Does the difference between the definition of probability in physics and probability in mathematics really matter?

Question

The realistic definition of transition probability in physics is well defined and constrains the probability to rational numbers. The abstract definition of probability in mathematics is also well defined, but it allows probability to be an arbitrary real number element of [0,1], whether rational or irrational. The difference is enormous and the conclusions diverge widely. The question is who do we track and when? Additional clarification required: We can provide two specific published examples among many others,

1. Numerical resolution of the 3D PDE of heat diffusion as a function of time in its most general case using the physical definition of probability.

2. Solve the statistical numerical integration for an arbitrary number of free nodes using the physical definition of probability. The trapezoidal ruler and the FDM-based Simpson ruler would be just a special case.

Answer 1

I would not say there is a difference between the definition of probability in Mathematics and Physics. Probably, what you are referring to as the definition in Physics is the interpretation of probability theory.

I assume that what you mention as the Mathematical definition is the theory as summarized by Kolmogoroff's axioms (or equivalent). That is a special chapter of Measure Theory in Mathematics. However, we have to notice two important things on the mathematical side.

1. Kolmogoroff's theory is a general scheme consistent with many interpretations of the theory. As an axiomatic theory, it does not define the basic objects it works on; like in axiomatic geometry, we do not define what a point, a line, or a plane are. Then, for application to the real world, we have to provide an interpretation fixing how we can assign probabilities to the events. It turns out that many interpretations can be used in connection and consistent with Kolmogoroff's axioms (frequentistic, classical "a priori," subjective Bayesian, ...).
2. There are other mathematical frameworks accommodating probability theories quite different from Kolmogoroff's version. It is possible to find an extended, even though not up-to-date, report in the classical book by T.L. Fine Theories of Probability, Academic Press (1973). Comparative Probability, Complexity-based Probability, and Carnap's Theory of Logical Probability are a few of them.

In Physics, the axioms by Kolmogoroff are usually used. In many cases, the frequentistic approach prevails regarding the interpretation, although, in different contexts, the Bayesian point of view has been advocated as the most appropriate.

After this summary of the situation, the point about rational or irrational values for probability seems to raise a marginal issue.

First, in a frequentistic approach, one can assume that frequencies are ideally measured over infinite sequences of experiments. But, even without stressing this point of view, one can safely use a theory based on real numbers to approximate rational-number-based measures satisfactorily. That is common in Physics and any application of Mathematics to the real world. Strictly speaking, even measuring a length in the real world can only provide rational number outcomes. This fact does not prevent from using geometrical relations based on real numbers.