In the preceding two chapters, we have discussed the concepts of natural systems, which belong primarily to science, and formal systems, which belong to mathematics. We now turn to the fundamental question of establishing relations between the two classes of systems. The establishment of such relations is fundamental to the concept of a model, and indeed, to all of theoretical science. In the present chapter we shall discuss such relations in a general way, and consider a wealth of specific illustrative examples in Chap. 3 below.
In a sense, the difficulty and challenge in establishing such relations arises from the fact that the entities to be related are fundamentally different in kind. A natural system is essentially a bundle of linked qualities, or observables, coded or named by the specific percepts which they generate, and by the relations which the mind creates to organize them. As such, a natural system is always incompletely known; we continually learn about such a system, for instance by watching its effect on other systems with which it interacts, and attempting to include the observables rendered perceptible thereby into the scheme of linkages established previously. A formal system, on the other hand, is entirely a creation of the mind, possessing no properties beyond those which enter into its definition and their implications. We thus do not “learn” about a formal system, beyond establishing the consequences of our definitions through the application of conventional rules of inference, and sometimes by modifying or enlarging the initial definitions in particular ways. We have seen that even the study of formal systems is not free of strife and controversy; how much more strife can be expected when we attempt to relate this world to another of a fundamentally different character? And yet that is the task we now undertake; it is the basic task of relating experiment to theory. We shall proceed to develop a general framework in which formal and natural systems can be related, and then we shall discuss that framework in an informal way.
The essential step in establishing the relations we seek, and indeed the key to all that follows, lies in an exploitation of synonymy. We are going to force the name of a percept to be also the name of a formal entity; we are going to force the name of a linkage between percepts to also be the name of a relation between mathematical entities; and most particularly, we are going to force the various temporal relations characteristic of causality in the natural world to be synonymous with the inferential structure which allows us to draw conclusions from premises in the mathematical world. We are going to try to do this in a way which is consistent between the two worlds; i.e. in such a way that the synonymies we establish do not lead us into contradictions between the properties of the formal system and those of the natural system we have forced the formal system to name. In short, we want our relations between formal and natural systems to be like the one Goethe postulated as between the genius and Nature: what the one promises, the other surely redeems.
Another way to characterize what we are trying to do here is the following: we seek to encode natural systems into formal ones in a way which is consistent, in the above sense. Via such an encoding, if we are successful, the inferences or theorems we can elicit within these formal systems become predictions about the natural systems we have encoded into them; consistency then means that these predictions will be verified in the natural world when appropriately decoded into linkage relations in that world. And as we shall see, once such a relation between natural and formal systems has been established, a host of other important relations will follow of themselves; relations which will allow us to speak precisely about analogy, similarity, metaphor, complexity, and a spectrum of similar concepts.
If we successfully accomplish the establishment of a relation of this kind between a particular natural system and some formal system, then we will obtain thereby a composite structure whose character is crudely indicated in Fig. 2.1 below:
In this figure, the arrows labelled “encoding” and “decoding” represent correspondences between the observables and linkages comprising the natural system and symbols or propositions belonging to the formal system. Linkages between these observables are also encoded into relations between the corresponding propositions in the formal system. As we noted earlier, the rules of inference of the formal system, by means of which we can establish new propositions of that system as implications, must be re-interpreted (or decoded) in the form of specific assertions pertaining to the observables and linkages of the natural system; these are the predictions. If the assertions decoded in this fashion are verified by observation; or what is the same thing, if the observed behavior of the natural system encodes into the same propositions as those obtained from the inferential rules of the formal system, we shall say that (to that extent) the relation between the two systems which we have established, and which is diagrammed in Fig. 2.1, is a modeling relation. Under these circumstances, we shall also say that the formal system of Fig. 2.1, modulo the encoding and decoding rules in question, is a model of the natural system to which it is related by those rules.