Arkady Plotnitsky

Reality Without Realism: On the Ontological and Epistemological Architecture of Quantum Mechanics

First, this article considers the nature of quantum reality (the reality responsible for quantum phenomena) and the concept of realism (our ability to represent this reality) in quantum theory, in conjunction with the roles of locality, causality, and probability and statistics there. Second, it offers two interpretations of quantum mechanics, developed by the authors of this article, the second of which is also a different (from quantum mechanics) theory of quantum phenomena. Both of these interpretations are statistical. The first interpretation, by A. Plotnitsky, “the statistical Copenhagen interpretation,” is nonrealist, insofar as the description or even conception of the nature of quantum objects and processes is precluded. The second, by A. Khrennikov, is ultimately realist, because it assumes that the quantum-mechanical level of reality is underlain by a deeper level of reality, described, in a realist fashion, by a model, based in the pre-quantum classical statistical field theory, the predictions of which reproduce those of quantum mechanics. Moreover, because the continuous fields considered in this model are transformed into discrete clicks of detectors, experimental outcomes in this model depend on the context of measurement in accordance with N. Bohr’s interpretation and the statistical Copenhagen interpretation, which coincides with N. Bohr’s interpretation in this regard.

Are quantum-mechanical-like models possible, or necessary, outside quantum physics?

This article examines some experimental conditions that invite and possibly require recourse to quantum-mechanical-like mathematical models (QMLMs), models based on the key mathematical features of quantum mechanics, in scientific fields outside physics, such as biology, cognitive psychology, or economics. In particular, I consider whether the following two correlative features of quantum phenomena that were decisive for establishing the mathematical formalism of quantum mechanics play similarly important roles in QMLMs elsewhere. The first is the individuality and discreteness of quantum phenomena, and the second is the irreducibly probabilistic nature of our predictions concerning them, coupled to the particular character of the probabilities involved, as different from the character of probabilities found in classical physics. I also argue that these features could be interpreted in terms of a particular form of epistemology that suspends and even precludes a causal and, in the first place, realist description of quantum objects and processes. This epistemology limits the descriptive capacity of quantum theory to the description, classical in nature, of the observed quantum phenomena manifested in measuring instruments. Quantum mechanics itself only provides descriptions, probabilistic in nature, concerning numerical data pertaining to such phenomena, without offering a physical description of quantum objects and processes. While QMLMs share their use of the quantum-mechanical or analogous mathematical formalism, they may differ by the roles, if any, the two features in question play in them and by different ways of interpreting the phenomena they considered and this formalism itself. This article will address those differences as well.

Mysteries without Mysticism and Correlations without Correlata: On Quantum Knowledge and Knowledge in General

Following Niels Bohrs interpretation of quantum mechanics as complementarity, this article argues that quantum mechanics may be seen as a theory of, in N. David Mermins words, “correlations without correlata,” understood here as the correlations between certain physical events in the classical macro world that at the same time disallow us to ascertain their quantum-level correlata.

A Matter of Principle: The Principles of Quantum Theory, Dirac’s Equation, and Quantum Information

This article is concerned with the role of fundamental principles in theoretical physics, especially quantum theory. The fundamental principles of relativity will be addressed as well, in view of their role in quantum electrodynamics and quantum field theory, specifically Dirac’s work, which, in particular Dirac’s derivation of his relativistic equation of the electron from the principles of relativity and quantum theory, is the main focus of this article. I shall also consider Heisenberg’s earlier work leading him to the discovery of quantum mechanics, which inspired Dirac’s work. I argue that Heisenberg’s and Dirac’s work was guided by their adherence to and their confidence in the fundamental principles of quantum theory. The final section of the article discusses the recent work by D’Ariano and coworkers on the principles of quantum information theory, which extend quantum theory and its principles in a new direction. This extension enabled them to offer a new derivation of Dirac’s equations from these principles alone, without using the principles of relativity.

Experimenting with Ontologies: Sets, Spaces, and Topoi with Badiou and Grothendieck

The paper explores the ontology and logic of the irreducibly multiple in set theory and in topos theory by considering the differences between Badious logical and Grothendiecks ontological approach to topos theory. It argues that Grothendiecks ontological program for topos theory leads to a more radical concept of the multiple than does the set-theoretical ontology, which defines Badious view of ontology even in his later, more topos theoretically oriented work. Extending Grothendiecks way of thinking to other fields enables one to give ontological multiplicities—no longer bound by the set-theoretical ontology or ultimately by any mathematical ontology, even in mathematics—a great diversity and richness. It follows that the set-theoretical ontology is not sufficiently rich to accomplish what Badiou thinks it could accomplish even in mathematics itself, let alone elsewhere; and Badiou wants it to work elsewhere—indeed, wherever it is possible to speak of ontology. I shall also consider, in closing, some implications of the arguments for the workings of the multiple in ethics, politics, and culture.

Prediction, repetition, and erasure in quantum physics: experiment, theory, epistemology

The article considers the epistemology of quantum phenomena and the quantum eraser experiments as reflecting certain essential features of these phenomena, and uses these experiments to introduce a new concept, that of ‘erasure’. This concept is defined by the fact that a given quantum measurement destroys, ‘erases’, the usefulness of actual or even possible information associated with a given quantum system prior to this measurement, for the purposes of predictions concerning the experiments performed on this system after this measurement. The concept of ‘erasure’ allows one to capture certain fundamental aspects of quantum phenomena more sharply than previously, and to differentiate classical and quantum phenomena, and theories in a new way.

“This is an Extremely Funny Thing, Something Must be Hidden Behind That”: Quantum Waves and Quantum Probability with Erwin Schrödinger

The aim of this article is to reassess the significance of quantum waves and of Erwin Schrodinger’s work by extending Max Born’s 1926 interpretation of Schrodinger’s wave function in terms of probability to the viewpoint of the modern‐day quantum information theory, which, I argue, was anticipated by Schrodinger in his cat paradox paper, “Die gegenwartige Situation in der Quantenmechanik” [The Present Situation in Quantum Mechanics], published in 1935.

Comprehending the Connection of Things: Bernhard Riemann and the Architecture of Mathematical Concepts

This chapter is an essay on the conceptual nature of Riemann’s thinking and its impact, as conceptual thinking, on mathematics, physics, and philosophy. In order to fully appreciate the revolutionary nature of this thinking and of Riemann’s practice of mathematics, one must, this chapter argues, rethink the nature of mathematical or scientific concepts in Riemann and beyond. The chapter will attempt to do so with the help of Deleuze and Guattari’s concept of philosophical concept. The chapter will argue that a fundamentally analogous concept of concept is also applicable in mathematics and science, specifically and most pertinently to Riemann, in physics, and that this concept is exceptionally helpful and even necessary for understanding Riemann’s thinking and practice, and creative mathematical and scientific thinking and practice in general.

The visualizable, the representable and the inconceivable: realist and non-realist mathematical models in physics and beyond.

The project of this article is twofold. First, it aims to offer a new perspective on, and a new argument concerning, realist and non-realist mathematical models, and differences and affinities between them, using physics as a paradigmatic field of mathematical modelling in science. Most of the article is devoted to this topic. Second, the article aims to explore the implications of this argument for mathematical modelling in other fields, in particular in cognitive psychology and economics.

On foundational thinking in fundamental physics, from Riemann to Einstein to Heisenberg

This paper considers the nature of foundational thinking in fundamental physics, most especially in quantum mechanics. By “fundamental physics” I mean those areas of experimental and theoretical physics that deal with the ultimate constitution of nature, for example, as defined by the so-called elementary particles in the case of quantum physics. By “foundational thinking” I mean thinking that concerns fundamental physics itself. First, I argue, following Riemann, that our foundational thinking is based on hypotheses that we form and test. Second, I argue that foundational thinking in physics is defined by concepts, and that in modern physics foundational concepts always contains physical, mathematical, and philosophical components. Third, finally, I argue that the relationships between these components and, hence, our foundational thinking, are different in quantum mechanics than they are in classical physics and relativity. In these theories mathematics describes, by way of idealized models, physical re...