This book, explores the conceptual foundations of Einstein's theory of relativity: the fascinating, yet tangled, web of philosophical, mathematical, and physical ideas that is the source of the theory's enduring philosophical interest.
Originally published in 1983.
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Foundations of Space-Time Theories
Relativistic Physics and Philosophy of Science
By Michael Friedman
PRINCETON UNIVERSITY PRESSCopyright © 1979 Princeton University Press
All rights reserved.
Introduction: Relativity Theory and Logical Positivism
The relationship between the development of relativity theory and twentieth-century philosophy of science is both fascinating and complex. On the one hand, relativity theory, perhaps more so than any other scientific theory, developed against a background of explicitly philosophical motivations. As is well known, both Leibnizean relationalism and Machian empiricism figured prominently in Einstein's thought. On the other hand, twentieth-century philosophy of science, and logical positivism in particular, is almost inconceivable without relativity, for relativity theory was second only to Principia Mathematica as an intellectual model for the positivists. It appeared to realize all their most characteristic ideals, from a general distrust of non-observational, "metaphysical" notions (such as absolute space and absolute simultaneity) to a specific program for dividing science into "factual" statements on the one side and "definitions" or "conventions" on the other. In short, Einstein himself was influenced by positivist and empiricist philosophy, and the logical positivists used both this fact and the theories that resulted as central sources of support for their philosophical views.
This book is an attempt to sort out some of the many different elements in this complicated relationship of mutual influence. To what extent does relativity theory actually realize Einstein's philosophical motivations? And, consequently, to what extent does relativity support the philosophical ideals of logical positivism? My answer is that both Einstein and the positivists were wrong. The theories Einstein created do not vindicate his Leibnizean-Machian motivations. Relativity theory neither supports nor embodies a general positivistic point of view. But this conclusion is of only secondary interest to me. The moral I would most like to convey is this: one cannot hope to understand adequately either the development of relativity theory or twentieth-century positivism without a detailed account of their reciprocal interaction. In particular, we cannot hope to advance beyond the positivists if we ignore the roots of their doctrines in the scientific practice of their time. Logical positivism's intimate involvement with twentieth-century physics is one of the main sources of its considerable force and vitality. Any alternative philosophy of science must come to terms with this physics in an equally vital way.
It seems to me that contemporary criticisms of logical positivism suffer because they ignore this moral, and this is especially true of popular attacks on the observational/theoretical distinction. That is not to say that these attacks are unwarranted. The positivists' use of the observational/theoretical distinction was often naive: they assumed to be clear and unproblematic a distinction that is at best both vague and highly complicated; what is worse, they appeared to be unaware of the many ways in which changes in theory can alter what counts as observational and non-observational (in popular parlance, they ignored the "theory-ladenness" of observation). The trouble is that all this is much too easy to say. We know that the observational/theoretical distinction is shaky and problematic, but we want to know more about its role in the positivists' general epistemological program and in the revolutionary physical theories that inspired that program. Criticisms of positivism that avoid these matters are liable to appear superficial and unsatisfying. We can throw away the observational/theoretical distinction if we like, but we will have no idea how to construct a better epistemology or a better account of twentieth-century physics.
If we look at the actual development of relativity theory, the observational/theoretical distinction appears to play a central role. It is a key element, in fact, in Einstein's arguments for both the special principle of relativity and the general principle of relativity. In outline, these arguments go as follows. Classical physics makes use of absolute motion, that is, motion of physical bodies with respect to absolute space. But only relative motion — motion of physical bodies with respect to other physical bodies — is observable. Therefore, appeals to absolute motion should be eliminated from physical theory. Thus, the special principle of relativity allows us to dispense with the notions of absolute velocity and absolute rest. All systems moving with constant velocity are "equivalent," and velocity makes sense only relative to one or another such system. Accordingly, Einstein criticized classical electrodynamics for invoking something more than relative velocity:
The observable phenomenon here depends only on the relative motion of the conductor and the magnet, whereas the customary view draws a sharp distinction between the two cases in which either the one or the other of the two bodies is in motion. (, 37)
On the other hand, the special principle of relativity does not go far enough, because special relativity still makes use of an absolute distinction between inertial systems (systems moving with constant velocity) and accelerated systems. Absolute velocity is indeed eliminated, but absolute acceleration is retained. Einstein again criticized this distinction by appealing to what is observable:
But the privileged space R1 of Galileo [= an inertial frame], thus introduced, is a merely factitious cause, and not a thing that can be observed. (, 113)
The general theory of relativity is supposed to overcome this defect in the special principle by dispensing with the distinction between inertial systems and accelerated systems, just as special relativity dispenses with the distinction between systems at rest and systems moving with (nonzero) constant velocity. According to the general principle of relativity, all states of motion are "equivalent."
In short, the development of relativity theory appears to have the following structure:
This development appears to illustrate the fruitfulness of both the observational/theoretical distinction and the positivists' general distrust of nonobservational entities and properties. It is no wonder, then, that the positivists were so attracted to the picture expressed in the following passage from Reichenbach:
The physicist who wanted to understand the Michelson experiment had to commit himself to a philosophy for which the meaning of a statement is reducible to its verifiability, that is, he had to adopt the verifiability theory of meaning if he wanted to escape a maze of ambiguous questions and gratuitous complications. It is this positivist, or let me rather say, empiricist commitment which determines the philosophical position of Einstein. It was not necessary for him to elaborate on it to any great extent; he merely had to join a trend of development characterized, within the generation of physicists before him, by such names as Kirchhoff, Hertz, Mach, and to carry through to its ultimate consequences a philosophical evolution documented at earlier stages in such principles as Occam's razor and Leibniz's identity of indiscernibles. (, 290–291)
Similarly, we can also understand why the positivists were attracted to conventionalism. The essence of conventionalism is a doctrine of "equivalent descriptions": alternative, seemingly incompatible theoretical descriptions are declared to be equivalent when they agree on all observations. Consequently, theoretical assertions are not "objectively" true or false: they have truth-value only relative to one or another arbitrarily chosen "equivalent description." This doctrine also appears to be borne out by relativity theory, where alternative ascriptions of motion are not "absolutely" true or false but are true or false only relative to one or another arbitrarily chosen reference system. Moreover, all reference systems are "equivalent"; they provide equally good descriptions of the same observable facts.
This way of looking at the story is still relatively superficial, however. It is not just that relativity theory provides a model or exemplar for the empiricist and verificationist prejudices of twentieth-century positivism; rather, the historical development of the former plays an integral role in the historical development of the latter. Logical positivism did not start out as a version of empiricism or verificationism à la Hume or Mach. Contrary to popular wisdom, the influence of Kant was much more important. Twentieth-century positivism began as a neo-Kantian movement whose central preoccupation was not the observational/theoretical distinction but the form/content distinction. Without arguing the point here, I believe that this theme comes through clearly in Carnap's Aufbau (1928), in Schlick's General Theory of Knowledge (1918) and Space and Time in Contemporary Physics (1917), and, most explicitly, in Reichenbach's The Theory of Relativity and A Priori Knowledge (1920). Central to all these works is the idea that natural knowledge has two different elements: a form, which is in some sense conceptual or mind-dependent, and a content, which is contributed by the world or experience. Moreover, all these works side with Kant against empiricism in emphasizing the necessity and importance of such formal elements in our knowledge of nature. Where Kant went wrong was in characterizing these formal elements as unrevisable, synthetic a priori truths; they are better described as "conventions" or "arbitrary definitions."
Kant's version of the form/content distinction drew much of its strength from eighteenth-century mathematical physics and in particular from the Euclidean-Newtonian picture of a space-time framework or "container" filled with matter (content) obeying deterministic laws of motion. Here, according to the early positivists, the Kantian view is limited by its too intimate connection with outmoded mathematics and physics. The task of twentieth-century philosophy is to develop a version of the Kantian form/content distinction that is properly sensitive to developments in modern mathematics and modern physics. A brief review of these developments will give us a preliminary sense of the issues and complexities involved in the relationship between positivism and relativity theory. We shall see also how the form/content distinction actually becomes transformed into the theoretical/observational distinction.
The most important mathematical development from this point of view is an extreme generalization of the concepts of space and geometry. This process begins with Gauss's theory of surfaces of 1827. But here already there is a twofold generalization that is the source of much later confusion: a generalization in the use of coordinates, and a generalization to non-Euclidean geometries. First, Gauss considers the use of general (Gaussian) coordinates x1, x2 in place of the familiar rectangular or Cartesian coordinates. In Cartesian coordinates the "line element" takes the familiar Pythagorean form
ds2 = dx21 + dx22. (1)
In arbitrary coordinates it takes the more complicated form
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
where the gij's are real-valued functions of the coordinates. (For example, in polar coordinates ds2 = dr2 + r2 dθ2; so g11 = 1, g12 = g21 = 0, and g22 = r2.) Secondly, however, Gauss also considers the geometry of arbitrary curved surfaces (for example, the surface of a sphere or of a hyperboloid of revolution). He shows that the geometry on such a surface is in general non-Euclidean. For example, the geometry on the surface of a sphere is a geometry of positive curvature in which there are no parallels; the geometry on a hyperboloid of revolution is a geometry of negative curvature in which there are many parallels.
These non-Euclidean surfaces can also be covered by Gaussian coordinates x1, x2, and they also have a line element given by (2). Moreover, as Gauss shows, the coefficients of the line element in (2) — the gij's — give us complete information about the curvature, and hence the geometry, of our surface. It is extremely important, however, to distinguish a mere change in coordinates from an actual change in geometry. In particular, the more complicated form (2) of the line element does not necessarily signal a non-Euclidean geometry: we can use non-Cartesian coordinates on a flat, Euclidean plane as well (as our example of polar coordinates clearly shows). What distinguishes a flat, Euclidean geometry is the existence of coordinates satisfying (1): on a Euclidean plane we can always transform the more complicated form (2) into the Pythagorean form (1) (for example, by transforming polar coordinates into Cartesian coordinates). By contrast, on a curved, non-Euclidean surface it is impossible to perform such a transformation: Cartesian coordinates simply do not exist.
We should distinguish, then, between intrinsic and extrinsic features of a surface. Intrinsic features characterize the geometrical structure of a surface — its curvature, Euclidean or non-Euclidean character, and so on — and are completely independent of any particular coordinatization of the surface. Extrinsic features, on the other hand, correspond to particular coordinatizations of the surface; accordingly, they vary as we change from one coordinate system to another. Thus, the actual form of the line element (2) — obtained by substituting particular functions for the gij's — is an extrinsic feature of the surface, for these functions change when we transform our coordinate system. The connection between intrinsic features and extrinsic features is this: a given intrinsic feature of a surface corresponds to the existence of coordinate systems with certain extrinsic features. For example, a flat, Euclidean structure corresponds to the existence of Cartesian coordinates in which g11g22 = 1, g12 = g21 = 0; it corresponds to the simple form (1) for the line element.
In the remainder of the nineteenth century these ideas underwent further generalization. The high point of this process was Riemann's Inaugural Address of 1854. Riemann considers arbitrary spaces or manifolds of any number of dimensions: points in such manifolds can be uniquely represented by n-tuples of real numbers. Given a particular coordinatization x1, x2, ..., xn of an arbitrary n-dimensional manifold, we can define an n-dimensional line element or metric tensor by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
subject to the conditions of symmetry (gij = gji) and positive-definiteness (ds2 > 0). (If we drop the positive-definiteness requirement, we obtain a semi-Riemannian metric, which is important in relativity.) Riemann shows how to define the notion of curvature for such a manifold and shows that the special case of a flat or Euclidean manifold is characterized by the existence of coordinates in which the matrix of coefficients (gij) in (3) takes the form diag(1,1, ..., 1): that is, gij = 1 for i = j, gij = 0 for i ≠ j. (If our metric is semi-Riemannian and flat, this matrix of coefficients can be put in the form (gij) = diag ([+ or -] 1, [+ or -] 1, ..., [+ or -]1).)
Once again, however, we must distinguish changes in coordinates from changes in geometrical structure, extrinsic from intrinsic. A mere change in coordinates from x1, x2, ... xn to [bar.x]1, [bar.x]2, ..., [bar.x]n results in a change in the coefficients in (3) from gij to [bar.g]ij, but, of course, the line element itself is preserved:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Excerpted from Foundations of Space-Time Theories by Michael Friedman. Copyright © 1979 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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Table of Contents
- FrontMatter, pg. i
- Contents, pg. vii
- Preface, pg. xi
- I. Introduction: Relativity Theory and Logical Positivism, pg. 1
- II. Space-Time Theories, pg. 32
- III. Newtonian Physics, pg. 71
- IV. Special Relativity, pg. 125
- V. General Relativity, pg. 177
- VI. Relationalism, pg. 216
- VII. Conventionalism, pg. 264
- Appendix: Differential Geometry, pg. 340
- Bibliography, pg. 368
- Index, pg. 377