An essay on the foundations of geometry. Bertrand Russell
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meaning and extent will appear later—is also proved, consisting roughly in the substitution of hyperbolic for circular functions.
14. Very similar is the system of Johann Bolyai, so similar, indeed, as to make the independence of the two works, though a well-authenticated fact, seem all but incredible. Johann Bolyai first published his results in 1832, in an appendix to a work by his father Wolfgang, entitled; "Appendix, scientiam spatii absolute veram exhibens: a veritate aut falsitate Axiomatis XI. Euclidei (a priori haud unquam decidenda) independentem; adjecta ad casum falsitatis, quadratura circuli geometrica." Gauss, whose bosom friend he became at college and remained through life, was, as we have seen, the inspirer of Wolfgang Bolyai, and used to say that the latter was the only man who appreciated his philosophical speculations on the axioms of Geometry; nevertheless, Wolfgang appears to have left to his son Johann the detailed working out of the hyperbolic system. The works of both the Bolyai are very rare, and their method and results are known to me only through the renderings of Frischauf and Halsted[13]. Both as to method and as to results, the system is very similar to Lobatchewsky's, so that neither need detain us here. Only the initial postulates, which are more explicit than Lobatchewsky's, demand a brief attention. Frischauf's introduction, which has a philosophical and Newtonian air, begins by setting forth that Geometry deals with absolute (empty) space, obtained by abstracting from the bodies in it, that two figures are called congruent when they differ only in position, and that the axiom of Congruence is indispensable in all determination of spatial magnitudes. Congruence was to refer to geometrical bodies, with none of the properties of ordinary bodies except impenetrability (Erdmann, Axiome der Geometrie, p. 26). A straight line is defined as determined by two of its points[14], and a plane as determined by three. These premisses, with a slight exception as to the straight line, we shall hereafter find essential to every Geometry. I have drawn attention to them, as it is often supposed that non-Euclideans deny the axiom of Congruence, which, here and elsewhere, is never the case. The stress laid on this axiom by Bolyai is probably due to the influence of Gauss, whose work on the curvature of surfaces laid the foundation for the use made of congruence by Helmholtz.
15. It is important to remember that, throughout the period we have just reviewed, the purpose of hyperbolic Geometry is indirect: not the truth of the latter, but the logical independence of the axiom of parallels from the rest, is the guiding motive of the work. If, by denying the axiom of parallels while retaining the rest, we can obtain a system free from logical contradictions, it follows that the axiom of parallels cannot be implicitly contained in the others. If this be so, attempts to dispense with the axiom, like Legendre's, cannot be successful; Euclid must stand or fall with the suspected axiom. Of course, it remained possible that, by further development, latent contradictions might have been revealed in these systems. This possibility, however, was removed by the more direct and constructive work of the second period, to which we must now turn our attention.
Second Period.
16. The work of Lobatchewsky and Bolyai remained, for nearly a quarter of a century, without issue—indeed, the investigations of Riemann and Helmholtz, when they came, appear to have been inspired, not by these men, but rather by Gauss[15] and Herbart. We find, accordingly, very great difference, both of aim and method, between the first period and the second. The former, beginning with a criticism of one point in Euclid's system, preserved his synthetic method, while it threw over one of his axioms. The latter, on the contrary, being guided by a philosophical rather than a mathematical spirit, endeavoured to classify the conception of space as a species of a more general conception: it treated space algebraically, and the properties it gave to space were expressed in terms, not of intuition, but of algebra. The aim of Riemann and Helmholtz was to show, by the exhibition of logically possible alternatives, the empirical nature of the received axioms. For this purpose, they conceived space as a particular case of a manifold, and showed that various relations of magnitude (Massverhältnisse) were mathematically possible in an extended manifold. Their philosophy, which seems to me not always irreproachable, will be discussed in Chapter II.; here, while it is important to remember the philosophical motive of Riemann and Helmholtz, we shall confine our attention to the mathematical side of their work. In so doing, while we shall, I fear, somewhat maim the system of their thoughts, we shall secure a closer unity of subject, and a more compact account of the purely mathematical development. But there is, in my opinion, a further reason for separating their philosophy from their mathematics. While their philosophical purpose was, to prove that all the axioms of Geometry are empirical, and that a different content of our experience might have changed them all, the unintended result of their mathematical work was, if I am not mistaken, to afford material for an à priori proof of certain axioms. These axioms, though they believed them to be unnecessary, were always introduced in their mathematical works, before laying the foundations of non-Euclidean systems. I shall contend, in Chapter III., that this retention was logically inevitable, and was not merely due, as they supposed, to a desire for conformity with experience. If I am right in this, there is a divergence between Riemann and Helmholtz the philosophers, and Riemann and Helmholtz the mathematicians. This divergence makes it the more desirable to trace the mathematical development apart from the accompanying philosophy.
17. Riemann's epoch-making work, "Ueber die Hypothesen, welche der Geometrie zu Grande liegen[16]", was written, and read to a small circle, in 1854; owing, however, to some changes which he desired to make in it, it remained unpublished till 1867, when it was published by his executors. The two fundamental conceptions, on whose invention rests the historic importance of this dissertation, are that of a manifold, and that of the measure of curvature of a manifold. The former conception serves a mainly philosophical purpose, and is designed, principally, to exhibit space as an instance of a more general conception. On this aspect of the manifold, I shall have much to say in Chapter II.; its mathematical aspect, which alone concerns us here, is less complicated and less fruitful of controversy. The latter conception also serves a double purpose, but its mathematical use is the more prominent. We will consider these two conceptions successively.
18. (1) Conception of a manifold[17]. The general purpose of Riemann's dissertation is, to exhibit the axioms as successive steps in the classification of the species space. The axioms of Geometry, like the marks of a scholastic definition, appear as successive determinations of class-conceptions, ending with Euclidean space. We have thus, from the analytical point of view, about as logical and precise a formulation as can be desired—a formulation in which, from its classificatory character, we seem certain of having nothing superfluous or redundant, and obtain the axioms explicitly in the most desirable form, namely as adjectives of the conception of space. At the same time, it is a pity that Riemann, in accordance with the metrical bias of his time, regarded space as primarily a magnitude[18], or assemblage of magnitudes, in which the main problem consists in assigning quantities to the different elements or points, without regard to the qualitative nature of the quantities assigned. Considerable obscurity thus arises as to the whole nature of magnitude[19]. This view of Geometry underlies the definition of the manifold, as the general conception of which space forms a special case. This definition, which is not very clear, may be rendered as follows.
19. Conceptions of magnitude, according to Riemann, are possible there only, where we have a general conception, capable of various determinations (Bestimmungsweisen). The various determinations of such a conception together form a manifold, which is continuous or discrete, according as the passage from one determination to another is continuous or discrete. Particular bits of a manifold, or quanta, can be compared by counting when discrete, and by measurement when continuous. "Measurement consists in a superposition of the magnitudes to be compared. If this be absent, magnitudes can only be compared when one is part of another, and then only the more or less, not the how much, can be decided" (p. 256). We thus reach the general conception of a manifold of several dimensions, of which space and colours are mentioned as special cases. To the absence of this conception Riemann attributes the "obscurity" which, on the subject of the axioms, "lasted from Euclid to Legendre" (p. 254). And Riemann certainly has succeeded, from an algebraic