An essay on the foundations of geometry. Bertrand Russell

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       Bertrand Russell

      An essay on the foundations of geometry

      Published by Good Press, 2019

       [email protected]

      EAN 4057664648976

Table of Contents

       INTRODUCTION. OUR PROBLEM DEFINED BY ITS RELATIONS TO LOGIC, PSYCHOLOGY AND MATHEMATICS.

       CHAPTER I. A SHORT HISTORY OF METAGEOMETRY.

       First Period.

       Second Period.

       Third Period.

       CHAPTER II. CRITICAL ACCOUNT OF SOME PREVIOUS PHILOSOPHICAL THEORIES OF GEOMETRY.

       Kant.

       Riemann.

       Helmholtz.

       Erdmann.

       Lotze.

       Delbœuf.

       CHAPTER III.

       Section A. THE AXIOMS OF PROJECTIVE GEOMETRY.

       Section B. THE AXIOMS OF METRICAL GEOMETRY.

       CHAPTER IV. PHILOSOPHICAL CONSEQUENCES.

INTRODUCTION.

       OUR PROBLEM DEFINED BY ITS RELATIONS TO LOGIC, PSYCHOLOGY AND MATHEMATICS.

       Table of Contents

      1. Geometry, throughout the 17th and 18th centuries, remained, in the war against empiricism, an impregnable fortress of the idealists. Those who held—as was generally held on the Continent—that certain knowledge, independent of experience, was possible about the real world, had only to point to Geometry: none but a madman, they said, would throw doubt on its validity, and none but a fool would deny its objective reference. The English Empiricists, in this matter, had, therefore, a somewhat difficult task; either they had to ignore the problem, or if, like Hume and Mill, they ventured on the assault, they were driven into the apparently paradoxical assertion that Geometry, at bottom, had no certainty of a different kind from that of Mechanics—only the perpetual presence of spatial impressions, they said, made our experience of the truth of the axioms so wide as to seem absolute certainty.

      Here, however, as in many other instances, merciless logic drove these philosophers, whether they would or no, into glaring opposition to the common sense of their day. It was only through Kant, the creator of modern Epistemology, that the geometrical problem received a modern form. He reduced the question to the following hypotheticals: If Geometry has apodeictic certainty, its matter, i.e. space, must be à priori, and as such must be purely subjective; and conversely, if space is purely subjective, Geometry must have apodeictic certainty. The latter hypothetical has more weight with Kant, indeed it is ineradicably bound up with his whole Epistemology; nevertheless it has, I think, much less force than the former. Let us accept, however, for the moment, the Kantian formulation, and endeavour to give precision to the terms à priori and subjective.

2. One of the great difficulties, throughout this controversy, is the extremely variable use to which these words, as well as the word empirical, are put by different authors. To Kant, who was nothing of a psychologist, à priori and subjective were almost interchangeable terms[1]; in modern usage there is, on the whole, a tendency to confine the word subjective to Psychology, leaving à priori to do duty for Epistemology. If we accept this differentiation, we may set up, corresponding to the problems of these two sciences, the following provisional definitions: à priori applies to any piece of knowledge which, though perhaps elicited by experience, is logically presupposed in experience: subjective applies to any mental state whose immediate cause lies, not in the external world, but within the limits of the subject. The latter definition, of course, is framed exclusively for Psychology: from the point of view of physical Science all mental states are subjective. But for a Science whose matter, strictly speaking, is only mental states, we require, if we are to use the word to any purpose, some differentia among mental states, as a mark of a more special subjectivity on the part of those to which this term is applied.

      Now the only mental states whose immediate causes lie in the external world are sensations. A pure sensation is, of course, an impossible abstraction—we are never wholly passive under the action of an external stimulus—but for the purposes of Psychology the abstraction is a useful one. Whatever, then, is not sensation, we shall, in Psychology, call subjective. It is in sensation alone that we are directly affected by the external world, and only here does it give us direct information about itself.

      3. Let us now consider the epistemological question, as to the sort of knowledge which can be called à priori. Here we have nothing to do—in the first instance, at any rate—with the cause or genesis of a piece of knowledge; we accept knowledge as a datum to be analysed and classified. Such analysis will reveal a formal and a material element in knowledge. The formal element will consist of postulates which are required to make knowledge possible at all, and of all that can be deduced from these postulates; the material element, on the other hand, will consist of all that comes to fill in the form given by the formal postulates—all that is contingent or dependent on experience, all that might have been otherwise without rendering knowledge impossible. We shall then call the formal element à priori, the material element empirical.

      4. Now what is the connection between the subjective and the à priori? It is a connection, obviously—if it exists at all—from the outside, i.e. not deducible directly from the nature of either, but provable—if it can be proved—only by a general view of the conditions of both. The question, what knowledge is à priori, must, on the above definition, depend on a logical analysis of knowledge, by which the conditions of possible experience may be revealed; but the question, what elements of a cognitive state are subjective, is to be investigated by pure Psychology, which has to determine what, in our perceptions, belongs to sensation, and what is the work of thought or of association. Since, then, these two questions belong to different sciences, and can be settled independently, will it not be wise to conduct the two investigations separately? To decree that the à priori shall always be subjective, seems