The Foundations of Science · Henri Poincaré
SPACE AND GEOMETRY
These imaginary beings will therefore like ourselves be led to classify the phenomena they witness and to distinguish among them the 'changes of position' susceptible of correction by a correlative voluntary movement. If they construct a geometry, it will not be, as ours is, the study of the movements of our rigid solids; it will be the study of the changes of position which they will thus have distinguished and which are none other than the 'non-Euclidean displacements'; _it will be non-Euclidean geometry_. Thus beings like ourselves, educated in such a world, would not have the same geometry as ours. THE WORLD OF FOUR DIMENSIONS.--We can represent to ourselves a four-dimensional world just as well as a non-Euclidean. The sense of sight, even with a single eye, together with the muscular sensations relative to the movements of the eyeball, would suffice to teach us space of three dimensions. The images of external objects are painted on the retina, which is a two-dimensional canvas; they are _perspectives_. But, as eye and objects are movable, we see in succession various perspectives of the same body, taken from different points of view. At the same time, we find that the transition from one perspective to another is often accompanied by muscular sensations. If the transition from the perspective _A_ to the perspective _B_, and that from the perspective _A'_ to the perspective _B'_ are accompanied by the same muscular sensations, we liken them one to the other as operations of the same nature.
Studying then the laws according to which these operations combine, we recognize that they form a group, which has the same structure as that of the movements of rigid solids. Now, we have seen that it is from the properties of this group we have derived the notion of geometric space and that of three dimensions. We understand thus how the idea of a space of three dimensions could take birth from the pageant of these perspectives, though each of them is of only two dimensions, since _they follow one another according to certain laws_. Well, just as the perspective of a three-dimensional figure can be made on a plane, we can make that of a four-dimensional figure on a picture of three (or of two) dimensions. To a geometer this is only child's play. We can even take of the same figure several perspectives from several different points of view. We can easily represent to ourselves these perspectives, since they are of only three dimensions. Imagine that the various perspectives of the same object succeed one another, and that the transition from one to the other is accompanied by muscular sensations. We shall of course consider two of these transitions as two operations of the same nature when they are associated with the same muscular sensations. Nothing then prevents us from imagining that these operations combine according to any law we choose, for example, so as to form a group with the same structure as that of the movements of a rigid solid of four dimensions.
Here there is nothing unpicturable, and yet these sensations are precisely those which would be felt by a being possessed of a two-dimensional retina who could move in space of four dimensions. In this sense we may say the fourth dimension is imaginable. CONCLUSIONS.--We see that experience plays an indispensable rôle in the genesis of geometry; but it would be an error thence to conclude that geometry is, even in part, an experimental science. If it were experimental, it would be only approximative and provisional. And what rough approximation! Geometry would be only the study of the movements of solids; but in reality it is not occupied with natural solids, it has for object certain ideal solids, absolutely rigid, which are only a simplified and very remote image of natural solids. The notion of these ideal solids is drawn from all parts of our mind, and experience is only an occasion which induces us to bring it forth from them. The object of geometry is the study of a particular 'group'; but the general group concept pre-exists, at least potentially, in our minds. It is imposed on us, not as form of our sense, but as form of our understanding. Only, from among all the possible groups, that must be chosen which will be, so to speak, the _standard_ to which we shall refer natural phenomena.
Experience guides us in this choice without forcing it upon us; it tells us not which is the truest geometry, but which is the most _convenient_. Notice that I have been able to describe the fantastic worlds above imagined _without ceasing to employ the language of ordinary geometry_. And, in fact, we should not have to change it if transported thither. Beings educated there would doubtless find it more convenient to create a geometry different from ours, and better adapted to their impressions. As for us, in face of the _same_ impressions, it is certain we should find it more convenient not to change our habits.
Text from Project Gutenberg, public domain in the United States.