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]'; a.s.sume this. At the instant [alpha], the object _A_ occupied a certain point _M_ of s.p.a.ce. 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 antic.i.p.ation 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 a.s.sertion 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 mult.i.tude 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 a.s.sertion. I content myself with affirming an experimental fact which is ordinarily verified.

At the end of the preceding chapter we a.n.a.lyzed visual s.p.a.ce; we saw that to engender this s.p.a.ce 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 s.p.a.ce 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 s.p.a.ce,' of only two dimensions. On the other hand, consider tactile s.p.a.ce, limiting ourselves to the sensations of a single finger, that is in sum to the a.s.semblage of positions this finger can occupy.

This tactile s.p.a.ce that we shall a.n.a.lyze in the following section and which consequently I ask permission not to consider further for the moment, this tactile s.p.a.ce, I say, has three dimensions. Why has s.p.a.ce properly so called as many dimensions as tactile s.p.a.ce and more than simple visual s.p.a.ce? It is because touch does not operate at a distance, while vision does operate at a distance. These two a.s.sertions have the same meaning and we have just seen what this is.

Now I return to a point over which I pa.s.sed 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 s.p.a.ce_

Thus I know how to recognize the ident.i.ty 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 s.p.a.ce. 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 s.p.a.ce; 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 const.i.tute 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 const.i.tute a continuum of more than three dimensions.

In fact, let there be in s.p.a.ce 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 const.i.tute 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 mult.i.tude of different att.i.tudes. 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 s.p.a.ce 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 cla.s.s 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 s.p.a.ce 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 s.p.a.ce, as identical with geometric s.p.a.ce.

But one reflection before going further; according to the foregoing, we know the points of s.p.a.ce, 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 s.p.a.ce.

4. _Ident.i.ty of the Different s.p.a.ces_

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 s.p.a.ce, 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 s.p.a.ce 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 s.p.a.ce 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 s.p.a.ce, 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 s.p.a.ce. 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 s.p.a.ce, since to any series of muscular sensations corresponds, as we have said, a point, whether in the first s.p.a.ce 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 s.p.a.ce to two series [Sigma] and [Sigma]', are identical, so also are the two corresponding points of the second s.p.a.ce _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 a.s.sociative, but not commutative.

That fixed, in order that [Sigma] and [Sigma]' should correspond to the same point _M_ = _M'_ of the first s.p.a.ce, 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 s.p.a.ce. Q.E.D.

Our two s.p.a.ces 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 s.p.a.ce and the second at the point _N_ of the second s.p.a.ce, these two points are identical. This is how we are led to regard these two s.p.a.ces as identical; or better, this is what we mean when we say that they are identical.

What we have just said of the ident.i.ty of the two tactile s.p.a.ces makes unnecessary our discussing the question of the ident.i.ty of tactile s.p.a.ce and visual s.p.a.ce, which could be treated in the same way.

5. _s.p.a.ce 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 role of experience and to a.n.a.lyze the experimental facts which intervene in the genesis of s.p.a.ce 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 s.p.a.ce has often three dimensions, but not always.