Minkowski spacetime and Lorentz invariance: The cart and the horse or two sides of a single coin?

Michel Janssen and Harvey Brown have driven a prominent recent debate concerning the direction of an alleged arrow of explanation between Minkowski spacetime and Lorentz invariance of dynamical laws in special relativity. In this article, I critically assess this controversy with the aim of clarifying the explanatory foundations of the theory. First, I show that two assumptions shared by the parties—that the dispute is independent of issues concerning spacetime ontology, and that there is an urgent need for a constructive interpretation of special relativity—are problematic and negatively affect the debate. Second, I argue that the whole discussion relies on a misleading conception of the link between Minkowski spacetime structure and Lorentz invariance, a misconception that in turn sheds more shadows than light on our understanding of the explanatory nature and power of Einstein׳s theory. I state that the arrow connecting Lorentz invariance and Minkowski spacetime is not explanatory and unidirectional, but analytic and bidirectional, and that this analytic arrow grounds the chronogeometric explanations of physical phenomena that special relativity offers.     

Intuitionism, transformational generative grammar and mental acts

A remarkable philosophical affinity may be observed between the intuitionistic conception of mathematics and the transformational generative approach to the study of language: both disciplines profess a mentalistic ontology, both posit an idealized subject, and both insist on their autonomy with respect to other disciplines. This philosophical parallel is formalized in terms of a generalization of the intuitionistic notion of creative subject; resulting are the foundations of a unified theory of mental (propositional) acts based on intuitionistic logic — capturing, inter alia, similarities between proof acts and speech acts. As an application of the theory, it is then shown how the notion of mental act may provide for an insightful formalization of various hypotheses pertaining to the linguistic dependence or relativity of mathematics.     

Einstein's reinterpretation of the Fizeau experiment: How it turned out to be crucial for special relativity

In this article, we aim at clarifying the role played by Fizeau’s 1851 experiment, both in the context of discovery and in the context of justification of the special theory of relativity. In 1907 Laue proved that Fresnel's formula was a consequence of the relativistic composition of velocities; since then, Einstein regarded Fizeau's experiment as confirmatory evidence for his theory, and even as a crucial experiment in favor of the relativistic addition of velocities. On the other hand, in the 1920's Einstein stated that this experiment was decisive in the path that led him to the discovery of his theory before 1905, but he did not explain why. We survey all the available evidence on this subject and conclude that the original ether-drag experiment was reinterpreted within a new conceptual framework in which the meaning of the very concept of velocity undergoes a radical change.     

Structuralism and the conformity of mathematics and nature

Structuralists typically appeal to some variant of the widely popular ‘mapping’ account of mathematical representation to suggest that mathematics is applied in modern science to represent the world’s physical structure. However, in this paper, I argue that this realist interpretation of the ‘mapping’ account presupposes that physical systems possess an ‘assumed structure’ that is at odds with modern physical theory. Through two detailed case studies concerning the use of the differential and variational calculus in modern dynamics, I show that the formal structure that we need to assume in order to apply the mapping account is inconsistent with the way in which mathematics is applied in modern physics. The problem is that a realist interpretation of the ‘mapping’ account imposes too severe of a constraint on the conformity that must exist between mathematics and nature in order for mathematics to represent the structure of a physical system.     

The quantum vacuum and the cosmological constant problem

The cosmological constant problem arises at the intersection between general relativity and quantum field theory, and is regarded as a fundamental problem in modern physics. In this paper, we describe the historical and conceptual origin of the cosmological constant problem which is intimately connected to the vacuum concept in quantum field theory. We critically discuss how the problem rests on the notion of physically real vacuum energy, and which relations between general relativity and quantum field theory are assumed in order to make the problem well-defined.     

On miracles and spacetime

What have recently been dubbed two ‘miracles’ of general relativity—(1) that all non-gravitational interactions are locally governed by Poincaré invariant dynamical laws; and (2) that, in the regime of experimental practice in which curvature effects may be ignored, the local Poincaré symmetries of the dynamical laws governing matter fields coincide with the local Poincaré symmetries of the dynamical metric field—remain unaccounted for in that theory. In this paper, I demonstrate that these two ‘miracles’ admit of a natural explanation in one particular successor theory to general relativity—namely, perturbative string theory. I argue that this point has important implications when considering both the ‘chronogeometricity’ (that is, the object in question being surveyed by rods and clocks built from matter fields) and spatiotemporal status of the dynamical metric field in both general relativity and perturbative string theory.     

Ernst Cassirer's transcendental account of mathematical reasoning

Cassirer's philosophical agenda revolved around what appears to be a paradoxical goal, that is, to reconcile the Kantian explanation of the possibility of knowledge with the conceptual changes of nineteenth and early twentieth-century science. This paper offers a new discussion of one way in which this paradox manifests itself in Cassirer's philosophy of mathematics. Cassirer articulated a unitary perspective on mathematics as an investigation of structures independently of the nature of individual objects making up those structures. However, this posed the problem of how to account for the applicability of abstract mathematical concepts to empirical reality. My suggestion is that Cassirer was able to address this problem by giving a transcendental account of mathematical reasoning, according to which the very formation of mathematical concepts provides an explanation of the extensibility of mathematical knowledge. In order to spell out what this argument entails, the first part of the paper considers how Cassirer positioned himself within the Marburg neo-Kantian debate over intellectual and sensible conditions of knowledge in 1902–1910. The second part compares what Cassirer says about mathematics in 1910 with some relevant examples of how structural procedures developed in nineteenth-century mathematics.     

Drawing the line between kinematics and dynamics in special relativity

Special relativity is preferable to those parts of Lorentz's classical ether theory it replaced because it shows that various phenomena that were given a dynamical explanation in Lorentz's theory are actually kinematical. In his book, Physical Relativity, Harvey Brown challenges this orthodox view. I defend it. The phenomena usually discussed in this context in the philosophical literature are length contraction and time dilation. I consider three other phenomena in the same class, each of which played a role in the early reception of special relativity in the physics literature: the Fresnel drag effect, the velocity dependence of electron mass, and the torques on a moving capacitor in the Trouton–Noble experiment. I offer historical sketches of how Lorentz's dynamical explanations of these phenomena came to be replaced by their now standard kinematical explanations. I then take up the philosophical challenge posed by the work of Harvey Brown and Oliver Pooley and clarify how those kinematical explanations work. In the process, I draw attention to the broader importance of the kinematics–dynamics distinction.     

Heuristics versus norms: On the relativistic responses to the Kaufmann experiments

The aim of this article is to provide a historical response to Michel Janssen’s (2009) claim that the special theory of relativity establishes that relativistic phenomena are purely kinematical in nature, and that the relativistic study of such phenomena is completely independent of dynamical considerations regarding the systems displaying such behavior. This response will be formulated through a historical discussion of one of Janssen's cases, the experiments carried out by Walter Kaufmann on the velocity-dependence of the electron's mass. Through a discussion of the different responses formulated by early adherents of the principle of relativity (Albert Einstein, Max Planck, Hermann Minkowski and Max von Laue) to these experiments, it will be argued that the historical development of the special theory of relativity argues against Janssen's historical presentation of the case, and that this raises questions about his general philosophical claim. It will be shown, more specifically, that Planck and Einstein developed a relativistic response to the Kaufmann experiments on the basis of their study of the dynamics of radiation phenomena, and that this response differed significantly from the response formulated by Minkowski and Laue. In this way, it will be argued that there were, at the time, two different approaches to the theory of relativity, which differed with respect to its relation to theory, experiment, and history: Einstein's and Planck's heuristic approach, and Minkowski's and Laue's normative approach. This indicates that it is difficult to say, historically speaking, that the special theory of relativity establishes the kinematical nature of particular phenomena. Instead, it will be argued that the theory of relativity should not be seen as a theory but rather as outlining an approach, and that the nature of particular scientific phenomena is something that is open to scientific debate and dispute.     

The Russell Archives: Some new light on Russell's logicism

This paper describes the materials in the Russell Archives relevant to Russell's work on logic and the foundations of mathematics, and suggests the kinds of information that may and may not be drawn about the historical development of his ideas. By way of illustration, a couple of episodes are described. The first (arts. 2–4) concerns a logical system closely related to his theory of denoting, which preceeds the system used in Principia mathematics, while the second (arts. 5) describes a delay in publishing the second volume of that work due to the discovery by Whitehead of a conceptual error.     

Socially conditioned mathematical change: the case of the French Revolution

This paper examines a historical case of conceptual change in mathematics that was fundamental to its progress. I argue that in this particular case, the change was conditioned primarily by social processes, and these are reflected in the intellectual development of the discipline. Reorganization of mathematicians and the formation of a new mathematical community were the causes of changes in intellectual content, rather than being mere effects. The paper focuses on the French Revolution, which gave rise to revolutionary developments in mathematics. I examine how changes in the political constellation affected mathematicians both individually and collectively, and how a new professional community—with different views on the objects, problems, aims, and values of the discipline—arose. On the basis of this account, I will discuss such Kuhnian themes as the role of the professional community and normal versus revolutionary development.     

Reconceptualising equilibrium in Boltzmannian statistical mechanics and characterising its existence

In Boltzmannian statistical mechanics macro-states supervene on micro-states. This leads to a partitioning of the state space of a system into regions of macroscopically indistinguishable micro-states. The largest of these regions is singled out as the equilibrium region of the system. What justifies this association? We review currently available answers to this question and find them wanting both for conceptual and for technical reasons. We propose a new conception of equilibrium and prove a mathematical theorem which establishes in full generality – i.e. without making any assumptions about the system׳s dynamics or the nature of the interactions between its components – that the equilibrium macro-region is the largest macro-region. We then turn to the question of the approach to equilibrium, of which there exists no satisfactory general answer so far. In our account, this question is replaced by the question when an equilibrium state exists. We prove another – again fully general – theorem providing necessary and sufficient conditions for the existence of an equilibrium state. This theorem changes the way in which the question of the approach to equilibrium should be discussed: rather than launching a search for a crucial factor (such as ergodicity or typicality), the focus should be on finding triplets of macro-variables, dynamical conditions, and effective state spaces that satisfy the conditions of the theorem.     

A beautiful sea: P. A. M. Dirac's epistemology and ontology of the vacuum

This paper charts P.A.M. Dirac's development of his theory of the electron, and its radical picture of empty space as an almost-full plenum. Dirac's Quantum Electrodynamics famously accomplished more than the unification of special relativity and quantum mechanics. It also accounted for the ‘duplexity phenomena’ of spectral line splitting that we now attribute to electron spin. But the extra mathematical terms that allowed for spin were not alone, and this paper charts Dirac's struggle to ignore or account for them as a sea of strange, negative-energy, particles with positive ‘holes’. This work was not done in solitude, but rather in exchanges with Dirac's correspondence network. This social context for Dirac’s work contests his image as a lone genius, and documents a community wrestling with the ontological consequences of their work. Unification, consistency, causality, and community are common factors in explanations in the history of physics. This paper argues on the basis of materials in Dirac's archive that --- in addition --- mathematical beauty was an epistemological factor in the development of the electron and hole theory. In fact, if we believe that Dirac's beautiful mathematics captures something of the world, then there is both an epistemology and an ontology of mathematical beauty.     

Maimon's criticism of Kant's doctrine of mathematical cognition and the possibility of metaphysics as a science

The aim of this paper is to discuss Maimon's criticism of Kant's doctrine of mathematical cognition. In particular, we will focus on the consequences of this criticism for the problem of the possibility of metaphysics as a science. Maimon criticizes Kant's explanation of the synthetic a priori character of mathematics and develops a philosophical interpretation of differential calculus according to which mathematics and metaphysics become deeply interwoven. Maimon establishes a parallelism between two relationships: on the one hand, the mathematical relationship between the integral and the differential and on the other, the metaphysical relationship between the sensible and the supersensible. Such a parallelism will be the clue to the Maimonian solution to the Kantian problem of the possibility of metaphysics as a science.     

Mesh and measure in early general relativity

In the early days of general relativity, several of Einstein׳s readers misunderstood the role of coordinates or “mesh-system” in ways that threatened the basic predictions of the theory. This confusion largely derived from intrinsic defects of Einstein׳s first systematic exposition of his theory. A few of Einstein׳s followers, including Arthur Eddington, Hermann Weyl, and Max von Laue, identified the interpretive difficulties and solved them by combining a deeply geometrical understanding of the theory with detailed attention to the concrete conditions of measurement.     

The diffusion of scientific innovations: A role typology

How do scientific innovations spread within and across scientific communities? In this paper, we propose a general account of the diffusion of scientific innovations. This account acknowledges that novel ideas must be elaborated on and conceptually translated before they can be adopted and applied to field-specific problems. We motivate our account by examining an exemplary case of knowledge diffusion, namely, the early spread of theories of rational decision-making. These theories were grounded in a set of novel mathematical tools and concepts that originated in John von Neumann and Oskar Morgenstern's Theory of Games and Economic Behavior (1944/1947) and subsequently spread widely across the social and behavioral sciences. Introducing a network-based diffusion measure, we trace the spread of those tools and concepts into distinct research areas. We furthermore present an analytically tractable typology for classifying publications according to their roles in the diffusion process. The proposed framework allows for a systematic examination of the conditions under which scientific innovations spread within and across preexisting and newly emerging scientific communities.     

Deflationary representation,inference, and practice

This paper defends the deflationary character of two recent views regarding scientific representation, namely RIG Hughes' DDI model and the inferential conception. It is first argued that these views' deflationism is akin to the homonymous position in discussions regarding the nature of truth. There, we are invited to consider the platitudes that the predicate “true” obeys at the level of practice, disregarding any deeper, or more substantive, account of its nature. More generally, for any concept X, a deflationary approach is then defined in opposition to a substantive approach, where a substantive approach to X is an analysis of X in terms of some property P, or relation R, accounting for and explaining the standard use of X. It then becomes possible to characterize a deflationary view of scientific representation in three distinct senses, namely: a “no-theory” view, a “minimalist” view, and a “use-based” view—in line with three standard deflationary responses in the philosophical literature on truth. It is then argued that both the DDI model and the inferential conception may be suitably understood in any of these three different senses. The application of these deflationary ‘hermeneutics’ moreover yields significant improvements on the DDI model, which bring it closer to the inferential conception. It is finally argued that what these approaches have in common—the key to any deflationary account of scientific representation—is the denial that scientific representation may be ultimately reduced to any substantive explanatory property of sources, or targets, or their relations.     

Strategic planning and forecasting

This paper examines the role of forecasting in strategic planning through the use of analytical portfolio models and corporate simulation models. It also outlines a conceptual framework on which it may be possible to develop a theory of strategic planning—namely, microeconomic theory. It, finally, describes a practical application of this theoretical framework, known as the strategy matrix. It concludes by noting the importance of forecasting as an input to the strategy matrix.     

William Thomson's dynamical theory: An insight into a scientist's thinking

William Thomson, later Lord Kelvin, played a major role in the nineteenth century in changing scientific theory from the statical view, associated with imponderables, to the dynamical view which conceived of energy as a separate and convertible entity. Thomson's conversion from the statical to the dynamical view of nature was due to the influence of experimentalists, Michael Faraday and James Prescott Joule. It was Thomson's use of mathematical metaphor that enabled him to interpret on a theoretical level the physical explanation given by Faraday and Joule.