We outline a framework for analyzing episodes from the history of science in which the application of mathematics plays a constitutive role in the conceptual development of empirical sciences. Our starting point is the inferential conception of the application of mathematics, recently advanced by Bueno and Colyvan. We identify and discuss some systematic problems of this approach. We propose refinements of the inferential conception based on theoretical considerations and on the basis of a historical case study. We demonstrate the usefulness (...) of the refined, dynamical inferential conception using the well-researched example of the genesis of general relativity. Specifically, we look at the collaboration of the physicist Einstein and the mathematician Grossmann in the years 1912--1913, which resulted in the jointly published ``Outline of a Generalized Theory of Relativity and a Theory of Gravitation,'' a precursor theory of the final theory of general relativity. In this episode, an independently developed mathematical theory, the theory of differential invariants and the absolute differential calculus, was applied in the process of physical theorizing aiming at finding a relativistic theory of gravitation. We argue that the dynamical inferential conception not only provides a natural framework to describe and analyze this episode, but it also generates new questions and insights. We comment on the mathematical tradition on which Grossmann drew, and on his own contributions to mathematical theorizing. We argue that the dynamical inferential conception allows us to identify both the role of heuristics and of mathematical resources as well as the systematic role of problems and mistakes in the reconstruction of episodes of conceptual innovation and theory change. (shrink)

This contribution gives an overview of Einstein's work on unified field theory. It characterizes this work from four perspectives, by looking at its conceptual, representational, biographical, and philosophical dimensions.

This volume offers an integrated understanding of how the theory of general relativity gained momentum after Einstein had formulated it in 1915. Chapters focus on the early reception of the theory in physics and philosophy and on the systematic questions that emerged shortly after Einstein's momentous discovery. They are written by physicists, historians of science, and philosophers, and were originally presented at the conference titled Thinking About Space and Time: 100 Years of Applying and Interpreting General Relativity, held at the (...) University of Bern from September 12-14, 2017. By establishing the historical context first, and then moving into more philosophical chapters, this volume will provide readers with a more complete understanding of early applications of general relativity and of related philosophical issues. Because the chapters are often cross-disciplinary, they cover a wide variety of topics related to the general theory of relativity. These include: Heuristics used in the discovery of general relativity Mach's Principle The structure of Einstein's theory Cosmology and the Einstein world Stability of cosmological models The metaphysical nature of spacetime The relationship between spacetime and dynamics The Geodesic Principle Symmetries Thinking About Space and Time will be a valuable resource for historians of science and philosophers who seek a deeper knowledge of the uses of general relativity, as well as for physicists and mathematicians interested in exploring the wider historical and philosophical context of Einstein's theory. (shrink)

Beyond Einstein: Perspectives on Geometry, Gravitation, and Cosmology explores the rich interplay between mathematical and physical ideas by studying the interactions of major actors and the roles of important research communities over the course of the last century.

In this note, we point attention to and briefly discuss a curious manuscript of Einstein, composed in 1938 and entitled “Unified Field Theory,” the only such writing, published or unpublished, carrying this title without any further specification. Apparently never intended for publication, the manuscript sheds light both on Einstein′s modus operandi as well as on the public role of Einstein′s later work on a unified field theory of gravitation and electromagnetism.

The history of the publication of the gravitational field equations of general relativity in November 1915 by Einstein and Hilbert is briefly reviewed. An analysis of the internal structure and logic of Hilbert's theory as expounded in extant proofs and in the published version of his relevant paper is given with respect to the specific question what information would have been found on a missing piece of Hilbert's proofs. The existing texts suggest that the missing piece contained the explicit form (...) of the Riemann curvature scalar in terms of the Ricci tensor as a specification of the axiomatically underdetermined Lagrangian in Hilbert's action integral. An alternative reading that the missing piece of the proofs already may have contained the Einstein tensor, i.e. an explicit calculation of the gravitational part of Hilbert's Lagrangian is argued to be highly implausible. (shrink)

In this article, we analyze the third of three papers, in which Einstein presented his quantum theory of the ideal gas of 1924–1925. Although it failed to attract the attention of Einstein’s contemporaries and although also today very few commentators refer to it, we argue for its significance in the context of Einstein’s quantum researches. It contains an attempt to extend and exhaust the characterization of the monatomic ideal gas without appealing to combinatorics. Its ambiguities illustrate Einstein’s confusion with his (...) initial success in extending Bose’s results and in realizing the consequences of what later came to be called Bose–Einstein statistics. We discuss Einstein’s motivation for writing a non-combinatorial paper, partly in response to criticism by his friend Ehrenfest, and we paraphrase its content. Its arguments are based on Einstein’s belief in the complete analogy between the thermodynamics of light quanta and of material particles and invoke considerations of adiabatic transformations as well as of dimensional analysis. These techniques were well known to Einstein from earlier work on Wien’s displacement law, Planck’s radiation theory and the specific heat of solids. We also investigate the possible role of Ehrenfest in the gestation of the theory. (shrink)

A formulation by Einstein of the Einstein-Podolsky-Rosen incompleteness argument found in his scientific manuscripts is presented and briefly commented on. It is the only known version in which Einstein discussed the argument for spin observables. The manuscript dates, in all probability, from late 1954 or early 1955 and hence also represents Einstein's latest version of the incompleteness argument and one of his last statements on quantum theory in general. A puzzling formulation raises the question of Einstein's interpretation of space quantization (...) and the non-classical spin degree of freedom. (shrink)

Sixteen years after his “Foundations of Geometry,” Hilbert published a communication that bears a similar and, by use of the definite article, even less mistakable title: “The Foundations of Physics.” In the opening paragraph of this article, Hilbert announced his intention self-confidently:In the following, I should like to set up — following the axiomatic method — a new system of fundamental equations of physics, constructed essentially from two simple axioms; equations that are of ideal beauty and in which, as I (...) believe, is contained the solution of both Einstein’s and Mie’s problems. (shrink)

Albert Einstein ist für seine Arbeiten in der Physik weltberühmt. Nur wenige wissen jedoch, dass Einstein selbst auch philosophische Arbeiten publiziert hat und seine Erkenntnisse weitreichende Folgen für die Philosophie haben. Oder haben „Raum“ und „Zeit“ nichts mit Wissen zu tun?

Different or conflicting accounts of the same episode in the history of science may arise from viewing that episode from different perspectives. The metaphor suggests that conflicting accounts can be seen as complementary, constructing a multi-dimensional understanding, if the different perspectives can be coordinated. As an example, I discuss different perspectives on the Stern-Gerlach experiment. In a static interpretation, the SGE has been viewed as an experiment that allows the determination of the magnetic moment of silver atoms. Based on the (...) concept of magnetic momentum arising from orbital angular momentum, the original experiment was designed in 1922 as an experimentum crucis to decide between Bohr’s quantum theory and classical electromagnetic theory, and its outcome was interpreted as a confirmation of the Bohr-Sommerfeld quantum postulates. After the advent of quantum mechanics, the SGE was reinterpreted in terms of magnetic moment arising from the electron’s spin angular momentum. In a dynamical interpretation, physicists have asked for the physical mechanism responsible for the quantization of the angular momentum with respect to the direction of the magnetic field. Although different suggestions were explored, none was ever accepted as fully satisfactory. Today this difficulty is seen as a paradigmatic instance of the unsolved quantum measurement problem. (shrink)

Einstein's early thoughts about superconductivity are discussed as a case study of how theoretical physics reacts to experimental findings that are incompatible with established theoretical notions. One such notion that is discussed is the model of electric conductivity implied by Drude's electron theory of metals, and the derivation of the Wiedemann-Franz law within this framework. After summarizing the experimental knowledge on superconductivity around 1920, the topic is then discussed both on a phenomenological level in terms of implications of Maxwell's equations (...) for the case of infinite conductivity, and on a microscopic level in terms of suggested models for superconductive charge transport. Analyzing Einstein's manuscripts and correspondence as well as his own 1922 paper on the subject, it is shown that Einstein had a sustained interest in superconductivity and was well informed about the phenomenon. It is argued that his appointment as special professor in Leiden in 1920 was motivated to a considerable extent by his perception as a leading theoretician of quantum theory and condensed matter physics and the hope that he would contribute to the theoretical direction of the experiments done at Kamerlingh Onnes' cryogenic laboratory. Einstein tried to live up to these expectations by proposing at least three experiments on the phenomenon, one of which was carried out twice in Leiden. Compared to other theoretical proposals at the time, the prominent role of quantum concepts was characteristic of Einstein's understanding of the phenomenon. The paper concludes with comments on Einstein's epistemological reflections on the problem. (shrink)

Books about Einstein abound but they sell. Perhaps more than with other subjects, if you want to publish a book about Einstein, you need to delimit your subject matter and target a sizeable audience. Topobiographies, as one might call them, that is, biographies with a focus on a specific location, are a popular way to meet this challenge. You are cutting down your subject matter to manageable proportions and you are addressing a naturally defined readership. With Einstein, topobiographical works almost (...) constitute a genre.Let me mention some examples. Carl Seelig wrote a book about Einstein in Switzerland (Seelig 1952). Max Flückiger (1974) followed his example with a book specifically about “Albert Einstein in Bern.” For Einstein’s Berlin years, not a biography, but a collection of sources was presented in the year of the hundredth anniversary of his birth by Christa Kirsten and Hans-Jürgen Treder (1979). Less topographically constrained, Jamie Sayen (1985) wrote about “Einstein in Ame .. (shrink)

We discuss Einstein’s knowledge of projective geometry. We show that two pages of Einstein’s Scratch Notebook from around 1912 with geometrical sketches can directly be associated with similar sketches in manuscript pages dating from his Princeton years. By this correspondence, we show that the sketches are all related to a common theme, the discussion of involution in a projective geometry setting with particular emphasis on the infinite point. We offer a conjecture as to the probable purpose of these geometric considerations.

Einstein’s early calculations of gravitational lensing, contained in a scratch notebook and dated to the spring of 1912, are reexamined. A hitherto unknown letter by Einstein suggests that he entertained the idea of explaining the phenomenon of new stars by gravitational lensing in the fall of 1915 much more seriously than was previously assumed. A reexamination of the relevant calculations by Einstein shows that, indeed, at least some of them most likely date from early October 1915. But in support of (...) earlier historical interpretation of Einstein’s notes, it is argued that the appearance of Nova Geminorum 1912 (DN Gem) in March 1912 may, in fact, provide a relevant context and motivation for Einstein’s lensing calculations on the occasion of his first meeting with Erwin Freundlich during a visit in Berlin in April 1912. We also comment on the significance of Einstein’s consideration of gravitational lensing in the fall of 1915 for the reconstruction of Einstein’s final steps in his path towards general relativity. (shrink)

Inspired by a question that Einstein had asked him, Piaget analyzed the child's conception of time with a series of experiments that were published in book form in 1946. I briefly recapitulate Piaget's analysis as an interpretation of the conception of absolute time in classical physics. Piaget's suggestions as to how the analysis would carry over to a genetic understanding of time in the special theory of relativity are reviewed. In light of Piaget's work, some observations are made about Einstein's (...) 1905 paper on the `Electrodynamics of Moving Bodies.' The specific transformational operations that mediate between the viewpoints of different inertial observers are characterized as a basis for the cognitive restructuring of spatio-temporal concepts in the relativistic context. (shrink)

This volume collects reflections on the role of philosophy in case studies in the history of science. Case studies have played a prominent role in recent history and philosophy of science. They have been used to illustrate, question, explore, or explicate philosophical points of view. Even if not explicitly so, historical narratives are always guided by philosophical background assumptions. But what happens if different philosophies lead to different narratives of the same historical episodes? Can historical case studies decide between competing (...) philosophical viewpoints? What are the criteria that a case study has to fulfill in order to be philosophically relevant? Bringing together leading practitioners in the fields of history and philosophy of the physical and the life sciences, this volume addresses this methodological problem and proposes ways of rendering explicit philosophical assumptions of historical work. (shrink)

The publication of the first two volumes of the Collected Papers of Albert Einstein in the years 1987 and 1989 marks a watershed in the history of Einstein scholarship. These volumes put together all available documents relevant to Einstein’s early years up to his move to Berne, and they present all his published writings up to 1909, when he would take up his first proper academic appointment at Zurich university. The initiator of the editorial enterprise and editor of these first (...) two volumes, John Stachel, was well aware of the significance of this endeavour. Together with Don Howard, active in the editorial project as well, he also founded the Einstein Studies series whose first volume came out around the same time, in 1989. The series provides a forum for Einstein research, with an emphasis, however, on the history and philosophy of general relativity in those volumes that have appeared up to now. The present volume, number 8 in the series, focuses on the young Einstein and his early work. It has been long in the making. Some papers originated at a conference on Einstein’s early years held in 1990, in an attempt to harvest and digest the fruits of the publication of the first volumes of Einstein’s Collected Papers. Other contributions were written especially for the volume, one paper was published before and is reprinted here. It was the editors’ intention to offer a selection of some of the best recent scholarly studies of Einstein’s early years, and the outcome certainly justifies this claim. (shrink)

This book develops a unified theory of moral progress. The author argues that there are mechanisms in place that consistently drive societies towards moral improvement and that a sophisticated, naturalistically respectable form of teleology can be defended. The book's main aim is to flesh out the process of moral progress in more detail, and to show how, when the right mechanisms and institutions of moral progress are matched together, they create pressure for the desired types of moral gains to manifest. (...) The first part of the book deals with two issues: the conceptual one about what moral progress is, and the broadly empirical one whether it is possible. It shows that cultural evolution successfully explains the origins of modern forms of morally welcome change. The second part argues that there is logical space for a moderate, scientifically credible form of teleology, and that the converse case for moral decline is weak. It addresses the types, drivers and institutions of moral progress that allow for the storage, transmission and cumulative improvement of our normative infrastructure over time. Finally, the third part demonstrates why moral progress cannot be accounted for in metaethically realist terms. Moral Teleology will be of interest to researchers and advanced students working in ethics, moral epistemology and moral psychology. (shrink)

Many accounts of analogy based on sentential semantics owe their continued popularity more to a lack of theoretical specificity than to their superior explicative power. I examine a recent attempt to remedy this situation.Conclusion: Once the sentential semantics account of analogy is spelled out in sufficient detail to permit its systematic application to a variety of cases, it quickly becomes apparent why it must fail, and why we should give preference to a multi-constraint theory of cognitive process instead.

Die Religionsschrift und die Predigten Schleiermachers sind sowohl theoretische Entfaltung als auch literarisch-rhetorischer Vollzug von Religionskommunikation. Beiden Aspekten geht Mirjam Sauer nach. Im Licht frühromantischer Literaturproduktion und spätaufklärerischer Predigten interpretiert sie die erschlossenen Kommunikationsmodelle als Beiträge zu einer Theologie religiöser Rede an der Schwelle zur Moderne.

In hindsight, it is primarily Susanne K. Langer’s theory of act, and only secondarily her theory of art, that is central to the conception of Vitality Semiotics. It focuses on affective, semiotically relevant forms that constitute our world experience, human social interaction, and ultimately art experience. Thus, this somewhat unusual distinction between these two aspects of Langer’s work is not only important for art and our understanding of the world, but can also be seen as fundamental to social interaction and, (...) furthermore, to the universal formation of culture(s). It is, as the article suggests, to be regarded as the basis of the conditio humana. (shrink)

Die Beurteilung von Handlungen als ethisch erlaubt gehort zum moralischen Bewusstsein und zur moralischen Verstandigung. In der theologischen Ethik gab es jedoch Kritik an der Kategorie des ethisch Erlaubten, am grundsatzlichsten bei Friedrich Schleiermacher. Die Kritiker vertraten die These, dass sich alle moglichen Handlungen als entweder moralisch geboten oder verboten ausweisen liessen und dass eine mittlere Kategorie des bloss Erlaubten die Klarheit ethischer Verbindlichkeit trube. Fuss analysiert die Argumentationen reprasentativer evangelischer Ethiker vor allem des 19. und fruhen 20. Jahrhunderts, die (...) einerseits fur die Kritik am Begriff des Erlaubten und andererseits fur seine Rehabilitierung stehen. Dabei wird das Erlaubnisproblem in seiner Beziehung zu anderen ethischen Kontroversthemen gesehen, wie etwa den Problemen des supererogatorischen Handelns oder der Adiaphora. Das Schlusskapitel beschreibt den Begriff des Erlaubten im Kontext einer Lehre von elementaren Pflichten. (shrink)