CHAPTER IITHE BIRTH OF METRICAL GEOMETRY
IN the preceding chapter, we mentioned that the concept of continuity was suggested by our sense experiences; and that our understanding of space as a three-dimensional continuum had arisen from a synthesis of our various sense impressions. In the present chapter, we shall be concerned more especially with the geometry of the space continuum; and we shall show how measurement in turn would appear to have arisen from experience, more particularly visual experience.
For this purpose, let us consider the case of a motionless observer, rooted to the earth ever since his birth—a species of man-plant. Viewing the world whence he stood, he would notice that whereas certain visual impressions manifested a property of what he would recognise as “permanence,” others would appear as squirming forms moving across his field of vision. In order to simplify this discussion, we shall omit to take into consideration any awareness of focussing efforts, on the part of our observer, as also any appreciation on his part of the convergence of his eyes. Under the circumstances, his visual perceptions would reveal a world of two dimensions, “up and down, right and left”; the third dimension with which we are all familiar,i.e., “away from and towards,” would be lacking. As a result, the squirming forms passing through his field of vision would be interpreted as betraying a two-dimensional world of changing forms, which would in no wise be connected with the existence of a third dimension. In particular, there would be no reason for him to attribute these changes to the variations in the distances of rigid material objects from his post of observation.
But suppose, now, that concomitant with the activity of his will, our observer were to become aware of certain muscular exertions that accompanied a variation in the shapes of those forms which had hitherto remained fixed and undeformed in his field of vision. In ordinary parlance, our observer would be displacing his post of observation, that is to say, “walking.” He would no longer remain fixed like a tree. As a result of these displacements, which he would end by recognising as such, not only would forms erstwhile fixed in shape appear to vary, but, vice versa, certain forms hitherto squirming could be made to maintain an unchanging appearance. Eventually, he would recognise that in those cases where variations of shape and size could be counteracted by suitable displacements of his post of observation, he had been observing rigid bodies varying their distances along a third dimension with respect to him. In this way, there would arise an understanding both of rigid bodies and of a third dimension. Furthermore, owing tohis being able to repeat his experimentshereasthere, a realisation of the homogeneity of space would ensue.
We see, then, that the three-dimensional space of experience appears to have arisen as the result of a synthesis of private views, each one of which would be that of an observer unable to move from a certain fixed spot. This synthesis would be extremely complex; unfortunately we have no time to mention the various conditions that would have to be taken into consideration. Suffice it to say that our senses of sight, of touch, of muscular effort, of sound and of smell, to which should also be added the action of the semi-circular canals, would all play a part, dovetailing one into the other. Further considerations would also show that there is nothing mysterious in the fact that these various data should yield concordant results, rather than an incompatible set of conflicting spaces.
All we wish to point out is that by the physical space of experience, we do not merely wish to imply space with the objects located therein, such as it would appear from some definite point of observation. We do not mean the private vision in which rails converge and distant objects appear smaller; we mean a synthesis of these private perspectives, yielding us a common public space.
One private perspective with its converging rails taken by itself and considered without reference to other perspectives could not contain sufficient data to enable us to conceive of three-dimensional space, homogeneous and isotropic.[6]That this synthesis has been arrived at without the conscious effort of reason is granted. Nevertheless, though instinctive, the co-ordination of private experiences and perspectives is of great complexity; and it would not be impossible to conceive of this co-ordination as having followed other lines, just as an aggregate of books may be arranged in alphabetical order, or in order of size or of content, and so forth. With a change in our ordering relation we might have obtained a space of a greater number of dimensions. Undoubtedly, however, when account is taken of the facts of experience, the three-dimensional co-ordination is by far the simplest; hence there is no reason to be surprised at its having imposed itself with such force. These too brief indications must suffice for the present.
And now let us return to the problem of measurement. We have mentioned that in certain cases it would be possible, by changing our post of observation, to counteract the apparent modifications in the visual shapes of bodies that had suffered a displacement in our field of vision.
For instance, if we received a visual impression corresponding to a circle, and if this impression were followed by one corresponding to a triangle, and if it were impossible for us to re-establish the circular impression, we should have to assume that the body hadchanged in shape; whereas, if it were possible to re-establish the circular impression by exerting certain efforts (which would finally be interpreted as a displacement of our point of observation), we should end by assuming that we had witnessed a partial rotation of a rigid cone-shaped object.
This discovery of rigid objects in nature is of fundamental importance. Without it, the concept of measurement would probably never have arisen and metrical geometry would have been impossible. But with the discovery of objects which were recognised as rigid, hence as maintaining the same size and shape wherever displaced, it was only natural to appeal to them as standards of spatial measurement.
Measurements conducted in this way would soon have proved that between any two points a certain species of line called the straight line would yield the shortest distance; and this in turn would have suggested the use of straight measuring rods. Henceforth, two straight rods would be considered equal or congruent if, when brought together, their extremities coincided. As for a physical definition of straightness, it could have been arrived at in a number of ways, either by stretching a rope between two points or by appealing to the properties of these rigid bodies themselves. For instance, two rods would be recognised as straight if, after coinciding when placed lengthwise, they continued to coincide when one rod was turned over on itself. Finally, parallelograms would be constructed by forming a quadrilateral with four equal rods, and parallelism would thus have been defined.
Equipped in this way, the first geometricians (those who built the Pyramids, for instance) were able to execute measurements on the earth’s surface and later to study the geometry of solids, or space-geometry. Thanks to their crude measurements, they were in all probability led to establish in an approximate empirical way a number of propositions whose correctness it was reserved for the Greek geometers to demonstrate with mathematical accuracy. Thus there is not the slightest doubt that geometry in its origin was essentially an empirical and physical science, since it reduced to a study of the possible dispositions of objects (recognised as rigid) with respect to one another and to parts of the earth. In fact, the very word geometry proves this point conclusively.
Now, an empirical science is necessarily approximate, and geometry as we know it to-day is an exact science. It professes to teach us that the sum of the three angles of a Euclidean triangle is equal to 180°, and not a fraction more or a fraction less. Obviously no empirical determination could ever lay claim to such absolute certitude. Accordingly, geometry had to be subjected to a profound transformation, and this was accomplished by the Greek mathematicians Thales, Democritus, Pythagoras, and finally Euclid.
The difficulty that Euclid had to face was to succeed in defining exactly what he meant by a straight line and by the equality of two distances in space. So long as geometry was in its empirical stage these definitions were easy enough. All that men had to say was, “Two solid rods will be recognised as straight if after turning one of themover they still remain in perfect contact,” or again, “The distance between the two extremities of a material rod remains the same by definition wherever we may transport the rod.”
Euclid, however, could not appeal to such approximate empirical definitions; for perfect rigour was his goal. Accordingly he was compelled to resort to indirect methods. By positing a system of axioms and postulates, he endeavoured to state in an accurate way properties which were presented only in an approximate way by the solids of nature. Euclid’s geometry was thus the geometry of perfectly rigid bodies, which, though idealised copies of the bodies commonly regarded as rigid in the world of experience, were yet defined in such a manner as to be untainted by the inaccuracies attendant on all physical measurements.
But this empirical origin of Euclid’s geometrical axioms and postulates was lost sight of, indeed was never even realised. As a result Euclidean geometry was thought to derive its validity from certain self-evident universal truths; it appeared as the only type of consistent geometry of which the mind could conceive. Gauss had certain misgivings on the matter, but did not have the courage to publish his results owing to his fear of the “outcry of the Bœotians.” At any rate, the honour of discovering non-Euclidean geometry fell to Lobatchewski and Bolyai.
To make a long story short, it was found that by varying one of Euclid’s fundamental assumptions, known as theParallel Postulate, it was possible to construct two other geometrical doctrines, perfectly consistent in every respect, though differing widely from Euclidean geometry. These are known as the non-Euclidean geometries of Lobatchewski and of Riemann.
Euclid’s parallel postulate can be expressed by stating that through a point in a plane it is always possible to trace one and only one straight line parallel to a given straight line lying in the plane. Lobatchewski denied this postulate and assumed that an indefinite number of non-intersecting straight lines could be drawn, and Riemann assumed that none could be drawn.
From this difference in the geometrical premises important variations followed. Thus, whereas in Euclidean geometry the sum of the angles of any triangle is always equal to two right angles, in non-Euclidean geometry the value of this sum varies with the size of the triangles. It is always less than two right angles in Lobatchewski’s, and always greater in Riemann’s. Again, in Euclidean geometry, similar figures of various sizes can exist; in non-Euclidean geometry, this is impossible.
It appeared, then, that the universal absoluteness of truth formerly credited to Euclidean geometry would have to be shared by these two other geometrical doctrines. But truth, when divested of its absoluteness, loses much of its significance, so this co-presence of conflicting universal truths brought the realisation that a geometry was true only in relation to our more or less arbitrary choice of a system of geometrical postulates. From a purely rational point ofview, there was no means of deciding which of the several consistent sets was true. The character of self-evidence which had been formerly credited to the Euclidean axioms was seen to be illusory.
However, there are a number of rather delicate points to be considered, and these we shall now proceed to investigate. Euclid’s parallel postulate and the alternative non-Euclidean postulates reduce to indirect definitions of what we intend to call astraight linein the respective geometries mentioned. If there existed such a universal asabsolute straightness, represented, let us say, by a Euclidean straight line, we might claim that Euclidean geometry constituted the true geometry, since its straight line conformed to the ideal of absolute straightness. But this existence of a universal representing absolute straightness is precisely one of the metaphysical cobwebs of which the discovery of non-Euclidean geometry has purged science. To illustrate this point more fully, let us assume that we think we know what is implied by a straight line. Whether we merely imagine a straight line or endeavour to realise one concretely, we are always faced with the same difficulty. For instance, we consider that a rod is straight when it can be turned over and superposed with itself, or else we place our eye at one of its extremities and note that no bumps are apparent. Again, we may realise straightness by stretching a string, viewing a plumb line or the course of a billiard ball. We may also execute measurements with our rigid rods; the straight line between any two points will then be defined by the shortest distance. But whatever method we adopt, it is apparent that our intuitive recognition of straightness in any given case will always be based on physical criteria dealing with the behaviour of light rays and material bodies. We may close our eyes and think of straightness in the abstract as much as we please, but ultimately we should always be imagining physical illustrations.
Suppose, then, that material bodies, including our own human body, were to behave differently when displaced. If corresponding adjustments were to affect the paths of light rays, we should be led to credit rigidity to bodies which from the Euclidean point of view would be squirming when set in motion. As a result, our straight line, that is, the line defined by a stretched rope, our line of sight, the shortest path between two points, would no longer coincide with a Euclidean straight line. From the Euclidean standpoint our straight line would be curved, but from our own point of view it would be the reverse; the Euclidean straight line would now manifest curvature both visually and as a result of measurement. A super-observer called in as umpire would tell us that we were arguing about nothing at all. He would say: “You are both of you justified in regarding as straight that which appears to you visually as such and that which measures out accordingly. It will be to your advantage, therefore, to reserve your definitions of straightness for lines which satisfy these conditions. But you are both of you wrong when you attribute any absolute significance to the concept, for you must realise that your opinions will always becontingent on the nature of the physical conditions which surround you.”
Incidentally, we are now in a position to understand why the Euclidean axioms appeared self-evident or at least imposed by reason. They represented mathematical abstractions derived from experience, from our experience with the light rays and material bodies among which we live. We shall return to these delicate questions in a subsequent chapter. For the present, let us note that since our judgment of straightness is contingent on the disclosures of experience, even the geometry of the space in which we actually live cannot be decided upona priori. To a first approximation, to be sure, this geometry appears to be Euclidean; but we cannot prophesy what it may turn out to be when nature is studied with ever-increasing refinement. It was with this idea in view that Gauss, who had mastered in secret the implications of non-Euclidean geometry, undertook triangulations with light rays over a century ago. Furthermore, even were the geometry to be established for one definite region of space, we could not assert that our understanding of straightness, hence of geometry, might not vary from place to place and from time to time; hence we cannot assert with Kant that the propositions of Euclidean geometry possess any universal truth even when restricting ourselves to this particular world in which we live.
Such discussions might have appeared to be merely academic a few years ago; and non-Euclidean geometry, though of vast philosophical interest, might have seemed devoid of any practical importance. But to-day, thanks to Einstein, we have definite reasons for believing that ultra-precise observation of nature has revealed our natural geometry arrived at with solids and light rays to be slightly non-Euclidean and to vary from place to place. So although the non-Euclidean geometers never suspected it (with the exception of Gauss, Riemann and Clifford), our real world happens to be one of the dream-worlds whose possible existence their mathematical genius foresaw.
Now, all these investigations initiated by attempts to prove the correctness of the parallel postulate led mathematicians to further discoveries.
A more thorough study of Euclid’s axioms and postulates proved them to be inadequate for the deduction of Euclid’s geometry. Euclid himself had never been embarrassed by the incompleteness of his basic premises, for the simple reason that although he failed to express the missing postulates explicitly, he appealed to them implicitly in the course of his demonstrations. The great German mathematician Hilbert and others succeeded in filling the gap by stating explicitly a complete system of postulates for Euclidean and non-Euclidean geometries alike. Among the postulates missing in Euclid’s list was the celebrated postulate of Archimedes, according to which, by placing an indefinite number of equal lengths end to end along a line, we should eventually pass any point arbitrarily selected on the line. Hilbert, by denying this postulate, just as Lobatchewski and Riemann had denied Euclid’s parallel postulate, succeeded in constructing a new geometry knownas non-Archimedean. It was perfectly consistent but much stranger than the classical non-Euclidean varieties. Likewise, it was proved possible to posit a system of postulates which would yield Euclidean or non-Euclidean geometries of any number of dimensions; hence, so far as the rational requirements of the mind were concerned, there was no reason to limit geometry to three dimensions.
Incidentally, we see to what rigour of analysis and to what profound introspection the mathematical mind must submit; for the implicit postulates appealed to unconsciously by Euclid are so inconspicuous that it is only owing to the dialectics of modern mathematicians that their presence was finally disclosed and the deficiency remedied by their explicit statement.
From all this rather long discussion on the subject of postulates and axioms we see that the axioms or postulates of geometry are most certainly not imposed upon usa prioriin any unique manner. We may vary them in many ways and, as regards real space, our only reason for selecting one system of postulates rather than another (hence one type of geometry in preference to another) is because it happens to be in better agreement with the facts of observation when solid bodies and light rays are taken into consideration. Our choice is thus dictated by motives of a pragmatic nature; and the Kantians were most decidedly in the wrong when they assumed that the axioms of geometry constituted a priori synthetic judgments transcending reason and experience.