An essay on the Foundations of Geometry: Chapter III - Section B, Part II

An Essay on the Foundations of Geometry, by Bertrand Russell is part of HackerNoon’s Book Blog Post series. The Table of Links for this book can be found here. Chapter III - Section B, Part II: The Axiom of Dimensions

II. The Axiom of Dimensions

158. We have seen, in discussing the axiom of Free Mobility, that all position is relative, that is, a position exists only by virtue of relations. It follows that, if positions can be defined at all, they must be uniquely and exhaustively defined by some finite number of such relations. If Geometry is to be possible, it must happen that, after enough relations have been given to determine a point uniquely, its relations to any fresh known point are deducible from the relations already given. Hence we obtain, as an à priori condition of Geometry, logically indispensable to its existence, the axiom that Space must have a finite integral number of Dimensions. For every relation required in the definition of a point constitutes a dimension, and a fraction of a relation is meaningless. The number of relations required must be finite, since an infinite number of dimensions would be practically impossible to determine. If we remember our axiom of Free Mobility, and remember also that space is a continuum, we may state our axiom, for metrical Geometry, in the form given by Helmholtz (v. Chap. I. § 25): "In a space of n dimensions, the position of every point is uniquely determined by the measurement of n continuous independent variables (coordinates)."

159. So much, then, is à priori necessary to metrical Geometry. The restriction of the dimensions to three seems, on the contrary, to be wholly the work of experience. This restriction cannot be logically necessary, for as soon as we have formulated any analytical system, it appears wholly arbitrary. Why, we are driven to ask, cannot we add a fourth coordinate to our x, y, z, or give a geometrical meaning to x4? In this more special form, we are tempted to regard the axiom of dimensions, like the number of inhabitants of a town, as a purely statistical fact, with no greater necessity than such facts have.

Geometry affords intrinsic evidence of the truth of my division of the axiom of dimensions into an à priori and empirical portion. For while the extension of the number of dimensions to four, or to n, alters nothing in plane and solid Geometry, but only adds a new branch which interferes in no way with the old, some definite number of dimensions is assumed in all Geometries, nor is it possible to conceive of a Geometry which should be free from this assumption.

160. Let us, since the point seems of some interest, repeat our proof of the apriority of this axiom from a slightly different point of view. We will begin, this time, from the most abstract conception of space, such as we find in Riemann's dissertation, or in Erdmann's extents. We have here, an ordered manifold, infinitely divisible and allowing of Free Mobility. Free Mobility involves, as we saw, the power of passing continuously from any one point to any other, by any course which may seem pleasant to us; it involves, also, that, in such a course, no changes occur except changes of mere position, i.e., positions do not differ from one another in any qualitative way. (This absence of qualitative difference is the distinguishing mark of space as opposed to other manifolds, such as the colour- and tone-systems: in these, every element has a definite qualitative sensational value, whereas in space, the sensational value of a position depends wholly on its spatial relation to our own body, and is thus not intrinsic, but relative.) From the absence of qualitative differences among positions, it follows logically that positions exist only by virtue of other positions; one position differs from another just because they are two, not because of anything intrinsic in either. Position is thus defined simply and solely by relation to other positions. Any position, therefore, is completely defined when, and only when, enough such relations have been given to enable us to determine its relation to any new position, this new position being defined by the same number of relations. Now, in order that such definition may be at all possible, a finite number of relations must suffice. But every such relation constitutes a dimension. Therefore, if Geometry is to be possible, it is à priori necessary that space should have a finite integral number of dimensions.

161. The limitation of the dimensions to three is, as we have seen, empirical; nevertheless, it is not liable to the inaccuracy and uncertainty which usually belong to empirical knowledge. For the alternatives which logic leaves to sense are discrete—if the dimensions are not three, they must be two or four or some other number—so that small errors are out of the question. Hence the final certainty of the axiom of three dimensions, though in part due to experience, is of quite a different order from that of (say) the law of Gravitation. In the latter, a small inaccuracy might exist and remain undetected; in the former, an error would have to be so considerable as to be utterly impossible to overlook. It follows that the certainty of our whole axiom, that the number of dimensions is three, is almost as great as that of the à priori element, since this element leaves to sense a definite disjunction of discrete possibilities.

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This book is part of the public domain. Bertrand Russell. (1897). AN ESSAY ON THE FOUNDATIONS OF GEOMETRY. Urbana, Illinois: Project Gutenberg. Retrieved June 2022, from https://www.gutenberg.org/files/52091/52091-h/52091-h.htm#N158

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