Slutsky and probability theory
Furthermore, we deal with some relation that can pair-wise interface any two subjects with each other, and the result of this operation will be some third subject from the same set. Under these conditions the theory of groups develops a complex, highly elegant mathematics and is very important for various applications of the set of theorems, the material content of which within the limits of the theory remains indefinite. In this uncertainty also resides one of its strengths – formal purity and the variety of applications. If a group consists of a series of natural numbers, and the basic operation, giving from two subjects a third, is addition, there will be one interpretation; if multiplication, – then there will be another; if we make the group from all possible permutations of several numbers, then it will be still a new interpretation, if from all possible rotations of some regular polyhedron, – that once again is new, and so on. We have something similar also in our case. The formal concept of valency can have not only one significance; therefore also the significance of the theorems, long since familiar to us in their classical form, in essence have multiple meanings, only their character remains hidden, participating in the world only in the form of disputes, in significant measure fruitless, about the concept of probability itself. Let us try to outline in a few strokes some possible interpretations of the calculus of alternatives. First of all, of course, its classical form. To it we come, substituting in place of equi-valence – equi-probability, in place of valency – probability. This statement can be considered from two points of view. The first – purely formal, the other – material. With the purely formal, which first of all can only interest mathematicians, concepts are introduced purely conditionally. Let us suppose, that the ‘possibility’ of an event can either be greater, or equal to the ‘possibility’ of another event. Let us suppose, that two events, each composed from an identical number of other uniquely possible, incompatible and equally likely events, necessarily must be equipossible. Then irrespective of any more definite significance of the concept of equipossibility and independent of its conditions it becomes possible to introduce by the usual path the concept of ‘probability’ in its purely mathematical aspect. Or, still another way, more closely adjoining to the above train of thought, it is possible to state as follows: suppose, that ‘possibility’ can be expressed by means of a number. Then suppose that the ‘possibility’ of any event is equal to the sum of ‘possibilities’ of those unique and incompatible events on which it is composed. Then, etc., etc. All this train of thought is equivalent to the following. We have already a formal mathematical calculus with concepts and axioms. By way of application we take some concept, for example, as in this case, the concept of ‘possibility’, and we suppose, that for it those axioms, which lie at the foundation of formal disjunctive calculus, are valid, i.e. that ‘possibilities’ are expressed by numbers, that with these numbers there are unequivocally connected all members of a given alternative in a one-to-one manner, that these numbers are subordinated to those formal correspondences, which we have introduced for the numbers, connected with members of the alternative, by the relation of valency. From the purely mathematical point of view, it as though it is quite sufficient to make the transition from the calculus of alternatives to the calculus of probabilities.