# EXPERIMENT AND GEOMETRY

February 1, 2022
1. The principles of geometry are not experimental facts. Euclid’s postulate cannot be proved by experiment.
1. Think of a material circle, measure its radius and circumference, and see if the ratio of the two lengths is equal to π. What have we done?

We have made an experiment on the properties of the matter with which this roundness has been realised, and of which the measure we used is made.

1. Geometry and Astronomy

The same question may also be asked in another way. If Lobatschewsky’s geometry is true, the parallax of a very distant star will be finite. If Riemann’s is true, it will be negative. These are the results which seem within the reach of exper- iment, and it is hoped that astronomical observations may enable us to decide between the three geometries.

But what we call a straight line in astronomy is simply the path of a ray of light. If, therefore, we were to discover negative parallaxes, or to prove that all parallaxes are higher than a certain limit, we should have a choice between two conclusions: we could give up Euclidean geometry, or modify the laws of optics, and suppose that light is not rigorously propagated in a straight line.

It is needless to add that every one would look upon this solution as the more advantageous. Euclidean geometry, therefore, has nothing to fear from fresh experiments.

1. Can we maintain that certain phenomena which are possible in Euclidean space would be impossible in non-Euclidean space, so that experiment in establishing these phenomena would directly contradict the non-Euclidean hypothesis?

I think that such a question cannot be seriously asked. To me it is exactly equivalent to the following, the absurdity of which is obvious:—There are lengths which can be expressed in metres and cen- timetres, but cannot be measured in toises, feet, and inches; so that experiment, by ascertaining the existence of these lengths, would directly contradict this hypothe- sis, that there are toises divided into six feet. Let us look at the question a little more closely. I assume that the straight line in Euclidean space possesses any two prop- erties, which I shall call A and B; that in non-Euclideanexperiment and geometry.

1. Will you say that if the experiments have refer- ence to the bodies, they at least have reference to the geometrical properties of the bodies. First, what do you understand by the geometrical properties of bodies? I assume that it is a question of the relations of the bodies to space. These properties therefore are not reached by experiments which only have reference to the relations of bodies to one another, and that is enough to show that it is not of those properties that there can be a ques- tion. Let us therefore begin by making ourselves clear as to the sense of the phrase: geometrical properties of bodies. When I say that a body is composed of several parts, I presume that I am thus enunciating a geometri- cal property, and that will be true even if I agree to give the improper name of points to the very small parts I am considering. When I say that this or that part of a cer- tain body is in contact with this or that part of another body, I am enunciating a proposition which concerns the mutual relations of the two bodies, and not their rela-experiment and geometry.

tions with space. I assume that you will agree with me that these are not geometrical properties. I am sure that at least you will grant that these properties are indepen- dent of all knowledge of metrical geometry. Admitting this, I suppose that we have a solid body formed of eight thin iron rods, oa, ob, oc, od, oe, of , og, oh, connected at one of their extremities, o. And let us take a second solid body—for example, a piece of wood, on which are marked three little spots of ink which I shall call α β γ. I now suppose that we find that we can bring into con- tact αβγ with ago; by that I mean α with a, and at the same time β with g, and γ with o. Then we can suc- cessively bring into contact αβγ with bgo, cgo, dgo, ego, f go, then with aho, bho, cho, dho, eho, f ho; and then αγ successively with ab, bc, cd, de, ef , f a. Now these are observations that can be made without having any idea beforehand as to the form or the metrical properties of space. They have no reference whatever to the “geometri- cal properties of bodies.” These observations will not be possible if the bodies on which we experiment move in a group having the same structure as the Lobatschewskian group (I mean according to the same laws as solid bodies in Lobatschewsky’s geometry). They therefore suffice to prove that these bodies move according to the Euclideanscience and hypothesis

group; or at least that they do not move according to the Lobatschewskian group. That they may be compati- ble with the Euclidean group is easily seen; for we might make them so if the body αβγ were an invariable solid of our ordinary geometry in the shape of a right-angled triangle, and if the points abcdef gh were the vertices of a polyhedron formed of two regular hexagonal pyramids of our ordinary geometry having abcdef as their common base, and having the one g and the other h as their ver- tices. Suppose now, instead of the previous observations, we note that we can as before apply αβγ successively to ago, bgo, cgo, dgo, ego, f go, aho, bho, cho, dho, eho, f ho, and then that we can apply αβ (and no longer αγ) successively to ab, bc, cd, de, ef , and f a. These are ob- servations that could be made if non-Euclidean geometry were true. If the bodies αβγ, oabcdef gh were invariable solids, if the former were a right-angled triangle, and the latter a double regular hexagonal pyramid of suit- able dimensions. These new verifications are therefore impossible if the bodies move according to the Euclidean group; but they become possible if we suppose the bodies to move according to the Lobatschewskian group. They would therefore suffice to show, if we carried them out, that the bodies in question do not move according to theexperiment and geometry.

Euclidean group. And so, without making any hypothe- sis on the form and the nature of space, on the relations of the bodies and space, and without attributing to bod- ies any geometrical property, I have made observations which have enabled me to show in one case that the bod- ies experimented upon move according to a group, the structure of which is Euclidean, and in the other case, that they move in a group, the structure of which is Lo- batschewskian. It cannot be said that all the first ob- servations would constitute an experiment proving that space is Euclidean, and the second an experiment proving that space is non-Euclidean; in fact, it might be imagined (note that I use the word imagined ) that there are bod- ies moving in such a manner as to render possible the second series of observations: and the proof is that the first mechanic who came our way could construct it if he would only take the trouble. But you must not conclude, however, that space is non-Euclidean. In the same way, just as ordinary solid bodies would continue to exist when the mechanic had constructed the strange bodies I have just mentioned, he would have to conclude that space is both Euclidean and non-Euclidean. Suppose, for in- stance, that we have a large sphere of radius R, and that its temperature decreases from the centre to the surfacescience and hypothesis 96 of the sphere according to the law of which I spoke when I was describing the non-Euclidean world. We might have bodies whose dilatation is negligible, and which would behave as ordinary invariable solids; and, on the other hand, we might have very dilatable bodies, which would behave as non-Euclidean solids. We might have two dou- ble pyramids oabcdef gh and o 0 a 0 b 0 c 0 d 0 e 0 f 0 g 0 h 0 , and two tri- angles αβγ and α 0 β 0 γ 0 . The first double pyramid would be rectilinear, and the second curvilinear. The trian- gle αβγ would consist of undilatable matter, and the other of very dilatable matter. We might therefore make our first observations with the double pyramid o 0 a 0 h 0 and the triangle α 0 β 0 γ 0 . And then the experiment would seem to show—first, that Euclidean geometry is true, and then that it is false. Hence, experiments have reference not to space but to bodies. supplement.

1. To round the matter off, I ought to speak of a very delicate question, which will require considerable devel- opment; but I shall confine myself to summing up what I have written in the Revue de Métaphysique et de Morale and in the Monist. When we say that space has threeexperiment and geometry.

dimensions, what do we mean? We have seen the impor- tance of these “internal changes” which are revealed to us by our muscular sensations. They may serve to char- acterise the different attitudes of our body. Let us take arbitrarily as our origin one of these attitudes, A. When we pass from this initial attitude to another attitude B we experience a series of muscular sensations, and this series S of muscular sensations will define B. Observe, however, that we shall often look upon two series S and S 0 as defining the same attitude B (since the initial and fi- nal attitudes A and B remaining the same, the intermedi- ary attitudes of the corresponding sensations may differ). How then can we recognise the equivalence of these two series? Because they may serve to compensate for the same external change, or more generally, because, when it is a question of compensation for an external change, one of the series may be replaced by the other. Among these series we have distinguished those which can alone compensate for an external change, and which we have called “displacements.” As we cannot distinguish two dis- placements which are very close together, the aggregate of these displacements presents the characteristics of a physical continuum. Experience teaches us that they are the characteristics of a physical continuum of six dimen-science and hypothesis

sions; but we do not know as yet how many dimensions space itself possesses, so we must first of all answer an- other question. What is a point in space? Every one thinks he knows, but that is an illusion. What we see when we try to represent to ourselves a point in space is a black spot on white paper, a spot of chalk on a black- board, always an object. The question should therefore be understood as follows:—What do I mean when I say the object B is at the point which a moment before was occupied by the object A? Again, what criterion will enable me to recognise it? I mean that although I have not moved (my muscular sense tells me this), my finger, which just now touched the object A, is now touching the object B. I might have used other criteria—for in- stance, another finger or the sense of sight—but the first criterion is sufficient. I know that if it answers in the affirmative all other criteria will give the same answer. I know it from experiment. I cannot know it à priori. For the same reason I say that touch cannot be exercised at a distance; that is another way of enunciating the same experimental fact. If I say, on the contrary, that sight is exercised at a distance, it means that the criterion fur- nished by sight may give an affirmative answer while the others reply in the negative.experiment and geometry.

To sum up. For each attitude of my body my finger determines a point, and it is that and that only which defines a point in space. To each attitude corresponds in this way a point. But it often happens that the same point corresponds to several different attitudes (in this case we say that our finger has not moved, but the rest of our body has). We distinguish, therefore, among changes of attitude those in which the finger does not move. How are we led to this? It is because we often remark that in these changes the object which is in touch with the finger remains in contact with it. Let us arrange then in the same class all the attitudes which are deduced one from the other by one of the changes that we have thus dis- tinguished. To all these attitudes of the same class will correspond the same point in space. Then to each class will correspond a point, and to each point a class. Yet it may be said that what we get from this experiment is not the point, but the class of changes, or, better still, the corresponding class of muscular sensations. Thus, when we say that space has three dimensions, we merely mean that the aggregate of these classes appears to us with the characteristics of a physical continuum of three dimen- sions. Then if, instead of defining the points in space with the aid of the first finger, I use, for example, another finger, would the results be the same?

That is by no means à priori evident. But, as we have seen, experi- ment has shown us that all our criteria are in agreement, and this enables us to answer in the affirmative. If we recur to what we have called displacements, the aggre- gate of which forms, as we have seen, a group, we shall be brought to distinguish those in which a finger does not move; and by what has preceded, those are the dis- placements which characterise a point in space, and their aggregate will form a sub-group of our group. To each sub-group of this kind, then, will correspond a point in space. We might be tempted to conclude that experiment has taught us the number of dimensions of space; but in reality our experiments have referred not to space, but to our body and its relations with neighbouring objects. What is more, our experiments are exceeding crude. In our mind the latent idea of a certain number of groups pre-existed; these are the groups with which Lie’s theory is concerned.

Which shall we choose to form a kind of standard by which to compare natural phenomena?

When this group is chosen, which of the sub-groups shall we take to characterise a point in space?

Experiment has guided us by showing us what choice adapts itself best to the properties of our body; but there its rôle ends.