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 of view, 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 a straight line in the respective geometries mentioned. If there existed such a universal as absolute 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 be contingent 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 upon a 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 known as 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.