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 s.p.a.ce; 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 s.p.a.ce are not experimental truths, properly so called. If we had wished to verify other laws, we could have succeeded also, by giving other a.n.a.logous 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 s.p.a.ce 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 s.p.a.ce, which is a physical continuum, never with geometric s.p.a.ce, which is a mathematical continuum. At the very most it would appear to tell us that it is convenient to give to geometric s.p.a.ce three dimensions, so that it may have as many as representative s.p.a.ce.

The empiric question may be put under another form. Is it impossible to conceive physical phenomena, the mechanical phenomena, for example, otherwise than in s.p.a.ce 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 s.p.a.ce 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="">

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 s.p.a.ce, are in no wise imposed upon mechanics with an invincible force.

6. _Mind and s.p.a.ce_

Experience, therefore, has played only a single role, it has served as occasion. But this role was none the less very important; and I have thought it necessary to give it prominence. This role would have been useless if there existed an _a priori_ form imposing itself upon our sensitivity, and which was s.p.a.ce of three dimensions.

Does this form exist, or, if you choose, can we represent to ourselves s.p.a.ce 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 s.p.a.ce of four, nor s.p.a.ce of three, dimensions; we can not first represent them to ourselves empty, and no more can we represent to ourselves an object either in s.p.a.ce of four, or in s.p.a.ce of three, dimensions: (1) Because these s.p.a.ces are both infinite and we can not represent to ourselves a figure _in_ s.p.a.ce, that is, the part _in_ the whole, without representing the whole, and that is impossible, because it is infinite; (2) because these s.p.a.ces are both mathematical continua, and we can represent to ourselves only the physical continuum; (3) because these s.p.a.ces are both h.o.m.ogeneous, and the frames in which we enclose our sensations, being limited, can not be h.o.m.ogeneous.

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 s.p.a.ce 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 mult.i.tude of a.s.sociations of ideas, which are the fruit of a long personal experience and of the still longer experience of the race.

Is it these a.s.sociations (or at least those of them that we have inherited from our ancestors), which const.i.tute 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 a.n.a.lysis 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 a.s.sociated 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 a.s.sociation 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 a.s.sociation and we can conceive that it may be broken; so that we may not say that sensation can not enter consciousness without entering in s.p.a.ce, but that in fact it does not enter consciousness without entering in s.p.a.ce, which means, without being entangled in this a.s.sociation.

No more can I understand one's saying that the idea of time is logically subsequent to s.p.a.ce, 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 s.p.a.ce, 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 s.p.a.ce 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 a.s.sociated together in a very complex woof, our series are _cla.s.sed_. 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 cla.s.sed in three cla.s.ses corresponding to the three dimensions of s.p.a.ce. Of course this cla.s.sification 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 cla.s.s.

But as _all_ my muscular sensations have already been cla.s.sed in one of the three pre-existent cla.s.ses, I can only represent to myself a series belonging to one of these three cla.s.ses, 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 cla.s.sification and replace it by a new one in which the series of muscular sensations should have been distributed into four cla.s.ses. The difficulty would have disappeared.

It is presented sometimes under a more striking form. Suppose I am enclosed in a chamber between the six impa.s.sable 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 s.p.a.ce 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 s.p.a.ce of four and s.p.a.ce of three dimensions. What then is the role of experience?

It gives the indications following which the choice is made.

Another thing: whence does s.p.a.ce get its quant.i.tative character? It comes from the role which the series of muscular sensations play in its genesis. These are series which may _repeat themselves_, and it is from their repet.i.tion that number comes; it is because they can repeat themselves indefinitely that s.p.a.ce is infinite. And finally we have seen, at the end of section 3, that it is also because of this that s.p.a.ce is relative. So it is repet.i.tion which has given to s.p.a.ce its essential characteristics; now, repet.i.tion supposes time; this is enough to tell that time is logically anterior to s.p.a.ce.

7. _Role of the Semicircular Ca.n.a.ls_

I have not hitherto spoken of the role of certain organs to which the physiologists attribute with reason a capital importance, I mean the semicircular ca.n.a.ls. Numerous experiments have sufficiently shown that these ca.n.a.ls are necessary to our sense of orientation; but the physiologists are not entirely in accord; two opposing theories have been proposed, that of Mach-Delage and that of M. de Cyon.

M. de Cyon is a physiologist who has made his name ill.u.s.trious by important discoveries on the innervation of the heart; I can not, however, agree with his ideas on the question before us. Not being a physiologist, I hesitate to criticize the experiments he has directed against the adverse theory of Mach-Delage; it seems to me, however, that they are not convincing, because in many of them the _total_ pressure was made to vary in one of the ca.n.a.ls, while, physiologically, what varies is the _difference_ between the pressures on the two extremities of the ca.n.a.l; in others the organs were subjected to profound lesions, which must alter their functions.

Besides, this is not important; the experiments, if they were irreproachable, might be convincing against the old theory. They would not be convincing _for_ the new theory. In fact, if I have rightly understood the theory, my explaining it will be enough for one to understand that it is impossible to conceive of an experiment confirming it.

The three pairs of ca.n.a.ls would have as sole function to tell us that s.p.a.ce has three dimensions. j.a.panese mice have only two pairs of ca.n.a.ls; they believe, it would seem, that s.p.a.ce has only two dimensions, and they manifest this opinion in the strangest way; they put themselves in a circle, and, so ordered, they spin rapidly around. The lampreys, having only one pair of ca.n.a.ls, believe that s.p.a.ce has only one dimension, but their manifestations are less turbulent.

It is evident that such a theory is inadmissible. The sense-organs are designed to tell us of _changes_ which happen in the exterior world. We could not understand why the Creator should have given us organs destined to cry without cease: Remember that s.p.a.ce has three dimensions, since the number of these three dimensions is not subject to change.

We must, therefore, come back to the theory of Mach-Delage. What the nerves of the ca.n.a.ls can tell us is the difference of pressure on the two extremities of the same ca.n.a.l, and thereby: (1) the direction of the vertical with regard to three axes rigidly bound to the head; (2) the three components of the acceleration of translation of the center of gravity of the head; (3) the centrifugal forces developed by the rotation of the head; (4) the acceleration of the motion of rotation of the head.

It follows from the experiments of M. Delage that it is this last indication which is much the most important; doubtless because the nerves are less sensible to the difference of pressure itself than to the brusque variations of this difference. The first three indications may thus be neglected.

Knowing the acceleration of the motion of rotation of the head at each instant, we deduce from it, by an unconscious integration, the final orientation of the head, referred to a certain initial orientation taken as origin. The circular ca.n.a.ls contribute, therefore, to inform us of the movements that we have executed, and that on the same ground as the muscular sensations. When, therefore, above we speak of the series _S_ or of the series [Sigma], we should say, not that these were series of muscular sensations alone, but that they were series at the same time of muscular sensations and of sensations due to the semicircular ca.n.a.ls.

Apart from this addition, we should have nothing to change in what precedes.

In the series _S_ and [Sigma], these sensations of the semicircular ca.n.a.ls evidently hold a very important place. Yet alone they would not suffice, because they can tell us only of the movements of the head; they tell us nothing of the relative movements of the body or of the members in regard to the head. And more, it seems that they tell us only of the rotations of the head and not of the translations it may undergo.

PART II

THE PHYSICAL SCIENCES

CHAPTER V

a.n.a.lYSIS AND PHYSICS

I

You have doubtless often been asked of what good is mathematics and whether these delicate constructions entirely mind-made are not artificial and born of our caprice.

Among those who put this question I should make a distinction; practical people ask of us only the means of money-making. These merit no reply; rather would it be proper to ask of them what is the good of acc.u.mulating so much wealth and whether, to get time to acquire it, we are to neglect art and science, which alone give us souls capable of enjoying it, 'and for life's sake to sacrifice all reasons for living.'

Besides, a science made solely in view of applications is impossible; truths are fecund only if bound together. If we devote ourselves solely to those truths whence we expect an immediate result, the intermediary links are wanting and there will no longer be a chain.

The men most disdainful of theory get from it, without suspecting it, their daily bread; deprived of this food, progress would quickly cease, and we should soon congeal into the immobility of old China.

But enough of uncompromising practicians! Besides these, there are those who are only interested in nature and who ask us if we can enable them to know it better.

To answer these, we have only to show them the two monuments already rough-hewn, Celestial Mechanics and Mathematical Physics.

They would doubtless concede that these structures are well worth the trouble they have cost us. But this is not enough. Mathematics has a triple aim. It must furnish an instrument for the study of nature. But that is not all: it has a philosophic aim and, I dare maintain, an esthetic aim. It must aid the philosopher to fathom the notions of number, of s.p.a.ce, of time. And above all, its adepts find therein delights a.n.a.logous to those given by painting and music. They admire the delicate harmony of numbers and forms; they marvel when a new discovery opens to them an unexpected perspective; and has not the joy they thus feel the esthetic character, even though the senses take no part therein? Only a privileged few are called to enjoy it fully, it is true, but is not this the case for all the n.o.blest arts?

This is why I do not hesitate to say that mathematics deserves to be cultivated for its own sake, and the theories inapplicable to physics as well as the others. Even if the physical aim and the esthetic aim were not united, we ought not to sacrifice either.