1. 1 [2] “On the Logical Positivists' Theory of Truth,” Analysis 2 (1935), 49–59.

2. 2 [6] “Some Remarks on ‘Facts’ and Propositions,” Analysis 2 (1935), 93–6.

3. 3 [7] “Some Remarks on Empiricism,” Analysis 3 (1936), 33–40.

4. 4 [12] “Le problème de la vérité,” Theoria 3 (1937), 206–46. Translated here by the author as “The Problem of Truth.”

5. 5 [107] “The Signification of the Concept of Truth for the Critical Appraisal of Scientific Theories,” Nuova Civilità delle Macchine 8 (1990), 109–13. Here, “signification” is a misprint for “significance,” which is itself a wateringdown of “irrelevance” in the original typescript. This was reprinted under the title “The Significance of the Concept of Truth for the Critical Appraisal of Scientific Theories” (but it was listed in the table of contents as “Evidence and Truth in Scientific Inquiry”) in William R. Shea and Antonio Spadafora, eds., Interpreting the World (Canton, MA: Science History Publications, 1992), pp. 121–9. Here, I follow the original typescript.

Hempel's defense of Neurath's and Carnap's physicalism in essays 1–3 testifies to the presence of certain “postmodern” themes in logical empiricism (= logical positivism):

Introduction: localization and Lorentz-invariant Quantum Theory

The current status of localization and related concepts, especially localized statevectors and position operators, within Lorentz-invariant Quantum Theory (LIQT) is ambiguous and controversial. Ever since the early work of Newton & Wigner (1949), and the subsequent extensions of their work, particularly by Hegerfeldt (1974, 1985), it has seemed impossible to identify localized statevectors or position operators in LIQT that were not counterintuitive – strange – in one way or another; the most striking strange property being the superluminal propagation of the localized states.

The ambiguous and controversial status of these concepts arises from the varied reactions that workers have to the strange properties. Some regard them, particularly the superluminal propagation, as utterly unacceptable, and conclude that no precise concepts of localized statevectors and position operators exist in LIQT (Wigner 1973, pp. 325–7; 1983, pp. 310–13; Malament 1996). Others downplay the whole issue, on the grounds that current theoretical and experimental practice has no need of sharply formulated concepts of localization, localized states etc. (Birrell & Davies 1984, pp. 48–59; Haag 1992, p. 34).

Still others, including ourselves, believe that the superluminal propagation does not lead to causal contradictions, and is not in conflict with available empirical data: so that it should not be ruled out. We also believe that the other strange properties are merely unfamiliar novelties of LIQT which we must simply learn to accept. (Like the superluminal propagation, they do not lead to causal contradictions, nor conflict with available data.) So in this essay, we aim to help remove the ambiguous and controversial status of localization concepts in LIQT.

Introduction

The title of my essay, ‘Can the fundamental laws of nature be the results of evolution?’, may suggest to some people that I shall discuss the history of science. This is not my intention. We have attained our present state of knowledge (presumably not the final state) of fundamental laws by an intricate historical process, and it is interesting to inquire how accurately, and with what reservations, the term ‘evolution’ in the Darwinian sense applies to the historical process of scientific development. (Tangentially, I remark that I have strong reservations about characterizing the development of the natural sciences as ‘evolutionary’, since that development is driven by a final cause – to find out the truth about nature – while the elimination of teleology is central to Darwinism.) My intention, however, is to discuss the laws themselves, as matters of fact concerning nature, rather than the knowledge of these laws by human beings. I wish to honour Michael Redhead by sharing his priorities: giving precedence to questions about the constitution of the world over questions about human knowledge.

But many people are baffled by this clarification. They willingly admit that secondary or derivative laws of nature may be the products of evolution, but cannot see how the basic laws themselves can evolve. Thus, the genetic code is certainly a law of biology, but it is presumably the result of billions of years of trial and error on the part of interacting molecular species; such, at any rate, is the thesis of prebiotic evolution.

Introduction

It is a pleasure to be able to contribute to this volume in honour of Professor Michael Redhead. In the foundations of physics community, his influence has been felt both through his Incompleteness, Nonlocality and Realism (1987), which has largely denned the terms of discourse in this field for the last decade, and by the new ‘Cambridge School’ of the philosophy of physics that he established there during his Professorship in the History and Philosophy of Science Department. Michael Redhead's work illustrates well what has always seemed to be a central tenet of his: that philosophers of science should use the conceptual background and the mathematical formalism of modern, successful scientific theories – especially quantum mechanics and quantum field theory – as a guide to understanding the ontology of the world (Redhead 1983, 1990) and to reexamining traditional philosophical questions (French & Redhead 1988; Redhead & Teller 1991, 1992). In his Tarner Lectures (1995) he used this type of work to anchor a general position that can be characterized as structural realism, which holds that the structures – mathematical formalism, models, analogies – of a successful scientific theory eventually define the real, even as regards central, unobservable entities.

While Professor Redhead has done an outstanding job of pressing forward with this general programme, it remains arguable that he and some of his co-workers, who have, in the main, stayed within the framework of standard quantum theory and relativity, may have effectively bracketed a different and potentially fruitful alternative theory.

Introduction

Michael Redhead began his Tarner Lectures by allowing that ‘many physicists would dismiss the sort of question that philosophers of physics tackle as irrelevant to what they see themselves as doing’ (1995, p. 1). He argued that, on the contrary, philosophy has much to offer physics: presenting examples and arguments from many parts of physics and philosophy, he led his audience towards his ultimate conclusion that physics and metaphysics enjoy a symbiotic relationship.

By way of tribute to Michael we would like to undertake a related project: convincing philosophers of physics themselves that the philosophy of space and time has something to offer contemporary physics. We are going to discuss the relationship between the interpretative problems of quantum gravity, and those of general relativity. We will argue that classical and quantum theories of gravity resuscitate venerable philosophical questions about the nature of space, time, and change; and that the resolution of some of the difficulties facing physicists working on quantum theories of gravity appears to require philosophical as well as scientific creativity. These problems have received little attention from philosophers. Indeed, scant attention has been paid to recent attempts to quantize gravity. As a result, most philosophers have been unaware of the problem of time in quantum gravity, and its relationship to the knot of philosophical and technical problems surrounding the general covariance of general relativity – so that it has been all too easy to dismiss this latter set of problems as philosophical contrivances. Consequently, philosophical discussion of space and time has suffered.

This point is best illustrated by attending to the contrast between what philosophers and physicists have to say about the significance of Einstein's hole argument.

Most of the essays in this collection were presented as papers at a two-day conference, held in the History and Philosophy of Science Department, Cambridge University, in June 1997. The editors are very grateful to the Department, to the British Society for the Philosophy of Science, and to the Mind Association for their generous financial support of the conference.

The conference was held on the occasion of Michael Redhead's retirement from the Professorship of the History and Philosophy of Science in the University of Cambridge; and the authors and editors are delighted to honour him with this volume. A bibliography of his writings is included.

Introduction

The relationship between mathematics and science is clearly of fundamental concern in both the philosophy of mathematics and the philosophy of science. How this relationship should be represented is a crucial issue in this area. One possibility is to employ a model-theoretic framework in which ‘physical’ structures are regarded as embedded in ‘mathematical’ ones. In section 2 I will briefly outline a form of this type of account which offers a function space analysis of theories (Redhead 1975). This function space analysis is then used to represent the relationship between theoretical and mathematical structures. In subsequent sections I will consider the role of group theory in physics from within this meta-theoretical framework and then draw some conclusions for realism in the philosophy of science.

Function spaces and the model-theoretic approach

According to Redhead, it is an ‘empirical-historical fact’ that theories in physics can be represented as mathematical structures (Redhead 1975). This then allows the possibility of representing the relation of mathematics to physics in terms of embedding a theory T in a mathematical structure M′, in the usual set-theoretic sense of there existing an isomorphism between T and a sub-structure M of M′. M′ is then taken to be a non-simple conservative extension of M. There is an immediate question regarding the nature of T. To be embedded in M′ it must already be ‘mathematized’ in some form or other. Thus, the issue here is not so much Wigner's inexplicable utility of mathematics in science, in the sense of its being the indispensable language in which theories are expressed, but rather the way in which new theoretical structure can be generated via this embedding of a theory, which is already mathematized, into a mathematical structure.

Within the foundations of physics in recent years, Bell's theorem has played the role of what Thomas Kuhn calls a ‘paradigm’: that is, an exemplary piece of work that others learn from, imitate and develop. Following a period of articulation and consolidation, the first generation of developments of the Bell theorem was initiated by Heywood and Redhead (1983). They produced a nonlocality result in the algebraic style of the Bell–Kochen–Specker theorem (Bell 1966; Kochen and Specker 1967), moving away from the probabilistic relations characteristic of the Bell theorems proper. More recently a second generation develops results by Peres (1990), Greenberger–Horne–Zeilinger (1990), and Hardy (1993). In addition to moving away from probabilities, this generation tries to dispense with the limiting inequalities of the Bell theorem to yield socalled ‘Bell theorems without inequalities’. With respect to probabilities, however, Hardy is a half-way house. It requires no inequalities but the result contradicts quantum mechanics under certain locality assumptions only if the statistical predictions of quantum mechanics hold in at least one case.

I want to examine the Hardy theorem and its interpretation. Initially, I intend to ignore respects in which it dispenses with probabilities because I want to point out the interesting significance of the theorem in a probabilistic context. We will see that when probabilities are restored, so are inequalities. Then we will see what the theorem has to contribute on the topic of locality.

Physics is a vast discipline, rich with topics inviting philosophical analysis. In undertaking that analysis, the philosophy of physics can be pursued in different styles. There is a spectrum: ranging from the ‘theorem-proof’ style as used in expounding a piece of mathematical physics (though here the theorems will be motivated by philosophical ideas), through philosophical-cum-physical argument that adverts in detail to pieces of physics (e.g. in displayed equations), to purely philosophical argument that is in some way based on, or influenced by, physics. For the most part, the essays in this collection are examples of the second of these three styles: they aim to combine detailed presentation of some technicalities in theoretical physics with the dialectical rigour of analytic philosophy.

The essays also have, with one exception, a common subject-matter: the philosophy, or if you prefer foundations, of quantum theory. The first six essays fall squarely in that subject-matter. The seventh and eighth broaden the discussion: they address, respectively, philosophical aspects of (i) the search for a theory of quantum gravity, and (ii) quantum theory's use of group theory. The final essay, by Abner Shimony, is much more speculative: it assesses the conjecture that the fundamental laws of nature are products of evolution.

Arthur Fine's essay starts the volume, on a central topic in philosophy of quantum theory: proofs of quantum nonlocality. Fine analyses a proof by Lucien Hardy, which as Fine says, is a characteristic example of a new generation of proofs that apparently make less use of probabilities and thereby dispense with inequalities. However, Fine shows that this appearance is misleading.

In the North Atlantic countries that have been the main stage of our story, the nineteenth century was a time of enormous production, not only of manufactured goods but also of art and literature, science, and philosophy. From the great wealth of new ideas in nineteenth-century mathematical physics we can consider only a very small part, chosen mainly for their impact on twentieth-century physics and philosophy. I begin with the new geometries (§4.1), whose emergence is often treated as a chapter in the history of mathematics and its philosophy but which in fact attracted some of the mathematicians who developed them – notably Riemann – chiefly for their potential significance for physics (which Einstein subsequently made good in a wholly unexpected way). The next two sections deal with the concept of field, especially in electrodynamics (§4.2), and with the introduction of chance into thermal physics (§4.3). Finally, we shall take a glance at some nineteenth-century philosophies, which set the tone of twentiethcentury debates (§4.4)

Geometries

Euclid's Fifth Postulate and Lobachevskian Geometry

I can still recall my frustration when, early in my first term of high school geometry, the teacher “proved” – such was his word – Euclid's Theorem 1.29 “by parallel transport”, in effect by sliding a wooden square along a steel ruler pressed on the blackboard.