The Foundations of Science · Henri Poincaré
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.
Text from Project Gutenberg, public domain in the United States.