An Essay on the Foundations of Geometry An Essay on the Foundations of Geometry was first published in 1897 when Bertrand Russell was 25 years old. It marks his first major foray into analytic philos- ophy, a movement in which Russell is one of the founding members and figurehead. It provides a brilliant insight into Russell’s early philosophical thought and an engaging and authoritative introduction to the philosophical and logical foundations of geometry – a version of which was fundamental to Einstein’s theory of relativity. Russell explores and introduces the concepts of geometry and their philo- sophical implications, including a historical overview of geometrical theory, making it an invaluable resource not only for students of philosophy but any- one interested in the origins of the thought of one of the twentieth century’s most important and widely-read philosophers. This Routledge Classics edition includes a new Foreword by Michael Potter. Bertrand Russell (1872–1970). A celebrated mathematician and logician, Russell was and remains one of the most genuinely widely read and popular philosophers of modern times. ‘Routledge Classics is more than just a collection of texts… it embodies and circulates challenging ideas and keeps vital debates current and alive.’ – Hilary Mantel The Routledge Classics series contains the very best of Rou- tledge’s publishing over the past century or so, books that have, by popular consent, become established as classics in their field. Drawing on a fantastic heritage of innovative writ- ing published by Routledge and its associated imprints, this series makes available in attractive, affordable form some of the most important works of modern times. For a complete list of titles visit: https://www.routledge.com/Routledge-Classics/book- series/SE0585 Bertrand Russell An Essay on the Foundations of Geometry With a new Foreword by Michael Potter London and New York Cover image: billnoll / Getty Images First published in Routledge Classics 2023 by Routledge 4 Park Square, Milton Park, Abingdon, Oxon OX14 4RN and by Routledge 605 Third Avenue, New York, NY 10158 Routledge is an imprint of the Taylor & Francis Group, an informa business © 2008, 2023 Bertrand Russell Peace Foundation Ltd Foreword © 2023 Michael Potter The right of the Bertrand Russell Peace Foundation Ltd to be identified as author of this work has been asserted by them in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. First published 1897 by Cambridge University Press British Library C ataloguing-i n-P ublication Data A catalogue record for this book is available from the British Library Library of Congress C ataloging-i n-P ublication Data A catalog record has been requested for this book ISBN: 9 78-1 -0 32-3 1264-4 (h bk) ISBN: 9 78- 1- 032- 31226-2 (p bk) ISBN: 9 78-1 -0 03-3 0887-4 (e bk) DOI: 10.4324/9 781003308874 Typeset in Joanna by codeMantra Contents FOREWORD TO THE ROUTLEDGE CLASSICS EDITION ix PREFACE xvii Introduction: Our Problem Defined by its Relations to Logic, Psychology and Mathematics 1 1 A Short History of Metageometry 8 2 Critical Account of Some Previous Philosophical Theories of Geometry 55 3 Section A: The Axioms of Projective Geometry 117 Section B: The Axioms of Metrical Geometry 144 4 Philosophical Consequences 177 INDEX 201 Foreword to the Routledge Classics Edition As a student at Cambridge, Bertrand Russell read mathematics for three years and moral sciences (philosophy) for one, sitting his fi- nal exams in 1894. As was usual for anyone wishing to pursue an academic career (there being at that time no PhD degree), he then worked on a Fellowship dissertation without formal supervision, re- ceiving only brief initial guidance from two of his undergraduate su- pervisors, the philosopher James Ward and the mathematician A. N. Whitehead. The subject he chose, the philosophical foundations of geometry, was a development of one of the essays he had written for Ward as an undergraduate. He submitted the dissertation at Trinity College in August 1895 and was elected to a Fellowship on that ba- sis shortly thereafter. He revised the dissertation over the next year (partly, it seems, in response to criticisms Whitehead had made of its focus on metrical rather than descriptive geometry) and the Essay on the Foundations of Geometry was published in May 1897. The book received some attention, especially in France, where it was the subject