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Universft/ Microfilms International 300 N. ZEEB RD.. ANN ARBOR. Ml 48106 8129020
H u m p h r e y , St e v e n Fr e d e r ic k
AN ANTI-REALIST CONCEPTION OF THEORIES OF MATHEMATICAL PHYSICS
The Ohio State University PH.D. 1981
University Microfilms International 300 N. Zceb Road. Ann Arbor, MI 48106 AN ANTI-REALIST CONCEPTION
OF
THEORIES OF MATHEMATICAL PHYSICS
DISSERTATION
Presented in Partial Fulfillment of the Requirements for
the Degree Doctor of Philosophy in the Graduate
School of The Ohio State University
By Steven Frederick Humphrey, D.A., M.A.
The Ohio State University
1981
Reading Committee: Approved By
Ronald Laymon
Marshall Swain --- Advis I would like to acknowledge the patient assistance of my adviser, Ron Laymon, whose guidance and criticism helped form and refine this work. I would also like to acknowledge the contribution of G.F.R. Ellis, who suggested many of the examples from astrophysics which I have used in this disser~ tation. ii VITA January 8, 1951.... Born - Lynwood, California. 1973...... B.A., University of California at Los Angeles, Los Angeles, California. 1977...... M.A., The Ohio State University, Columbus, Ohio. 1975-1981...... Teaching Associate, Department of Philosophy, The Ohio State University, Columbus, Ohio. FIELDS OF STUDY Major Fields: Philosophy of Physics Philosophy of Science Minor Fields: Philosophy of Language Philosophy of Mathematics Mathematical Logic iii TABLE OF CONTENTS Page ACKNOWLEDGMENTS...... ii VITA...... iii LIST OF FIGURES...... vi Chapter I. INTRODUCTION...... 1 1. Goals...... 1 2. Summary...... 2 II. METHODOLOGICAL PRELIMINARIES...... 6 1. Motivations...... 6 2. Criteria...... 8 3. Scientific Realism...... 15 4. Realism in General...... 42 III. GENERAL PROBLEMS WITH THE RECEIVED VIEW.. 49 1. The "Time-Reversed Solution" Case.. 49 2. Approximation and Idealization 61 IV. THE GENERAL THEORY OF RELATIVITY...... 64 1. The Special Theory of Relativity... 64 2. The General Theory of Relativity... 78 3. The Schwarzschild Solution...... 93 V. MORE PROBLEMS WITH THE RECEIVED VIEW 101 1. The Singularity Case...... 101 2. The "Other Universe" Case...... 124 3. Idealization and Approximation 131 VI. AN ALTERNATIVE VIEW...... 141 1. The Received View Revisited...... 141 2. General Picture...... 151 iv Page 3. Specific Picture...... 158 4. An Example...... 169 5. Amplification...... 185 VII. CONSEQUENCES OF THE ALTERNATIVE VIEW 198 1. Theory Testing...... 198 2. Time-Reversed Solution Case...... 201 3. The Singularity Case...... 206 4. The Uses of Theories...... 211 5. Epistemology...... 217 6. Ontic Commitment...... 224 7. Conclusion...... 235 APPENDIX: DIFFERENTIAL GEOMETRY...... 236 1. Manifolds...... 236 2. Vectors and Tensors...... 242 3. The Affine Connection...... 250 4. The Metric...... 259 NOTES...... 264 BIBLIOGRAPHY...... 270 v LIST OF FIGURES Figure Page 1. Trajectory of a Falling Ball, 54 2. Schwarzschild Space-Time, 126 3. Reisner-Nordstrom Space-Time. 129 4. Schwarzschild Space-Time, 129 5. Photographic Plates 173 6. Photon Deflection. 173 7. The Manifold. 239 vi I. INTRODUCTION 1. Goals: This essay is a discussion of the logic and structure of theorizing in mathematical physics. In it, I try to attain two main goals. One, I argue that a tradi tional, very popular, account of the logical structure of physical theories is inadequate, and two, I present and defend an alternative view. Uy aims are somewhat restricted, in that I am concerned almost exclusively with theories of mathematical physics. While no hard and fast division can be made between mathematical physics and other kinds of physics, still it seems that physical theorizing has come to rely more and more heavily upon complex and sophisticated systems of mathematics. It is this "mathematization" of physics that distinguishes the theorizing of the twentieth century from that of earlier periods, and which introduces the philosophical puzzles and problems that force us to re-evaluate and, I argue, abandon our rather naive and out moded conception of physical theorizing. Thus, for the moqt* part, the examples I adduce in support of the various claims I make are taken from the three major physical theories of this century: the Special Theory of Relativity (STR), the General Theory of Relativity (GTR), and Quantum Mechanics (QM).1 Though I address myself only to theories of 1 mathematical physics, I suspect that the account of theor- i izlng that I propose and defend will work for any scientific # * theory which relies heavily upon mathematical techniques. 2. Summary: Chapter II contains a discussion of the purposes for giving and criteria to be used in evaluating ac counts of the logic and structure of theorizing in the phys ical sciences. The accounts which have been offered are of two basic kinds; those which embody scientific realism, and those which are anti-realist, or instrumentalist. In Chap ter II, I also describe and give a brief history of the trad itional account of the structure of scientific theorizing which I will be criticizing throughout much of this essay. It is a realist account which I refer to as "the Received o View". I end this chapter with descriptions of the general versions of scientific realism and instrumentalism, making explicit the variety of views which fall under each general beading, and finally arguing that some kinds of scientific realism are compatible with some kinds of instrumentalism. In Chapter II, I argue that one important adequacy condition for an account of physical theorizing is that determinations made on the basis of a theory, construed according to that account, must agree with determinations made by current scientists working on that theory. Thus, my general strategy for criticizing the Received View is to find cases in physics where determinations made by physicists are at variance with those made according to the Received View. The cases I describe can be divided into two groups; those which involve, in some essential fashion, the STR, GTR, or QM, and those which involver older, "classical" physical theories. In Chapter III, I describe some of the cases, taken from classical physics, which generate problems for the Received View. In Chapter IV, I give a brief, and rather sketchy, textbook description of the GTR. Inasmuch as the GTR is a generalization of the STR, I begin Chapter IV with a description of the Special Theory. A complete treatment of these theories would require the elaboration of the rather complex and sophisticated mathematical apparatus of differential geometry, and in fact, all textbooks dealing with the GTR begin with a chapter or two on differential geometry. Rather than force my reader through this dense mathematical undergrowth, I have tried, wherever possible, to explain the mathematical concepts used in the GTR in a non-technical, intuitive way. Unfortunately, this is not always possible, so I have included, as an appendix, a rather terse, but not impossibly difficult, discussion of the apparatus of differential geometry. A complete under standing of the GTR would require a careful reading of the appendix, as well as of Chapter IV, but I think that most of the interesting philosophical aspects of the GTR can be understood if one reads only Chapter IV and perhaps a few of the definitions given in the appendix. I have chosen to concentrate upon the STR and GTR, rather than QM, because roost writers on thiB issue have taken the hulk of their data from QM, and some very interesting, and hereto fore ignored, puzzles arise when we consider the theories of relativity. So as not to prejudice any issues or beg any questions, I give what might* be called a "textbook" treatment of these theories, rather than couch them in r terms of any particular account of physical theorizing. In Chapter V, I return to my criticisms of the Received View, taking my cases now from the STR and GTR. I am especially Interested in the so-called "singularities" which are apparently predicted by the GTR. The positive part of this essay begins in Chapter VI, where I describe an alternative account of theorizing in mathematical physics. The account I provide is similar 3 to those offered by philosophers such as F. Suppe, 4 5 6 P. Suppes, J. Sneed, and B. van Fraassen, and, like those views, can be regarded as an instrumentalist, or anti-realist account of physical theorizing. In Chapter VII, I show bow this proposed alternative can be used to solve the puzzles and avoid the problems raised against the Received View. I end the chapter with a brief discussion of the scientific epistemology which arises as a result of the adoption of the view I propose. II. METHODOLOGICAL PRELIMINARIES 1. Motivations: There are two general motivations for Investigating the logic and structure of physical theorizing. The first, and more important of the two, is epistemological in character, while the second involves more metaphysical issues. Broadly speaking, the epistemological motivation is to find an answer to the question "How does science extend our knowledge?", and is similar to the Rationllsts' moti- 1 vation for investigating Euclidean geometry. According to the Rationalists, remember, the paradigmatic example of a system of knowledge was Euclidean geometry. Pre-analyti- cally, geometrical knowledge (i.e., knowledge of geometri cal truths) was regarded as the ideal form of knowledge. In order to fully understand the concept of knowledge (in what it consisted, how it could be attained, of what one could have knowledge), one had to understand geometry. That is, a thorough investigation of geometric theorizing was essential to constructing a Rationalist theory of epis- temology. Many a Rationalist epistemology began with an answer to the question "Given that we have geometrical knowledge, how is such knowledge possible?" Plato's Theory of Ideas and Kant's concept of the synthetic a priori are two of the best known fruits of this Rationalist program. Both were invented originally to explain how one could come to know, with certainty and prior to experience, the axioms of Ehclldean geometry. The Empiricists, on the other hand, rejected the Rationalists' claim that certainty and deductive validity were necessary for knowledge, and erected, in place of geometry, a new paradigm: science, and especially physics. The Empiricists maintained that it was science that provided the best example of a system of knowledge, and that it was scientific reasoning after which other researches should be patterned. Further, in order to fully understand the con cept of knowledge, we must ask after the logic and structure of physical theorizing. As should be clear, these different starting points yield radically different epistemologiesi For example, if geometry is taken as jhe paradigm of rationality, then only deductively valid arguments provide the justification necessary for knowledge, and inductive arguments, which do not guarantee the truth of their con clusions, do not provide adequate justification. If, on the other hand, science is taken as the paradigm of ration ality, then, since many of the arguments used to derive scientific claims are inductive, an epistemology based upon Empiricism will accord inductive argumentation full justificatory powers. Thus, the epistemological motivation for investigating the logic and structure of physical theorizing is to find an answer to the question "Given that physical science does extend our knowledge of the world, how does it do it?" The metaphysical motivation for investigating * physical theorizing is less well defined, though I think that it is important to describe it because, if for no other reason, it will help us to avoid confusing the meta physical with the epistemological motivation. Many recent empiricist philosophers have adopted the principle that metaphysical, and especially ontological, issues should be settled, not by philosophy, but by science. In slogan form, this view can be expressed as "physics determines metaphysics", or "what there is is what physics says there is". Now, if one accepts this dictum, and if one is inter ested in discovering what there is, then one must discover what physics says there is. That is, one must investigate the logic and structure of physical theorizing. 2. Criteria: As I see it, the task of giving an account of the logic and structure of physical theorizing is very similar, broadly speaking, to a number of philo sophical activities. Some examples are the attempt to give a formal syntax for a natural language, the "Harmon- Davidson Generative Semantics" program, and the attempt to find the logic of certain locutions occurring in ordi nary language, such as the logic of the alethic modalities, or of deontic operators. In all of these cases, the philosopher begins with some well-known, but vaguely under stood, phenomenon, and tries to give a more precise analy sis, or description, of that phenomenon. We start with the verbal behavior of some community of speakers, for example, notice that their activity seems to be rule governed, and then try to find explicit statements of the rules that are at work. This is very much like what the philosopher investigating the logic and structure of physical theorizing does. Theorizing in the physical sciences consists of a number of different activities. Scientists describe and invent theories, test theories, derive consequences and predictions from theories, explore the limits of theories, make inferences within theories, design experiments and experimental apparatus, accept and reject theories, and modify theories. Unfortunately, most scientists learn the methodologies they use in performing these tasks in much the same way as we learn our native language. Most physi cal theorists do their work unreflectively, and are .as unable to explicitly state the rules of their game as most native speakers of some language are to state explicitly the 10 rules of grammar that they are following. In providing an account of the logic and structure of physical theorizing, then, the philosopher of science must explicitly describe, or reconstruct, in a precise way, the rules that physical theorists follow. There is one Important difference, however, between the task of the philosopher of science and, for example, that of the linguist. When we investigate the logic and structure of physical theorizing, remember, we start with the assumption that science is rational, and provides ade quate justification for its claims. In the case of the linguist searching for a grammar, no such assumption is made. Our assumption of the epistemological adequacy of science constrains the philosopher to find, not only a descriptively adequate account of physical theorizing, but also orfe'according to which science proceeds rationally. That is, there are two distinct criteria to be used in evaluating accounts of the logic and structure of physical theorizing; they should be compatible with the behavior of practicing theorists, and should be such that science can be seen to be rational. Let me be more explicit about these criteria. Con sider the first: Our account roust be descriptively accurate. It must be faithful to the actual inferences made by prac ticing theorists. Our only access to the process of physical theorizing is through physical theorists. To understand science, we must observe scientists. That is, it would not be adequate simply to describe the scientifi cally naive philosopher's conception of science; we must examine real science. Uany mistakes have been made by philosophers who were unwilling, or unable, to investigate real, current, science, and were content to base their philosophical theories upon a naive, simplistic, conception of some out-of-date physical theory. (For example, Hempel's 2 account of scientific explanation is based upon an over simplified picture of Newtonian mechanics, so, though it works fairly well for that theory, it is woefully inadequate as an account of explanation in modern physics.) Since physical theorizing is what is done by physical theorists, to understand the former we must carefully observe the latter, and, to be adequate, an account of the logic and structure of physical theorizing must be faithful to actual scientific practice. For example, if, according to some proposed account, some piece of evidence serves to discon- firm a particular theory, yet no practicing theorist regards that evidence as being relevant to that theory, then that account is Inadequate. The determinations made according an adequate account of physical theorizing will coincide with those made by physical theorists. Now, to say that such an account must be descriptively accurate is not to 12 say that the inference patterns described by the account must be recognizable by the theorists, no more than the rules of grammar described by an adequate theory of syntax must be recognizable as such by native speakers. What is important is that the conclusions reached according to the account of physical theorizing coincide with those reached by the physical theorists. As mentioned above, as well as a desire to be descriptively accurate, our search for an account of the structure of science is also motivated by a desire to show how science extends our knowledge. Thus, not only should an adequate account of physical theorizing make the logical inferences of physical theorists explicit, but these infer ences should be such that they can be seen to be valid or reliable. That is, we also want those inferences to be of a sort recognized as providing justification adequate for knowledge. Consider, for example, the Hypothetico-Deductive account of the logic of theory testing. According to this view, the logic of theory acceptance, or preference, on the basis of empirical adequacy is seen to be a rather simple sort of induction, one which is recognized as being reliable. If scientific reasoning actually did proceed as described by the H-D account, then we would have an answer to the ques tion "How does science give us knowledge?" 13 We have, then, two distinct desiderata for our account of the logic and structure of physical theorizing; it should be at least compatible with the behavior and determination of practicing theorists, and it should be such that according to it, science is rational. Now, it should be clear that these criteria can provide a conflict. It is possible that according to the view of scientific methodology that best captures actual scientific practice, science does not provide us with justified beliefs. This is exactly the point at which the issue of descriptive vs. normative philosophy of science arises. If one is con vinced that science is rational (i.e., that it does provide justification for some of our beliefs), and one has an account of the structure of scientific theorizing according to which such theories are rational, then it is easy to see how such a person might feel justified in insisting that where actual practice diverges from the prescriptions made by his account, it is the scientist that is mistaken. That is, the account of the logic and structure of scientific theorizing tells us how science ought to be done (in order to be rational). Such a philosopher would be using his theory of knowledge, or justification, to evaluate scien tific methodology. On the other hand, if someone were antecedently convinced that science was rational, and that scientific knowledge was the paradigmatic form of knowledge, 14 then, given a conflict between actual scientific practice and some theory of justification, he would be led .to con clude that it was the theory of justification that was inadequate. Though I cannot hope to resolve this issue conclu sively, it is my contention that the normative elements of philosophizing about science should be kept to a minimum. While it is certainly likely that individual scientists occasionally make irrational inferences, and it is possible, though unlikely, that a great number of theorists make the same mistake in reasoning, it would be both Impudent and imprudent for philosophers to try to use their favorite account of justification, or their favorite account of scientific theorizing, to decide hard questions of science. In constructing an account of physical theorizing, our first priority must be to describe accurately, or reflect, the behavior of actual physical theorists. For example, if, in the community of relativistic astrophysicists, there is genuine dispute over some element of theory, then our reconstruction of that science should reflect that dis- 3 pute, not settle it. As Kuhn has pointed out, it is dispute, controversy, and confusion that stimulates much of the greatest progress that occurs in science. Our accounts of these theories should leave the solutions to 15 these problems open, to be solved by the participants. Such disputes and confusion are an essential part of the dynam ics of scientific theorizing, and should be regarded as such by any adequate account of the logic and structure of scientific theorizing. In some sense, the reconstruction 4 of physical theorizing is like David Lewis* analysis of counterfactuals; the vagueness of the similarity relation reflects the vagueness of the truth value of counter- factuals, and any account of these locutions which arbi trarily settled these matters would be wrong. Thus, in constructing and evaluating accounts of the logic and structure of theorizing in mathematical physics, I shall place the most emphasis on their descriptive adequacy. For example, though the traditional picture of scientific theorizing does succeed in making explicit justifiable inference patterns, it is inadequate because they are not the inference patterns actually used by scientists. How ever, though I will be most concerned with descriptive adequacy, I will argue that the alternative view I advocate also demonstrates the rationality of physical theorizing. 3. Scientific Realism: In this dissertation I will be criticizing a particular account of the structure and interpretation of scientific theories. This account is generally known as "scientific realism", and is an extension, or modification of the old Positivist picture of 16 scientific theorizing. In this section, I will describe the best version of the Positivist view (which, following F. Suppe, I call "the Received View"), discuss the problems which led to its abandonment, and show how it was modified to yield modern scientific realism.*’ The final version of the Received View was the result of years of philosophical work. It began with the early Positivists of the late nineteenth and early twen tieth centuries as an account of scientific methodology, underwent almost continuous modification through the first half of this century, and, through the work of Carnap, Hempel, and others, reached its apex during the 1950's. Despite the general abandonment of the Positivist program by most philosophers, the Received View has, until rather recently, been accepted by many as an adequate account of scientific theories. Let us turn to a description of the final version of the Received View. According to this picture, a physical theory can be given a canonical formulation in a first-order, formal language L (which may or may not contain modal operators). That is, in order to get at the "logical form" of a physi cal theory, we begin by "translating" it from the language used by scientists to describe the theory into a simpler, more precise, artificial language, whose quantifiers range only over individual variables. Further, for each theory 17 for which a canonical formulation is sought, there is a logical calculus K defined in terms of L. This logical calculus must be strong enough to generate the mathematical apparatus needed for the theory in question. For example, Newtonian mechanics makes essential use of the differential calculus, so in giving the canonical formulation of Newton ian mechanics, we must begin with a language rich enough to contain the axioms, definitions and theorems of the differ ential calculus. The primitive vocabulary of L contains both logical and nonlogical, or descriptive, constants and variables. The nonlogical primitive constants are divided into two disjoint classes; the set VQ , which contains just the "observation" terms, and the set V^., which contains the nonobservation, or "theoretical" terms. (This syntactic distinction can, in the construction of L, be made arbi trarily. That is, in setting out the vocabulary of L, we simply provide two disjoint stocks of predicates, singular terms, and variables. Problems arise only when we try to translate the scientist's vocabulary into L. For example, should the term 'electron', used by the physicist, be translated into an observation or theoretical predicate?) The language L is divided into three disjoint sub languages; the observation language L0 is the set of sen tences of L which contain only the terms of VQ , the 18 theoretical language Lt is the set of sentences of L which contain only terms in Vt, and the mixed language 1^ is the set of sentences of L which contain at least one member each of VQ and V f The logical calculus K is divided into corresponding subcalculi: K0 is the restriction of K to Lq , Kt is the restriction of K to Lt , and is the restric tion of K to h a - A scientific theory, formulated in L, consists of a set of theoretical postulates (i.e., the axioms of the theory), T, which is a subset of L t, a set of correspondence 4 rules (coordinative definitions), C, which are mixed sen tences, and their logical consequences. A theory TC is closed under the operations of K. Now, the Received View, as described above, is an account of the logical (i.e., syntactical) structure of scientific theories. Clearly, though, in order to gain a complete understanding of scientific methodology, and to provide solutions to the puzzles which motivated the search for such an account, the issue of the interpretation, or semantics, of such syntactic structures must be addressed. For example, one of the most important desideratum for a full account of scientific theories is to explain how theoretical terms are introduced and given meanings, and this obviously requires a discussion of the interpretation of scientific theories and the languages in which they are 19 formulated. There are two distinct ways of interpreting such a language, both of which are consistent with the Received View of the logical structure of a theory formu lated in that language: an instrumentalist interpretation and a realist interpretation. Let me discuss the instru mentalist version of the Received View first. According to this view, the language of the theory is interpreted as follows. Only the observation language is given a semantic interpretation, which meets the following condition. The domain of the interpretation consists of concrete observable events, things, etc., and the values given to the variables of the observation language by the semantic interpretation have to be designated by terms of the observation language. Thus, only the sentences of LQ are empirically true-or-false, since the terms of V-j. do not refer to any nonobservable entities which really exist. (Instrumentalism, then, seems to embody a nihilism about nonobservable entities. However, I will argue in the next section that one can hold a realist view of nonobservables while holding an instrumentalist view of scientific theories.) Scientific theories, which consist of the con junction of T with C and their logical consequences, are not empirically (i.e., synthetically) true-or-false. Rather, the value of a theory is primarily a function of its empirical adequacy; its ability to generate true 20 sentences of L0 . On this view, the members of C, if true at all, are analytically true, or true by convention. According to the final version of the Received View, the correspondence rules are reduction sentences (i.e., partial definitions) which serve to connect the theory with the observable phenomena. The "truth" of the theoretical pos tulates of a theory depend upon (a) which correspondence rules are selected, and (b) the empirical truth of the observation sentences generated by TC. The truth of these observation sentences is both a necessary and sufficient condition for the adequacy of the theory. According to the realist interpretation of scien tific theories, the empirical truth of the observation sen tences generated by the theory is a necessary but not a sufficient condition for the truth of the theory. On this interpretation, as on the instrumentalist, only the obser vation language is given a direct semantic interpretation, but on the realist view, the interpretation of the obser vation language is taken as providing a partial semantic interpretation of L. The theoretical terms and theoretical language are provided a partial interpretation through tho theoretical postulates and correspondence rules of the theory. On the instrumentalist view, the meaning of a theoretical term is simply a function of the role it plays in generating observation claims. On the realist version, 4 21 on the other hand, theoretical terms are introduced and their meanings are partially specified by the correspon dence rules of the theory. The theoretical terms of a theory refer to real but unobservable entities or their properties. A theory is true just in case the laws T of the theory are empirically true generalizations about the behavior and properties of the theoretical entities referred to by the members of Vt . Thus, the observation/ theoretical term dichotomy Implies another dichotomy, that of observable versus-{theoretical, or nonobservable, entity. The observable manifestations of the theoretical entity are described by the correspondence rules, and this serves to interpret partially the theoretical terms. A full inter pretation of these terms would require a description of the relevant theoretical entities' nonobservable properties and behavior, as well. According to earlier versions of the Received View, at least some of the correspondence rules of a theory were regarded as being analytic. Later treatments, due primarily to Carnap, remove the analyticlty from the correspondence rules by introducing analytic "meaning postulates" for the terms of both VQ and V-£. The meaning postulates reflect the intentions of the author of the formal language as to the use of the terms of that language. For example, if I want the predicates lB ‘ and ‘U' of my language to designate the properties bachelor and married, 22 respectively, and if I believe that these properties are incompatible, then I would lay down, as part of my formal system, the postulate '(x)(Bx D a* Mx)'. However, which postulates I lay down is entirely up to me, and thus they are analytic. The essential features of the Received View, in both its instrumentalist and realist versions, are the following. (1) A distinction is drawn between the observa tion and the theoretical terms of the language in which the canonical formulation of a theory is given. Since the adequacy of the Received View depends, in part, upon our ability to translate a theory into our formal language in such a way as to reflect accurately actual scientific prac tice, the observation/theoretical term dichotomy in the formal language implies that we should be able to determine unequivocally which of the scientist's terms are to be translated into observation terms and which are to be trans-* lated into theoretical terms. (2) An equally important distinction is drawn between analytic and synthetic sen tences. On the Received View, analyticity plays an essen tial role in the introduction and interpretation of theroetical terms, either through analytic correspondence rules or analytic meaning postulates. Further, analytic sentences are necessary for determining the truth or falsity of theoretical sentences by use of factual information. 23 That is, factual information, described by observation statements, is conjoined with analytic correspondence rules (and/or analytic meaning postulates) to yield theoretical statements. The death knell for the Received View sounded when it was realized, in the 1950's and 1960's, that these two 6 essential distinctions were untenable. Without an obser vation/theoretical term distinction, all of the terms occurring in the canonical formulation of a theory must be given the same semantic treatment. The untenability of the analytic/synthetic distinction forces us to look for another account of the introduction and interpretation of theoretical, or novel scientific, terms. Abandonment of these distinctions requires us to modify the positivist picture considerably. Further changes result from the realization that there is no guarantee that every physical theory can be reformulated as a first-order logical system, using as a logical calculus the predicate calculus, set theory, and arithmetic. The mathematics used in current physics, for example, is quite complex, and it would be extremely difficult, if not practically impossible, to give the canonical, first-order, formulation of any interesting physical theory. However, we can modify the traditional Received View in such a way as to preserve its spirit while avoiding the above-mentioned problems. The following, then, 24 is the version of the Received View that I will be addressing myself to in the rest of this essay.7 We begin with a formal, symbolic, language L. L may be a first-order language, or it may contain quanti fiers which range over predicate variables. L may or may not contain modal operators. The vocabulary of L consists of five kinds of symbols: (1) Logical constants, which include at least the logical constants of the first-order predicate calculus with identity; (2) variables, individual if the language is first-order, individual and predicate if the language is higher-order; (3) individual constants, or singular terms like proper names; (4) predicates of an arbitrary number of places (if n is a positive integer, then an n-place predicate is a symbol which, when accom panied by n individual constants, generates a sentence; for example, in ordinary mathematical language the symbol '<* is a two-place predicate); (5) operation symbols of an arbitrary number of places (if n is a positive integer, then an n-place operation symbol is a symbol which, when accom panied by n individual constants, generates a name; for example "+" is a two-place operation symbol). The precise vocabulary of L depends upon the richness and complexity of the theory we are formulating in L. While there is some philosophical interest in constructing the simplest language with the fewest primitives possible, it seems to be somewhat 25 restrictive to insist that all scientific theories be given a canonical formulation in a first-order language containing only the apparatus of the predicate calculus and perhaps set theory and arithmetic. Though it is possible to derive from this meager basis all of higher mathematics, it is much simpler to construct a language rich enough in which to formulate the mathematical apparatus directly. For example, Newtonian mechanics makes essential use of the differential and integral calculus. While lt is possible to give the canonical formulation of this theory in a language con taining only the apparatus of the first-order predicate calculus, it would be much simpler to begin with the vocab ulary and axiomatic basis of the calculus. If we were interested in establishing certain metamathematical proper ties of a scientific theory, such as soundness or complete ness, then it would be helpful to formulate the theory in as simple a language as possible. However, for the pur poses of this dissertation, such simplicity is unnecessary. On this version of the Received View, then, a physical theory, in its canonical formulation, is a deduc tively closed set of sentences of a formal language L, where L is rich enough to contain the axioms and theorems of the mathematical apparatus used in the theory. The sentences of the theory are derived from two kinds of axioms. For example, the canonical formulation of Newtonian 26 mechanics would contain the axioms and definitions (of, for « example, the derivative operator and the integral) of the calculus, together with the physical laws of Newtonian mechanics, such as Newton's first and second laws and Newton's law of universal gravitation. The above description captures the essence of the logical structure of physical theories according to the modified Received View, in the sense that it gives the syntax, or logical form, of such a theory. However, there is more to giving a logical reconstruction of physical theories than simply describing syntactic structures. In addition, semantic considerations must be taken into account. As in the traditional version of the Received View, theories may be interpreted instrumentally or realistically. Later in the dissertation, I will be describing, in some detail, and defending an instrumentalist interpretation of the General Theory of Relativity, and arguing that a realist interpretation of this theory is inadequate. For now, then, I will describe only the realist interpretation of scien tific theories, as construed by the modified Received View. According to the realist version of the modified Received View (which is often called "modern scientific realism"), physical theories are intended to provide literally true descriptions of an independently existing physical reality. That is, a physical theory is regarded 27 as being "about", or "true of", the world In the sense- that a Tarski-style truth theory (which embodies a corres pondence view of truth) is used in giving the semantics for a theory. Further, according to the view we are dis cussing, acceptance of a physical theory Involves belief that that theory is true. Theory testing is done in order to help scientists discover which among competing theories is true, and we are warranted in believing that a highly confirmed theory is true. Let us take a closer look at the semantical inter pretation given to scientific theories, canonically formu lated, according to modern scientific realism. On this view, truth is construed as a function of extension and logical form. Very roughly, that means that a sentence is true just in case the things to which the singular terms in the sentence refer stand in the relations indicated by the predicate terms in the sentence. For example, the sentence 'Jimmy Carter is a peanut farmer* is true just' in case the referent of the term 'Jimmy Carter' is a member of the class of things which make up the extension of the predicate 'is a peanut farmer'. To make the semantic concepts used in modern scien tific realism precise, the apparatus of extensional model- theoretic semantics is usually mobilized. To provide such Q a semantics for a physical theory, we proceed as follows. 28 The language, L, within which our theory is given its canonical formulation, is given a model-theoretic interpretation. An interpretation for a language is an ordered pair 'F' is a one-place first-order predicate, then I,('F?) is a subset of D, and if 'G* is a n-place, first-order predicate, or relation, symbolic then I('G') is a set of ordered n- tuples of elements of D. If 'H' is a one-place, second- order predicate, then I('H') is a subset of the power set of D. In other words, I assigns to predicate and relation symbols their extensions. Finally, if ,U* is an n-place operation symbol (e.g., or then I assigns to *M' a mapping from some set of ordered n-tuples of elements of D into D, where such a mapping is regarded as a set of ordered n+l-tuples. For example I('d') might be the set { This is not a function, because for any x e the non-negative reals, there will exist two real numbers y and z such that both interpretation for a language is an interpretation of that language according to which the logical constants and operation symbols receive their usual, interpretations. For example, the standard interpretation for a language containing ' + ' would assign to that symbol the set of ordered triples of real numbers in which the third member is the sum of the first two. Truth under an interpretation is defined roughly and recursively as follows. An atomic sentence 'Fa* is true under the interpretation I('a') e I('F'). The atomic sentence 'a ■ b' is true under The quantified sentence '(3x)Fx" is true under A model for a theory is an interpretation for the language of that theory under which all of the sentences of the theory are true. Finally, a permissible model for a theory is a model for that theory according to which all of 30 the familiar terms of the language of the theory which occur in the theory receive their usual Interpretations. In the definition of a permissible model, two expressions appear which require explanation: ’familiar term* and 'usual interpretation*. Let me begin with the latter. Assuming that there are paradigmatically familiar terms, such as 'the Sun', 'Ronald Reagan', or 'color', which have clear, determinate, theory-independent exten sions, then if, in giving the canonical formulation of some theory, one such term is represented, in the formal language L, by the term 't', then a permissible model for that theory assigns to 't' the ordinary extension of the term in English which *t* represents. For example, there might be two different models for the canonical formula tion of classical mechanics, one of which assigns mass to the term 'm1, which represents 'mass', while' the other assigns velocity to the term 'm'. Only the former model could be a permissible model for classical mechanics. (It is standard practice in many of the sciences, and in particular, in mathematical physics, to formulate theories in purely symbolic mathematical languages, languages which contain no terms of English. However, if this is done, it is necessary to specify which terms in English (or other natural language) the terms in the formal language are intended to represent. Thus, it is common to find, in physics textbooks, metalinguistic claims such as ft,m' represents 'mass'*'. Often, these claims are far from trivial, and require considerable argumentation for their justification. For example, if, in working within the mathematical formulation of a theory, some new term is defined or Introduced, lt is often a non-trivial matter to determine which natural language term it represents. On the other hand, it is hard to Imagine giving up some of the more basic of these claims. What would cause us to abandon the claim that 'm' represents 'mass' in some stan dard formulation of classical mechanics? Thus, the ques tion: "What is the logical status of these metalinguistic claims?" is a difficult one to answer. Uuch of the problem can be avoided, however, by simply allowing our formal language to be a suitably enriched fragment of English (or other natural language), and translating the theorist's English terms homophonically into.the formal language. In this case, a permissible model is a model for a theory which assigns to all of the familiar terms occurring in the theory their usual interpretations. The distinction between familiar and unfamiliar terms has to do with the question of how new terms of science are introduced and given a meeting. Most modern scientific realists adopt some version of the causal theory Q of reference. According to this theory (or theory schema), 32 the reference of a term, uttered by a speaker on an occasion, is a function of the existence of certain kinds of causal chains extending from some (hypothetical) dubbing ceremony, when the term was first introduced, to the event of the speaker's utterance. One important part of such theories of reference concerns the method whereby the referent of a term is fixed. Let us turn to a brief discussion of various theories of reference fixing. One early theory was associat ed with the Positivist's version of the Received View.1® Ac cording to this theory, terms have intensions, or meanings, which determine the extensions of the terms, and are given by analytic definitions or meaning postulates. For example, the meaning of the name 'Hoses' might be 'the man who led the children of Israel oui; of Egypt1, and hence the claim that 'Moses led the children of Israel out of Egypt1 is ana lytic and necessarily true. Versions of this theory were I. distinguished by whether It was claimed that all of a thing's properties contributed to the meaning of the term denoting it. (Leibniz, for example, might be construed as holding the view that all of a person's properties were part of his "individ ual concept", which constituted the intension of that person's name, and thus all true statements about a person were neces sarily true.) The analytic correspondence rules (in the form of explicit definitions or reduction sentences) of the Positi vists' version of the Received View served to (partially) 33 specify the meanings of theoretical terms introduced vithin * • the context of some theory. The reference of a term whose meaning was given in this way was whatever satisfied the analytic correspondence rule. A varient of this theory of reference was the cluster theory, according to which it was a cluster of properties which constituted the intension of a singular term, some weighted number of which determined the extension of the term*. On this view, someone might not have led the children of Israel out of Egypt, and yet still have been Moses, because he satisfied enough of the other properties which make up the intension of the name 'Moses1. Both of these views suffered from the fact that according to them, there are claims about individuals which are necessarily true, and yet which seem Intuitively to be only contingently true. The causal theory of reference provides a way of avoiding this problem. On this view, though definite descriptions and open sentences may be used to fix the referent of some term, they are not part of the meaning, or intension, of that term. When a term is introduced, its reference is fixed via some dubbing ceremony, which, in the paradigm cases, involves some act of ostension. "I hereby dub this child lying before me 'Melvin'." Since this dubbing does not contribute to the meaning of 'Melvin', the sentence "Melvin is lying before me" is not necessarily true. Theoretical terms, or, rather, novel scientific terms, are introduced, and their referents are fixed, in a similar way, though the act of ostension is often missing. For example, quarks are Identified, in quantum electrody namics QED), as those entities which make up hadrons, whose properties determine the properties of hadrons, and which have fractional charge. (The last characteristic is perhaps the most important, since virtually all of the strategies devised to detect free quarks involve the measurement of charge via elaborations of the classic Milllkan experiment.)11 Thus, certain of the claims of the theory serve to fix the referent of the novel term. How ever, unlike the "implicit definition" accounts of the reference of theoretical terms, these claims are not part of the meaning of the novel term, and thus the notion of analyticity need not be appealed to. Further, within the context of QED, terms such as 'hadron', 'spin', and 'charge' are familiar terms, in the sense that their referents are fixed via the claims of more basic and general theories, such as general Quantum Field Theory. Familiarity and..unfamiliarity are theory-relative notions. 'Quark' is an unfamiliar term relative to QED, but familiar relative to quantum chromodynamics (QCD). (QED is a quantum theory of electro-magnetism, while QCD is a quantum theory of the strong nuclear force.) Thus, 35 novel, or unfamiliar, scientific terms are introduced, and their significance established, wihtin the context of some theory. This is done by using the claims of the theory to fix the referent of the novel term and the familiar terms used in the theory. (M o r e precisely, using the formal apparatus introduced above, a term is introduced into the symbolic language in which the theory is formulated, and claims are made in that theory which connect that term with others in the language. The extension of the novel symbolic term is fixed via its relation to symbolic terms which represent familiar terms of English. Theorists then invent a novel term of English to be represented by the novel symbolic term Introduced in the theory, and the extension of the novel English term is identical with that of the novel symbolic term.) The notion of a permissible model for a theory allows us to talk of the universality of physical laws and theories. If we were to define the truth of a theory as there being a model for the theory, then any theory of, say, particle mechanics would be true as long as some set of particles, no matter how small that set was, behaved as described by the theory. If we define the truth of a theory as there being a permissible model for that theory, then, since the usual interpretation of the predicate 'is a particle* is the class of all particles, only that theory 36 that correctly described the behavior of all particles would be true. Let us turn to a discussion of the notion of ontic commitment that is Implied by modern scientific realism. Inasmuch as a theory of physics is construed, on this view, as being "about" the physical world, it is assumed that if a theory is true, then the physical world itself, together with an appropriate assignment function, constitutes a permissible model for the theory. That is, there is an interpretation of the language of the theory whose domain contains both material and abstract objects, whose assign ment function assigns to each familiar term in the theory its ordinary extension, and according to which all of the sentences of the theory are true. Now, if the theory in question contains unfamiliar terms (i.e., terms which are introduced within the context of the theory), then there must be elements of the domain of a permissible model for the theory which correspond to these terms. For example, in any permissible model for QED, there must be some sub set of the domain of the model which is assigned by the assignment function to the predicate 'is a quark' as its extension, and assigned in such a way as to make all of the sentences containing the predicate true. If no such sub class of the domain of the model were to correspond to this predicate, then it would lack an extension under that » 37 interpretation, and some of the sentences of the theory containing that term vould be false. For example, consider the following claim from QED. "Protons consist of two up quarks and one down quark." Vhere 'P* represents is a proton', 'Q' represents 'is a quark', 'U* represents 'has upness', 'D* represents 'has downness', and 'C' represents the operation term 'the combination of ;__, , an d , the symbolic rendition of this claim might be (x)(Px D (2y#z,w)(Qy * Qz • Qw * UY • Uz • Dw • x ** Cyzw)). Clearly, such a claim can be true only if 'Q* has an exten sion. If a model for a theory is an interpretation which makes all of the sentences of that theory true, then no term of the theory can lack an extension according to that inter pretation. Now, since physical theories are construed as being about the physical world, the truth of QED implies the existence of a new class of physical objects, which are referred to by the term 'quarks'. More generally, if the permissible model for a theory includes in its domain some new class of entities, then, since every element of that domain must exist in order to make all of the sentences of the theory true, the truth of the theory implies the exis tence of the new class of entities. To say that a theory is ontically committed to some class of entities is to say that if the theory were true, then that class of entities 38 would have to exist. Further, to be rational, a person who accepts a theory as being true must also accppt the ontic commitments of that theory, in the sense that he must believe that these entities exist. A theory is tested by comparing predictions made on the basis of the theory with observations. Predictions are derived in the following way. Through observation, the determinations of another theory, or even determinations of the theory at issue, a description of initial conditions is given. For example, if the theory contains an equation which describes the evolution of some physical system, then the relevant initial conditions would be some state of the system. Next, a set of boundary conditions are given. These boundary conditions are usually simply assumptions, and often embody simplifying idealizations. For example, we might assume that, during the evolution of our physical system, it will remain isolated, i.e., unaffected by any • outside forces, or that the evolution takes place in a gravity-free environment. Now, the statement of initial conditions, the statement of boundary conditions, and the theory together yield as a logical consequence a statement describing some other state of the system. That is, if we know some initial state of a physical system, we can use some theory to predict later states of that system. Many physical theories contain dynamic equations, which are 39 alleged to describe the evolution of, or change in, some physical system. For* example, classical mechanics contains Newton's laws of motoin, and quantum mechanics contains the Schrddinger equation. These equations are time-dependent, and, given a description of some state of a system, will yield a description of a later state of the system. Other equations are "place-dependent", in the sense that, given a description of conditions at one place, they will yield a description of conditions at another place. In any event, to derive a prediction of a theory, statements of initial conditions and boundary conditions are necessary. Now, it should be noted that predictions, derived in the manner described above, are not actually part of a theory. They are consequences of the theory' and statements of initial and boundary conditions. However, if the initial and boundary condition descriptions are true, then the truth of the theory implies the truth of its predic tions. We must distinguish the genuine from the spurious predictions of a theory. Let T be some physical theory, i.e., a deductively closed set of sentences formulated in some formal language. Craig's Theorem^ tells us that the set of sentences of the form T9(Oi3 Op) 40 are true, where 0^ is a statement of Initial and boundary conditions, and 0p is a prediction. Since T is deductively closed, T contains the set of "conditional predictions" having the form ( 0^3 0p ). Now, as should be clear, unless some statement of initial and boundary conditions is derived from the theory itself, no non-conditional predic tion is contained as a part of the theory. Thus, no model of the theory need make the statements of the form 0p true. In this sense, the predictions derived from the theory are not part of the theory, and need not be true according to models of the theory. However, if some statement of initial and boundary conditions is true, then the prediction Implied by the theory and that statement of Initial and boundary conditions is to be regarded as a genuine prediction of the theory, while those predictions which follow from false statements of initial and boundary conditions are merely spurious predictions of the theory. This fact is relevant to determinations of ontic commitment, as well. Not only is a theory committed to the entities needed to make the theory itself true, but is also committed to the entities needed to make its genuine predictions true. That is, the genuine predictions of a theory provide some of a theory's total ontic commitment. In my discussion of the General Theory of Relativity, I will describe a case whbre the theory apparently predicts • 41 the existence of a very extraordinary entity, whose exis tence Is not necessary to make the theory proper true. In such a case, theorists usually try to show that the prediction results from false descriptions of initial or boundary conditions. Modern scientific realism (i.e., the realist inter pretation of scientific theories construed according to the modified Received View) is based, essentially, upon four principles. Cl) The essence of physical theorizing lies in physical theories. Thus, for example, the ontology deter mined by physics is a function of the ontic commitments of our best physical theories. (2) Physical theories are linguistic entities; i.e., they are deductively closed sets of sentences formulated in some formal language rich enough to include the mathematical apparatus used in the theory. (3) Theories are intended to provide literally true descriptions of an independently existing physical reality, in the sense that all of the sentences of a theory are intended to be literally true and about the physical world. (4) Acceptance of a theory involves believing that it is true.13 It is this view of scientific theorizing which is presupposed in, for example, John Earman's argument for the existence of substantival space time, to wit:^ The covariant formulation of the GTR is our best physical theory of gravitation; the covariant formulation of the GTR is ontically committed to substantival space-time; thus, we are warranted in believing that substantival space-time exists. The form of this argument is standardly used to justify belief in abstract or theoretical entities, and, as should be readily seen, the falsity of any of the above four principles would be sufficient to vitiate this argument. 4. Realism in General; in this essay, I will be criticizing a widely-held account of the structure and interpretation of scientific theories, which I described in the last section, and which is usually referred to, in the literature, as 'modern scientific realism'. I maintain that this is an unfortunate choice of terminology, because the term 'scientific realism' is also used by many other philosophers to refer to a wide variety of metaphysical and epistemological views which have little or nothing to do with either the structure or the interpretation of scienti fic theories. Modern scientific realism has two basic components: structural and interpretive. As mentioned above, the structural part is a modification, or develop ment, of an old Positivist account of the structure of theories which has been called 'the Received View'. What I propose to do in the remainder of this essay, in order to avoid (but not, I hope, engender) confusion, is to use the term 'scientific realism* to refer to a class of general metaphysical and eplstemological views, which I will describe in this section, and use the term 'the Received View' to refer to what is usually called 'modern scientific realism'. Thus, according to my terminology, what F. Suppe calls 'the Received View' is a precursor of the structural part of what I will be calling 'the Received View', Let us start with simple realism. Two important points about realism must be made Immediately. (1) There are, in general, three different kinds of realism: onto logical, metaphysical, and eplstemological, and (2) when one is a realist, one is a realist about something (e.g., the material world, mathematical objects, possible worlds, etc.). Ontological realism about objects of type X is the view that objects of type X exist independently of their perception or apprehension by any mind. That is, they would exist even if there were no perceivers. Metaphysical realism about objects of type X is the view that all of the characteristics, or properties, of objects of type X are had determinately and independently of any perceiver. Kant might be taken as one who holds that while the existence of the material, or phenomenal, world is mind-independent, insofar as its existence depends upon the noumenal world, the nature of the world of experience is entirely 44 mind-dependent. EplstemoXogical realism about objects of type X is the view that we can gain knowledge about objects of type X. For each of these types of realism there is a cor responding type of idealism. Ontological idealism about objects of type X Is the view that objects of type X exist only when and only because they are perceived, or appre hended, by some percelver, finite or infinite. Metaphysical idealism about objects of type X is the view that at least some of the characteristics of objects of type X depend upon the perception of such an object. Locke's "secondary qualities" might be examples of mind-dependent properties. Epistemological idealism is the view that we can gain knowledge only of entities which can be perceived. Finally, there are nihilism and skepticism, two rather extreme views. Contrasted with ontological realism and ontological idealism about objects of type X, there is nihilism about objects of type X, which is the view that objects of type X do not exist at all. Skepticism about objects of type X is contrasted with epistemological realism and epistemological idealism, and is the:view that we can gain no knowledge of objects of type X. Scientific realism is essentially a version of epistemological realism about the physical world. It con sists primarily of the claim that science can provide knowledge about the unobserved physical world. That is, science extends our knowledge beyond that provided by sense experience alone. There are stronger and weaker versions of scientific realism, depending upon whether it is claimed that only science can give us such knowledge. Scientific realism, so characterized, is compatible with a wide variety of ontological and metaphysical positions. For example, one might be an ontological realist about physical, moral, and mathematical objects; a metaphysical idealist about physi cal and mathematical objects; and a scientific realist about physical objects. Such a person would thus believe that physical, moral, and mathematical objects have a mind- independent existence, but the characteristics of the physical and mathematical realms depend upon our conceptual schemes, and science can give us knowledge about thp physi cal world, in part by providing the necessary conceptual scheme. Now, nothing that has been said so far commits us to any particular view of the structure and interpretation of physical theories. As generally described, scientific realism simply maintains that science can (and does) give us knowledge of the physical world, but it does not tell us how this knowledge is provided. To answer this further question, we need a "theory of theories", an account of the structure and interpretation of scientific theories. One such theory Is that vhich I am calling 'the Received View', and is the best known and most widely accepted such account. According to this view, science provides knowl edge through scientific theories in the following way. Scientific theories are Intended to be literally true descriptions of what the world is like, and we are justi fied in believing that a theory is true if the theory is highly confirmed. So, we are Justified in believing the prediction, determinations, and claims of a highly con firmed theory, and thus can acquire knowledge of the world beyond our sense experience. It is Important to realize that the general version of scientific realism does not imply the account described briefly above. That is, one could hold a variety of posi tions regarding the structure and interpretation of scien tific tyeories and still maintain that science gives us knowledge about the unobserved physical world. For example, 'instrumentalism* denotes a class of such accounts, all of which claim that scientific theories are not intended to be literally true, but rather are used as mere computational devices in generating predictions and determinations. On these views, theory testing serves not to confirm or dis- confirm a theory, but to indicate the range of its empiri cal adequacy. To say that a theory is empirically adequate is to say that it generates literally true predictions and determinations. Thus, an instrumentalist may be a scien tific realist, in the general sense, because one can be an instrumentalist and still maintain that scientific theories can be used to justify some of our beliefs about the physi cal world. Of course, it is incumbent upon the instrumen talist to explain exactly how this justification is pro vided, and different versions of Instrumentalism give different explanations. Further, though the Received View has often been used to justify ontological and metaphysical realism about some class of theoretical entities, and instrumentalism has often been used to support nihilism about theoretical entities, one can be an instrumentalist about scientific theories and still be an ontological, metaphysical, and epistemological realist about theoretical entities such as electrons and quarks. Instrumentalism is a view about scientific theories, and may include the claim that there is more to doing science than constructing and testing theories. One can deny that scientific theories determine ontology without thereby denying that science is relevant to determining the metaphysical structure of the world. As I shall argue later in this dissertation, the fact that our best theory is apparently committed to some novel entity does not warrant our belief in the exis tence of that entity. Other, extra-theoretical, considera tions are relevant to justification of our existential beliefs. Thus, since scientific realism is a general view about science, while instrumentalism is a more specific vi6w about scientific theories, the two are not necessarily incompatible. III. GENERAL PROBLEMS WITH THE RECEIVED VIEW In this section, I will describe' a number of diffi culties which beset the Received View, as I have described it in Chapter III. Generally, all of these problems are of the same sort; the Received View, taken as an account of the logic and structure of physical theorizing, does not do Justice to the actual behavior, attitudes and practice of working physicists. I will be addressing the question "Can we, In the context of the Received .View, give an account, or explanation, of the actual behavior of prac ticing physicists?" 1. The "Time-Reversed Solution" Case: The first case we shall examine in our critical discussion of the Received View is a rather simple one. It is a case where, according to the Received View, some theory makes a number of false predictions, and should thereby be disconfirmed. However, no physicist regards these claims as genuine pre dictions of the theory, in the sense that they do not regard their falsity as disconfirmatory. This first case involves the so-called "time-reversed solutions" to the dynamical equations of a theory. All of the laws of motion, both classical and relativistic, and many other laws besides, have the form of equations which are of second 49 50 order in the time variable. That is, these laws are either linear in the second derivatives of position (or any quan tity that changes over time) with respect to time, or quad ratic in the first derivatives. When combined with the field equations, which describe the characteristics of some field of force (e.g., electromagnetic or gravitational), the equations of motion yield an expression which describes the trajectory of a body through that field of force. Suppose we1-want to solve the following problem, which might appear in any introductory physics text. Someone is standing on a platform 40 meters above the ground, and drops a lead ball straight down with no initial velocity (that is, the person does not throw the ball down). How long does it take for the ball to reach the ground? Before we can begin to solve this problem, we must make some assumptions about the boundary conditions of this experiment. For one thing, since the ball is made of lead, the effects of air resistance upon its trajectory will be minimal, and may be neglected. That is, we may, with no significant risk of error, treat the ball as though it'-were falling in a vacuum. Secondly, since the speeds reached by the ball during its fall will be small relative to the velocity of light, we may neglect relativistic effects, and use the classical equations of motion. Finally, since the gravitational field near the surface of the earth is very 51 weak, relatively speaking, we may, with no significant risk of error, use Newtonian gravitational theory, rather.than the GTR. Ve know, from Newtonian gravitational theory, that the force exerted upon a body near the surface of the earth by the gravitational attraction of the earth is given by the equation (1) F * GmM/r2 " m g where G is Newton's gravitational constant, and is equal to 6.673 x 10~8 cm3/g sec2 , m is the mass of the body in ques tion, M is the mass of the earth, which has been determined to be 5.077 x 1027 g, r is the radius of the earth, which 6 2 is 6.371 x 10 cm, and g « GM/r is called the "accelera tion of gravity near the surface of the earth", and is 2 easily calculated to be -9.81 m/sec . (This quantity is negative because the force of gravity upon the body is toward the center of the earth, i.e., in the negative radial direction.) Newton's second law of motion is (2) F » ma so we have that (3) a ■ g. 52 In order to solve the problem we have put before ourselves, we must find the equation which expresses the height of the ball above the ground as a function of time. That is, we need to find the equation of the curve, parameterized by t, that represents the trajectory of the ball. We know that, according to Newtonian mechanics, acceleration is the second derivative of position with respect to time. Thus, from (3) together with our value for g, we get (4) d2h(t)/dt2 « -9.81 where h(t) is the height of the ball above the ground at time t. Integrating this twice gives us (5) h(t) *> xQ + vQt -f (-9.81/2)t2 where x q and v q are constants of integration which repre sent, in this problem, the initial height and initial velocity of the ball, respectively. (We arrive at this interpretation of the constants of integration as follows. v q appears when we integrate (4) the first time, and get dh(t)/dt ** -9.81t + Vq . Since dh(t)/dt has the units of velocity (m/sec), so must Vq . Further, when t ■ 0, dh(t)/dt «* Vq , so Vq must be the initial velocity. The Interpretation of x q is gotten in similar fashion.) Equation (5) is the equation that we were looking for, which gives us the height of the ball above the ground as a 53 function of time. To generate a prediction as to when the ball will hit the ground, we set h(t) equal to zero, plug in the appropriate values for x0 and V q , and solve this equation for t. Inasmuch as the value thus arrived at for t will have been logically (mathematically) derived from the laws of Newtonian mechanics together with true (or approximately true) statements of Initial and boundary con ditions, the solution of equation (5) must count, according to the Received View, as a genuine prediction of Newtonian mechanics. However, notice the following Interesting fact. There are, not one, but two solutions, or roots, to this equation; 2.8557 seconds and -2.8557 seconds, as can be readily seen from Figure 1 on the next page. What are we to make of this? Both of the above solutions to equation (5), it seems, must count equally as genuine predictions of Newtonian mechanics, yet no one would seriously maintain that classical mechanics predicts that the ball hits the ground at two times, once almost three seconds before it was dropped and once again nearly six seconds later. That is, physicists ignore, as physic ally insignificant, one of the solutions to equation (5). How can the Received View accommodate this fact? Prima facie, at least, it looks as if the Received View is in trouble. If, within the context of the Received View, we cannot justify ignoring the time reversed solution to this b(t) * \ 50 m 40 m 30 m 20 m 10 m 10 m -20 m -30 m -40 m 50 m Figure 1. Trajectory ol a Falling Ball 55 equation, then,, since physicists go ignore it, and do not regard the failure of this event to occur as disconfinning classical mechanics, it would seem that there is something wrong with the Received View of the logic and structure of physical theories. There are a number of responses that one might make in defense of the Received View in this case, which fall into two general categories: syntactic and semantic. The syntactic defenses, and there are several, may be generally characterized as claiming that in deriving the prediction that the ball hits the ground 2.8557 seconds before it is released, the formulation of classical mechanics used is not the correct, or canonical, formulation of that theory. The semantic defense consists of the claim that the proble matic predictions arise as a result of the wrong interpre tation of classical mechanics. Let us begin with an examination of the syntactic defenses. The strategy of the syntactic defenses of the Received View is to show how, within the context of the Received View, the derivation of the anomalous predictions can be blocked. For example, one might suggest that the canonical formulation of classical mechanics contains the claim that no "predictions" of events occurring at negative times can be true; i.e., that nothing happens at negative times. There are a couple of problems with this. In the 56 first place, the fact that, in our problem, one of the predicted landings is said to occur at >2.8557 seconds is simply a function of our choice of coordinate system. Our problem was set up so that the ball was released at t**0. However, we could equally well have set it up so that the ball was released at t»3, in which case the landings would be predicted to occur at t=.1443 and +.=5.8557. The fact that neither of these numbers is nega> tive is of no help. Further, equation (5) can be used to solve a number of problems, including the following one: A projectile is shot from ground level and reaches its maximum height of 40 meters at t=0. Vhen was it launched and when will it land? If the theory itself is construed as containing the claim that only one solution to equation (5) is physically significant, then we will not be able to use classical mechanics to answer one of these questions. Clearly, though, classical mechanics can be used to answer both. A defender of the Received View might respond to this saying "I admit I was wrong in suggesting that the claim necessary to prevent the derivation of the anomalous prediction is contained in the theory. Now I can see that it is actually one of the boundary conditions used in solving the problem we have put before ourselves. Thus, only in this problem, and others like it, will we be unable 57 to derive the troublesome prediction." Plausible though this sounds, it is open to a crushing objection. In order to get any derivation from the boundary conditions and a theory, the boundary conditions must be formulated in the language of the theory. If one of the boundary conditions is a claim to the effect that the ball does not hit the ground at t“-2.8557 seconds (or at any time before it is released), then we can derive a contradiction from the boundary conditions together with the theory. The rest of the boundary conditions, together with the initial condi tions and the theory, yield, as we have seen, the predic tion that the ball does hit the ground at t--2.8557 seconds, and the suggested boundary condition yields the claim that it does not. Thus, from that set of boundary conditions and that theory, everything can be derived as a prediction, since contradictions imply everything. Clearly, this is intolerable. The persistent defender of the Received View might continue in his syntactic defense by suggesting that we modify the mathematics used by, and contained in, classical mechanics. This might be done in a couple of ways. First, we might interpret the claim that quadratic equations have two roots disjunctively. That is, the theory is to be regarded as predicting that either the ball hits the ground 2.8557 seconds before it is dropped or it hits 2.8557 seconds 58 after it is released, where the disjunction is Inclusive. In this way, we can put the claim that the ball does not hit the ground before it is dropped into the boundary con ditions and not generate a contradiction. Further, given that the disjunction is inclusive, we can use the theory to solve other problems, in which both solutions are to be taken seriously. Attractive though this proposal may seem, it does have its drawbacks. To see this, let us take a closer look at this proposal. The thrust of the proposal consists of the suggestion to change the mathematics used by the theory, or, rather, to give another account of the mathematics used in the theory. That is, we need differ ent formulation of algebra. We retain the assumption that solutions to equations describing the behavior of some physical system generate predictions of future, past, or distant behavior of that system. The proposed modification of the mathematics is to the effect that there is only one solution, or root, to a quadratic equation, though that solution may be either of two numbers. In the case described above, on this view, the equation of motion yields the claim that either -2.8557 is the solution to the equation or 2.8557 is the solution to the equation, or both. The set of boundary conditions contains the claim that -2.8557 is not the solution to equation (5), so the theory has, as a genuine prediction, only the claim that the ball hits the ground 2.8557 seconds after it is dropped. But what is the general definition of the expression 'is a solution to a quadratic equation1? It has to be something like the following: Real number a is a root to equation E containing one variable, x, iff the result of substituting a for x in E is an identity. In this case, however, both -2.8557 and 2.8557 are roots of equation (5), and the boundary conditions together with the theory again yield a contradiction. That is, the nor mal definition of a root of a quadratic equation makes it clear that we cannot interpret the claim that a quadratic equation has two roots disjunctively. A second way of modifying the mathematics used in a theory so as to prevent the derivation of anomalous predictions might be simply to eliminate negative numerals from the vocabulary of the language of the theory. This proposal, however, is borne of desperation, and can easily be seen to be inadequate, primarily because it would be extremely difficult to do any interesting physics without negative numbers. Further, as I have pointed out before, negative numbers are really red herrings in this issue. It is a simple matter to change the case so that the anomalous prediction involves only positive numbers. In the syntactic defense of the Received View, the defender either reformulates the theory Itself, or 60 restates the initial and/or boundary conditions used to derive the anomalous prediction, or tries to modify the mathematics used In the theory. As we have seen, none of these ploys will work, in the sense that none of them give us a way of accommodating this particular piece of behavior of actual theorists within the context of the Received View. That is, all of these attempts to save one tenet of the Received View end up violating another. The semantic defense of the Received View is that we limit the domain of our interpretation of the theory so as to exclude the anomalous solutions. This suggestion, however, is obviously inadequate. Though it is true that different physical theories make use of different mathema tical theories, and thus require different domains for their models, this is of no help to the defender of the Received View in our case. For one thing, it is not at all clear what we would exclude. As I have said before, simply excluding negative numbers would be of no use. Further, if we limit our domain yet do not change the mathematics, there will be sentences in the theory that will not be assigned a truth value. That is, the logical system comprised by a physical theory will be unsound, in that it will contain sentences which are provable but untrue.* 61 In presenting a final comment upon the criticisms I have leveled against the Received View, and the defense of that view, let me offer a reminder of what it is that the defender of the Received View is up against. As the time-reversed solution case clearly shows, practicing physicists routinely disregard, as physically insignificant, claims about the behavior of physical systems which can apparently be derived from some well-known physical theory, as that theory is construed upon the Received View. To save the Received View, its defender must show how the physicists' behavior can be justified, or even explained, within the context of the Received View. 2. Approximation and Idealization: Let us turn now to another issue which has proved troublesome for the Received View, which is the role of approximation and idealization in physics. No theory is regarded as being precisely true; all theories make essential use of simpli fying idealizations. Boyle's gas laws apply only to idealized gases. The Special Theory of Relativity applies only to ideal clocks and rods. The cosmologies developed on the basis of the General Theory of Relativity are highly idealized. £ven the little problem of the falling lead ball I described above involved a degree of idealization; we neglected air resistance; we used Newton's laws of motion, which are known to be strictly false; and we used 62 Newton's theory of gravitation, which is also known to be strictly false. It often happens that theories which are strictly false, and have been superseded by better and more precise theories, are still used and developed. For example, Newtonian gravitation theory has been replaced, in some sense, by the GTR, yet, for most purposes, it is the old, Newtonian, picture that is used. For instance, when computing the planned trajectory of some spacecraft (such as Pioneer, Voyager, or even Apollo), the more accu rate GTR is rarely used. Newtonian mechanics is regarded as being approximately true in some regimes, and is good enough to be used for many purposes. Finally, highly idealized initial and boundary conditions are often used to derive predictions of a theory which are used to test that theory. As we shall see later, all of the classic tests of the GTR arise from highly idealized boundary con ditions. It is not at all clear how, according to the Received View, the role of idealization and approximation in physics is to be explained. The Received View has it that theories of physics are intended to be literally true descriptions of reality, not approximately true ones. How is this basic tenet of the Received View to be squared with the pervasive use of approximation and idealization in physics? 63 Thus, for the above-mentioned general reasons, it seems that the Received View is inadequate as an account of the logic and structure of physical theorizing. In the next section, I will take a close look at the Special and General Theories of Relativity, and show why the Received View is inadequate as an account of those theories, in particular. IV. THE GENERAL THEORY OF RELATIVITY1 The General Theory of 'Relativity (GTR) is a theory of gravitation. It is, in a very real sense, a generaliza tion of the Special Theory of Relativity (STR), which is a theory of kinematics. Roughly speaking, the STR tells us how bodies move in the absence of external forces, and the GTR tells us how they behave under the Influence of gravity. Before entering a detailed discussion of the GTR, I will give a brief presentation of the STR.2 1. The Special Theory of Relativity. We begin with our ordinary, pre-analytic notions of space and time. The "points" which make up space are places, and the points of time are moments, or times. To facilitate discussion of places and times, and to allow us to re-identify such points, we place a coordinate system upon a region of space and an interval of time. A coordinate system for space is a mapping x from places, or spatial points, into R3 (i.e., the Cartesian cube of the real numbers), and a coordinate system for times is a mapping t from moments into R. Thus, a point in space is identified with an ordered triple of real numbers, and a moment with a single real number. Given the three-dimensional character of space, reflected 3 by the fact that the range of x is R , the coordinate 64 65 system x can be regarded as an ordered triple of mappings, each of which maps a place into R and represents the "axis" of the coordinate system. Thus, where p is a point l o o in space, x(p) *= (x (p), x (p), x (p)) « (a,b,c), and where q is a moment, t(q) «* d, where a,b,c,d e R. Now, there is more than one way to coordinatize some region of space and interval of time. For example, we can introduce other coordinate systems x' = (x ^ jX'^jX*3) and t* to cover the same regions of space and time covered by x and t. Thus, a single point p has two different coordinate representations, x(p) and x'(p). The relation ship between two coordinate systems is given by a coordi nate transformation; a set of equations which tell us how to determine the coordinate values of a point in one coor dinate system given the coordinate values of that point in another coordinate system. We will restrict our atten tion to rectilinear, Cartesian, coordinate systems. Two such systems, defined over the same region of space, may differ in any, or all, of three ways. The axes of one coordinate system may be rotated with respect to the axes of the other, one coordinate system may be in motion rela tive to the other, and the origin of one system may be displaced from the origin of the other at some reference time. There are (at least) two different kinds of coordinate transformations which can relate one coordinate system to another* A Galilean transformation is given by the equations (6) x'1 « R*xJ + v*t + d1 <7) t* - t + T, where i, j run from 1 to 3, Rj is a rotation (i.e., a uni- modular orthogonal matrix with determinant equal to 1), the v1 are constants equal to dx*/dt and represent the velocity of the primed coordinate system with respect to the unprimed system, the d* are constants which represent the displacement of the origin of the primed system from that of the unprlmed system at t=0, and x is a constant which reflects the difference in choice of origin for t and t'. Any index, like j in equation (6) that appears twice, once as a subscript and once as a superscript, is understood to be summed over unless otherwise noted; that is, equation (6) is an abbreviation for (8) x ’1 ** r Jx 1 + r |x 2 + Rjx3 + v*t + d1 . Consider, as an example, two coordinate systems x and x' which are at rest relative to one another, whose origins coincide at t^t'^O, and which are such that the primed 67 coordinate system is rotated by an angle of a around the x3 (*x’3) axis. Then Rj is familiar and is * . Pcosa sino 0*1 (9) Ri ■ -sina cosa 0 L 0 0 1J 1 1 9 2 1 Thus, in this case, x' ** x cosa + x^sina, x' *» -x sina + 2 3 1 x cosa, and x f “ x . If the two coordinate systems are not rotated with respect to one another, then 6j is called the "Kronecker delta". A Lorent2 transformation is given by the equations (11) x ,A *» AjX^ + A^t + d* (12) t ’ - Aj j + A*t + x Where A^ is a 4x4 orthogonal matrix, i,J run only from 1 to 3, and d* and t are constants. Ap is restricted by the conditions (13) AgA|nop * ny6 68 where 10 0 0 0 10 0 (14) na6 0 0 10 L0 0 o - u Ag embodies both a rotation and a translation (or "boost"). If we collapse our two coordinate systems x and t into one four-dimensional coordinate system covering a region of "space-time" and let t*x4 and x«d^, then the Lorentz trans formation equations may be expressed* as (15) x ,ot - Agx® + da where a ,8 run from 1 to 4. Kinematics, both classical (I.e., Newtonian) and relativistic, is the study of the motion of bodies in the absence of any external forces. That is, it is the study of inertial motion, where a-body is moving inertially if, in the absence of external forces, it either remains at rest or travels in a straight line with uniform velocity. An inertial reference frame is a region of space and time in which all free bodies move inertially with respect to the frame. Clearly, neither a frame at rest in a gravitational field nor a frame experiencing any acceleration will be an inertial frame. On the other hand, different inertial frames may be in uniform translatory (i.e., non-rotating) motion relative to one another. Now, central to any theory of kinematics is the general principle of relativity, which states that the laws of motion should he the same in every inertial frame. In other words, free bodies behave exactly the same in different inertial frames, as can be seen by comparing the behavior of your drink as you fly at 30,000 feet with a velocity of 500 mph to that of a similar drink as you sit at home. That is, without looking out of the window, no experiment you can perform will tell you whether you are at rest or moving with a uniform velocity. All of the laws of physics, then, should have the same form in every inertial frame. The principle of relativity is relevant to our discussion of coordinate transformations for the following reason. Let A and B be two inertial reference ‘frames moving with a uniform velocity with respect to one another. Let A be the frame regarded as being at "rest" and B the frame in "motion". Let us further provide Cartesian coor dinate systems for these frames, coordinate systems which 12 3 satisfy the following constraints. Let x»(x , x , x ,t) 12 3 cover A and x ,8*(x* ,x* ,x' ,t' ) cover B, and let x and x' be set up so that at t"*t'**0, x and x' coincide and so that the origin of x' is moving along the x^-axis, in the nega tive x^-direction, with a velocity v^-dx^/dt. Any law governing the behavior of bodies relative to A and expressed 70 in terms of x will also govern the behavior of those bodies relative to B just in case that law retains its form under a coordinate transformation from x to x'. That is, the law must have the same form when re-expressed in terms of x'. In order to satisfy the principle of relativity, the laws of a theory of kinematics must be invariant under a coordinate transformation. The question thus arises: "Which coordinate transformation equations should we use?" Since a law which is Invariant under one transformation may not be invariant under another, the choice of transformation equations is of utmost importance. Which coordinate trans formations express the correct relation between different inertial coordinate systems (i.e., coordinate systems covering different inertial frames)? In the situation described above, the Galilean transformation equations are simply x 11 » x* - vt x'2 = x2 (16> X ',3 “ X J3 t ' * t and, intuitively, would seem to be the natural choice. Indeed, all physicists up until the time of Einstein found these to be obviously correct. Consider any equation of Newtonian mechanics, for example 3 * where F Is the force acting on a body and p 1 ("mE (dx,x/dt')> i-1 is the momentum of the body as measured in inertial frame B. It is an easy matter to show that this equation takes the same form when written in terms of a coordinate system related to the first by a Galilean transformation. If we use equation (16), equation (17) becomes < 1 8 > p = " 3dt t [L dt # - v + — dt + — dt 1J Since v is a constant, this is equivalent to which clearly has the same form as does equation (17). Any equation which retains its form under a Galilean transforma tion is said to be Galilean invariant, and the claim, impli cit in Newtonian mechanics, that the laws of physics are invariant under Galilean transformation is called the Prin ciple of Galilean Relativity. Classical mechanics, then, is based upon the claim that the Galilean transformation equations express the correct relation between inertial coordinate systems. 72 Thus, it was held that the equations of any ade quate theory of mechanics or dynamics must be Galilean* invariant. However, the theory of electrodynamics pre sented by Jmaes Clerk Maxwell in 1864 did not satisfy the principle of Galilean relativity. For one thing, Maxwell's equations predict that the speed of light in a vacuum is a universal constant c, but if this is true in one coordinate system, then it will not be true in a "moving" coordinate system defined by a Galilean transformation. Maxwell's equations were taken to be true only in inertial frames at rest with respect to absolute space. Thus, one should be able to determine the velocity of the earth through abso lute space by measuring the velocity of light in the earth's reference frame and comparing it to the constant c. The most famous experiment designed to measure the differ ence between these two values was performed by Michelson and Morley in 1887. Their null result provoked controversy within the community of physicists. On one hand, Newton's laws of mechanics were considered to be highly confirmed, and were Galilean invariant. On the other, Maxwell's theory was also well-confirmed, powerful, and elegant, but was not Galilean invariant. Einstein's solution to this dilemma, presented in 1905, was to abandon the principle of Galilean relativity. He noted that Maxwell's equations were invariant under Lorentz transformation, and suggested that the laws of motion be modified so that they, too, were Lorentz invariant. This suggestion, a cornerstone of the STR, is called the Principle of Special Relativity. Let us see how the demand for Lorentz invariance affects the laws of motion. In the situation involving inertial frames A and B described above, the Lorentz equations are x ’1 ■= X ^ l - v2 )-l/2 + Vt(l - v2 )“^/2 x ’2 - X 2 X'3 •= X 3 t' = vxX(l - v 2 )“1/2 + t(l - V2)-1/2 Brief calculation reveals that equation (17) is not invar iant under a Lorentz transformation. However, if we replace equation (17) with dp1 d T 3 dx'1 _ -1/2 (21) F = --- « -— m E ---- (1 - vz ) dt 1 dt * |_ i»l d t 1 where p' = m(l - v2)2 -' V 2 E (dx'Vdt') i is the relativistic i=l expression for momentum, then we get an equation which is Lorentz invariant. The fundamental property that distinguishes the Lorentz transformations is that they leave invariant the "proper time" dx, defined by 74 (22) dx2 = dt2 - (dx1)2 - (dx2 )2 - (dx3 )2 In a new coordinate system x' , the coordinate differentials are given by (23) dx* » A®dxY so the new proper time will be (24) dx'2 « Jijj^dx^dx'3 (25) » na6A^A3dxYdx6 (26) - ny6dxYdx6 and therefore (27) dx'2 = dx2 . It is this property that accounts for the observation by Michelson and Morley that the speed of light is the same in all inertial systems. If we let dx2 = (dx1 )2 + (dx2 )2 + (dx2 )2, then dt2 “ dt2 - dx2 . A light wave front will have belocity (|dx/dt|) equal to the speed of light, which in standard units is unity; hence the propagation of light is described by the statement that (28) dx = 0, 75 Performing a Lorentz transformation does not change dx, so dx'2 * 0, and therefore [dx'/dt1| « l; that is, the speed of light in the new coordinate system is still unity. An examination of the particular Lorentz transfor mation, equation (20), described above, yields an inter esting feature of such transformations. In performing this transformation, we have chosen to use coordinate units which are such that the velocity of light in any system is unity. Any velocity less than that of light, then, is expressed as a dltnensionless fraction between 0 and 1. Thus, to transform any velocity expressed in ordinary (meters/sec) units into our units, we must divide it by the constant c *» 2.997925 x 10® meters/sec. The 'v' terms in (20), then, are equivalent to 'vcon/c', where vcon is the velocity of one coordinate system with respect to the other expressed in conventional units. Suppose, now, as Newton did, that the velocity of light is infinitely large. Then equation (20) collapses into equation (16). Clearly, what made Galilean relativity so attractive and obvious was the extremely large value of c. In his original paper, Einstein based his view that it was Lorentz, and not Galilean, invariance that was important upon two principles, or postulates: (1) The velocity of light is independent of the velocity of its source, and (2) the laws of nature should take the same 76 form In different inertial reference frames. Once con- . vinced that the Principle of Special Relativity was correct, it remained only to modify the kinematical and mechanical equations of Newton so that they were Lorentz invariant. In its essence, then, the STR is simply the claim that the laws of nature are Lorentz Invariant, and not Galilean Invariant. Finding the relativistic versions of the familiar laws of nature provided the means whereby the STR could be tested. The STR is a theory of kinematics; a theory of how physical objects behave in the absence of external forces in non-accelerating reference frames. This theory is extended to describe various deviations from this. For example, relativlstlc electrodynamics is a theory about how electromagnetic forces affect physical objects. In order to describe how a force affects some object, we must first know how the object behaves in the absence of that force. The STR provides that knowledge. Originally, as Einstein formulated it, the STR, i.e., the Lorentz transformation equations, gave us a way of determining what some measured spatial or temporal interval between events would look like to a variety of observers moving inertially. For example, if one observer measures the length of some body in some inertial reference frame, the STR tells us what value for the length of that body an observer experiencing uniform translatory motion relative to that frame will arrive at. In the original, so-called "3+1", formulation, space and time were regarded as being separate and distinct, each being represented by a different manifold. Space was represented by the three- dimensional C°°-manlfold R^, and time by the one dimensional R . In 1908, however, H. Minkowski suggested that space and time could be regarded as being part of a larger unity, which he called space-time, which consisted of all places- at-times (point-events). That is, rather than regarding the set of placeB as one thing and the set of moments as something entirely different, Minkowski suggested that we collapse the two into a single set of place-times. The set of such events, idealized to geometrical points, can be represented by a four-dimensional manifold, and the history of some particle can be represented by a curve on the mani fold. A four-dimensional coordinate system x ** (x1, x2, Q A x , xw ) can be defined on the manifold, and relative to such a coordinate system, the hyper-place x® » 0 can be 1 2 3 regarded as representing space, and the line x » x » x ■ 0 as representing time. The spatial and temporal intervals between' events in one coordinate system will differ from those in another coordinate system related to the first by a Lorentz transformation. However, the proper time inter val between events is invariant under a Lorentz transforma tion. Ve can reflect the Principle of Special Relativity, 78 as formulated by Einstein, by endowing the space-time manifold with the Minkowski metric, tv.3 The proper time interval between events on the manifold is then given by (29) dT2 « na gdxadxe and any coordinate transformation that leaves dr invariant must be a Lorentz transformation. Minkowski thus recast the STK in geometrical terms. According to this version of the STR, it is the geometrical structure of space-time itself that accounts for the behavior of free bodies. Inertial reference frames are regions of space-time endowed with a Minkowski metric. It is this version of the STR that we are interested in here, primarily because it is this version that Einstein used in formulating the GTR in 1915. 2. The General Theory of Relativity;. The GTR is a theory of how gravitational forces affect physical objects. It is based upon the STR in the same way that relativistic electrodynamics is based upon the STR. A finite reference frame in a gravitational field is not an inertial frame; it is not a flat, Minkowskiian, region of space-time. The GTR was designed as a theory of the physics in such non-inertial frames. The basis of the GTR is the Principle of Equivalence, which states that at every space-time point in an arbitrary gravitational field it is possible to choose a "locally inertial coordinate system" such that, within a suffic iently small region of the point in. question, the laws of nature take the same form as in unaccelerated Cartesian coordinate systems in the absence of gravitation. It was this principle that the famous elevator gedanken experiment of Einstein was designed to illustrate. Consider a closed elevator in free fall in a uniform gravitational field. No test performed by an occupant of this elevator would tell him that the elevator was not at rest in an inertial, gravitation-free, reference frame. Similarly, the occupant of a closed elevator could not tell whether he was being accelerated through a gravitation-free region of space or whether he was at rest in a uniform gravitational field. Let us take a brief excursion into differential geometry. Differential geometry is the study of the geo metrical structure of n-dimenslonal manifolds. A differ entiable manifold is a separable Hausdorff space which is topologically locally Euclidean. A topological space U is said to be a separable Hausdorff space if it satisfies the Hausdorff separation axiom: whenever p and q are two distinct points in U, there exist disjoint open sets U and V in M such that pcU and q e V. To say that a space is topologically locally Euclidean is to say that the space can be covered by coordinate systems that map regions 80 (whence comes the local character of this requirement) of the space onto regions of Euclidean n-space (i.e., Rn « endowed with its usual topology). A space would be topologically globally Euclidean only if a single coordi nate system could map the entire space onto Euclidean n- space. Now, compare the above statement of the Principle of Equivalence with the claim, central to differential geometry, that a manifold is locally Euclidean. The former states that, in effect, space-time is locally inertial, while the latter states that the manifold representing space-time is locally flat. Ve know from Minkowski's version of the STR that an Inertial reference frame is a flat region of space-time, and further that flat space-time is Minkowski space-time. Thus, the Principle of Equivalence is often characterized as stating that space-time is locally Minkowskiian. That is, at any point of the space-time manifold, we can find a coordinate system covering a neighborhood of that point in which the components of the 4 global connection defined on the manifold vanish and in which the components of the metric tensor at that point are equal to the Minkowski metric. It should be easy to see how the close analogy between the Principle of Equiva lence (space-time is locally Minkowskiian) and the defini tion of a manifold (manifolds are locally flat) has led 81 many theorists to maintain that gravitational effects were in fact a function of the geometrical structure of an underlying space-time manifold. That is, a deviation from inertial motion caused by gravitational forces is equivalent to a deviation of the geometry of the underlying space-time manifold from Minkowskiian. Let us take a closer and more detailed look at the GTR, formulated as a theory of the geometry of space-time. In the end, I will want to argue that this geometrical interpretation of the GTR is mistaken. As is the case with any field theory, the GTR con sists of two sets of equations; one which describes the effect of the field upon test particles, and one which describes the relationship between the field and its source. We can derive the first set straightforwardly from the Principle of Equivalence, but the second set requires i additional assumptions. Let us begin by discussing the equations of motion for a body in a gravitational field. In Newtonian gravitational theory, the equations of motion are the familiar F = ma, where the gravitational force o acting on a body is given by F 8 -GMm/r . Thus, the equa tions describing the behavior of a body under the influence of gravitational, and only gravitational, forces are (30) -GM/r2 « d2x*/dt2 82 The presence of a massive body generates a force field which serves to deviate test particles from their ordinary, * unaccelerated, straight-line motion. The background space is assumed to be Euclidean, with particles free of all forces obeying the equations (31) d2xA/dt2 = 0. The situation in the GTR is rather different. The Principle of Equivalence tells us that an equation governs the behavior of particles in a general gravitational field just in case two conditions are met: (A) The equation holds in the absence of gravitation; that is, it agrees with the laws of the STR when the metric tensor gQ^ equals the Minkowski tensor nap and when the affine connection vanishes. (B) The equation is generally covariant; i.e., it preserves its form under a general coordinate transformation x x'. Consider a particle moving freely under the influence of purely gravitational forces. According to the Principle of Equivalence, there is a freely falling (i.e., locally Minkowskiian) coordinate system £a in which its equation of motion is that of a straight line in space-time, that is, (32) d2 £a/dT2 * 0 83 with dt the proper time dr2 ■ + na3dSadS6. Since is a logically flat coordinate system, the connec tion coefficients vanish in this system, and (32) is equivalent to (33) d25°/dT2 + rgY(d56/dT)(deY/dT) - 0 Now let xp be any other coordinate system related to £a by the general transformation tY (xM ) ■ $J[xp . (xP need not be an inertial coordinate system, and Lorentz transformation.) The coordinates £a are thus func tions of the xp , and (32) becomes d H a dxp (34) 0 ■ — dx 3xp dx 35a d2xp .a2.^ dxM dxv (35) 3xy dx2 3xp3xv dx dx If we multiply this by 3x*/3£a , and use the product rule (36) (3?°/3xp )(3xX/3ea) *= 6* we get (37) d2x*/dx2 + TpV(dxp/dx)(dxp/dx) *» 0 where 84 (38) rjv s C9xX/3ea ) 0 25Y/3xV3xV)- Thus, since (37) correctly describes the motion of a parti cle free of non-gravitational forces in an inertial refer ence frame (because it is equivalent to equation (32)), and since (37) is generally covariant, the Principle of Equi valence tells us that it is the equation of motion for that particle in any coordinate system. But (37) Just is the equation for a geodesic of a manifold endowed with the connection so we can conclude that particles free of non-gravitational forces follow geodesics of the space-time manifold. Further, since rjv will in general be non-zero in an arbitrary gravitational field, the geometry of space- tlme is in general non-Euclidean. Thus, rather than the gravitational field exerting a force upon a test body which deviates it from straight-line motion, according to the GTR, the gravitational field is in fact equivalent to the curva ture of space-time itself (i.e., to a deviation of the geometry of space-time from flat). According to the GTR, the gravitational field is a geometrical field, and not a field of force, as it is on the Newtonian theory. We turn now to a discussion of the equations which describe the effect of mass-energy distribution upon the geometrical structure of space-time. The Einstein Field Equations were originally derived from the Principle of 85 Equivalence together with a number of additional assump tions. The Field Equations relate mass-energy distribution with the metric. Thus, we must first find a way to incor porate the contributing sources of the gravitational field into our mathematical formalism. This is accomplished through the definition of the energy-momentum tensor (some- times called the "stress-energy tensor"), T. From the STR, we learn that the Lorentz Invariant 2 2 form of Newton’s second law is fa ■ m(d xa/dx ). This suggests that we can define an energy-momentum vector. (39) pa = m(dxa/dx). We know that (40) dx = (dt2 - dx2 )1/2 « (1 - v2)1/2dt. in some inertial coordinate system xa , where (41) v e dx/dt Then the space components of jp form the momentum vector (42) p ® rayv and its time component is the energy (43) p° = E = my 86 • where (44) y = dt/dx « Cl - V2)"1/2 For small v, these definitions give (45) p ■ mv + 0(v3 ) (46) E « m + l/2mv2 + O(V^) in agreement with the nonrelativistic formulas, except for the term m in E. (Note: for a body at rest in an inertial coordinate system, v ** 0, so E ■ ri. We are here giving mass in units of energy, which requires multiplication of 2 mass given in conventional units by a factor of c . Thus, we have the famous equation E ** mconvc2 .) The energy momentum tensor field assigns to each point p in space-time an energy-momentum tensor T(p), of » type (2,0). T(p) tells us how much energy-momentum is flowing through any given 3-dimensional volume ("3-cell") at p from the point of view of any observer. Let A, B, and C be three vectors in flat space-time which form the edges of some paralleloplped. We define the volume one-form £ of this paralleloplped as follows. (47) I - e(A,B,C) with components, relative to some inertial coordinate basis, C48) - + ^ aB/ ° B 6C^ where +1 if uaBy is an even permutation of 0123 -1 if yafjy is an odd permutation of 0123 1 0 if any index is repeated Tp(..,£) then gives the total energy-momentum $ crossing the 3-cell with volume one-form Z, and T(w,i>) ** w • j> is the projection of the energy-momentum along the vector Where u is the 4-velocity of some observer at p, Tp(..,Q) gives the density of energy-momentum (i.e., the energy- momentum per unit of three-dimensional volume) as measured in the observer's Lorentz frame at p. Given an inertial coordinate basis {§a >, (50) T ** a e0 . The components, Ta®, of the energy-momentum tensor are interpreted as follows. * (51) .= mass-energy density as measured by co**moving observer, s energy flux into the k-direction. Tk0 ® density of k-momentum = T^k. T** = pressure (l = shear tress (i^j) (Note that T is symmetric). 88 Every source of energy-momentum contributes to the energy-momentum tensor field: matter, radiation, particle velocities, electromagnetic and other force fields, etc. The energy-momentum tensor obeys the following conservation laws. (52) V ■ T ■ G or, relative to some Lorentz frame, (53) 3TaB/3xa * G 6, where G® is the density of the external force f® acting on some system. (For an isolated system, G& ■ 0.) Remember that Ta^ has been defined only relative to a Lorentz coordinate system. We define T^v as a contra- variant tensor that reduces to the special relativistic T p in the absence of gravitation. Then the generally covariant equation that agrees with (53) is (54) T^v = Gv . IP In deriving the field equations, we are guided by the following consideration. In most cases, the equations describing the relation between a field and its source are linear, because the field itself does not contribute to itself. For example, the electromagnetic field is deter mined by charge, and the electromagnetic field does not 89 carry any charge* Thus, Maxwell's equations are linear. Gravitational fields, however, do carry energy and momen- * turn, and must therefore contribute to their own source. So, the gravitational field equations will have to be nonlinear partial differential equations, the nonlinearity representing the effect of gravitation upon itself. We are also guided by the so-called "correspondence principle", which states that, in the weak field limit, the GTR must agree with Newtonian gravitation theory. This implies that (55) g00 « -(1 + 2<|>) where 4> is the Newtonian potential, which at a distance r from the center of a spherical body of mass M takes the form (56) 4> - -(GM/r), and which is determined by Poisson's equation (57) V24> - 4irGp where G is Newton's gravitational constant, ? is the ordinary flat-space gradiant, and p is the mass density. Furthermore, the mass-energy density Tq q for slowly moving i' (i.e., nonrelativistic) matter in a weak gravitational field is just equal to its mass density, (58) T q q « p Thus, we have that (59) V2g00 - -8ltOT00 This field equation is only supposed to hold.for weak static fields generated by nonrelativistic matter, and is not even Lorentz invariant as it stands. However, it leads us to guess that the weak-field equations for a general distribution T0g of energy and momentum take the form <60) Ga(J = -8„GTa6 where Gag*is a linear combination of the metric and its first and second derivatives. It follows then from the Principle of Equivalence that the equations which govern gravitational fields of arbitrary strength must take the form (61) Gyv = -8h GTmv where G11%, is a tensor which reduces to G„« for weak fields. llv up Ve have the following constraints upon (A) By definition, G^v is a tensor. 91 (B) By assumption, GyV consists only of terms with two derivatives of the metric; that is, Gyv contains only terms that are either linear in the second derivatives or quadratic in the first derivatives of the metric. (C) Since T^v is symmetric, so is G^v . (D) Since T^v is conserved, so is G^v : i.e., « 0. From these constraints, the final version of Einstein's field equations can be derived. They are (62) Ricci - l/2gR - \g - -8irGT, or, in component notation, (63) Rm v - l/Rg^R - Xg)JV - -SnGT^. • • where Ricci is a contraction of Riemann, and R, the curvature scalar, is the contraction of Ricci. Now we have both halves of the GTR; the equations of motion, which describe gravitational fields and their effects upon arbitrary physical systems, and the field equations, which describe the relation between the gravi tational fields and their sources. The GTR, then, is a theory of gravitation, according to which gravitational phenomena are a function of the geometrical structure of space-time. Strictly speaking, the GTR is not a theory of the geometrical structure of space-time, as some people have claimed. Theories of the geometry of space-time can be derived from the GTR, by plugging into the field equa tions some value for the energy-momentum tensor field, and solving them for g^v . That is, using astronomical data, laboratory data, and educated guesses, the relativistic • astrophysicist arrives at values for the components of T in some selected coordinate system. These values include con tributions from all known and suspected matter and energy fields in some region of space-time. For example, the following are the currently accepted best-guesses for average mass-energy densities throughout the universe. (64) Intergalactlc dust: Pid St 10"28 g/cm3 Rest mass of galaxies: Prm et lO”31*1 g/cm3 ct Cosmic rays: Per 10-34 g/cm3 Zero rest-mass •*. -n ’.•» radiation: \ Electromagnetic 1 S! 1 0 -3 3 radiation: > Prad g/cm3 Gravitational ) radiation: J CO < 10-35 -V. Magnetic fields: pmag Microwave radiation: pmw « 4x10-34 g/cm3 As our beliefs about these values change, our theories of space-time change with them. For example, it has been 93 recently suggested that neutrinos are not massless, as was previously believed. This additional mass, which was unsuspected before, could make a substantial contribution to the total mass-energy In the universe, and thus could cause a significant change in our current theories of the geometry of space-time. Astrophysicists also search for local solutions to the field equations; i.e., theories of the geometrical structure of regions of space-time based upon guesses as to the energy-momentum density in those regions. These exact solutions to the gravitational field equations depend upon more than just assumptions about mass-energy distribution. As we shall see in the next section, where we will examine in some detail the deriva tion of the Schwarzschild solution, a number of additional assumptions roust be made in order to arrive at a unique theory of space-time. 3. The Schwarzschild Solution: All of the classi cal tests of the GTE (at least those that test the field equations) make use of the Schwarzschild solution to the field equations. Discovered in 1916 by K. Schwarzschild, the Schwarzschild solution describes the gravitational field external to a star meeting certain conditions. The fact that it is the external gravitational field that we are concerned with means that this is a so-called "vacuum" 94 solution, i.e., one according to which the energy-momentum tensor is zero. Inside the star, of course, the geometry will be considerably different,- as it will have to take into account the change in energy-tnomentum density as we go from the center to the surface of the star. Exterior to the star, however,, these densities are constant and equal to zero. Thus, the field equations for such a region of space-time are (65) Rwv « 0 (Here and hereafter, I assume that X, the "cosmological constant", is zero.) Simply specifying a zero value for the energy- momentum tensor is not sufficient to uniquely determine a solution to the field equations. In fact, there are many solutions to equation (65), including Minkowski space time. The Schwarzschild solution arises from (65) together with a number of additional assumptions about the gravi tational field in question, all of which are simplifying idealizations. First, we assume that the gravitational field, and hence the star which produces it, is static and isotropic (i.e., that it remains constant through time and is constant under rotational transformation). These two assumptions probably (it has not yet been proved) imply 95 that the field is spherically symmetric. We shall assume spherical symmetry. We further assume that at infinity the metric becomes Lorentzian, and that at large dis- . tances from the star (in the weak-field limit) Newtonian gravitational theory is correct. That these assumptions are idealizations should be clear. Consider our Sun. The Schwarzschild solution has been used to describe its gravitational field. However, the Sun is not static (it rotates, it pulsates), nor is it isotropic (it is not even exactly spherical). Further, the notion of the metric "at infinity" is already an idealiza tion, and the gravitational fields of other stars, planets, the interstellar medium, etc., prevent the metric from ever becoming precisely Mlnkowskiian. Finally, Newtonian theory is only approximately correct, even in the weakest field. Thus, strictly speaking, the Schwarzschild solution describes the geometrical structure external to a star that doesn't exist. However, for most purposes, the com plications that more realistic assumptions would introduce are not worth the trouble. (An exception to that comes when we test the GTR by putting a gyroscope into orbit around the earth and look for it to process. Such preces sion is predicted when we abandon our staticity assumption, and allow for rotation.) Let us go on and examine the derivation of the Schwarzchild solution. 96 Ve begin by using our iirst two assumptions (stati cal ty and isotropy) to determine the general form of the metric we are looking for. To say that the field is static is to say that it must be possible to find a set of "quasi-Uinkowskiian" coordinates (coordinates according to which the metric has signature +2) x1, x2, x3, x° = t, such that the invariant proper time dx2 = -gpVdx^dxv does not depend on t (i.e., 3g^v/3t «= 0). By "isotropic" we mean * 1 2 3 ^ thatthe metric depends on x(=(x , x , x )) and dx only through the rotational invariants dx2, x * dx, and x2. The most general form of the metric (expressed as a line ele ment) is then (66) dx2 = -F(r)dt2 + 2E(r)dtx • dx + D(r)(x • dx)2 + C(r)dx2 where F, E, D, and C are unknown functions of (67) r = (£ • $)1/2 It is convenient to replace x with spherical polar coordinates r, 0, 1 o 3 (68) x A « rsin0cos<|> x « rsin0sin<}> x « rcosB Equation (66) then becomes (69) dx2 *» -F(r)dt2 + 2rE(r)dtdr + r2D(r)dr2 + C(r)(dr2 + r2de2 + r2sin20d<}>2) We are free to reset our clocks by defining a new time coordinate (70) t' = t + 4>(r) with $ an arbitrary function of r. This allows us to eliminate the off-diagonal element gtr by setting (71) d*/dr = -(rE(r)/F(r)) Equation (69) then becomes (72) dx2 - -F(r)dt'2 + G(r)dr2 + C(r)(dr2 + r2de2 + r2sin2ed<}>2) where (73) G(r) = r2 We are also free to redefine the radius r, and thereby impose one further relation on the functions F, and C. For instance, suppose that we define 98 Then (72) takes what is called the standard form (75) dt2 - -B(r')dt'2 + A(r')dr'2 + r'2 (d62 + sin2ed$2) where (76) B(r’) = F(r) [ G(r)"| I" r. , dC(r) (77) A(r’) I + C(r)J I + 2C(r) dr (We drop primes from now on.). The metric tensor has the nonvanishing components (78) grr = A(r) ge0 - r2 g^ - r 2 sin2() gtt “ B(r > with functions A(r) and B(.r) that are to be determined by solving the field equations. To solve equation (65), we must find the values of the components of Ryu in this coordinate system. We can derive these values from the equations arA arA (79) r x s — H £ + r!Jlcrx - r!]v r x UKV axv axK I1* vn Kr> (80) Rpv 5 * \ Xv To derive the necessary values for r X .-, we use the equation yv ( 8 i) r x - i/2bxp!55J1 + ^«pv _ f fv v P'1 ' 3xv axP ax” 99 Setting all of the components of RpV equal to zero yields the following equations (82) B'/B - -A'/A (A prime now means differentiation with respect to r.) d (83) — (rB(r)) - rB'(r) + B(r) » 1 dr Now, (82) is equivalent to (84) A*(r)B(r) + A(r)B'(r) = 0 which is equivalent to (85) (A(r)B(r))' - 0 which implies (80) A(r)B(r) ** constant Recalling our assumption that forr*e°, the metric tensor must approach the Minkowski tensor, we have that (87) lim A(r) « lim B(r) = 1 From (86) and (87) we have then (88) A(r) = 1/B(r) The solution to (83) is 100 (89) rB(r) « r + constant To fix the constant of integration, we recall our assump tion that in the weak-field limit, Newton's theory becomes correct. That is, at great distances from a central mass M, the component gtt = -B must approach -(1 + 24>), where ♦ is the Newtonian potential -MG/r. Hence the constant of integration is -2MG, and our final solution is 2MG j (90) B(r) - |\ - 2MG j -1 A(r) - fl - The full Schwarzschild metric is then given by - r^d0^ - r^sin^Od#^. We identified the integration constant U with the mass of the sun by comparison with Newton's theory. In fact, it can be shown that M is precisely equal to the total energy p° of the sun and its gravitational field. V. MORE PROBLEMS WITH THE RECEIVED VIEW 1, The Singularity Case: Let us recall our ear lier discussion of the bearing of the Received View on the issue of ontic commitment. According to this picture, a theory contains, as a part, the axioms and theorems of the mathematical apparatus used in the theory. In the case of the GTR, this includes the definition of a manifold, upon which all of the geometrical objects which determine its geometrical structure are defined. Thus, clearly, the GTR is ontlcally committed to the space-time manifold. Fur ther, as I mentioned above, a theory can acquire ontic commitments in a somewhat different way. If, given some true statement of initial and boundary conditions, the theory predicts the existence of some novel entity, then, even though the theory proper does not require that entity to exist in order to be true, the theory can be said to be committed to that entity. This commitment is dependent upon the truth of the statement of initial conditions. In what follows, I shall describe a case, taken from current gravitation theory, where it has been demonstrated that the GTR is committed, in this latter sense, to a class of entities which very few theorists working in the field are willing to countenance. 101 102 Now, there are two distinct, though related, prin ciples here. First, according to the Received View, the only grounds for accepting or rejecting commitment to some new class of entities are that some highly confirmed physi cal theory imply the existence or non-existence of those entities. That is, questions of ontology are to be decided by science, not philosophy. Metaphysical princi ples are not to be used to decide what there is; physics is. Secondly, if, for whatever reason, we conclude that some entity whose existence is predicted by some theory does not in fact exist, then, according to the Received View, that theory must be false, and thus abandoned. The fact that a theory makes false predictions should either cause the theory to be abandoned, or else to precipitate a crisis in physics. (For example, around the turn of the century, it was known that many of the predictions made on the basis of classical mechanics were incorrect. Unfortunately, at that time there were no alternative theories in favor of which Newton's theory could be abandoned. However, as Kuhn has pointed out, until the introduction of relativity theory and quantum mechanics, the field of physics was in a profound state of crisis.) In the example I am about to describe, the theory in question clearly implies the exis tence of some new class of entities, yet most theorists 103 neither accept that commitment, nor do they abandon the theory as being Incorrect, or bad, nor do they see It as anything to get particularly upset about. * These case In question, as mentioned above, comes from research in relativistic cosmology, and the novel entities involved are called "singularities."* Let me begin my discussion of this case with the mathematical definition of a singularity. A manifold (endowed with a metric tensor field) is said to be non-singular if the metric is non-singular at every point on the manifold. Recall that the metric (regarded as a matrix of coordinate dependent components) is non-singular if it has an inverse; i.e., if there is a matrix such that the product of it and the metric equals the identity (unit) matrix. In a singular manifold, there will be distinct points which have an unde fined separation. The points on the manifold at which the metric becomes singular are called "singular points". There are a number of solutions to the Einstein Field Equations (i.e., relativistic space-times) which contain singular points. As an example, consider the four dimensional manifold endowed with the Schwarzschild metric, expressed in Schwarzschild coordinates. Expressed as a line element, the metric is given in equation (91) of the last chapter, where 2MG is expressed in renormalized units 104 of length, and Is equal to 2MG/c2 in conventional units. Notice that the component matrix 2MG (92) 1 0 0 0 r 2MG i 0 (i - _ ) - A 0 0 r 0 0 0 0 0 r2sin2e _ is singular at the point r « 0 and the points r ■ 2MG. Since the components of the metric tensor are coor dinate-dependent, finding a coordinate system in which the component matrix is singular does not imply that the metrix is singular. For this to be true, it would have to be the case that relative to any .coordinate system, the component matrix of the metric tensor is singular. Thus, we can try to eliminate these apparent singularities by finding a coordinate system in which the component matrix is non singular. It turns out that we can find such a coordinate system for the points r = 2MG, thus Indicating that the singularity there is only a "coordinate singularity", resulting from an infelicitous choice of coordinate system. Unfortunately, the same cannot be said for the singular point at r B 0. No recoordinatization will prevent the 105 metric from becoming singular at this point. Thus, Schwarzschild space-time is a singular manifold. Now, we can make our manifold non-singular by simply "cutting out" the singular points that it contains. In the Schwarzschild space-time, the metric is singular only at the point r » 0. For all points on the manifold approaching r *» 0, the metric is non-singular. Thus, by removing the point r = 0 from the manifold, we obtain a non-singular manifold. The hole left by this procedure, while it is not a singular point, is regarded as a singu larity. The presence of such a singularity is Indicated by the existence of incomplete geodesics. An incomplete geodesic is a geodesic whose affine parameter has a finite upper bound, and yet which lacks an endpoint. Recall that a curve is a mapping from some interval of the real line into the manifold. An incomplete geodesic curve then is a mapping which is defined only over a finite interval of the real line, and whose value for the upper bound of the affine parameter is not on the manifold. A space-time is said to be geodesically complete if no geodesic curve in the space-time is incomplete. A manifold is singularity- free if it is geodesically complete. (Actually, this is too simple. Cases could arise (e.g., Gddel space-time) where the manifold was geodesically complete, and yet was 106 ' not singularity-free. However, this complication is not relevant to the Issues at hancL) One peculiarity of this sort of singularity is that the curvature of the manifold becomes unboundedly large near the singularity. (Insofar as singular points have been cut out of the manifold, a definition of 'near1 is needed. One can say that points on an incomplete geodesic are near the singularity if they correspond to values of its affine parameter which are near the upper bound of that parameter.) If singular points were included in the manifold, then the radius of curva ture at the singularity would be zero; i.e., the curvature would be infinitely large. This would imply, in a rela tivistic space-time, that the energy-momentum density at that point would be infinitely large. However, since such singular points have been cut out of the manifold, as a singularity is neared, the curvature becomes arbitrarily large. As should be clear, providing a physical interpre tation of these formal notions is problematic. What would the physical manifestation of a singularity be like? They have been described as "the edge of space and the end of time", and as holes in space-time. Further, when a free- falling particle, or light-ray, represented by a geodesic curve, "falls into the hole left by the singular point", 107 Its existence comes to an end, and It Is no longer repre sented by a world-line on the manifold.' Finally, the notion of infinite curvature, or arbitrarily large tidal gravitational effects, is difficult to conceive. To achieve this, a finite amount of energy-momentum (in the form of mass-energy) would have to occupy an arbitrarily small volume of space. Ordinarily, the need to provide a physical inter pretation for these mathematical oddities is easily avoided, because the space-times that contain them are not physically realistic (i.e., they are idealized in some way). For example, the Schwarzschild solution represents the spherically symmetric empty space-time outside a spherically symmetric massive body with radius r ** R. The metric inside the body (r < R) has a different form determined by the energy-momentum tensor of the matter in the body. That is, the Schwarzschild metric correctly describes the geometry of space-time only outside r « R, and so the singularity of r « 0 does not arise. Now, if bodies such as stars are regarded as point-masses, then the singularity does occur, but the assumption that a star is a point-mass is an obvious Idealization. Another, similar, case arose in the early 1920's, and has vexed theorists ever since. In 1922, Alexandre Friedman found the so-called Frledman-Robertson-Walker solution to Einstein's field equations. This was a "global" solution, in that it was supposed to represent the geometrical structure of the entire universe. It stemmed from a number of simplifying idealizations, including the assumptions that the mass-energy of the universe was a perfect fluid, and that the universe was globally Isotropic and homogeneous. The FRW metric is a dynamic solution (i.e., it is time-dependent) which indi cated an expansion of the universe, and there is a point on the FRW space-time manifold at which the mass-energy of the universe is concentrated into a region of zero volume; a singularity. This is the first so-called "big bang" model of the universe. At the time, however, the prejudice was for a static universe, thus the FRW solution was regarded as being physically unrealistic, and so the singularity Involved was ignored. Even when Hubble's dis covery of a linear distance-red shift relation indicated that the static picture of the universe was incorrect, few theorists were prepared to accept the FRW cosmological model and the singular beginning it entailed. They felt that the extreme symmetry conditions assumed in the deri vation of the FRW metric were unrealistic, and that in solutions that were more realistic, the singularity would disappear. For many years, then, research into global solutions of the field equations consisted in the search for non-singular space-times which were good models of the available observational evidence. The search has so far proved to be fruitless. Singularities have appeared in all of the more realistic global solutions found to date, and most theorists have arrived at the conclusion that additional realism will not prevent their occurrence. Further, in the mid-60's, Roger Penrose and Stephen o Hawking proved a series of theorems which demonstrated that, in a relativistic space-time (i.e., one in which the Einstein field equations held), a singularity would form if a small number of rather plausible conditions were met. Thus, it seems extremely unlikely that a non-singular space-time will result from realistic initial and boundary conditions, and so it appears as if the GTR is committed to the existence of singularities. In response, most theorists have taken the view that because GTR does demand a singularity, the theory must fall to describe nature at some point. The fact that Einstein's equations predict the existence of a singularity indicates that the theory "breaks down" for some regions of the universe. To accept Einstein's theory as completely adequate would be to accept something never before known in physics; the concept of a true singularity actually achieved in nature, and the unbounded curvature and tidal gravitational forces that implies. The view of the 110 majority of relativists can be characterized generally as follows. GTR does correctly describe most of the gravita tional phenomena in the universe, and correctly describes conditions in most regions of space-time. In regions of extreme curvature and matter density, none of our current theories are adequate. For many years it was thought that a marriage of quantum mechanics and the GTR would yield a theory which would avoid prediction of singularities and thus be ade quate for all of the universe. Unfortunately, after 40 years of research and despite many heroic attempts and partial successes, no such quantum mechanical theory of gravity has been developed. In fact, if Uisner and 3 Wheeler are correct, then even if such a theory could be developed, it would predict the same sorts of singularities that are predicted by the GTR. Further, it seems very unlikely that any adequate theory could be developed and tested to the degree that would make it a plausible candi date, for the following reasons. According to today's physics, there are four basic forces: weak nuclear, strong nuclear, electromagnetic, and gravitational. Of the four, gravity has the smallest influence on sub-atomic processes in ordinary situations, though it does exert its influence over extremely large Ill distances. (Electromagnetic forces are also Inverse square, but due to the fact that electric charge is dis tributed evenly between positive and negative, its effects are negligible In macro-phenomena.) However, since the strength of gravitational forces is a function solely of mass-energy, there are situations in which the effects of gravity upon nuclear processes are non-negligible. Further, given enough mass, gravitational forces will overcome even 4 the strong nuclear forces. In such a situation, according to modern physics, there are no more equilibrium states for matter, and a body will collapse under its own weight to a zero volume, leaving an Infinite matter density and region of infinite curvature (or, rather, since the body leaves the manifold when it reaches this point, it leaves a region of unbounded matter density and curvature). To say that one is unprepared to accept the existence of a real singularity is to say that one believes that there is another equilibrium state for matter; that there is another force which has not yet been discovered and which will balance out the gravitational forces at work and prevent total collapse. Further, this new force would have to be such that it could not be overcome by any amount of gravitational force; I.e., no matter how massive (the universe Itself), no body could experience total gravita tional collapse. 112 Unfortunately, there are some rather Impressive barriers to the discovery of such a force; i.e., to the construction of a theory which would correctly describe conditions in regions of extremely high curvature. In the first place, obviously, such conditions cannot be dupli cated in the laboratory. We cannot subject ordinary matter to such extraordinary forces and observe the processes, because we cannot create such regions of extreme curvature artificially. Further, even if we could, it is not clear that it would be of any help. Though it has yet to be proved, most relativists believe that the so-called "cosmic censorship principle" is true. In the current jargon, this principle states that there are no naked singularities; i.e., there are no singularities that can be seen from infinity. All singularities will be bidden by some sort of particle, or event, horizon, similar to that surrounding the curvature singularity in the Schwarzschild model, Thus, if the principle is correct, then only those observers brave (or foolhardy) enough to venture through the event horizon of a black hole would be in a position to observe conditions in the regions of extreme curvature we are interested in. Unfortunately, neither they nor their reports could ever escape back through the event horizon to enlighten their less adventurous comrades. 113 Finally, even if there were singularities for which there were no event horizons, there would still be veils standing between us and the regions we are interested in. For example, it might be maintained that there is no such event horizon between us and the initial, cosmological, singularity. However, it would be extremely difficult, if not impossible, for any information from that earliest of times to reach us. For example, some theorists have said that the discovery of the 3°K cosmic microwave background radiation confirms the hypothesis that the universe sprang from the initial singularity with a "big bang", since this background radiation is the "lingering echo of that primordial explosion". This interpretation, however, is somewhat misleading. In fact, the most we can conclude is this. If the radiation background discovered by Penzias and Wilson in 1964 is in fact black-body radiation of tem perature 3°K, and if the universe is expanding at the rate determined by current calculations, then there was a time (about 700,000 years after that time which current calcu lation says was the time of the "big bang") at which (a) the radiation in the universe was in thermal equilibrium with the matter in the universe, (b) the temprature of this system was about 3,000°K, (c-) prior to this time, the universe was opaque to radiation, and (d) a short time later, as temperatures fell to the point where free 114 electrons could join with nuclei to form atoms, the uni verse became transparent to radiation. That is, when temperatures were above 3,000°K, and the universe was in thermal equilibrium, the presence of a larger number of free electrons made it extremely unlikely that any photon could travel an appreciable distance without being absorbed or scattered.) In this sense, the universe was opaque to radiation, and the moment of "recombination", when the free electrons Joined with nuclei, is the earliest time about which we could get information in the form of radiation. Since that time, nothing of any significance has happened to the radiation that existed at that time, except that, due to the cosmological red-shift, its wavelength has increased to the point that the temperature has fallen to 3°K. Now we know from statistical mechanics that the pro perties of any system in thermal equilibrium are entirely determined by the temperature and the densities of a few conserved quantities. Thus, the universe preserves only a limited memory of its initial conditions, and the radiation background and its apparent homogeneity actually serves to confirm only the hypothesis that the universe had a hot, dense, early phase. That is, the microwave radiation dis covered by Penzias and Wilson can be used to decide between "Big Bang" and "Steady State" cosmologies, but can tell 115 us nothing ahout the existence, or character, of the ini- 5 tlal singularity. Presumably, the same would be true of any "naked" singularity, since it, too, would involve conditions of extreme temperature. Thus, a veil is drawn across every region of extreme curvature; either in the form of an event horizon or in a form similar to that which hides the ini tial singularity from our view. The conclusions to be drawn from the above discus sion is, I feel, that no plausible theory (in the sense that it can be tested or that it has plausible physical underpinnings) can be constructed to fill in the gaps left by the GTE. The faith that most theorists seem to share that some force exists that will prevent the formation of real singularities in those regions of space-time in which the GTR breaks down is based solely upon the conviction that actual singularities cannot exist in nature. Though the forces necessary to prevent formation.of a singularity may in fact exist, it is very unlikely that a theory describing those forces could ever be discovered and adequately tested. This would be theoretical physics in its most extreme form. Thus, we have the following situation. On the assumption of any physically realistic initial and boundary conditions, the GTR predicts the formation of singularities, 1 1 6 many of which are interesting in that they involve the curvature of some region of space-time becoming unbounded. The GTR is, unquestionably, the best theory of gravitation that we have available. Most relativity theorists, how ever, refuse to accept commitment to the physical reality of space-time singularities, and conclude that the theory "breaks down" (i.e., no longer describes nature correctly) in these regions of extreme curvature. ' Rather than abandon the GTR, these theorists continue to test and confirm the theory, even though they know that finding a theory to plug the holes left by the GTR is extremely unlikely. Now, there are many interesting features of this situation, but the ones we are most interested in here are its implications for the Received View of the structure of the GTR. The Received View would seem to imply the following. If a theory T predicts the existence of objects of type X, and if T Is our most highly confirmed, simplest and most elegant theory, then either objects of type X exist or T is false. If one holds that such objects do not exist, and, hence, that T is false, then what does the Received View say about one's attitudes toward T? If the point of constructing a physical theory is to provide a literally true description of some aspect of the physical world, then if that goal cannot be achieved by some theory, then that theory should be abandoned. That is, on this 117 picture of scientific theorizing, false theories are. worthless theories* The prediction of singularities has not led anyone to abandon the GTR, however. In fact, more theorists are working to test.lt now than have ever before. If the point of physical theorizing is to give literally true theories and if a theory is known to be false, then the point of doing further research on that theory would be lost. In the case of the GTR, howeyer, much exciting research is now in progress. (Currently, it is the search for the gravitational radiation predicted by the theory that occupies many theorists.) Construed as an account of the logical structure of the overall activity of physical theorizing, the Received View leaves something to be desired. There is another interesting feature of the singu larity case; one that involves ontic commitment. According to the Received View, the ontic commitments of physical theories are determined in a straightforward syntactic way, and the ontic determinations made by science come from the ontic commitments of our best theories. Since the GTR is our best theory of gravitation, and since it predicts the formation of singularities, then, without further ado, physics is committed to the physical existence of singulari ties. However, as we have seen, most relativity theorists refuse to accept these commitments. Since it is theorists, 118 and not theories, that we must pay closest attention to, it seems that the Received View is Inadequate again. What principles could theorists be appealing to in order to justify their refusal to countenance those entities to which their theory so clearly commits them, according to the Received View? If the Received View were correct, then no such principles would exist. As we have seen above, the theorist's refusal seems to stem from a reluctance to accept entities the likes of which have never before been found in the physical world. Infinite, or unbounded, physical magnitudes have never been encountered. Another interesting characteristic of singularities is that, if the cosmic censorship principle Is correct, we would not expect to be able to see singularities. That is, they have no observable effects. Thus, the rejection Of singularities by theorists is not based upon their failure to see some thing they would expect to be able to see. Rather, their . .. .. ,« * attitude toward singularities is based upon philosophical, rather than empirical, objections. "Quantities of unbounded magnitude are physically impossible". Thus, at the very least we must conclude that the apparent ontic commitments of our best theories can be overridden by metaphysical prin ciples. There could be no observational evidence against the hypothesis that space-time singularities exist, our best theory is committed to their existence, and yet 119 theorists refuse to believe that they exist. Clearly this is a*case where the determinations made by practicing theorists run counter to those made according to the Received View, thus indicating the inadequacy of that account of scientific theorizing. It is also interesting to note what those few theorists who are willing to accept commitment to singulari ties have to say. The most notable physicist to hold this position is Charles Misner, which he defends in Misner [1969b]. His position has been noticed by other philos ophers, who have had some rather strange things to say about it. John Earman, a scientific realist who appar ently accepts the Received View, characterizes Misner's view as follows ". . . C. Misner argues that since obser vational evidence together with Einstein's theory suggests that time in our universe has a beginning, we had better g get used to the notion that time has an 'absolute zero1." That is, Earman characterizes Misner as arguing the straight realist line; since the GTR is our best theory and the GTR predicts the existence of singularities, singulari ties must exist. Actually, however, Misner's argument is rather more subtle. Progress in physics can proceed both from tolerance and from intolerance. One could be intolerant of classical models of the atom because atoms were observed not to suffer 120 catastrophic radiative decay. But Einstein and Bohr were tolerant of Planck's theory of radia tion (in spite of its singular discontinuities) which violated no observation. Relativistic cosmology has had reasonable success (the expansion of the Universe, the microwave back ground) tending in the direction of support for its most dramatic theoretical novelty (the ini tial singularity), and there are no observational indications of, say, an era of contraction pre ceding the present expansion. Since the objections to a singularity are conceptual, rather than observational, then, I judge it as a situation in which tolerance is indicated. Ve should stretch our minds, find some more acceptable set of words to describe the mathematical situation now iden tified as "singular", and then proceed to incor porate this singularity into our physical thinking until observational difficulties force revisions on us.? Thus, Uisner's argument depends upon the principle that we ought to be tolerant of those apparent ontlc commitments that have no observational consequences. In fact, Misner feels compelled to offer further argumentation. He points out that incorporation of the initial singularity into our picture of the physical world would help us to do cosmology in that it would explain how the observed conditions of the universe (the observed isotropy, in particular) could have evolved from a wide range of initial conditions. (.Without such a singularity, it is extremely unlikely that conditions in the early universe would have been homogeneous enough to have evolved into the presently observed globally homogeneous universe. Thus, according to Misner, the observed isotropy of the universe can best be explained by 121 the presence of the Initial singularity, which guarantees o a high degree of symmetry in the early universe.) Thus, Misner justifies his acceptance of singularities further on grounds of their explanatory power. But this argument is a far cry from the simple realist line described by Earman. Even here, where a theorist is willing to accept commitment to singularities, he justifies his position by appeal to principles which are independent of the theory at issue. Thus, not only is it true that most theorists disagree with the ontic commitments of the best theory, as determined by the Received View, but all theorists involved arrive at their ontlc commitments in ways that differ significantly from those prescribed by the Received View. Thus, we have good reason to believe that the Received View is inadequate. Let me say a few words about why the Received View, and its attendant criterion of ontic commitment, have been so popular. Perhaps the motivation can best be illustrated by the following passage from Bernard Cohen's interview with Albert Einstein, which describes a conversation between Einstein and Ernst Mach on the subject of the atomic theory. This conversation took place in 1911, shortly before Mach's death. 122 Einstein asked Mach what his position would be if it proved possible to predict a property of a gas .by assuming the existence of atoms— some property that could not be predicted without the assumption of atoms and yet one that could be observed. Einstein said he had always believed that the invention of scientific concepts and the building of theories upon them was one of the great creative properties of the human mind. His own view was thus opposed to Mach's, because Mach assumed that the laws of science were only an economical way of describing a large collection of facts. Could Mach accept the hypothesis of atoms under the circum stances Einstein had stated, even if it meant very complicated computations? Einstein told me how delighted he was when Mach replied affirmatively. If an atomic hypothesis would make it possible to connect by logic some observable properties which would remain unconnected without this hypothesis, then, Mach said, he would have to accept it. Under these circumstances it would be "economical" to assume that atoms may exist because then one could derive relations between observations. Einstein had been satisfied:* indeed more than a little pleased.9 Clearly, in this exchange, Einstein is holding a version of scientific realism and Mach an extreme form of instrumen talism and nihilism about theoretical entities. (Mach later recanted on his agreement that atoms may exist.) It is also clear what is motivating Einstein to adopt his realist position. If, by assuming the existence of some unobservable, or theoretical, entity (in this case, atoms), we are able to predict, or explain, certain observable phenomena which we would otherwise be unable to predict, or explain, then we have good reason to believe that those theoretical entities do in fact exist. This principle is one with which it would be hard to find fault. However, in the Received View’s attempt to give this principle a rigorous formulation, it underwent a subtle change. According to the Received View, the principle is that if a theory is the best theory available in terms of its predic tive and explanatory power and if it is committed, formally, to the existence of some set of theoretical entitles, then we have good reason to believe that these theoretical entities exist. The difference between these two formula tions can be brought out by finding a theory which is the best theory available in terms of its predictive and expla natory power and which is committed, formally, to the existence of some unobservable entity which plays no pre dictive or explanatory role in the theory whatsoever. That is, suppose we find the GTR, for example, committed to some entity whose postulated existence contributes nothing to our abilities to predict or explain any observable phenomena at all, let alone any observable phenomena we otherwise wouldn't be able to predict or explain. In such a case, the first formulation of our principle would not justify our belief that the entity in question actually existed, while the second formulation would justify that belief. The singularity cases described above provides us with just this sort of example. Not only are relativity theorists almost unanimous in their rejection of singularities, but 124 it is also true that it is extremely unlikely that singu larities could have any observable consequences. Now, it is Ulsner's contention that by assuming the existence of the initial (cosmological) singularity we are in a better position to explain the observed homogeneity of the uni verse than we would be if that assumption is denied. How ever, the arguments advanced by Misner have recently been called into question. (See note 8, above.) In any event, there is another example, again from the GTR, of a theoretical entity whose existence is predicted by the formal, canonical formulation of the theory, but which plays absolutely no explanatory or predictive role. 2. The "Other Universe” Case: Consider again the Schwarzschild solution to the Field Equations. The metric of this solution is (93) dx2 - -(1 - 2MG/r)dt2 + (1 - 2MG/r)-1dr2 + r2dfi2 in Schwarzschild coordinates (where dn2 ■ de2 + sin2ed<|>2) As we have noted before, this metric, as written in these coordinates, is singular at the points r «* 0 and r ® 2MG. Further, the singularity at r ** 2MG is a coordinate singu larity, and can be eliminated by a suitable coordinate transformation. There are a number of other coordinate systems within which to write the Schwarzschild metric, but 125 but the best o£ them is the Kruskal-Szerkeres coordinate system. The *two coordinate systems are related by (94) u « (r/2MG - 1)1/2er/4MGcosh(t/4MG) v a (r/2MG - l)1/2er/4MGsinh(t/4MG) * u « (i - r/2MG)1/2er/4MGsinh(t/4MG) i /•> v / A w r when r < 2MG v a (i - r/2MG)2e ^ cosh(t/4MG) where u and v are. the dimensionless radial and time coor- dinates of the Kruskal-Szekeres coordinate system and r and t are the radial and time coordinates of the Schwarzchild coordinate system. By making this change of coordinate systems, the Schwarzschild metric becomes (95) dx2 » 32(MG)3/r"r/2MG - dv2 + du2 + r2dn2 Now, in this coordinate system, the metric is non-singular at r » 2MG. However, the singularity at r a o is located, 2 2 in the Kruskal-Szekeres coordinate system, at v* - u* a i. Thus, there are actually two singularities, not one; both v a +(i + u2)1/2 and v a -{i + u2)*'2 correspond to r = o. Further, notice that r » 2MG (the region of space-time far outside the gravitational radius at r « 2MG) is given by u2 >> v2 . Thus, there are actually two exterior regions; both u >> + |v| and u >> — |v| correspond to r >> 2HG■ To illustrate these points, consider the Kruskal-Szekeres diagram on the next page. Regions IX and IV represent the 126 t®-® t— 2.5M / t-2.5M t**-M 4 Si n h « I m rKD-nri !f "r^j it 11 1 j L U t"-2.5M t“2.5M r»2M -v Figure 2. Schwarzschild Space-Time areas inside the gravitational radii of the two singulari ties, and regions 1 and III represent two distinct, asymptotically flat universes in which r > 2MG. in the Kruskal-Szekeres diagram, as in a Minkowski diagram, light rays are represented by 45-degree lines. Ve can see, from looking at the diagram, that no time-like or null curve can travel from region I into region III, or from region III into region I. Thus, no particle or light ray can pass from one of these asymptotically flat universes into the other. Assuming that we live in one of these universes, this means that no information about the other universe could ever reach us; i.e., though the other universe is not strictly disconnected from ours (space-like world lines can pass from one to the other), it is physically inacces sible to us. The physical processes in our universe go on unaffected by anything in the other universe, as the processes in that universe are unaffected by anything in ours.*0 In fact, as is usually the case, things are a little more complicated than this. As my reader has no doubt already discerned, the Schwarzschild solution arises from rather highly idealized initial and boundary condi tions. For example, the characteristics of the geometry surrounding a Schwarzschild black hole depend solely upon the mass of the body that underwent gravitational collapse. It might be thought that taking into account such addi tional factors as rotation (angular momentum) and charge would have a significant effect upon the space-time geo metry around a black hole, and this has been shown to be true. We have, for example, the Kerr solution, which represents the stationary axisymmetrlc asymptotically flat field outside a rotating massive object, and the Reissner- Nordstrbm solution, whidh represents the space-time outside a spherically symmetric charged body carrying an electric charge. It might further be hoped that these more realis tic solutions would avoid prediction of regions of the universe which are causally disconnected from ours, and, in a certain sense, this is also true. Consider the Penrose diagrams on the following page. In the Reissner-Nordstrdm space-time, since the singularities are time-like, two distinct universes are connectlble by a time-like world- line. This has led to speculation that matter could travel through such black holes into these other universes.** If this is possible, it might imply the possibility of matter appearing explosively in our region of space-time through "white holes", and this would imply that at least some of these other universes might have observational effects. However, it has been argued that, even though these dis tinct regions are connected by time-like curves, no matter or radiation could actually pass through the black hole r=0 (singularity) r=0 (singularity) A — -:----- r = 2^ \ 11 /r=2M i n I r=0 r=0 \ t = ° (singularity) (singularity) / IV / t = “ r=0 (singularity) Figure 4 Schwarzschild Space-Time Figures 3 and 4 are Penrose diagrams of Reisner-Nordstrom space-time and Schwarz schild space-time, respectively. Notice that in the former, the singularities are time-like, whereas in the latter, they are space-like. These diagrams are taken from Hawking and Ellis (1973), p. 154 and p. 158. r=0 (singularity) Figure 3 6ZX Riesner-Nordstrom Space-Time from one universe into the other. (I.e., the scenario of the Disney movie "The Black Hole" is impossible) Simpson and Penrose^ argue that the interior of such a black hole would be smashed by unbounded blue shift effects associated with classical matter falling along the so-called "inner horizon" (r = *■__)• In this way, the space "bridge" would be destroyed by back reaction of the gravitational field due to the energetic matter. Birrell and Davies ". . . demonstrate that quantum mechanical effects, similar in origin to the attraction energy between two electrically neutral conducting plates (Casimir effect), also cause an unbounded back-reaction which would smash the idealised interior geometry, even if no actual matter were falling 1 q into the hole." Thus, it seems that even though these "other universes" are connected, in the mathematical sense, with ours, no information can pass from them into our universe. In other words, the hypothesis that such uni verses exist generates no observational consequences by which to test that hypothesis. This case is interesting for a couple of reasons. For one thing, it shows that the attempted formal charac terization of the principle used by scientific realists such as Einstein to Justify their belief in the existence of entities they could not observe which is given by the Received View is Inadequate, in that it does not capture 131 the essence of this principle. It shows that a theory can be committed, in the sense of the Received View, to enti ties which possess no explanatory or predictive power what soever. It is also interesting to note what theorists have to say, or rather what they don't have to say, about the existence of the "other universes". Consider the following passage from a letter from G. F. R. Ellis. "A bit different . . . is the.attitude taken by physicists to the universe beyond the event horizon. This is a topic that appears to worry rather few people; however, I have written on it to some extent . . . In this case the theory as generally accepted predicts something which may or may not exist; its properties are unverifiable . . . By and 14 large, people prefer not to ask questions about this." It seems, then, that the issue of the existence of the "other universes" associated with black holes is regarded as an open question among relativity theorists. According to the Received View, however, it is an open and shut case; the GTR predicts their existence, the GTR is our best theory, therefore we are warranted in believing that they exist. This provides us with yet another reason for concluding that the Received View is not an adequate account of the structure of the GTR. 3. Idealization and Approximation: Let us turn now to a discussion of yet another shortcoming of the Received View, seen as an account of theorizing in the field of relativlstlc astrophysics. As we have seen in the derivation of the Schwarzschild solution, Idealization and approximation play a large, and essential, role in theorizing in astrophysics. None of the exact solutions to the field equations are even purported to be precisely accurate descriptions of the space-time geometry. In local solutions, such as the Schwarzschild, Kerr, and Relssner- NordstrtJm space-times, the manifold is regarded as being asymptotically flat, implying that the gravitational sources at the center of the space-time are the only ones present. The assumption that, at infinity, space-time is Minkowskiian is used repeatedly in the derivation and interpretation of these local solutions. Further, this idealization, as well as others, is of vital importance in deriving the predic tions which have been classically regarded as providing tests of the GTR. For example, in the derivation of the degree of deflection experienced by a light ray as it grazes the limb of the sun, the assumption that, at infinity, space-time is Minkowskiian, is essential in finding the equation of motion describing the trajectory of 15 that light ray through Schwarzschild space-time. As Ellis points out ". . .in the majority of studies . . . physicists consciously construct an artifact that they know does not resemble the real universe, namely an asymptotically flat space time, and proceed to make deductions on this basis without worrying about it at all.' This is of course quite unlike the real universe; the asymptotically flat region (which extends to infinity) is a construct used to 0 make predictions. Its relation to the real universe has 16 not been discussed in depth anywhere.1* Further, global solutions, such as the Friedman-Robertson-Walker space- times, are derived from even more highly idealized assump tions. For example, in deriving these solutions, it is assumed that the energy-momentum in the universe is distri buted homogeneously and has the characteristics of a per fect fluid.-1-7 That these assumptions are idealizations can be seen.from the fact that such inhomogeneities as stars, galaxies, and galaxy clusters constitute part of the subject matter of the discipline known as astrophysics. Clearly, then, idealization plays a crucial role in astro- physical theorizing, yet it is not at all clear how such idealization might be accommodated by the Received View, which maintains that the point of constructing a physical theory is to provide a literally true description of the physical world. If physical theories are intended to be literally true, and acceptance of a theory involves believing that it is true, then why is it that all of the evidence supporting the GTR, currently the most widely accepted theory of gravitational phenomena, is based upon 134 highly idealized models of the physical universe? In other words, how can success or failure of predictions based upon idealized (i.e., strictly false) initial and boundary conditions be relevant to the truth of any theory? Another problem with the Received View that also Involves idealization comes from a consideration of the Special Theory of Relativity and its relation to the GTR. The STR is widely accepted by physicists, and there is good reason to think that if it is regarded as being true at all, it is regarded as being true of our physical uni verse. For example, certain phenomena are explained by appeal to relativistic kinematics; phenomena such as the fact that particles that normally exist for a very short time survive for much longer periods of time when they are created in a high energy linear accelerator. Such particles, travelling, as they do, at speeds approaching that of light, experience a relativistic time dilation, as measured in the laboratory rest frame. Now, if adequate explanation of physical phenomena is achieved only through appeal to true theories, as would seem to be implied by the Received View, then the fact that physicists find the above explanation of time dilation effects adequate commits the defender of the Received View to finding a way to acco mmodate the literal truth of the STR. This, however, proves to be problematic. 135 The STR has heen regarded as a theory of kinematics; that is, as a theory of the behavior of bodies in the absence of acceleration and gravitation, or, in still other words, as a theory of the behavior of bodies in inertial reference frames. An inertial reference frame is a region of space-time which is experiencing neither acceleration nor gravitational forces. The STR, then, can be regarded as the claim that in inertial reference frames, free bodies obey the relativistic laws of motion. The difficulty in showing how this claim could be literally true comes from considerations of the GTR. According to the GTR, remember, an inertial frame is a flat region of the space- time manifold. However, given the presence of field sources in space-time, the GTR predicts that space-time is globally curved. That is, even at great distances from any field source, space-time will have negligible (for all practical purposes) but non-zero curvature. Thus, even though the equivalence principle guarantees the existence of inertial frames at every point of space-time, it does so only in the following extended sense. A "flat region of space-time" is actually a vector space consisting of the class of tangent vectors at that point. Such a tangent space is not on the manifold, which is why we have to use the affine connection to define parrallel transport of a vector from one tangent space into another,3-** The GTR, 136 ; then, Implies that finite inertial reference frames are not in space-time* Thus, according to the GTR, finite inertial reference frames, which are the proper subject matter of the STR, do not exist. If there are no inertial frames, then the STR cannot be non-vacuously true. (Regarding its vacuous truth as truth enough will not suffice, for, since Newtonian kinematics can be interpreted as the claim that in inertial reference frames, free bodies obey the classical laws of motion, Newtonian kinematics is also vacuously true in the absence of inertial frames.) It might be suggested that the STR be interpreted as dealing with the existence of Inertial reference frames only counterfactually. That is, we can take the STR as claiming that if there were any inertial reference frames, then free bodies in them would obey the relativistic laws of motion. Unfortunately, as in the last case, on this interpretation the STR can be made only vacuously true. Since the absence of actual inertial frames is caused by the presence of field sources in space-time, inertial frames could exist only if there were no field sources. But, according to the GTR, everything is a field source, including radiation and gravitational fields themselves. Thus, if there were any inertial reference frames, then they would contain no free bodies, and, thus, all of the 137 free bodies In them would obey any set of laws you choose. So, again, the STR can be made only vacuously true. 19 Now, it has been suggested that theories of kinematics be formulated covariantly and regarded as theories of the geometrical structure of space-time. The STR, then, is to be regarded as claiming that space-time is Uinkowskilan. This can be Interpreted in two ways: that space-time is globally Uinkowskilan or that it -is locally Uinkowskilan. The global interpretation is obviously not intended, for on that interpretation, the STR is clearly false. The GTR, together with the presence of field sources in space-time, guarantees that space-time is not globally Uinkowskilan. If we are to interpret the STR as claiming that space-time is locally Minkowskiian, the question arises as to what "local" means in this context. The standard view is that the STR is claiming that inertial reference frames are regions of space-time over which the Minkowski metric is defined. That is, STR is a theory of the local structures of space-time in the sense that iner tial frames are not global. This suffers from the problem, mentioned above, that there are no inertial reference frames. If the STR is interpreted as the claim that iner tial reference frames are regions of Uinkowskl space-time, then it is at best only vacuously true. Another way of interpreting the phrase "locally Uinkowskilan" is in the 138 same sense it Is used in the claim that space-time is locally flat; that for every point on the manifold, there is an infinitesimal neighborhood which is homeomorphic to Euclidean 4-space. Thus, the STR'could be regarded as claiming that infinitesimal regions of space-time are Minkowskilan. However, if this is how the STR is to be Interpreted, then it cannot be used to explain time dila tion phenomena, or any physical phenomena, since these do not occur in infinitesimal regions of space-time. If the STR is to have any explanatory or predictive force at all, it must be about the geometrical structure of finite regions of space-time. But the GTR implies that no finite regions of space-time are Minkowskilan, so it is difficult to see how the STR could be non-vacuously true. The only option remaining seems to be to interpret the STR as claiming that if there were any inertial refer ence frames, then they would be Minkowskian. I find this claim rather hard to understand, and find it even more difficult to explain how the STR, so formulated, could have anything to do with any actual phenomena. If we use a possible worlds interpretation of counterfactuals, then, since any world in which inertial reference frames exist is a world in which either no field sources exist or the GTR is false, the worlds we must look at to evaluate the 139 truth of the STR are so dissimilar from ours that the STR loses any explanatory or predictive force for phenomena in our world. Thus, we can make the STR literally true only by dissociating it completely from the actual world. That is, the STR is not, as the Received View would have it, a literally true description of the physical universe. In this chapter we have examined four cases from the General and Special Theories of Relativity which serve to point up the inadequacies of the Received View of the logic and structure of theorizing in the physical sciences. The singularity and "other universe" cases are instances where a theory, as seen by the Received View, predicts the existence of some novel entity, yet most theorists either are unwilling to accept the existence of that entity or prefer to remain neutral on the question. Further, theorists continue to accept the theories in question as being the best available and think it unlikely that any new theory which will avoid these novel predictions is forthcoming. The examples involving the highly idealized exact solutions to the field equations and the relationship between the STR and the GTR demonstrate the essential role that idealization and approximation play in physical theorizing. The Received View, with its insistence that physical theories are intended to be literally true, is unable to give a plausible account of this vital part of 140 theorizing in the physical sciences. Thus, I conclude, the Received View, which so many realist philosophers of science presuppose and base their realist and anti-conventionalist arguments upon, is Inadequate as an account of the structure of real science. * VI. AN ALTERNATIVE VIEW 1. The Received View Revisited: In this chapter, I shall describe an alternative account of the logical structure of theories of mathematical physics, one which I claim is superior to the Received View in accurately characterizing the actual behavior of practicing theorists, especially relativistic astrophysicists. Before I begin, however, let me review what I have been calling the "Received View". In the first part of this century, scientists and philosophers of science were puzzled by certain aspects of scientific theorizing. They noticed the following facts. There is a prima facie difference between purely theoreti cal claims and claims describing the evidence which is used to test those theoretical claims. That is, theories are seen as extending our knowledge beyond the realm of the immediately experienced, yet these theories must be tested against that world of experience. One of the ways in which scientific theories seem to extend our knowledge is through the introduction of novel terms, terms which are not used to describe the relevant evidence, but are used in explaining that evidence. However, it was also noticed that some of these explanations were suspect, in that the 141 142 theories generating them could not be falsified. The early Positivists regarded such "metaphysical" theories as unscientific, and were concerned to give an account of science which excluded them from serious consideration Thus, the Positivists addressed themselves to the following questions: When are novel, or "theoretical", terms legi timately introduced into scientific discourse? That is, when are sentences containing such terms cognitively meaningful? How do such terms gain their meanings? Finally, how does the truth of a sentence which describes empirical evidence bear upon the truth of a sentence con taining theoretical terms? The old Positivist account of the structure of scientific theories* functioned as a framework against which these questions could be given fruitful answers. Though it was discussed at length in Chapter XI, let me review its salient features. According to this account, A. There is a first-order language L (possibly augmented by modal operators) and a logical calculus K defined in terms of L. B. The nonlogical vocabulary of L is bifurcated into two disjoint classes; VQ , which contains just the obser vation terms, and Vt, which contains the theoretical, or nonobservation,9 terms. * 143 C. The language L is divided into three sublanguages: L o , which is the restriction of L to V o . L., t which is the restriction of L to V^., and 1^,, the class of sen tences of 1) which contain at least one term from each of VQ and (these last are called "mixed sentences"). D. The language is Interpreted in the following way. The terms in V0 are assigned elements of a domain D, this domain containing only concrete observable events and things. The interpretation of LQ is construed as a partial semantic interpretation of L, A partial interpretation of the elements of Vt is provided by the theoretical postulates T (the axioms of the theory) and by the correspondence rules C, which are mixed sentences. E. A theory is regarded as the conjunction TC of T and C. There were a number of ways in which this general framework was fleshed out by the Positivists. As should be clear, within the context of this picture of scientific theories, the nature of the correspondence rules is of vital importance in answering the questions the Positivists posed for themselves, for it is through the correspondence rules that theoretical terms are interpreted and evidence, expressed in L0 , bears upon theory. The two most important views of the nature of correspondence rules held by Positivists were that Cl) they were definitions, which served to explicitly define elements of Vt solely in terms of VQ , and (2) they were analytic reduction sentences, which served to partially define the novel terms. The first foundered upon our inability to give such explicit definitions for dispositional terms, and so the second became the most widely accepted account of correspondence rules. A sentence containing a theoretical term is meaningful just in case that term is capable of introduc tion through chains of true reduction sentences. That is, a theoretical term is legitimately introduced by providing a chain of reduction sentences which connect it to the observational vocabulary, and acquires (at least partial) meaning through that connection. In response to the question of how evidence is relevant to theory, four major accounts of the logic of theory confirmation were given in the context of this o view of the structure of scientific theories. One such set of strategies involved the elimination of theories altogether, which was accomplished in one of two ways. Either theories were reduced entirely to sets of senten'ces containing only observation terms, or else theories were regarded as uninterpreted formal systems, whose sentences were neither true nor false. These instrumentalist 145 accounts answered the question of relevance by avoiding it. Either theoretical terms are dispensed with altogether, or they are refused Interpretation. In either case, the issue of how the truth, of statements of evidence bear upon the truth of theories disappears. Another set of confirmation strategies that were proposed were the deductive accounts. Theories are tested by using observation claims, correspondence rules, and theoretical claims to deduce further observation claims, which are directly tested. Successful predictions con firmed a theory, unsuccessful ones disconfirmed it. Where as in the elimination strategies, correspondence rules were used to define theoretical terms away, in deductive stra tegies, they are used to derive statements of evidence from statements of theory. Relevance is gained through deduc tive logic and correspondence rules. Bootstrap accounts provided another theory of confirmation. According to these pictures, observation claims, correspondence rules, and perhaps theoretical claims were used to generate positive (or negative) instances of other theoretical claims, which served to con firm (or disconfirm) those theoretical claims. We take a measurement and thus obtain a value for an observation term, use a correspondence rule containing that observation term to obtain a value for a theoretical term, and then use that 146 value, together with values for other theoretical terms obtained in the same way, to provide either a positive or negative instance of some theoretical claim. On deduc tive accounts, we use correspondence rules in two ways, to get initial conditions into the theory and then to get predictions out. On bootstrap accounts, on the other hand, we use them only to get values into theory; we don't need to get them back out. Finally, probabilistic strategies were proposed. On these accounts, correspondence rules, as such, are superfluous. Theory and evidence are connected by condi tional probability statements. The mystery here is where these conditional probability statements come from, and how their truth is to be evaluated. Perhaps these proba bilistic strategies should be understood as construing correspondence rules as conditional probability statements, thus preserving the framework of the Received View. As can be seen from this brief discussion, the nature of the correspondence rules is of fundamental importance, and, as I have mentioned, they were generally g .regarded as being analytic truths. (A possible exception is the probabilistic strategy. It is not clear what the status of conditional probability statements is, but regarding them as analytic is not implausible.) This 147 analyticlty has led to some Interesting things. For example, as Reichenbach pointed out, if the correspondence rules are analytic, then they are definitional in charac ter, and, if true at all, only conventionally true. But if they are conventionally true, we are at liberty to adopt a different set of conventions, the only constraint being logldal consistency. However, since it is these con ventions which connect theory with evidence, the truth of a theory will depend upon which set of conventions we adopt. One set may lead to confirmation,while another may generate unsuccessful predictions. Thus, theories, if true at all, are only conventionally true. This conclusion is alleged to fall right out of the analytic, or defini tional, character of the correspondence rules. The Positivist's conception of scientific theories foundered upon the following. It was pointed out that no precise distinction could be given between an observation and a theoretical vocabulary, and, more importantly, it was discovered that the analytic/synthetic distinction could not be maintained. Two of the cornerstones in the foundation of the Positivist's view were thus kicked out from under it. The modifications which were forced upon this account led to what is known as modern scientific realism. (I maintain that, given the variety of ways in 148 which the term 'scientific realism' Is used, its use engenders more confusion than illumination. Thus, 1 have been calling this modern realist account "the Received View".) The dissolution of these distinctions does not make the puzzles posed by the Positivists disappear. Theoreti cal, or novel, terms are Introduced in the context of some theory, and acquire meaning in some way or other, and this still needs to be understood. Further, theory does out strip evidence, so we still need an account of the rele vance of evidence to theory, even though no clear distinc tion can be drawn between their respective languages. For example, theoretical claims are made which cannot (in some practical sense of 'cannot') be tested directly. An hypothesis as to the conditions in the Earth's core might be such a claim. It is in principle possible to test such an hypothesis directly, but it is practically impossible. Thus, we must rely upon other claims which can be directly tested to help us evaluate the truth of our hypothesis, and it is the relationship between the evidence claims and the hypothesis, in virtue of which the truth of the former bears upon the truth of the latter, which must be explained. The Received View (i.e., modern scientific realism) is an attempt to solve these puzzles while 149 retaining as much of the Positivist's view as possible, and is roughly as follows. Theories are still deductively closed sets of sen tences formulated in some language rich enough to contain the mathematics used in the theory, though the restriction to a first-order language is often relaxed.4 Theories still contain theoretical postulates and sentences which play the role of correspondence rules, but on this account, g they are all synthetic. Given the breakdown of the obser- vation-term/theoretical-term distinction, the language of science is not broken into sublanguages. Further, we cannot provide a complete interpretation of only part of the language. That is, if a semantic interpretation is to be provided, it must be a complete interpretation of the entire language, every term being assigned an extension from the domain of the interpretation. This view is called "scientific realism" for at least a couple of reasons. For one thing, our Inability to identify a special class of analytic sentences in a theory prevents the sort of conventionalism advocated by Reichenbach. If a theory is true, it is synthetically and objectively true. For another thing, on this view, a theory is ontically committed to those things which must exist in order to make the whole theory true. Thus, in 150 both the conventionallst/reallst and instrumentalist/ realist debates, this view supports realism. Now, in Chapter V of this dissertation, I tried to show that the Received View, as described above, was inadequate as an account of the Special and General Theories of Relativity. That is, I described cases where there are discrepancies between the determinations made according to those theories construed realistically and those made by practicing relativists, and concluded that the Received View needs modification. The examples I described seem to indicate that the problems arise from the fact that, according to the Received View, (1) theories are intended to be literally true, and (2) sophisticated mathematics is used to derive consequences of the theories. The fact that certain of the consequences so derived from the GTR are widely regarded as being false, yet the theory itself is not regarded as being inadequate, implies that the trouble with the Received View lies in this conjunction of principles. Further, the fact that the realist view cannot account for the widespread use of superseded theories, such as Newtonian gravitational theory, indicates that it is the claim that theories are to be interpreted as being literally true-or-false that is at the bottom of the inadequacy of the view. In what follows, I shall construct an alternative account of theories of mathematical physics, 151 and use the inadequacies of modern scientific realism as a guide (as one uses the rotting hulks of stranded ships as a guide in negotiating a treacherous channel). For these reasons, in constructing an alternative account of physical theorizing, I have begun by denying the claim that physical theories be taken as literally true-or-false. The view I will be articulating and defending is similar to those "semantic approaches" advo cated by Suppe, Suppes, van Fraassen, Sneed, et al. One important difference between my view and theirs is that mine is tailored to theorizing in relativity theory, while theirs is designed to accommodate quantum mechanics. I will begin with a very general, sketchy description of the major points in this account, concentrating upon the points at which it departs from the old Positivist view and from the more modern Received view. 2. General Picture: Basically, my alternative account is based upon the following four claims. (1) Theory construction and evaluation constitutes only a part of theorizing in the physical sciences. Theories are simply tools, or instruments, used by theorists to generate cor rect predictions and determinations. In addition to theories, physical scientists have recourse to various theory-independent, methodological principles which serve to guide them in their theorizing. (2) As in both the Received View and the Positivists view, a theory is still regarded as a linguistic entity, a deductively closed set of sentences formulated in some language. However, on my view, this language is a purely mathematical language, con taining only terms which refer, if at all, only to mathe matical objects. For example, the GTR (covariantly formu lated) is formulated in the langauge of differential geometry, and contains only terms which are part of that mathematical apparatus, such as^'manifold', 'metric tensor field', 'affine connectic/n*, 'tangent vector', etc. No terra used in the theory proper refers, or is intended to refer, to any physical object or event, observable or otherwise. Further, the theory contains the relevant mathematical apparatus, in the sense that the definitions of the relevant mathematical objects and the rules for manipulating these entities are part of the theory. Such a theory is deductively closed if it is closed under the operations admissible according to the mathematical appara tus. Admittedly, this is counterintuitive, but, as I shall defend later, it is a very fruitful way of looking at physical theories. (3) Theories are intended only to be empirically adequate, where the empirical adequacy of a theory is a function of its ability to generate successful 153 predictions and determinations. This Is in contrast to the Received View and scientific realism, which insist that theories are intended as literally true descriptions of reality. On my view, we might say that theories are Intended to provide (in the sense of "generate") literally true descriptions of physical reality, even though they are not intended to be literally true descriptions of reality.6 (4) Finally, acceptance of a theory involves only the conviction that that theory is, and will continue to be, empirically adequate. Let me expand upon these principles, beginning with number (1). Consider the unwillingness of most astro physicists to accept the physical existence of space-time singularities, in spite of their prediction by the GTR. Clearly, this embodies an extra-theoretical principle of methodology, which might be something like "Regard as physically insignificant any theoretical element the likes of which has never before been encountered". Inasmuch as actual Infinities (or unbounded quantities) have never been experienced in nature, we may treat them as artifacts of the mathematical apparatus used in the GTR. Even Misner's accepting attitude is based upon another suggested methodological principle, to wit: "One ought to be toler ant of singularities which violate no observation", and he argues for the adoption of this principle by pointing out I 154 a case in the history of physics where its use led to 7 progress in the development of quantum mechanics. As another example of the use of extra-theoretical principles, consider the history of high-energy particle physics. Theories are continually being developed which predict the existence of novel particles; theories such as quantum electrodynamics, quantum chromodynamics, the "electroweak" theory of Weinberg, Glashow, and Salam, and the new SU(5) g ''Grand Unification" theory of Glashow and Georgi. However, theorists are generally skeptical of the actual existence of the predicted particles until they are "found" in accelerator experiments. Finally, consider the time- reversed solution case discussed in Chapter III. Here Is another case where theorists appear to be using an extra- theoretical principle to rule out, as physically insignifi cant, mathematically derived predictions of a theory. My claim number (2) provides the principal differ ence between my account of physical theorizing and that of van Fraassen, Suppes, etc. According to their view, espec- 9 10 ially as described by Suppes and Sneed, theory construc tion proceeds through the definition of a set-theoretic predicate. For example, classical mechanics would be presented as 'x is a classical particle mechanics iff . . where the dots are replaced by a set-theoretic predicate.'1'1 A theory is then identified with the class of structures which satisfy this set^theoretic predicate. Thus, van Fraassen says "To present a theory is to specify a family of structures, its models; . . . He goes on to argue that this way of looking at theories is supported by the actual form in which such theories are presented in the physical literature. In particular, he considers a pro posed "axiomatization" of quantum mechanics, and suggests that it is best regarded as the specification of a class of models. While this approach to the structure of scien tific theories may work quite well for classical (i.e., non-relativistlc) quantum mechanics, I maintain that it is not an adequate picture of the structure of the GTR. The GTR is traditionally presented, not by specifying a class of models, or relativistic space-times, but by laying out the mathematical apparatus of differential geometry, and then formulating the key principles and laws in terms of that apparatus. Thus, after setting up the rules and definitions of differential geometry, the traditional presentation of the GTR goes on to give the Equations of Motion (the geodesic equations), the Principle of General Covariance (which is a rule for finding the covariant form of physical laws), conservation laws (e.g., ■ 0), and the Field Equations. The GTR, then, may be considered to be the set consisting of the axioms and definitions of 156 differential geometry, the above mathematically formulated principles,and their logical consequences. Now, this distinction between my view and that of van Fraassen, et al.. is not of monumental importance. While I deny that the GTE is to be identified with a set of models, it is certainly true that the GTR has a set of models, in the semantic senBe. A semantic model for the GTR is a set of mathematical entities (affine connections, metric tensor fields, and the like) defined on a manifold which stand in the relations described by the principles of the GTR. On the view of van Fraassen, et al., such mathematical structures are models of the GTR in the sense that they satisfy the relevant set-theoretic predicate, whatever it turns out to be. On>either construal of the GTR, it is an open question whether any portion of the physical world qualifies as a model of that theory. One very important point of agreement between my account and that of van Fraassen, et al.. involves my claim number (3). On both views, the GTR can be empirically adequate, and hence as good as any theory need be, even if no portion of the physical world qualifies as a model for the GTR. The empirical adequacy of a theory depends, on both approaches, upon certain relations holding between some of the models and the results of measuring and observing various features of the physical world. To state it quite generally, a 157 theory Is empirically adequate just in case some of its semantic models, which are, in the case of the GTR, mathe matical structures, are ''iconic" models13 of portions of the physical world. I will give a detailed account of the 'iconic modeling* relation in the next section. (It turns out that there are significant differences between my account of this relation and that of van Fraassen, et al.) At this point, let me simply say that the empirical ade quacy of a physical theory is a function of that theory's ability to generate models which can be used to make successful predictions and determinations. Theory testing, then, does not give us any evidence as to the truth value of a theory, but only evidence for, or against, its empirical adequacy. If we accept a theory as being empirically adequate, we are willing to rely upon that theory to generate models that yield successful pre dictions for cases not yet tested. That is, though theory acceptance does not involve the belief that the theory is true, it does involve the conviction that that theory will yield correct predictions and determinations in future and novel situations. Further, theory acceptance comes in degrees. For example, for the purpose of predicting astro nomical phenomena involving the solar system, I might accept Newton's Theory of Gravitation, because it is easier to use in such circumstances than the GTR and because it has 158 proved to be just as good as the GTR in describing that circumscribed set of celestial phenomena. Overall, how ever* I might accept the GTR to a greater degree, because it is adequate for a wider range of phenomena than is Newton's theory. Thus, since empirical adequacy, unlike truth, comes in degrees, the utility, and frequent use, of "false" theories is not puzzling. 3. Specific Picture: Let me turn now to a more detailed discussion of the alternative account of physical theorizing that I am proposing. The key notion, of course, is that of the empirical adequacy of a physical theory, which is to be explained as follows. A theory has a set of semantic models, which are mathematical structures that interpret the theory so as to make all of its sentences true. Appearances are structures which are described in reports of observation, measurement, and experiment. In the simplest cases, they are portions of the physical world. Roughly speaking, a theory is empirically adequate if every appearance is represented by a subset, or empirical sub structure, of some model for the theory. I will explain this more fully by giving detailed accounts of the notions of a model, an appearance, and the representation relation. On the account that I am proposing, remember, a theory is a deductively closed set of sentences, formulated in a purely mathematical language and containing the 159 mathematical apparatus in the theory. A model for such a set is a mathematical structure consisting of a set of mathematical objects together with an interpretation func tion which assigns to each term in the language of the theory an element or subset of the set of mathematical objects in such a way that all of the sentences of the theory are made true. (The fact that the domains of these models contain only mathematical objects reflects my con viction that no non-mathematical objects could satisfy the definitions and laws of the relevant mathematical apparatus. Force fields are not vector fields, mass is not a real valued function, and acceleration is not a vector, though many of the salient features of the physical quantities are nicely represented by the corresponding mathematical quantities.) For example, the models for the GTR are called 'relativistic space-times*. The domain of a rela- tlvistic space-time, S, is a set {M,V,V, g,P,T}, where M is a connected, 4-dimensional C°°-manifold, V is the set of tangent vectors on M, V is an affine connection defined on M, g is a metric tensor field defined on M, r is the set * of geodesic curves on M, and T is a second-rank tensor field defined on U. To say that this set is a model for the GTR is to say that the mathematical objects in the set « obey the laws of the GTR. That is, Vg; » 0, VT ■ 0, for all curves c(X) (where c is a function from an interval of 160 R into H and vc is the tangent vector field along c(X), i.e., vc :c(X)-*-V), c (X)g T iff ^vcvc " 0* an(* B an(* ^ are related according to the Field Equations. Clearly, there are many models for theories such as the GTR. The task of finding new models for the GTR is an ongoing one, and con- sists of making assumptions as to a value for T and then looking for a value for g which is related to that value for T according to the Field Equations. In fact, given the non-linearity of the Field Equations, for any value proposed for T, there are many values for g which satisfy the Field Equations. A "solution" to the Field Equations, then, consists of a specification of a value for g, which is used in the construction of a relativistic space-time. Space-tlines, such as Schwarzschlld space-time, Gtidel space-time, Kerr-Newman space-time, etc., are known by their metrics (the Schwarzschlld metric, etc.). Some relativistic space-times, such as the GtJdel space-time, are completely unrealistic, in the sense that they bear no resemblance to any portion of the physical world. Nonethe less, such space-times are models for the GTR. An appearance is a structure that can be described in experimental and measurement reports. In a sense, it is a semantic model for a set of sentences reporting the results of measurements, observations, experiments, etc. These sentences are formulated in a suitably rich fragment 161 of some natural language, and, most of the time, are Intended to be literally true descriptions of some portion of the physical world. (X will discuss exceptions to this later.) The relation between an appearance and the world is often very tenuous and uncertain, but a description of an appearance is as close to a description of the world as the scientist can, or needs to, get. The description of an appearance will contain statements of measured values of physical quantities, such as the masses of various bodies, or the measured distances between bodies. It will also contain descriptions of observed, or inferred, phenomena, such as the trajectory of a photon. There are a number of important, and Interesting, features of appearances, but let me mention only one here. Uany reports of the results of measurements indicate a range of a values for a physical quantity, rather than a unique, precise value for that quantity. For example, the mass of the Earth is given as (5.977 + 0.004) x 1 0 ^ grams. Such ranges are often indicative of experimental, or measurement uncertainty, and are a function of the limits 14 of precision of our measurement apparatus. Advances in technology can serve to reduce these ranges, but it seems quite unlikely that they could ever be eliminated. Another source of such ranges in appearances are unknown contravening factors. For example, in measuring the 162 deflection of starlight due to the gravitational field of the Sun, the small, but unknown, effects of the solar corona upon the trajectory of a photon must be taken into account. The presence of such contravening factors forces us to broaden the range of values attributed to some physical quantity. In the case of the deflection of star light passing near the Sun, experiments performed during eclipses during the last 62 years have yielded values for the deflection due to gravitational effects ranging from 15 1.61 + 0.40 arc seconds to 2.73 + 0.31 arc seconds. This represents measurement uncertainties of up to 25%, but in many cases, that is as good as can be done. In constructing and evaluating physical theories, then, there are two kinds of structures which are relevant; mathematical structures, which are semantic models for the physical theories, and empirical structures, or appearances, which are semantic models for the sets of sentences reporting the results of observations, measurements, experi ments, etc. The empirical adequacy of a theory, as I men tioned above, is a function of the relationships holding between these two kinds of structures. Van Fraassen suggests that "... the theory is empirically adequate if it has some model such that all appearances are isomorphic to empirical substructures of that model."’16 An empirical 163 substructure of a model is simply the restriction of the model to a subset of that model. The most Important aspect of this picture is that appearances need not be isomorphic to any complete model. However, I have a number of objec tions to this analysis of empirical adequacy, as it stands. Discussion of these objections will lead us to a better definition. In the first place, I maintain that isomorphism is much too restrictive, and does not adequately reflect the actual practice of physicists. To see this, let us take a closer look at appearances and their descriptions. These descriptions are going to contain terms which may be iden tified syntactically as operators, as opposed to singular terms or predicates. An n-place operator is a term which, when accompanied by n singular terms, generates a singular term. For example "+" is a two-place operator. Since semantics rides piggy-back on syntax, operators are interpreted, in a model, in a certain way. While singular terms are assigned, by an interpretation function, elements of the domain of the model, and predicates are assigned subsets of the domain, n-place operators are assigned func tions from ordered n-tuples of elements of the domain into elements of the domain. For example, "+" is ordinarily assigned that function from pairs of real numbers into a real number such that the value of the function is equal 164. to the sum of the arguments of the function. Now, many of the operators which occur in the description of an appear ance which is relevant to physics are real-valued opera tors, in the sense that these operators, when accompanied by singular terms, generate the names of real numbers, and are assigned functions which map elements of the domain into the real numbers. Thus, a description of an appearance will contain operators, and the appearance, being a model for that description, will contain functions. Similar things are true of theories and their models. Theories, sets of sentences, contain operators, and the models for the theories contain functions. Now, to say that an appearance is isomorphic to an empirical substructure of some model is to say that the elements of the appearance can be mapped, one-one, onto the elements of the empirical substructure of the model, in such a way that all properties and relations are preserved. In particular, if an appearance were isomorphic to some subset of a model, then there would exist an isomorphism h such that for each n-place, real-valued function f in the appearance, there would exist an n-place, real-valued function f in the empirical substructure such that for all elements e in the appearance for which f is defined, f'(h(e)) = h(f(e)). In simpler terms, an isomorphism between an appearance and an empirical substructure would 165 imply that the operation of every function in the appear-, ance would be exactly mirrored in the empirical substruc ture. It is my claim that this in fact does not happen. For example, consider the term 'mass' which occurs in the descriptions of many appearances. In the description of some appearance, we might find the term 'the mass of the Earth1 (or, more formally, 'mass(Earth)'), and the claim 'the mass of the Earth is (5.977 + 0.004) x 1027 grams'. Thus, in the appearance, we have the Earth, a function which is the extension of the term 'mass', and a value for that function when applied to the Earth. Notice, however, that that value is not a unique real number, but is, instead, an interval of the real line. On the other hand, the corresponding function in the empirical substructure does yield unique real numbers, as in fact it must. For in order to appear in a model of a theory, i.e., in order to be used to make the tehory true, the values of that function must be related to the values of other functions in precise ways. Consider for example a theory which con tains Newton's version of Kepler's third law, M « w2a2. Only a model in which M, w, and a all yield precise numeri cal values will this sentence come out true. Thus, if M is the function in the model which corresponds to mass in the appearance, the conditions necessary for these struc tures to be isomorphic cannot be met. 166 Secondly, the claim that all appearances must stand in the relevant relationship, whatever it turns out to be, to empirical substructures of one model is far too strong. There are cases where some appearances stand in the appro priate relation to empirical substructures of one model, while other appearances stand in that relation to empirical substructures of another model of the same theory. For example, there are, at last count, five tests of the GTR. In four of them, the reports of the results of observations and measurements describe an appearance which stands in the appropriate relation to empirical substructures of a Schwarzschlld space-time, while in the fifth case, it is a Kerr-Newman space-time which is used to model the relevant gravitational field. In the first four cases, the gravita tional field is regarded as being spherically symmetric, but in the fifth, it is assumed that the source of the gravitational field is rotating. According to the GTR, this rotation should cause gyroscopes in an equatorial orbit to process. The experiment, which, so far as I know, has not yet been performed, involves the placing of such a gyroscope in an orbit around the Earth, and measuring its * precession. Thus, the relevant appearance, if the pre dicted effect occurs, will contain a value for this precession. But such an appearance will not stand in the required relation to any subset of a Schwarzschlld 167 space-time,.empirical adequacy cannot require that all appearances stand in this relation to empirical substruc tures of one model. Finally, van Fraassen*s suggested analysis does not reflect the fact that empirical adequacy comes in degrees. According to his definition, either a theory is empirically adequate, or it is not. But this does not seem to reflect actual practice. For example, Newtonian gravitation theory is regarded as being empirically ade quate to a certain degree. There are many appearances which do stand in the required relation to subsets of models of Newton's theory, which is indicated by the fact that Newton's theory is so often used. Non-negligible discrepancies between appearances and Newtonian mechanics occur only when particles with extremely high velocities or extremely massive bodies are considered. For roost prac> tical problems, Newton’s theory is entirely adequate. Einstein's theories are preferred, overall, because they can accommodate a wider range of appearances than can Newton's theories. Thus, in defining empirical adequacy, we should perhaps have two definitions, one for minimal empirical adequacy and one for maximal empirical adequacy; definitions which give us a way of determining relative degrees of empirical adequacy. 168 We are now ready to consider the definition of empirical adequacy. I will lead up to it by defining a preliminary notion first. (1) A model M for a theory T represents an appearance A just in case (a) there exists a mapping h such that for all elements x in A, h(x) e M, and * (b) for every function P in A and every element x in A for which P(x) is defined, there exists a function P' defined over hCAD such that P ‘(h(x)) e h(P(x)). The image of h is what van Fraassen calls an "empirical substructure". Further, it is to be expected that for any real number r, h(r) *» r. Thus, if P and P* are real valued functions, the last condition will be that P'(h(x)) e P(x). If (b) attributed an identity between P'(h(x)) and P(x), h would be an isomorphism from A onto an empirical substruc ture. By allowing the functions in an appearance to adopt ranges of values for a single argument, and then requiring only that the value of the corresponding function in the model for the corresponding object in the model be in the relevant range, we paint a more accurate picutre of the actual behavior of physical theorists. Empirical adequacy is then defined as follows. 169 (2) A physical theory T is minimally empirically adequate iff there exists an appearance A and a model II for T such that M represents A. (3) A physical theory T is maximally empirically adequate iff for every appearance A there exists a model M for T such that M represents A. If scientists are searching for the final, complete, over all theory which accommodates all phenomena, then, clearly, only that theory could ever be maximally empirically ade quate. These definitions give us a straightforward way of comparing competing theories. Theory T is more empirically adequate than is theroy T 1 if the class of appearances represented by models of T' is a proper subset of the class of appearances represented by models of T. 4. An Example: In this section, I will be examining, in some detail, one of the classic tests of the GTR: the deflection of light rays passing through the gravitational field of the Sun. This case illustrates, in a fairly clear way, many of the Interesting features of the alternative account of the structure of the GTR that I am proposing. The structure of this section will be as follows. I will begin by describing the class of relevant appearances, and then describe the model that is used to 170 predict the deflection. Finally, I will specify how the model is "attached" to the appearances. Before I begin, let me make one point about this example. Historically, it proceeded as follows. Einstein presented the GTR in 1916 in a paper entitled "The Founda- tion of the General Theory of Relativity". In the last section of this paper, S22, Einstein suggested three tests of his theory: the gravitational red-shift, the deflection of starlight passing through the gravitational field of a massive body, and the precession of the perihelion of the orbit of Uercury. In the first two of these suggested tests, the predicted effects were unexpected (hence it was a genuine prediction), while the precession of Mercury's orbit was well-known (hence the "prediction" of this effect by the GTR was in fact merely a determination, or explana tion). The difference between a prediction and a determina tion is an epistemic one. If I were interested in accur ately reflecting the epistemic situation involved in the deflection case, I would first describe those parts of the relevant appearances that were known prior to the invention of the GTR, then describe the relativistic model, show how it is used to predict the deflection of starlight, and finally describe the results of the observations and meas urements of that predicted deflection, i.e., the rest of 171 the relevant appearances. However, I do not feel that such accuracy is very Important, from a logical point of view. That is, there is no significant logical difference between the deflection prediction and the precession determination. Thus, in describing the deflection experiment, I will describe the entire appearance at once, rather than splitting it into initial conditions and predicted, or final, conditions. The appearances relevant to the deflection test contain the following objects: the Sun, certain distant stars, the Earth, and certain photons travelling from the stars, past the Sun, to the Earth. Actually, it is the trajectories of these photons in which we are most inter ested, rather than the photons themselves. Information about these trajectories is obtained through an examination of the images of the distant stars on photographic plates. It is assumed that, in the absence of gravitation, photons travel in straight lines, and it is also assumed that there are no important gravitational forces operating between the distant stars and the Sun. The characteristics of these objects which are contained in the relevant appearances are the mass of the Sun, determined by use of Kepler's third law17 to be (1.98874 ± 0.000027) x 1033 grams; the radius of the Sun (6,9598 + 0.0007) x 1010cm; the positions of the various distant stars; and their distances from the 172 Sun, which for the purposes at hand, are regarded as being infinite; the position of the Earth, and its dis tance from the Sun, which is also regarded as being infinite; the mass of the Earth, which is, in this case, regarded as being negligible; the endpoints of the photon trajectories, which are the various stars and the Earth; the distances of the trajectories from the Sun at their points of closest approach, determined through an examina tion of photographic plates; the measured deflection of the photon trajectories, determined through a comparison of photographic plates exposed during an eclipse with those exposed six months later at night; and finally, the deflection of a hypothetical photon trajectory which Just grazes the limb of the Sun, calculated using the actually measured deflections and the equation (96) 0O « ( r 0 /Ro )L$ where R q is the radius of the Sun, r^ is the point of closest approach of a photon trajectory, and is the measured deflection of that photon trajectory. As should be clear, the deflection experiment consists of taking pictures of distant stars beyond the Sun during a solar eclipse. The relevant characteristics of the electromagnetic radiation emitted by those stars are determined by examining the photographic plates exposed 173 during the eclipse and comparing them with plates exposed when the Earth is on the opposite side of the Sun. The deflection of the starlight can he calculated using this data and the ordinary rules of geometric optics. Let us look at how this is done. Consider the following diagrams. a b • t •a ’ O b * Sun Figure 5. Photographic Plates a 1 ~ o Earth Sun Figure 6. Photon Deflection Figure 5 represents two photographic plates, taken of the same region of the sky: one, A, exposed during a solar eclipse; and the other, B, exposed six months later at night, a and b represent the real positions of two stars in that region, while a' and b' represent their virtual positions as seen during the eclipse. Figure 6 represents * 174 the same situation, as seen from above. The dotted line between the distant star, represented by a, and the Earth represents the trajectory of a photon when the Sun is not in the way, while the solid curved line represents a photon trajectory during an eclipse. Again, a represents the real position of the star and a* its virtual position. It is assumed that, in the absence of any massive bodies, photons travel in straight lines as they- pass through the region represented in Figure 6. That is, it is assumed that lines ae, a'e, and ac are all straight. Angle represents the deflection experienced by the photon, but, clearly, cannot be measured directly. Angle a can be determined in the following way. If the scale of the photographs represented in Figure 5 is known (I.e., if it is known what portion of the sky is covered by the photograph), then by measuring the difference between the real and virtual positions of a distant star, we can calculate the angle a. (For example, if a photograph twelve inches across covers six degrees of the "celestial sphere", and the real and virtual positions of a star are measured to be one inch apart, then the angle 18 a is 0.5 degrees of arc.) Angle can be calculated from this data in the following way. We know that the sum of the angles a,5 , and y is equal to tt radians, and further than the sum of angles y and A That is, our assumption that the distant stars are an infinite distance from the Sun allows us to determine the deflection of a photon trajectory passing the Sun simply by measuring angle a. This is not quite all there is to the relevant appearance, however. We have still to calculate the deflection experienced by a hypothetical photon trajectory which just grazes the limb of the Sun. (As a matter of practical fact, observations cannot be carried closer to the Sun's disk than about two solar radii. Thus, this deflection must be extrapolated, rather than measured.) To get this value, 0o , we use equation (96). By solving this equation for a number of photon trajectories from a number of sources, we may arrive at a value for 6q which we will use in testing the adequacy of the GTR. Let us turn now to a description of the relevant model, which will be a Schwarzschild-space-time.19 In describing such a model, we begin with the manifold. Like all relativistic space-times, our model contains a con nected, 4-dimensional c^-manifold. We define a Schwarzscbild coordinate system (t,r,0,4>) over the manifold, 176 with r going f?om 0 to ». (All components will hereafter be expressed in this coordinate system.) The metric for our space-time has the general form (97) dT2 *» -B(r)dt2 + A(r)dr2 + r2de2 + r2sih20d«*.2 where B(r) » 1 - 2MG/r and A(r) “ (1 - 2MG/r)- 1 . An affine connection, V, is defined on the manifold in the following way. The components of the connection (the connection coefficients), are defined in terms of the metric. (98) r j v - i/2 g Ap(g pii v + gpv>p - spV(p>. where gyV»p * 38yv/3xP* “a and ®8 are basis vectors of our coordinate system, (98) v5oSe - oprgB. Where S is a third-rank tensor and u is a vector, (100) V5S - s “By ; 5uSea ■ i!6 ■ SiY. where ea is a basis vector and and are basis 1-forms. Sapy; is defined as ( 101) s“By;s - s«ByiS + s“llYr - '1B 4 - From equations (97) and (98), we can calculate the non vanishing components of the affine connection. 177 r (102) rjr 1 dA(r) 2A(r) dr A(r) rsln29 rr Q 1 dB(r) H ~ A(r) tt 2A(r) dr e r ■ -sinGcosG ■ cot 9 $r r 1 dB(r) 2B(r) dr The equations of free fall (i.e., the geodesic equations are (103) d2x* dxv dx v ______X PA : o- + T ------» 0 dp2 dp dp where p is a parameter describing the curve. Equation (103) comes from our general, coordinate-independent, geodesic equation, Vq u « 0, and our definitions of the affine connection. A curve is a geodesic just in case it satisfies (103). Inserting the nonvanishing components of the affine connection, given in (102), into (103) yields the following four equations. (104) 178 d2e , 2 de dr (105) 0 * sln6cos8 dp2 r dp dp (106) + 2cote (107) dp B(r) dp dp (A prime denotes d/dr.) These equations describe a geo desic curve passing through a space-time endowed with a Schwarzschild metric and a Schwarzschlld coordinate system in standard form. All interesting features of such a curve can be discovered by solving these equations. In this case, the feature we are most Interested in is the shape of geodesic orbit, which is described by equations which express 6 and $ as a function of r. Thus, our task is to find, from the above equations, the equations which describe the shape of a geodesic curve. We do this by looking for constants of the motion. Since the metric field of our space-time is assumed to be isotropic, we may restrict our attention to those geodesics whose orbits are confined to the equatorial plane of the coordinate system, that is, 6 « i t/2 . Equation (105) is immediately satisfied, and we may forget about 6 as a dynamic variable* Dividing (106) and (107) by d$/dp and dt/dp, respectively, we find (108) 0 « (d2 (109) 0 « (d2t/dp2 )(dp/dt) + (B» d dt (111) — (In — + In B(r)) - 0 dp dp 2 respectively. Thus, we know that In r (d In B(r)(dt/dp) are constants of the motion. We normalize p so that the solution of (111) is (112) dt/dp » 1/B(r) The other constant we label J. I.e., (113) r2 (d<|>/dp) « J (constant). Inserting ' $ ° ti/2', (112), and (113) into (104) yields d2r A'(r)/drf J2 B'(r) (114) 0 «— g + I — I ---- +------dpz 2A(r) \dp J r3A(tf) 2A(r)Bz(r) Multiplying this equation by 2A(r)dr/dp yields dr d2r ? 2J2 dr B'(r) dr (115) 0 - 2A(r) v + A'(r dp dpz r3 dp B2(r) dp which we may write as (116) 180 and our last constant of motion Is therefore (117) A(r)(dr/dp)2 + J2/r2 - 1/B(r) ■* -E (constant) The shape of geodesic orbit can be obtained directly by using (112) to eliminate dp from (113) and (117). This gives (118) which can be written as d The solution to this equation may be determined by the integral (120) This is the general expression for the ^-position of a point on:a geodesic curve as a function of its r-position. To find the solution, we must find the values of the con stants J and E. Now, we know that at infinity the metric becomes Minkowskiian; i.e., A(«) » B(») « 1. Further, using (112) to eliminate dp from (117) yields 181 A(r) (1 2 1 )pr',“‘I) \ & t } r — B(r) “~E Thus, at Infinity, we have that (122) (dr/dt)2 - 1 - E - V2 . Ve express J in terms of ro» the minimum r-value of the geodesic. At rQ , dr/d (123) J “ rQ ({l/B(r0 )J - 1 + V2 )1/2 Now, in the case of a null geodesic, V *» 1, and the orbit is described by inserting (122) and (123) into (120). This yields (124) We could integrate this numerically by inserting for A(r) and B(r) the values obtained from the Schwarzschild solution and expanding in the small parameters MG/r and MG/r. However, it is easier to expand before integrating, 20 using for A(r) and B(r) the Robertson expansions (125) A(r) « 1 + 2y(MG/r) + . . . (126) B(r) = 1 - 2MG/r + . . . 182 Where y is an unknown dimensionless parameter which, according to the GTR, has the value 1. (In contrast, the 21 Brans-Dlcke theory gives y the value y *= (to + l)/(w + 2), where to is the unknown dimensionless parameter of this theory.) The integral then becomes elementary, and gives, to first order in MG/rg (127) Using this equation, we can compute the deflection experienced by a geodesic as it travels from infinity to its minimum value rQ and then increases again to infinity. The total change in (128) A Inserting (127) into (128) gives us a value for the deflec tion of a null geodesic of 4MG /1 + y (129) A* = I ----- r0 \ 2 In the model we are constructing, H is defined as (130) m(r) 4Trr2T00(r)dr M « m(R^) 183 where T00 is the time-time component of the energy-momentum tensor and Rq ■ 6.9598 2c 1010 cm. T is defined so that 0 3 * If « 1.98874 x 10 . For a null geodesic coming from infinity to Tq *■ R0 and then going to infinity, (129) gives a value for the deflection of this geodesic of A<|> » 8.486677 x 10”® radians = 4.862508 x 10"^ degrees ■ 1.750503 arc seconds. To complete our discussion of this example, it remains only to describe the relation which holds between the appearance and the model. That is, we must answer the question "Does the model represent the appearance?" Let us see. The appearance contains the following entities: the Sun, the Earth, several stars, and several photon trajectories. It also contains a number of hypothetical photon trafectories, each of which grazes the limb of the sun. These entities have a number of relevant character istics in the appearance, such as the mass and radius of the Sun, the distance from the Earth to the Sun and the distance from the Sun to the various stars (which, in the appearance, are regarded as being infinite), the measured angle of deflection of each of the real photon trajectories, and the calculated angle of deflection of the hypothetical photon trajectories. Each of these entitles can be mapped onto some element of the model. The Sun is represented 184 by a spherical region of the manifold, and the stars and the Earth by points on the manifold. The photon trajec tories are represented by null geodesics. In particular, the trajectory of a photon which travels from star a, past the Sun, to the Earth is represented by a geodesic whose endpoints are the points representing star a and the Earth. Finally, the trajectories of our hypothetical photons are represented by geodesics which graze the edge of the region of the manifold which represents the Sun. Thus, clause (a) of our definition of the representation relation is satisfied. There is a function which maps each element of the appearance onto some element of the model. Let us turn to an examination of the characteristics of these entities. In the appearance, the Sun, has a mass of (1.98874 + 0.000027) x 105 g and a radius of (6.9598 + 0.0007) x 1010 cm, while in the model, M, which is defined in terms of the energy-momentum tensor and-the region of the manifold representing the Sun, is equal to 1.98874 x 105 g and R0 , the radius of this region, is equal to 6.9598 x 1010 cm. Further, according to the metric defined on our manifold, the interval between the point representing the Earth and r ■ 0, and the intervals between the points representing the stars and r » 0, are infinite. Finally, and most importantly, in the appearance, the average deflec tion of the hypothetical, Sun-grazing photon trajectories 185 is 1.90 + 0.21", and in the model, the deflection of a geodesic grazing the edge of the region representing the Sun is computed to be 1.750503", well within the required interval. Thus clause (b) of our definition of the repre sentation relation is also satisfied. So, we know that our model for the GTR, our Schwarzschild space-time, represents our appearance, the results of our observations and measure ments. 5. Amplification: Having described my alternative account of theories of mathematical physics, in a general way, and having given an example of how it works, I will now discuss some of the more intersting features of this view. Recall that, according to this account, physical theories are not to be regarded as being literally true. That is, even though a theory may be true relative to a model, and, in fact, true relative to many models, it will not be true simpliciter. In other words, no portion of the physical world constitutes a model for a theory such as the GTR. (This fact should not be at all surprising. Plato knew that no physical system exactly satisfied the laws of geometry, and this is as true of Riemannian geometry as it is of Euclidean geometry.) Now, remember, from our dis cussion of ontic commitment, that truth is vital in deter mining ontology. For example, a scientific realist, who believes that the GTR is true, may look at the mathematical character of that theory, and the geometrical structure of its models, and conclude that a 4-dimensional C^-manifold, endowed with a metric tensor field and affine connection, and containing a non-denumerable number of geodesic curves, exists as a part of the physical world. (It is not for nothing that this sort of realism is sometimes called "Platonism".) Vhen such a realist hears the physicist's claim that force is a vector quantity, he interprets it literally; force is a vector. In fact, one might notice that all of what we intuitively regard as the physical world is described by highly sophisticated mathematical theories, and conclude that physical objects are nothing over and above the solutions to certain matter-field equa tions, and perhaps even that only mathematical entities are fundamentally real. This sort of "hyper-Pythagoreanism" has recently been associated with Professor Quine. I take it as a good reductio ad absurdum argument against this sort of scientific realism. The view I am proposing, with its abandonment of the claim that physical theories are intended to be literally true, allows us to avoid this bizarre metaphysi cal consequence. On my view, the claim that force is a vector quantity is interpreted as the claim that force can be represented by, or is analogous to, a vector. That is, all of the Important features of a force field, for example, are represented by a vector field; the magnitude of the vec tors represents the strength of the field, and the direction of each vector represents the direction in which a test parti cle is deflected as it passes through the field. The claim that photons travel along null geodesics does not commit us to two kinds of things; photon, trajectories and null geode sics. Rather, it is simply that geodesics capture the impor tant characteristics of particle and photon trajectories. If a model represents an appearance, elements of themodel will represent elements of the appearance, in the sense that there is an analogy between the model and the appearance. This is just the notion of an iconic model. described by Mary Hesse. Put simply, my view consists of the claim that semantic models for theories serve as iconic models for appearances. The only difference between using a styrofoam ball-toothplck structure as a model for an atom and using a relativistic space-time as a model for some portion of the universe is that in the first case, the model is mechanical, or physical, while in the second, the model is mathematical. In both cases, the relation between the iconic model and the physi cal system being modelled is the same; the model captures the important features of the system. Further, just as there are features of a physical model which are analogous to nothing in the system being modelled, so, too, there are- 188 features of relativistic space-times which have no physical analogue. Models are not Intended to mirror the world exactly. Let us turn from talk of models to a discussion of appearances. As mentioned above, many of the quantities ascribed to entities in an appearance are intervals of the real line, rather than unique real numbers. In my earlier discussion of this fact, I indicated that the primary reason for this was limits of precision of our measuring apparatus. However, there are a number of other reasons why ranges of values occur in appearances, reasons that have little to do with error. For example, sometimes the ranges of values attributed to some entity in the descrip tion of an appearance are reflections of tolerances, rather than of limits of precision. It may be, for the purposes at hand, that the precision needed is less that that which our measurement apparatus is capable of providing. Ah example of this might arise in the case of navigating inter-planetary space probles. To bring such a probe to the points in the solar system we desire (for instance, in a Jupiter fly-by mission, where we want the probe to pass within a certain distance of Jupiter), we must have avail able values for the distances between and velocities of various planets. However, in such a circumstance, the tolerance for error may be relatively large (the probe 189 doesn't have to pass exactly 5 x 105 km from Jupiter; (5 + 0.5) x 105 km would be perfectly adequate). In such a case, the values attributed to physical quantities in the appearance may be rather large Intervals of real numbers. Another Interesting feature of appearances is that their descriptions may contain highly theoretical state ments, statements derived, at least in part, from rather speculative principles. For example, in constructing a global model of physical space-time, we need data as to the distances between ourselves and the distant celestial objects. These distances are usually calculated in one of two ways. Either we use principles relating apparent luminosity to distance, or we use Hubble's principle of a linear red-shift-distance relation. In the first case, we measure the luminosity of a distant star and use its rela tive brightness to judge its relative distance, and in the second case, we measure the red shift of the electromagnetic spectrum from a distant object and judge its distance from that. In both cases, we start with the results of an observation, but then use a highly theoretical, but widely accepted, principle to obtain a value for the distance between ourselves and some distant object. This value becomes part of an appearance which must be accommodated by any theory of gravitation. Such theoretical principles, though generally accepted, are defeasible. (In fact, 190 Hubble'a principle has recently been contested.22) Even so, It is against such appearances as result from the use of such principles that any theory of cosmology must be judged. In searching for a theory, we must have available a stock of data, a store of phenomena (or their descript tions) which the theory is intended to accommodate. An appearance is a stock of data. It embodies our current best estimates of what the phenomena are. Often these best estimates are more akin to best guesses. In constructing a theory of cosmology, we begin with cosmological phenomena. But it is extraordinarily difficult to get information about this phenomena. (It is for this reason that, for the first fifty years of its existence, the GTR was regarded as "a theorist's paradise, but an experimentalist’s hell", and was felt by most physicists to be only a mathematical curiosity.) Thus, we must settle for the best information that we can get, even if it is extremely imprecise and speculative. Advances in technology which allow us to gain more information (e.g., development of radio and X-ray astronomy, and launching of rocket-based telescopes) make the appearances we use more precise and less reliant upon tendentious principles, and this has the effect of weeding out certain theories. For example, prior to the discovery, by Penzias and Wilson in 1964, of the 2.7 degree background 191 microwave radiation, the appearances used by cosmologlsts to test their theories could not be used to decide between Big Bang and Steady State theories of the universe. The discovery of this isotropic radiation, and its inclusion « in the appearances used by cosmologlsts, led to the abandon ment of the Steady State theories, for they could not be used to predict its existence. However, at least in the case of general relativity, we may never be able to avoid using some highly theoretical principles in describing igjpearances. For example, in constructing global relativ- * istic models, it is invariably assumed that, on a very large scale, mass-energy is distributed homogeneously throughout the universe. However, as E llis^ points out, this claim is derived from the observed isotropy of the universe together with the "Copernican Principle", which states that we do not occupy a privileged position in the universe. This principle, as Ellis shows, cannot be obser- vationally tested, but it is used in generating virtually all of the appearances used in cosmological research. Descriptions of appearances also contain approxi mations and idealizations, which have, in the appearance, the same status as reports of measurements and observations. There are cases in which the exact data we would like is unavailable to us. For example, in finding global solu tions for the Field Equations, we need global values for the energy-momentum tensor field. As ve have seen, it is usually assumed that the universe is spatially homogeneous, but, given the presence of such inhomogeneities as stars, galaxies, and galaxy clusters, this is clearly a simpli fying idealization. However, even if we knew the global distribution of roatter-energy, which we don't and never, will, it would be impossibly complex to use that informa tion in constructing a precise cosmological model. That is, even if this data was available to us, we wouldn't be able to use it. There are other cases where certain data is available to us, we could use it, but its usefulness would be negligible. An example of this arose in our dis cussion of the deflection experiment. We can obtain fairly good values for the distance between ourselves and certain stars. However, we assume that they are at infinity, because for one thing, it makes the calculation of the deflection angle from the observed separation angle between the real and virtual positions of a star much simpler, and for another, the information gained from more realistic distance assumptions would be of negligible significance. In constructing, and testing the adequacy oft some physical theories* then, we compare the predictions and determina tions made on the basis of the theory, not with the world as it is, but with a simplified, idealized picture of the world. 193 A question naturally arises here: What Is the relation between appearances and the real world? That is, how does science gives us knowledge about the physical world if the models are connected only with idealized, inaccurate, approximate pictures of that world? Let me leave that question unanswered for the moment, and come back to it in the next chapter, where I will address myself to the general issue of the scientific epistemology implied by the view I am proposing. Having discussed models and appearances, let us turn now to the connections between them. I have suggested that a model represents an appearance just in case there is a function which maps the elements of the appearance into the model in a certain way. These mappings serve a func tion very similar to that served by correspondence rules (coordinative definitions, meaning postulates) in the Received View; they connect the language of theory with that of evidence. However, on my view, they are neither analytic nor synthetic truths; in fact, they are not true- or-false at all. Rather, they are to be taken as prescrip tions, some of which are better (i.e., more fruitful) than others). For example, consider the claim, which occurs in every text on relativity theory, that light rays follow null geodesics. On my view, this is to be interpreted as 194 the prescription, "let photon trajectories In an appear ance be represented by null geodesics In a model". There are a variety of such prescriptions that may be suggested. One might let photon trajectories be represented by time like geodesics, or by non-geodesic curves. However, it is very unlikely that any relativistic space-time will repre sent any appearance when conntected by these mappings. That is, while no "correspondence rule" may be said to be true or false, some are better than others at achieving the representation, by a model, of an appearance. There is a degree of arbitrariness in the selection of the mappings which connect a model with an appearance. In principle, any element of an appearance may be mapped onto any element of a model. In practice, however, only a small number of mappings have even the slightest chance of connecting a model to an appearance in the appropriate way. The possibility remains that a relativistic space-time, for example, represents some appearance which has nothing to do with gravitational phenomena. Riemannian geometry has been used to construct models of world economics, and this could not be accomplished by mapping photon trajec tories onto null geodesics. Catastrophe theory is a mathematical theory in search of applications. Applying a mathematical theory to a physical phenomenon consists in finding a mapping that generates the appropriate relation 195 between a mathematical model and some physical system. It Is not Inconceivable that the GTR has applications outside the realm of gravitational phenomena. However, this does seem unlikely. Though arbitrary to some degree, selection of these mappings is also constrained to a degree. Virtually every theory of modern physics contains what are called "correspondence principles", which serve to preserve the unity of physics by connecting newer, more sophisticated theories with older, simpler theories. For example, Newtonian mechanics is recovered from the STR in the "correspondence limit" in which all relevant velocities are negligibly small compared to the speed of light. That is, appearances containing only bodies of low velocity are represented by both classical and relativistic models. Classical mechanics is recovered from quantum mechanics in the "correspondence limit" in which the quantum numbers of the quantum states in question are so large that wave and diffraction phenomena make negligible changes in the predictions of classical mechanics. Vhen appearances con tain only particles larger than a hydrogen atom, they can be represented by both classical and quantum models. In these examples and others, the newer, more sophisticated theory is better than its predecessor because it gives a good description of a more extended domain of physics, or 196 a more accurate description of the same domain, or both. Correspondence principles give us the power to recover an older theory from a newer one, and thus explain the successes of the older theory, and, in many cases, guide the development of the newer theory. These correspondence principles also serve to constrain our choice of mappings of appearances into models. When some element of a model is found whose physical analogue is not obvious, corres pondence principles are used to find an interpretation. For example, it is the correspondence of the GTR with Newtonian gravitation theory that indicates that the quan tity M (M «■ m(r) for r > R, i.e., outside the region repre senting the SUN) in a Schwarzschild space-time represents the mass that governs the Keplerian motions of the planets. Correspondence principles serve as clues Indicating how to connect models with appearances so as to achieve empirical adequacy. To conclude this chapter, let me point out one more interesting feature of the view I am proposing. The rela tion between my view and the Received View, especially the scientific realists version of the Received View, is analogous to the relation between the STR and classical mechanics. As mentioned above, classical mechanics may be regarded as a special case of the STR, in the sense that it is valid for a set of appearances which is a proper subset 197 of the set of appearances for which the STR is valid. Similarly, the Received View can be seen as a -special case of my alternative view. Ab the mathematics used in a theory gets simpler and simpler, empirical substructures get closer and closer to being Identical with whole models. That is, if the relation between appearances and models is isomorphism, then my view essentially turns into scientific realism. It is not until we look at theories which make use of very sophisticated mathematical apparati that the differences between my view and the Received View become clear and Important. It is my contention that we can recast even those theories for which the Received View works quite well into the framework of my proposed alter native account. Thus, the successes of the Received View, and its corresponding popularity, can be explained even though it is, strictly speaking, incorrect even for the simplest theories. VII. CONSEQUENCES OF THE ALTERNATIVE VIEW 1. Theory Testing; On my alternative view of the structure of theorizing in the mathematical sciences, theories are not to be regarded as being true or false, but as being more or less adequate at generating models that represent the physical world. Theory testing, then, does not serve to confirm or disconfirm a theory, in the sense of providing evidence for its truth or falsity. In testing a theory, we are searching for the limits of its empirical adequacy. On the Received View, remember, a theory was tested by generating a prediction from the theory together with statements of initial and boundary conditions, and then performing experiments or observations to determine the truth-value of the prediction. The predictive character of such a determination is a reflection of our epistemic posi tion. We know the initial and boundary conditions but not the outcome of our experiments and observations. On my view, this epistemic situation is logically irrelevant. The appearance which we are trying to represent contains the results of our experiments and observations as well as the initial and boundary conditions. Our epistemic posi tion is represented by the fact that there are two 198 199 different appearaches to be dealt vlth. In general, on my view, theory testing proceeds as follows.1 Ve begin with an appearance containing only what are usually called "initial and boundary conditions". For example, such an appearance might contain the Sun, the Earth, a few stars, the mass and radius of the Sun, and the distances from the Earth to the Sun and from the Earth and Sun to the stars. Such an appearance would not contain any photon trajectories. To test a theory, we find a model for that theory which represents that appearance. For example, if we are testing the GTR, we use a Schwarzschild space-time to represent our appearance. (Note that there are classical, (i.e., non-relatlvistic) models that repre sent the appearance described above). Next, we examine the model, looking for features which may have physical analogues. For example, in the Schwarzschild model we notice that null geodesics passing close to the region of the manifold representing the Sun experience a deflection. Past successes in testing the STR suggest to us that null geodesics in space-time models can be used to represent the trajectories of photons. Thus, we suspect that the deflection of these null geodesics may be indicative of a deflection of photon trajectories passing near the Sun. In essence, we use an argument by analogy. The appearance and the model have several features in common, the model has an extra feature, thus, the appearance may veil have that feature also. We then look for that feature in the 2 world. The result of this hunt, positive or null, is added to our original appearance to form a new appearance. If the new appearance is represented by the model, then we have extended the range of applicability of the theory we are testing. That is, there is a new appearance, or class of appearances, which is represented by a model of the theory in question. If the new appearance is not represented by the model, then we have found a limit to the adequacy of the theory, in the form of an appearance, or class of appearances, which is not represented by any model for the theory. (This claim sounds somewhat stronger than we are entitled to make. One might suggest that we are warranted only in concluding that we have found an appearance that is not represented by one particular model for the theory, and are not warranted in concluding that no model for the theory represents the appearance. However, a moment's reflection shows that the stronger claim can be supported. If there were two models for a theory, both of which represent the original appearance, but only one of which represents the final appearance while the other does not, then the theory would be inconsistent.) For example, consider again the deflection case. We look for the deflection of photons passing near the Sun, and find them. 201 Our new appearance, then, contains the contents of the original appearance together with a few photon trajectories and their measured deflections. As we have seen, the deflection of the geodesics in the model is within the range of the measured deflection of the photon trajectories in the appearance. Thus, the model represents the appear ance. Further, this constitutes evidence that other appearances involving the deflection of photon trajectories passing near an approximately spherically symmetric, static massive body can be represented by models for the GTR. On the other hand, though there is a model for Newton's gravitational theory which represents our original appear ance, no classical space-time represents our final appear ance. We have found a limit to the empirical adequacy of Newton's theory. We can conclude, then, that the GTR is superior to Newton's theory, because the GTR can be used to represent more appearances than can Newton's theory. Let us turn now to a discussion of the cases described in my criticisms of the Received View. It is incumbent upon me to show how my alternative picture can solve or avoid these puzzles. 2. Time-Reversed Solution Case: The first chal lenge I raised against the Received View involved using the classical laws of motion to describe what happens to a ball dropped from a height and allowed to hit the ground. Recall that the problem arose from the fact that the equation describing the path of the ball was quadratic, and thus generated two predictions as to the time at which the ball hit the ground. Redescribed in terms of my alter native picture, the case Is as follows. There is an appearance which contains a ball which is allowed to fall from a height and hit the ground. The relevant features of the ball and its path contained in the appearance are the height of the ball above the ground at the moment of release, the time at which it was released, the time at which it landed, the value of the ’’acceleration of gravity” at the surface of the Earth, and the effect of air resis tance, regarded in this case as being negligible. The model, which is a model for classical mechanics, contains a curve which is described by a quadratic equation para meterized by t. This equation gives the position of the points on the curve as a function of t. The hypersurface t *■ 0 represents the time at which the ball is dropped and the hypersurface at h » 0 represents the ground. In this model, the curve reaches h ** 0 at two values for t, t *» 2.8557 and t = -2.8557, and the parabola extends from h *• _w at t ** -«*>, through h n 40 at t = 0, and h * -» at t 8 +». Since we can map the trajectory of the ball, which occurs in the appearance, onto a segment of the 203 parabola extending from t •» 0 to t ■ 2.8557, and since all of the relevant characteristics of the trajectory are adequately represented by features of the parabola, there is an empirical substructure of our model which stands in the appropriate relation to the appearance. Thus, our model represents the results of our observation of the ball and its path. Notice that this same model can be used to represent a slightly different appearance, one in which the ball is thrown upwards from the ground with an initial velocity of 28.0143 m/sec. In this case, the relevant empirical substructure contains a segment of the parabola extending from t ■ -2.8557 to t - +2.8557. This case presented a problem for the Received View because we could find no way to justify ignoring the fact that the ball does not hit the ground 2.8557 seconds before it is released. That is, on the Received View, classical mechanics makes a false prediction and should thus be dis- confirmed. On my account, on the other hand, it is not required that an entire model stand in the appropriate relation to some appearance, but only that some empirical substructure of that model does. Thus, even though a model contains elements which reflect no element of an appearance, the model may still adequately represent that appearance. The fact that, in the model described above, the parabola crosses h « 0 at t =-2.8557 is of no « 204 significance at all in evaluating the adequacy of the model or the theory which generates the model. A critic of this view might wonder, at this point, if any theory can ever he found Inadequate by generating a false prediction. If it is only required that a subset of a model stand in the appropriate relation to an appearance in order for the model to represent the appearance, then all extraneous elements of a model may safely be Ignored. The only way a theory can fail to generate a model which represents an appearance is if the appearance contains some element or characteristic which does not correspond to any element or characteristic in any model for that theory. That is, models can be too impoverished but cannot be too rich. Strictly speaking, this is true. However, recall taht on the view I am defending, neither models nor theories are confirmed or disconfirmed. Theories are judged by how many different kinds of appearances can be represented by their models. Theory evaluation is theory comparison. The best theory is the theory which is most empirically adequate. Further, it turns out that the richness of a model can count against the theory which generates that model, but only under certain circumstances. Consider the following example. Let A be an appearance, T a theory, U a model for T which represents A, and E the relevant empirical substructure of M. Suppose there is some element e in E which has a characteristic P for which there is no corresponding characteristic in A. That is, e's having P is in M-E. Ve might take this as suggesting that the element a of A which is represented by e has some physical characteristic P* which has gone unnoticed, and begin looking for such a characteristic. If we find such a novel characteristic, then we have a new appearance B, which is simply the union of A with the results of our search (i.e., B is the result of adding the fact that a has P' to A), and B is represented by M, since there is a sub set F of M (which is simply the union of E with the fact that e has P), which stands in the appropriate relation to B. Thus, in this case, we have one more appearance which is represented by U. On the other hand, suppose our serach for P* is fruitless. Then we have yet another appearance, C, which is the result of adding the fact that a does not have P 1 to A, and C is not represented by M. Now, if there is some other theory T' which has a model M' which represents both A and C, then ceteris paribus. T' is a better theory than T, and it is the presence of extraneous elements in a model for T that Is responsible for this. Thus, models can be too rich. Returning to our falling ball example, we have a new appearance, which results from our original appearance- 206 by adding the fact that the ball is not on the ground 2.8557 seconds before it is released. However, there is no theory of mechanics available which has a model that does represent this new appearance, so the inability of 3 our classical model to do so is not significant. 3. The Singularity Case: This example, taken from current astrophysics, proved troublesome for the Received View for the following reasons. Remember, the GTR, as construed by the Received View, predicts the exis tence of a class of novel physical entitles, space-time singularities. These are regions of space-time where the curvature scalars, and hence the tidal gravitational forces, become unbounded. Their existence is predicted by the GTR in the sense that they appear in all of the more realistic solutions to the Field Equations of the GTR. However, few theorists are willing to countenance their physical exis tence. ' This presents at least two problems for the Received View. In the first place, this is a case where a theory makes a prediction which is widely regarded as being false, and yet this fact seems to have very little bearing upon the acceptability of the theory. Secondly, according to the Received View, the ontology of the physical world is determined by the ontic commitments of our best physical theories, and yet here we have a case where our best physi cal theory of gravitation is ontically committed to 207 space-time singularities, but few theorists are willing to accept that commitment. Thus, it is difficult to make sense, using the Received View, of the attitudes and beha vior of practicing theorists vis-a-vis space-time singu larities. Let us redescrlbe this case in terms of the alter native account of physical theorizing which I am defending. There are many mathematical models for the GTR (i.e., relativistic space-times) which are not singularity-free. In fact, the singularity theorems of Penrose and Hawkins show that singularities will occur in any relativistic space-time in which a few very plausible conditions are satisfied. These conditions are very plausible in the sense that they are part of empirical substructures of many realistic models. For example, one of the conditions is the existence of a closed, trapped surface, which is a surface inside of which ingoing and outgoing null geodesics converge. Any model which represents a region of the universe containing a black hole will be one in which a closed, trapped surface exists. Since roost astrophysicists believe in the physical existence of black holes, and since any space-time which can represent a black hole will con tain a singularity, many of the interesting and important models for the GTR will not be singularity-free. However, as Ellis points out, "... the working relativist or 208 astrophysicist talks around [singularities] as something that may 'go away* when a full Quantum Theory of Gravity is available .... Thus these singularities are generally 4 regarded as artifacts of the theory . . . ." That is, most theorists are unwilling to believe that regions of unbounded mass-energy density and unbounded gravitational forces exist. Thus, though we have very little under standing of the nature of regions of extremely high gravl tational fields, most relativists feel that no model for the GTR could represent such a region. As Hawking and Ellis suggest It seems to be a good principle that the predic tion of a singularity by a physical theory indicates that the theory has broken down, i.e., it no longer provides a correct description of observations. The question is; when does General Relativity break down? One would expect it to break down anyway when quantum gravitational effects become important; from dimensional arguments it seems that this should not happen until the radius of curvature becomes of the order of 10”33cm. This would corres pond to a density of lO^gm cm”3. However one might question whether a Lorentz manifold is an appropriate model for space-time on length scales of this order. So far experiments have shown that assuming a manifold structure for lengths greater than 10“15cra gives predictions in agreement with observations, . . . but it may be that a breakdown occurs for lengths between 10”15 and 10”33cm. A radius of 10”13cm corresponds to a density of 1 0 5 8gm cm”3 which for all practical purposes could be regarded as a singularity. Thus maybe one should construct a surface . . . around regions where the radius of curvature is less than, say, 10”15cm. On our side of this surface a manifold picture of space- time would be appropriate, but on the other side an 209 as yet unknown quantum description would be necessary. Matter crossing the surface could be thought of as entering or leaving the universe, and there would be no reason why that entering should balance that leaving.5 The fact that theorists do not believe in the physi cal existence of singularities indicates that there are appearances which cannot be represented by any relativistic space-time. The fact that these appearances are described negatively, as regions of space-time which contain extreme gravitational forces but do not contain singularities, is unimportant. They are regarded as significant and relevant to theories of gravitation.6 However, no theory currently available generates models which can represent these appearances. In fact, if, as mentioned in Chapter VI, Misner and Wheeler are correct in claiming that any quantum theory of gravitation will predict the same sorts of singu larities predicted by the GTR, then the hopes of Hawking and Ellis, cited above, that some quantum theory will save the day are otiose. Thus, it seems likely that no theory will fare better than the GTR in generating models which represent regions of extremely high gravitational field intensity, and so the GTR will remain the most empirically adequate theory available. The fact that it is incomplete, or "idealized" (in the sense that singularities, which are idealizations of regions of extremely high, but finite, 210 curvature, cannot be avoided), though other than we might wish, is not a reason for condemning the theory. As Tipler, Clarke, and Ellis point out, "General relativity is not the ultimate truth; however, the realization that a theory is idealized does not lead one to abandon it, but rather to discover its limits."7 Sentiments such as these can be understood only if it is empirical adequacy that is important, and not truth. That is, we can make far better sense of the attitudes of current theorists if we construe their activity according to the account of physical theorizing I am proposing, rather than according to the Received View. It is also interesting to note that we can capture much of the spirit of Misner's suggestion that we learn to O tolerate space-time singularities. On my view, Uisner can be taken as denying the principle of Hawking and Ellis that the prediction of a singularity signals a breakdown of the theory, and replacing it with the principle that it is only when the singularity has observable effects, which should be, but are not, observed, that a breakdown is indicated. That is, Uisner suggests that we ignore the appearances which result from the assumption that physical singularities do not exist, and attend only to those which result from observation and measurement.® Since space-time singulari ties have no observable effects, their presence may be 2 1 1 tolerated In our relativistic space-times with no adverse consequences. It is only when observational difficulties arise that we are forced to revise our theory. Many theorists seem t6 have a similar attitude toward the so-called "other universes" which occur in some relativistic space-times. As mentioned above, these "disconnected" regions appear in Schwarzschild, Reissner- Nordstrom, and Kerr-Newman space-times, and their presence in these models is tolerated even though they have no observable effects. That is, no appearance currently under consideration requires the Inclusion of these disconnected regions in the empirical substructure of any model, and if the claims made by Simpson and Penrose, and Birrell and Davies are correct, then no appearance will ever require their inclusion in an empirical substructure. However, no harm is done by their presence in these models, and so they are safely ignored. Again, the alternative account of physical theorizing that I am defending provides an insight into the attitudes of current theorists that is not available on the Received View. 4. The Uses of Theories; On the Received View, a theory is acceptable, and accepted, only if there is good evidence that it is true. Recall that two of the principles fundamental to the Received View are that theories are intended to provide literally true descrip* tions of an independently existing, external physical reality and that the acceptance of a theory involves the belief that it is true. Classical, i.e., Newtonian, theories of kinematics, mechanics, and gravitation have been convincingly disconfirmed, and shown to be signifi cantly inferior to their relativistic and quantum competi tors. That is, we have good evidence that these theories are false, and so should be unacceptable. However, all of these theories are still used to solve a wide range of physical problems and make reliable predictions in a great variety of situations. If, as is claimed by the Received View, the point, or object, of, doing science is to con struct literally true theories, then why are these false theories so widely used? The Received View, with its emphasis on literal truth, cannot account for this. On the other hand, if we accept the alternative account which 1 am proposing, the mystery disappears. According to this view, the point of doing science is to make successful predictions, and thus, the empirical adequacy of a theory is far more important than its literal truth. When we compare Newtonian Gravitational Theory with the GTR, for example, we find that the latter is far more empirically adequate than is the former, which is to 213 say that more appearances are represented by relativistic space-times than are represented by classical space-times. However, there are many, important, appearances for which the classical theory is entirely adequate. Any system involving only weakly gravitating bodies and relatively slow (v << 1) particles can be adequately represented by a classical model. Further, it is often easier to use classical models in such cases, because the mathematics of Newton's theory is far simpler than that used by the GTR. Thus, when we are dealing with such an appearance, it makes more sense to use a classical model rather than a rela tivistic one. In such cases, the differences between the predictions generated by the classical model and those derived from the relativistic model will be negligible, and dwarfed by other, background, effects, such as the imprecision of the measuring instruments used to arrive at the appearance. Similar to the puzzle of the use of superseded theories is the puzzle involving the acceptance of the # STR. As we saw in Chapter VI, the STR is a superseding theory, widely used and appealed to for explanations of a variety of physical phenomena, and yet it is difficult to see how it could be construed as literally true. It turns out that no actually existing physical objects obey the laws of relativistic kinematics or mechanics, nor is any finite region of physical space-time precisely Uinkowskiian. Thus, again, the actual behavior of prac ticing theorists, in accepting the STR, is in conflict with the principles of the Received View. However, on the alternative account I am defending, again, the puzzle disappears. It is the empirical adequacy of the STR that is important, and there are uncounted appearances which can be represented by models generated by the STR. The key here is the notion of an inertial frame. The STR is a theory of the physics of bodies in inertial frames, and the Received View foundered upon the nonexistence of any inertial frames. However, consider the following defini tion of "inertial frame", given by Taylor and Wheeler A reference frame is said to be inertial in a certain region of space and time when, throughout that region of spacetime, and within some specified accuracy,every test particle tfiat is initially at rest remains at rest, and every test particle that is Initially in motion continues that motion with out change in speed or in direction. 0 (emphasis, mine) This definition serves to identify the elements in an appearance which are to be represented by Uinkowskiian regions of space-time in a model, in the sense that geo desics in Uinkowskl space-time will be, used to represent particle trajectories In the inertial frames defined above. For example, the region of space and time containing the 215 target area of a high energy linear accelerator during a collision experiment may be regarded as inertial within the limits of measurement error, even though the accelera tor lies in the gravitational field of the Earth. (This is primarily because of the extremely short time it takes for a muon, for example, to travel from the point at which the particle beam collided with the target to a particle detector.) We can predict the features of this muon trajectory, including the lifetime of the muon, by examining the relevant geodesic in a region of Minkowski space-time, and, it has been found, these predictions are well within the measurement "window". Thus, Minkowski space-time does represent the relevant appearance, which lends support for the claim that the STR is empirically adequate. The fact that many such appearances can be represented by models generated by the STR explains why that theory is so widely accepted and used. Considerations such as the above point the way to an understanding of the role, and importance, of idealiza tion and approximation in modern physics. Recall that, using the Received View, it was difficult to exlpain the fact that even our best theories are idealized to some degree. Many theories are at best approximately true, and are intended to be that way. If, as is claimed by the 216 Received View, the point of doing science is to construct literally true theories, the role of Idealization and approximation is indeed a mystery. If the main point of doing science is to construct theories which are empirically adequate, however, ideali zation and approximation have an obvious importance. Idealizing features of the world, where we can, makes the task of generating successful predictions much easier. In many cases, deviations of entities from ideal behavior are non-negligible only in extreme regimes. For example, the behavior of real gases cannot be adequately modelled, for most purposes, by Boye's theory of ideal gases only when a gas is subjected to temperature extremes, as in supercooling. As another example, the Schwarzschild model of the Sun is Inadequate only when we are Interested in gyroscopic precession experiments. In these cases, the deviation of some entity from ideal behavior is swamped by other background effects and/or measurement imprecision. It is only when such deviations become important that we must find more realistic accounts, but even these will con tain idealizations. Further, in many cases, such as con structing global models of the universe, we never will eliminate idealization. Finally, the nonlinearity of the Einstein Field Equations forces us to use systematic approximation methods such as the post-Newtonian approxi mation method and the weak-field approximation method to arrive at anything resembling a complete account of any celestial system.Thus, not only are idealization and approximation useful tools in doing science, in many cases they are essential, and my alternative view, but not the Received View, can account for this fact. 5. Epistemology: In the second chapter of this dissertation, I suggested that one of the principal moti vations for constructing an account of the logic and struc ture of physical theorizing is to explain how science gives us knowledge. Recall that, on the Empiricist program, scientific knowledge is regarded as the paradigmatic form of knowledge, and an adequate theory of epistemology should be based upon an understanding of the justification pro vided by scientific theorizing. According to the Received View, scientific justification is provided in the following two ways. On the one hand, confirmation of a theory, in the form of successful predictions or of positive instances of the laws of the theory, justifies our belief that the theory is true, and, on the other hand, the truth of a theory can be used to justify our belief that further predictions, derived from initial and boundary conditions we are justified in believing, together with the theory, are also true. That Is, If we are Justified in believing that some theory T is true, and if we are further Justi- fled in believing that statements of initial and boundary conditions 1 and B are true, then if the conjunction of T, I, and B entails some statement P, we are Justified in believing P. If, however, we abandon the view that physi cal theories are to be regarded as empirically true-or- false, then we certainly are not Justified in believing that any such theory is true, and if this is the case, we seem to lose our Justification of any consequences of such a theory. Thus, the Received View seems to explain scien tific Justification, while my proposed alternative account does not. Let us take a closer look at this issue, beginning with the alleged Justification provided according to the Received View. On this view the generation, by a theory, of true predictions and determinations provides Justifica tion for the belief in the truth of the theory.The truth of the predictions and determinations is established through experiment, measurement, and observation. However, as we have seen, the results of such empirical research are often not precise values, or statements, but involve ranges of values. Theories, on the other hand, generate precise statements of unique values. If a theory generates the prediction that some physical quantity Q will have a value of n, but measurement and observation yield a value for Q of m + r, then, even if |m - n| < r, how strong is our justification for the claim that the prediction is literally true? It seems that all we are Justified in claiming is that the predicted value is within the range of values determined experimentally. But how does this justify our belief that the theory used to generate the prediction is true? Further, even if we are justified in believing that the prediction is literally true, since, as we saw above, the auxiliary hypotheses (statements of initial and boundary conditions) used to generate this prediction are often idealized (sometimes essentially so), the truth of even a large number of predictions does not justify our belief in the truth of the theory. If the conjunction of T, I, and B implies P, and I and/or B are idealized (i.e., literally false), then clearly the truth of P is irrelevant to the truth of T. Thus, even if the logical structure of scientific theories is as the Received View maintains, it is not at all clear how we are to gain justification for the claim that some theory is literally true. And if we are not justified in believing the truth of a theory, then we are not justified in believing the truth of the consequences of the theory. Further, even if 220 ve were justified In believing that some theory was true, the fact that the auxiliary hypotheses (which must be used in generating predictions.) are often Idealized prevents our being justified in believing the consequences of the theory and those auxiliary hypotheses. Thus, the Received View does not explain scientific justification nearly so well as it first appeared. Let us turn to a discussion of justification according to my alternative account of scientific theor izing. Clearly, if theories are neither true nor false, then we cannot be Justified in believing that they are true. However, if a theory is empirically adequate, then it can be used to justify our beliefs that the predictions and determinations made on the basis of the theory are true (or, more precisely, the empirical adequacy of a theory can justify our beliefs that certain experiments, measurements and observations will yield particular results). Theory testing serves to establish the degree of empirical adequacy of a theory. The range of applicability of a theory is discovered by constructing models for the theory and comparing predictions and determinations made on the basis of those models with various appearances. Notice that on this view we can account for the fact that scientists feel it is important to test a theory over a 221 wide range of phenomena. As philosophers such as Glymour*1 have pointed out, theories are judged not only by how many successful predictions they generate, but also by how many different kinds of phenomena they can accommodate. In fact, Glymour counts the fact that his theory of confirmation ("boostrapping") can explain this attitude among scientists as one of the strongest points in its favor. But this attitude can be equally well explained if my alternative account is correct. The range of applicability of a theory is a function of how many different kinds of appearance it can accommodate. For example, the GTR is more empirically adequate than is Newton's theory because there are relati- vistic models which can represent every kind of appearance that can be represented by a classical model, and there are some kinds of appearance, namely, those involving high particle velocities and strong gravitational fields, which can be represented by some relativistic model but by no classical model. Testing over a wide range of appearances gives us information about the limits of adequacy of a theory. That is, it tells us for what kinds of appearance we can use a theory to generate predictions we can be justified in believing. Suppose that some class of appearances, A, all the members of which are similar in certain important respects 4 (e.g., they all involve approximately spherically symmet ric, static massive bodies), is in the range of applica bility of some theory T. This is determined by finding models for T which represent some of the members of A. Given a new appearance a in A, we have good reason to believe that we can find a model for T which represents a. Further, having found that model, we can use it to make predictions about the results of experiments, measurements, and observations. For example, we know that relativistic space-times, specifically, Schwarzschild space-tlmes, can be used to represent the regions surrounding approximately spherically symmetric, static massive bodies. Thus, we can calculate, using the appropriate Schwarzschild space- time, the deflection experienced by a photon trajectory grazing the limb of the planet Jupiter. (The appropriate Schwarzschild space-time will be much like the space-time which represents the region around the Sun, differing only in the values of M and R, which represent the mass and radius of the gravitating body, respectively.) That is, given the fact that Jupiter is approximately spherically symmetric and static, and has a particular mass, we are justified in believing that a Schwarzschild space-time represents the region surrounding Jupiter, and further, that the measured deflection of a photon trajectory grazing the limb of the planet will be about 0.02". In general, if a theory generates models which represent a certain class of appearances, then, given, an appearance in that class and a model which represents that appearance, we can make predictions and be justified in believing them. This fact allows us to gain knowledge, or justification, in situations where it would be absent according to the Received View. For example, according to the Received View, Newton's theory of gravitation, being'false, could not be used to justify our belief in the success of any of its predictions. But clearly, given the success of the space program, which uses Newton's theory for navigational pur poses, we can gain knowledge from this superseded theory. On my alternative view, it is easy to see how. Newton's theory is empirically adequate for most appearances involving low particle (spacecraft) velocities and weak gravitational fields, such as are encountered in the solar system. Thus, for any problem involving navigation through the solar system, we can count upon the predictions made on the basis of Newton's theory of gravitation. Let me end this section by saying something about the relation between appearances and the world. Most of the time, the appearances used by theorists embody their most accurate conception of the physical world. Sometimes, however, they contain known idealizations and 224 simplifications. How can a model which represents such an appearance tell us anything about the physical world? The answer is simple. If similarly idealized appearances are represented by a certain kind of model for a theory, and predictions based upon those models proved to be accurate (i.e., if the predicted values fell within the range of measured values), then we have good reason to believe that the model of that kind which represents the idealized appearance at issue will generate similarly accurate predictions about the results of measurements and obser vations. If an appearance is in the range of applicability of a theory, then even if it is idealized, the theory can give us knowledge about the world. 6. Ontic Commitment: On the view I am proposing, since theories are not to be regarded as being literally true-or-false, they have no ontic commitments, in the ordinary sense. That is, if a theory is ontically com mitted to a particular sort of entity Just in case that sort of entity must exist in order for the theory to be true, then if theories are neither true nor false, then the notion of ontic commitment becomes spurious. This might be taken as indicating an advantage for the Received View. Science clearly does direct our existential beliefs, but without the notion of ontic commitment, how is this to 225 be explained? That Is, how, on my alternative account, can science justify our existential beliefs? On my view, this question breaks down into two parts. How does a scientific theory suggest existential claims to the theorist, and how does the theorist justify his belief in the existential claims he accepts? That is, given that science suggests many existential claims, some of which are accepted by theorists and some of which are not (e.g., the singularity case), how is this done, and how are the decisions to accept or reject the suggestions made? The answer to the first question is easy to provide. Consider a theory T which is empirically adequate to some degree, and let A be an appearance which is represented by a model U for T. M contains many mathematical objects, some of which are in the empirical substructure of M (for A) and some of which are not. A theory is taken as sug gesting that the mathematical entities not appearing in the empirical substructure have physical correlates. That is, the theory suggests that the model as a whole serves as an empirical substructure in representing some portion of the physical world. These suggestions may be stronger or weaker. At one extreme, the extraneous mathematical entities occur in a highly idealized model, which repre sents a limited class of appearances, and do not appear in more realistic models. (In this context, a model is said to be more or less realistic depending upon the realism of the initial and boundary conditions used to generate the model. Global vacuum solutions to the Field Equations are highly unrealistic, while local solutions arising from assumptions of spherical symmetry and static- ity are somewhat less so.) As an example, consider a Schwarzschild space-time with a point field source at r ■ 0. In such a relativistic model, the point at r ■ 0 is a singular point, but this singularity disappears when we move to a model with a finite field source with radius r > 2MG. Thus, in the more realistic model, this parti cular mathematical object is absent, and the suggestion . that it has a physical analogue, based upon the Idealized model, is rather weak. (There is also the case where the novel entities occur in models which are so idealized that they represent no appearance at all. Gddel space-time is an example of this. Gil del space-time is a global (i.e., cosmological) solution based upon the assumption that the universe is homogeneous but not isotropic. The observed isotropy of the cosmic microwave background radiation indicates that this is a highly idealized model. Gttdel space-time contains closed timelike curves which, if taken seriously, would imply that physical particles could run into their earlier selves. The highly idealized character of this solution makes this suggestion extremely weak.) At the other extreme, which is much more interesting, novel entities occur in the most realistic models. For example, the singularity theorems of Penrose and Hawking imply that even in models based upon completely accurate descriptions of the state and distribution of matter-energy throughout the universe, singularities will occur. That is, these novel mathematical entities will occur in even the best possible models for the GTR. In cases such as this, the suggestion that a physical analogue of a mathematical singularity exists is very strong indeed. If theories are not Intended to be literally true, however, we are not compelled to accept even the strongest suggestions of our best physical theories. How, then, do theorists decide upon their existential beliefs? It might be maintained that, in spite of the fact that theories are not to be regarded as being true-or-false, they do have genuine ontic commitments, in the sense that the empirical adequacy of a theory, together with the fact that it is the best available, gives us good reason to believe that all of the elements of a completely realistic model for the theory have physical correlates. One might claim, for instance, that though theories are not intended to provide literally true descriptions of reality, they are 228 Intended' to generate mathematical models which are isomor phic with some portion of the physical world. (Though no element of a model is identical with any feature of the physical world, there is, it is hoped, a one-one relation holding between our best models and portions of the world.) Thus, in the ideal limit of realistic models for a maxi mally empirically adequate theory, every mathematical object would be expected to have a physical correlate. The empirical adequacy of a theory is then explained by the fact that its most realistic models exactly mirror the workings of nature. In fact, there are cases in the history of physics in which something like this seems to be going on. Con sider, for example, the discovery and development of quan tum chromodynamics ("quark theory"). In 1964, Murray Gell- Mann proposed the quark model of subnuclear particles as a means of keeping track of the properties of hadrons, that class of elementary particles which includes protons and neutrons. Each of these particles is regarded as a composite of either three quarks or one quark and one * antiquark. Each quark is assigned an electric charge of either -1/3 or +2/3; the antiquarks have the opposite charge. For example, u quarks have a charge of +2/3, d quarks have a charge of -1/3, and protons are made up of two u quarks and one d quark, and thus have a charge of 220 +1. The electric charge of any hadron can be found by merely adding up the charges of the eonstltuent quarks, and several other properties of hardons can be handled In the same way. Vhen he first proposed the quark model, Gell-Man did not intend it as a literally true description of anything in the physical world. In fact, Gell-Mann wrote: "It is fun to speculate about the way quarks would behave if they were physical particles . . . instead of purely mathematical entities.Since then, however, most par ticle physicists have become convinced that quarks do have a physical existence, and the reason seems to be the spectacular successes of the quark model at generating extraordinarily accurate predictions. The model has been used to predict the existence of an entirely family of new hadrons. The first of these, labeled psi or J, was discovered in 1974, and the dozen or so charmed particles that have been observed since then all fit into assigned niches in the quark model. Thus, though quarks themselves are notoriously difficult to observe directly ("free" quarks may be impossible to come by; the force holding quarks together increases as the distance between them increases, and the energy necessary to free a quark may be sufficient to create new ones which then bind to the old), 230 the quark model has been extraordinary successful. This success could be explained (in some sense) If the quarks in the model had material correlates. Thus, it is con cluded, we have good reason to believe that quarks exist as physical objects. The criterion of ontic commitment being suggested here, then, is the following: A theory is committed to the existence of physical correlates of every mathematical entity occurring in the most realistic models possible for that theory. This criterion has the same effect as does the criterion proposed on the Received View, but the truth of the theory is not required. However, as might be expected, this new criterion suffers from the same problems mentioned in connection with the old one. Remember, the success of a model is strictly a function of the accuracy of the observational predictions made on the basis of the model, and, as we have seen, there are models which are quite successful and yet which contain elements or regions which demonstrably do not contribute to any observational prediction (e.g., singularities, the universe beyond the event horizon of a Reissner-Nordstrttm black hole). That is, there is some evidence that the proposed new criterion of ontic commitment is violated by practicing theorists in deciding their existential beliefs. Let us search further for their Justifications. 231 When, In the course of constructing relativistic models, some mathematical object appears whose physical analogue, if it existed, would be quite unlike anything thus far observed, the standard strategy is to blame its occurrence on the use of idealization in constructing the model, and then try to find a more realistic model in which the offending mathematical entity is absent. Finding such a model is sufficient to conclude that the entity in ques tion, or rather its physical analogue, is not predicted by the theory used to generate the models. That is, the empirical adequacy of the theory does not justify our belief in that physical analogue. We must be careful here. The above considerations imply that a necessary condition for the empirical adequacy of a theory to justify our belief in the existence of some novel physical object is that the most realistic models generated by the theory contain the relevant mathematical entity. It is not at all clear, however, that this is a sufficient condition. There are cases wherein some mathematical object appears in some very successful model, there is no way to eliminate the mathematical object from the model, and yet most theorists would be unprepared to say that the mathematical object has a physical correlate. In general it is only the adequacy of those models in which the entity in question contributes to the observational predictions that justifies belief in the physical existence of the novel entity. That is, ve a r e . justified in believing only that those mathematical elements of one of the most realistic models for our best theory which are related to some element of an empirical substructure of that model via some relation which occurs in that empirical substructure have physical correlates. For example, though singularities appear in the most realistic models for the GTH, our best theory of gravita tion, no time-like or null curves can be drawn from the singularity to any point in any empirical substructure of the model, and time-like or null connectlbility in the model represents causal connectlbility in the world. Thus, we are not justified in believing in the physical existence of singularities. Black holes provide another excellent example of mathematical entities occurring in realistic models whose physical existence was not generally accepted until it could be shown that physical black holes would have observable effects and those effects were observed. Though black holes (i.e., closed, trapped surfaces resulting from gravitational collapse) appear in many of the most realistic relativistic models, most theorists were unprepared to accept their physical existence. The notion of a star or other massive body collapsing all of mass into a point and hiding Itself beyond an event horizon was a bit much for some physicists to swallow. Uany argued that, while the existence of such entities was compatible with the GTR, its actual physical existence would be prevented by some as-yet-unknown physical process. For example, many argued that in the process of gravitational collapse, enough mass would always be thrown off so that the resulting pieces would have masses below the Volkoff- Oppenheimer limit. It was soon realized that black holes, if they existed, would have observational consequences. For example, if they occurred in binary star systems, we should be able to detect them by their effect upon their companion star. That is, if no trace of one of the com panions is found, and if the behavior of the other star is such that the calculated mass of the unseen companion is greater than about 3.5 solar masses, then we would have good reason to think that the unseen companion is in fact a black hole. Further, as material from the luminous companion falls onto the accretion disc of the black hole, it should give off rather intense radiation at x-ray frequencies. Thus, if the unseen companion in a binary star system is a strong x-ray source, we have additional evidence that it is a black hole. An astronomical search for such an object yielded one excellent candidate: Cygnus X-l. The discovery of this object has convinced most theorists that black holes do in fact exist, while the Bimple occurrence of closed, trapped surfaces in many of the most realistic relativistlc models was inadequate justification for belief in these tinlque objects. The singularity case and the "other universe" case are instances where a theory seems to "predict" the existence of some novel class of entitles, yet theorists refuse to countenance their physical existence because they contri bute to no observational predictions. The black hole case is an instance where, again, a theory seems to "predict" the existence of some novel entity, but it was not until it was shown that black holes could have observational consequences, and further, that there was observational evidence for their existence, that theorists were willing to accept commitment to their existence. Thus, it seems that three things are needed to justify belief in the physical analogue of some novel mathematical entity: (1) that it can be shown not to disappear as models are made more and more realistic; (2) that it can be shown to contribute to observational predictions in models which represent appearances; and (3) that nothing else can plausibly be said to explain the observations, consistent with our best theory. These conditions also seem to be jointly sufficient, at least if the behavior of practicing theorists is any guide. 235 7. Conclusion; In this dissertation, I have tried to accomplish two things. I have tried to show that a widely accepted theory of the logic and structure of scien tific theorizing and the role of scientific theories is inadequate when applied to current physics, especially relativistic astrophysics and cosmology. I argued that its Inadequacy stems primarily from the fact that it does not provide an accurate description of the behavior of practicing theorists. I have also presented an alternative account of these matters, and tried to show that it is superior to the traditional view, by showing that we are better able to understand and explain certain features of theorizing in modern mathematical physics. APPENDIX: DIFFERENTIAL GEOMETRY 1. Manifolds: The fundamental entity with which differential geometry is concerned is the differentiable manifold. Intui tively, an n-dimensional differentiable manifold is a set of "points" tied together continuously and differentiably so that it is locally isomorphic to Euclidean n-space. To make these notions more precise, let us begin with the abstract concept of a space. A space is simply a set of "points". The original use of these concepts was to discuss the struc ture of physical space, or perhaps perceptual space. They have been generalized, however, so that just about anything may pount as a space. For example, buppose we can completely characterize the state of some body by specifying its temper ature and its charge. We can represent the set of possible states of'this body as a space, the points of which represent the possible combinations of temperature and charge. "Phase" spaces like this are used in quantum mechanics. The dimen sionality of a space is determined by the number of parameters necessary to uniquely pick out a point in the space. Our space of possible states of the body is two-dimensional. Sup pose we wanted to represent the set of possible states of a system of N bodies. To specify the state of such a system, we would have to specify the temperature and charge of each 237 of the N bodies. Thus, 2N parameters must be specified to uniquely fix a point in the space. This space would thus be 2N-dimensional. It is often useful to attach additional structure to a space. We can give it topological structure in the following way. Let M be a space, and let T be a class of subsets of II. T is a topology on M iff T satisfies the following axioms. (A) M and $ belong to T. (B) The union of any number of sets in T belongs to T. (C) The intersection of any two sets in T belongs to T. The members of T are called the open sets of M, and the pair (U,T) is a topological space. For example, consider the real number line. On the "usual topology" for R (there are many other topologies which may be defined for R), the open sets are the open intervals of the real line. A region S of an n-dimensional space M can be covered n with a coordinate system by defining a mapping x:S*R which maps each point in S into the set of ordered n-tuples of real numbers. The two-dimensional real space R2 can be coordina- tized by I, the identity function. However, it can also be coordinatized in a variety of other ways. Consider our Bpace of possible states of some body. We are assuming that the parameters are real properties of the body in question. That is, whether we measure the temperature of the body in degrees Kelvin, Celsius, or Fahrenheit, it has but one temperature. Thus, we may define the following coordinate system for our 238 space; x=(t,c), where t and c are coordinate functions which map points of the space into R (i.e., for some point p in our space, x(p)» "slot" function uA . Where f»(f1,...,fQ), u^®f*fA. Once a coordinate system has been defined for a space, it is a simple matter to change to a different coordinate system. Such a change is called a coordinate transformation. and operates like a function from Rn into Rn. Given an n- dimensional space M and a coordinate system xB(x},...,xn ) defined on a region S of If, we can define a coordinate trans formation i •••»£n)» where each is a function of x^. A new coordinate system y»(y ,...,yn) can be simply defined as y°£*x (i.e., y i“^ 1*x 1» etc*)* Returning to our example, suppose that our coordinate function t gives us the temper ature of the body in degrees Fahrenheit. We can easily de fine a coordinate transformation which will get us to a coor dinate system in which temperature is given in degrees Cel sius. Let £(x)=5(x-32)/9, and let y(p)Mgox(p). A map f from an open set ACRn into R is continuous at a point peA iff for any open set V containing f(p) there exists an open set U containing p such that fCUDcV. A function is a continuous function if it is continuous at every point. A continuous map is called a C° map. A map f from an open set AcRn into R is called Cr on A if it possesses continuous partial derivatives on A of all orders £r. If f is Cr on A 239 for all r, then f is C* on A. A map f from an open set ACRn into Rk (k an integer >1) is Cr on A if each of its slot func- tions fj*u^»f is Cr on A for i**l,...,k. We are now ready to define a manifold. A n-dimen- Bional manifold M is a set U together with a c£ atlas {Ua »4>a}* A Cr atlas is a collection of charts (Ua »4i0) where the Ua are subsets of M and the $a are one-one maps of the corresponding to open sets in Rn such that (D) the U~ cover M, i.e., M= a (E) if UaHU^ is non-empty, then the map V*B~’; ♦B For an illustration of condition (E),- consider the drawing below. Figure 7. The Manifold 240 Perhaps the simplest example of a manifold is Rn toge ther with the atlas {Ua ,I}, where the Ua cover Rn and 1 is the Identity function. Is our space of possible states of a body, together with an atlas whose $a maps are coordinate r maps, a C manifold for any r? If the issue is whether a space of physical quantities or properties is a manifold, then there are constraints places upon the 4>a maps. For example, the fact that one of the parameters which character izes the points of our space is the charge of a body is very Important. Since charge is not distributed continuously, but comes in quanta, the value of the second coordinate function of any coordinate system an any point in the space must be an integral multiple of some constant. That is, for any coor dinate system x»(t,c) defined on any open set A in our space and for any point peA, c(p)»ik, where i is an integer and k is some constant. Consider any two charts (U, A.) But neither $ nor As stated in Chapter IV, differentiable manifolds can be defined alternatively as separable Hausdorff spaces which are topologically locally Euclidean. A topological space U is said to be a Hausdorff space if it satisfies the Hausdorff separation axiom: whenever p and q are two distinct points in II, there exist disjoint open sets U and V in M such that peU and qeV. To say that a space is topologically locally Euclidean is to say that the space can be covered by coor- • dinate systems that map regions (whence comes the local char acter of this requirement) of the space onto regions of Euclidean n-space (i.e., Rn endowed with its usual topology. A space would be topologically globally Euclidean only if a single coordinate system could map the entire space onto Euclidean n-space. Finally, let us extend the notion of a map being Cr . We have defined the notion of a map from an open subset of Rn into R being Cr and the notion of a map from an open sub set of Rn into Rm being Cr. Now we will define the Cr-ness of maps from an open subset of a Cs n-manifdldainto R and k . into an open subset of a C d-manifold. Let A be the domain of a function f and assume A is an open subset of the Cs n-manifold M. If f is real valued, then f is Cr on A if 242 f*4“ * is Cr on $(AOU) for every chart (U^) on U. If N is a Ck d-manifold and f is N-valued, then f is Cr on A if f is continuous and for every real valued function g, that is Cr on an open domain in N, the composite g»f is Cr on ArtF”1(domain of g). We are now in a position to define a number of interest ing geometrical objects on a manifold. Many can be defined strictly in terms of the manifold structure of a space, while others require the provision of additional structure. Among the former are curves, vectors, and tensors. The notions of geodesic and curvature will have to wait until we can intro duce that additional structure. 2. Vectors and Tensors: Though many of the objects we will be discussing can be defined on spaces which are not mani folds, from this point on, we will be confining our attention to C n-manifolds. A c£ curve A(t) on a manifold II is a Cr map of an open interval of R into 11. t is called the "curve parameter". Relative to a coordinate system x, the curve is given by x(A(t)). The tangent vector to the Cr curve A(t) at the point X(t0), expressed as (131) I = = 3ft = A 110 aZ t-t0 is an operator which maps each Cr function f at the point A(t0) into the number (df/dt)jJt. (dA/dt)|t-t^ tells us how a point moving along A(t) is displaced, as a function of an infinitesimal change in t, in the neighborhood of A(t0), and (df/dt)^|t tells us what happens to f in this displacement. 243 Now, where (x*,...,xn) are local coordinates in a neighbor hood of A(t0), we have that d I 5 dxiCACt))l 3 dx1 d I (132) * L . > ^ ^ (Here, and throughout this dissertation, we adopt the summa tion convention whereby a repeated index implies summation over all values of that index.) The (3/3x*)|p ,...,(3/3xn)|p are the coordinate derivatives at a point p. The set of all tangent vectors at a point p constitutes a vector space, called the tangent vector space at £, denoted by Tp . Any vector in Tp can be expressed as a linear combination of the coordinate derivatives at p. Thus, any such set of coordi nate derivatives at p (so long as they are linearly indepen dent; i.e., non-coplanar) constitute a set of basis vectors which span the space Tp . Note that (dx^/3x^) “ 6j, where 6j is the Kronecker delta. Any vector v in a tangent space Tp at p can, given a coordinate system (x1,...^11) in the neighborhood of p, be expressed as v*(3/3x*), where the v1 are called the components of the vector vuin the coordinate system ( x x n), and can be represented as the ordered n- tuple (vl,...,vn). The components v* « v(x*) «* (dx*/dt)|D are the derivatives of the coordinate functions x* Along-»the curve whose tangent vector is v at p. A one-form (covariant vector) u at p is a real valued linear function on the space Tp of vectors at p. If x is a vector at p, the number into which w maps x will be written in Tp such that if plane, and if < a basis (&a) of vectors at p, one can define a unique set of n one-forms {ea} by the condition: e* maps any vector x to the number x* (the i**1 component of x with respect to the basis {ea>). Then in particular, gard {ea} as a basis of one-forms since any one-form u at p can be expressed as u ■ u^e*- where the numbers by ■ tell us which vector or one-form we are talking about, not which component of some vector or one-form.) The set of all one-forms at p forms an n-dimensional vector space at dual space T* to the tangent space T . The basis {ea> is P P the 1 dual■ — basis 1 ■■—1 — to the basis (e_) a of vectors. For any —toeT*_ p and xeTp, one can express the number components x* of to, x with respect to dual basds (ea), {ea> by the relations (133) Each function f on U defines a one-form df at p by the rule: for each vector 5t, (134) coordinates, the set of differentials (dx1,...,dxn) at p form the basis of one-forms dual to the basis (3/Sx1,...,3/3xn) of vectors at p, since this basis, the differential df of an arbitrary function f is given by (135) df « (df/dx^dx1 . One may think of df as a normal to the surface f^constant at p. If afO, adf will also be a normal to this surface. Prom the spaces Tp of vectors at p and T*^ of one-forms at p, we can form the Cartesian product (136) II® « T*p X ... X T*p X Tp X ... X Tp , i.e., the ordered r+s-tuple of vectors and one-forms (rj_l,.. ., # * • • *?s ) where the pa and i}S are arbitrary vectors and one-forms respectively. A tensor of type (r,s) at p is a function on n® which is linear in each argument. If § is a tensor of type (r,s) at p, we write the number into which S maps the element (ij.1,... ,... ,ys ) of n® as S(ji1 , • • • »Hr * ?!».*..y8>- We now define a number of operations which involve ten- * sors. The set of tensors of type (r,s) at p forms a vector space TB(p). Elements of T^(p) are r-contra tensors and ele ments of T®(p) are s-co tensors. In particular, Tj(p) ■ Tp and T$(p) ■ T*p* *s ta*cen to be R b^ definition. If 6eRg(p) and AcT„(p), the tensor product of 0 and A, written §0A, is defined to be the element of T8^ ( p ) such that §«A(al»•..»ar+m*?i».••*yg+n > ■ g(tij ..,n r , p l,...,ys)A(ar+1,. . . • • •*^s+n** For example, if Q is a vector at p and m is a one-form at p, then u0u maps the pair (^,v) into R, where \ is an arbitrary 2 46 one-form and v an arbitrary vector. That is, (138) (uGwXX^v) * If {e_} and {ea} are dual bases of T_ and T* respec- H i & a tively, then (e. 9...9e_ 9e 9...0e° } (a*, bJ run from 1 to a i ar a n) will be a basis for Tg(p). An arbitrary tensor SeTg(p) can be expressed in terms of this basis as (139) 8 - Sa '-’*arb ...b ea 9...95a «ebl« . ..9ebs x s i r where S rb ...b are the components of S with respect s to the bases {3a)» (ea ) and are given by (140) Sa‘‘ - S(ea i,...,ear,8. ,...,gb ). I s I S ' The contraction of a tensor S of type (r,s) with compo nents _ with respect to bases {§„}, (ea), on the OX * * »5 «* "" first contravariant and first covariant indices is defined to be the tensor 6 of type (r-l,s-l) whose components with ftb H respect to the same basis are S i.e., • (141) C » Sab*’*daf eb9 . ..9ed9ef® . ..9eg . For example, if 0 is a tensor of type (1,2) whose components, & 9 relative to some basis, are 0 b c , then the tensor, A, which results from the contraction of the first contravariant and first covariant indices of £ will have, relative to the same basis, the components Ac = ©aac ® 011C + ••• + ®nnc* A tensor is symmetric on two of its "slots" if its out put is unaffected by an interchange of the vectors or one- forms in those slots. For example, the tensor £ of type (2,2) is symmetric on its first and third slots if ■* d(u,u,v,A_). In component notation, § is symmetric on its t. rl first and third indices if, relative to some basis, c “ 0_b_dw & . Similarly, a tensor is antisymmetric on its first and third slots if §(v,w,u,X.) * -0(0,«,v,X.)• Given two vectors u and v, their wedge product. the "bi- vector" uAv, is defined by (142) uAv = u8v - v8Q. Similarly, the "two-form" wAA, constructed from two one-forms is (143) uAA. H w8A - X8m. Any tensor formed using the wedge product will be completely antisymmetric, i.e., antisymmetric on all of its indices. A set of local coordinates (x*) on an open set U in U defines a basis {(d/dx-**)^} of vectors and a basis ((dx*) jp} of one-forms at each point p of U, and so defines a basis of tensors of type (r,s) at each point of U. Such a basis of tensors will be called a coordinate basis. A tensor field S of type (r.s) on a set Veit is an assignment of an element of t 5(p ) to each point peV such that the components of S with respect to any coordinate basis defined on an open subset of lr V are C functions. Curves, vectors, one-forms and tensors are all coordi- nate-independent objects, in the sense that their existence and character on a manifold are independent of any coordinate system. However, coordinate basis vectors and one-forms, in asmuch as they explicitly depend upon a coordinate system, 248 do vary as we go from one coordinate system to another. That is, a coordinate transformation induces a transformation of basis vectors and one-forms. Further, since many different combinations of vectors and one-forms can provide bases for their respective spaces, we need a transformation of basis vectors and one-forms in general. Ve define these transfor mations in the following way. If {6a) and {ea} are a pair of dual bases for Tp and T*p, and {5a i> and {ea '} are another pair of dual bases for theBe spaces, then the latter can be represented in terms of the former by (144) Sft, - where 4_,a is an nxn non-singular matrix. Similarly, El (145) ea ' - 4a 'aea El * where $ a is another nxn non-singular matrix. Since {§_,}, (e& '} are dual bases «b 'a> “ <£b '.Sa .> “ <*b 'b2.b >*al*®a> (146) . ht h « h . ■ 4 .4 . 6 k = 4 a^b va' v bua va' v a i.e., 4ft,a and 4a a are inverse matrices, and 6ab « 4ab ,4b b . In particular, if (x ,...,xn ) is a coordinate system in the neighborhood of p on a manifold U which induces the coordi nate vector basis (3/3x^) on Tp, and (x1^ ,...»x'n ) is another coordinate system in the neighborhood of p on H which induces * the basis O/dx'^) on Tp , then, since x' * x'(x), we have that (147) O/ax^) - (3Xj/3x,1)(3 /3xj) where (3xj/3xf^) is the Jacobi&n matrix. 249 Since vectors are coordinate independent objects, if v is a vector in Tp , then, where {ea> is a basis for Tp , v may be expressed in component form as v * v*e^. If {ea '} is also a basis for Tp , then v may also be expressed as v * v^'ej,. So, ** From the above discussion of transformation of basis vectors, we know that e^ ■ 4 ^ ' e . ,, 1' 1 1 * 4' i 1' so vJ ej, ■ v 4,^ ej,; thus v* ■ v 4^ . This is the gen eral transformation law for the components of vectors. Sim ilar comments can be made about the transformation of the components of one-forms. Since the basis of a tensor space at p is determined by the bases for the vector and dual spaces at p, the trans formation law for the basis of a tensor space is arrived at by a simple extension of the transformation laws for vector q I n t and one-form bases.: The components S x** ry.t hi of a X * * *D 8 tensor S with respect to the dual bases {8 ,), (ea } are ft “ given by a I a I B '• r„. b- ” (148) s S(ea *,...,ea r,6 ). a a 9 They are related to the components SBi**,Brb ^ of S with respect to the dual bases {80) , {ea } by ft S ft^«i*«ft^t* H kt kli (149) . b i***b © Saxii.ar K 4a i ...4a r 4., b x...4Kt bs b i•**bs i r 1 s where the 4. , bi are the vector transformation matrices and b Vi a* 4 1 the 4 * are the one-form transformation matrices. ai 250 3. The Affine Connection: Differentiable manifolds possess considerable topological structure, but they have no intrin sic geometry. We cannot yet speak meaningfully of the dis tance between points on the manifold or of the curvature of the manifold. In order to discuss the geometrical structure of a space, we will have to introduce additional geometrical objects, objects which cannot be defined solely in terms of the topological objects which are already on the manifold. Two such objects must be added in order to make use of the full apparatus of differential geometry. They are the affine connection and the metric. In this section I will discuss the affine connection. A connection V at a point p of U is a rule which assigns to each vector field 5 at p a differential operator Vj which maps an arbitrary Cr (r>l) vector field £ into a vector field where (150) vfx+gyX (151) Vs (a?+32) ■ a V ^ + (152) V5(fy) - x(f)y + fV5y where x, y, and z are any vector fields, f and g are any functions, and a and 3 are any members of R. Then is the covariant derivative (with respect to V) of £ in the direc tion x at £. We can also define Vy, the covariant derivative of jr, as that tensor field of type (1,1) which, when contrac ted with x, produces the vector V-y. Then we have (153) (152) -M- V(fy) - df«? + fVy. 251 These notions are simply generalizations of familiar concepts used in ordinary, Euclidean geometry. To allow for the pos sibility of a curved space, many familiar geometrical objects must be generalized in this fashion. For example, our def inition of a vector is a generalization of the ordinary "tail-to-tip arrow" notion of a vector. In flat space, a vector is a bi-local object, existing in the space. In a curved space, however, a vector exists in a tangent space located at a point. V? is a generalization of our familiar notion of the gradient of and is a generalization of the notion of the directional derivative of ? (in the 5t dir ection). Ve further have that (154) Vf = df Vsf = 3~f H G(f) = The definition of a covariant derivative can be extend ed to any Cr tensor field, if r£l, by the rules: (A) if § is a Cr tensor field of type (q,s), then VS is a r_1 C tensor field of type (q,s+l), (B) V is linear and commutes with contractions, (C) for arbitrary tensor fields §, T, V(SfiT) - VSST + S«VT. Thus, V-S(ul,. .. ,... ,VS ) ■ VS(u*,...,u»r,Vl,...,Vs ,Q). Given any vector basis (6a ) and dual one-form basis (ea) on a neighborhood U, we write the components of V? as ya .b . so (155) V? B ya ;bS.b* sa* 252 % T a The connection is determined on U by n* C functions r bc defined by <1S6> r\c ' <-a,7Sb5c> ~ V8C - rabc eb*ea. Given that, for any function f, df ■ f t nea in the basis {ea} where f,a “ “ ®a^f)* we can calculate y a . b as follows. Vy *= V(yaea ) “ (7ya )8ea + yaVeft (157) « d y ^ S a + y ar cba eb8§c ■ ya ,b £b®5a + y c r abc eb®8a “ (yatb + yCrabc> sb®sa Thus, ya .b * ya h + ycrabc, where ya b - O y a/3xb ). In general, if {3/3xa } is a coordinate vector basis and {dxa } a dual one-form basis, then the components of VS for any tensor S are sef:::g;h - *sft:::p»xh * rahJsJS:;:J + (an uPPer (158) . j indices) - r hesjf!!!g ~ ^a11 l°wer indices.) The "connection coefficients", I’apy» ftre correction factors which quantify the twisting and turning of a field of basis vectors. In a flat space, no such correction is needed, thus r“By - o, ands“b;;;^.h -s £ ;;;2 (b. The components of V^S can be computed in a similar fashion. Where fl ■ d/dt is a vector tangent to a curve parameterized by t, and S * Sabc 8a®eb8ec in bases (5a). (ea ) t (159) V0£ - (DSabc/dt) Sa«eb«e°; (160) (DSabt/dt) - 8abo;dud 253 Let A be a curve on M with tangent vector field v. If T is a tensor field on ^ f is parallel along A. if V^T * 0 on X. Given a curve X with endpoints p, q, one obtains a unique tensor at q by parallely transporting any given ten sor from p along X. In a flat space, vectors may be compared by simply moving them around in the space, being careful to keep their orientations constant. In a curved space, how-> > ever, the vectors reside in tangent spaces, one for each point in the space. Parallel transport is a way to compare vectors in different tangent spaces. Consider a vector field v (which assigns to points on the manifold a vector in the tangent spaces at those points) and a vector u tangent to a curve X at a point p. Suppose X connects the points p and q. By parallely transporting the vector v(p) along X to q, we can compare it with the vector v(q). V~v tells us by how much the vector field v deviates from constant along the curve X. Let X be a curve on M with tangent vector field $. X is a geodesic if *» 0; i.e., a geodesic is a curve along which its own tangent vector field is parallely transported. Thus defined, a geodesic curve is the "straightest" curve connecting two points. Given any coordinate basis, the com ponent form of the geodesic equation » 0 is <&£+ra dxiM (161> da2 + rtW By do s dor = °- where a is the curve parameter for X. 254 A vector £L0 given only at one point p0 suffices to com pute the derivative CLCfD s 3- f, which is simply a number 9 «o associated with the point pQ. In contrast, a vector field fl provides a vector CL(p)— which is a differential operator 9ft(p)— at eacb Poin-t P in some region of a manifold. This vector field operates on a function f to produce not just a number, but another function HCf] = 3^f* A second vector field $ can just as well operate on this new function, to produce yet another function (162) ?{acfD} * a * o af). Does this function agree with the result of applying V first and then Q? Equivalently, does the commutator (163) ca,socf3 s wcf3> - sxacfD} vanish? The simplese special case is when Q and 9 are basis vectors of a coordinate system, tl « 3/3xa , ^ » 3/3x&. Then the commutator does vanish, because partial derivatives al ways commute: (164) C3/3xa ,3/3xBDCf3 - 32f/3xB3xa - 32f/3xa3xB « 0. But in general the commutator is nonzero, as one sees from the following: ca.^DCfD » ua (165) Notice that the commutator like t& and is itself a vector field. One important application of commutators is the distinc- tion between a coordinate-induced basis, (ea) ■ {3/3xa}, and 255 a non-coordinate basis. Because partial derivatives always commute, (106) CSa ,eeD “ C3/3xa ,8/3x03 - 0 in any coordinate basis. Thus, for any field of basis vec tors {ea(p)J in Tp, {§a(p)> is a coordinate-induced basis iff Ce0 ,eg3 “ 0 for all §a and e^. Given a Cr connection V, one can define a Cr-1 tensor field T of type (1,2) by the relation (167) T(x,y) * V~y - V-x - Cx,yD where x, y are arbitrary Cr vector, fields. This tensor is called the torsion tensor. Using a coordinate basis, its components are ( 168) t1^ - r1Jk - r1^ In what follows we shall be dealing only with torsion-free connections, i.e., we shall assume that *» 0. In this case, the coordinate components of the connection obey » so such a connection is often called a symmetric connection. A connection is torsion-free iff f.jj " **ji for a11 func“ tions f. From the geodesic equation it follows that a tor sion-free connection is completely determined by a knowledge of the geodesics on H. Further, when the torsion vanishes, we have that (169) C5i,?3 - V-y - V-x. If one starts from a given point p and parallely trans ports a vector Xp along a curve X that ends at p again, one will obtain a vector x'p which is in general different from 256 xp ; if one chooses a different curve X', the new vector one obtains at p will in general be different from xp and x'p. This is due to the fact that the covariant derivatives do not generally commute. The Riemann (curvature) tensor gives a measure of this non-commutation. Given C vector fields x, y, and z, a C1*-1 vector field ft(x,y)z is defined by a Cr connection V by (170) R(5c,?)z « Vj(V-z) - V-.(V~z) - Then ft is a Cr“* tensor field of type (3,1). To write this expression in component form, we define the second covariant derivative W z of the vector z as the covariant derivative V(Vz) of Vz; it has components (171) z“ .bc - Cza .b);o. Then equation (170) can be written RabcdxCydzb ° Cza;dyd);cxC " Cza;cxC);dyd (172) - za ;dCyd ;cxc - *d.cyc> " (zft;do - zaiod)xC5,d' where the Riemann tensor components with respect to dual bases (5a )i {gd) are defined by Rabcd - As x, are arbitrary vectors, <173> . « \ d o ' z a !Cd " Rabodzb expresses the non-commutation of second covariant derivatives of z in terms of the Riemann tensor. Choosing the bases as coordinate bases, one finds the expression <174> «abcd ■ -1^ - T-jf + rV*db - rV*cb 257 for the coordinate components of the Riemann tensor, in terms of the coordinate components of the connection. It now turns out that parallel transport of an arbi trary vector along an arbitrary closed curve is locally in- tegrable (i.e., x'p is necessarily the same as xp for each pell) only if Rabcd " 0 at all points of U. In this case we say that the connection is flat. By contracting the curva ture tensor, we can define the Ricci tensor as the tensor of type (0,2) with components Rp^ » Rttbad* It can be shown that there is a coordinate system around p U for which the components of a particular connection van ish iff the curvature tensor vanishes at p (the Implication from left to right follows trivially from the above equation giving the components of the Riemann tensor in terms of the components of the connection; for the other direction see, e.g., Laugwltz (1065), pp. 109-110). For such a coordinate system the geodesic equation becomes (175) d2x°/da2 - 0, so geodesics have the equations (176) xa * aaa + b°, i.e., they are "straight lines". Thus, a region of the mani fold for which the curvature tensor vanishes is called flat, or semi-Euclidean. The introduction of a connection onto a manifold pro vides the manifold with geometrical structure additional to the topological structure which arises from its manifold character. The connection determines the covariant deri- % vative, what is to count as parallel transport, the class of geodesic curves, and the curvature of the manifold. None of these theings can be meaningfully said to be had by a space prior to the endowment of that space with a connection. In like fashion, there are still many interesting properties which can be had by and relationships which can hold between points of a space which cannot be meaningfully attributed to a space possessing only topological and affine structure. For example, though the connection gives us a way to compare the magnitudes of parallel vectors in different tangent spaces, it does not give us a way to measure those magni tudes, nor to compare the magnitudes of non-parallel vectors. Further, there are two ways to Intuitively characterize a geodesic curve; as the "straightest" curve connecting two points, and as the "shortest" curve connecting two points. Without additional geometrical structure placed on a mani fold, only the first can be given full mathematical rigor. This stems, of course, from the fact that we have no way of determining the "length" of a vector or the "distance" be tween two points of a manifold. (We can meaningfully speak of the "distance" between two points on a manifold if those points lie on a single curve. For example, if A(t) is a curve such that X(t0) ** p and X(tQ) ■ q, then we can define the distance between p and q as being |t^ - t0|. This tech nique will not work for points not lying on a single curve, 259 however, and, since different curves may be parameterized differently, this does not give us a definition of "distance" which is consistent over the whole manifold.) The general mathematical definition of these notions requires the intro duction of additional geometrical structure onto the mani fold, which is done by defining on the manifold a tensor field, known as the metric tensor field. 4. The Metric: A metric tensor g at a point peM is a sym metric tensor of type (0,2) at p, so a Cr metric on U is a Cr symmetric tensor field g(p). The metric g at p assigns a "magnitude" (|g(x,x)|)^ to each vector xeTp and defines the "cosine angle" gCS.y) ( |g(x,x)*g(y,?)|)* between any vectors X,yeTp such that g(X,x)mg( 9 ,9 ) / 0; vec tors x, y will be said to be orthogonal if g(5t,?) ■ 0. (Thus, the metric is, in some sense, a generalization of the familiar scalar (or "dot") product of vectors.) The components of g with respect to a basis {8a) are (177) gab - g(Ba ,5b ) - g(&b,Ba ), i.e., the components are simply the scalar products of the basis vectors eQ . If a coordinate basis {a/8xa > is used, then (178) g - gab dxHid*b . The magnitudes, defined by the metric, of vectors in tangent spaces are related to magnitudes (distances) on the 260 manifold by the definition: the path length between points p ■ A(a) and q » A(b) along a C° curve A(t) with tangent vector d/dt is the quantity (179) L - /g( |g(d/dt.d/dt )!)*«. That is, to find the length of a curve between two points we integrate over the tangent vectors to the curve at all points on the curve between the two points in question. Ve may symbolically express the relations (178) and (170) in the form (180) ds2 «* gjjdx*dx*J used to represent the length of the "infinitesimal" arc de termined by the coordinate displacement x1 -*■ x1 + dx1. The metric is said to be non-degenerate at p if there is no non-zero vector xeTp such that g(x,$F) “ 0 for all vec tors yeTp . in terms of components, the metric is non-degen erate if the matrix (gafe) of components of g is non-singular (i.e., if there is a matrix A such that (gajj)A ■ A(g&b) ■ I, where I is the identity, or unit, matrix. A is then uniquely determined and is called the inverse matrix of (ga^)>)< Ve shall from now on always assume the metric tensor is non degenerate. Then we can define a unique symmetric tensor of type (2,0) with components gal> with respect to the basis {ea> dual to the basis {e°}, by the relations (181) eab*bc - «ac . i.e., the matrix (g ) of components is the Inverse of the ab matrix (g&b). It follows that the matrix (g ) is also 261 non-singular, so the tensors gab, gftb can be used to give an isomorphism between any covariant tensor argument and any contravariant argument, or to "raise and lower indices". Thus, if xa are the components of a contravariant vector, then xa are the components of a uniquely associated covar iant vector (one-form), where xa ■ 8abx b > xa " Babxb* sim" ilarly, to a tensor T&b of type (0,2) we can associate unique tensors Tab - ga°Tcb, Tab - Bb°Tao, Tab - gaogbdTcd. We shall in general regard such associated covariant and contravariant tensors as representations of the same geo metrical object (so, in particular, gftb, 6aa and gab may be though of as representations (with respect to dual basds) of the same geometric object g). The signature of g at p is the number of positive eigenvalues of the matrix (ga^) at p, minus the number of negative ones. For example, if g is a metric for a 3-dim ensional manifold whose components, with respect to dual bases, are 1 0 0 (182) (gab) 0 1 0 0 0-1 then the signature of g is (? - 1) « 1. If g is non-degen erate and continuous, the signature will be constant on H. By suitable choice of the basis {&a>, the metric components can at any point p be brought to the form £(n+s)texms i(n-s) terms (183) gab ** diag(+ 1 *+l »•*•»+l»-l»•••»-!)* 262 where s is the signature of g and n is the dimension of II. In this case the basis vectors {e0) form an orthonormal set at p, i.e., each is a unit vector orthogonal to every other basis vector. A metric whose signature is n is called a positive def inite metric: for such a metric, g(x,x) « 0 -*■ X ** 0, and the canonical form is n terms (184) gab - diagC+rrf^T+l) A metric whose signature is (n-2) is called a Lorentz metric; the canonical form is (n-1) terms (185) gab ® diag($-l,.^.,+f,-l). With a Lorentx metric on M, the non-zero vectors at p can be divided into three classes: a vector xeTp being said to be. . , timelike, null. or spacelike according to whether g(x,x) is negative, zero, or positive, respectively. So far, the metric tensor and connection have been in troduced as separate structures on M. However, given a met ric g on M, there is a unique torsion-free connection V on M defined by the condition: the covariant derivative of & is zero, i .e ., (186) V& « 0, or, in component notation, with respect to dual bases, (»’) Babjc " 0> With this connection, parallel transport of vectors preserves 263 scalar products defined by g, so in particular magnitudes of vectors are invariant (coordinate-independent). From (186) it follows that V~(g(?,2S)) - V5g <189> rabc - H3gab/axc + 3gac/9xb - 3gbo/3xa}. The quantity on the right-hand side is usually called the "Christoffel relations". As we saw in our discussion of the manifold, fundamental to differential geometry is the assumption that the spaces with which we will be working will be topologically locally Euclidean. This means that in order to apply the tools of differential (Riemannian) geometry to a space, that space must be locally flat. A space is locally flat Just in case one can, at any point p in the space, construct a coordinate system in which raau vanishes. Given a metric tensor field pt g defined on this space, (189) together with the vanishing of r® (p) implies that, in this coordinate system, g „ (p)** pY op,u 0. The condition Vg B 0 guarantees that the geodesics of the overall manifold (determined by the global connection) will coincide with the straight lines of the Ideally flat re gions of the manifold (determined by the local behavior of g). NOTES CHAPTER I ?-Under the general heading of "Quantum Mechanics" I Include Relativistic Quantum Mechanics, Quantum Electrodynamics, Quantum Chromodynamics, Quantum Flavourdynamics, etc. 2 This term comes from Suppe (1077), and is somewhat midlead ing. The essence of this view constitutes what van Fraas- sen, in his (1980), calls "scientific realism", but, as will be come apparent in my discussion of the variety of positions falling under that general heading, this term is potentially misleading. The term "the Received View" is further misleading in that it Implies that the view it names is almost universally held, which is not quite true. In fact, this whole issue (realism vs. anti-realism) seems to be a very lively one, with each side claiming to be in the minority. 3Suppe (1977), p. 222; (1967), Ch. 2; (1972a); (1972b); (1973). 4Suppes (1967a); (1962); (1967b). 5Sneed (1971). 6van Fraassen (1968); (1970); (1972); (1980). CHAPTER II ^For a discussion of these motivations, see Reichenbach (1951), Chapters 3 and 5. 3Cf. Hempel (1965). 3Cf, Kuhn (1962). 4Cf. Lewis (1973). 5 My description of the Recieved View is taken primarily from Suppe (1977), which provides an excellent overview of the history of Philosophy of Science. 6 The chief bell-ringers were Quine, in (1953), Putnam, in (1962a) and (1962b), and Achinstein, in (1965). 264 265 ^The model I used for giving the logical basis for a scien tific theory comes from Montague (1974), Chapter 11. ®The semantic apparatus I describe comes from a variety of sources, such as Montague (1974), Thomason (1970), Church (1956), and course lectures given by Church, D. Kaplan, D. Kalish, and T. Burge, at UCLA in 1972-73. ®For elaborations of the causal theory of reference, see Kripke (1972), and Putnam (1973). Perhaps the best example of a realist philosopher of science using the cau sal account to solve philosophical puzzles occurs in Friedman (1977) and (1972). *®This theory is, of course, the so-called "RuSsell-Searle" account of reference, which is described and criticized at length in Kripke (.1972). 11Quark search experiments have recently been reviewed by Jones (1977) and Lyons (1980). A recent, very promising experiment is described in LaRue, et al., (1981). *2Craig's result actually assumes a first-order theory, but analogous claims would seem to hold for any formal lan guage sufficient for science. 13Cf. van Fraassen (1980), pp. 6-9. Inversions of this argument occur Earman (1970), Earman (1969), and Earman and Friedman (1973), CHAPTER III *It might be suggested that the problem may be avoided by providing classical mechanics with only a partial seman tics, as was suggested by the old-style Positivists. How ever, this would be to simply abandon the Received View, realistically construed, and adopt a form of instrumental ism. In fact, this is exactly what I think should be done, but it is certainly not a move open to a defneder of the Received View (i.e., modern scientific realism). CHAPTER IV *Thd material in this chapter is drawn from the following sources, Misner, et al., (1973), Hawking and Ellis (1973), Zeldovich and Novikov (1971), Veinberg (1972), Taylor and Wheeler (1966), Einstein::(1952a) and (.1 9 5 2 b), and Feynman (1963). 2In this discussion of the theories of relativity, I try to keep things as non-technical as possible. Wherever 266 mathematical concepts are used, I will refer the reader to those pages in the appendix where the notions are defined and explained. * 3For further discussion of metrics, and in particular the Minkowski metric, see the Appendix, Section 4. 4Cf. Appendix, Section 3. CHAPTER V 3-The sources for the discussion of singularities comes from Hawking and Ellis (1973),' Mlsner, et al. (1973), and es pecially Tipler, et al. (1980). 2Cf. Penrose (1965), Hawking and Penrose (1970), Hawking (1967). 3Cf. Mlsner (1969a), Vheeler (1977). 4This is because the strong nuclear force is finitely large and has an upper bound, while gravity is limited only by the amount of mass present. ®The above discussion of the background microwave radiation and its relevance is drawn from Weinberg (1977) and Tayler (1979). ^Earman (1977), p. 125. 7Misner (1969b), p. 1329. ®For opposing viewpoints, see Ellis (1979) and MacCallum (1979). 9Cited in Clark (1971), p. 204). l°For further discussion of the other universe associated with the Schwarzschild space-time, see Mlsner, et al. (1973), pp. 833-835, and Hawking and Ellis (19737, PP* 153-156. ^ S e e Hawking and Ellis (1973), pp. 156-158, especially pp. 158-159. 12Simpson and Penrose (1973). l^Birrel and Davies (1978), p. 35. 14 G.F,R, Ellis, personal communication. *5See my detailed discussion of this case in Chapter VI, 267 which is taken primarily from Weinberg (1972), Chapter 8. Notice especially equations 8.4.30, and 8.5.1 through 8.5.4. 16 G.F.R. Ellis, personal communication. 17 For a full discussion of these global solutions, see Mis- ner, et al. (1973), Part VI, Hawking and Ellis (1973), pp. 134-149, and Weinberg (1972), Part 5. 18 See appendix, p. 253. 18By J. Earman, over and over again. Cf. Earman and Friedman (1973). CHAPTER VI *That is, the view called, by F. Suppe, "the Received View", which is a precursor of the view I have been calling by that name. ^The following discussion of confirmation strategies is taken, for the most part, from Glymour's discussion in Glymour (1980), Chapter II. Q Of course, these "reduction sentences" did, in a sense, have empirical consequences, so there were constraints operating on the choice of correspondence rules. For example, as Reichenbach pointed out in his (1958), once a geometry (an axiomatic system) is selected, the choice of correspondence rules which serve to attach the mathematical system to the world is constrained by the empirical conse quences of the conjunction of the mathematical system and the correspondence rules. 4 For example, see Friedman (1972), p. 1. g That is, all of the claims are defeasible, or subject to change. 6 That is, through the theory itself is not intended to be literally true, it is used to derive empirical claims about parts of the physical world. 7Misner (1969b). p. 1129. ®Cf. Georgi (1980), and Georgi (1981). 9In Suppes (1967) and (1962), and in UcKinsey, et al. (1953) 10In Sneed (1971). 268 13lbid., Chapter VI. 12 van Fraassen (1980), p. 64. 13The term "iconic model" is taken from Mary Hesse (1966). 14I am Ignoring here the use of statistical methods (e.g., standard deviations) to arrive at these ranges of values. There is some debate over whether such statistical methods lead to accurate estimates of true errors. For example, see Muller's warning in his (1981). 1SThis data is to be found in Weinberg (1972), p. 193. 13van Fraassen (1980), p. 64. 17Cf. Misner, et al. (1973), p. 636. 18Things are in fact a little more complicated that this. For a discussion of some of the complications, see Weinberg (1972), pp. 191-192. 19The description of this model is taken primarily from Wein berg (1972), Chapter 8. 20Cf. Robertson (1962), p. 228. 21Cf. Brans and Dicke (1961), and Dicke (1962). 22See Nicoll, et al. (1980). 23Ellis (1979). CHAPTER VII ^This, of course, is a rational reconstruction of the logic of theory testing, and is not intended to be an accurate description of what experimentalists actually do. o Null "facts" may well be part of an appearance. For exam ple, the fact that a photon trajectory experiences no de flection may be very important in evaluating some theory. Q The equations of motion of any theory of mechanics will yield a parabolic space-time trajectory for the mathemat ical object which represents the ball. Before the ball is released, it is subjected to a variety of forces (e.g., it is carried, lifted, juggled, etc.), and though the ball's history might be represented by some set of models which are "joined" together in some appropriate way, no single model for any theory of mechanics can represent that his tory. 269 4Ellis, G.F.R., personal communication. ^Hawking and Ellis (1973), pp. 363-364. 6Recall that appearances often contain best guesses and es timates, as well as the results of measurement and obser vation. (See my discussion of appearances on pp. 188- 193.) If relativists do not believe that physical singu larities exist, then no appearance used in evaluating the GTR will contain physical singularities. On the other hand, since virtually all astrophysicists believe that the universe, on a global scale, is homogeneous, the appear ances against which we test our global models contain homogeneous distributions of mass-energy. Again, theories are tested against appearances, but appearances represent our best, or at least our current, view of the world. 7Tipler, et al. (1980), p. 180. ®Recall the discussion in Chapter V, pp. 94-96. Cf. Misner (1969b). ®See note 6, above. ^Tay lor and Wheeler (1963), pp. 9-10. **Cf. Weinberg, (1972), Chapters 9 and 10, and Uisner, et al. (1973), Chapter 39. 12Though I specifically discuss the hypothetico-deductive view of theory confirmation, the points I make apply as well to other confirmation strategies, such as Glymour's "bootstrapping" account. ■*‘2In Glymour (1980). 14In Gell-Mann (1964). BIBLIOGRAPHY Achinstein, P., American Philosophical Quarterly 2, 193 (1965). Birrell, N.D. and Davies, P.C.W., Nature 272, 35 (1978). Brans, C.H. and Dicke, R.H., Phys. Rev. 124, 925 (1961). Church, A., Introduction to Mathematical Logic (Princeton University Press, Princeton, 1956). Clark, R.W., Einstein: The Life and Times (Vorld Publishing Co., New York, 1971). Dicke, R.H., Phys. 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