Part II Space
Chapter III
The Non-euclidean Geometries
Every conclusion supposes premises; these premises themselves either are self-evident and need no demonstration, or can be established only by relying upon other propositions, and since we can not go back thus to infinity, every deductive science, and in particular geometry, must rest on a certain number of undemonstrable axioms. All treatises on geometry begin, therefore, by the enunciation of these axioms. But among these there is a distinction to be made: Some, for example, ‘Things which are equal to the same thing are equal to one another,’ are not propositions of geometry, but propositions of analysis. I regard them as analytic judgments a priori, and shall not concern myself with them.
But I must lay stress upon other axioms which are peculiar to geometry. Most treatises enunciate three of these explicitly:
1º Through two points can pass only one straight;
2º The straight line is the shortest path from one point to another;
3º Through a given point there is not more than one parallel to a given straight.
Although generally a proof of the second of these axioms is omitted, it would be possible to deduce it from the other two and from those, much more numerous, which are implicitly admitted without enunciating them, as I shall explain further on.
It was long sought in vain to demonstrate likewise the third axiom, known as Euclid’s Postulate. What vast effort has been wasted in this chimeric hope is truly unimaginable. Finally, in the first quarter of the nineteenth century, and almost at the same time, a Hungarian and a Russian, Bolyai and Lobachevski, established irrefutably that this demonstration is impossible; they have almost rid us of inventors of geometries ‘sans postulatum’; since then the Académie des Sciences receives only about one or two new demonstrations a year.
The question was not exhausted; it soon made a great stride by the publication of Riemann’s celebrated memoir entitled: Ueber die Hypothesen welche der Geometrie zu Grunde liegen. This paper has inspired most of the recent works of which I shall speak further on, and among which it is proper to cite those of Beltrami and of Helmholtz.
The Bolyai-Lobachevski Geometry.—If it were possible to deduce Euclid’s postulate from the other axioms, it is evident that in denying the postulate and admitting the other axioms, we should be led to contradictory consequences; it would therefore be impossible to base on such premises a coherent geometry.
Now this is precisely what Lobachevski did.
He assumes at the start that: Through a given point can be drawn two parallels to a given straight.
And he retains besides all Euclid’s other axioms. From these hypotheses he deduces a series of theorems among which it is impossible to find any contradiction, and he constructs a geometry whose faultless logic is inferior in nothing to that of the Euclidean geometry.
The theorems are, of course, very different from those to which we are accustomed, and they can not fail to be at first a little disconcerting.
Thus the sum of the angles of a triangle is always less than two right angles, and the difference between this sum and two right angles is proportional to the surface of the triangle.
It is impossible to construct a figure similar to a given figure but of different dimensions.
If we divide a circumference into n equal parts, and draw tangents at the points of division, these n tangents will form a polygon if the radius of the circle is small enough; but if this radius is sufficiently great they will not meet.
It is useless to multiply these examples; Lobachevski’s propositions have no relation to those of Euclid, but they are not less logically bound one to another.
Riemann’s Geometry.—Imagine a world uniquely peopled by beings of no thickness (height); and suppose these ‘infinitely flat’ animals are all in the same plane and can not get out. Admit besides that this world is sufficiently far from others to be free from their influence. While we are making hypotheses, it costs us no more to endow these beings with reason and believe them capable of creating a geometry. In that case, they will certainly attribute to space only two dimensions.
But suppose now that these imaginary animals, while remaining without thickness, have the form of a spherical, and not of a plane, figure, and are all on the same sphere without power to get off. What geometry will they construct? First it is clear they will attribute to space only two dimensions; what will play for them the rôle of the straight line will be the shortest path from one point to another on the sphere, that is to say, an arc of a great circle; in a word, their geometry will be the spherical geometry.
What they will call space will be this sphere on which they must stay, and on which happen all the phenomena they can know. Their space will therefore be unbounded since on a sphere one can always go forward without ever being stopped, and yet it will be finite; one can never find the end of it, but one can make a tour of it.
Well, Riemann’s geometry is spherical geometry extended to three dimensions. To construct it, the German mathematician had to throw overboard, not only Euclid’s postulate, but also the first axiom: Only one straight can pass through two points.
On a sphere, through two given points we can draw in general only one great circle (which, as we have just seen, would play the rôle of the straight for our imaginary beings); but there is an exception: if the two given points are diametrically opposite, an infinity of great circles can be drawn through them.
In the same way, in Riemann’s geometry (at least in one of its forms), through two points will pass in general only a single straight; but there are exceptional cases where through two points an infinity of straights can pass.
There is a sort of opposition between Riemann’s geometry and that of Lobachevski.
Thus the sum of the angles of a triangle is:
Equal to two right angles in Euclid’s geometry;
Less than two right angles in that of Lobachevski;
Greater than two right angles in that of Riemann.
The number of straights through a given point that can be drawn coplanar to a given straight, but nowhere meeting it, is equal:
To one in Euclid’s geometry;
To zero in that of Riemann;
To infinity in that of Lobachevski.
Add that Riemann’s space is finite, although unbounded, in the sense given above to these two words.
The Surfaces of Constant Curvature.—One objection still remained possible. The theorems of Lobachevski and of Riemann present no contradiction; but however numerous the consequences these two geometers have drawn from their hypotheses, they must have stopped before exhausting them, since their number would be infinite; who can say then that if they had pushed their deductions farther they would not have eventually reached some contradiction?
This difficulty does not exist for Riemann’s geometry, provided it is limited to two dimensions; in fact, as we have seen, two-dimensional Riemannian geometry does not differ from spherical geometry, which is only a branch of ordinary geometry, and consequently is beyond all discussion.
Beltrami, in correlating likewise Lobachevski’s two-dimensional geometry with a branch of ordinary geometry, has equally refuted the objection so far as it is concerned.
Here is how he accomplished it. Consider any figure on a surface. Imagine this figure traced on a flexible and inextensible canvas applied over this surface in such a way that when the canvas is displaced and deformed, the various lines of this figure can change their form without changing their length. In general, this flexible and inextensible figure can not be displaced without leaving the surface; but there are certain particular surfaces for which such a movement would be possible; these are the surfaces of constant curvature.
If we resume the comparison made above and imagine beings without thickness living on one of these surfaces, they will regard as possible the motion of a figure all of whose lines remain constant in length. On the contrary, such a movement would appear absurd to animals without thickness living on a surface of variable curvature.
These surfaces of constant curvature are of two sorts: Some are of positive curvature, and can be deformed so as to be applied over a sphere. The geometry of these surfaces reduces itself therefore to the spherical geometry, which is that of Riemann.
The others are of negative curvature. Beltrami has shown that the geometry of these surfaces is none other than that of Lobachevski. The two-dimensional geometries of Riemann and Lobachevski are thus correlated to the Euclidean geometry.
Interpretation of Non-Euclidean Geometries.—So vanishes the objection so far as two-dimensional geometries are concerned.
It would be easy to extend Beltrami’s reasoning to three-dimensional geometries. The minds that space of four dimensions does not repel will see no difficulty in it, but they are few. I prefer therefore to proceed otherwise.
Consider a certain plane, which I shall call the fundamental plane, and construct a sort of dictionary, by making correspond each to each a double series of terms written in two columns, just as correspond in the ordinary dictionaries the words of two languages whose significance is the same:
Space: Portion of space situated above the fundamental plane.
Plane: Sphere cutting the fundamental plane orthogonally.
Straight: Circle cutting the fundamental plane orthogonally.
Sphere: Sphere.
Circle: Circle.
Angle: Angle.
Distance between two points: Logarithm of the cross ratio of these two points and the intersections of the fundamental plane with a circle passing through these two points and cutting it orthogonally. Etc., Etc.
Now take Lobachevski’s theorems and translate them with the aid of this dictionary as we translate a German text with the aid of a German-English dictionary. We shall thus obtain theorems of the ordinary geometry. For example, that theorem of Lobachevski: ‘the sum of the angles of a triangle is less than two right angles’ is translated thus: “If a curvilinear triangle has for sides circle-arcs which prolonged would cut orthogonally the fundamental plane, the sum of the angles of this curvilinear triangle will be less than two right angles.” Thus, however far the consequences of Lobachevski’s hypotheses are pushed, they will never lead to a contradiction. In fact, if two of Lobachevski’s theorems were contradictory, it would be the same with the translations of these two theorems, made by the aid of our dictionary, but these translations are theorems of ordinary geometry and no one doubts that the ordinary geometry is free from contradiction. Whence comes this certainty and is it justified? That is a question I can not treat here because it would require to be enlarged upon, but which is very interesting and I think not insoluble.
Nothing remains then of the objection above formulated. This is not all. Lobachevski’s geometry, susceptible of a concrete interpretation, ceases to be a vain logical exercise and is capable of applications; I have not the time to speak here of these applications, nor of the aid that Klein and I have gotten from them for the integration of linear differential equations.
This interpretation moreover is not unique, and several dictionaries analogous to the preceding could be constructed, which would enable us by a simple ‘translation’ to transform Lobachevski’s theorems into theorems of ordinary geometry.
The Implicit Axioms.—Are the axioms explicitly enunciated in our treatises the sole foundations of geometry? We may be assured of the contrary by noticing that after they are successively abandoned there are still left over some propositions common to the theories of Euclid, Lobachevski and Riemann. These propositions must rest on premises the geometers admit without enunciation. It is interesting to try to disentangle them from the classic demonstrations.
Stuart Mill has claimed that every definition contains an axiom, because in defining one affirms implicitly the existence of the object defined. This is going much too far; it is rare that in mathematics a definition is given without its being followed by the demonstration of the existence of the object defined, and when this is dispensed with it is generally because the reader can easily supply it. It must not be forgotten that the word existence has not the same sense when it refers to a mathematical entity and when it is a question of a material object. A mathematical entity exists, provided its definition implies no contradiction, either in itself, or with the propositions already admitted.
But if Stuart Mill’s observation can not be applied to all definitions, it is none the less just for some of them. The plane is sometimes defined as follows:
The plane is a surface such that the straight which joins any two of its points is wholly on this surface.
This definition manifestly hides a new axiom; it is true we might change it, and that would be preferable, but then we should have to enunciate the axiom explicitly.
Other definitions would suggest reflections not less important.
Such, for example, is that of the equality of two figures; two figures are equal when they can be superposed; to superpose them one must be displaced until it coincides with the other; but how shall it be displaced? If we should ask this, no doubt we should be told that it must be done without altering the shape and as a rigid solid. The vicious circle would then be evident.
In fact this definition defines nothing; it would have no meaning for a being living in a world where there were only fluids. If it seems clear to us, that is because we are used to the properties of natural solids which do not differ much from those of the ideal solids, all of whose dimensions are invariable.
Yet, imperfect as it may be, this definition implies an axiom.
The possibility of the motion of a rigid figure is not a self-evident truth, or at least it is so only in the fashion of Euclid’s postulate and not as an analytic judgment a priori would be.
Moreover, in studying the definitions and the demonstrations of geometry, we see that one is obliged to admit without proof not only the possibility of this motion, but some of its properties besides.
This is at once seen from the definition of the straight line. Many defective definitions have been given, but the true one is that which is implied in all the demonstrations where the straight line enters:
“It may happen that the motion of a rigid figure is such that all the points of a line belonging to this figure remain motionless while all the points situated outside of this line move. Such a line will be called a straight line.” We have designedly, in this enunciation, separated the definition from the axiom it implies.
Many demonstrations, such as those of the cases of the equality of triangles, of the possibility of dropping a perpendicular from a point to a straight, presume propositions which are not enunciated, for they require the admission that it is possible to transport a figure in a certain way in space.
The Fourth Geometry.—Among these implicit axioms, there is one which seems to me to merit some attention, because when it is abandoned a fourth geometry can be constructed as coherent as those of Euclid, Lobachevski and Riemann.
To prove that a perpendicular may always be erected at a point A to a straight AB, we consider a straight AC movable around the point A and initially coincident with the fixed straight AB; and we make it turn about the point A until it comes into the prolongation of AB.
Thus two propositions are presupposed: First, that such a rotation is possible, and next that it may be continued until the two straights come into the prolongation one of the other.
If the first point is admitted and the second rejected, we are led to a series of theorems even stranger than those of Lobachevski and Riemann, but equally exempt from contradiction.
I shall cite only one of these theorems and that not the most singular: A real straight may be perpendicular to itself.
Lie’s Theorem.—The number of axioms implicitly introduced in the classic demonstrations is greater than necessary, and it would be interesting to reduce it to a minimum. It may first be asked whether this reduction is possible, whether the number of necessary axioms and that of imaginable geometries are not infinite.
A theorem of Sophus Lie dominates this whole discussion. It may be thus enunciated:
Suppose the following premises are admitted:
1º Space has n dimensions;
2º The motion of a rigid figure is possible;
3º It requires p conditions to determine the position of this figure in space.
The number of geometries compatible with these premises will be limited.
I may even add that if n is given, a superior limit can be assigned to p.
If therefore the possibility of motion is admitted, there can be invented only a finite (and even a rather small) number of three-dimensional geometries.
Riemann’s Geometries.—Yet this result seems contradicted by Riemann, for this savant constructs an infinity of different geometries, and that to which his name is ordinarily given is only a particular case.
All depends, he says, on how the length of a curve is defined. Now, there is an infinity of ways of defining this length, and each of them may be the starting point of a new geometry.
That is perfectly true, but most of these definitions are incompatible with the motion of a rigid figure, which in the theorem of Lie is supposed possible. These geometries of Riemann, in many ways so interesting, could never therefore be other than purely analytic and would not lend themselves to demonstrations analogous to those of Euclid.
On the Nature of Axioms.—Most mathematicians regard Lobachevski’s geometry only as a mere logical curiosity; some of them, however, have gone farther. Since several geometries are possible, is it certain ours is the true one? Experience no doubt teaches us that the sum of the angles of a triangle is equal to two right angles; but this is because the triangles we deal with are too little; the difference, according to Lobachevski, is proportional to the surface of the triangle; will it not perhaps become sensible when we shall operate on larger triangles or when our measurements shall become more precise? The Euclidean geometry would thus be only a provisional geometry.
To discuss this opinion, we should first ask ourselves what is the nature of the geometric axioms.
Are they synthetic a priori judgments, as Kant said?
They would then impose themselves upon us with such force that we could not conceive the contrary proposition, nor build upon it a theoretic edifice. There would be no non-Euclidean geometry.
To be convinced of it take a veritable synthetic a priori judgment, the following, for instance, of which we have seen the preponderant rôle in the first chapter:
If a theorem is true for the number 1, and if it has been proved that it is true of n + 1 provided it is true of n, it will be true of all the positive whole numbers.
Then try to escape from that and, denying this proposition, try to found a false arithmetic analogous to non-Euclidean geometry—it can not be done; one would even be tempted at first blush to regard these judgments as analytic.
Moreover, resuming our fiction of animals without thickness, we can hardly admit that these beings, if their minds are like ours, would adopt the Euclidean geometry which would be contradicted by all their experience.
Should we therefore conclude that the axioms of geometry are experimental verities? But we do not experiment on ideal straights or circles; it can only be done on material objects. On what then could be based experiments which should serve as foundation for geometry? The answer is easy.
We have seen above that we constantly reason as if the geometric figures behaved like solids. What geometry would borrow from experience would therefore be the properties of these bodies. The properties of light and its rectilinear propagation have also given rise to some of the propositions of geometry, and in particular those of projective geometry, so that from this point of view one would be tempted to say that metric geometry is the study of solids, and projective, that of light.
But a difficulty remains, and it is insurmountable. If geometry were an experimental science, it would not be an exact science, it would be subject to a continual revision. Nay, it would from this very day be convicted of error, since we know that there is no rigorously rigid solid.
The axioms of geometry therefore are neither synthetic a priori judgments nor experimental facts.
They are conventions; our choice among all possible conventions is guided by experimental facts; but it remains free and is limited only by the necessity of avoiding all contradiction. Thus it is that the postulates can remain rigorously true even though the experimental laws which have determined their adoption are only approximative.
In other words, the axioms of geometry (I do not speak of those of arithmetic) are merely disguised definitions.
Then what are we to think of that question: Is the Euclidean geometry true?
It has no meaning.
As well ask whether the metric system is true and the old measures false; whether Cartesian coordinates are true and polar coordinates false. One geometry can not be more true than another; it can only be more convenient.
Now, Euclidean geometry is, and will remain, the most convenient:
1º Because it is the simplest; and it is so not only in consequence of our mental habits, or of I know not what direct intuition that we may have of Euclidean space; it is the simplest in itself, just as a polynomial of the first degree is simpler than one of the second; the formulas of spherical trigonometry are more complicated than those of plane trigonometry, and they would still appear so to an analyst ignorant of their geometric signification.
2º Because it accords sufficiently well with the properties of natural solids, those bodies which our hands and our eyes compare and with which we make our instruments of measure.
Chapter IV
Space and Geometry
Let us begin by a little paradox.
Beings with minds like ours, and having the same senses as we, but without previous education, would receive from a suitably chosen external world impressions such that they would be led to construct a geometry other than that of Euclid and to localize the phenomena of that external world in a non-Euclidean space, or even in a space of four dimensions.
As for us, whose education has been accomplished by our actual world, if we were suddenly transported into this new world, we should have no difficulty in referring its phenomena to our Euclidean space. Conversely, if these beings were transported into our environment, they would be led to relate our phenomena to non-Euclidean space.
Nay more; with a little effort we likewise could do it. A person who should devote his existence to it might perhaps attain to a realization of the fourth dimension.
Geometric Space and Perceptual Space.—It is often said the images of external objects are localized in space, even that they can not be formed except on this condition. It is also said that this space, which serves thus as a ready prepared frame for our sensations and our representations, is identical with that of the geometers, of which it possesses all the properties.
To all the good minds who think thus, the preceding statement must have appeared quite extraordinary. But let us see whether they are not subject to an illusion that a more profound analysis would dissipate.
What, first of all, are the properties of space, properly so called? I mean of that space which is the object of geometry and which I shall call geometric space.
The following are some of the most essential:
1º It is continuous;
2º It is infinite;
3º It has three dimensions;
4º It is homogeneous, that is to say, all its points are identical one with another;
5º It is isotropic, that is to say, all the straights which pass through the same point are identical one with another.
Compare it now to the frame of our representations and our sensations, which I may call perceptual space.
Visual Space.—Consider first a purely visual impression, due to an image formed on the bottom of the retina.
A cursory analysis shows us this image as continuous, but as possessing only two dimensions; this already distinguishes from geometric space what we may call pure visual space.
Besides, this image is enclosed in a limited frame.
Finally, there is another difference not less important: this pure visual space is not homogeneous. All the points of the retina, aside from the images which may there be formed, do not play the same rôle. The yellow spot can in no way be regarded as identical with a point on the border of the retina. In fact, not only does the same object produce there much more vivid impressions, but in every limited frame the point occupying the center of the frame will never appear as equivalent to a point near one of the borders.
No doubt a more profound analysis would show us that this continuity of visual space and its two dimensions are only an illusion; it would separate it therefore still more from geometric space, but we shall not dwell on this remark.
Sight, however, enables us to judge of distances and consequently to perceive a third dimension. But every one knows that this perception of the third dimension reduces itself to the sensation of the effort at accommodation it is necessary to make, and to that of the convergence which must be given to the two eyes, to perceive an object distinctly.
These are muscular sensations altogether different from the visual sensations which have given us the notion of the first two dimensions. The third dimension therefore will not appear to us as playing the same rôle as the other two. What may be called complete visual space is therefore not an isotropic space.
It has, it is true, precisely three dimensions, which means that the elements of our visual sensations (those at least which combine to form the notion of extension) will be completely defined when three of them are known; to use the language of mathematics, they will be functions of three independent variables.
But examine the matter a little more closely. The third dimension is revealed to us in two different ways: by the effort of accommodation and by the convergence of the eyes.
No doubt these two indications are always concordant, there is a constant relation between them, or, in mathematical terms, the two variables which measure these two muscular sensations do not appear to us as independent; or again, to avoid an appeal to mathematical notions already rather refined, we may go back to the language of the preceding chapter and enunciate the same fact as follows: If two sensations of convergence, A and B, are indistinguishable, the two sensations of accommodation, A´ and B´, which respectively accompany them, will be equally indistinguishable.
But here we have, so to speak, an experimental fact; a priori nothing prevents our supposing the contrary, and if the contrary takes place, if these two muscular sensations vary independently of one another, we shall have to take account of one more independent variable, and ‘complete visual space’ will appear to us as a physical continuum of four dimensions.
We have here even, I will add, a fact of external experience. Nothing prevents our supposing that a being with a mind like ours, having the same sense organs that we have, may be placed in a world where light would only reach him after having traversed reflecting media of complicated form. The two indications which serve us in judging distances would cease to be connected by a constant relation. A being who should achieve in such a world the education of his senses would no doubt attribute four dimensions to complete visual space.
Tactile Space and Motor Space.—‘Tactile space’ is still more complicated than visual space and farther removed from geometric space. It is superfluous to repeat for touch the discussion I have given for sight.
But apart from the data of sight and touch, there are other sensations which contribute as much and more than they to the genesis of the notion of space. These are known to every one; they accompany all our movements, and are usually called muscular sensations.
The corresponding frame constitutes what may be called motor space.
Each muscle gives rise to a special sensation capable of augmenting or of diminishing, so that the totality of our muscular sensations will depend upon as many variables as we have muscles. From this point of view, motor space would have as many dimensions as we have muscles.
I know it will be said that if the muscular sensations contribute to form the notion of space, it is because we have the sense of the direction of each movement and that it makes an integrant part of the sensation. If this were so, if a muscular sensation could not arise except accompanied by this geometric sense of direction, geometric space would indeed be a form imposed upon our sensibility.
But I perceive nothing at all of this when I analyze my sensations.
What I do see is that the sensations which correspond to movements in the same direction are connected in my mind by a mere association of ideas. It is to this association that what we call ‘the sense of direction’ is reducible. This feeling therefore can not be found in a single sensation.
This association is extremely complex, for the contraction of the same muscle may correspond, according to the position of the limbs, to movements of very different direction.
Besides, it is evidently acquired; it is, like all associations of ideas, the result of a habit; this habit itself results from very numerous experiences; without any doubt, if the education of our senses had been accomplished in a different environment, where we should have been subjected to different impressions, contrary habits would have arisen and our muscular sensations would have been associated according to other laws.
Characteristics of Perceptual Space.—Thus perceptual space, under its triple form, visual, tactile and motor, is essentially different from geometric space.
It is neither homogeneous, nor isotropic; one can not even say that it has three dimensions.
It is often said that we ‘project’ into geometric space the objects of our external perception; that we ‘localize’ them.
Has this a meaning, and if so what?
Does it mean that we represent to ourselves external objects in geometric space?
Our representations are only the reproduction of our sensations; they can therefore be ranged only in the same frame as these, that is to say, in perceptual space.
It is as impossible for us to represent to ourselves external bodies in geometric space, as it is for a painter to paint on a plane canvas objects with their three dimensions.
Perceptual space is only an image of geometric space, an image altered in shape by a sort of perspective, and we can represent to ourselves objects only by bringing them under the laws of this perspective.
Therefore we do not represent to ourselves external bodies in geometric space, but we reason on these bodies as if they were situated in geometric space.
When it is said then that we ‘localize’ such and such an object at such and such a point of space, what does it mean?
It simply means that we represent to ourselves the movements it would be necessary to make to reach that object; and one may not say that to represent to oneself these movements, it is necessary to project the movements themselves in space and that the notion of space must, consequently, pre-exist.
When I say that we represent to ourselves these movements, I mean only that we represent to ourselves the muscular sensations which accompany them and which have no geometric character whatever, which consequently do not at all imply the preexistence of the notion of space.
Change of State and Change of Position.—But, it will be said, if the idea of geometric space is not imposed upon our mind, and if, on the other hand, none of our sensations can furnish it, how could it have come into existence?
This is what we have now to examine, and it will take some time, but I can summarize in a few words the attempt at explanation that I am about to develop.
None of our sensations, isolated, could have conducted us to the idea of space; we are led to it only in studying the laws, according to which these sensations succeed each other.
We see first that our impressions are subject to change; but among the changes we ascertain we are soon led to make a distinction.
At one time we say that the objects which cause these impressions have changed state, at another time that they have changed position, that they have only been displaced.
Whether an object changes its state or merely its position, this is always translated for us in the same manner: by a modification in an aggregate of impressions.
How then could we have been led to distinguish between the two? It is easy to account for. If there has only been a change of position, we can restore the primitive aggregate of impressions by making movements which replace us opposite the mobile object in the same relative situation. We thus correct the modification that happened and we reestablish the initial state by an inverse modification.
If it is a question of sight, for example, and if an object changes its place before our eye, we can ‘follow it with the eye’ and maintain its image on the same point of the retina by appropriate movements of the eyeball.
These movements we are conscious of because they are voluntary and because they are accompanied by muscular sensations, but that does not mean that we represent them to ourselves in geometric space.
So what characterizes change of position, what distinguishes it from change of state, is that it can always be corrected in this way.
It may therefore happen that we pass from the totality of impressions A to the totality B in two different ways:
1º Involuntarily and without experiencing muscular sensations; this happens when it is the object which changes place;
2º Voluntarily and with muscular sensations; this happens when the object is motionless, but we move so that the object has relative motion with reference to us.
If this be so, the passage from the totality A to the totality B is only a change of position.
It follows from this that sight and touch could not have given us the notion of space without the aid of the ‘muscular sense.’
Not only could this notion not be derived from a single sensation or even from a series of sensations, but what is more, an immobile being could never have acquired it, since, not being able to correct by his movements the effects of the changes of position of exterior objects, he would have had no reason whatever to distinguish them from changes of state. Just as little could he have acquired it if his motions had not been voluntary or were unaccompanied by any sensations.
Conditions of Compensation.—How is a like compensation possible, of such sort that two changes, otherwise independent of each other, reciprocally correct each other?
A mind already familiar with geometry would reason as follows: Evidently, if there is to be compensation, the various parts of the external object, on the one hand, and the various sense organs, on the other hand, must be in the same relative position after the double change. And, for that to be the case, the various parts of the external object must likewise have retained in reference to each other the same relative position, and the same must be true of the various parts of our body in regard to each other.
In other words, the external object, in the first change, must be displaced as is a rigid solid, and so must it be with the whole of our body in the second change which corrects the first.
Under these conditions, compensation may take place.
But we who as yet know nothing of geometry, since for us the notion of space is not yet formed, we can not reason thus, we can not foresee a priori whether compensation is possible. But experience teaches us that it sometimes happens, and it is from this experimental fact that we start to distinguish changes of state from changes of position.
Solid Bodies and Geometry.—Among surrounding objects there are some which frequently undergo displacements susceptible of being thus corrected by a correlative movement of our own body; these are the solid bodies. The other objects, whose form is variable, only exceptionally undergo like displacements (change of position without change of form). When a body changes its place and its shape, we can no longer, by appropriate movements, bring back our sense-organs into the same relative situation with regard to this body; consequently we can no longer reestablish the primitive totality of impressions.
It is only later, and as a consequence of new experiences, that we learn how to decompose the bodies of variable form into smaller elements, such that each is displaced almost in accordance with the same laws as solid bodies. Thus we distinguish ‘deformations’ from other changes of state; in these deformations, each element undergoes a mere change of position, which can be corrected, but the modification undergone by the aggregate is more profound and is no longer susceptible of correction by a correlative movement.
Such a notion is already very complex and must have been relatively late in appearing; moreover it could not have arisen if the observation of solid bodies had not already taught us to distinguish changes of position.
Therefore, if there were no solid bodies in nature, there would be no geometry.
Another remark also deserves a moment’s attention. Suppose a solid body to occupy successively the positions α and β; in its first position, it will produce on us the totality of impressions A, and in its second position the totality of impressions B. Let there be now a second solid body, having qualities entirely different from the first, for example, a different color. Suppose it to pass from the position α, where it gives us the totality of impressions A´, to the position β, where it gives the totality of impressions B´.
In general, the totality A will have nothing in common with the totality A´, nor the totality B with the totality B´. The transition from the totality A to the totality B and that from the totality A´ to the totality B´ are therefore two changes which in themselves have in general nothing in common.
And yet we regard these two changes both as displacements and, furthermore, we consider them as the same displacement. How can that be?
It is simply because they can both be corrected by the same correlative movement of our body.
‘Correlative movement’ therefore constitutes the sole connection between two phenomena which otherwise we never should have dreamt of likening.
On the other hand, our body, thanks to the number of its articulations and muscles, may make a multitude of different movements; but all are not capable of ‘correcting’ a modification of external objects; only those will be capable of it in which our whole body, or at least all those of our sense-organs which come into play, are displaced as a whole, that is, without their relative positions varying, or in the fashion of a solid body.
To summarize:
1º We are led at first to distinguish two categories of phenomena:
Some, involuntary, unaccompanied by muscular sensations, are attributed by us to external objects; these are external changes;
Others, opposite in character and attributed by us to the movements of our own body, are internal changes;
2º We notice that certain changes of each of these categories may be corrected by a correlative change of the other category;
3º We distinguish among external changes those which have thus a correlative in the other category; these we call displacements; and just so among the internal changes, we distinguish those which have a correlative in the first category.
Thus are defined, thanks to this reciprocity, a particular class of phenomena which we call displacements.
The laws of these phenomena constitute the object of geometry.
Law of Homogeneity.—The first of these laws is the law of homogeneity.
Suppose that, by an external change α, we pass from the totality of impressions A to the totality B, then that this change α is corrected by a correlative voluntary movement β, so that we are brought back to the totality A.
Suppose now that another external change α´ makes us pass anew from the totality A to the totality B.
Experience teaches us that this change α´ is, like α, susceptible of being corrected by a correlative voluntary movement β´ and that this movement β´ corresponds to the same muscular sensations as the movement β which corrected α.
This fact is usually enunciated by saying that space is homogeneous and isotropic.
It may also be said that a movement which has once been produced may be repeated a second and a third time, and so on, without its properties varying.
In the first chapter, where we discussed the nature of mathematical reasoning, we saw the importance which must be attributed to the possibility of repeating indefinitely the same operation.
It is from this repetition that mathematical reasoning gets its power; it is, therefore, thanks to the law of homogeneity, that it has a hold on the geometric facts.
For completeness, to the law of homogeneity should be added a multitude of other analogous laws, into the details of which I do not wish to enter, but which mathematicians sum up in a word by saying that displacements form ‘a group.’
The Non-Euclidean World.—If geometric space were a frame imposed on each of our representations, considered individually, it would be impossible to represent to ourselves an image stripped of this frame, and we could change nothing of our geometry.
But this is not the case; geometry is only the résumé of the laws according to which these images succeed each other. Nothing then prevents us from imagining a series of representations, similar in all points to our ordinary representations, but succeeding one another according to laws different from those to which we are accustomed.
We can conceive then that beings who received their education in an environment where these laws were thus upset might have a geometry very different from ours.
Suppose, for example, a world enclosed in a great sphere and subject to the following laws:
The temperature is not uniform; it is greatest at the center, and diminishes in proportion to the distance from the center, to sink to absolute zero when the sphere is reached in which this world is enclosed.
To specify still more precisely the law in accordance with which this temperature varies: Let R be the radius of the limiting sphere; let r be the distance of the point considered from the center of this sphere. The absolute temperature shall be proportional to R2 − r2.
I shall further suppose that, in this world, all bodies have the same coefficient of dilatation, so that the length of any rule is proportional to its absolute temperature.
Finally, I shall suppose that a body transported from one point to another of different temperature is put immediately into thermal equilibrium with its new environment.
Nothing in these hypotheses is contradictory or unimaginable.
A movable object will then become smaller and smaller in proportion as it approaches the limit-sphere.
Note first that, though this world is limited from the point of view of our ordinary geometry, it will appear infinite to its inhabitants.
In fact, when these try to approach the limit-sphere, they cool off and become smaller and smaller. Therefore the steps they take are also smaller and smaller, so that they can never reach the limiting sphere.
If, for us, geometry is only the study of the laws according to which rigid solids move, for these imaginary beings it will be the study of the laws of motion of solids distorted by the differences of temperature just spoken of.
No doubt, in our world, natural solids likewise undergo variations of form and volume due to warming or cooling. But we neglect these variations in laying the foundations of geometry, because, besides their being very slight, they are irregular and consequently seem to us accidental.
In our hypothetical world, this would no longer be the case, and these variations would follow regular and very simple laws.
Moreover, the various solid pieces of which the bodies of its inhabitants would be composed would undergo the same variations of form and volume.
I will make still another hypothesis; I will suppose light traverses media diversely refractive and such that the index of refraction is inversely proportional to R2 − r2. It is easy to see that, under these conditions, the rays of light would not be rectilinear, but circular.
To justify what precedes, it remains for me to show that certain changes in the position of external objects can be corrected by correlative movements of the sentient beings inhabiting this imaginary world, and that in such a way as to restore the primitive aggregate of impressions experienced by these sentient beings.
Suppose in fact that an object is displaced, undergoing deformation, not as a rigid solid, but as a solid subjected to unequal dilatations in exact conformity to the law of temperature above supposed. Permit me for brevity to call such a movement a non-Euclidean displacement.
If a sentient being happens to be in the neighborhood, his impressions will be modified by the displacement of the object, but he can reestablish them by moving in a suitable manner. It suffices if finally the aggregate of the object and the sentient being, considered as forming a single body, has undergone one of those particular displacements I have just called non-Euclidean. This is possible if it be supposed that the limbs of these beings dilate according to the same law as the other bodies of the world they inhabit.
Although from the point of view of our ordinary geometry there is a deformation of the bodies in this displacement and their various parts are no longer in the same relative position, nevertheless we shall see that the impressions of the sentient being have once more become the same.
In fact, though the mutual distances of the various parts may have varied, yet the parts originally in contact are again in contact. Therefore the tactile impressions have not changed.
On the other hand, taking into account the hypothesis made above in regard to the refraction and the curvature of the rays of light, the visual impressions will also have remained the same.
These imaginary beings will therefore like ourselves be led to classify the phenomena they witness and to distinguish among them the ‘changes of position’ susceptible of correction by a correlative voluntary movement.
If they construct a geometry, it will not be, as ours is, the study of the movements of our rigid solids; it will be the study of the changes of position which they will thus have distinguished and which are none other than the ‘non-Euclidean displacements’; it will be non-Euclidean geometry.
Thus beings like ourselves, educated in such a world, would not have the same geometry as ours.
The World of Four Dimensions.—We can represent to ourselves a four-dimensional world just as well as a non-Euclidean.
The sense of sight, even with a single eye, together with the muscular sensations relative to the movements of the eyeball, would suffice to teach us space of three dimensions.
The images of external objects are painted on the retina, which is a two-dimensional canvas; they are perspectives.
But, as eye and objects are movable, we see in succession various perspectives of the same body, taken from different points of view.
At the same time, we find that the transition from one perspective to another is often accompanied by muscular sensations.
If the transition from the perspective A to the perspective B, and that from the perspective A´ to the perspective B´ are accompanied by the same muscular sensations, we liken them one to the other as operations of the same nature.
Studying then the laws according to which these operations combine, we recognize that they form a group, which has the same structure as that of the movements of rigid solids.
Now, we have seen that it is from the properties of this group we have derived the notion of geometric space and that of three dimensions.
We understand thus how the idea of a space of three dimensions could take birth from the pageant of these perspectives, though each of them is of only two dimensions, since they follow one another according to certain laws.
Well, just as the perspective of a three-dimensional figure can be made on a plane, we can make that of a four-dimensional figure on a picture of three (or of two) dimensions. To a geometer this is only child’s play.
We can even take of the same figure several perspectives from several different points of view.
We can easily represent to ourselves these perspectives, since they are of only three dimensions.
Imagine that the various perspectives of the same object succeed one another, and that the transition from one to the other is accompanied by muscular sensations.
We shall of course consider two of these transitions as two operations of the same nature when they are associated with the same muscular sensations.
Nothing then prevents us from imagining that these operations combine according to any law we choose, for example, so as to form a group with the same structure as that of the movements of a rigid solid of four dimensions.
Here there is nothing unpicturable, and yet these sensations are precisely those which would be felt by a being possessed of a two-dimensional retina who could move in space of four dimensions. In this sense we may say the fourth dimension is imaginable.
Conclusions.—We see that experience plays an indispensable rôle in the genesis of geometry; but it would be an error thence to conclude that geometry is, even in part, an experimental science.
If it were experimental, it would be only approximative and provisional. And what rough approximation!
Geometry would be only the study of the movements of solids; but in reality it is not occupied with natural solids, it has for object certain ideal solids, absolutely rigid, which are only a simplified and very remote image of natural solids.
The notion of these ideal solids is drawn from all parts of our mind, and experience is only an occasion which induces us to bring it forth from them.
The object of geometry is the study of a particular ‘group’; but the general group concept pre-exists, at least potentially, in our minds. It is imposed on us, not as form of our sense, but as form of our understanding.
Only, from among all the possible groups, that must be chosen which will be, so to speak, the standard to which we shall refer natural phenomena.
Experience guides us in this choice without forcing it upon us; it tells us not which is the truest geometry, but which is the most convenient.
Notice that I have been able to describe the fantastic worlds above imagined without ceasing to employ the language of ordinary geometry.
And, in fact, we should not have to change it if transported thither.
Beings educated there would doubtless find it more convenient to create a geometry different from ours, and better adapted to their impressions. As for us, in face of the same impressions, it is certain we should find it more convenient not to change our habits.
Chapter V
Experience and Geometry
1. Already in the preceding pages I have several times tried to show that the principles of geometry are not experimental facts and that in particular Euclid’s postulate can not be proven experimentally.
However decisive appear to me the reasons already given, I believe I should emphasize this point because here a false idea is profoundly rooted in many minds.
2. If we construct a material circle, measure its radius and circumference, and see if the ratio of these two lengths is equal to π, what shall we have done? We shall have made an experiment on the properties of the matter with which we constructed this round thing, and of that of which the measure used was made.
3. Geometry and Astronomy.—The question has also been put in another way. If Lobachevski’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 results which seem within the reach of experiment, and there have been hopes that astronomical observations might enable us to decide between the three geometries.
But in astronomy ‘straight line’ means simply ‘path of a ray of light.’
If therefore negative parallaxes were found, or if it were demonstrated that all parallaxes are superior to a certain limit, two courses would be open to us; we might either renounce Euclidean geometry, or else modify the laws of optics and suppose that light does not travel rigorously in a straight line.
It is needless to add that all the world would regard the latter solution as the more advantageous.
The Euclidean geometry has, therefore, nothing to fear from fresh experiments.
4. Is the position tenable, that certain phenomena, possible in Euclidean space, would be impossible in non-Euclidean space, so that experience, in establishing these phenomena, would directly contradict the non-Euclidean hypothesis? For my part I think no such question can be put. To my mind it is precisely equivalent to the following, whose absurdity is patent to all eyes: are there lengths expressible in meters and centimeters, but which can not be measured in fathoms, feet and inches, so that experience, in ascertaining the existence of these lengths, would directly contradict the hypothesis that there are fathoms divided into six feet?
Examine the question more closely. I suppose that the straight line possesses in Euclidean space any two properties which I shall call A and B; that in non-Euclidean space it still possesses the property A, but no longer has the property B; finally I suppose that in both Euclidean and non-Euclidean space the straight line is the only line having the property A.
If this were so, experience would be capable of deciding between the hypothesis of Euclid and that of Lobachevski. It would be ascertained that a definite concrete object, accessible to experiment, for example, a pencil of rays of light, possesses the property A; we should conclude that it is rectilinear, and then investigate whether or not it has the property B.
But this is not so; no property exists which, like this property A, can be an absolute criterion enabling us to recognize the straight line and to distinguish it from every other line.
Shall we say, for instance: “the following is such a property: the straight line is a line such that a figure of which this line forms a part can be moved without the mutual distances of its points varying and so that all points of this line remain fixed”?
This, in fact, is a property which, in Euclidean or non-Euclidean space, belongs to the straight and belongs only to it. But how shall we ascertain experimentally whether it belongs to this or that concrete object? It will be necessary to measure distances, and how shall one know that any concrete magnitude which I have measured with my material instrument really represents the abstract distance?
We have only pushed back the difficulty.
In reality the property just enunciated is not a property of the straight line alone, it is a property of the straight line and distance. For it to serve as absolute criterion, we should have to be able to establish not only that it does not also belong to a line other than the straight and to distance, but in addition that it does not belong to a line other than the straight and to a magnitude other than distance. Now this is not true.
It is therefore impossible to imagine a concrete experiment which can be interpreted in the Euclidean system and not in the Lobachevskian system, so that I may conclude:
No experience will ever be in contradiction to Euclid’s postulate; nor, on the other hand, will any experience ever contradict the postulate of Lobachevski.
5. But it is not enough that the Euclidean (or non-Euclidean) geometry can never be directly contradicted by experience. Might it not happen that it can accord with experience only by violating the principle of sufficient reason or that of the relativity of space?
I will explain myself: consider any material system; we shall have to regard, on the one hand, ‘the state’ of the various bodies of this system (for instance, their temperature, 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 shall, moreover, distinguish the mutual distances of these bodies, which define their relative positions, from the conditions which define the absolute position of the system and its absolute orientation in space.
The laws of the phenomena which will happen in this system will depend on the state of these bodies and their mutual distances; but, because of the relativity and passivity 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 instant will depend solely on the state of these same bodies and on their mutual distances at the initial instant, but will not at all depend on the absolute initial position of the system or on its absolute initial orientation. This is what for brevity I shall call the law of relativity.
Hitherto I have spoken as a Euclidean geometer. As I have said, an experience, whatever it be, admits of an interpretation on the Euclidean hypothesis; but it admits of one equally on the non-Euclidean hypothesis. Well, we have made a series of experiments; we have interpreted them on the Euclidean hypothesis, and we have recognized 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 experiments, interpreted in this new manner, still be in accord with our ‘law of relativity’? And if there were not this accord, should we not have also the right to say experience had proven the falsity of the non-Euclidean geometry?
It is easy to see that this is an idle fear; in fact, to apply the law of relativity in all rigor, it must be applied to the entire universe. For if only a part of this universe were considered, and if the absolute position of this part happened to vary, the distances to the other bodies of the universe would likewise vary, their influence on the part of the universe considered would consequently augment or diminish, which might modify the laws of the phenomena happening there.
But if our system is the entire universe, experience is powerless to give information about its absolute position and orientation in space. All that our instruments, however perfected they may be, can tell us will be the state of the various parts of the universe and their mutual distances.
So our law of relativity may be thus enunciated:
The readings we shall be able to make on our instruments at any instant will depend only on the readings we could have made on these same instruments at the initial instant.
Now such an enunciation is independent of every interpretation of experimental facts. If the law is true in the Euclidean interpretation, it will also be true in the non-Euclidean interpretation.
Allow me here a short digression. I have spoken above of the data which define the position of the various bodies of the system; I should likewise have spoken of those which define their velocities; I should then have had to distinguish the velocities with which the mutual distances of the different bodies vary; and, on the other hand, the velocities of translation and rotation of the system, that is to say, the velocities with which its absolute position and orientation vary.
To fully satisfy the mind, the law of relativity should be expressible thus:
The state of bodies and their mutual distances at any instant, as well as the velocities with which these distances vary at this same instant, will depend only on the state of those bodies and their mutual distances at the initial instant, and the velocities with which these distances vary at this initial instant, but they will not depend either upon the absolute initial position of the system, or upon its absolute orientation, or upon the velocities with which this absolute position and orientation varied at the initial instant.
Unhappily the law thus enunciated is not in accord with experiments, at least as they are ordinarily interpreted.
Suppose a man be transported to a planet whose heavens were always 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. Yet this man could become aware that it turned, either by measuring its oblateness (done ordinarily by the aid of astronomic observations, but capable of being done by purely geodetic means), or by repeating the experiment of Foucault’s pendulum. The absolute rotation of this planet could therefore be made evident.
That is a fact which shocks the philosopher, but which the physicist is compelled to accept.
We know that from this fact Newton inferred the existence of absolute space; I myself am quite unable to adopt this view. I shall explain why in Part III. For the moment it is not my intention to enter upon this difficulty.
Therefore I must 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, this difficulty is the same for Euclid’s geometry as for Lobachevski’s; I therefore need not trouble myself with it, and have only mentioned it incidentally.
What is important is the conclusion: experiment can not decide between Euclid and Lobachevski.
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.
Ancestral Experience
It has often been said that if individual experience could not create geometry the same is not true of ancestral experience. But what does that mean? Is it meant that we could not experimentally demonstrate Euclid’s postulate, but that our ancestors have been able to do it? Not in the least. It is meant that by natural selection our mind has adapted itself to the conditions of the external world, that it has adopted the geometry most advantageous to the species: or in other words the most convenient. This is entirely in conformity with our conclusions; geometry is not true, it is advantageous.