CHAPTER X

WHAT IS GEOMETRY?

Ambiguity of the expression 'Properties of Space '.-Geometry as the science of Configuration.-The alleged incertitude of the Axiom of Parallels in relation to the conceivability of different Kinds of Space.—Are the fundamental notions of Geometry particular ideas or general ideas? Particular ideas and general ideas necessarily involve one another, and neither class can be more fundamental than the other.-Technical use of the terms 'Definition' and 'Indefinables' by mathematicians.

It is still customary, though perhaps less so than it was a few years ago, to define geometry as the science or the investigation of the properties of space. The definition is itself, however, of comparatively modern origin; it was not inherited from the geometers of antiquity; it has been remarked that the term space itself is not to be found either in Euclid or in Archimedes. If we look into textbooks or treatises of geometry we find that the talk is of points, lines, surfaces, solids; straight lines, circles, conic sections; planes, cylinders, cones, &c. in general, of shapes or figures simple by comparison with those of the vast mass of common objects which surround us-so that if it were deemed desirable to give some brief definition of geometry, one would suppose that such a definition, at once brief and intelligible, would be that geometry is the science or investigation of shape or configuration in some of its simpler aspects. This is a sufficient indication, to begin with, of the subject of thought; for, in the end, definition of any subject of thought waits upon the development of the subject.

The science or investigation of the properties of space must always have appeared, to some thinkers at least, somewhat of an absurdity—even a contradiction, in so far as their conception of space, in contrast with that of body or matter, involved the negation of properties. How far this must in general be the case may be judged if we reflect that the ether of space was invented in order to escape the inconceivability of 'action at a distance' (an inconceivability which is quite genuine save when the term 'action' is emptied of its meaning, which is what

J. S. Mill did in his discussion of the question); and this inconceivability is at bottom none other than that involved in assigning properties to space. I do not know when this definition first came into vogue, but I should imagine we owe it principally to the peculiar development of geometrical speculation which began with Gauss, and which received from him the name 'nonEuclidean'. In the course of that development, geometry, as it was known to Euclid, became one of several possible geometrical systems', each of which was associated with, was involved in or involved, a particular conception of space. Such phrases as 'kinds of space ', ' species of space,' ' varieties of space' (whether expressive of real conceptual development or of mystical illusions) once admitted, would naturally lead to the use of the term properties' in relation to space, for already in the use of such terms as 'kinds', 'species,'' varieties,' space has been assimilated, however distantly, with body. If it is urged that these terms are thus employed metaphorically, the answer is inevitable; metaphor in the use of words expresses either the awareness of an analogy in thought, or the belief that we have, or ought to have, this awareness. In neither case can we escape the consequences, though we may sometimes succeed in shutting our eyes to them. They are very obvious in Helmholtz's essay ' On the Origin and Significance of Geometrical Axioms', especially where, in order to illustrate the differences between the Euclidean and each of the two principal non-Euclidean spaces, he admits the supposition of a moderately rigid body transferred from one kind of space to another (though the kinds cannot, of course, co-exist), and considers the changes of shape to which it would thus be subjected—an illustration which involves an obscure and yet persistent notion of space acting on body, which is a stultification of the notion of space itself.

Different self-consistent systems of geometry, if each system is to be exclusive of the others, appear to be possible only on the condition that we have several different notions of space,1 each one exclusive of the others, and the fons et origo of one of these systems of geometry; for it is clear that not only the individual's first obscure awareness of space, but also the idea or notion arising from the perpetual process of comparison which goes 1 Or, again, several different notions of metrical relation (Cayley's Theory of Distance).

on in this developing awareness, are pre-conditions of his geometrizing. From this point of view the emergence in the first half of the nineteenth century of different systems of geometry is not a little enigmatic, for, as a matter of fact, the origin of these non-Euclidean geometrical systems is attributed by the originators themselves (e. g. by Lobatschewsky in his Studies on the Theory of Parallels) not to any modification in the conception of space, but to the alleged incertitude involved in Euclid's so-called Axiom of Parallels. These systems then led, or were supposed to lead, to different notions of space. It may indeed be held that the uncertainty alleged to be involved in Euclid's axiom is itself the result of an indefiniteness in the notion of space, and that the development of self-consistent but mutually exclusive systems of geometry was a clearing-up of this indefiniteness, a vanishing of it in the establishment of definite but differing notions of space. To discuss that question is, in large measure, to discuss the philosophy of modern geometry.

A brief yet comprehensive view of geometry as it presents itself to a considerable number of contemporary mathematicians is given by L. Couturat in his Principles of Mathematics.1 Mr. Couturat expressly disclaims any pretensions to originality, and this circumstance is, in his eyes, precisely what should recommend his work to the reader. He means by this that he gives expression therein not to an isolated set of opinions, but in the main to the opinions elaborated and largely held in common by a number of brilliant as well as patient modern investigators of mathematical principles. In original intention, as Mr. Couturat explains in his preface, the work was to be merely an account of the magistral ouvrage of Mr. Bertrand Russell,2 but in the course of this exposition Mr. Couturat found himself gradually drawn to include in his account a brief analysis of the main part of the work of contemporary mathematicians on the same subject. We have thus in this useful work, and within the compass of some 200 pages, the pith and marrow of the new views on mathematical principles, the remaining pages of the volume being assigned to an interesting and damaging criticism of Kant's philosophy of mathematics.

Geometry'-says Mr. Couturat-' is still commonly regarded 1 Les Principes des Mathématiques, Paris, Félix Alcan, 1905.

2 The Principles of Mathematics, vol. i, Cambridge University Press, 1903.

as being the science of space. According to sound method then, it would seem that we ought to begin with a definition of space. Now in the first place such a definition is very difficult and complicated, and in the next it is perfectly useless: the idea of and the very word space are not to be found either in Euclid or in Archimedes. The same may be said of the notions of the line and the surface, which Euclid himself attempts to define at the outset of his Elements. The definition of these general notions requires very great tact, and they become rigorous only by the aid of the integral calculus; and this is equivalent to saying that their place is neither in the elements nor in the principles of Geometry. It must not then be supposed that if these three notions cannot be defined at the outset of geometry this is because they are primary, fundamental, and simple; on the contrary, it is because they are very complex, and we can constitute geometry perfectly well without them, as will presently be seen. Geometry is founded, not upon the general and vague ideas of space, of surface, and of line, but upon the particular and precise ideas of the straight line, the plane, and especially the point; and it is among these that we find the primary and indefinable notions of this science. The point especially is the indefinable element of all systems of geometry. Points are the individual terms of all the relations the study of which constitutes the several geometries; and if space can be defined at the outset of geometry, it must be as the aggregate of points' 1 (l'ensemble des points).

It is clear from this passage that the terms 'definition' and indefinable' are not employed in accordance with general custom, but in a technical sense especially relevant to mathematics, approaching to, if not actually identical with, what is commonly understood by calculation or determination (cf. the allusion to the integral calculus in relation to the definitions of the line and the surface). If we were to agree that the idea of one' is simple and indefinable, we might then also agree that all finite integral numbers are definable by means of this indefinable, together with the idea (whether indefinable or not) of addition. But to force an interpretation upon a term which already has a recognized meaning, when there is another term whose recognized meaning is that forced upon the first, is a procedure which tends ultimately to confusion of ideas. And this does in some measure show itself, no matter which of the new expositions of elementary geometry we may turn to. The geometer finds himself obliged to give some definitions in the 1 Op. cit., pp. 126, 127.