What does space having 3 dimensions mean?

When we say that space has 3 dimensions, what do we mean?

We have seen the importance of these “internal changes” which are revealed to us by our muscular sensations.

They may serve to characterise the different attitudes of our body.

Let us take arbitrarily as our origin one of these attitudes, A.

When we pass from this initial attitude to another attitude B we experience a series of muscular sensations, and this series S of muscular sensations will define B.

Observe, however, that we shall often look upon two series S and S 0 as defining the same attitude B (since the initial and final attitudes A and B remaining the same, the intermediary attitudes of the corresponding sensations may differ).

How then can we recognise the equivalence of these 2 series?

Because they may serve to compensate for the same external change, or more generally, because, when it is a question of compensation for an external change, one of the series may be replaced by the other.

Among these series we have distinguished those which can alone compensate for an external change, and which we have called “displacements.” As we cannot distinguish two displacements which are very close together, the aggregate of these displacements presents the characteristics of a physical continuum. Experience teaches us that they are the characteristics of a physical continuum of 6 dimensions.

But we do not know as yet how many dimensions space itself possesses, so we must first of all answer another question. What is a point in space?

Every one thinks he knows, but that is an illusion. What we see when we try to represent to ourselves a point in space is a black spot on white paper, a spot of chalk on a black- board, always an object. The question should therefore be understood as follows:—What do I mean when I say the object B is at the point which a moment before was occupied by the object A? Again, what criterion will enable me to recognise it?

I mean that although I have not moved (my muscular sense tells me this), my finger, which just now touched the object A, is now touching the object B. I might have used other criteria—for instance, another finger or the sense of sight—but the first criterion is sufficient.

I know that if it answers in the affirmative all other criteria will give the same answer.

I know it from experiment.

I cannot know it à priori. For the same reason I say that touch cannot be exercised at a distance; that is another way of enunciating the same experimental fact.

If I say, on the contrary, that sight is exercised at a distance, it means that the criterion fur- nished by sight may give an affirmative answer while the others reply in the negative.experiment and geometry.

To sum up. For each attitude of my body my finger determines a point, and it is that and that only which defines a point in space. To each attitude corresponds in this way a point. But it often happens that the same point corresponds to several different attitudes (in this case we say that our finger has not moved, but the rest of our body has). We distinguish, therefore, among changes of attitude those in which the finger does not move.

How are we led to this? It is because we often remark that in these changes the object which is in touch with the finger remains in contact with it. Let us arrange then in the same class all the attitudes which are deduced one from the other by one of the changes that we have thus distinguished.

To all these attitudes of the same class will correspond the same point in space. Then to each class will correspond a point, and to each point a class.

Yet it may be said that what we get from this experiment is not the point, but the class of changes, or, better still, the corresponding class of muscular sensations.

Thus, when we say that space has three dimensions, we merely meanthat the aggregate of these classes appears to us with the characteristics of a physical continuum of three dimensions.

Then if, instead of defining the points in space with the aid of the first finger, I use, for example, another finger, would the results be the same?

That is by no means à priori evident.

But, as we have seen, experiment has shown us that all our criteria are in agreement. This enables us to answer in the affirmative.

If we recur to what we have called displacements, the aggregate of which forms, as we have seen, a group, we shall be brought to distinguish those in which a finger does not move; and by what has preceded, those are the displacements which characterise a point in space, and their aggregate will form a sub-group of our group.

To each sub-group of this kind, then, will correspond a point in space.

We might be tempted to conclude that experiment has taught us the number of dimensions of space; but in reality our experiments have referred not to space, but to our body and its relations with neighbouring objects.

What is more, our experiments are exceeding crude.

In our mind the latent idea of a certain number of groups pre-existed. These are the groups with which Lie’s theory is concerned.

Which shall we choose to form a kind of standard by which to compare natural phenomena?

When this group is chosen, which of the sub-groups shall we take to characterise a point in space?

Experiment has guided us by showing us what choice adapts itself best to the properties of our body; but there its rôle ends.