An Essay on the Prerequisites for the Probability Theory

The probability calculus and statistics as well permeate nearly every discipline and professional sector, while no theories underpinning this wide spreading field reached universal consensus so far. The probability interpretations present irreconcilable traits, so the concept of probability is still substantially unclear. Purpose of this work: The present paper intends to demonstrate how the different models of probability constitute the facial problem which conceals another hidden and more fundamental question. Method: We show how authors do not agree with the concept of probability P and moreover they have different ideas about the precise object qualified by P, which has priority from the point of logic. It is clear how the element X measured by P(X) influences its meaning. In consequence of the conflicting opinions, theorists tend toward a compromise. They use the outcome or result of an experiment as the argument X of P(X) and represent X as a subset of the event space. This paper suggests replacing the outcome-subset with the event-triad E, which provides a comprehensive mathematical support. Results: The last section shows how the triadic model is formally consistent with the conventional theories and can integrate the conflicting views on probability. This unifying result can help mathematicians to go beyond the present theoretical deadlock. In summary, this paper advocates a more explicit notation system for probability and points out how probability can be ambiguous without rigorous specification of the sample space and the experiment in general.

NON-ARCHIMEDEAN PROBABILITY: FREQUENCY AND AXIOMATICS THEORIES

We propose a new theory of probability based on the general principle of the statistical stabilization of relative frequencies. According to this principle it is possible to consider the statistical stabilization not only with respect to the standard real topology on the field of rational numbers Q but also with respect to an arbitrary topology on Q. The case of p-adic (and more general non-Archimedean) topologies is the most important. Our frequency theory of Probability is a fruitful extension of the frequency theory of R. von Mises[18]. It's well known that the axiomatic theory of Kolmogorov uses the frequency theory as one of the foundations. And a new general frequency theory can be considered as the base for the general axiomatic theory of probability (Kolmogorov's theory is a particular case of this theory which corresponds to the real topology of the statistical stabilization on Q). The situation in the theory of probability becomes similar to that in modern geometry. The Kolmogorov axiomatics (as the Euclidean) is only one of the possibilities, and we have generated a great number of different non-Kolmogorov theories of probability.The applications to p-adic quantum mechanics and field theory are considered.