In his book `Physics and Philosophy', Heisenberg suggested that the quantum world is one of ``potentialities or possibilities'' and that the classical realm is one of ``things or facts''. After ascertaining that his categories most naturally have the structure of ontological equivalence classes, we show that they cannot be seriously introduced into the quantum formalism without rendering it incoherent, as under some circumstances the formalism permits an equality between members of distinct equivalence classes. This is labeled the `equality of inequivalents problem'. Three possible reactions are discussed: First, one could deny the problem by challenging some of the assumptions that underlie its formulation; second, one could accept the problem and take it to be grounds for dismissing Heisenberg's or similar distinctions altogether; third, one could accept the problem but take its source to be not Heisenberg's distinction but the quantum formalism. A plausibility argument in support of the third reaction is given by analogy: it is shown that axiomatic probability suffers from a similar problem if one wishes to seriously distinguish between things which are possibilities and things which are not. An enriched axiomatization of probability is proposed which captures the concept of probability as a measure over possibilities and thereby overcomes the equality of inequivalents problem in axiomatic probability.
In his book Physics and Philosophy', Heisenberg suggested that the quantum world is one of ``potentialities or possibilities'' and that the classical realm is one of ``things or facts''. After ascertaining that his categories most naturally have the structure of ontological equivalence classes, we show that they cannot be seriously introduced into the quantum formalism without rendering it incoherent, as under some circumstances the formalism permits an equality between members of distinct equivalence classes. This is labeled the equality of inequivalents problem'. Three possible reactions are discussed: First, one could deny the problem by challenging some of the assumptions that underlie its formulation; second, one could accept the problem and take it to be grounds for dismissing Heisenberg's or similar distinctions altogether; third, one could accept the problem but take its source to be not Heisenberg's distinction but the quantum formalism. A plausibility argument in support of the third reaction is given by analogy: it is shown that axiomatic probability suffers from a similar problem if one wishes to seriously distinguish between things which are possibilities and things which are not. An enriched axiomatization of probability is proposed which captures the concept of probability as a measure over possibilities and thereby overcomes the equality of inequivalents problem in axiomatic probability.
Abstract Apparent paradoxes in Quantum Physics demand a sharp distinction between a "real part of the world" and the realm of possibilities. In the former we may distinguish individual elements (coarse events) relating to space-time, with causal connections between them respecting the relativistic locality principle. The latter, quantitatively described by probability assignments conditioned on existing facts and depending on the definition of equivalence classes of situations, cannot be subdivided in space-time categories. There are global correlations (Pauli principle...). The cut between the realms of facts and possibilities implies an evolutionary picture of the "real world" in which the (generalized) arrow of time assumes basic significance. Some deficiencies of existing theory are pointed out.
THE conceptual problems raised by the discovery of quantum effects have not been fully resolved after half a century. Part of the reason for this is undoubtedly to be found in the mutual ignorance which prevails between physicists and philosophers. In his book Heisenberg brings together a philosophically inclined temperament with an unrivalled knowledge of physics. The result is a book of very great interest, however much one might disagree with his conclusions. The collection of essays of which the other book is made up is intended to bring crucial conceptual problems in physics especially to the notice of philosophers. The design and intention of the collection is admirable for this is the proper way to overcome the departmentalization of interest that has left the deep conceptual problems still unresolved. Men like Heisenberg who combine philosophical temperament with scientic knowledge of great distinction are rare, and yet the difficulties can only be cleared up by the application of philosophical techniques of analysis to problems which are an intimate and difficult part of physics. A great deal of the interest of Heisenberg's book lies in the very comprehensive account he gives of the origin and growth of the Copenhagen school, and the way the new phenomena forced a largely unwelcome picture of nature upon Bohr and his friends. The most important chapter from a philosophical point of view is that in which Heisenberg defends this interpretation against the attacks to which it has been subjected. The essence of his argument for the interpretation can be summed up in two propositions. (i) The Copenhagen interpretation of physical theory, which depends upon the acceptance of the so-called Principle of Complementarity, is an inevitable and final interpretation. The reason for this is that there are only two possible general models for understanding given physical phenomena; a particle model or a wave model. These are exhaustive and exclusive because the former depends upon discontinuous and the latter upon continuous (using substance here in the traditional philosophical sense). If we find, as we do, that the two fundamental constituents of the universe, matter and radiation, each exhibit two classes of phenomena, one explainable only on a particle model and one on a wave model, then we are compelled to adopt a complementarity hypothesis and give up the idea of finding a single and uniform model for the understanding of phenomena. There are only two kinds of models; they are mutually exclusive; both are required; hence we either despair before an insoluble problem or adopt the Copenhagen interpretation. (2) Any serious criticism of this interpretation must offer an alternative
The article discusses approaches to understanding the nature of probability in mathematics and physics with the consequences of these approaches, as well as the difficulties that arise. The ontological status of probability is discussed. It is shown that probability theory, contrary to R. Mises, should be classified as a mathematical science, using the definition formulated by A.N. Kolmogorov. The application of this definition in practice is provided within the framework of the classical approach, provided that the connection with physical reality is defined through the concept of equal opportunity. To do this, the concepts of simple and complex events are introduced, and the method of decomposing complex events into simple ones is explained. The point of view is substantiated, according to which a mathematical probability determined and related to physical reality in this way can be considered an objective probability, that is, existing independently of any tests. Various interpretations of quantum mechanics and approaches to understanding probability are also revealed. It is shown that in physics today the existing approaches to understanding the nature of probability contradict each other. In the Bohm interpretation, probability is an analogue of the classical approach – statistical objective probability, which has a mathematical formalization, and in quantum Bayesianism, probability quantifies the degree of reliability and depends on the subject conducting the experiment. The Copenhagen interpretation of quantum mechanics tries to eliminate the difficulties associated with combining the wave and corpuscular properties of light and electron beams into a single picture by giving the ontological status of probability, which leads to the problem of understanding physical reality. The article argues for the need for a new connection between the mathematical description of probability and physical reality. The empirical (frequency) approach of R. Mises is analyzed, which causes a lot of criticism, as it does not have a clear consistent formalization. The article argues that this approach is inconsistent with quantum mechanics. The probability given by M. Born in the Copenhagen interpretation of quantum mechanics is also considered, which formulates a new definition of probability, which does not exclude the difficulties associated with the physical interpretation of probability. Born’s approach is based on the classical approach and is reduced to a formal axiomatic definition, which assumes the interpretation of probability as a mathematical construct. The connection of probability with physical reality and the consistency of the statistical method of finding probability values remain unresolved problems, which creates a dualism of understanding probability: in mathematics – one thing, in physics – another. If in the mathematical approach, the solution of these problems is carried out through the connection of probability with physical properties, then in quantum mechanics, in the Copenhagen interpretation, problems persist, which is caused by giving the wave function an ontological status.
Einstein's article on the EPR paradox is the most cited of his works, but not many know that it was not fully representative of the way he thought about the incompleteness of the quantum formalism. Indeed, his main worry was not Heisenberg's uncertainty principle, which he accepted, but the experimental non-separability of spatially separate systems. The same problem was also recognized, years later, by one of us, as part of an axiomatic analysis of the quantum formalism, which revealed an unexpected structural limitation of the quantum formalism in Hilbert space, preventing the description of separate systems. As we will explain, this limitation does not manifest at the level of the states, but of the projectors describing the properties, in the sense that there are not enough properties in the formalism to describe separate systems. The question remains whether separability is a possibility at the fundamental level and if a formalism should integrate it into its mathematical structure, as a possibility. To aid our intuition, we offer a reflection based on a powerful analogy between physical systems and human conceptual entities, as the question of separability also arises for the latter.
The discovery of quantum physics opened a broad debate on the epistemological status of the principle of causality. In this article, I analyze the problem of causality in Heisenberg by distinguishing two levels of philosophical explanation in his thought. As a philosopher of science, his understanding of causality is closely linked to Moritz Schlick, although he will later adopt a neo-Kantian view of the a priori. As a philosopher of nature, Heisenberg recovers the notion of potency to understand the wave function. My thesis is that this notion of potency could have allowed him to elaborate a realistic and nondeterministic theory of causality, but the identification of causality and determinism (in Schlick’s line) did not lead him to such a conclusion.
This paper investigates the correspondence between unactualized certainties, interpreted ontologically, and actualities. It does this first through the lens of a recently proposed enrichment of axiomatic probability which makes it possible to distinguish between actualities and unactualized possibilities, including those which are certain. Two kinds of unactualized certainties are considered: those due to the sample space being a singleton, and those involving a sample space with more than one element. After comparing standard axiomatic probability with the enrichment in regards to how they represent the distinction, attention is then focused on quantum mechanics. There, the correspondence will be examined through the lens of a recently proposed modification of the standard formalism, the Heisenberg Interpretation which, unlike the standard quantum formalism but like the enrichment of probability, also permits distinctions between unactualized possibilities and actualities. Two situations are found to exemplify the correspondence there: one involving partially measured entangled systems and the other involving the Born rule. Finally, it will be shown how the correspondence clarifies the very concept of an unactualized possibility.