consisting of propositions which could actually be true — only we did not care whether they were or not. But this is not correct of our "pure geometry".
Sentences or formulae in which the words or symbols have no definite meaning are, of course, no propositions at all; they are "propositional functions", i.e. empty forms which will become propositions as soon as certain definite significations are assigned to the symbols of which they are composed. As long as no signification is assigned the symbols are really nothing but simple marks to indicate empty places which must be filled with meaning in order to get a proposition. There is, of course, the condition that wherever the same sign occurs, it must be given the same meaning. Such signs indicating empty places for significant symbols, are called variables, and the significant symbols by which they are replaced are called concepts.
What has been done in the case of geometry can be done for any other science in so far as it is really scientific, i.e. consists of logically connected propositions: by disregarding the meaning of the symbols we can change the concepts into variables, and the result is a system of propositional functions which represents the pure structure of science, leaving out its content, separating it altogether from reality. When we speak of science, we shall, for the reasons given above, always have in mind theoretical physics, at least for the time being.
A purely deductive system of the kind described has been called (the term was first used by Pieri, I think) a hypothetical-deductive system. It is called "hypothetical" with respect to its possible use in science. It will, evidently, be useful in all cases where we find entities in nature, which, when substituted for the variables of the system, will change all its propositional functions into true propositions. (Perhaps I ought not to say that the entities themselves could be substituted for the variables; I mean of course that the variables are replaced by symbols signifying those entities.) We may express this by saying: If the symbols of our system stand for entities for which the axioms hold, then all the propositions of the system will be true of those entities. Or, in other words: If entities can be found which satisfy the axioms of the system, then the system will be science of these entities. It is on account of the "if" at the beginning of these sentences that the deductive system is called "hypothetical".
