1 Probability as a physical concept

From the analysis in DMP follows that the notion of probability must be given a physical significance of its own. But is it really meaningful to consider probability as physical quantity? Is not probability a descriptive quantity of purely mathematical origin? In other words, can actually the notion of probability be applied as a concept in a physics language?

Let us address this issue by considering a person, we may call him Boris, who plays Russian roulette. We can take it for granted that Boris has some physiologic reactions like frequent pulse and perspiration and, and this is essential, this behavior of Boris is not because of the revolver and the single bullet but because of the possibility (probability!) that the bullet appears in the crucial location in the revolver. Clearly a physiologic reaction is in the usual sense a physical phenomenon and, if we accept the common concept of causality in the world we presently consider, this physiologic effect on Boris must have physical cause. Hence, the probability we for the moment entertain must be physical, as much as perspiration exists physically (an analogy between probability and force is not far away).

So we conclude that there is nothing remarkable in ascribing probability a physical significance of its own; in fact, often it is almost inevitable that it is done. One may thus say that a particle moves or an electric bulb breaks because the object is acted upon by some transition probability.

2 Primary concepts

As discussed in DMP (Section 0.3), in order to contemplate phenomena of the surrounding nature and to communicate the results, one must possess some form of a language where one can recognize two kind of concepts, namely primary and secondary concepts. The former sort of concepts are not defined but understood intuitively, whereas the latter are defined in terms of primary concepts or of other secondary concepts already defined.

A reasonable example would be the basic part of classic mechanics where a possible set of primary concepts could be particle, mass of particle, time, particle’s location as a function of time. The Newton equation of motion (mass)(acceleration) = (force) can then be applied as a definition of the concept of force that thus comes out as a secondary concept; note that “acceleration” is a secondary concept because it is defined(!) in terms of the primary concept of “location” (the second time derivative thereof).

The notion of primary concepts is essential because one must start from somewhere, though sometimes it is applied simply for convenience. For instance, one might daily speak of “apple” without going into a definition in biologic terms (e.g. an apple is a fruit, which is a statement that demands some definition of the term “fruit”…). In a scientific context one tries to keep the number of primary concepts as few as possible and, moreover, one tries to select them in a way that makes them general in the sense that they are applicable to a variety of systems (situations). A frequently useful rule of thumb is that primary concepts are introduced prior to the analysis where they are used (one must start from somewhere with something).

3 Probability as a primary concept

From Section 1 above we conclude that it is possible to accept probability as a physical quantity, and from the analysis in DMP we learn that it is occasionally also necessary that this is done. For formal reasons it is however also necessary that probability is in addition accepted as a primary concept. To see why this is so, let us first note that the origin of probability theory is almost habitually based on the idea of toss of a coin. So assume that we toss a coin several times in the usual manner and let N be the number of tosses, and let be the number of those tosses that result in head. In the old theory, the probability of receiving head in a single toss was often defined (Sic!) as , which among other things would seemingly imply that this probability comes out as a secondary concept. However, the ratio is a stochastic variable and the usual notion of “lim” does not apply. This problem is solved by the Bernoulli theorem, which is one of the most fundamental things in probability theory. The theorem states that the ratio converges in probability to and thereby provides a way out of the formal difficulty. But, clearly, that theorem would be totally meaningless if the probability concept is not introduced prior to the contemplation and the derivation of the very theorem: the notion of probability must be introduced as a primary concept.

The Bernoulli theorem not only solves a formal problem, but it also provides the very basis for the connection between probability theory and the “real” world (data). Of importance is that this theorem also gives basic advice concerning design of experiment I have particularly in mind the elementary matter that the larger N is the “better” is the ratio as an estimator of the corresponding probability (this is the vary basis for the today self-evident matter that, whenever possible, the samples should be large).

4 Summary

It is natural and occasionally even necessary to accept the notion of probability as a physical entity. But it is then important that it enters the theory as a primary concept, and this goes for any theory where use is made of probability theory.