Introduction To Geometry

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. The Foundations of Geometry: Introduction

PREFACE.

The present work is based on a dissertation submitted at the Fellowship Examination of Trinity College, Cambridge, in the year 1895. Section B of the third chapter is in the main a reprint, with some serious alterations, of an article in Mind (New Series, No. 17). The substance of the book has been given in the form of lectures at the Johns Hopkins University, Baltimore, and at Bryn Mawr College, Pennsylvania.

My chief obligation is to Professor Klein. Throughout the first chapter, I have found his "Lectures on non-Euclidean Geometry" an invaluable guide; I have accepted from him the division of Metageometry into three periods, and have found my historical work much lightened by his references to previous writers. In Logic, I have learnt most from Mr Bradley, and next to him, from Sigwart and Dr Bosanquet. On several important points, I have derived useful suggestions from Professor James's "Principles of Psychology."

My thanks are due to Mr G. F. Stout and Mr A. N. Whitehead for kindly reading my proofs, and helping me by many useful criticisms. To Mr Whitehead I owe, also, the inestimable assistance of constant criticism and suggestion throughout the course of construction, especially as regards the philosophical importance of projective Geometry.

Haslemere.

May, 1897.

INTRODUCTION.

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

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[2] 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[3] 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 dangerous, when we reflect that such a view places our results, as to the à priori, at the mercy of empirical psychology. How serious this danger is, the controversy as to Kant's pure intuition sufficiently shows.

   
   

5. I shall, therefore, throughout the present Essay, use the word à priori without any psychological implication. My test of apriority will be purely logical: Would experience be impossible, if a certain axiom or postulate were denied? Or, in a more restricted sense, which gives apriority only within a particular science: Would experience as to the subject-matter of that science be impossible, without a certain axiom or postulate? My results also, therefore, will be purely logical.

   
   

   If Psychology declares that some things, which I have declared à priori, are not subjective, then, failing an error of detail in my proofs, the connection of the à priori and the subjective,[4] so far as those things are concerned, must be given up. There will be no discussion, accordingly, throughout this Essay, of the relation of the à priori to the subjective—a relation which cannot determine what pieces of knowledge are à priori, but rather depends on that determination, and belongs, in any case, rather to Metaphysics than to Epistemology.

   
   

6. As I have ventured to use the word à priori in a slightly unconventional sense, I will give a few elucidatory remarks of a general nature. The à priori, since Kant at any rate, has generally stood for the necessary or apodeictic element in knowledge. But modern logic has shown that necessary propositions are always, in one aspect at least, hypothetical. There may be, and usually is, an implication that the connection, of which necessity is predicated, has some existence, but still, necessity always points beyond itself to a ground of necessity, and asserts this ground rather than the actual connection. As Bradley points out, "arsenic poisons" remains true, even if it is poisoning no one. If, therefore, the à priori in knowledge be primarily the necessary, it must be the necessary on some hypothesis, and the ground of necessity must be included as à priori. But the ground of necessity is, so far as the necessary connection in question can show, a mere fact, a merely categorical judgment. Hence necessity alone is an insufficient criterion of apriority.

   To supplement this criterion, we must supply the hypothesis or ground, on which alone the necessity holds, and this ground will vary from one science to another, and even, with the progress of knowledge, in the same science at different times. For as knowledge becomes more developed and articulate, more and more necessary connections are perceived, and the merely categorical truths, though they remain the foundation of apodeictic judgments, diminish in relative number.

   
   

   Nevertheless, in a fairly advanced science such as Geometry, we can, I think, pretty completely supply the appropriate ground, and establish, within the limits of the isolated science, the distinction between the necessary and the merely assertorical.

   
   

7. There are two grounds, I think, on which necessity may be sought within any science. These may be (very roughly) distinguished as the ground which Kant seeks in the[5] Prolegomena, and that which he seeks in the Pure Reason. We may start from the existence of our science as a fact, and analyse the reasoning employed with a view to discovering the fundamental postulate on which its logical possibility depends; in this case, the postulate, and all which follows from it alone, will be à priori.

   
   

   Or we may accept the existence of the subject-matter of our science as our basis of fact, and deduce dogmatically whatever principles we can from the essential nature of this subject-matter. In this latter case, however, it is not the whole empirical nature of the subject-matter, as revealed by the subsequent researches of our science, which forms our ground; for if it were, the whole science would, of course, be à priori. Rather it is that element, in the subject-matter, which makes possible the branch of experience dealt with by the science in question[2]. The importance of this distinction will appear more clearly as we proceed[3].

   
   

8. These two grounds of necessity, in ultimate analysis, fall together. The methods of investigation in the two cases differ widely, but the results cannot differ. For in the first case, by analysis of the science, we discover the postulate on which alone its reasonings are possible. Now if reasoning in the science is impossible without some postulate, this postulate must be essential to experience of the subject-matter of the science, and thus we get the second ground. Nevertheless, the two methods are useful as supplementing one another, and the first, as starting from the actual science, is the safest and easiest method of investigation, though the second seems the more convincing for exposition.

   
   

9. The course of my argument, therefore, will be as follows: In the first chapter, as a preliminary to the logical analysis of Geometry, I shall give a brief history of the rise and development of non-Euclidean systems. The second chapter will prepare the ground for a constructive theory of Geometry, by a criticism of some previous philosophical views; in this chapter, I shall[6] endeavour to exhibit such views as partly true, partly false, and so to establish, by preliminary polemics, the truth of such parts of my own theory as are to be found in former writers.

   
   

   A large part of this theory, however, cannot be so introduced, since the whole field of projective Geometry, so far as I am aware, has been hitherto unknown to philosophers. Passing, in the third chapter, from criticism to construction, I shall deal first with projective Geometry. This, I shall maintain, is necessarily true of any form of externality, and is, since some such form is necessary to experience, completely à priori. In metrical Geometry, however, which I shall next consider, the axioms will fall into two classes: (1) Those common to Euclidean and non-Euclidean spaces. These will be found, on the one hand, essential to the possibility of measurement in any continuum, and on the other hand, necessary properties of any form of externality with more than one dimension. They will, therefore, be declared à priori. (2) Those axioms which distinguish Euclidean from non-Euclidean spaces. These will be regarded as wholly empirical.

   
   

   The axiom that the number of dimensions is three, however, though empirical, will be declared, since small errors are here impossible, exactly and certainly true of our actual world; while the two remaining axioms, which determine the value of the space-constant, will be regarded as only approximately known, and certain only within the errors of observation[4]. The fourth chapter, finally, will endeavour to prove, what was assumed in Chapter III., that some form of externality is necessary to experience, and will conclude by exhibiting the logical impossibility, if knowledge of such a form is to be freed from contradictions, of wholly abstracting this knowledge from all reference to the matter contained in the form.

   I shall hope to have touched, with this discussion, on all the main points relating to the Foundations of Geometry.

   FOOTNOTES:

   
   

[1] Cf. Erdmann, Axiome der Geometrie, p. 111: "Für Kant sind Apriorität und ausschliessliche Subjectivität allerdings Wechselbegriffe."

[2] I use "experience" here in the widest possible sense, the sense in which the word is used by Bradley.

[3] Where the branch of experience in question is essential to all experience, the resulting apriority may be regarded as absolute; where it is necessary only to some special science, as relative to that science.

[4] I have given no account of these empirical proofs, as they seem to be constituted by the whole body of physical science. Everything in physical science, from the law of gravitation to the building of bridges, from the spectroscope to the art of navigation, would be profoundly modified by any considerable inaccuracy in the hypothesis that our actual space is Euclidean. The observed truth of physical science, therefore, constitutes overwhelming empirical evidence that this hypothesis is very approximately correct, even if not rigidly true.

About HackerNoon Book Series: We bring you the most important technical, scientific, and insightful public domain books.

This book is part of the public domain. Bertrand Russell. (1897). AN ESSAY ON THE FOUNDATIONS OF GEOMETRY. Urbana, Illinois: Project Gutenberg. Retrieved DATE, from https://www.gutenberg.org/cache/epub/52091/pg52091-images.html.

This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever.  You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org, located at https://www.gutenberg.org/policy/license.html.