# Space And Geometry

February 8, 2022

Let us begin with a little paradox.

Beings whose minds were made as ours, and with senses like ours, but without any preliminary education, might receive from a suitably-chosen external world impressions which would lead them to construct a geometry other than that of Euclid, and to localise the phenomena of this external world in a non-Euclidean space, or even in space of four dimensions.

As for us, whose education has been made by our actual world, if we were suddenly transported into this new world, we should have no difficulty in referring phenomena to our Euclidean space.

Perhaps somebody may appear on the scene some day who will devote his life to it, and be able to represent to himself the fourth dimension.

## Geometrical Space and Representative Space

The images we form of external objects are localised in space, and even that they can only be formed on this condition.

This space, which thus serves as a kind of framework ready prepared for our sensations and representations, is identical with the space of the geometers, having all the properties of that space. To all clear-headed men who think in this space and geometry.

way, the preceding statement might well appear extraordinary

but it is as well to see if they are not the victims of some illusion which closer analysis may be able to dissipate.

What are the properties of space?

Space is:

• continuous
• infinite
• of 3 dimensions
• homogeneous—all its points are identical
• isotropic.

Compare this now with the framework of our representations and sensations, which I may call representative space.

## Visual Space.

A purely visual impression is caused by an image formed on the back of the retina.

This image as continuous, but as possessing only two dimensions, which already distinguishes purely visual from what may be called geometrical space.

On the other hand, the image is enclosed within a limited framework; and there is a no less important difference: this pure visual space is not homogeneous.

All the points on the retina, apart from the images which may be formed, do not play the same rôle.

The yellow spot can in no way be regarded as identicalscience and hypothesis

with a point on the edge of the retina. Not only does the same object produce on it much brighter impressions, but in the whole of the limited framework the point which occupies the centre will not appear identical with a point near one of the edges.

Closer analysis no doubt would show us that this continuity of visual space and its two dimensions are but an illusion.

It would make visual space even more different than before from geometrical space, but we may treat this remark as incidental.

However, sight enables us to appreciate distance, and therefore to perceive a third dimension. But every one knows that this perception of the third dimension reduces to a sense of the effort of accommodation which must be made, and to a sense of the convergence of the two eyes, that must take place in order to perceive an object distinctly.

These are muscular sensations quite different from the visual sensations which have given us the concept of the two first dimensions.

The third dimension will therefore not appear to us as playing the same rôle as the two others. What may be called complete visual space is not therefore an isotropic space. It has, it is true, exactly three dimensions; which means that the elements of our visual sensations (those at least which concur in forming the concept of extension) will be completely defined if wespace and geometry.

know three of them; or, in mathematical language, they will be functions of three independent variables. But let us look at the matter a little closer.

The third dimension is revealed to us in 2 ways:

1. by the effort of accommodation
2. the convergence of the eyes.

These two indications are always in harmony;

there is between them a constant relation; or, in mathe- matical language, the two variables which measure these two muscular sensations do not appear to us as indepen- dent. Or, again, to avoid an appeal to mathematical ideas which are already rather too refined, we may go back to the language of the preceding chapter and enunciate the same fact as follows:

If two sensations of convergence A and B are indistinguishable, the two sensations of ac- commodation A 0 and B 0 which accompany them respec- tively will also be indistinguishable.

But that is, so to speak, an experimental fact. Nothing prevents us à pri- ori from assuming the contrary, and if the contrary takes place, if these two muscular sensations both vary inde- pendently, we must take into account one more indepen- dent variable, and complete visual space will appear to us as a physical continuum of four dimensions. And so in this there is also a fact of external experiment. Nothing prevents us from assuming that a being with a mind likescience and hypothesis

ours, with the same sense-organs as ourselves, may be placed in a world in which light would only reach him af- ter being passed through refracting media of complicated form.

The two indications which enable us to appreciate distances would cease to be connected by a constant relation. A being educating his senses in such a world would no doubt attribute four dimensions to complete visual space.

## Tactile and Motor Space

“Tactile space” is more complicated still than visual space, and differs even more widely from geometrical space. It is useless to repeat for the sense of touch my remarks on the sense of sight.

But outside the data of sight and touch there are other sensations which contribute as much and more than they do to the genesis of the concept of space. They are those which everybody knows, which accompany all our movements, and which we usually call muscular sensations.

The corresponding framework constitutes what may be called motor space. Each muscle gives rise to a special sensation which may be increased or diminished so that the aggregate of our muscular sensations will depend upon as many variables as we have muscles.

From this point of view motor space would have as many dimensions as we have muscles. I know that it is said that ifspace and geometry.

the muscular sensations contribute to form the concept of space, it is because we have the sense of the direction of each movement, and that this is an integral part of the sensation.

If this were so, and if a muscular sense could not be aroused unless it were accompanied by this geometrical sense of direction, geometrical space would certainly be a form imposed upon our sensitiveness. But I do not see this at all when I analyse my sensations.

What I do see is that the sensations which correspond to movements in the same direction are connected in my mind by a simple association of ideas.

It is to this association that what we call the sense of direction is reduced. We cannot therefore discover this sense in a single sensation.

This association is extremely complex, for the contraction of the same muscle may correspond, according to the position of the limbs, to very different movements of direction.

Moreover, it is evidently acquired; it is like all associations of ideas, the result of a habit. This habit itself is the result of a very large number of experiments, and no doubt if the education of our senses had taken place in a different medium, where we would have been subjected to different impres- sions, then contrary habits would have been acquired, and our muscular sensations would have been associatedscience and hypothesis

according to other laws.

Characteristics of Representative Space

Thus representative space in its triple form—visual, tactile, and motor. It differs essentially from geometrical space.

It is neither homogeneous nor isotropic; we cannot even say that it is of three dimensions. It is often said that we “project” into geometrical space the objects of our exter- nal perception; that we “localise” them. Now, has that any meaning, and if so what is that meaning? Does it mean that we represent to ourselves external objects in geometrical space?

Our representations are only the reproduction of our sensations; they cannot therefore be arranged in the same framework—that is to say, in rep- resentative space.

It is also just as impossible for us to represent to ourselves external objects in geometri- cal space, as it is impossible for a painter to paint on a flat surface objects with their three dimensions.

Representative space is only an image of geometrical space, an image deformed by a kind of perspective, and we can only represent to ourselves objects by making them obey the laws of this perspective.

Thus we do not represent to ourselves external bodies in geometrical space, but we reason about these bodies as if they were situated in geometrical space.

When it is said, on the other hand, that we “lo-space and geometry.

calise” such an object in such a point of space, what does it mean?

It simply means that we represent to ourselves the movements that must take place to reach that object. And it does not mean that to represent to ourselves these movements they must be projected into space, and that the concept of space must therefore pre-exist. When I say that we represent to ourselves these movements, I only mean that we represent to ourselves the muscular sensations which accompany them, and which have no geometrical character, and which therefore in no way imply the pre-existence of the concept of space.