3. Tactile Space
Thus I know how to recognize the identity of two points, the point occupied by A at the instant α and the point occupied by B at the instant β, but only on one condition, namely, that I have not budged between the instants α and β. That does not suffice for our object. Suppose, therefore, that I have moved in any manner in the interval between these two instants, how shall I know whether the point occupied by A at the instant α is identical with the point occupied by B at the instant β? I suppose that at the instant α, the object A was in contact with my first finger and that in the same way, at the instant β, the object B touches this first finger; but at the same time my muscular sense has told me that in the interval my body has moved. I have considered above two series of muscular sensations S and S´, and I have said it sometimes happens that we are led to consider two such series S and S´ as inverse one of the other, because we have often observed that when these two series succeed one another our primitive impressions are reestablished.
If then my muscular sense tells me that I have moved between the two instants α and β, but so as to feel successively the two series of muscular sensations S and S´ that I consider inverses, I shall still conclude, just as if I had not budged, that the points occupied by A at the instant α and by B at the instant β are identical, if I ascertain that my first finger touches A at the instant α, and B at the instant β.
This solution is not yet completely satisfactory, as one will see. Let us see, in fact, how many dimensions it would make us attribute to space. I wish to compare the two points occupied by A and B at the instants α and β, or (what amounts to the same thing since I suppose that my finger touches A at the instant α and B at the instant β) I wish to compare the two points occupied by my finger at the two instants α and β. The sole means I use for this comparison is the series Σ of muscular sensations which have accompanied the movements of my body between these two instants. The different imaginable series Σ form evidently a physical continuum of which the number of dimensions is very great. Let us agree, as I have done, not to consider as distinct the two series Σ and Σ + S + S´, when S and S´ are inverses one of the other in the sense above given to this word; in spite of this agreement, the aggregate of distinct series Σ will still form a physical continuum and the number of dimensions will be less but still very great.
To each of these series Σ corresponds a point of space; to two series Σ and Σ´ thus correspond two points M and M´. The means we have hitherto used enable us to recognize that M and M´ are not distinct in two cases: (1) if Σ is identical with Σ´; (2) if Σ´ = Σ + S + S´, S and S´ being inverses one of the other. If in all the other cases we should regard M and M´ as distinct, the manifold of points would have as many dimensions as the aggregate of distinct series Σ, that is, much more than three.
For those who already know geometry, the following explanation would be easily comprehensible. Among the imaginable series of muscular sensations, there are those which correspond to series of movements where the finger does not budge. I say that if one does not consider as distinct the series Σ and Σ + σ, where the series σ corresponds to movements where the finger does not budge, the aggregate of series will constitute a continuum of three dimensions, but that if one regards as distinct two series Σ and Σ´ unless Σ´ = Σ + S + S´, S and S´ being inverses, the aggregate of series will constitute a continuum of more than three dimensions.
In fact, let there be in space a surface A, on this surface a line B, on this line a point M. Let C0 be the aggregate of all series Σ. Let C1 be the aggregate of all the series Σ, such that at the end of corresponding movements the finger is found upon the surface A, and C2 or C3 the aggregate of series Σ such that at the end the finger is found on B, or at M. It is clear, first that C1 will constitute a cut which will divide C0, that C2 will be a cut which will divide C1, and C3 a cut which will divide C2. Thence it results, in accordance with our definitions, that if C3 is a continuum of n dimensions, C0 will be a physical continuum of n + 3 dimensions.
Therefore, let Σ and Σ´ = Σ + σ be two series forming part of C3; for both, at the end of the movements, the finger is found at M; thence results that at the beginning and at the end of the series σ the finger is at the same point M. This series σ is therefore one of those which correspond to movements where the finger does not budge. If Σ and Σ + σ are not regarded as distinct, all the series of C3 blend into one; therefore C3 will have 0 dimension, and C0 will have 3, as I wished to prove. If, on the contrary, I do not regard Σ and Σ + σ as blending (unless σ = S + S´, S and S´ being inverses), it is clear that C3 will contain a great number of series of distinct sensations; because, without the finger budging, the body may take a multitude of different attitudes. Then C3 will form a continuum and C0 will have more than three dimensions, and this also I wished to prove.
We who do not yet know geometry can not reason in this way; we can only verify. But then a question arises; how, before knowing geometry, have we been led to distinguish from the others these series σ where the finger does not budge? It is, in fact, only after having made this distinction that we could be led to regard Σ and Σ + σ as identical, and it is on this condition alone, as we have just seen, that we can arrive at space of three dimensions.
We are led to distinguish the series σ, because it often happens that when we have executed the movements which correspond to these series σ of muscular sensations, the tactile sensations which are transmitted to us by the nerve of the finger that we have called the first finger, persist and are not altered by these movements. Experience alone tells us that and it alone could tell us.