CHAPTER IV

SPACE AND ITS THREE DIMENSIONS

1. _The Group of Displacements_

Let us sum up briefly the results obtained. We proposed to investigate what was meant in saying that space has three dimensions and we have asked first what is a physical continuum and when it may be said to have _n_ dimensions. If we consider different systems of impressions and compare them with one another, we often recognize that two of these systems of impressions are indistinguishable (which is ordinarily expressed in saying that they are too close to one another, and that our senses are too crude, for us to distinguish them) and we ascertain besides that two of these systems can sometimes be discriminated from one another though indistinguishable from a third system. In that case we say the manifold of these systems of impressions forms a physical continuum _C_. And each of these systems is called an _element_ of the continuum _C_.

How many dimensions has this continuum? Take first two elements _A_ and _B_ of _C_, and suppose there exists a series [Sigma] of elements, all belonging to the continuum _C_, of such a sort that _A_ and _B_ are the two extreme terms of this series and that each term of the series is indistinguishable from the preceding. If such a series [Sigma] can be found, we say that _A_ and _B_ are joined to one another; and if any two elements of _C_ are joined to one another, we say that _C_ is all of one piece.

Now take on the continuum _C_ a certain number of elements in a way altogether arbitrary. The aggregate of these elements will be called a _cut_. Among the various series [Sigma] which join _A_ to _B_, we shall distinguish those of which an element is indistinguishable from one of the elements of the cut (we shall say that these are they which _cut_ the cut) and those of which _all_ the elements are distinguishable from all those of the cut. If _all_ the series [Sigma] which join _A_ to _B_ cut the cut, we shall say that _A_ and _B_ are _separated_ by the cut, and that the cut _divides_ _C_. If we can not find on _C_ two elements which are separated by the cut, we shall say that the cut _does not divide_ _C_.

These definitions laid down, if the continuum _C_ can be divided by cuts which do not themselves form a continuum, this continuum _C_ has only one dimension; in the contrary case it has several. If a cut forming a continuum of 1 dimension suffices to divide _C_, _C_ will have 2 dimensions; if a cut forming a continuum of 2 dimensions suffices, _C_ will have 3 dimensions, etc. Thanks to these definitions, we can always recognize how many dimensions any physical continuum has. It only remains to find a physical continuum which is, so to speak, equivalent to space, of such a sort that to every point of space corresponds an element of this continuum, and that to points of space very near one another correspond indistinguishable elements. Space will have then as many dimensions as this continuum.

The intermediation of this physical continuum, capable of representation, is indispensable; because we can not represent space to ourselves, and that for a multitude of reasons. Space is a mathematical continuum, it is infinite, and we can represent to ourselves only physical continua and finite objects. The different elements of space, which we call points, are all alike, and, to apply our definition, it is necessary that we know how to distinguish the elements from one another, at least if they are not too close. Finally absolute space is nonsense, and it is necessary for us to begin by referring space to a system of axes invariably bound to our body (which we must always suppose put back in the initial attitude).

Then I have sought to form with our visual sensations a physical continuum equivalent to space; that certainly is easy and this example is particularly appropriate for the discussion of the number of dimensions; this discussion has enabled us to see in what measure it is allowable to say that 'visual space' has three dimensions. Only this solution is incomplete and artificial. I have explained why, and it is not on visual space but on motor space that it is necessary to bring our efforts to bear. I have then recalled what is the origin of the distinction we make between changes of position and changes of state. Among the changes which occur in our impressions, we distinguish, first the _internal_ changes, voluntary and accompanied by muscular sensations, and the _external_ changes, having opposite characteristics. We ascertain that it may happen that an external change may be _corrected_ by an internal change which reestablishes the primitive sensations. The external changes, capable of being corrected by an internal change are called _changes of position_, those not capable of it are called _changes of state_. The internal changes capable of correcting an external change are called _displacements of the whole body_; the others are called _changes of attitude_.

Now let [alpha] and [beta] be two external changes, [alpha]' and [beta]' two internal changes. Suppose that a may be corrected either by [alpha]' or by [beta]', and that [alpha]' can correct either [alpha] or [beta]; experience tells us then that [beta]' can likewise correct [beta]. In this case we say that [alpha] and [beta] correspond to the _same_ displacement and also that [alpha]' and [beta]' correspond to the _same_ displacement. That postulated, we can imagine a physical continuum which we shall call _the continuum or group of displacements_ and which we shall define in the following manner. The elements of this continuum shall be the internal changes capable of correcting an external change. Two of these internal changes [alpha]' and [beta]' shall be regarded as indistinguishable: (1) if they are so naturally, that is, if they are too close to one another; (2) if [alpha]' is capable of correcting the same external change as a third internal change naturally indistinguishable from [beta]'. In this second case, they will be, so to speak, indistinguishable by convention, I mean by agreeing to disregard circumstances which might distinguish them.

Our continuum is now entirely defined, since we know its elements and have fixed under what conditions they may be regarded as indistinguishable. We thus have all that is necessary to apply our definition and determine how many dimensions this continuum has. We shall recognize that it has _six_. The continuum of displacements is, therefore, not equivalent to space, since the number of dimensions is not the same; it is only related to space. Now how do we know that this continuum of displacements has six dimensions? We know it _by experience_.

It would be easy to describe the experiments by which we could arrive at this result. It would be seen that in this continuum cuts can be made which divide it and which are continua; that these cuts themselves can be divided by other cuts of the second order which yet are continua, and that this would stop only after cuts of the sixth order which would no longer be continua. From our definitions that would mean that the group of displacements has six dimensions.

That would be easy, I have said, but that would be rather long; and would it not be a little superficial? This group of displacements, we have seen, is related to space, and space could be deduced from it, but it is not equivalent to space, since it has not the same number of dimensions; and when we shall have shown how the notion of this continuum can be formed and how that of space may be deduced from it, it might always be asked why space of three dimensions is much more familiar to us than this continuum of six dimensions, and consequently doubted whether it was by this detour that the notion of space was formed in the human mind.

2. _Identity of Two Points_

What is a point? How do we know whether two points of space are identical or different? Or, in other words, when I say: The object _A_ occupied at the instant [alpha] the point which the object _B_ occupies at the instant [beta], what does that mean?

Such is the problem we set ourselves in the preceding chapter, §4. As I have explained it, it is not a question of comparing the positions of the objects _A_ and _B_ in absolute space; the question then would manifestly have no meaning. It is a question of comparing the positions of these two objects with regard to axes invariably bound to my body, supposing always this body replaced in the same attitude.

I suppose that between the instants [alpha] and [beta] I have moved neither my body nor my eye, as I know from my muscular sense. Nor have I moved either my head, my arm or my hand. I ascertain that at the instant [alpha] impressions that I attributed to the object _A_ were transmitted to me, some by one of the fibers of my optic nerve, the others by one of the sensitive tactile nerves of my finger; I ascertain that at the instant [beta] other impressions which I attribute to the object _B_ are transmitted to me, some by this same fiber of the optic nerve, the others by this same tactile nerve.

Here I must pause for an explanation; how am I told that this impression which I attribute to _A_, and that which I attribute to _B_, impressions which are qualitatively different, are transmitted to me by the same nerve? Must we suppose, to take for example the visual sensations, that _A_ produces two simultaneous sensations, a sensation purely luminous _a_ and a colored sensation _a'_, that _B_ produces in the same way simultaneously a luminous sensation _b_ and a colored sensation _b'_, that if these different sensations are transmitted to me by the same retinal fiber, _a_ is identical with _b_, but that in general the colored sensations _a'_ and _b'_ produced by different bodies are different? In that case it would be the identity of the sensation _a_ which accompanies _a'_ with the sensation _b_ which accompanies _b'_, which would tell that all these sensations are transmitted to me by the same fiber.

However it may be with this hypothesis and although I am led to prefer to it others considerably more complicated, it is certain that we are told in some way that there is something in common between these sensations _a_ + _a'_ and _b_ +_b'_, without which we should have no means of recognizing that the object _B_ has taken the place of the object _A_.

Therefore I do not further insist and I recall the hypothesis I have just made: I suppose that I have ascertained that the impressions which I attribute to _B_ are transmitted to me at the instant [beta] by the same fibers, optic as well as tactile, which, at the instant [alpha], had transmitted to me the impressions that I attributed to _A_. If it is so, we shall not hesitate to declare that the point occupied by _B_ at the instant [beta] is identical with the point occupied by _A_ at the instant [alpha].

I have just enunciated two conditions for these points being identical; one is relative to sight, the other to touch. Let us consider them separately. The first is necessary, but is not sufficient. The second is at once necessary and sufficient. A person knowing geometry could easily explain this in the following manner: Let _O_ be the point of the retina where is formed at the instant [alpha] the image of the body _A_; let _M_ be the point of space occupied at the instant [alpha] by this body _A_; let _M'_ be the point of space occupied at the instant [beta] by the body _B_. For this body _B_ to form its image in _O_, it is not necessary that the points _M_ and _M'_ coincide; since vision acts at a distance, it suffices for the three points _O_ _M_ _M'_ to be in a straight line. This condition that the two objects form their image on _O_ is therefore necessary, but not sufficient for the points _M_ and _M'_ to coincide. Let now _P_ be the point occupied by my finger and where it remains, since it does not budge. As touch does not act at a distance, if the body _A_ touches my finger at the instant [alpha], it is because _M_ and _P_ coincide; if _B_ touches my finger at the instant [beta], it is because _M'_ and _P_ coincide. Therefore _M_ and _M'_ coincide. Thus this condition that if _A_ touches my finger at the instant [alpha], _B_ touches it at the instant [beta], is at once necessary and sufficient for _M_ and _M'_ to coincide.

But we who, as yet, do not know geometry can not reason thus; all that we can do is to ascertain experimentally that the first condition relative to sight may be fulfilled without the second, which is relative to touch, but that the second can not be fulfilled without the first.

Suppose experience had taught us the contrary, as might well be; this hypothesis contains nothing absurd. Suppose, therefore, that we had ascertained experimentally that the condition relative to touch may be fulfilled without that of sight being fulfilled and that, on the contrary, that of sight can not be fulfilled without that of touch being also. It is clear that if this were so we should conclude that it is touch which may be exercised at a distance, and that sight does not operate at a distance.

But this is not all; up to this time I have supposed that to determine the place of an object I have made use only of my eye and a single finger; but I could just as well have employed other means, for example, all my other fingers.

I suppose that my first finger receives at the instant [alpha] a tactile impression which I attribute to the object _A_. I make a series of movements, corresponding to a series _S_ of muscular sensations. After these movements, at the instant [alpha]', my _second_ finger receives a tactile impression that I attribute likewise to _A_. Afterward, at the instant [beta], without my having budged, as my muscular sense tells me, this same second finger transmits to me anew a tactile impression which I attribute this time to the object _B_; I then make a series of movements, corresponding to a series _S'_ of muscular sensations. I know that this series _S'_ is the inverse of the series _S_ and corresponds to contrary movements. I know this because many previous experiences have shown me that if I made successively the two series of movements corresponding to _S_ and to _S'_, the primitive impressions would be reestablished, in other words, that the two series mutually compensate. That settled, should I expect that at the instant [beta]', when the second series of movements is ended, my _first finger_ would feel a tactile impression attributable to the object _B_?

To answer this question, those already knowing geometry would reason as follows: There are chances that the object _A_ has not budged, between the instants [alpha] and [alpha]', nor the object _B_ between the instants [beta] and [beta]'; assume this. At the instant [alpha], the object _A_ occupied a certain point _M_ of space. Now at this instant it touched my first finger, and _as touch does not operate at a distance_, my first finger was likewise at the point _M_. I afterward made the series _S_ of movements and at the end of this series, at the instant [alpha]', I ascertained that the object _A_ touched my second finger. I thence conclude that this second finger was then at _M_, that is, that the movements _S_ had the result of bringing the second finger to the place of the first. At the instant [beta] the object _B_ has come in contact with my second finger: as I have not budged, this second finger has remained at _M_; therefore the object _B_ has come to _M_; by hypothesis it does not budge up to the instant [beta]'. But between the instants [beta] and [beta]' I have made the movements _S'_; as these movements are the inverse of the movements _S_, they must have for effect bringing the first finger in the place of the second. At the instant [beta]' this first finger will, therefore, be at _M_; and as the object _B_ is likewise at _M_, this object _B_ will touch my first finger. To the question put, the answer should therefore be yes.

We who do not yet know geometry can not reason thus; but we ascertain that this anticipation is ordinarily realized; and we can always explain the exceptions by saying that the object _A_ has moved between the instants [alpha] and [alpha]', or the object _B_ between the instants [beta] and [beta]'.

But could not experience have given a contrary result? Would this contrary result have been absurd in itself? Evidently not. What should we have done then if experience had given this contrary result? Would all geometry thus have become impossible? Not the least in the world. We should have contented ourselves with concluding _that touch can operate at a distance_.

When I say, touch does not operate at a distance, but sight operates at a distance, this assertion has only one meaning, which is as follows: To recognize whether _B_ occupies at the instant [beta] the point occupied by _A_ at the instant [alpha], I can use a multitude of different criteria. In one my eye intervenes, in another my first finger, in another my second finger, etc. Well, it is sufficient for the criterion relative to one of my fingers to be satisfied in order that all the others should be satisfied, but it is not sufficient that the criterion relative to the eye should be. This is the sense of my assertion. I content myself with affirming an experimental fact which is ordinarily verified.

At the end of the preceding chapter we analyzed visual space; we saw that to engender this space it is necessary to bring in the retinal sensations, the sensation of convergence and the sensation of accommodation; that if these last two were not always in accord, visual space would have four dimensions in place of three; we also saw that if we brought in only the retinal sensations, we should obtain 'simple visual space,' of only two dimensions. On the other hand, consider tactile space, limiting ourselves to the sensations of a single finger, that is in sum to the assemblage of positions this finger can occupy. This tactile space that we shall analyze in the following section and which consequently I ask permission not to consider further for the moment, this tactile space, I say, has three dimensions. Why has space properly so called as many dimensions as tactile space and more than simple visual space? It is because touch does not operate at a distance, while vision does operate at a distance. These two assertions have the same meaning and we have just seen what this is.

Now I return to a point over which I passed rapidly in order not to interrupt the discussion. How do we know that the impressions made on our retina by _A_ at the instant [alpha] and _B_ at the instant [beta] are transmitted by the same retinal fiber, although these impressions are qualitatively different? I have suggested a simple hypothesis, while adding that other hypotheses, decidedly more complex, would seem to me more probably true. Here then are these hypotheses, of which I have already said a word. How do we know that the impressions produced by the red object A at the instant [alpha], and by the blue object _B_ at the instant [beta], if these two objects have been imaged on the same point of the retina, have something in common? The simple hypothesis above made may be rejected and we may suppose that these two impressions, qualitatively different, are transmitted by two different though contiguous nervous fibers. What means have I then of knowing that these fibers are contiguous? It is probable that we should have none, if the eye were immovable. It is the movements of the eye which have told us that there is the same relation between the sensation of blue at the point _A_ and the sensation of blue at the point _B_ of the retina as between the sensation of red at the point _A_ and the sensation of red at the point _B_. They have shown us, in fact, that the same movements, corresponding to the same muscular sensations, carry us from the first to the second, or from the third to the fourth. I do not emphasize these considerations, which belong, as one sees, to the question of local signs raised by Lotze.

3. _Tactile Space_

Thus I know how to recognize the identity of two points, the point occupied by _A_ at the instant [alpha] and the point occupied by _B_ at the instant [beta], but only _on one condition_, namely, that I have not budged between the instants [alpha] and [beta]. 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 [alpha] is identical with the point occupied by _B_ at the instant [beta]? I suppose that at the instant [alpha], the object _A_ was in contact with my first finger and that in the same way, at the instant [beta], 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 [alpha] and [beta], 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 [alpha] and by _B_ at the instant [beta] are identical, if I ascertain that my first finger touches _A_ at the instant [alpha], and _B_ at the instant [beta].

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 [alpha] and [beta], or (what amounts to the same thing since I suppose that my finger touches _A_ at the instant [alpha] and _B_ at the instant [beta]) I wish to compare the two points occupied by my finger at the two instants [alpha] and [beta]. The sole means I use for this comparison is the series [Sigma] of muscular sensations which have accompanied the movements of my body between these two instants. The different imaginable series [Sigma] 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 [Sigma] and [Sigma] + _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 [Sigma] will still form a physical continuum and the number of dimensions will be less but still very great.

To each of these series [Sigma] corresponds a point of space; to two series [Sigma] and [Sigma]' 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 [Sigma] is identical with [Sigma]'; (2) if [Sigma]' = [Sigma] + _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 [Sigma], 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 [Sigma] and [Sigma] + [sigma], where the series [sigma] 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 [Sigma] and [Sigma]' unless [Sigma]' = [Sigma] + _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 C_{0} be the aggregate of all series [Sigma]. Let C_{1} be the aggregate of all the series [Sigma], such that at the end of corresponding movements the finger is found upon the surface _A_, and C_{2} or C_{3} the aggregate of series [Sigma] such that at the end the finger is found on _B_, or at _M_. It is clear, first that C_{1} will constitute a cut which will divide C_{0}, that C_{2} will be a cut which will divide C_{1}, and C_{3} a cut which will divide C_2. Thence it results, in accordance with our definitions, that if C_{3} is a continuum of _n_ dimensions, C_{0} will be a physical continuum of _n_ + 3 dimensions.

Therefore, let [Sigma] and [Sigma]' = [Sigma] + [sigma] be two series forming part of C_{3}; 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 [sigma] the finger is at the same point _M_. This series [sigma] is therefore one of those which correspond to movements where the finger does not budge. If [Sigma] and [Sigma] + [sigma] are not regarded as distinct, all the series of C_{3} blend into one; therefore C_{3} will have 0 dimension, and C_{0} will have 3, as I wished to prove. If, on the contrary, I do not regard [Sigma] and [Sigma] + [sigma] as blending (unless [sigma] = _S_ + _S'_, _S_ and _S'_ being inverses), it is clear that C_{3} 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 C_{3} will form a continuum and C_{0} 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 [sigma] where the finger does not budge? It is, in fact, only after having made this distinction that we could be led to regard [Sigma] and [Sigma] + [sigma] 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 [sigma], because it often happens that when we have executed the movements which correspond to these series [sigma] 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.

If we have distinguished the series of muscular sensations _S_ + _S'_ formed by the union of two inverse series, it is because they preserve the totality of our impressions; if now we distinguish the series [sigma], it is because they preserve _certain_ of our impressions. (When I say that a series of muscular sensations _S_ 'preserves' one of our impressions _A_, I mean that we ascertain that if we feel the impression _A_, then the muscular sensations _S_, we _still_ feel the impression _A_ _after_ these sensations _S_.)

I have said above it often happens that the series [sigma] do not alter the tactile impressions felt by our first finger; I said _often_, I did not say _always_. This it is that we express in our ordinary language by saying that the tactile impressions would not be altered if the finger has not moved, _on the condition_ that _neither has_ the object _A_, which was in contact with this finger, moved. Before knowing geometry, we could not give this explanation; all we could do is to ascertain that the impression often persists, but not always.

But that the impression often continues is enough to make the series [sigma] appear remarkable to us, to lead us to put in the same class the series [Sigma] and [Sigma] + [sigma], and hence not regard them as distinct. Under these conditions we have seen that they will engender a physical continuum of three dimensions.

Behold then a space of three dimensions engendered by my first finger. Each of my fingers will create one like it. It remains to consider how we are led to regard them as identical with visual space, as identical with geometric space.

But one reflection before going further; according to the foregoing, we know the points of space, or more generally the final situation of our body, only by the series of muscular sensations revealing to us the movements which have carried us from a certain initial situation to this final situation. But it is clear that this final situation will depend, on the one hand, upon these movements and, _on the other hand, upon the initial situation_ from which we set out. Now these movements are revealed to us by our muscular sensations; but nothing tells us the initial situation; nothing can distinguish it for us from all the other possible situations. This puts well in evidence the essential relativity of space.

4. _Identity of the Different Spaces_

We are therefore led to compare the two continua _C_ and _C'_ engendered, for instance, one by my first finger _D_, the other by my second finger _D'_. These two physical continua both have three dimensions. To each element of the continuum _C_, or, if you prefer, to each point of the first tactile space, corresponds a series of muscular sensations [Sigma], which carry me from a certain initial situation to a certain final situation.[8] Moreover, the same point of this first space will correspond to [Sigma] and [Sigma] + [sigma], if [sigma] is a series of which we know that it does not make the finger _D_ move.

[8] In place of saying that we refer space to axes rigidly bound to our body, perhaps it would be better to say, in conformity to what precedes, that we refer it to axes rigidly bound to the initial situation of our body.

Similarly to each element of the continuum _C'_, or to each point of the second tactile space, corresponds a series of sensations [Sigma]', and the same point will correspond to [Sigma]' and to [Sigma]' + [sigma]', if [sigma]' is a series which does not make the finger _D'_ move.

What makes us distinguish the various series designated [sigma] from those called [sigma]' is that the first do not alter the tactile impressions felt by the finger _D_ and the second preserve those the finger _D'_ feels.

Now see what we ascertain: in the beginning my finger _D'_ feels a sensation _A'_; I make movements which produce muscular sensations _S_; my finger _D_ feels the impression _A_; I make movements which produce a series of sensations [sigma]; my finger _D_ continues to feel the impression _A_, since this is the characteristic property of the series [sigma]; I then make movements which produce the series _S'_ of muscular sensations, _inverse_ to _S_ in the sense above given to this word. I ascertain then that my finger _D'_ feels anew the impression _A'_. (It is of course understood that _S_ has been suitably chosen.)

This means that the series _S_ + [sigma] + _S'_, preserving the tactile impressions of the finger _D'_, is one of the series I have called [sigma]'. Inversely, if one takes any series [sigma]', _S'_ + [sigma]' + _S_ will be one of the series that we call [sigma]'.

Thus if _S_ is suitably chosen, _S_ + [sigma] + _S'_ will be a series [sigma]', and by making [sigma] vary in all possible ways, we shall obtain all the possible series [sigma]'.

Not yet knowing geometry, we limit ourselves to verifying all that, but here is how those who know geometry would explain the fact. In the beginning my finger _D'_ is at the point _M_, in contact with the object _a_, which makes it feel the impression _A'_. I make the movements corresponding to the series _S_; I have said that this series should be suitably chosen, I should so make this choice that these movements carry the finger _D_ to the point originally occupied by the finger _D'_, that is, to the point _M_; this finger _D_ will thus be in contact with the object _a_, which will make it feel the impression _A_.

I then make the movements corresponding to the series [sigma]; in these movements, by hypothesis, the position of the finger _D_ does not change, this finger therefore remains in contact with the object a and continues to feel the impression _A_. Finally I make the movements corresponding to the series _S'_. As _S'_ is inverse to _S_, these movements carry the finger _D'_ to the point previously occupied by the finger _D_, that is, to the point _M_. If, as may be supposed, the object _a_ has not budged, this finger _D'_ will be in contact with this object and will feel anew the impression _A'_.... _Q.E.D._

Let us see the consequences. I consider a series of muscular sensations [Sigma]. To this series will correspond a point _M_ of the first tactile space. Now take again the two series _S_ and _S'_, inverses of one another, of which we have just spoken. To the series _S_ + [Sigma] + _S'_ will correspond a point _N_ of the second tactile space, since to any series of muscular sensations corresponds, as we have said, a point, whether in the first space or in the second.

I am going to consider the two points _N_ and _M_, thus defined, as corresponding. What authorizes me so to do? For this correspondence to be admissible, it is necessary that if two points _M_ and _M'_, corresponding in the first space to two series [Sigma] and [Sigma]', are identical, so also are the two corresponding points of the second space _N_ and _N'_, that is, the two points which correspond to the two series _S_ + [Sigma] + _S'_ and _S_ + [Sigma]' + _S'_. Now we shall see that this condition is fulfilled.

First a remark. As _S_ and _S'_ are inverses of one another, we shall have _S_ + _S'_ = 0, and consequently _S_ + _S'_ + [Sigma] = [Sigma] + _S_ + _S'_ = [Sigma], or again [Sigma] + _S_ + _S'_ + [Sigma]' = [Sigma] + [Sigma]'; but it does not follow that we have _S_ + [Sigma] + _S'_ = [Sigma]; because, though we have used the addition sign to represent the succession of our sensations, it is clear that the order of this succession is not indifferent: we can not, therefore, as in ordinary addition, invert the order of the terms; to use abridged language, our operations are associative, but not commutative.

That fixed, in order that [Sigma] and [Sigma]' should correspond to the same point _M_ = _M'_ of the first space, it is necessary and sufficient for us to have [Sigma]' = [Sigma] + [sigma]. We shall then have: _S_ + [Sigma]' + _S'_ = _S_ + [Sigma] + [sigma] + _S'_ = _S_ + [Sigma] + _S'_ + _S_ + [sigma] + _S'_.

But we have just ascertained that _S_ + [sigma] + _S'_ was one of the series [sigma]'. We shall therefore have: _S_ + [Sigma]' + _S'_ = _S_ + [Sigma] + _S'_ + [sigma]', which means that the series _S_ + [Sigma]' + _S'_ and _S_ + [Sigma] + _S'_ correspond to the same point _N_ = _N'_ of the second space. Q.E.D.

Our two spaces therefore correspond point for point; they can be 'transformed' one into the other; they are isomorphic. How are we led to conclude thence that they are identical?

Consider the two series [sigma] and _S_ + [sigma] + _S'_ = [sigma]'. I have said that often, but not always, the series [sigma] preserves the tactile impression _A_ felt by the finger _D_; and similarly it often happens, but not always, that the series [sigma]' preserves the tactile impression _A'_ felt by the finger _D'_. Now I ascertain that it happens _very often_ (that is, much more often than what I have just called 'often') that when the series [sigma] has preserved the impression _A_ of the finger _D_, the series [sigma]' preserves at the same time the impression _A'_ of the finger _D'_; and, inversely, that if the first impression is altered, the second is likewise. That happens _very often_, but not always.

We interpret this experimental fact by saying that the unknown object _a_ which gives the impression _A_ to the finger _D_ is identical with the unknown object _a'_ which gives the impression _A'_ to the finger _D'_. And in fact when the first object moves, which the disappearance of the impression _A_ tells us, the second likewise moves, since the impression _A'_ disappears likewise. When the first object remains motionless, the second remains motionless. If these two objects are identical, as the first is at the point _M_ of the first space and the second at the point _N_ of the second space, these two points are identical. This is how we are led to regard these two spaces as identical; or better, this is what we mean when we say that they are identical.

What we have just said of the identity of the two tactile spaces makes unnecessary our discussing the question of the identity of tactile space and visual space, which could be treated in the same way.

5. _Space and Empiricism_

It seems that I am about to be led to conclusions in conformity with empiristic ideas. I have, in fact, sought to put in evidence the rôle of experience and to analyze the experimental facts which intervene in the genesis of space of three dimensions. But whatever may be the importance of these facts, there is one thing we must not forget and to which besides I have more than once called attention. These experimental facts are often verified but not always. That evidently does not mean that space has often three dimensions, but not always.

I know well that it is easy to save oneself and that, if the facts do not verify, it will be easily explained by saying that the exterior objects have moved. If experience succeeds, we say that it teaches us about space; if it does not succeed, we hie to exterior objects which we accuse of having moved; in other words, if it does not succeed, it is given a fillip.

These fillips are legitimate; I do not refuse to admit them; but they suffice to tell us that the properties of space are not experimental truths, properly so called. If we had wished to verify other laws, we could have succeeded also, by giving other analogous fillips. Should we not always have been able to justify these fillips by the same reasons? One could at most have said to us: 'Your fillips are doubtless legitimate, but you abuse them; why move the exterior objects so often?'

To sum up, experience does not prove to us that space has three dimensions; it only proves to us that it is convenient to attribute three to it, because thus the number of fillips is reduced to a minimum.

I will add that experience brings us into contact only with representative space, which is a physical continuum, never with geometric space, which is a mathematical continuum. At the very most it would appear to tell us that it is convenient to give to geometric space three dimensions, so that it may have as many as representative space.

The empiric question may be put under another form. Is it impossible to conceive physical phenomena, the mechanical phenomena, for example, otherwise than in space of three dimensions? We should thus have an objective experimental proof, so to speak, independent of our physiology, of our modes of representation.

But it is not so; I shall not here discuss the question completely, I shall confine myself to recalling the striking example given us by the mechanics of Hertz. You know that the great physicist did not believe in the existence of forces, properly so called; he supposed that visible material points are subjected to certain invisible bonds which join them to other invisible points and that it is the effect of these invisible bonds that we attribute to forces.

But that is only a part of his ideas. Suppose a system formed of n material points, visible or not; that will give in all 3_n_ coordinates; let us regard them as the coordinates of a _single_ point in space of 3_n_ dimensions. This single point would be constrained to remain upon a surface (of any number of dimensions < 3_n_) in virtue of the bonds of which we have just spoken; to go on this surface from one point to another, it would always take the shortest way; this would be the single principle which would sum up all mechanics.

Whatever should be thought of this hypothesis, whether we be allured by its simplicity, or repelled by its artificial character, the simple fact that Hertz was able to conceive it, and to regard it as more convenient than our habitual hypotheses, suffices to prove that our ordinary ideas, and, in particular, the three dimensions of space, are in no wise imposed upon mechanics with an invincible force.

6. _Mind and Space_

Experience, therefore, has played only a single rôle, it has served as occasion. But this rôle was none the less very important; and I have thought it necessary to give it prominence. This rôle would have been useless if there existed an _a priori_ form imposing itself upon our sensitivity, and which was space of three dimensions.

Does this form exist, or, if you choose, can we represent to ourselves space of more than three dimensions? And first what does this question mean? In the true sense of the word, it is clear that we can not represent to ourselves space of four, nor space of three, dimensions; we can not first represent them to ourselves empty, and no more can we represent to ourselves an object either in space of four, or in space of three, dimensions: (1) Because these spaces are both infinite and we can not represent to ourselves a figure _in_ space, that is, the part _in_ the whole, without representing the whole, and that is impossible, because it is infinite; (2) because these spaces are both mathematical continua, and we can represent to ourselves only the physical continuum; (3) because these spaces are both homogeneous, and the frames in which we enclose our sensations, being limited, can not be homogeneous.

Thus the question put can only be understood in one way; is it possible to imagine that, the results of the experiences related above having been different, we might have been led to attribute to space more than three dimensions; to imagine, for instance, that the sensation of accommodation might not be constantly in accord with the sensation of convergence of the eyes; or indeed that the experiences of which we have spoken in § 2, and of which we express the result by saying 'that touch does not operate at a distance,' might have led us to an inverse conclusion.

And then yes evidently that is possible; from the moment one imagines an experience, one imagines just thereby the two contrary results it may give. That is possible, but that is difficult, because we have to overcome a multitude of associations of ideas, which are the fruit of a long personal experience and of the still longer experience of the race. Is it these associations (or at least those of them that we have inherited from our ancestors), which constitute this _a priori_ form of which it is said that we have pure intuition? Then I do not see why one should declare it refractory to analysis and should deny me the right of investigating its origin.

When it is said that our sensations are 'extended' only one thing can be meant, that is that they are always associated with the idea of certain muscular sensations, corresponding to the movements which enable us to reach the object which causes them, which enable us, in other words, to defend ourselves against it. And it is just because this association is useful for the defense of the organism, that it is so old in the history of the species and that it seems to us indestructible. Nevertheless, it is only an association and we can conceive that it may be broken; so that we may not say that sensation can not enter consciousness without entering in space, but that in fact it does not enter consciousness without entering in space, which means, without being entangled in this association.

No more can I understand one's saying that the idea of time is logically subsequent to space, since we can represent it to ourselves only under the form of a straight line; as well say that time is logically subsequent to the cultivation of the prairies, since it is usually represented armed with a scythe. That one can not represent to himself simultaneously the different parts of time, goes without saying, since the essential character of these parts is precisely not to be simultaneous. That does not mean that we have not the intuition of time. So far as that goes, no more should we have that of space, because neither can we represent it, in the proper sense of the word, for the reasons I have mentioned. What we represent to ourselves under the name of straight is a crude image which as ill resembles the geometric straight as it does time itself.

Why has it been said that every attempt to give a fourth dimension to space always carries this one back to one of the other three? It is easy to understand. Consider our muscular sensations and the 'series' they may form. In consequence of numerous experiences, the ideas of these series are associated together in a very complex woof, our series are _classed_. Allow me, for convenience of language, to express my thought in a way altogether crude and even inexact by saying that our series of muscular sensations are classed in three classes corresponding to the three dimensions of space. Of course this classification is much more complicated than that, but that will suffice to make my reasoning understood. If I wish to imagine a fourth dimension, I shall suppose another series of muscular sensations, making part of a fourth class. But as _all_ my muscular sensations have already been classed in one of the three pre-existent classes, I can only represent to myself a series belonging to one of these three classes, so that my fourth dimension is carried back to one of the other three.

What does that prove? This: that it would have been necessary first to destroy the old classification and replace it by a new one in which the series of muscular sensations should have been distributed into four classes. The difficulty would have disappeared.

It is presented sometimes under a more striking form. Suppose I am enclosed in a chamber between the six impassable boundaries formed by the four walls, the floor and the ceiling; it will be impossible for me to get out and to imagine my getting out. Pardon, can you not imagine that the door opens, or that two of these walls separate? But of course, you answer, one must suppose that these walls remain immovable. Yes, but it is evident that I have the right to move; and then the walls that we suppose absolutely at rest will be in motion with regard to me. Yes, but such a relative motion can not be arbitrary; when objects are at rest, their relative motion with regard to any axes is that of a rigid solid; now, the apparent motions that you imagine are not in conformity with the laws of motion of a rigid solid. Yes, but it is experience which has taught us the laws of motion of a rigid solid; nothing would prevent our _imagining_ them different. To sum up, for me to imagine that I get out of my prison, I have only to imagine that the walls seem to open, when I move.

I believe, therefore, that if by space is understood a mathematical continuum of three dimensions, were it otherwise amorphous, it is the mind which constructs it, but it does not construct it out of nothing; it needs materials and models. These materials, like these models, preexist within it. But there is not a single model which is imposed upon it; it has _choice_; it may choose, for instance, between space of four and space of three dimensions. What then is the rôle of experience? It gives the indications following which the choice is made.

Another thing: whence does space get its quantitative character? It comes from the rôle which the series of muscular sensations play in its genesis. These are series which may _repeat themselves_, and it is from their repetition that number comes; it is because they can repeat themselves indefinitely that space is infinite. And finally we have seen, at the end of section 3, that it is also because of this that space is relative. So it is repetition which has given to space its essential characteristics; now, repetition supposes time; this is enough to tell that time is logically anterior to space.

7. _Rôle of the Semicircular Canals_

I have not hitherto spoken of the rôle of certain organs to which the physiologists attribute with reason a capital importance, I mean the semicircular canals. Numerous experiments have sufficiently shown that these canals are necessary to our sense of orientation; but the physiologists are not entirely in accord; two opposing theories have been proposed, that of Mach-Delage and that of M. de Cyon.

M. de Cyon is a physiologist who has made his name illustrious by important discoveries on the innervation of the heart; I can not, however, agree with his ideas on the question before us. Not being a physiologist, I hesitate to criticize the experiments he has directed against the adverse theory of Mach-Delage; it seems to me, however, that they are not convincing, because in many of them the _total_ pressure was made to vary in one of the canals, while, physiologically, what varies is the _difference_ between the pressures on the two extremities of the canal; in others the organs were subjected to profound lesions, which must alter their functions.

Besides, this is not important; the experiments, if they were irreproachable, might be convincing against the old theory. They would not be convincing _for_ the new theory. In fact, if I have rightly understood the theory, my explaining it will be enough for one to understand that it is impossible to conceive of an experiment confirming it.

The three pairs of canals would have as sole function to tell us that space has three dimensions. Japanese mice have only two pairs of canals; they believe, it would seem, that space has only two dimensions, and they manifest this opinion in the strangest way; they put themselves in a circle, and, so ordered, they spin rapidly around. The lampreys, having only one pair of canals, believe that space has only one dimension, but their manifestations are less turbulent.

It is evident that such a theory is inadmissible. The sense-organs are designed to tell us of _changes_ which happen in the exterior world. We could not understand why the Creator should have given us organs destined to cry without cease: Remember that space has three dimensions, since the number of these three dimensions is not subject to change.

We must, therefore, come back to the theory of Mach-Delage. What the nerves of the canals can tell us is the difference of pressure on the two extremities of the same canal, and thereby: (1) the direction of the vertical with regard to three axes rigidly bound to the head; (2) the three components of the acceleration of translation of the center of gravity of the head; (3) the centrifugal forces developed by the rotation of the head; (4) the acceleration of the motion of rotation of the head.

It follows from the experiments of M. Delage that it is this last indication which is much the most important; doubtless because the nerves are less sensible to the difference of pressure itself than to the brusque variations of this difference. The first three indications may thus be neglected.

Knowing the acceleration of the motion of rotation of the head at each instant, we deduce from it, by an unconscious integration, the final orientation of the head, referred to a certain initial orientation taken as origin. The circular canals contribute, therefore, to inform us of the movements that we have executed, and that on the same ground as the muscular sensations. When, therefore, above we speak of the series _S_ or of the series [Sigma], we should say, not that these were series of muscular sensations alone, but that they were series at the same time of muscular sensations and of sensations due to the semicircular canals. Apart from this addition, we should have nothing to change in what precedes.

In the series _S_ and [Sigma], these sensations of the semicircular canals evidently hold a very important place. Yet alone they would not suffice, because they can tell us only of the movements of the head; they tell us nothing of the relative movements of the body or of the members in regard to the head. And more, it seems that they tell us only of the rotations of the head and not of the translations it may undergo.