Probabilistic Events and Physical Reality: A Complete Algebra of Probability

First the crucial but very confidential fact is brought into evidence that, as Kolmogorov himself repeatedly claimed, there exists no abstract theory of probabilities, simply because the factual concept of probability is itself unachieved: it is nowhere specified how to construct the factual probability law to be asserted on a given physical random phenomenon. Then an algorithm of semantic integration is built that permits to identify this factual probability law.

It is argued that the Copenhagen Interpretation of Quantum Mechanics, founded ontologically on the concept of probability, may be questionable in view of the fact that within Probability Theory itself the ontological status of the concept of probability has always been, and is still under discussion.

We propose two distinct interpretations of extended probabilities which are realistic for the physical world.

A popular scientific contribution should not contradict any established facts and ought to be understandable. I complied with both these requirements and am offering a sufficiently full introduction to probability theory. Furthermore, I enlivened my text with much information about the history of probability and commentaries on the occurring notions which do not belong to that theory. Indeed, a student is not a container to be filled but a torch to be kindled. I hope therefor...

The theory of probability and the quantum theory, the one mathematical and the other physical, are related in that each admits a number of very different interpretations. It has been proposed that the conceptual problems of the quantum theory could be, if not resolved, at least mitigated by a proper interpretation of probability. We rather show, through a historical and analytical overview of probability and quantum theory, that if some interpretations of the one and the othe...

In the footsteps of the book \textit{Measure Theory and Integration By and For the Learner} of our series in Probability Theory and Statistics, we intended to devote a special volume of the very probabilistic aspects of the first cited theory. The book might have assigned the title : From Measure Theory and Integration to Probability Theory. The fundamental aspects of Probability Theory, as described by the keywords and phrases below, are presented, not from experiences as in...

The crucial but very confidential fact is brought into evidence that, as Kolmogorov himself repeatedly claimed, the mathematical theory of probabilities cannot be applied to physical, factual probabilistic situations because the factual concept of probability is not defined : it is nowhere specified how to construct, for a given physical random phenomenon, the specific numerical distribution of relative frequencies of outcomes from the universe of elementary events produced b...

This paper argues for the status of formal probability theory as a mathematical, rather than a scientific, theory. David Freedman and Philip Stark's concept of model based probabilities is examined and is used as a bridge between the formal theory and applications.

Criticisms of so called `subjective probability' come on the one hand from those who maintain that probability in physics has only a frequentistic interpretation, and, on the other, from those who tend to `objectivise' Bayesian theory, arguing, e.g., that subjective probabilities are indeed based `only on private introspection'. Some of the common misconceptions on subjective probability will be commented upon in support of the thesis that coherence is the most crucial, unive...

This work has been prompted by the surprising lack of mathematical coherence in the common usage of some of the fundamental entities in the theory of probability, with an inherent risk of contradiction. While disentangling the intricacies, we realized that the same issue has been raised many times, with only partial solutions, notably by Boole, Hilbert, De Finetti and Renyi, among others. In particular, a restoration of foundational coherence in the usage of probability theor...