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Ideas and Ideals

On the Nature of Probability in Mathematics and Physics

Kirill Mikhaylov

Published Sep 25, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.17212/2075-0862-2026-18.3.2-476-492

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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.
Abstract In this chapter I revisit the frequency interpretation of probability of von Mises, and argue that it contains the key elements of any coherent interpretation of scientific probability—notwithstanding the heavy criticisms von Mises collected from mathematicians and philosophers alike. I propose a modified definition that is not based on von Mises’ ‘infinite collectives’ but retains an essential ingredient of his interpretation, namely that probability can only quantitatively be defined for events that are, or can be, repeated in similar conditions and that exhibit frequency stabilisation. New is that the mentioned ‘conditions’ should be ‘partitioned’. Thus, I will partition probabilistic systems into object and environment, or in object, initiating, and probing subsystem, and show that such partitioning solves a series of classic problems. Other consistent interpretations can be unified with frequentism, e.g. Bayesianism à la Jaynes. I explore in detail the tantalising subjective or operationalist touch of probability, and argue that it can be put on objective grounds. One further upshot of the model is that several quantum puzzles can be seen as deriving from an imprecise interpretation of probability.
The aim of the article is to analyze the ontological foundations of quantum mechanics and the relationship of various interpretations of quantum mechanics with the question of truth in scientific knowledge. The theoretical material (various interpretations of quantum mechanics) was analyzed. The research methods were analysis, comparative-historical and institutional approaches. On the basis of the study, it can be concluded that physicists are only able to predict the probabilities of physical phenomena or processes, without any knowledge of the structure of entity in itself. The study helps to complement the theoretical picture of the ontology of quantum mechanics and to get an idea about the development prospects of the methodology of physical sciences. We should not ascribe properties that are unobservable in experience to the entity because science can truly interpret only the representation of entity in itself in experience and the ways of dealing with it, not future events or possible states of the entity. Otherwise, science “slides” down to the level of dialectical (rhetorical) exercises based not on true premises, but on generally accepted or “authoritative” biases, which gives room to extreme forms of relativism and takes natural scientists away from studying nature to inventing empirically unsupported hypotheses. When, in accordance with the Cartesian ontology, the scientist assumes that the entity is as the “subject” observes it (that is, the “reflection” of the entity in consciousness is identified with entity in itself), this assumption leads the scientist to attempts to “scientifically” describe the future events or the possible states of the entity. The criterion for the truth of a scientific theory put forward by Einstein in his dispute with Bohr remains a fundamentally unattainable ideal. Cognition of nature is probabilistically and fundamentally incomplete since a person can consciously comprehend natural phenomena only within the framework of their representation in the existing being. The probabilistic nature of quantum mechanics is due to the impossibility of knowing things as they exist by themselves, that is, as they exist outside of representation in the mind of the observer. This does not mean that physical entities did not exist before their knowledge: they existed potentially, but were not valid for the observer, that is, conscious. Quantum physics is as incomplete as any other physical theory when compared with an unattainable ideal, but this fact does not negate its truth for modern scientists since quantum theory is based on the results of verified experiments.
Abstract Many theories and models from physics are probabilistic. This observation raises several philosophical questions: What are probabilities in physics? Do they reflect objective chances which exist independently of the human mind? Or do they only express subjective credences and thus capture our own uncertainty about the world? Finally, which metaphysical lessons, if at all, can one draw from the largely probabilistic character of physics? The chapters collected in this volume address these questions and provide a detailed philosophical appraisal of the status of probabilities in all of physics. Particular emphasis is laid upon statistical physics and quantum mechanics. Many chapters reflect a desire to understand probabilities from physics as objective chances. These chances are characterized, e.g., as time-averages, as probabilities from a best system in the terms of David Lewis, or using the Boltzmannian typicality approach. Other chapters are sympathetic to a Bayesian view of probabilities in physics. The chapters about quantum mechanics elucidate the peculiar characteristics of quantum correlations and discuss strategies to justify the Born Rule. Finally, the chapters of this volume demonstrate how closely interpretive issues about probabilities are entangled with other foundational problems of physics such as the Reversibility Paradox, the ontology of the quantum world and the question whether the world is deterministic.
This article analyzes the Copenhagen interpretation of quantum mechanics with regard to its historical role, logical consistency, and ontological assumptions. It argues that the Copenhagen interpretation has been extraordinarily successful as a practical framework connecting the mathematical formalism of quantum mechanics with experimental results, but that its positivist restriction to observable quantities does not provide a complete physical explanation of quantum wave behaviour. The analysis of other interpretations suggests that no interpretation of quantum mechanics, including the Copenhagen interpretation, can avoid some form of hidden entity. The central argument is based on the double-slit experiment: probability or mere possibility cannot by themselves physically cause interference, and therefore some real wave-like entity must exist whose effect is only probabilistically described by the wave function. The wave function is therefore not understood as a physical wave itself, but as an epistemic description of the interaction between a particle and a deeper hidden wave structure. The Copenhagen interpretation need not be simply rejected; rather, it can be understood as a successful simplification of a deeper interpretation. It accurately describes measurable results, but it does not explain the real mechanism that produces them. The article suggests that inspiration for such a deeper view can be found in quantum field theory: particles are perturbations of fields, and quantum entanglement may indicate the existence of a deeper nonlocal field of entanglement.
In the present article the author will deal with an inquiry of the ontological structure of quantum mechanics, and he will analyze the evolution of the heno-ontological equation (the identity of being one and of being) in the shape of the new arrangement prepared by the orthodox interpretation of the theory.Hence the article will particularly focus on the Copenhagen interpretation and the lack of intuition [Anschaulichkeit] in the representation of physical external objects; as far as the latter topic is concerned, the author will also consider the classic representation of Heisenberg's philosophy given by Heelan.The author will then show the impossibility of holding the old physical body-representation which worked for classical mechanics as well, because quantum mechanics denies an absolute localizability of physical entities.A non-defective determination of localizability was in fact the very postulate which lays at the basis of the heno-ontological equation as Aristotle had envisaged it: what is not localizable, is not one being.It also has to be considered a specific lack as regards the chronological permanence of the identity of the quantum object, which appears to be constantly modified by its interaction with the devices and with other particles.Accordingly, the author will explain the peculiar definition of physical entity in the Copenhagen interpretation of quantum mechanics as a non-substantial entity, as a non-body, considering the argument that henceforth the actual substance, in physics, will be only the whole totality.Arguing about the latter point, the author will develop a definition of individuality which could not be extraneous to the most recent interpretations of quantum mechanics.He will indeed aim for a non strictly ontological definition of individuality, if one has to figure out the individuum as a body of daily experience.Rather than a body, the quantum individuum is almost a being whose ontological content is only relative to the transformations of a primitive context: this implies that his ontological charge cannot be considered permanent.From this point of view, one can argue that quantum mechanics is very close to the old representation of nature as being by the ancient Greek physicists, which tended to distinguish between the ontological definition of individual entities and the unity of being as a whole.Hence the article will show, through Cassirer's review, how it is possible to depict a new setting for the relationship between one and being, as well as trying to shed some light on the basic assumptions of the ontology of Copenhagen interpretation and of modern quantum mechanics.
I review the role of probability in contemporary physics and the origin of probabilistic time asymmetry, beginning with the pre-quantum case (both stochastic mechanics and classical statistical mechanics) but concentrating on quantum theory. I argue that quantum mechanics radically changes the pre-quantum situation and that the philosophical nature of objective probability in physics, and of probabilistic asymmetry in time, is dependent on the correct resolution of the quantum measurement problem.