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PROBABILITY AND SCIENTIFIC INFERENCE 1 PROBABILITY AND SCIENTIFIC INFERENCE G. SPENCER BROWN Research Lecturer of Christ Church, Oxford LONGMANS, GREEN AND CO LONDON • NEW YORK • TORONTO LONGMANS, GREEN AND CO LTD 6 & 7 CLIFFORD STREET LONDON W I THIBAULT U°USH THIBAULT SQUARE CAPE TOWN (y0$-6ll LONSDALE STREET MELBOURNE C I LONGMANS, GREEN AND CO INC 55 FIFTH AVENUE NEW YORK 3 LONGMANS, GREEN AND CO 20 CRAN1IELD ROAD TORONTO 16 ORIENT LONGMANS PRIVATE LTD CALCUTTA BOMBAY MADRAS DELHI HYDERABAD DACCA First published 1957 PRINTED IN GREAT BRITAIN BY SPOTTISWOODE, II ALLAN TYNE AND CO LTD LONDON AND COLCHESTER PREFACE Philosophers arc sometimes thought to engage themselves in hair-splitting activities with no other purpose than to exercise the disciplines needed to perform them. It is not always suspected that the most prized advancements in scientific and other knowledge come through the application of these disciplines. Briefly, to philosophize is to criticize what is said in its relation to what is seen, and we should, therefore, set about such criticism whenever we begin to suspect that what we see is being distorted or occulted by what we say. A well-known example of word-made obscurity lies in the concept of simultaneity fundamental to Newtonian mechanics. Einstein’s mechanics begin, in the best tradition, with a destruc tive analysis of this concept. All concepts arc destructible and it is not always obvious which to destroy. I have been concerned with the destruction of one such concept which has, I believe, obscured and occulted to a distressing extent certain facts of observation which would otherwise have been clearly seen. Some philosophers, struck with the recent discovery that no one set of metaphysical assumptions is necessary to communication, have fallen into the fallacy of thinking that we can communicate without any metaphysics at all. I.ockc fell into the same kind of error when he supposed that because what he called primary qualities were dependent for their perception upon no one sense, they were therefore qualities which matter possessed independently of sense. It is one of the greatest, if not the greatest, of discoveries in 2000 VI PK PFACli years to have seen that no one system of metaphysics is neces sary; but it is a simple mistake to suppose that none is. What must be clone in the field of Probability is to replace an old metaphysical system with a new one. This means undertaking two tasks: first, to show where old assumptions were inadequate; and then, to work out better ones. Each of these tasks is a major one, and I do not think any purpose would be served in attempting them both at once. An advance in scientific theory depends ultimately upon carrying forward informed opinion, and it is an ordinary biological fact that opinion, however informed, cannot be carried through too many stages at once. The primary task of this book, therefore, has been one of analysis; and though I have indicated how the synthesis must follow, I have not completed it. That, if the opportunity be given, will be my next task. But if anyone is impatient he has only to follow me to the end of the book and lie will be ready to make the synthesis himself. I have been criticized for not publishing before now my v evidence from random number counts. This delay occurred first because of illness, after which I gave priority to the task of discovering why certain things had happened rather than to the task of trying to convince people at this stage that they had happened. I hope my colleagues will forgive this slight departure from current practice. But I should, in defence, point out that my claim, first published in 1953, was a scientific prediction and therefore public property. It has since then been open to verification or falsification by anyone willing to spend a few weeks counting. Perhaps one of the fruits of the initial withholding of the details was the subsequent striking confirmation of the prediction by a relatively hostile observer. PREFACE Vil When I answered Mr Oratn’s paper I still envisaged publishing the results of my counts on the Tippett,.tables. I don’t think I shall do this now, for, though they arc in some ways more striking than the rest, I do not feel, now that we know why such results occur, that it is necessary to establish by repetition of similar instances that they occur. In any case, observations by independent observers should carry more weight than my own. Anyone who has worked with chance machines knows very well how difficult it is not to observe certain oddities in their behaviour; it is only that classical probability, not having a place for them, has always prevented our talking about these oddities in terms of chance. The fact that the machine does something noticeably improbable has not been connected with the fact that it logically cannot do anything else. I am grateful to many friends for their help in making this book. Some of these'I have already thanked in the text, but there are others, perhaps not mentioned there, to whom I am equally grateful; to Sir Ronald Fisher for his early encourage ment, and to Professor A. C. Hardy for enabling me to begin work on this thesis in his Department; to Mrs Eileen J. Garrett of the Parapsychology Foundation Incorporated, and Mrs K. M. Goldncy of the Society for Psychical Research, for their generous practical help despite the gradual divergence of my opinions from the tradition of their field; to Mr Robin Farquharson and to my supervisor, Mr W. C. Kncalc, with whom many of the ideas here presented were first discussed and clarified; to Dr Handel Davies and Mr Garry Arnott for checking some of the algebra; to the Rev. Dr Eric Mascall and Mr R. II. Dundas for kindly reading the proofs; and to Mr John Howard Barton for making the index. Vlll PREFACE I should like, finally, to thank the Governing Body of Christ Church whose patronage has enabled me to put together and publish the work I have done. G. S. B. Oxford, Dcccmber"i956 Acknowledgements ' The thesis in Chapter X was originally given in a paper to the Mathematical and Psychological Sections of the British Association in September 1954. I am grateful to the organizers for encouraging me to present it in a publishable form. The gist of Chapter XIV is taken from a paper read to the Third London Symposium on Information Theory and published by Messrs. Butterworth & Co. (Publishers) Ltd. whose permission to re-present the argument here I acknowledge with thanks. I also acknowledge with thanks the permission of the Editor of the Journal of the Society for Psychical Research and Mr A. T. Oram to reproduce the paper ‘An Experiment with Random Numbers’ and subsequent correspondence. CONTENTS CMAPT1 1R PAGE I 'The Real World I II Worlds and Alodels 5 III The Classical Problem 15 IV Dissolution of the Classical Problem 20 V Truth 26 VI Measuring Probability 32 VII Probability as an Indicator 35 VIII. The Random Series 43 IX The Paradoxes o f Probability 57 X Critical Series 67 XI Bias and Stretch 82 XII Bernoulli’s Theorem 88 XIII Some Practical Considerations 92 XIV The Chance Machine 100 XV The Diminishing Field 106 Appendix I On Miracles 109 Appendix II On Practice T 13 Commentary 136 Index 151 IX