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)
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?
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.
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)
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)
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)
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)
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)
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)
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)
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 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)
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)