to know it, we do not need to actually do these movements, we just have to represent them. This sense of space that the intelligence would be powerless to express, could only be sopme sort of force that would reside in the depths of the unconscious, and then this force could be known to us only by the acts it causes; and these acts are precisely the movements of which I have just spoken. The sense of space is thus reduced to a constant association between certain sensations and certain movements, or to the representation of these movements. (Is it necessary, in order to avoid an ever-recurring equivocation, despite my repeated explanations, to repeat once more that by this I mean not the representation of these movements in space, but the representation of the sensations which accompany them?)

Why now and to what extent is space relative? It is clear that if all the objects around us and our body itself, as well as our measuring instruments, were transported to another region of space, without their mutual distances varying, we would not perceive them, and it is indeed what happens, since we are trained without suspecting it by the movement of the Earth. If the objects were all enlarged in the same proportion, and so were our measuring instruments, we would not notice any more. Thus not only can we know the absolute position of an object in space, so this word, “absolute position of an object,” has no meaning and it is only necessary to speak of its position relative to other objects; but the word “absolute magnitude of an object,” “absolute distance of two points,” has no meaning; we must speak only of the ratio of two magnitudes, of the ratio of two distances. But there is more: suppose that all the objects are deformed according to a certain law, more complicated than the preceding ones, according to a quite arbitrary law and that at the same time our measuring instruments are deformed according to the same law; neither should we be able to see it, so that space is much more relative than we usually think. We can only notice changes in shape of objects that differ from the simultaneous changes in shape of our measuring instruments.

Our measuring instruments are solid bodies; or they are formed of several solid bodies movable relative to each other and whose relative displacements are indicated by marks placed on these bodies, by indexes moving on graduated scales, and it is precisely by reading these indications that the instrument is used. We therefore know whether our instrument has or has not been moved in the manner of an invariable solid, since in this case the indications in question have not changed. Our instruments also include glasses with which we make sightings, so we can say that the light ray is also one of our instruments.

Will our intuition of space teach us more about it? We have just seen that it is reduced to a constant association between certain sensations and certain movements. This means that the members with whom we make these movements also play, so to speak, the role of measuring instruments. These instruments, which are less precise than those of the scientist, suffice us for everyday life, and it is with them that the child, and the primitive man, has measured space or, to put it better, has built himself the space of which he is satisfied for the needs of his daily life. Our body is our first measuring instrument; Like the others, it is composed of several solid pieces moving relative to each other, and certain sensations warn us of the relative movements of these parts, so that, as in the case of artificial instruments, we know if our body is moved or not like an invariable solid. In short, our instruments, those which the child owes to nature, those which the scientist owes to his genius, have as basic elements the solid body and the luminous ray.

In these conditions does space have geometric properties independent of the instruments used to measure it? It may, we have said, suffer some form of deformation without warning us of it, if our instruments also undergo it. In reality, it is amorphous, it is a flaccid form, without rigidity, which can apply to everything; he has no properties of his own; to do geometry is to study the properties of our instruments, that is, the solid body.

But then, as our instruments are imperfect, geometry should change each time they perfect; builders should be able to put on their flyers: “I provide a space well above that of my competitors, much simpler, much more convenient, much more comfortable.” We know that it is not so; we would be tempted to say that geometry is the study of the properties that the instruments would have if they were perfect. But for that it would be necessary to know what it is that a perfect instrument, and we do not know it since there is none, and that we could define the ideal instrument only by the geometry, which is a vicious circle. And then we will say that geometry is the study of a set of laws little different from those actually obeyed by our instruments, but much simpler, laws that do not actually govern any natural object, but are conceivable for the mind. In this sense, geometry is a convention, a sort of odd side between our love of simplicity and our desire not to deviate too much from what we learn our instruments. This convention defines both space and the perfect instrument.

What we said about space applies also to time; I do not wish to speak here of the time as conceived by the disciples of Bergson, of that duration which, far from being a pure quantity devoid of all quality, is, so to speak, the very quality and of which the various parts, of which elsewhere mutually penetrate each other, are distinguished qualitatively from each other. This duration could not be an instrument for scientists; it has been able to play this role only by undergoing a profound transformation, than by spatialising itself, as Bergson says. It had to be measurable; what can not be measured can not be an object of science. Now, the measurable time is also essentially relative. If all the phenomena slow down, and if it was the same with the functioning of our clocks we would not notice; and that whatever the law of this slowdown, provided that it is the same for all kinds of phenomena and for all clocks. The properties of time are therefore only those of clocks, just as the properties of space are only those of measuring instruments.

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