This chapter concerns dimensions as the term is used in the physical sciences today. Some key points made are: Quantities of the same kind have the same dimension; but that two quantities have the same dimension does not necessarily mean they are of the same kind. The dimension of a quantity is not determined for a single quantity in isolation, but relative to a system of quantities and the relations that hold between them. Dimensions, units, and quantities are distinct notions. (...) In this article, I explain how dimensions, units, and quantities are involved in the design of coherent systems of units; the account involves the equations of physics. When the use of a coherent system of units can be presumed, dimensional analysis is a powerful logico-mathematical method for deriving equations and relations in physics, and for parameterizing equations in terms of dimensionless parameters, which allows identifying physically similar systems. The source of the information yielded by dimensional analysis is not yet well understood in philosophy of physics. This chapter aims to reveal the role of dimensions not only in applications of dimensional analysis to obtain information by involving the principle of dimensional homogeneity, but to the role of dimensions in encoding information about physical relationships in the language of dimensions, specifically via the feature of coherence of a system of units. Philosophers of mathematics and philosophers of science have been concerned to address the question of the effectiveness of mathematics in science. It is argued here that no philosophical analysis of the question of the applicability of mathematics to science is complete without including dimensions and dimensional analysis in the picture. (shrink)

Arguments for the effectiveness, and even the indispensability, of mathematics in scientific explanation rely on the claim that mathematics is an effective or even a necessary component in successful scientific predictions and explanations. Well-known accounts of successful mathematical explanation in physical science appeals to scientists’ ability to solve equations directly in key domains. But there are spectacular physical theories, including general relativity and fluid dynamics, in which the equations of the theory cannot be solved directly in target domains, and yet (...) scientists and mathematicians do effective work there (McLarty 2023, Elder 2023). Building on extant accounts of structural scientific explanation (Bokulich 2011, Leng 2021), I argue that philosophical accounts of the role of equations in scientific explanation need not rely on scientists’ ability to solve equations independently of their understanding of the empirical or experimental context. For instance, the process of formulating solutions to equations can involve significant appeal to information about experimental contexts (Curiel 2010) or of physically similar systems (Sterrett 2023). Working from a close analysis of work in fluid mechanics by Martin Bazant and Keith Moffatt (2005), I propose an account of heuristic structural explanation in mathematics (Einstein 1921, Pincock 2021), which explains how physical explanations can be constructed even in domains where basic equations cannot be solved directly. (shrink)

This paper develops a theory of analog representation. We first argue that the mark of the analog is to be found in the nature of a representational system’s interpretation function, rather than in its vehicles or contents alone. We then develop the rulebound structure theory of analog representation, according to which analog systems are those that use interpretive rules to map syntactic structural features onto semantic structural features. The theory involves three degree-theoretic measures that capture three independent ways in which (...) a system can be more or less analog. We explain how our theory improves upon prior accounts of analog representation, provides plausible diagnoses for novel challenge cases, extends to hybrid systems that are partially analog and partially symbolic, and accounts for some of the advantages and disadvantages of representing analogically versus symbolically. (shrink)

Mechanisms consist of component parts and processes organized in a specific way to produce changes that may give rise to one or more phenomena. I aim to examine the generative mechanism of generative material models in the production of new material models. A generative material model in biology is a living material model that is capable of generating new material models. I contend that generative mechanisms of a generative material model are not to be conflated with biological mechanisms: the former (...) is a mechanism manipulated and altered by the modeler while the latter is a naturally occurring mechanism. I examine the ways in which idealization may contribute to the generative mechanism in generating new model organisms. I articulate that the component parts and activities in a generative mechanism are to a large extent shaped by idealization in generative material modeling. Although biological mechanisms do have a role to play in the generation of model organisms, I propose that we shall view the generative mechanism as a hybrid of biological mechanisms and modeling activities. I shall argue that the notion of generative mechanisms as hybrid mechanisms is different from the extant conceptions of mechanisms in various aspects. (shrink)

This paper defends three claims about concrete or physical models: these models remain important in science and engineering, they are often essentially idealized, in a sense to be made precise, and despite these essential idealizations, some of these models may be reliably used for the purpose of causal explanation. This discussion of concrete models is pursued using a detailed case study of some recent models of landslide generated impulse waves. Practitioners show a clear awareness of the idealized character of these (...) models, and yet address these concerns through a number of methods. This paper focuses on experimental arguments that show how certain failures to accurately represent feature X are consistent with accurately representing some causes of feature Y, even when X is causally relevant to Y. To analyse these arguments, the claims generated by a model must be carefully examined and grouped into types. Only some of these types can be endorsed by practitioners, but I argue that these endorsed claims are sufficient for limited forms of causal explanation. (shrink)

While the predominant focus of the philosophical literature on scientific modeling has been on single-scale models, most systems in nature exhibit complex multiscale behavior, requiring new modeling methods. This challenge of modeling phenomena across a vast range of spatial and temporal scales has been called the tyranny of scales problem. Drawing on research in the geosciences, I synthesize and analyze a number of strategies for taming this tyranny in the context of conceptual, physical, and mathematical modeling. This includes several strategies (...) that can be deployed in physical modeling, even when strict dynamical scaling fails. In all cases, I argue that having an adequate conceptual model—given both the nature of the system and the particular purpose of the model—is essential. I draw a distinction between depiction and representation, and use this research in the geosciences to advance a number of debates in the philosophy of modeling. (shrink)

In this paper, I enrich the context of Wittgenstein's Tractatus given over a decade ago in my book Witttgenstein Flies A Kite (and related earlier works dating from 2000). I've since located a sketch reprinted from a 1914 Paris magazine showing a lawyer using a model bus and dolls to depict a traffic accident; I present it here along with a discussion of the modelmaker movement of that time. The modelmaker movement was a movement at the intersection of popular culture (...) and technical expertise that really needs to be understood and recognized in discussing Wittgenstein's use of Modell and Bild. I discuss its role in relation to experimental models used in scientific research. Other new aspects presented here include: the very special role of model-fl ying clubs (known in Germany as Modell-Flugverein); the use of scientific forensics in courts of law, really just beginning then (c. 1914), and a part of popular culture as well; the significance of more recent work by others on Boltzmann's personal interest in flight, and on the widespread but now-forgotten discussion of dimensional analysis in the history of physics. I conclude that all these lend support to the views on the Tractatus I laid out in my book, and summarize and elaborate on some of them here, inasmuch as space permits. More generally, I argue that the philosophical community interested in interpreting Wittgenstein's early works stands to gain from becoming better acquainted with the scientific and technological developments of the milieu in which they were conceived. (shrink)

The proposed revision to the International System of Units contains two features that are bound to be of special interest to those concerned with foundational questions in philosophy of science. These are that the proposed system of international units can be defined without drawing a distinction between base units and derived units, and without restricting the means by which the value of the quantities associated with the units are to be established. In this paper, I address the question of the (...) role of base units in light of the New SI: Do the "base units" of the SI play any essential role anymore, if they are neither at the bottom of a hierarchy of definitions themselves, nor the only units that figure in the statements fixing the numerical values of the "defining constants"? The answer I develop and present also shows why it is important to retain the distinction between dimensions and quantities. I argue for an appreciation of the role of dimensions in understanding issues related to systems of units. (shrink)

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. (shrink)

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. (shrink)