In the articles I have heretofore devoted to space I have above all emphasized the problems raised by non-Euclidean geometry, while leaving almost completely aside other questions more difficult of approach, such as those which pertain to the number of dimensions. All the geometries I considered had thus a common basis, that tridimensional continuum which was the same for all and which differentiated itself only by the figures one drew in it or when one aspired to measure it.
In this continuum, primitively amorphous, we may imagine a network of lines and surfaces, we may then convene to regard the meshes of this net as equal to one another, and it is only after this convention that this continuum, become measurable, becomes Euclidean or non-Euclidean space. From this amorphous continuum can therefore arise indifferently one or the other of the two spaces, just as on a blank sheet of paper may be traced indifferently a straight or a circle.
In space we know rectilinear triangles the sum of whose angles is equal to two right angles; but equally we know curvilinear triangles the sum of whose angles is less than two right angles. The existence of the one sort is not more doubtful than that of the other. To give the name of straights to the sides of the first is to adopt Euclidean geometry; to give the name of straights to the sides of the latter is to adopt the non-Euclidean geometry. So that to ask what geometry it is proper to adopt is to ask, to what line is it proper to give the name straight?
It is evident that experiment can not settle such a question; one would not ask, for instance, experiment to decide whether I should callABorCDa straight. On the other hand, neither can I say that I have not the right to give the name of straights to the sides of non-Euclidean triangles because they are not inconformity with the eternal idea of straight which I have by intuition. I grant, indeed, that I have the intuitive idea of the side of the Euclidean triangle, but I have equally the intuitive idea of the side of the non-Euclidean triangle. Why should I have the right to apply the name of straight to the first of these ideas and not to the second? Wherein does this syllable form an integrant part of this intuitive idea? Evidently when we say that the Euclidean straight is atruestraight and that the non-Euclidean straight is not a true straight, we simply mean that the first intuitive idea corresponds to amore noteworthyobject than the second. But how do we decide that this object is more noteworthy? This question I have investigated in 'Science and Hypothesis.'
It is here that we saw experience come in. If the Euclidean straight is more noteworthy than the non-Euclidean straight, it is so chiefly because it differs little from certain noteworthy natural objects from which the non-Euclidean straight differs greatly. But, it will be said, the definition of the non-Euclidean straight is artificial; if we for a moment adopt it, we shall see that two circles of different radius both receive the name of non-Euclidean straights, while of two circles of the same radius one can satisfy the definition without the other being able to satisfy it, and then if we transport one of these so-called straights without deforming it, it will cease to be a straight. But by what right do we consider as equal these two figures which the Euclidean geometers call two circles with the same radius? It is because by transporting one of them without deforming it we can make it coincide with the other. And why do we say this transportation is effected without deformation? It is impossible to give a good reason for it. Among all the motions conceivable, there are some of which the Euclidean geometers say that they are not accompanied by deformation; but there are others of which the non-Euclidean geometers would say that they are not accompanied by deformation. In the first, called Euclidean motions, the Euclidean straights remain Euclidean straights and the non-Euclidean straights do not remain non-Euclidean straights; in the motions of the second sort, or non-Euclidean motions, the non-Euclidean straights remain non-Euclidean straightsand the Euclidean straights do not remain Euclidean straights. It has, therefore, not been demonstrated that it was unreasonable to call straights the sides of non-Euclidean triangles; it has only been shown that that would be unreasonable if one continued to call the Euclidean motions motions without deformation; but it has at the same time been shown that it would be just as unreasonable to call straights the sides of Euclidean triangles if the non-Euclidean motions were called motions without deformation.
Now when we say that the Euclidean motions are thetruemotions without deformation, what do we mean? We simply mean that they aremore noteworthythan the others. And why are they more noteworthy? It is because certain noteworthy natural bodies, the solid bodies, undergo motions almost similar.
And then when we ask: Can one imagine non-Euclidean space? That means: Can we imagine a world where there would be noteworthy natural objects affecting almost the form of non-Euclidean straights, and noteworthy natural bodies frequently undergoing motions almost similar to the non-Euclidean motions? I have shown in 'Science and Hypothesis' that to this question we must answer yes.
It has often been observed that if all the bodies in the universe were dilated simultaneously and in the same proportion, we should have no means of perceiving it, since all our measuring instruments would grow at the same time as the objects themselves which they serve to measure. The world, after this dilatation, would continue on its course without anything apprising us of so considerable an event. In other words, two worlds similar to one another (understanding the word similitude in the sense of Euclid, Book VI.) would be absolutely indistinguishable. But more; worlds will be indistinguishable not only if they are equal or similar, that is, if we can pass from one to the other by changing the axes of coordinates, or by changing the scale to which lengths are referred; but they will still be indistinguishable if we can pass from one to the other by any 'point-transformation' whatever. I will explain my meaning. I suppose that to each point of one corresponds one point of the other and only one, and inversely; and besides that thecoordinates of a point are continuous functions,otherwise altogether arbitrary, of the corresponding point. I suppose besides that to each object of the first world corresponds in the second an object of the same nature placed precisely at the corresponding point. I suppose finally that this correspondence fulfilled at the initial instant is maintained indefinitely. We should have no means of distinguishing these two worlds one from the other. The relativity of space is not ordinarily understood in so broad a sense; it is thus, however, that it would be proper to understand it.
If one of these universes is our Euclidean world, what its inhabitants will call straight will be our Euclidean straight; but what the inhabitants of the second world will call straight will be a curve which will have the same properties in relation to the world they inhabit and in relation to the motions that they will call motions without deformation. Their geometry will, therefore, be Euclidean geometry, but their straight will not be our Euclidean straight. It will be its transform by the point-transformation which carries over from our world to theirs. The straights of these men will not be our straights, but they will have among themselves the same relations as our straights to one another. It is in this sense I say their geometry will be ours. If then we wish after all to proclaim that they deceive themselves, that their straight is not the true straight, if we still are unwilling to admit that such an affirmation has no meaning, at least we must confess that these people have no means whatever of recognizing their error.
All that is relatively easy to understand, and I have already so often repeated it that I think it needless to expatiate further on the matter. Euclidean space is not a form imposed upon our sensibility, since we can imagine non-Euclidean space; but the two spaces, Euclidean and non-Euclidean, have a common basis, that amorphous continuum of which I spoke in the beginning. From this continuum we can get either Euclidean space or Lobachevskian space, just as we can, by tracing upon it a proper graduation, transform an ungraduated thermometer into a Fahrenheit or a Réaumur thermometer.
And then comes a question: Is not this amorphous continuum, that our analysis has allowed to survive, a form imposed upon our sensibility? If so, we should have enlarged the prison in which this sensibility is confined, but it would always be a prison.
This continuum has a certain number of properties, exempt from all idea of measurement. The study of these properties is the object of a science which has been cultivated by many great geometers and in particular by Riemann and Betti and which has received the name of analysis situs. In this science abstraction is made of every quantitative idea and, for example, if we ascertain that on a line the pointBis between the pointsAandC, we shall be content with this ascertainment and shall not trouble to know whether the lineABCis straight or curved, nor whether the lengthABis equal to the lengthBC, or whether it is twice as great.
The theorems of analysis situs have, therefore, this peculiarity, that they would remain true if the figures were copied by an inexpert draftsman who should grossly change all the proportions and replace the straights by lines more or less sinuous. In mathematical terms, they are not altered by any 'point-transformation' whatsoever. It has often been said that metric geometry was quantitative, while projective geometry was purely qualitative. That is not altogether true. The straight is still distinguished from other lines by properties which remain quantitative in some respects. The real qualitative geometry is, therefore, analysis situs.
The same questions which came up apropos of the truths of Euclidean geometry, come up anew apropos of the theorems of analysis situs. Are they obtainable by deductive reasoning? Are they disguised conventions? Are they experimental verities? Are they the characteristics of a form imposed either upon our sensibility or upon our understanding?
I wish simply to observe that the last two solutions exclude each other. We can not admit at the same time that it is impossible to imagine space of four dimensions and that experience proves to us that space has three dimensions. The experimenter puts to nature a question: Is it this or that? and he can not putit without imagining the two terms of the alternative. If it were impossible to imagine one of these terms, it would be futile and besides impossible to consult experience. There is no need of observation to know that the hand of a watch is not marking the hour 15 on the dial, because we know beforehand that there are only 12, and we could not look at the mark 15 to see if the hand is there, because this mark does not exist.
Note likewise that in analysis situs the empiricists are disembarrassed of one of the gravest objections that can be leveled against them, of that which renders absolutely vain in advance all their efforts to apply their thesis to the verities of Euclidean geometry. These verities are rigorous and all experimentation can only be approximate. In analysis situs approximate experiments may suffice to give a rigorous theorem and, for instance, if it is seen that space can not have either two or less than two dimensions, nor four or more than four, we are certain that it has exactly three, since it could not have two and a half or three and a half.
Of all the theorems of analysis situs, the most important is that which is expressed in saying that space has three dimensions. This it is that we are about to consider, and we shall put the question in these terms: When we say that space has three dimensions, what do we mean?
I have explained in 'Science and Hypothesis' whence we derive the notion of physical continuity and how that of mathematical continuity has arisen from it. It happens that we are capable of distinguishing two impressions one from the other, while each is indistinguishable from a third. Thus we can readily distinguish a weight of 12 grams from a weight of 10 grams, while a weight of 11 grams could be distinguished from neither the one nor the other. Such a statement, translated into symbols, may be written:
A = B, B = C, A < C.
This would be the formula of the physical continuum, as crude experience gives it to us, whence arises an intolerable contradictionthat has been obviated by the introduction of the mathematical continuum. This is a scale of which the steps (commensurable or incommensurable numbers) are infinite in number but are exterior to one another, instead of encroaching on one another as do the elements of the physical continuum, in conformity with the preceding formula.
The physical continuum is, so to speak, a nebula not resolved; the most perfect instruments could not attain to its resolution. Doubtless if we measured the weights with a good balance instead of judging them by the hand, we could distinguish the weight of 11 grams from those of 10 and 12 grams, and our formula would become:
A < B, B < C, A < C.
But we should always find betweenAandBand betweenBandCnew elementsDandE, such that
A = D, D = B, A < B; B = E, E = C, B < C,
and the difficulty would only have receded and the nebula would always remain unresolved; the mind alone can resolve it and the mathematical continuum it is which is the nebula resolved into stars.
Yet up to this point we have not introduced the notion of the number of dimensions. What is meant when we say that a mathematical continuum or that a physical continuum has two or three dimensions?
First we must introduce the notion of cut, studying first physical continua. We have seen what characterizes the physical continuum. Each of the elements of this continuum consists of a manifold of impressions; and it may happen either that an element can not be discriminated from another element of the same continuum, if this new element corresponds to a manifold of impressions not sufficiently different, or, on the contrary, that the discrimination is possible; finally it may happen that two elements indistinguishable from a third may, nevertheless, be distinguished one from the other.
That postulated, ifAandBare two distinguishable elements of a continuumC, a series of elements may be found,E1,E2, ...,En, all belonging to this same continuumCand such that each ofthem is indistinguishable from the preceding, thatE1is indistinguishable fromA, andEnindistinguishable fromB. Therefore we can go fromAtoBby a continuous route and without quittingC. If this condition is fulfilled for any two elementsAandBof the continuumC, we may say that this continuumCis all in one piece. Now let us distinguish certain of the elements ofCwhich may either be all distinguishable from one another, or themselves form one or several continua. The assemblage of the elements thus chosen arbitrarily among all those ofCwill form what I shall call thecutor thecuts.
Take onCany two elementsAandB. Either we can also find a series of elementsE1,E2, ...,En, such: (1) that they all belong toC; (2) that each of them is indistinguishable from the following,E1indistinguishable fromAandEnfromB; (3)and besides that none of the elements E is indistinguishable from any element of the cut. Or else, on the contrary, in each of the seriesE1,E2, ...,Ensatisfying the first two conditions, there will be an elementEindistinguishable from one of the elements of the cut. In the first case we can go fromAtoBby a continuous route without quittingCandwithout meeting the cuts; in the second case that is impossible.
If then for any two elementsAandBof the continuumC, it is always the first case which presents itself, we shall say thatCremains all in one piece despite the cuts.
Thus, if we choose the cuts in a certain way, otherwise arbitrary, it may happen either that the continuum remains all in one piece or that it does not remain all in one piece; in this latter hypothesis we shall then say that it isdividedby the cuts.
It will be noticed that all these definitions are constructed in setting out solely from this very simple fact, that two manifolds of impressions sometimes can be discriminated, sometimes can not be. That postulated, if, todividea continuum, it suffices to consider as cuts a certain number of elements all distinguishable from one another, we say that this continuumis of one dimension; if, on the contrary, to divide a continuum, it is necessary to consider as cuts a system of elements themselves forming one or several continua, we shall say that this continuum isof several dimensions.
If to divide a continuumC, cuts forming one or several continua of one dimension suffice, we shall say thatCis a continuumof two dimensions; if cuts suffice which form one or several continua of two dimensions at most, we shall say thatCis a continuumof three dimensions; and so on.
To justify this definition it is proper to see whether it is in this way that geometers introduce the notion of three dimensions at the beginning of their works. Now, what do we see? Usually they begin by defining surfaces as the boundaries of solids or pieces of space, lines as the boundaries of surfaces, points as the boundaries of lines, and they affirm that the same procedure can not be pushed further.
This is just the idea given above: to divide space, cuts that are called surfaces are necessary; to divide surfaces, cuts that are called lines are necessary; to divide lines, cuts that are called points are necessary; we can go no further, the point can not be divided, so the point is not a continuum. Then lines which can be divided by cuts which are not continua will be continua of one dimension; surfaces which can be divided by continuous cuts of one dimension will be continua of two dimensions; finally, space which can be divided by continuous cuts of two dimensions will be a continuum of three dimensions.
Thus the definition I have just given does not differ essentially from the usual definitions; I have only endeavored to give it a form applicable not to the mathematical continuum, but to the physical continuum, which alone is susceptible of representation, and yet to retain all its precision. Moreover, we see that this definition applies not alone to space; that in all which falls under our senses we find the characteristics of the physical continuum, which would allow of the same classification; that it would be easy to find there examples of continua of four, of five, dimensions, in the sense of the preceding definition; such examples occur of themselves to the mind.
I should explain finally, if I had the time, that this science, of which I spoke above and to which Riemann gave the name of analysis situs, teaches us to make distinctions among continua of the same number of dimensions and that the classification of these continua rests also on the consideration of cuts.
From this notion has arisen that of the mathematical continuum of several dimensions in the same way that the physical continuum of one dimension engendered the mathematical continuum of one dimension. The formula
A > C, A = B, B = C,
which summed up the data of crude experience, implied an intolerable contradiction. To get free from it, it was necessary to introduce a new notion while still respecting the essential characteristics of the physical continuum of several dimensions. The mathematical continuum of one dimension admitted of a scale whose divisions, infinite in number, corresponded to the different values, commensurable or not, of one same magnitude. To have the mathematical continuum ofndimensions, it will suffice to takenlike scales whose divisions correspond to different values ofnindependent magnitudes called coordinates. We thus shall have an image of the physical continuum ofndimensions, and this image will be as faithful as it can be after the determination not to allow the contradiction of which I spoke above.
It seems now that the question we put to ourselves at the start is answered. When we say that space has three dimensions, it will be said, we mean that the manifold of points of space satisfies the definition we have just given of the physical continuum of three dimensions. To be content with that would be to suppose that we know what is the manifold of points of space, or even one point of space.
Now that is not as simple as one might think. Every one believes he knows what a point is, and it is just because we know it too well that we think there is no need of defining it. Surely we can not be required to know how to define it, because in going back from definition to definition a time must come when we must stop. But at what moment should we stop?
We shall stop first when we reach an object which falls under our senses or that we can represent to ourselves; definition then will become useless; we do not define the sheep to a child; we say to him:Seethe sheep.
So, then, we should ask ourselves if it is possible to represent to ourselves a point of space. Those who answer yes do not reflect that they represent to themselves in reality a white spot made with the chalk on a blackboard or a black spot made with a pen on white paper, and that they can represent to themselves only an object or rather the impressions that this object made on their senses.
When they try to represent to themselves a point, they represent the impressions that very little objects made them feel. It is needless to add that two different objects, though both very little, may produce extremely different impressions, but I shall not dwell on this difficulty, which would still require some discussion.
But it is not a question of that; it does not suffice to representonepoint, it is necessary to representa certainpoint and to have the means of distinguishing it from anotherpoint. And in fact, that we may be able to apply to a continuum the rule I have above expounded and by which one may recognize the number of its dimensions, we must rely upon the fact that two elements of this continuum sometimes can and sometimes can not be distinguished. It is necessary therefore that we should in certain cases know how to represent to ourselvesa specificelement and to distinguish it from anotherelement.
The question is to know whether the point that I represented to myself an hour ago is the same as this that I now represent to myself, or whether it is a different point. In other words, how do we know whether the point occupied by the objectAat the instant α is the same as the point occupied by the objectBat the instant β, or still better, what this means?
I am seated in my room; an object is placed on my table; during a second I do not move, no one touches the object. I am tempted to say that the pointAwhich this object occupied at the beginning of this second is identical with the pointBwhich it occupies at its end. Not at all; from the pointAto the pointBis 30 kilometers, because the object has been carried along in the motion of the earth. We can not know whether an object, be it large or small, has not changed its absolute position in space, and not only can we not affirm it, but this affirmation has nomeaning and in any case can not correspond to any representation.
But then we may ask ourselves if the relative position of an object with regard to other objects has changed or not, and first whether the relative position of this object with regard to our body has changed. If the impressions this object makes upon us have not changed, we shall be inclined to judge that neither has this relative position changed; if they have changed, we shall judge that this object has changed either in state or in relative position. It remains to decide which of thetwo. I have explained in 'Science and Hypothesis' how we have been led to distinguish the changes of position. Moreover, I shall return to that further on. We come to know, therefore, whether the relative position of an object with regard to our body has or has not remained the same.
If now we see that two objects have retained their relative position with regard to our body, we conclude that the relative position of these two objects with regard to one another has not changed; but we reach this conclusion only by indirect reasoning. The only thing that we know directly is the relative position of the objects with regard to our body.A fortioriit is only by indirect reasoning that we think we know (and, moreover, this belief is delusive) whether the absolute position of the object has changed.
In a word, the system of coordinate axes to which we naturally refer all exterior objects is a system of axes invariably bound to our body, and carried around with us.
It is impossible to represent to oneself absolute space; when I try to represent to myself simultaneously objects and myself in motion in absolute space, in reality I represent to myself my own self motionless and seeing move around me different objects and a man that is exterior to me, but that I convene to call me.
Will the difficulty be solved if we agree to refer everything to these axes bound to our body? Shall we know then what is a point thus defined by its relative position with regard to ourselves? Many persons will answer yes and will say that they 'localize' exterior objects.
What does this mean? To localize an object simply means to represent to oneself the movements that would be necessary toreach it. I will explain myself. It is not a question of representing the movements themselves in space, but solely of representing to oneself the muscular sensations which accompany these movements and which do not presuppose the preexistence of the notion of space.
If we suppose two different objects which successively occupy the same relative position with regard to ourselves, the impressions that these two objects make upon us will be very different; if we localize them at the same point, this is simply because it is necessary to make the same movements to reach them; apart from that, one can not just see what they could have in common.
But, given an object, we can conceive many different series of movements which equally enable us to reach it. If then we represent to ourselves a point by representing to ourselves the series of muscular sensations which accompany the movements which enable us to reach this point, there will be many ways entirely different of representing to oneself the same point. If one is not satisfied with this solution, but wishes, for instance, to bring in the visual sensations along with the muscular sensations, there will be one or two more ways of representing to oneself this same point and the difficulty will only be increased. In any case the following question comes up: Why do we think that all these representations so different from one another still represent the same point?
Another remark: I have just said that it is to our own body that we naturally refer exterior objects; that we carry about everywhere with us a system of axes to which we refer all the points of space and that this system of axes seems to be invariably bound to our body. It should be noticed that rigorously we could not speak of axes invariably bound to the body unless the different parts of this body were themselves invariably bound to one another. As this is not the case, we ought, before referring exterior objects to these fictitious axes, to suppose our body brought back to the initial attitude.
I have shown in 'Science and Hypothesis' the preponderant rôle played by the movements of our body in the genesis of thenotion of space. For a being completely immovable there would be neither space nor geometry; in vain would exterior objects be displaced about him, the variations which these displacements would make in his impressions would not be attributed by this being to changes of position, but to simple changes of state; this being would have no means of distinguishing these two sorts of changes, and this distinction, fundamental for us, would have no meaning for him.
The movements that we impress upon our members have as effect the varying of the impressions produced on our senses by external objects; other causes may likewise make them vary; but we are led to distinguish the changes produced by our own motions and we easily discriminate them for two reasons: (1) because they are voluntary; (2) because they are accompanied by muscular sensations.
So we naturally divide the changes that our impressions may undergo into two categories to which perhaps I have given an inappropriate designation: (1) the internal changes, which are voluntary and accompanied by muscular sensations; (2) the external changes, having the opposite characteristics.
We then observe that among the external changes are some which can be corrected, thanks to an internal change which brings everything back to the primitive state; others can not be corrected in this way (it is thus that, when an exterior object is displaced, we may then by changing our own position replace ourselves as regards this object in the same relative position as before, so as to reestablish the original aggregate of impressions; if this object was not displaced, but changed its state, that is impossible). Thence comes a new distinction among external changes: those which may be so corrected we call changes of position; and the others, changes of state.
Think, for example, of a sphere with one hemisphere blue and the other red; it first presents to us the blue hemisphere, then it so revolves as to present the red hemisphere. Now think of a spherical vase containing a blue liquid which becomes red in consequence of a chemical reaction. In both cases the sensation of red has replaced that of blue; our senses have experienced the same impressions which have succeeded each other in the sameorder, and yet these two changes are regarded by us as very different; the first is a displacement, the second a change of state. Why? Because in the first case it is sufficient for me to go around the sphere to place myself opposite the blue hemisphere and reestablish the original blue sensation.
Still more; if the two hemispheres, in place of being red and blue, had been yellow and green, how should I have interpreted the revolution of the sphere? Before, the red succeeded the blue, now the green succeeds the yellow; and yet I say that the two spheres have undergone the same revolution, that each has turned about its axis; yet I can not say that the green is to yellow as the red is to blue; how then am I led to decide that the two spheres have undergone thesamedisplacement? Evidently because, in one case as in the other, I am able to reestablish the original sensation by going around the sphere, by making the same movements, and I know that I have made the same movements because I have felt the same muscular sensations; to know it, I do not need, therefore, to know geometry in advance and to represent to myself the movements of my body in geometric space.
Another example: An object is displaced before my eye; its image was first formed at the center of the retina; then it is formed at the border; the old sensation was carried to me by a nerve fiber ending at the center of the retina; the new sensation is carried to me byanothernerve fiber starting from the border of the retina; these two sensations are qualitatively different; otherwise, how could I distinguish them?
Why then am I led to decide that these two sensations, qualitatively different, represent the same image, which has been displaced? It is because Ican follow the object with the eyeand by a displacement of the eye, voluntary and accompanied by muscular sensations, bring back the image to the center of the retina and reestablish the primitive sensation.
I suppose that the image of a red object has gone from the centerAto the borderBof the retina, then that the image of a blue object goes in its turn from the centerAto the borderBof the retina; I shall decide that these two objects have undergone thesamedisplacement. Why? Because in both cases I shall have been able to reestablish the primitive sensation, andthat to do it I shall have had to execute thesamemovement of the eye, and I shall know that my eye has executed the same movement because I shall have felt thesamemuscular sensations.
If I could not move my eye, should I have any reason to suppose that the sensation of red at the center of the retina is to the sensation of red at the border of the retina as that of blue at the center is to that of blue at the border? I should only have four sensations qualitatively different, and if I were asked if they are connected by the proportion I have just stated, the question would seem to me ridiculous, just as if I were asked if there is an analogous proportion between an auditory sensation, a tactile sensation and an olfactory sensation.
Let us now consider the internal changes, that is, those which are produced by the voluntary movements of our body and which are accompanied by muscular changes. They give rise to the two following observations, analogous to those we have just made on the subject of external changes.
1. I may suppose that my body has moved from one point to another, but that the sameattitudeis retained; all the parts of the body have therefore retained or resumed the samerelativesituation, although their absolute situation in space may have varied. I may suppose that not only has the position of my body changed, but that its attitude is no longer the same, that, for instance, my arms which before were folded are now stretched out.
I should therefore distinguish the simple changes of position without change of attitude, and the changes of attitude. Both would appear to me under form of muscular sensations. How then am I led to distinguish them? It is that the first may serve to correct an external change, and that the others can not, or at least can only give an imperfect correction.
This fact I proceed to explain as I would explain it to some one who already knew geometry, but it need not thence be concluded that it is necessary already to know geometry to make this distinction; before knowing geometry I ascertain the fact (experimentally, so to speak), without being able to explain it. But merely to make the distinction between the two kinds of change, I do not need toexplainthe fact, it suffices meto ascertainit.
However that may be, the explanation is easy. Suppose thatan exterior object is displaced; if we wish the different parts of our body to resume with regard to this object their initial relative position, it is necessary that these different parts should have resumed likewise their initial relative position with regard to one another. Only the internal changes which satisfy this latter condition will be capable of correcting the external change produced by the displacement of that object. If, therefore, the relative position of my eye with regard to my finger has changed, I shall still be able to replace the eye in its initial relative situation with regard to the object and reestablish thus the primitive visual sensations, but then the relative position of the finger with regard to the object will have changed and the tactile sensations will not be reestablished.
2. We ascertain likewise that the same external change may be corrected by two internal changes corresponding to different muscular sensations. Here again I can ascertain this without knowing geometry; and I have no need of anything else; but I proceed to give the explanation of the fact, employing geometrical language. To go from the positionAto the positionBI may take several routes. To the first of these routes will correspond a seriesSof muscular sensations; to a second route will correspond another seriesS´´, of muscular sensations which generally will be completely different, since other muscles will be used.
How am I led to regard these two seriesSandS´´as corresponding to the same displacementAB? 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: α and β, 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 passing of blue into red and the other the passing of yellow into green. Consider, on the other hand, two series of internal changesSandS´´; like the others, they will have nothing in common. And yet I say that α and β correspond to the same displacement, and thatSandS´´correspond also to the same displacement. why? Simply becauseScan correct α as well as β and because α can be corrected byS´´as well as byS. And then a question suggests itself:
If I have ascertained thatScorrects α and β and thatS´´corrects α, am I certain thatS´´likewise corrects β? 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 space, because we should have no more interest in classifying 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 rôle of experience in all this. It has shown me that a certain law is approximately verified. It has not told mehowspace is, and that it satisfies the condition in question. I knew, in fact, before all experience, that space 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.
Although motor impressions have had, as I have just explained, an altogether preponderant influence in the genesis of the notion of space, which never would have taken birth without them, it will not be without interest to examine also the rôle of visual impressions and to investigate how many dimensions 'visual space' 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 whatgives 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 space.
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 class 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 analyze visual space 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 callC, 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 pass from one of these sensations to the other in a continuous way without passing at a certain moment through a sensation belonging to the cut.
If, therefore, the cut hasndimensions, the total manifold of my red sensations, or if you wish, the whole visual space, will haven+ 1.
Now, I distinguish the red sensations affecting a point of the cutC. The assemblage of these sensations will form a new cutC´. It is clear that this will divide the cutC, always giving to the word divide the same meaning.
If, therefore, the cutC´hasndimensions, the cutCwill haven+ 1 and the whole of visual spacen+ 2.
If all the red sensations affecting the same point of the retina were regarded as identical, the cutC´reducing to a single element would have 0 dimensions, and visual space 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 analyze 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 cutC´is itself a continuum and the cutChas 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 cutC´´with all those of the sensations of the cutC´, 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. ThereforeC´´will not be a continuum and will have 0 dimension; and asC´´dividesC´it will thence result thatC´has one,Ctwo andthe whole visual space 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 cutC´´, could nevertheless be distinguished since they would be accompanied by two different sensations of accommodation. ThereforeC´´would be in its turn a continuum and would have one dimension (atleast); thenC´would have two,Cthree andthe whole visual space would have four dimensions.
Will it then be said that it is experience which teaches us that space 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 glasses 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 space have four dimensions and that the optician who constructed them has given one more dimension to space? Evidently not; all we can say is that experience has taught us that it is convenient to attribute three dimensions to space.
But visual space is only one part of space, and in even the notion of this space there is something artificial, as I have explained at the beginning. The real space is motor space and this it is that we shall examine in the following chapter.
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 havendimensions. 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 continuumC. And each of these systems is called anelementof the continuumC.
How many dimensions has this continuum? Take first two elementsAandBofC, and suppose there exists a series Σ of elements, all belonging to the continuumC, of such a sort thatAandBare the two extreme terms of this series and that each term of the series is indistinguishable from the preceding. If such a series Σ can be found, we say thatAandBare joined to one another; and if any two elements ofCare joined to one another, we say thatCis all of one piece.
Now take on the continuumCa certain number of elements in a way altogether arbitrary. The aggregate of these elements will be called acut. Among the various series Σ which joinAtoB, 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 whichcutthe cut) and those of whichallthe elements are distinguishable from all those of the cut. Ifallthe series Σ which joinAtoBcut the cut, we shall say thatAandBareseparatedby the cut, and that the cutdividesC. If we can not find onCtwo elements which are separated by the cut, we shall say that the cutdoes not divideC.
These definitions laid down, if the continuumCcan be divided by cuts which do not themselves form a continuum, this continuumChas only one dimension; in the contrary case it has several. If a cut forming a continuum of 1 dimension suffices to divideC,Cwill have 2 dimensions; if a cut forming a continuum of 2 dimensions suffices,Cwill 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 betweenchanges of position and changes of state. Among the changes which occur in our impressions, we distinguish, first theinternalchanges, voluntary and accompanied by muscular sensations, and theexternalchanges, having opposite characteristics. We ascertain that it may happen that an external change may becorrectedby an internal change which reestablishes the primitive sensations. The external changes, capable of being corrected by an internal change are calledchanges of position, those not capable of it are calledchanges of state. The internal changes capable of correcting an external change are calleddisplacements of the whole body; the others are calledchanges of attitude.
Now let α and β be two external changes, α´ and β´ two internal changes. Suppose that a may be corrected either by α´ or by β', and that α´ can correct either α or β; experience tells us then that β´ can likewise correct β. In this case we say that α and β correspond to thesamedisplacement and also that α´ and β´ correspond to thesamedisplacement. That postulated, we can imagine a physical continuum which we shall callthe continuum or group of displacementsand 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 α´ and β´ shall be regarded as indistinguishable: (1) if they are so naturally, that is, if they are too close to one another; (2) if α´ is capable of correcting the same external change as a third internal change naturally indistinguishable from β'. 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 hassix. 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 itby experience.
It would be easy to describe the experiments by which wecould 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.
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 objectAoccupied at the instant α the point which the objectBoccupies at the instant β, 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 objectsAandBin 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 α and β 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 α impressions that I attributed to the objectAwere 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 β other impressions which I attribute to the objectBare transmitted to me, some bythis 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 toA, and that which I attribute toB, impressions which are qualitatively different, are transmitted to me by the same nerve? Must we suppose, to take for example the visual sensations, thatAproduces two simultaneous sensations, a sensation purely luminousaand a colored sensationa´, thatBproduces in the same way simultaneously a luminous sensationband a colored sensationb´, that if these different sensations are transmitted to me by the same retinal fiber,ais identical withb, but that in general the colored sensationsa´andb´produced by different bodies are different? In that case it would be the identity of the sensationawhich accompaniesa´with the sensationbwhich accompaniesb´, 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 sensationsa+a´andb+b´, without which we should have no means of recognizing that the objectBhas taken the place of the objectA.
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 toBare transmitted to me at the instant β by the same fibers, optic as well as tactile, which, at the instant α, had transmitted to me the impressions that I attributed toA. If it is so, we shall not hesitate to declare that the point occupied byBat the instant β is identical with the point occupied byAat the instant α.
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: LetObe the point of the retina where is formed at the instant α the image of the bodyA; letMbe the point of space occupied at the instant α by this bodyA; letM´be the point ofspace occupied at the instant β by the bodyB. For this bodyBto form its image inO, it is not necessary that the pointsMandM´coincide; since vision acts at a distance, it suffices for the three pointsOMM´to be in a straight line. This condition that the two objects form their image onOis therefore necessary, but not sufficient for the pointsMandM´to coincide. Let nowPbe 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 bodyAtouches my finger at the instant α, it is becauseMandPcoincide; ifBtouches my finger at the instant β, it is becauseM´andPcoincide. ThereforeMandM´coincide. Thus this condition that ifAtouches my finger at the instant α,Btouches it at the instant β, is at once necessary and sufficient forMandM´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 α a tactile impression which I attribute to the objectA. I make a series of movements, corresponding to a seriesSof muscular sensations. After these movements, at the instant α', mysecondfinger receives a tactile impression that I attribute likewise toA. Afterward, at the instant β, without my having budged, as my muscular sense tells me, this same second finger transmits to meanew a tactile impression which I attribute this time to the objectB; I then make a series of movements, corresponding to a seriesS´of muscular sensations. I know that this seriesS´is the inverse of the seriesSand 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 toSand toS´, the primitive impressions would be reestablished, in other words, that the two series mutually compensate. That settled, should I expect that at the instant β', when the second series of movements is ended, myfirst fingerwould feel a tactile impression attributable to the objectB?
To answer this question, those already knowing geometry would reason as follows: There are chances that the objectAhas not budged, between the instants α and α', nor the objectBbetween the instants β and β'; assume this. At the instant α, the objectAoccupied a certain pointMof space. Now at this instant it touched my first finger, andas touch does not operate at a distance, my first finger was likewise at the pointM. I afterward made the seriesSof movements and at the end of this series, at the instant α', I ascertained that the objectAtouched my second finger. I thence conclude that this second finger was then atM, that is, that the movementsShad the result of bringing the second finger to the place of the first. At the instant β the objectBhas come in contact with my second finger: as I have not budged, this second finger has remained atM; therefore the objectBhas come toM; by hypothesis it does not budge up to the instant β'. But between the instants β and β' I have made the movementsS´; as these movements are the inverse of the movementsS, they must have for effect bringing the first finger in the place of the second. At the instant β´ this first finger will, therefore, be atM; and as the objectBis likewise atM, this objectBwill 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 objectAhas moved between the instants α and α', or the objectBbetween the instants β and β'.
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 concludingthat 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 whetherBoccupies at the instant β the point occupied byAat the instant α, 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 byAat the instant α andBat theinstant β 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 α, and by the blue objectBat the instant β, 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 pointAand the sensation of blue at the pointBof the retina as between the sensation of red at the pointAand the sensation of red at the pointB. 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.
Thus I know how to recognize the identity of two points, the point occupied byAat the instant α and the point occupied byBat the instant β, but onlyon one condition, namely, that I have not budged between the instants α and β. 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 byAat the instant α is identical with the point occupied byBat the instant β? I suppose that at the instant α, the objectAwas in contact with my first finger and that in the same way, at the instant β, the objectBtouches 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 sensationsSandS´, andI have said it sometimes happens that we are led to consider two such seriesSandS´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 α and β, but so as to feel successively the two series of muscular sensationsSandS´that I consider inverses, I shall still conclude, just as if I had not budged, that the points occupied byAat the instant α and byBat the instant β are identical, if I ascertain that my first finger touchesAat the instant α, andBat the instant β.
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 byAandBat the instants α and β, or (what amounts to the same thing since I suppose that my finger touchesAat the instant α andBat the instant β) I wish to compare the two points occupied by my finger at the two instants α and β. The sole means I use for this comparison is the series Σ of muscular sensations which have accompanied the movements of my body between these two instants. The different imaginable series Σ 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 Σ and Σ +S+S´, whenSandS´are inverses one of the other in the sense above given to this word; in spite of this agreement, the aggregate of distinct series Σ will still form a physical continuum and the number of dimensions will be less but still very great.
To each of these series Σ corresponds a point of space; to two series Σ and Σ´ thus correspond two pointsMandM´. The means we have hitherto used enable us to recognize thatMandM´are not distinct in two cases: (1) if Σ is identical with Σ´; (2) if Σ´ = Σ +S+S´,SandS´being inverses one of the other. If in all the other cases we should regardMandM´as distinct, the manifold of points would have as many dimensions as the aggregate of distinct series Σ, that is, much more than three.
For those who already know geometry, the following explanation would be easily comprehensible. Among the imaginableseries 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 Σ and Σ + σ, where the series σ 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 Σ and Σ´ unless Σ´ = Σ +S+S´,SandS´being inverses, the aggregate of series will constitute a continuum of more than three dimensions.
In fact, let there be in space a surfaceA, on this surface a lineB, on this line a pointM. LetC0be the aggregate of all series Σ. LetC1be the aggregate of all the series Σ, such that at the end of corresponding movements the finger is found upon the surfaceA, andC2orC3the aggregate of series Σ such that at the end the finger is found onB, or atM. It is clear, first thatC1will constitute a cut which will divideC0, thatC2will be a cut which will divideC1, andC3a cut which will divideC2. Thence it results, in accordance with our definitions, that ifC3is a continuum ofndimensions,C0will be a physical continuum ofn+ 3 dimensions.