(One and the same system may possibly find many applications to reality. Many sets of entities may be discovered so that the axioms, and consequently the whole system, will be true for each set. All these sets will have those properties in common that are expressed by the axioms, but in all their other properties they may, of course, be entirely different from each other — so different that they may belong to entirely different realms of being (if I am permitted to use this old fashioned philosophical phrase): one may be a set of colours, another one a set of points in space, another one a set of economic values, and so forth, and yet each set may fit into the same frame, the purely formal relations between the elements may be the same within each set, so that they will all be interpretations of the same hypothetical deductive system).
All this is well known to anyone who has studied the subject, and it is generally recognised that science in its logical aspects has the character I have been trying to describe. But for our present purpose we must concentrate our whole interest on the question, "how is the empty structure of a hypothetical-deductive system actually filled with meaning?"
What is the stuff which must be added to the empty frame, in order to make a science of it ? — There seems to be only one possible answer to this question, namely, "the purely formal structure must be filled with content — it could not be anything else, because there is nothing else". (Indeed, did we not say ourselves, that all structure must be the structure of some content, and that content was nothing but that which had a certain structure? If we are to have concepts instead of mere variables, if we are to have real propositions instead of mere empty forms, if we are to have a science of some domain of reality instead of a mere hypothetical-deductive system, then our symbols must stand for real content, for if they stood for mere structure, we should again in the end, be left without meaning, for again there would, be the possibility of many different interpretations. But actual science deals with reality, which is unique, and not with possibilities only, of which there are many.
If this is the right answer it must appear difficult to reconcile it with our former insight that content never enters into our propositions and that all expression is done solely by means of pure structure.
