SPACE, TIME AND GEOMETR Y

Transcription

1 SPACE, TIME AND GEOMETR Y

2 SYNTHESE LIBRARY MONOGRAPHS ON EPISTEMOLOGY, LOGIC, METHODOLOGY, PHILOSOPHY OF SCIENCE, SOCIOLOGY OF SCIENCE AND OF KNOWLEDGE, AND ON THE MATHEMATICAL METHODS OF SOCIAL AND BEHAVIORAL SCIENCES Editors: DONALD DAVIDSON, The Rockefeller University and Princeton University JAAKKO HINTIKKA, Academy of Finland and Stanford University GABRIEL NUCHELMANS, University of Leyden WESLEY C. SALMON, Indiana University

3 SPACE, TIME AND GEOMETRY Edited by PATRICK SUPPES Stanford University D. REIDEL PUBLISHING COMPANY DORDRECHT-HOLLAND / BOSTON-U.S.A.

4 First printing.' December 1973 Library of Congress Catalog Card Number ISBN-13: I DOl: / I e-isbn-13: Published by D. Reidel Publishing Company, P.O. Box 17, Dordrecht, Holland Sold and distributed in the U.S.A., Canada and Mexico by D. Reidel Publishing Company, Inc. 306 Dartmouth Street, Boston, Mass , U.S.A. All Rights Reserved Copyright 1973 by D. Reidel Publishing Company, Dordrecht-Holland Sotlcover reprint of the hardcover I st edition 1973 No part of this book may be reproduced in any form, by print, photo print, microfilm, or any other means, without written permission from the publisher

5 TABLE OF CONTENTS PREFACE INTRODUCTION VII x PART I/CAUSALITY AND TIME ZOLTAN DOMOTOR / Causal Models and Space-Time Geometries 3 GEORGE BERGER / Temporally Symmetric Causal Relations in Minkowski Space-Time 56 JOHN EAR MAN / Notes on the Causal Theory of Time 72 BAS C. VAN FRAASSEN / Earman on the Causal Theory of Time 85 ROBER T PAL TER / Kant's Formulation of the Laws of Motion 94 ROBER T WEINGARD / On Travelling Backward in Time 115 P. J. ZWART / The Flow of Time 131 PART II/GEOMETRYOF SPACE AND TIME JULES VUILLEMIN / Poincare's Philosophy of Space 159 CRAIG HARRISON / On the Structure of Space-Time 178 CLARK GLYMOUR / Topology, Cosmology and Convention 193 MICHAEL FRIEDMAN / Griinbaum on the Conventionality of Geometry 217 AR THUR FINE / Reflections on a Relational Theory of Space 234 ADOLF GRUNBAUM / The Ontology of the Curvature of Empty Space in the Geometrodynamics of Clifford and Wheeler 268

6 VI TABLE OF CONTENTS MICHAEL FRIEDMAN / Relativity Principles, Absolute Objects and Symmetry Groups 296 ROB E R T W. LA T Z E R / N ondirected Light Signals and the Structure of Time 321 RICHARD H. HUDGIN / Coordinate-Free Relativity 366 PATRICK SUPPES / Some Open Problems in the Philosophy of Space and Time 383 ERNEST W. ADAMS / The Naive Conception of the Topology of the Surface of a Body 402

7 PREFACE The articles in this volume have been stimulated in two different ways. More than two years ago the editor of Synthese, laakko Hintikka, announced a special issue devoted to space and time, and articles were solicited. Part of the reason for that announcement was also the second source of papers. Several years ago I gave a seminar on special relativity at Stanford, and the papers by Domotor, Harrison, Hudgin, Latzer and myself partially arose out of discussion in that seminar. All of the papers except those of Griinbaum, Fine, the second paper of Friedman, and the paper of Adams appeared in a special double issue of Synthese (24 (1972), Nos. 1-2). I am pleased to have been able to add the four additional papers mentioned in making the special issue a volume in the Synthese Library. Of these four additional articles, only the one by Fine has previously appeared in print (Synthese 22 (1971), ); its relevance to the present volume is apparent. In preparing the papers for publication and in carrying out the various editorial chores of such a task, I am very much indebted to Mrs. Lillian O'Toole for her extensive assistance.

8 INTRODUCTION The philosophy of space and time has been of permanent importance in philosophy, and most of the major historical figures in philosophy, such as Aristotle, Descartes and Kant, have had a good deal to say about the nature of space and time. This classical tradition of analysis of spatial and temporal concepts has been relatively informal and accessible to the general philosopher. It is probably correct to say that the level of accessibility of the philosophy of mathematics and the philosophy of physics is about the same in the writings of important systematic philosophers from Aristotle to Kant. Beginning in the latter part of the nineteenth century with the work of Frege and others, the philosophy of mathematics took a sharp tum away from this broad general tradition and became a more technical and more deeply developed and specialized subject. Today, neither philosophers nor mathematicians who do not have some special interest in foundations of mathematics attempt to contribute to the large and continually developing literature. Until fairly recently, the same line of deeper development was not characteristic of the philosophy of physics. Important philosophers of physics like Mach and Bridgman, writing over the same time span that runs from Frege to G6del and Tarski, presented a body of ideas and methodological concepts that were clearly on a less satisfactory intellectual level than work in the philosophy of mathematics. Of course, there were notable exceptions to the relatively mediocre intellectual standard set by Mach and Bridgman. Analysis of some of that early work may be found in various articles in the present volume. For example, Domotor discusses the important work of Robb on the geometry of special relativity, and Vuillemin emphasizes aspects of Poincare's philosophy of space that are not as well known as they should be. By mentioning only Robb and Poincare, I do not mean to suggest that they are the only exceptions to the Mach-Bridgman standard. The literature on the philosophy of physics is enormous, and many good things can be found in that literature in the last half of the nineteenth century and the early decades of this century.

9 x INTRODUCTION All the same, when one looks at the development of the foundations of geometry over this period it is surprising and disappointing that this development has not had the impact on the philosophy of space and time that the axiomatic theory of the natural numbers and axiomatic set theory have had on the philosophy of mathematics. Fortunately, that time seems to have passed and in the articles contained in this volume alone, we can see a renaissance of interest in the philosophy of space and time among younger philosophers. What is most pleasing is their ability to deal with the geometry of space and time in a manner that does not avoid the technical issues and results, but that still places an emphasis on matters of philosophical interest. Except for Griinbaum, who is more or less the dean of American philosophers interested in space and time, Vuillemin, who represents a return of French philosophy to the intellectual traditions of Descartes and Poincare, and myself, all authors of the articles in this volume are, I believe, under forty or forty-five, and, with the exception of Hudgin and Latzer, all are philosophers. The seven articles of Part I deal with the many issues about causality and time that have much occupied recent literature of the philosophy of physics. In the first four articles of Domotor, Berger, Earman and van Fraassen, most of the issues of current interest are pushed back and forth in the implicit defense and criticism of the main views now current among philosophers about these matters. The article by Palter analyzes Kant's formulation of the laws of motion, and in doing so, brings attention to Kant's Metaphysical Foundations of Natural Science, which has been too much neglected in discussions of Kant's philosophy of space and time. The last two articles of Part I, by Weingard and Zwart, discuss some of the issues that have arisen about time reversibility and the flow of time. The longer Part II contains eleven papers on the geometry of space and time. Vuillemin's article on Poincare has already been mentioned. Harrison discusses problems about the structure of space-time that go back to Newton and end with contemporary discussions about the status of the relational theory of time in the theory of relativity. Glymour discusses conventionality in the philosophy of space and time from the standpoint of topology and, thereby, offers some new ways of analyzing this venerable topic. The first article by Friedman and the article by Fine

10 INTRODUCTION XI take a critical look at some of the issues that Griinbaum has discussed extensively in previous publications. Griinbaum's own article sets forth his views on the ontology of the curvature of empty space, especially in the geometrodynamics of Clifford and Wheeler. Friedman's second article has much to say about principles of relativity and the ontological status of objects. The articles of Latzer and Hudgin present some axiomatic results of foundational interest. Those who have followed the axiomatic efforts in connection with the geometry of special relativity will find especially interesting Latzer's alternative to Robb's approach in his use of a binary primitive for nondirected light signals. In the next article I formulate some open problems in the philosophy of space and time and, in this formulation, stress the desirability of a still closer interaction between the foundations of geometry and the philosophy of space and time. In the final article Adams analyzes the geometry and topology of the surfaces of physical objects. PATRICK SUPPES