Formalizing the separability condition in Bell's theorem

The nonseparability of physical systems is often invoked in philosophical analyses of what has come to be known as Bell's theorem. Until recently, the formalization of the notion of separability was assumed to be unproblematic, equivalent to that of outcome independence (Jarrett incompleteness). Although this equivalence has been called into question, an alternative has not yet been specified with sufficient precision, leading to confusion as to what kinds of models should be considered separable. I identify four plausible candidates for the proper formalization of the separability condition, understood in terms of part–whole determination, and then discuss the relative merits of each. I show that three of these are, in fact, equivalent to outcome independence under most conditions, and that one is not, raising the possibility of a new decomposition of Bell locality. I also question whether part–whole determination should be considered sufficient for separability.     

Abduction, tomography, and other inverse problems

Charles S. Peirce introduced in the late 19^th century the notion of abduction as inference from effects to causes, or from observational data to explanatory theories. Abductive reasoning has become a major theme in contemporary logic, philosophy of science, and artificial intelligence. This paper argues that the new growing branch of applied mathematics called inverse problems deals successfully with various kinds of abductive inference within a variety of scientific disciplines. The fundamental theorem about the inverse reconstruction of plane functions from their line integrals was proved by Johann Radon already in 1917. The practical applications of Radon’s theorem and its generalizations include computerized tomography which became a routine imaging technique of diagnostic medicine in the 1970s.     

The landing zone – Ground for model transfer in chemistry

This essay proposes a new notion - the landing zone - in order to identify conceptual features that allow modelers to transfer mathematical tools across disciplinary borders. This discussion refers to the transferable models as ‘templates’. Templates are functions, equations, or computational methods that are capable of being generalized from a particular subject matter. There are formal and conceptual prerequisites for the transfer of a template to a new domain. A landing zone is an ontology that contributes to the satisfaction of these conditions for successful transfer. This paper presents a case study on a model in chemistry - the Quantum Theory of Atoms in Molecules (QTAIM) - that makes use of transferred templates from physics - the virial theorem and the wave function. The landing zone in this case is a new ontological notion, that of the topological atom, which prepares ground for the use of the virial theorem and the wave function in chemistry. The virial theorem requires that there exists in-principle stability to the system that it represents, and the wave function requires transformation in its representation that is justified. The ontology of QTAIM - the landing zone for these templates - grounds the scientific use of these templates in the context of chemistry.     

Probability in 1919/20: the von Mises-Pólya-Controversy

The correspondence between Richard von Mises and George Pólya of 1919/20 contains reflections on two well-known articles by von Mises on the foundations of probability in the Mathematische Zeitschrift of 1919, and one paper from the Physikalische Zeitschrift of 1918. The topics touched on in the correspondence are: the proof of the central limit theorem of probability theory, von Mises' notion of randomness, and a statistical criterion for integer-valuedness of physical data. The investigation will hint at both the fruitfulness and the limits of several of von Mises' notions such as ``collective', ``distribution' and ``complex adjuncts' (characteristic functions) for further developments in probability theory and in ``directional statistics'. By pointing to the selectiveness of Pólya's criticism, the historical analysis shows the differing expectations of the two men with respect to the further development of the theory of probability and its applications. The paper thus gives a glimpse of the provisional state of the theory around 1920, before others such as P. Lévy (1886–1971) and A. N. Kolmogorov (1903–1987) stepped in and created a new paradigm for probability theory.     

On the historical origins of the contemporary notion of incommensurability: Paul Feyerabend’s assault on conceptual conservativism

This paper investigates the historical origins of the notion of incommensurability in contemporary philosophy of science. The aim is not to establish claims of priority, but to enhance our understanding of the notion by illuminating the various issues that contributed to its development. Kuhn developed his notion of incommensurability primarily under the influence of Fleck, Polanyi, and Köhler. Feyerabend, who had developed his notion more than a decade earlier, drew directly from Duhem, who had developed a notion of incommensurability in 1906. The idea is that in the course of scientific advance, when fundamental theories change, meanings change, which can result in a new conception of the nature of reality. Feyerabend repeatedly used this notion of incommensurability to attack various forms of conceptual conservativism. These include the logical positivists’ foundational use of protocol statements, Heisenberg’s methodological principle that established results must be presupposed by all further research, attempts to separate philosophical accounts of ontology from physics, Bohr’s principle of complementarity, and logical empiricist accounts of reduction and explanation. Focusing on the function of the notion of incommensurability common to Feyerabend’s various critiques explicates Feyerabend’s early philosophy as a series of challenges to forms of conceptual conservativism.     

The Lunar Theories of Tycho Brahe and Christian Longomontanus in the Progymnasmata and Astronomia Danica

Tycho Brahe's lunar theory, mostly the work of his assistant Christian Longomontanus, published in the Progymnasmata (1602), was the most advanced and accurate lunar theory yet developed. Its principal innovations are: the introduction of equant motion for the first inequality in order to separate the determination of direction and distance; a more accurate limit for the second inequality although requiring a more complex calculation; additional inequalities of the variation and, in place of the annual inequality in Tycho's earlier theory, a reduction in the equation of time; in the latitude theory a variation of the inclination of the orbital plane and an inequality of the motion of the nodes; a reduction in the range of variation of distance, parallax, and apparent diameter. Some of these were already present in Tycho's earlier lunar theory (1599), but all were changed in notable ways. Twenty years later Longomontanus published a modified version of the lunar theory in Astronomia Danica (1622), for the purpose of facilitating the calculation through new correction tables, and also explained his reasons for parts of the theory in the Progymnasmata. This paper is a technical study of both lunar theories.     

A summary of Euler’s work on the pentagonal number theorem

In this article, we give a summary of Leonhard Euler’s work on the pentagonal number theorem. First we discuss related work of earlier authors and Euler himself. We then review Euler’s correspondence, papers and notebook entries about the pentagonal number theorem and its applications to divisor sums and integer partitions. In particular, we work out the details of an unpublished proof of the pentagonal number theorem from Euler’s notebooks. As we follow Euler’s discovery and proofs of the pentagonal number theorem, we pay attention to Euler’s ideas about when we can consider a mathematical statement to be true. Finally, we discuss related results in the theory of analytic functions.     

On the status of the geodesic principle in Newtonian and relativistic physics

A theorem due to Geroch and Jang (1975) provides a sense in which the geodesic principle has the status of a theorem in General Relativity. I have recently shown that a similar theorem holds in the context of geometrized Newtonian gravitation (Newton–Cartan theory) (Weatherall, J.O., 2011). Here I compare the interpretations of these two theorems. I argue that despite some apparent differences between the theorems, the status of the geodesic principle in geometrized Newtonian gravitation is, mutatis mutandis, strikingly similar to the relativistic case.     

On the notion of free will in the Free Will Theorem

The (Strong) Free Will Theorem (fwt) of Conway and Kochen (2009) on the one hand follows from uncontroversial parts of modern physics and elementary mathematical and logical reasoning, but on the other hand seems predicated on an undefined notion of free will (allowing physicists to “freely choose” the settings of their experiments). This makes the theorem philosophically vulnerable, especially if it is construed as a proof of indeterminism or even of libertarian free will (as Conway & Kochen suggest).However, Cator and Landsman (Foundations of Physics 44, 781–791, 2014) previously gave a reformulation of the fwt that does not presuppose indeterminism, but rather assumes a mathematically specific form of such “free choices” even in a deterministic world (based on a non-probabilistic independence assumption). In the present paper, which is a philosophical sequel to the one just mentioned, I argue that the concept of free will used in the latter version of the fwt is essentially the one proposed by Lewis (1981), also known as ‘local miracle compatibilism’ (of which I give a mathematical interpretation that might be of some independent interest also beyond its application to the fwt). As such, the (reformulated) fwt in my view challenges compatibilist free will à la Lewis (albeit in a contrived way via bipartite epr-type experiments), falling short of supporting libertarian free will.     

The locus ceruleus: a possible neural focus for genetic differences in emotionality

Summary The Maudsley Reactive and Non-Reactive strains have been developed as a model for the study of individual variations in stress-reactivity, and many differences in biobehavioral systems have been found between them. This review discusses limitations of the emotionality construct in accounting for differences between the Maudsley strains and offers an alternative, theoretical approach. Amaral and Sinnamon have proposed that the locus ceruleus (LC) plays a stress-attenuating role in mediating behavioral, physiological and neuroendocrine response to prepotent, emergency-provoking stimuli and, building upon this formulation, it is proposed that the LC has been an important focus for gene action in the Maudsley model. It is suggested that the LC of the Non-Reactive strain is more strongly activated by stressful stimuli than the LC of Reactive rats, and is the basis of many of the behavioral and physiological differences between them. Behavioral and biochemical evidence consistent with this proposition is reviewed. Identification of the LC as a target for gene-action in the Maudsley model has an important advantage. It substitutes variations at a specific anatomic location in the brain for a loosely defined construct like emotionality, and the hypothesis is amenable to empirical tests by a variety of experimental approaches.Supported by MH-39210 from the National Institute of Mental Health to DAB     

Non-contextuality,finite precision measurement and the Kochen–Specker theorem

Meyer originally raised the question of whether non-contextual hidden variable models can, despite the Kochen–Specker theorem, simulate the predictions of quantum mechanics to within any fixed finite experimental precision (Phys. Rev. Lett. 83 (1999) 3751). Meyer's result was extended by Kent (Phys. Rev. Lett. 83 (1999) 3755). Clifton and Kent later presented constructions of non-contextual hidden variable theories which, they argued, indeed simulate quantum mechanics in this way (Proc. Roy. Soc. Lond. A 456 (2000) 2101).These arguments have evoked some controversy. Among other things, it has been suggested that the Clifton–Kent models do not in fact reproduce correctly the predictions of quantum mechanics, even when finite precision is taken into account. It has also been suggested that careful analysis of the notion of contextuality in the context of finite precision measurement motivates definitions which imply that the Clifton–Kent models are in fact contextual. Several critics have also argued that the issue can be definitively resolved by experimental tests of the Kochen–Specker theorem or experimental demonstrations of the contextuality of Nature.One aim of this paper is to respond to and rebut criticisms of the Meyer–Clifton–Kent papers. We thus elaborate in a little more detail how the Clifton–Kent models can reproduce the predictions of quantum mechanics to arbitrary precision. We analyse in more detail the relationship between classicality, finite precision measurement and contextuality, and defend the claims that the Clifton–Kent models are both essentially classical and non-contextual. We also examine in more detail the senses in which a theory can be said to be contextual or non-contextual, and in which an experiment can be said to provide evidence on the point. In particular, we criticise the suggestion that a decisive experimental verification of contextuality is possible, arguing that the idea rests on a conceptual confusion.     

Heuristic analogy in Ars Conjectandi: From Archimedes' De Circuli Dimensione to Bernoulli's theorem

This article investigates the way in which Jacob Bernoulli proved the main mathematical theorem that undergirds his art of conjecturing—the theorem that founded, historically, the field of mathematical probability. It aims to contribute a perspective into the question of problem-solving methods in mathematics while also contributing to the comprehension of the historical development of mathematical probability. It argues that Bernoulli proved his theorem by a process of mathematical experimentation in which the central heuristic strategy was analogy. In this context, the analogy functioned as an experimental hypothesis. The article expounds, first, Bernoulli's reasoning for proving his theorem, describing it as a process of experimentation in which hypothesis-making is crucial. Next, it investigates the analogy between his reasoning and Archimedes' approximation of the value of π, by clarifying both Archimedes' own experimental approach to the said approximation and its heuristic influence on Bernoulli's problem-solving strategy. The discussion includes some general considerations about analogy as a heuristic technique to make experimental hypotheses in mathematics.     

Mathematical instrumentalism, Gödel’s theorem, and inductive evidence

Mathematical instrumentalism construes some parts of mathematics, typically the abstract ones, as an instrument for establishing statements in other parts of mathematics, typically the elementary ones. Gödel’s second incompleteness theorem seems to show that one cannot prove the consistency of all of mathematics from within elementary mathematics. It is therefore generally thought to defeat instrumentalisms that insist on a proof of the consistency of abstract mathematics from within the elementary portion. This article argues that though some versions of mathematical instrumentalism are defeated by Gödel’s theorem, not all are. By considering inductive reasons in mathematics, we show that some mathematical instrumentalisms survive the theorem.     

Newton’s notion and practice of unification

In this paper I deal with a neglected topic with respect to unification in Newton’s Principia. I will clarify Newton’s notion (as can be found in Newton’s utterances on unification) and practice of unification (its actual occurrence in his scientific work). In order to do so, I will use the recent theories on unification as tools of analysis (Kitcher, Salmon and Schurz). I will argue, after showing that neither Kitcher’s nor Schurz’s account aptly capture Newton’s notion and practice of unification, that Salmon’s later work is a good starting point for analysing this notion and its practice in the Principia. Finally, I will supplement Salmon’s account in order to answer the question at stake.     

A new look at E.G. Björling and the Cauchy sum theorem

We give a new account of Björling’s contribution to uniform convergence in connection with Cauchy’s theorem on the continuity of an infinite series. Moreover, we give a complete translation from Swedish into English of Björling’s 1846 proof of the theorem. Our intention is also to discuss Björling’s convergence conditions in view of Grattan-Guinness’ distinction between history and heritage. In connection to Björling’s convergence theory we discuss the interpretation of Cauchy’s infinitesimals.     

Quotient Rings of Noncommutative Rings in the First Half of the 20th Century

A keystone of the theory of noncommutative noetherian rings is the theorem that establishes a necessary and sufficient condition for a given ring to have a quotient ring. We trace the development of this theorem, and its applications, from its first version for noncommutative domains in the 1930s to Goldies theorems in the late 1950s.     

The locus ceruleus: a possible neural focus for genetic differences in emotionality

The Maudsley Reactive and Non-Reactive strains have been developed as a model for the study of individual variations in stress-reactivity, and many differences in biobehavioral systems have been found between them. This review discusses limitations of the 'emotionality' construct in accounting for differences between the Maudsley strains and offers an alternative, theoretical approach. Amaral and Sinnamon have proposed that the locus ceruleus (LC) plays a stress-attenuating role in mediating behavioral, physiological and neuroendocrine response to prepotent, emergency-provoking stimuli and, building upon this formulation, it is proposed that the LC has been an important focus for gene action in the Maudsley model. It is suggested that the LC of the Non-Reactive strain is more strongly activated by stressful stimuli than the LC of Reactive rats, and is the basis of many of the behavioral and physiological differences between them. Behavioral and biochemical evidence consistent with this proposition is reviewed. Identification of the LC as a target for gene-action in the Maudsley model has an important advantage. It substitutes variations at a specific anatomic location in the brain for a loosely defined construct like emotionality, and the hypothesis is amenable to empirical tests by a variety of experimental approaches.     

Interpreting the Wigner–Eckart Theorem

The Wigner–Eckart theorem is central to the application of symmetry principles throughout atomic, molecular, and nuclear physics. Nevertheless, the theorem has a puzzling feature: it is dispensable for solving problems within these domains, since elementary methods suffice. To account for the significance of the theorem, I first contrast it with an elementary approach to calculating matrix elements. Next, I consider three broad strategies for interpreting the theorem: conventionalism, fundamentalism, and conceptualism. I argue that the conventionalist framework is unnecessarily pragmatic, while the fundamentalist framework requires more ontological commitments than necessary. Conceptualism avoids both defects, accounting for the theorem’s significance in terms of how it epistemically restructures the calculation of matrix elements. Specifically, the Wigner–Eckart theorem modularizes and unifies matrix element problems, thereby changing what we need to know to solve them.     

Reconsidering the Fresnel–Maxwell theory shift: how the realist can have her cake and EAT it too

This paper takes another look at a case study which has featured prominently in a variety of arguments for rival realist positions. After critically reviewing the previous commentaries of the theory shift that took place in the transition from Fresnel’s ether to Maxwell’s electromagnetic theory of optics, it will defend a slightly different reading of this historical case study. Central to this task is the notion of explanatory approximate truth, a concept which must be carefully analysed to begin with. With this notion properly understood, it will be finally argued, the popular Fresnel–Maxwell case study points towards a novel formulation of scientific realism.     

Theoretical construction in physics – The role of Leibniz for Weyl's ‘Philosophie der Mathematik und Naturwissenschaft’

This paper aims at closing a gap in recent Weyl research by investigating the role played by Leibniz for the development and consolidation of Weyl's notion of theoretical (symbolic) construction. For Weyl, just as for Leibniz, mathematics was not simply an accompanying tool when doing physics—for him it meant the ability to engage in well-guided speculations about a general framework of reality and experience. The paper first introduces some of the background of Weyl's notion of theoretical construction and then discusses particular Leibnizian inheritances in Weyl's ‘Philosophie der Mathematik und Naturwissenschaft’, such as the general appreciation of the principles of sufficient reason and of continuity. Afterwards the paper focuses on three themes: first, Leibniz's primary quality phenomenalism, which according to Weyl marked the decisive step in realizing that physical qualities are never apprehended directly; second, the conceptual relation between continuity and freedom; and third, Leibniz's notion of ‘expression’, which allows for a certain type of (surrogative) reasoning by structural analogy and which gave rise to Weyl's optimism regarding the scope of theoretical construction.