Experience and Geometry
Project Gutenberg EPUB3 edition
Poincaré, Henri and Halsted, George Bruce
To sum up, whichever way we look at it, it is impossible to discover in geometric empiricism a rational meaning.
6. Experiments only teach us the relations of bodies to one another; none of them bears or can bear on the relations of bodies with space, or on the mutual relations of different parts of space.
"Yes," you reply, "a single experiment is insufficient, because it gives me only a single equation with several unknowns; but when I shall have made enough experiments I shall have equations enough to calculate all my unknowns."
To know the height of the mainmast does not suffice for calculating the age of the captain. When you have measured every bit of wood in the ship you will have many equations, but you will know his age no better. All your measurements bearing only on your bits of wood can reveal to you nothing except concerning these bits of wood. Just so your experiments, however numerous they may be, bearing only on the relations of bodies to one another, will reveal to us nothing about the mutual relations of the various parts of space.
7. Will you say that if the experiments bear on the bodies, they bear at least upon the geometric properties of the bodies? But, first, what do you understand by geometric properties of the bodies? I assume that it is a question of the relations of the bodies with space; these properties are therefore inaccessible to experiments which bear only on the relations of the bodies to one another. This alone would suffice to show that there can be no question of these properties.
Still let us begin by coming to an understanding about the sense of the phrase: geometric properties of bodies. When I say a body is composed of several parts, I assume that I do not enunciate therein a geometric property, and this would remain true even if I agreed to give the improper name of points to the smallest parts I consider.
When I say that such a part of such a body is in contact with such a part of such another body, I enunciate a proposition which concerns the mutual relations of these two bodies and not their relations with space.
I suppose you will grant me these are not geometric properties; at least I am sure you will grant me these properties are independent of all knowledge of metric geometry.
This presupposed, I imagine that we have a solid body formed of eight slender iron rods, OA, OB, OC, OD, OE, OF, OG, OH, united at one of their extremities O. Let us besides have a second solid body, for example a bit of wood, to be marked with three little flecks of ink which I shall call α, β, γ. I further suppose it ascertained that αβγ may be brought into contact with AGO (I mean α with A, and at the same time β with G and γ with O), then that we may bring successively into contact αβγ with BGO, CGO, DGO, EGO, FGO, then with AHO, BHO, CHO, DHO, EHO, FHO, then αγ successively with AB, BC, CD, DE, EF, FA.
These are determinations we may make without having in advance any notion about form or about the metric properties of space. They in no wise bear on the 'geometric properties of bodies.' And these determinations will not be possible if the bodies experimented upon move in accordance with a group having the same structure as the Lobachevskian group (I mean according to the same laws as solid bodies in Lobachevski's geometry). They suffice therefore to prove that these bodies move in accordance with the Euclidean group, or at least that they do not move according to the Lobachevskian group.
That they are compatible with the Euclidean group is easy to see. For they could be made if the body αβγ was a rigid solid of our ordinary geometry presenting the form of a right-angled triangle, and if the points ABCDEFGH were the summits of a polyhedron formed of two regular hexagonal pyramids of our ordinary geometry, having for common base ABCDEF and for apices the one G and the other H.
Suppose now that in place of the preceding determination it is observed that as above αβγ can be successively applied to AGO, BGO, CGO, DGO, EGO, AHO, BHO, CHO, DHO, EHO, FHO, then that αβ (and no longer αγ) can be successively applied to AB, BC, CD, DE, EF and FA.
These are determinations which could be made if non-Euclidean geometry were true, if the bodies αβγ and OABCDEFGH were rigid solids, and if the first were a right-angled triangle and the second a double regular hexagonal pyramid of suitable dimensions.
Therefore these new determinations are not possible if the bodies move according to the Euclidean group; but they become so if it be supposed that the bodies move according to the Lobachevskian group. They would suffice, therefore (if one made them), to prove that the bodies in question do not move according to the Euclidean group.
Thus, without making any hypothesis about form, about the nature of space, about the relations of bodies to space, and without attributing to bodies any geometric property, I have made observations which have enabled me to show in one case that the bodies experimented upon move according to a group whose structure is Euclidean, in the other case that they move according to a group whose structure is Lobachevskian.
And one may not say that the first aggregate of determinations would constitute an experiment proving that space is Euclidean, and the second an experiment proving that space is non-Euclidean.
In fact one could imagine (I say imagine) bodies moving so as to render possible the second series of determinations. And the proof is that the first mechanician met could construct such bodies if he cared to take the pains and make the outlay. You will not conclude from that, however, that space is non-Euclidean.
Nay, since the ordinary solid bodies would continue to exist when the mechanician had constructed the strange bodies of which I have just spoken, it would be necessary to conclude that space is at the same time Euclidean and non-Euclidean.
Suppose, for example, that we have a great sphere of radius R and that the temperature decreases from the center to the surface of this sphere according to the law of which I have spoken in describing the non-Euclidean world.
We might have bodies whose expansion would be negligible and which would act like ordinary rigid solids; and, on the other hand, bodies very dilatable and which would act like non-Euclidean solids. We might have two double pyramids OABCDEFGH and O´A´B´C´D´E´F´G´H´ and two triangles αβγ and α´β´γ´. The first double pyramid might be rectilinear and the second curvilinear; the triangle αβγ might be made of inexpansible matter and the other of a very dilatable matter.
It would then be possible to make the first observations with the double pyramid OAH and the triangle αβγ, and the second with the double pyramid O´A´H´ and the triangle α´β´γ´. And then experiment would seem to prove first that the Euclidean geometry is true and then that it is false.
Experiments therefore have a bearing, not on space, but on bodies.
Supplement
8. To complete the matter, I ought to speak of a very delicate question, which would require long development; I shall confine myself to summarizing here what I have expounded in the Revue de Métaphysique et de Morale and in The Monist. When we say space has three dimensions, what do we mean?
We have seen the importance of those 'internal changes' revealed to us by our muscular sensations. They may serve to characterize the various attitudes of our body. Take arbitrarily as origin one of these attitudes A. When we pass from this initial attitude to any other attitude B, we feel a series of muscular sensations, and this series S will define B. Observe, however, that we shall often regard two series S and S´ as defining the same attitude B (since the initial and final attitudes A and B remaining the same, the intermediary attitudes and the corresponding sensations may differ). How then shall we recognize the equivalence of these two series? Because they may serve to compensate the same external change, or more generally because, when it is a question of compensating an external change, one of the series can be replaced by the other. Among these series, we have distinguished those which of themselves alone can compensate an external change, and which we have called 'displacements.' As we can not discriminate between two displacements which are too close together, the totality of these displacements presents the characteristics of a physical continuum; experience teaches us that they are those of a physical continuum of six dimensions; but we do not yet know how many dimensions space itself has, we must first solve another question.
What is a point of space? Everybody thinks he knows, but that is an illusion. What we see when we try to represent to ourselves a point of space is a black speck on white paper, a speck of chalk on a blackboard, always an object. The question should therefore be understood as follows:
What do I mean when I say the object B is at the same point that the object A occupied just now? Or further, what criterion will enable me to apprehend this?
I mean that, although I have not budged (which my muscular sense tells me), my first finger which just now touched the object A touches at present the object B. I could have used other criteria; for instance another finger or the sense of sight. But the first criterion is sufficient; I know that if it answers yes, all the other criteria will give the same response. I know it by experience, I can not know it a priori. For the same reason I say that touch can not be exercised at a distance; this is another way of enunciating the same experimental fact. And if, on the contrary, I say that sight acts at a distance, it means that the criterion furnished by sight may respond yes while the others reply no.
And in fact, the object, although moved away, may form its image at the same point of the retina. Sight responds yes, the object has remained at the same point and touch answers no, because my finger which just now touched the object, at present touches it no longer. If experience had shown us that one finger may respond no when the other says yes, we should likewise say that touch acts at a distance.
In short, for each attitude of my body, my first finger determines a point, and this it is, and this alone, which defines a point of space.
To each attitude corresponds thus a point; but it often happens that the same point corresponds to several different attitudes (in this case we say our finger has not budged, but the rest of the body has moved). We distinguish, therefore, among the changes of attitude those where the finger does not budge. How are we led thereto? It is because often we notice that in these changes the object which is in contact with the finger remains in contact with it.
Range, therefore, in the same class all the attitudes obtainable from each other by one of the changes we have thus distinguished. To all the attitudes of the class will correspond the same point of space. Therefore to each class will correspond a point and to each point a class. But one may say that what experience arrives at is not the point, it is this class of changes or, better, the corresponding class of muscular sensations.
And when we say space has three dimensions, we simply mean that the totality of these classes appears to us with the characteristics of a physical continuum of three dimensions.
One might be tempted to conclude that it is experience which has taught us how many dimensions space has. But in reality here also our experiences have bearing, not on space, but on our body and its relations with the neighboring objects. Moreover they are excessively crude.
In our mind pre-existed the latent idea of a certain number of groups—those whose theory Lie has developed. Which group shall we choose, to make of it a sort of standard with which to compare natural phenomena? And, this group chosen, which of its sub-groups shall we take to characterize a point of space? Experience has guided us by showing us which choice best adapts itself to the properties of our body. But its rôle is limited to that.