Friedman and Some of his Critics on the Foundations of General Relativity

This paper is an examination of Michael Friedman’s analysis of the conceptual structure of Einstein’s theory of gravitation, with a particular focus on a number of critical reactions to it. Friedman argues that conceptual frameworks in physics are stratified, and that a satisfactory analysis of a framework requires us to recognize the differences in epistemological character of its components. He distinguishes first-level principles that define a framework of empirical investigation from second-level principles that are formulable in that framework. On his account, the theory of Riemannian manifolds and the equivalence principle define the framework of empirical investigation in which Einstein’s field equations are an intellectual and empirical possibility. Friedman is a major interpreter of relativity and his view has provoked a number of critical reactions, nearly all of which miss the mark. I aim to free Friedman’s analysis of Einsteinian gravitation from a baggage of misconceptions and to defend the notion that physical theories are stratified. But I, too, am a critic and I criticize Friedman’s view on several counts, notably his account of a constitutive principle and that of the principle of equivalence

Newtonian Mechanics and its Philosophical Significance

Newtonian mechanics is more than just an empirically successful theory of matter in motion: it is an account of what knowledge of the physical world should look like. But what is this account? What is distinctive about it? To answer these questions, I begin by introducing the laws of motion, the relations among them, and the spatio-temporal framework that is implicit in them. Then I turn to the question of their methodological character. This has been the locus of philosophical discussion from Newton’s time to the present, and I survey the views of some of the major contributors. I show that while Newtonian mechanics motivates a number of philosophical ideas about force, mass, motion, and causality – and through this, ideas about space and time – the laws are themselves the outcome of a philosophical or critical conceptual analysis

Friedman and Some of his Critics on the Foundations of General Relativity

This paper is an examination of Michael Friedman’s analysis of the conceptual structure of Einstein’s theory of gravitation, with a particular focus on a number of critical reactions to it. Friedman argues that conceptual frameworks in physics are stratified, and that a satisfactory analysis of a framework requires us to recognize the differences in epistemological character of its components. He distinguishes first-level principles that define a framework of empirical investigation from second-level principles that are formulable in that framework. On his account, the theory of Riemannian manifolds and the equivalence principle define the framework of empirical investigation in which Einstein’s field equations are an intellectual and empirical possibility. Friedman is a major interpreter of relativity and his view has provoked a number of critical reactions, nearly all of which miss the mark. I aim to free Friedman’s analysis of Einsteinian gravitation from a baggage of misconceptions and to defend the notion that physical theories are stratified. But I, too, am a critic and I criticize Friedman’s view on several counts, notably his account of a constitutive principle and that of the principle of equivalence

Some disputed aspects of inertia, with particular reference to the equivalence principle

This thesis is a contribution to the foundations of space-time theories. It examines the proper understanding of the Newtonian and 1905 inertial frame concepts and the critical analysis of these concepts that was motivated by the equivalence principle. This is the hypothesis that it is impossible to distinguish locally between a homogeneous gravitational field and a uniformly accelerated frame. The three essays that comprise this thesis address, in one way or another, the criteria through which the inertial frame concepts are articulated. They address the place of these concepts in the conceptual framework of physics and their significance for our understanding of space and time. In Chapter 2, I examine two claims that arise in Brown’s (2005) account of inertia. Brown claims there is something objectionable about the way in which the motions of free particles in Newtonian theory and special relativity are coordinated. Brown also claims that since a geodesic principle can be derived in Einsteinian gravitation the objectionable feature is explained away. I argue that there is nothing objectionable about inertia and that, while the theorems that motivate Brown’s claim can be said to figure in a deductive-nomological explanation, their main contribution lies in their explication rather than their explanation of inertial motion. In Chapter 3, I examine Friedman’s recent approach to the analysis of physical theories (2001; 2010a; 2010b; 2011). Friedman argues that the identification of certain principles as ‘constitutive’ is essential to the correct methodological analysis of physics. I explicate Friedman’s characterisation of a constitutive principle and his account of the constitutive principles that Newtonian and Einsteinian gravitation presuppose for their formulation. I argue that something very close to Friedman’s view is defensible. In Chapter 4, I examine the so-called background-independence that Einsteinian gravitation is said to exemplify. This concept has figured in the work of Rovelli (2001; 2004), Smolin (2006), Giulini (2007), and Belot (2011), among others. I propose three ways of fixing the extension of background-independence, and I argue that there is something chimaerical about the concept. I argue, however, that there is a proposal that clarifies the feature of Einsteinian gravitation that motivates the concept

Computing, Modelling, and Scientific Practice: Foundational Analyses and Limitations

This dissertation examines aspects of the interplay between computing and scientific practice. The appropriate foundational framework for such an endeavour is rather real computability than the classical computability theory. This is so because physical sciences, engineering, and applied mathematics mostly employ functions defined in continuous domains. But, contrary to the case of computation over natural numbers, there is no universally accepted framework for real computation; rather, there are two incompatible approaches --computable analysis and BSS model--, both claiming to formalise algorithmic computation and to offer foundations for scientific computing. The dissertation consists of three parts. In the first part, we examine what notion of 'algorithmic computation' underlies each approach and how it is respectively formalised. It is argued that the very existence of the two rival frameworks indicates that 'algorithm' is not one unique concept in mathematics, but it is used in more than one way. We test this hypothesis for consistency with mathematical practice as well as with key foundational works that aim to define the term. As a result, new connections between certain subfields of mathematics and computer science are drawn, and a distinction between 'algorithms' and 'effective procedures' is proposed. In the second part, we focus on the second goal of the two rival approaches to real computation; namely, to provide foundations for scientific computing. We examine both frameworks in detail, what idealisations they employ, and how they relate to floating-point arithmetic systems used in real computers. We explore limitations and advantages of both frameworks, and answer questions about which one is preferable for computational modelling and which one for addressing general computability issues. In the third part, analog computing and its relation to analogue (physical) modelling in science are investigated. Based on some paradigmatic cases of the former, a certain view about the nature of computation is defended, and the indispensable role of representation in it is emphasized and accounted for. We also propose a novel account of the distinction between analog and digital computation and, based on it, we compare analog computational modelling to physical modelling. It is concluded that the two practices, despite their apparent similarities, are orthogonal

Leaving mathematics as it is: Wittgenstein’s later philosophy of mathematics

Wittgenstein’s later philosophy of mathematics has been widely interpreted to involve Wittgenstein’s making dogmatic requirements of what can and cannot be mathematics, as well as involving Wittgenstein dismissing whole areas (e.g. set theory) as not legitimate mathematics. Given that Wittgenstein promised to ‘leave mathematics as it is’, Wittgenstein is left looking either hypocritical or confused. This thesis will argue that Wittgenstein can be read as true to his promise to ‘leave mathematics as it is’ and that Wittgenstein can be seen to present coherent, careful and non-dogmatic treatments of philosophical problems in relation to mathematics. If Wittgenstein’s conception of philosophy is understood in sufficient detail, then it is possible to lift the appearance of confusion and contradiction in his work on mathematics. Whilst apparently dogmatic and sweeping claims figure in Wittgenstein’s writing, they figure only as pictures to be compared against language-use and not as definitive accounts (which would claim exclusive right to correctness). Wittgenstein emphasises the importance of the applications of mathematics and he feels that our inclination to overlook the connections of mathematics with its applications is a key source of a number of philosophical problems in relation to mathematics. Wittgenstein does not emphasise applications to the exclusion of all else or insist that nothing is mathematics unless it has direct applications. Wittgenstein does question the alleged importance of certain non-applied mathematical systems such as set theory and the logicist systems of Frege and Russell. But his criticism is confined to the aspirations towards philosophical insight that has been attributed to those systems. This is consonant with Wittgenstein’s promises in (PI, §124) to ‘leave mathematics as it is’ and to see ‘leading problems of mathematical logic’ as ‘mathematical problems like any other.’ It is the aim of this thesis to see precisely what Wittgenstein means by these promises and how he goes about keeping them