The informal aspects, arising in the interpretation of physical experiments, of the theory of probability and mathematical statistics are discussed. The conditions that verifying experiments must satisfy are presented and the role of heuristic (extralogical) assertions is analyzed using the example of mathematical expectation. The principal hypotheses implicit in experiments are enumerated: the principle of reproducibility (the past will be repeated in the future); the principle of reasonable sufficiency; and, the statistical principle (better to predict something rather than nothing). Considerable attention is devoted to Fisher and multisample confidence intervals. It is noted that Fisher confidence intervals are inconsistent. The arguments for introducing contrivances into practical calculations of probabilities are enumerated: incompleteness of any system of hypotheses; subjective estimates of probabilities; adjoining of statistical ensembles; nonstationariness and instability; rare phenomena; and, the use of classical probabilities and the law of large numbers. It is concluded that the relative frequency of appearance (empirical probability) is a normal physical quantity in the sense that it admits physical measurement. Its abnormality is manifested in the fact that it is burdened, more than other physical quantities, with conventions and hypotheses which must be specially checked (verified).
The informal aspects, arising in the interpretation of physical experiments, of the theory of probability and mathematical statistics are discussed. The conditions that verifying experiments must satisfy are presented and the role of heuristic (extralogical) assertions is analyzed using the example of mathematical expectation. The principal hypotheses implicit in experiments are enumerated: the principle of reproducibility (the past will be repeated in the future); the principle of reasonable sufficiency; and, the statistical principle (better to predict something rather than nothing). Considerable attention is devoted to Fisher and multisample confidence intervals. It is noted that Fisher confidence intervals are inconsistent. The arguments for introducing contrivances into practical calculations of probabilities are enumerated: incompleteness of any system of hypotheses; subjective estimates of probabilities; adjoining of statistical ensembles; nonstationariness and instability; rare phenomena; and, the use of classical probabilities and the law of large numbers. It is concluded that the relative frequency of appearance (empirical probability) is a normal physical quantity in the sense that it admits physical measurement. Its abnormality is manifested in the fact that it is burdened, more than other physical quantities, with conventions and hypotheses which must be specially checked (verified).
In this expository paper we discuss the role of probability in statistical physics: it should be that of a general tool guiding induction and must not be restricted only to the empirical interpretation coming from its evaluation by an observed frequency. It is true that in many cases the value of a probability can be sensibly expressed by a suitable frequency, but this does not entails that the latter can be taken as definition of probability. Moreover, in each situation in which one thinks that probabilities are physically real things and believes that something which is only “very probable” can be regarded as being “practically certain”, some “small” probabilities are actually ignored: this attitude of mind may render illegitimate also the probabilistic interpretation of the relevant physical laws. A “law” which is based on a probabilistic interpretation cannot aim at explaining why a fact that we forecast will occur, but rather at explaining why we forecast that this fact will occur. Finally, we emphasize the role of conditional probability, and we show that looking at it from the right perspective allows to avoids also classic paradoxical aspects of quantum mechanics, without the need to resort to the interpretation based on the dualism “particle-wave”.
In a deterministic theory, one can deduce the state of the universe at any one time from the dynamical laws that govern its evolution and its state at any other time.In particular, one can deduce from the conditions that obtained at some early boundary condition for the universe, its state at all subsequent times.It is a piece of well-received philosophical common sense that there is no room in a deterministic theory for objective probabilities.Arguments will be given below that this piece of philosophical common sense is mistaken, and that a probability measure over the space of physically possible trajectories is an indispensable component of any theory -deterministic or otherwise -that can be used as a basis for prediction or receive confirmation from the evidence.Phase space is the central representational space in physics.It is the space within which all of the states pertaining to a system are represented, and it encodes all of the general information that we have about those states.To take an obvious example (less obvious examples will be given below) the fact that a pair of quantities are represented by different dimensions in phase space encodes the fact that they represent separate degrees of freedom.Physically possible histories are represented by trajectories through phase space and in a deterministic theory, when dealing with a closed system, there is (no more than) one trajectory through each point.Since the universe as a whole is the only truly closed system, this means there is one trajectory through each point in the phase space of the universe.Probabilities may be invoked in applications of theory, when the state of a system or the values of external variables are not known with full precision, but they are eliminable given complete knowledge.And they don't appear in the statement of the laws.The laws serve simply to identify the physically allowable trajectories.About Newtonian dynamics, Laplace famously said "given for one instant an intelligence which could comprehend all the forces by which nature is animated and the respective situation of the beings who compose it -an intelligence sufficiently vast to submit these data to analysis -it would embrace in the same formula the movements of the greatest bodies of the universe and those of the lightest atom; for it, nothing would be uncertain and the future, and the past, would be present to its eyes."ii It is a short step from this to the idea that probabilities invoked in classical contexts are subjective rather than objective.If they represent anything, it is facts about our epistemic states, not facts about the world.iii Popper, and Lewis, and other influential figures have endorsed this view, categorically rejecting any role for objective probabilities indeterministic contexts.In Popper's words:
Albert Einstein once said that he does not believe in a God who plays dice. He was referring to the fundamental principle that quantum theory permits only statistical assertions concerning the values of physical quantities. This issue, namely, whether the description of Nature in statistical terms is complete or even consistent, was the subject of a very lively debate-dialogue between Einstein and Niels Bohr in the years 1920-1930. Nowadays there is no debate as the principles of quantum theory have been verified countless times in experiments. In this lecture (which requires no previous background) I shall discuss both classical and quantum probabilities. Although the meaning of probability remains the same in both contexts, the method of calculating them is different. I shall explain the strange but beautiful manner in which quantum probabilities are calculated, explaining the mathematical foundations in an elementary manner. The meaning of probability
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.
We report an inconsistency found in probability theory (also referred to as measure-theoretic probability). For probability measures induced by real-valued random variables, we deduce an "equality" such that one side of the "equality" is a probability, but the other side is not. For probability measures induced by extended random variables, we deduce an "equality" such that its two sides are unequal probabilities. The deduced expressions are erroneous only when it can be proved that measure-theoretic probability is a theory free from contradiction. However, such a proof does not exist. The inconsistency appears only in the theory rather than in the physical world, and will not affect practical applications as long as ideal events in the theory (which will not occur physically) are not mistaken for observable events in the real world. Nevertheless, unlike known paradoxes in mathematics, the inconsistency cannot be explained away and hence must be resolved. The assumption of infinite additivity in the theory is relevant to the inconsistency, and may cause confusion of ideal events and real events. As illustrated by an example in this article, since abstract properties of mathematical entities in theoretical thinking are not necessarily properties of physical quantities observed in the real world, mistaking the former for the latter may lead to misinterpreting random phenomena observed in experiments with microscopic systems. Actually the inconsistency is due to the notion of "numbers" adopted in conventional mathematics. A possible way to resolve the inconsistency is to treat "numbers" from the viewpoint of constructive mathematics.
Since its emergence, quantum mechanics has been a challenge for an understanding of reality which is based on our intuition in a classical world. Nevertheless, it has often been tried to impose this understanding of reality on quantum theory - with limited success. Instead, it might be a better alternative to redefine the meaning of physical reality. This is the objective of the paper. A consideration of the quantum measurement process, conditional probabilities and some well-known typical quantum physical experiments provides the reasoning for the following three hypotheses: (1) Prior to a first measurement, a physical system is not in a quantum state. (2) Physical reality is all that and only that about which (classical) information is available in the universe. (3) Information creation is an independent process and is not covered by the Schroedinger equation. It is the first step of the quantum measurement process and does not have a classical counterpart. The first hypothesis makes sense only if the quantum measurement process can be described without presupposing an initial state for the system under consideration. This becomes possible by the objective conditional probabilities which represent the transition probabilities between the outcomes of successive quantum measurements and have been introduced by the author in some recent papers. The second hypothesis holds as well in the classical case, but a certain incompleteness of reality is typical of quantum mechanics and the origin of many quantum phenomena. Classically, the existence of a complete reality is presumed, and hypothesis 3 has no meaning then.