How am I led to regard these two series _S_ and _S"_ as corresponding to the same displacement _AB_? It is because these two series are capable of correcting the same external change. Apart from that, they have nothing in common.

Let us now consider two external changes: [alpha] and [beta], which shall be, for instance, the rotation of a sphere half blue, half red, and that of a sphere half yellow, half green; these two changes have nothing in common, since the one is for us the pa.s.sing of blue into red and the other the pa.s.sing of yellow into green. Consider, on the other hand, two series of internal changes _S_ and _S"_; like the others, they will have nothing in common. And yet I say that [alpha] and [beta]

correspond to the same displacement, and that _S_ and _S"_ correspond also to the same displacement. Why? Simply because _S_ can correct [alpha] as well as [beta] and because [alpha] can be corrected by _S"_ as well as by _S_. And then a question suggests itself:

If I have ascertained that _S_ corrects [alpha] and [beta] and that _S"_ corrects [alpha], am I certain that _S"_ likewise corrects [beta]? Experiment alone can teach us whether this law is verified. If it were not verified, at least approximately, there would be no geometry, there would be no s.p.a.ce, because we should have no more interest in cla.s.sifying the internal and external changes as I have just done, and, for instance, in distinguishing changes of state from changes of position.

It is interesting to see what has been the role of experience in all this. It has shown me that a certain law is approximately verified. It has not told me _how_ s.p.a.ce is, and that it satisfies the condition in question. I knew, in fact, before all experience, that s.p.a.ce satisfied this condition or that it would not be; nor have I any right to say that experience told me that geometry is possible; I very well see that geometry is possible, since it does not imply contradiction; experience only tells me that geometry is useful.

6. _Visual s.p.a.ce_

Although motor impressions have had, as I have just explained, an altogether preponderant influence in the genesis of the notion of s.p.a.ce, which never would have taken birth without them, it will not be without interest to examine also the role of visual impressions and to investigate how many dimensions 'visual s.p.a.ce' has, and for that purpose to apply to these impressions the definition of -- 3.

A first difficulty presents itself: consider a red color sensation affecting a certain point of the retina; and on the other hand a blue color sensation affecting the same point of the retina. It is necessary that we have some means of recognizing that these two sensations, qualitatively different, have something in common. Now, according to the considerations expounded in the preceding paragraph, we have been able to recognize this only by the movements of the eye and the observations to which they have given rise. If the eye were immovable, or if we were unconscious of its movements, we should not have been able to recognize that these two sensations, of different quality, had something in common; we should not have been able to disengage from them what gives them a geometric character. The visual sensations, without the muscular sensations, would have nothing geometric, so that it may be said there is no pure visual s.p.a.ce.

To do away with this difficulty, consider only sensations of the same nature, red sensations, for instance, differing one from another only as regards the point of the retina that they affect. It is clear that I have no reason for making such an arbitrary choice among all the possible visual sensations, for the purpose of uniting in the same cla.s.s all the sensations of the same color, whatever may be the point of the retina affected. I should never have dreamt of it, had I not before learned, by the means we have just seen, to distinguish changes of state from changes of position, that is, if my eye were immovable. Two sensations of the same color affecting two different parts of the retina would have appeared to me as qualitatively distinct, just as two sensations of different color.

In restricting myself to red sensations, I therefore impose upon myself an artificial limitation and I neglect systematically one whole side of the question; but it is only by this artifice that I am able to a.n.a.lyze visual s.p.a.ce without mingling any motor sensation.

Imagine a line traced on the retina and dividing in two its surface; and set apart the red sensations affecting a point of this line, or those differing from them too little to be distinguished from them. The aggregate of these sensations will form a sort of cut that I shall call _C_, and it is clear that this cut suffices to divide the manifold of possible red sensations, and that if I take two red sensations affecting two points situated on one side and the other of the line, I can not pa.s.s from one of these sensations to the other in a continuous way without pa.s.sing at a certain moment through a sensation belonging to the cut.

If, therefore, the cut has _n_ dimensions, the total manifold of my red sensations, or if you wish, the whole visual s.p.a.ce, will have _n_ + 1.

Now, I distinguish the red sensations affecting a point of the cut _C_.

The a.s.semblage of these sensations will form a new cut _C'_. It is clear that this will divide the cut _C_, always giving to the word divide the same meaning.

If, therefore, the cut _C'_ has _n_ dimensions, the cut _C_ will have _n_ + 1 and the whole of visual s.p.a.ce _n_ + 2.

If all the red sensations affecting the same point of the retina were regarded as identical, the cut _C'_ reducing to a single element would have 0 dimensions, and visual s.p.a.ce would have 2.

And yet most often it is said that the eye gives us the sense of a third dimension, and enables us in a certain measure to recognize the distance of objects. When we seek to a.n.a.lyze this feeling, we ascertain that it reduces either to the consciousness of the convergence of the eyes, or to that of the effort of accommodation which the ciliary muscle makes to focus the image.

Two red sensations affecting the same point of the retina will therefore be regarded as identical only if they are accompanied by the same sensation of convergence and also by the same sensation of effort of accommodation or at least by sensations of convergence and accommodation so slightly different as to be indistinguishable.

On this account the cut _C'_ is itself a continuum and the cut _C_ has more than one dimension.

But it happens precisely that experience teaches us that when two visual sensations are accompanied by the same sensation of convergence, they are likewise accompanied by the same sensation of accommodation. If then we form a new cut _C"_ with all those of the sensations of the cut _C'_, which are accompanied by a certain sensation of convergence, in accordance with the preceding law they will all be indistinguishable and may be regarded as identical. Therefore _C"_ will not be a continuum and will have 0 dimension; and as _C"_ divides _C'_ it will thence result that _C'_ has one, _C_ two and _the whole visual s.p.a.ce three dimensions_.

But would it be the same if experience had taught us the contrary and if a certain sensation of convergence were not always accompanied by the same sensation of accommodation? In this case two sensations affecting the same point of the retina and accompanied by the same sense of convergence, two sensations which consequently would both appertain to the cut _C"_, could nevertheless be distinguished since they would be accompanied by two different sensations of accommodation. Therefore _C"_ would be in its turn a continuum and would have one dimension (at least); then _C'_ would have two, _C_ three and _the whole visual s.p.a.ce would have four dimensions_.

Will it then be said that it is experience which teaches us that s.p.a.ce has three dimensions, since it is in setting out from an experimental law that we have come to attribute three to it? But we have therein performed, so to speak, only an experiment in physiology; and as also it would suffice to fit over the eyes gla.s.ses of suitable construction to put an end to the accord between the feelings of convergence and of accommodation, are we to say that putting on spectacles is enough to make s.p.a.ce have four dimensions and that the optician who constructed them has given one more dimension to s.p.a.ce? Evidently not; all we can say is that experience has taught us that it is convenient to attribute three dimensions to s.p.a.ce.

But visual s.p.a.ce is only one part of s.p.a.ce, and in even the notion of this s.p.a.ce there is something artificial, as I have explained at the beginning. The real s.p.a.ce is motor s.p.a.ce and this it is that we shall examine in the following chapter.

CHAPTER IV

s.p.a.cE 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 s.p.a.ce 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 s.p.a.ce, of such a sort that to every point of s.p.a.ce corresponds an element of this continuum, and that to points of s.p.a.ce very near one another correspond indistinguishable elements. s.p.a.ce 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 s.p.a.ce to ourselves, and that for a mult.i.tude of reasons. s.p.a.ce is a mathematical continuum, it is infinite, and we can represent to ourselves only physical continua and finite objects. The different elements of s.p.a.ce, 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 s.p.a.ce is nonsense, and it is necessary for us to begin by referring s.p.a.ce to a system of axes invariably bound to our body (which we must always suppose put back in the initial att.i.tude).

Then I have sought to form with our visual sensations a physical continuum equivalent to s.p.a.ce; 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 s.p.a.ce' has three dimensions. Only this solution is incomplete and artificial. I have explained why, and it is not on visual s.p.a.ce but on motor s.p.a.ce 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 att.i.tude_.

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 circ.u.mstances 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 s.p.a.ce, since the number of dimensions is not the same; it is only related to s.p.a.ce. 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 s.p.a.ce, and s.p.a.ce could be deduced from it, but it is not equivalent to s.p.a.ce, 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 s.p.a.ce may be deduced from it, it might always be asked why s.p.a.ce 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 s.p.a.ce was formed in the human mind.

2. _Ident.i.ty of Two Points_

What is a point? How do we know whether two points of s.p.a.ce 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 s.p.a.ce; 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 att.i.tude.

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 ident.i.ty 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 s.p.a.ce occupied at the instant [alpha] by this body _A_; let _M'_ be the point of s.p.a.ce 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.