In spite of the many efforts made to clarify von Neumann’s methodology of science, one crucial point seems to have been disregarded in recent literature: his closeness to Hilbert’s spirit. In this paper I shall claim that the scientific methodology adopted by von Neumann in his later foundational reflections originates in the attempt to revaluate Hilbert’s axiomatics in the light of Gödel’s incompleteness theorems. Indeed, axiomatics continues to be pursued by the Hungarian mathematician in the spirit of Hilbert’s school. I (...) shall argue this point by examining four basic ideas embraced by von Neumann in his foundational considerations: a) the conservative attitude to assume in mathematics; b) the role that mathematics and the axiomatic approach have to play in all that is science; c) the notion of success as an alternative methodological criterion to follow in scientific research; d) the empirical and, at the same time, abstract nature of mathematical thought. Once these four basic ideas have been accepted, Hilbert’s spirit in von Neumann’s methodology of science will become clear. (shrink)

Reconstructions of quantum theory are a novel research program in theoretical physics which aims to uncover the unique physical features of quantum theory via axiomatization. I focus on Hardy’s “Quantum Theory from Five Reasonable Axioms” (2001), arguing that reconstructions represent a modern usage of axiomatization with significant points of continuity to von Neumann’s axiomatizations in quantum mechanics. In particular, I show that Hardy and von Neumann share similar methodological ordering, have a common operational framing, and insist on the empirical basis (...) of axioms. In the reconstruction programme, interesting points of discontinuity with historical axiomatizations include the stipulation of a generalized space of theories represented by a framework and the stipulation of analytic machinery at two levels of generality (first by establishing a generalized mathematical framework and then by positing specific formulations of axioms). In light of the reconstruction programme, I show that we should understand axiomatization attempts as being context–dependent, context which is contingent upon the goals of inquiry and the maturity of both mathematical formalism and theoretical underpinnings within the area of inquiry. Drawing on Mitsch (2022)’s account of axiomatization, I conclude that reconstructions should best be understood as provisional, practical, representations of quantum theory that are well suited for theory development and exploration. However, I propose my context–dependent re–framing of axiomatization as a means of enriching Mitsch’s account. (shrink)

We discuss the mathematical structures that underlie quantum probabilities. More specifically, we explore possible connections between logic, geometry and probability theory. We propose an interpretation that generalizes the method developed by R. T. Cox to the quantum logical approach to physical theories. We stress the relevance of developing a geometrical interpretation of quantum mechanics.

Berichte zur Wissenschaftsgeschichte, EarlyView.

Once one of the main protagonists of history of science, the historiography on quantum theory has recently gone through a process of reconfiguration of methods, research questions and epistemological framework. In this paper, I review the recent developments and propose some reflections on its future evolution.

The essential role of classical mechanics in the “old quantum theory” is well known. With the rise of a genuine quantum formalism, classical analogies remained a powerful heuristic tool. However, classical insights soon proved problematic, and in some cases, even counterproductive. The case of the implementation of quantum canonical transformations provides a distinguished case study for the historian studying the circumstances which led to the transformation theory of London, Dirac and Jordan. -/- The attempts to use canonical transformations in strict (...) analogy to classical practice met serious difficulties as soon as one tried to accommodate the action-angle scheme. At some point, the very consistency of quantization went addressed because of the unclear quantum equivalence of classically equivalent Hamiltonians. -/- The early attempts, often beset with ill-defined problems, are illustrative of the lack of understanding of the underlying mathematical scheme. The latter had to be properly appraised before the problem of the meaning of changing variables in quantum theory could be solved. This is why the latter proved instrumental in the discovery of transformation theory. -/- Some of the problems which arose then are still highly instructive from the foundational point of view, and worth to be rediscovered, beyond their historical interest. (shrink)

Traditionally, “the operator calculus of Born and Wiener” has been considered one of the four formulations of quantum mechanics that existed in 1926. The present paper reviews the operator calculus as applied by Max Born and Norbert Wiener during the last months of 1925 and the early months of 1926 and its connections with the rise of the new quantum theory. Despite the relevance of this operator calculus, Born–Wiener’s joint contribution to the topic is generally bypassed in historical accounts of (...) quantum mechanics. In this study, we analyse the paper that epitomises the contribution, and we explain the main reasons for the apparent lack of interest in Born and Wiener’s work. We argue that they did not solve the main problem for which the tool was intended, that of linear motion, because of their reluctance to use Dirac delta functions. (shrink)

Although it has become a common place to refer to the ׳sixth problem׳ of Hilbert׳s (1900) Paris lecture as the starting point for modern axiomatized probability theory, his own views on probability have received comparatively little explicit attention. The central aim of this paper is to provide a detailed account of this topic in light of the central observation that the development of Hilbert׳s project of the axiomatization of physics went hand-in-hand with a redefinition of the status of probability theory (...) and the meaning of probability. Where Hilbert first regarded the theory as a mathematizable physical discipline and later approached it as a ׳vague׳ mathematical application in physics, he eventually understood probability, first, as a feature of human thought and, then, as an implicitly defined concept without a fixed physical interpretation. It thus becomes possible to suggest that Hilbert came to question, from the early 1920s on, the very possibility of achieving the goal of the axiomatization of probability as described in the ׳sixth problem׳ of 1900. (shrink)

This paper re-evaluates the formative year of quantum mechanics—from Heisenberg’s first paper on matrix mechanics to Schrödinger’s equivalence proof—by focusing on the role of radiation in the emerging theory. We argue that the radiation problem played a key role in early quantum mechanics, a role that has not been taken into account in the standard histories. Radiation was perceived by the main protagonists of matrix and wave mechanics as a central lacuna in these emerging theories and continued to contribute to (...) the theoretical development and conceptual clarification of quantum mechanics. Studying the interplay between quantum mechanics and radiation, the paper provides an account of (a) how quantum mechanics was able to connect to its empirical basis in spectroscopy and (b) how Schrödinger’s equivalence proof emerged from his explorative calculations on the emission of radiation. (shrink)