Mathematics and science PDF
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MATHEMATICS and Science i This page is intentionally left blank MATHEMATICS and Science edited by Ronald E. Mickens Department of Physics Clark Atlanta University USA %b World Scientific wH Singapore • New Jersey • London • Hong Kong Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 9128 USA office: 687 Hartwell Street, Teaneck, NJ 07666 UK office: 73 Lynton Mead, Totteridge, London N20 8DH MATHEMATICS AND SCIENCE Copyright © 1990 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photo copying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. ISBN 981-02-0233-4 Printed in Singapore by JBW Printers & Binders Pte. Ltd. Preface The spectacular successes of the sciences is due in large measure to the use of mathematics in the creation and analysis of models for the pheno mena of interest. Clearly, mathematics is a very effective tool/language for the formulation of scientific theories. Why is this the case? The question has been considered by philosophers, mathematicians and scientists since the dawn of human understanding and appreciation of mathematics, and the study of natural systems.1 Mathematics seems to be an intellectual activity separate from the sciences. Its postulates may come from the pro cesses of pure thought or may be suggested by experience. In either case, once formulated, they form a basis for which the structure of a mathemat ical system can be developed independently of their genesis. In recent times, the essay of the physicist Eugene P. Wigner sparked renewed interest in this problem.2 He asserted that "the unreasonable ef fectiveness of mathematics* is a mystery whose understanding and solution is yet to come: The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure even though perhaps also to our bafflement, to wide branches of learning, (p. 14) The purpose of the present book is to present the opinions of a num ber of distinguished individuals who have given careful thought as to why mathematics is so "unreasonably effective" when applied to the analysis of the natural sciences. (A very broad view is taken as to what constitutes the "natural" sciences. Included are the physical, biological and social sci- V VI Preface ences.) Each essay discusses one or more of the following questions: 1. What is the nature of mathematical truth? 2. Why is mathematics the language of science? 3. Is mathematics a science? 4. What properties does an area of knowledge have to possess before it is considered as being a science? 5. How has mathematics influenced the direction, interpretation and analysis of (particular areas of) science? 6. How has science influenced the direction of mathematical research? It is not to be expected that the essays of this volume will settle, once and for all times, the "problem" of the effectiveness (unreasonable or not) of mathematics in the sciences. However, each essay, in its own right, provides a valuable contribution to a better understanding of the various issues of relevance for this topic. The essays of this volume represent the personal reflections, thoughts and styles of the various contributors. Consequently, as editor, I have made only trivial changes to the manuscripts as I received and reviewed them. I hope that readers will find its contents as interesting and stimulating as I do. At the end of this volume is a Bibliography. It contains a short list of books and papers that give further discussions on the relationship between mathematics and the sciences. I would like to thank Dr. K. K. Phua, Editor-in-Chief of World Scientific Publishing for his enthusiastically endorsing this publishing project. I also wish to thank my wife, Maria Mickens, for not only her typing responsibilities related to this book, but, also, for her understanding and patience. RONALD E. MICKENS May 1990 Atlanta, GA References 1. J. V. Graniner, Math. Magazine 61 (1988) 220-230. 2. E. P. Wigner, Commun. Pure Appl. Math. 13 (1960) 1-14. Contents Preface v On the Effectiveness and Limits of Mathematics in Physics 1 A. O. Barut Why is the Universe Knowable? 14 P. C. W. Davits Mathematics in Sociology: Cinderella's Carriage or Pumpkin? 34 Patrick Doreian Fundamental Roles of Mathematics in Science 55 Donald Greenspan Inner Vision, Outer Truth 64 Reuben Hersh Mathematics and the Natural Order 73 Wendell G. Holladay A Few Systems-Colored Views of the World 94 Yi Lin The Reasonable Effectiveness of Mathematical Reasoning 115 Saunders Mac Lane Vlll Contents Three Aspects of the Effectiveness of Mathematics in Science 122 Louts Narens and R. Duncan Luce Mathematics and Natural Philosophy 136 Robert L. Oldershaw Mathematics and the Language of Nature 154 F. David Peat The Reason Within and the Reason Without 173 John Polkinghorne The Modelling Relation and Natural Law 183 Robert Rosen Structure and Effectiveness 200 La Verne Shelton Psychology and Mathematics 223 James T. Towns end and Helena Kadlec Ariadne's Thread: The Role of Mathematics in Physics 249 Hans C. von Baeyer The Disproportionate Response 258 Bruce J. West The Unreasonable Effectiveness of Mathematics in the Natural Sciences 291 Eugene P. Wigner (reprint) The Effectiveness of Mathematics in Fundamental Physics 307 A. Zee Bibliography 325 Subject Index 329 MATHEMATICS and Science
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