Topics in the Foundation of Statistics PDF
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TOPICS IN THE FOUNDATION OF STATISTICS Edited by Bas C. van Fraassen Department ofP hilosophy, Princeton University Reprinted from Foundations of Science Volume 1, No.1, 1995/96 Springer-Science+Business Media, B.V. A C.I.P. Catalogue record for this book is available from the Library of Congress ISBN 978-90-481-4792-2 ISBN 978-94-015-8816-4 (eBook) DOI 10.1007/978-94-015-8816-4 Printed on acid-free paper Prepared with pennission of Oficyna Akademicka All Rights Reserved © 1997 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1997 Softcover reprint of the hardcover 1st edition 1997 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. CONTENTS FOUNDATIONS OF SCIENCE - DISCUSSION: 5-18 Bas van Fraassen A Philosophical Approach to Foundations of Science Patrie Suppes A Pluralistic View of Foundations of Science Arne Collen The Foundation of Science THE FOUNDATION OF STATISTICS: David Freedman Some Issues in the Foundation of Statistics 19-39 Comments by James Berger, E. L. Lehmann, Paul Holland, Clifford Clogg, Neil Henry 41-67 David Freedman Rejoinder 69-83 Diedrik Aerts Applications of Quantum Statistics in and Sven Aerts Psychological Studies of Decision Proc~sses 85-97 Maria Carla Operationism, Probability and Quantum Galavotti Mechanics 99-118 Paul Humphreys Computational Empiricism 119-130 VARIA: Joseph Agassi Blame Not the Laws of Nature 131-154 *** Biographical Notes 155-157 On AFOS (some history of the initiative and some recent developments) 159-160 Joseph Agassi Summary of AFOS Workshop, 1994 161-166 Foundations of Science 1 (1995/96), 5-18 FOUNDATIONS OF SCIENCE - DISCUSSION Bas van Fraassen Department of Philosophy Princeton University Princeton, N J 08544, USA A PHILOSOPHICAL APPROACH TO FOUNDATIONS OF SCIENCE Abstract. Foundational research focuses on the theory, but theories ar'! to be related also to other theories, experiments, facts in their domains, data, and to their uses in applications, whether of prediction, control, or explanation. A theory is to be identified through its class of models, but not so narrowly as to disallow these roles. The language of science is to be studied separately, with special reference to the relations listed above, and to the consequent need for resources other than for theoretical description. Peculiar to the foundational level are questions of completeness (specifically in the representation of measurement), and of interpretation (a topic beset with confusions of truth and evidence, and with inappropriate metalinguistic abstraction). In foundational research, focus is on the theory, that is, on the prod uct of theoretical activity in science, as opposed to, for example, on the processes of theory testing, choice, evaluation, confirmation, or historic and other contextual factors that may playa role in such theorizing. Nevertheless, such research pertains not only to the structure of the theory itself, but also to its relations to other theories, to the facts in the domain of the theory, to experiment, to data obtained from experiment and observation, and to its use in applications, whether of prediction, control, or explanation. 1. What is a theory? Here I am in agreement WIth Patrick Suppes, that we must distinguish the theory from its formulations. Thus a set of axioms or theorems cannot be the theory; the theory is something which admits B. C. van Fraassen (ed.), Topics in the Foundation of Statistics © Springer Science+Business Media Dordrecht 1997 6 Bas van Fraassen of many alternative formulations, which may indeed be given in alternative languages with different vocabularies and even different logical resources. For this reason it is is best to identify a theory through a class of mathematical structures, its models, which can be described in those various ways. At the same time, however, a theory gives information about the world, and may be believed or doubted. Therefore, although a theory may be identified through a class of models, and may even (in accordance with logical and mathematical practice) be identified with that class, there is clearly more to it. For a class is not the sort of thing that gives information or may be believed or doubted. Historically, this point has sometimes been made by saying that scien tific theories are not "uninterpreted objects" such as are studied in pure mathematics. That is not a good choice of words, it seems to me, and may be misleading. It suggests that a scientific theory could be instead a com posite, something abstract together with something else that interprets that abstract thing. But then what is that something else? Is it too something abstract, like a function? If so we have the same problem all over again - we have replaced one abstract thing by two. On the other hand if it is suggested that a function needs no interpretation, though it is an abstract thing, then why did the first abstract thing need interpretation? If on the third hand the something else is not an abstract thing at all, but something concrete, we shall have to study it (scientifically, in the way all concrete things are to be studied) - and how shall we do that, if not by constructing a mathematical model of it? Therefore I prefer to say that a theory can be identified through its class of models. These models are the main subject of study for foundational research, taken in its strictest sense now, pertaining to that theory. But the theory states that these models include a correct representation of what there is (or of what the theory is about). This statement (by which I do not mean any linguistic formulation, but rather what all formulations of the theory "say") is subject to assent or doubt. It is also subject to interpretation prior to any assent or doubt. To this role of interpretation I shall return below. We must also note here the importance and propriety of Suppes' call to make mathematics rather than meta-mathematics the tool of philosophy of science with respect to foundational discussions. The class of models is a class of mathematical structures, described in any way mathematics al lows. Mathematical description has its limits of course, in that mathematics describes its subject matter only up to isomorphism. 2. Logical positivism and logical empiricism began, somewhat unfortu- A Philosophical Approach to Foundations of Science. 7 nat ely, by assimilating the relation between a theory and its domain to the relation of theory to evidence. Both are important, and to some extent they overlap. Those relations of theory to experiment that fall under such epis temic headings as confirmation and testing I do not count as belonging to the area of foundations. However, when we consider the relation of the theory to fact (to its do main, to the world, however we want to phrase this), it is important to ask whether the theory itself models the processes of measurement or experi ment, or whether those are described in terms of some other theory. This becomes crucial for those theories in physics that are potentially universal theories, theories of everything, such as quantum mechanics and general rel ativity. When the adequacy of the theory requires it to have models which correctly represent certain phenomena, and this adequacy can be ascertained only through certain types of processes, we face a potential consistency prob lem. Does the theory represent not only those phenomena but also those processes which give experimental access to those phenomena? If so, do the two sorts of representations 'mesh'? This is the form of the measure ment problem in quantum mechanics, but may crop up in any foundational discussion, in some form or other. 3. Uses of a theory comprise prediction and explanation. Both of these are topics that tend to lead us from foundations into problems concerning the language of science. Prediction requires the user to relate himself to some model or models of the theory, in a way that is analogous to someone who specifies his own location on a map, or gives the co-ordinates of his location, or specifies the co-ordinate system which is 'rigidly attached' to him. The linguistic resources to do this do not belong to the purely theoretical language which is sufficient for the presentation of the theory, but to the language of applied science. Since the description of the models is mathematical, this co-ordination with the theory required for an application may involve more than specifying the user's own co-ordinates. For example, he may wish to use a certain formal theory to predict thermal phenomena, while the equations of that theory describe equally well e. g. gas diffusion. It seems to me that Toulmin's analogy between a theory and a map is quite apt here, even if he used it differently. A circuit diagram might, by coincidence, share some structure with a roadmap or railway map, or a map of one city might share some structure with that of another. In that case I can use one in place of the other, by ignoring its origin, paying attention only to the relevant structure, and locating myself on it in a different way. There is a certain limitation of language that is evident here, where theory 8 Bas van Fraassen and practice meet. For describing the act of self-location on the map is very different from doing it, and until it is done, conditions for the use of the map are not met. The use of a theory in explanation may require even more, at least if accounts of explanation in terms of the logic of questions are correct. Issues of completeness for theories - such as the much discussed completeness of quantum mechanics issue - relate solely to the expressive power of the the oreticallanguage, i. e. the language in which the models and the domain of the theory are described. Use of the theoretical description as explana tion, however, may require attention to very specific interests and concerns characterizing the context of inquiry. The very same description may be able to play the role of explanation in one context and not in another - so explanatory completeness does not follow from expressive completeness. 4. Some of the problems touched on under the above headings do not pertain to foundations of the sciences at all, although they belong to philoso phy of science or in some cases to science. In addition, some of the problems indicated were left dangling, so to speak. I will here add comments only about one. Among the relations of a theory to its domain there are those which determine whether or not the theory is true. Prior to the question of truth, however, are questions of interpretation and of possibility. The question of truth does not arise except relative to an interpretation. On a purely theoretical level we can investigate which interpretations the theory admits, and thereby answer the question: how could the world (domain, fact) possibly be the way this theory says it is? That it is not easy to answer this question in general is clearly due to two factors. The first is the diffi culty of developing any interpretation at all for sufficiently complex theories presented in ma.thematical form. The second is the further demand that the question be answered through an interpretation on which the theory is compatible with other accepted parts of science. In the case of quantum me chanics, for example, we have found that it admits no interpretations which cohere with certain traditional assumptions about causality, while on the other hand, interpretations that imply the existence of superluminal signals are not generally considered satisfactory either. In conclusion I would like to draw attention to ongoing work to elaborate on the conception of foundational research in the sense of section (1) above, which received its original impetus from the work of Suppes. Some history is provided by way of introduction in Fred Suppe (Suppe, 1989); a recent contribution to be noted is (da Costa and French, 1990). References Da Costa, N.C.A. and French, S. (1990), The model-theoretic approach in the philosophy of science, Philosophy of Science 57, 248-265. Suppe, F. (1989), The Semantic Conception of Theories and Scientific Re alism. Urbana: University of lllinois Press. Patrick Suppes Stanford University Stanford, CA 943005, USA A PLURALISTIC VIEW OF FOUNDATIONS OF SCIENCE Before writing this I read Bas van Fraassen's statement. I find little I can disagree with in what he says. Consequently, what I would like to do is address several clusters of concepts and issues that are often somewhat neglected in discussions of the philosophical foundations of science. I. Epistemology of Experiments Bas rightly emphasizes the importance of going beyond models of theories in analyzing the content of science. What I want to stress is how far the whole activity of experimentation, including its reporting, is from the standard theoretical model of science, which Bas and I substantially agree about. The first point is that experimental reports, much more than theoreti cal papers, are not generally understandable. They are like detailed sports reporting, intelligible only to the initiated. There is, of course, the insis tence of journals that reports be as brief as possible, and certainly there is no requirement they be generally intelligible. However, I think there is a deeper epistemological reason, which also applies to sports reporting. It is not possible to describe in ordinary language, even augmented by some technical terms, the backhand stroke of a tennis player with any accuracy. Similarly, it is not possible to describe in language the many activities of an experimentalist. This applies to his actions setting up and running experi ments, but also to his perceptions in digesting the results in their first "raw" References Da Costa, N.C.A. and French, S. (1990), The model-theoretic approach in the philosophy of science, Philosophy of Science 57, 248-265. Suppe, F. (1989), The Semantic Conception of Theories and Scientific Re alism. Urbana: University of lllinois Press. Patrick Suppes Stanford University Stanford, CA 943005, USA A PLURALISTIC VIEW OF FOUNDATIONS OF SCIENCE Before writing this I read Bas van Fraassen's statement. I find little I can disagree with in what he says. Consequently, what I would like to do is address several clusters of concepts and issues that are often somewhat neglected in discussions of the philosophical foundations of science. I. Epistemology of Experiments Bas rightly emphasizes the importance of going beyond models of theories in analyzing the content of science. What I want to stress is how far the whole activity of experimentation, including its reporting, is from the standard theoretical model of science, which Bas and I substantially agree about. The first point is that experimental reports, much more than theoreti cal papers, are not generally understandable. They are like detailed sports reporting, intelligible only to the initiated. There is, of course, the insis tence of journals that reports be as brief as possible, and certainly there is no requirement they be generally intelligible. However, I think there is a deeper epistemological reason, which also applies to sports reporting. It is not possible to describe in ordinary language, even augmented by some technical terms, the backhand stroke of a tennis player with any accuracy. Similarly, it is not possible to describe in language the many activities of an experimentalist. This applies to his actions setting up and running experi ments, but also to his perceptions in digesting the results in their first "raw" B. C. van Fraassen (ed.), Topics in the Foundation of Statistics © Springer Science+Business Media Dordrecht 1997 10 Patrick Suppes form. The written account can only hint at the main features of what is done or what is observed. The central epistemological point of these remarks is that this radical incompleteness of descriptions of experiments is not a mark of bad science, but is an essential, unremovable feature of almost all science. The drastic descriptive limitation of what we have to say about experiments is, in my view, a fundamental limitation of our scientific knowledge, possible or actual. Moreover, this radical incompleteness of the experimental reporting leads to more appeals to authority in experimental work than in theoretical work. It is common to hear, in every part of experimental science: "Well, we know those results are right and can be trusted because we know X and his lab." Of course, there is the answer that experiments can be repeated by others, and this is the great empirical check against being overwhelmed by authority. But it is still an important point, which can be easily amplified by various historical examples, that in many respects theoretical results can be evaluated for error much easier and more directly than experimental work. To bring this up to current science, computational experiments and simulations need to be included as well. Part of the epistemology of experiments recognized by everyone is the pres ence of experimental error, but the theory of error has not crept into the philosophical foundations of science, but remains on the ground floor of ac tual experimental work, with only an occasional philosophical nod to its importance. Yet error is a central concept of a proper general epistemology, and, on the other hand, has a long technical history of theoretical develop ment, at least since the early work of Simpson in the 18th century. Its con ceptual place in science, however, is, in my view, still not fully recognized. Let me give just one personal example. I have spent many years working on the foundations of measurement, and I recently went to a gathering of the measurement theorists' "clan" in Kiel, Germany. It was generally agreed that a really proper inclusion and analysis of error in foundational theories of measurement is the number one general problem. Finally, I emphasize, as I have before, the 'hierarchy of models used in the analysis of experimental data. Modern statistics has developed within the set-theoretical view of mathematics, as can easily be seen by perusing the pages of The Annals of Statistics and other leading journals. The many levels of data reduction usually needed to get to detailed statistical analysis is an epistemological problem, not a statistical problem as such, and needs more philosophical analysis, with close attention to the varying practices in different parts of science. Skepticism about explicit objectivity holding "all the way down" is certainly one of the reasons for the recent increase
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