The Foundations of Science · Henri Poincaré
IV
Why now have all these spaces three dimensions? Go back to the "table of distribution" of which we have spoken. We have on the one side the list of the different possible dangers; designate them by _A1_, _A2_, etc.; and, on the other side, the list of the different remedies which I shall call in the same way _B1_, _B2_, etc. We have then connections between the contact studs or push buttons of the first list and those of the second, so that when, for instance, the announcer of danger _A3_ functions, it will put or may put in action the relay corresponding to the parry _B4_. As I have spoken above of centripetal or centrifugal wires, I fear lest one see in all this, not a simple comparison, but a description of the nervous system. Such is not my thought, and that for several reasons: first I should not permit myself to put forth an opinion on the structure of the nervous system which I do not know, while those who have studied it speak only circumspectly; again because, despite my incompetence, I well know this scheme would be too simplistic; and finally because on my list of parries, some would figure very complex, which might even, in the case of extended space, as we have seen above, consist of many steps followed by a movement of the arm. It is not a question then of physical connection between two real conductors but of psychologic association between two series of sensations.
If _A1_ and _A2_ for instance are both associated with the parry _B1_, and if _A1_ is likewise associated with the parry _B2_, it will generally happen that _A2_ and _B2_ will also themselves be associated. If this fundamental law were not generally true, there would exist only an immense confusion and there would be nothing resembling a conception of space or a geometry. How in fact have we defined a point of space. We have done it in two ways: it is on the one hand the aggregate of announcers _A_ in connection with the same parry _B_; it is on the other hand the aggregate of parries _B_ in connection with the same announcer _A_. If our law was not true, we should say _A1_ and _A2_ correspond to the same point since they are both in connection with _B1_; but we should likewise say they do not correspond to the same point, since _A1_ would be in connection with _B2_ and the same would not be true of _A2_. This would be a contradiction. But, from another side, if the law were rigorously and always true, space would be very different from what it is. We should have categories strongly contrasted between which would be portioned out on the one hand the announcers _A_, on the other hand the parries _B_; these categories would be excessively numerous, but they would be entirely separated one from another. Space would be composed of points very numerous, but discrete; it would be _discontinuous_. There would be no reason for ranging these points in one order rather than another, nor consequently for attributing to space three dimensions.
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