Experiment And GeometryFebruary 2, 2022
- The principles of geometry are not experimental facts. Euclid’s postulate cannot be proved by experiment.
- Think of a material circle, measure its radius and circumference, and see if the ratio of the two lengths is equal to
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.
- 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.
- 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.
space it still possesses the property A, but no longer possesses the property B; and, finally, I assume that in both Euclidean and non-Euclidean space the straight line is the only line that possesses the property A. If this were so, experiment would be able to decide be- tween the hypotheses of Euclid and Lobatschewsky. It would be found that some concrete object, upon which we can experiment—for example, a pencil of rays of light— possesses the property A. We should conclude that it is rectilinear, and we should then endeavour to find out if it does, or does not, possess the property B. But it is not so. There exists no property which can, like this prop- erty A, be an absolute criterion enabling us to recognise the straight line, and to distinguish it from every other line. Shall we say, for instance, “This property will be the following: the straight line is a line such that a figure of which this line is a part can move without the mu- tual distances of its points varying, and in such a way that all the points in this straight line remain fixed”? Now, this is a property which in either Euclidean or non- Euclidean space belongs to the straight line, and belongs to it alone. But how can we ascertain by experiment if it belongs to any particular concrete object? Distances must be measured, and how shall we know that any concrete magnitude which I have measured with my material instrument really represents the abstract distance?
We have only removed the difficulty a little farther off.
In reality, the property that I have just enunciated is not a property of the straight line alone; it is a property of the straight line and of distance. For it to serve as an abso- lute criterion, we must be able to show, not only that it does not also belong to any other line than the straight line and to distance, but also that it does not belong to any other line than the straight line, and to any other magnitude than distance.
Now, that is not true, and if we are not convinced by these considerations, I chal- lenge any one to give me a concrete experiment which can be interpreted in the Euclidean system, and which cannot be interpreted in the system of Lobatschewsky. As I am well aware that this challenge will never be ac- cepted, I may conclude that no experiment will ever be in contradiction with Euclid’s postulate; but, on the other hand, no experiment will ever be in contradiction with Lobatschewsky’s postulate.
- But it is not sufficient that the Euclidean (or non- Euclidean) geometry can ever be directly contradicted by experiment. Nor could it happen that it can only agree with experiment by a violation of the principle of sufficient reason, and of that of the relativity of space. Let me explain myself. Consider any material system whatever.
We have to consider on the one hand the “state” of the various bodies of this system—for example, their temper- ature, their electric potential, etc.; and on the other hand their position in space. And among the data which enable us to define this position we distinguish the mutual dis- tances of these bodies that define their relative positions, and the conditions which define the absolute position of the system and its absolute orientation in space. The law of the phenomena which will be produced in this system will depend on the state of these bodies, and on their mutual distances; but because of the relativity and the inertia of space, they will not depend on the absolute position and orientation of the system. In other words, the state of the bodies and their mutual distances at any moment will solely depend on the state of the same bodies and on their mutual distances at the initial moment, but will in no way depend on the absolute initial position of the system and of its absolute initial orientation.
This is what we shall call, for the sake of abbreviation, the law of relativity.
So far I have spoken as a Euclidean geometer. But I have said that an experiment, whatever it may be, requires an interpretation on the Euclidean hypothesis; it equally requires one on the non-Euclidean hypothesis. Well, we have made a series of experiments.
We have interpreted them on the Euclidean hypothesis, and we have recognised that these experiments thus interpreted do not violate this “law of relativity.” We now interpret them on the non-Euclidean hypothesis. This is always possible, only the non-Euclidean distances of our different bodies in this new interpretation will not generally be the same as the Euclidean distances in the primitive interpretation. Will our experiment interpreted in this new manner be still in agreement with our “law of relativity,” and if this agreement had not taken place, would we not still have the right to say that experiment has proved the falsity of non-Euclidean geometry? It is easy to see that this is an idle fear.
In fact, to apply the law of relativity in all its rigour, it must be applied to the en- tire universe; for if we were to consider only a part of the universe, and if the absolute position of this part were to vary, the distances of the other bodies of the universe would equally vary; their influence on the part of the universe considered might therefore increase or dimin- ish, and this might modify the laws of the phenomena which take place in it. But if our system is the entireexperiment and geometry.
universe, experiment is powerless to give us any opinion on its position and its absolute orientation in space. All that our instruments, however perfect they may be, can let us know will be the state of the different parts of the universe, and their mutual distances.
Hence, our law of relativity may be enunciated as follows:—The readings that we can make with our instruments at any given mo- ment will depend only on the readings that we were able to make on the same instruments at the initial moment.
Now such an enunciation is independent of all interpretation by experiments. If the law is true in the Euclidean interpretation, it will be also true in the non-Euclidean interpretation.
Allow me to make a short digression on this point.
I have spoken above of the data which define the position of the different bodies of the system. I might also have spoken of those which define their velocities. I should then have to distinguish the velocity with which the mutual distances of the different bodies are chang- ing, and on the other hand the velocities of translation and rotation of the system; that is to say, the veloci- ties with which its absolute position and orientation are changing. For the mind to be fully satisfied, the law of relativity would have to be enunciated as follows:
The state of bodies and their mutual distances at any given moment, as well as the velocities with which those dis- tances are changing at that moment, will depend only on the state of those bodies, on their mutual distances at the initial moment, and on the velocities with which those distances were changing at the initial moment. But they will not depend on the absolute initial position of the system nor on its absolute orientation, nor on the ve- locities with which that absolute position and orientation were changing at the initial moment. Unfortunately, the law thus enunciated does not agree with experiments—at least, as they are ordinarily interpreted. Suppose a man were translated to a planet, the sky of which was con- stantly covered with a thick curtain of clouds, so that he could never see the other stars. On that planet he would live as if it were isolated in space. But he would notice that it revolves, either by measuring its ellipticity (which is ordinarily done by means of astronomical observations, but which could be done by purely geodesic means), or by repeating the experiment of Foucault’s pendulum. The absolute rotation of this planet might be clearly shown in this way. Now, here is a fact which shocks the philoso- pher, but which the physicist is compelled to accept. We know that from this fact Newton concluded the existence of absolute space. I myself cannot accept this way of looking at it.
I shall explain why in Part 3, but for the moment it is not my intention to discuss this difficulty. I must therefore resign myself, in the enunciation of the law of relativity, to including velocities of every kind among the data which define the state of the bodies. However that may be, the difficulty is the same for both Euclid’s geometry and for Lobatschewsky’s.
I need not therefore trouble about it further, and I have only mentioned it in- cidentally. To sum up, whichever way we look at it, it is impossible to discover in geometric empiricism a rational meaning.