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This thesis seeks to assess and develop the use of formal methods in philosophy of science. More specifically, I argue that the particular strand of philosophy of science that concerns itself with the formal structure of scientific knowledge has focused excessively on the structure of scientific theories. In response, I consider the prospects for the formal study of other, supratheoretical aspects of science, culminating in the proposal for a research programme centred around the formal study of, what I shall call, scientific knowledge-systems. This line of argument is laid out over the course of two parts, along with an extended introduction. In my extended introduction, I will situate myself with respect to the wider philosophy of science. This is necessary, because the body of work I seek to build on in this thesis is emblematic of a style of philosophy of science, referred to here as the architectonic style, which today has largely fallen out of favour. Accordingly, I will first offer some general arguments for the desirability and viability of this style of metascientific enquiry. In Part I of the thesis, I offer a critical appraisal of extant work on the formal structure of scientific knowledge, referred to there as architectonic metascience. Following an extensive survey of architectonic metascience (Chapter 1), I argue that it suffers from the problem of theory-centrism (Chapter 2). The upshot of this observation, I argue, is that frameworks for the formal study of scientific knowledge should adopt a supra-theoretical unit of analysis (or macro-unit, for short). I conclude my appraisal by discussing the few extant-but seldom acknowledged-examples of the formal study of macro-units (Chapter 3). Finally, in Part II, I seek to make a contribution to the formal study of macro-units. A recurring theme throughout the history of analytic philosophy is the idea that we may draw on the metatheoretical study of logic to inform the metatheoretical study of science. In line with this tradition, I first survey the movement of logical abstractivism, within which we find various frameworks for the systematic study of different systems of logic (Chapter 4). Following this is an intermezzo which presents an existing application of logical abstractivism to philosophy of science (Chapter 5), and a brief discussion on the explication of target systems in formal analyses of science (Chapter 6). Building on these reflections, I set out the programme of metascientific abstractivism for the study of scientific knowledge-systems (Chapter 7).

This thesis is about conceptual aspects of gauge theories. Gauge theories lie at the heart of modern physics: in particular, they constitute the standard model of particle physics. At its simplest, the idea of gauge is that nature is best described using a descriptively redundant language; the different descriptions are said to be related by a gauge symmetry. The over-arching question the thesis aims to answer is: how can descriptive redundancy be fruitful for physics? This question embraces many important topics in the philosophical literature on gauge theory, which I will address. This thesis has two main Parts. Part I provides technical and conceptual background. It relates the redundancies of gauge theories with the redundancies in the foundations of spacetime physics. In particular, to those of Einstein's theory of general relativity, that are more familar to the average philosopher. This Part provides a perspicuous, geometrical understanding of the physics of gauge theory, on a par with the chronogeometric understanding of general relativity. In Part II I will assess two surprising uses of and one contentious question about gauge symmetry. First, I will provide one answer to the question: "Why gauge theory?", that is: why introduce redundancies in our models of nature in the first place? This type of answer is pragmatic: because such redundancies are useful for model-building, in a particular way; and they allow us to focus our mathematical apparatus on different aspects of the same phenomena. Second, I present a choice of gauge that is related to a physically natural, and general, splitting of the electric field; which undermines the way one usually thinks of a choice of gauge as motivated by calculational convenience, or as completely arbitrary. Last, I will assess arguments and counter-arguments for the direct physical significance of gauge symmetries. The conclusion provides a second type of answer to the question of "Why gauge?". Namely: because we need it to couple subsystems.

This thesis addresses three contemporary debates in the philosophy of science: namely, scientific realism, emergence, and theoretical equivalence. The thesis brings logico-semantic tools of the analytic tradition—about syntactic and semantic construals of theories, and about extensions and intensions—to bear on these debates. The thesis has two parts: Part I (Chapters 1-3) lays out the overall framework about scientific theories, scientific realism, and emergence. Part II (Chapters 4-6) develops more detailed themes. Part I first gives a conception of a scientific theory (Chapter 1), using logico-semantic tools that will be used in the rest of the thesis. Chapter 2 then brings these tools to bear on the debate about scientific realism, by construing the continuity of theories as a matter of extensions. The resulting position is a modest scientific realism, according to which one is justified in believing what confirmed theories say about extensions but not, in general, about intensions. I dub it ‘extensional scientific realism’. Chapter 3 proposes an account of the distinction between ontological and epistemic emergence, based on an explication of the notion of ‘novel reference’. The ontological emergence of one theory from another is defined as the failure of an appropriate linkage map between the two theories to ‘‘mesh’’ with the two theories’ interpretations. In Part II, Chapter 4 first develops a notion of theoretical equivalence, and introduces duality in physics, as an appropriate isomorphism between theories. The Chapter discusses the relation between duality and theoretical equivalence in philosophy of science. Chapter 5 discusses the heuristic roles of dualities in theory construction. It develops a distinction between the theoretical and heuristic functions of scientific theories, and illustrates the heuristic function of duality in theory construction. Chapter 6 discusses how theories without a spacetime can lead to scientific understanding. To this end, the Chapter describes three theoretical tools that are often used in theory construction and which lead to understanding, both in cases with and cases without straightforward spacetime visualisation.

This thesis examines various reductive case studies in thermal physics. In particular, I argue that according to my account of reduction-as-construction, there are two suc- cessful examples of reduction. Thermodynamics reduces to statistical mechanics, and statistical mechanics reduces to the underlying microdynamics - be they quantum or classical. The reduction of a given theory alters that theory's scope, that is: its domain of applicability. The scope of thermodynamics will be central to this thesis - and I will argue for a narrower scope than some authors. This thesis consists of four Chapters, together with an introduction and a conclusion. In Chapter 1, I discuss how different levels of description relate to one another. I argue that a higher-level of description is reduced to the lower level, if the higher-level quantities and their behaviour can be constructed or captured by the lower-level theory. I claim that 'functionalism' can be helpful in securing reductions. In this Chapter I also argue that the aim of reduction is to vindicate, not eliminate, the higher-level theory. In Chapter 2, I tackle the reduction of thermodynamics to statistical mechanics. I articulate the functional, or nomological, role of various thermodynamic quantities that are implicitly defined by the zeroth, first and second laws of thermodynamics: temperature, energy and entropy respectively. I then argue that there are quantities in statistical mechanics that realise these roles: though finding them sometimes requires us to focus on quantum, rather than classical, statistical mechanics. In Chapter 3, I consider the reductive relationship between statistical mechanics and the underlying microdynamics. I demonstrate how the irreversible equations of statistical mechanics can be constructed from the underlying microdynamics using what I label the 'Zwanzig-Zeh-Wallace' framework. Yet this framework uses a procedure called 'coarse-graining' which has been heavily criticised in the literature; so in this Chapter I offer a justification of coarse-graining. One upshot is that the time-asymmetry in statistical mechanics is weakly emergent. In Chapter 4, I consider a question about the domain of applicability of thermal physics. Namely: does it apply to self-gravitating systems, such as elliptical galaxies? Much controversy surrounds this question: some argue yes, others argue no. I deflate the dispute by claiming that thermodynamics does not apply, but statistical mechanics does. Thus, my delineation of thermodynamics and statistical mechanics earlier in this thesis not only makes headway with the question of reduction, but also sheds light on this dispute. I argue that this situation - statistical mechanics, but without thermodynamics - can be understood in terms of a central notion in thermal physics: the thermodynamic limit. But as I also discuss: justifying this idealisation has been philosophically controversial.

This dissertation is about some of the most important philosophical aspects of chaos research, a famous recent mathematical area of research about deterministic yet unpredictable and irregular, or even random behaviour. It consists of three parts. First, as a basis for the dissertation, I examine notions of unpredictability in ergodic theory, and I ask what they tell us about the justification and formulation of mathematical definitions. The main account of the actual practice of justifying mathematical definitions is Lakatos's account on proof-generated definitions. By investigating notions of unpredictability in ergodic theory, I present two previously unidentified but common ways of justifying definitions. Furthermore, I criticise Lakatos's account as being limited: it does not acknowledge the interrelationships between the different kinds of justification, and it ignores the fact that various kinds of justification - not only proof-generation - are important. Second, unpredictability is a central theme in chaos research, and it is widely claimed that chaotic systems exhibit a kind of unpredictability which is specific to chaos. However, I argue that the existing answers to the question "What is the unpredictability specific to chaos?" are wrong. I then go on to propose a novel answer, viz. the unpredictability specific to chaos is that for predicting any event all sufficiently past events are approximately probabilistically irrelevant. Third, given that chaotic systems are strongly unpredictable, one is led to ask: are deterministic and indeterministic descriptions observationally equivalent, i.e., do they give the same predictions? I treat this question for measure-theoretic deterministic systems and stochastic processes, both of which are ubiquitous in science. I discuss and formalise the notion of observational equivalence. By proving results in ergodic theory, I first show that for many measure-preserving deterministic descriptions there is an observationally equivalent indeterministic description, and that for all indeterministic descriptions there is an observationally equivalent deterministic description. I go on to show that strongly chaotic systems are even observationally equivalent to some of the most random stochastic processes encountered in science. For instance, strongly chaotic systems give the same predictions at every observation level as Markov processes or semi-Markov processes. All this illustrates that even kinds of deterministic and indeterministic descriptions which, intuitively, seem to give very different predictions are observationally equivalent. Finally, I criticise the claims in the previous philosophical literature on observational equivalence.