The Foundations of Science · Henri Poincaré
SPACE AND ITS THREE DIMENSIONS
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.
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.
Text from Project Gutenberg, public domain in the United States.