A formal axiomatic system is like a parlour board game. It has playing pieces and rules defining possible moves, as well as a game space within which the action takes place. It consists of four sorts of things: symbols, formation rules, initial conditions or expressions, and transformation rules. To these elements are added theorems-- the conclusions, goals or "moves". Formation rules are the basic rules or assumptions determining how the symbols may and may not be strung together to form "expressions". Initial expressions are also called axioms or postulates-- the input of the system. Transformation rules are the algorithms by which expressions may be handled in order to derive theorems. Theorems are the output of the system.
Unlike the elements of a causal system, those of a formal system are intentional and unambiguous. They exist exactly and only as what they have been declared to be. Like the rules of a game, they are well-defined and true because we agree to them. Truth-by-definition is a process of useful idealization. For example, even though perfect right angles and dimensionless points cannot actually be drawn, their idealizations can be manipulated in thought with complete precision.
Formal systems contain explicitly only what has been defined for them and, implicitly, the logical consequences of these definitions. This gives formal systems a self-contained, tautological and "digital" character. Empirical facts do not have the formal validity which makes the truths of logic seem irresistibly necessary. On the other hand, logically necessary truths are devoid of information for this very reason. There is no news in them about the world, because they concern what is changeless.
A formal system hangs together purely on the threads of logical necessity until it is "interpreted" as a mapping of some portion of the real world-- as plane geometry can be interpreted to map the physical properties of the earth's surface . Then its premises may appear as truths about the world, its logical structure to mirror the organization of reality. But a formal system can also be viewed as a self-contained game played according to arbitrary rules.
The essence of the concept of formal system is that intuitive areas of thought are replaced with mechanical operations. A procedure is formalized for arriving at conclusions on the basis of prior specified assumptions and agreed-upon rules of reasoning. That is, a method of proof is defined. This precludes "just knowing" the truth, and also allows us to distinguish truth from either belief or provability. The relativized or subjectified version of truth is provability. The concept of truth implies something which exists in itself absolutely, but a proposition or belief requires a mind to formulate it and a method to prove it. Now, if proof is held to be merely an inconvenient detour to a truth that exists prior to and independent of any methods of proof, then reasoning and provability are dispensable, since the mind can justify its leaps by direct appeal to "intuition". Certainly the mind can conceive a theorem or proposition as true or false before a proof has been undertaken, just as it conceives that objects continue to exist when out of sight. Theorems may be intuited as true, even though unprovable in any known system, just as some physical entities may be suspected to exist, though never observed.
In contrast, provability is always relative to some particular formal system. By definition, it is decidable for all complete systems. Since proof means derivability from axioms, no system can prove its own axioms, but can only justify itself as convention or by appeal to some outside reality. Truth refers to such a reality beyond the pale of a given system. Proof is always proof-within-a-system, relying on an intentional acceptance of the premises of the system. Truth is absolute in that it refuses to specify its premises-- to be axiomatized, in other words-- since that would imply having to appeal to a domain beyond itself for justification.
To lie beyond a specific system is quite different than to lie beyond all possible systems. We can always find and specify a formal system larger than any given one. This constitutes the "reality" in which the propositions of the latter are interpreted as true or false. But our intuitive idea of truth is that it lies beyond all formality, beyond all limitation, beyond all human creation. Existing in and of itself, truth is transcendent reality. Such truth can be approached but never reached, because it can never be fully specified. No sooner does one attempt to circle the world-beyond-all-worlds in which this absolute truth is true, then in the same gesture a meta-system is defined outside the circle one has drawn.
The mind as formalism has the same dual nature as mathematics: it can indulge in pure gratuitous fancy or its creative efforts can be hitched to practical activity within the natural or social spheres. The map as self-contained system is a thing in its own right-- a work of art, perhaps-- as well as a sign pointing to an outer world. The mind is a map which structures the organism's relationship with the world, and it is also an arbitrary construct, a formalism, a game.
Cognitive structures are not models of the world in the sense that a model airplane is an imitation of a real aircraft, but rather in the sense that theories of the atom are models of entities not otherwise accessible to experience. A real airplane embodies a theory about what kind of machine could fly. It is at once a physical object and a conceptual system. This system is the design of the aircraft, based on an empirical knowledge of flight and a theory of aerodynamics. The fact that the airplane actually flies is proof of the theory and validation of the knowledge. The scale model (whether or not it actually flies) is a simplified version of the same conceptual system. As physical objects, both aircraft and model belong to a real analog domain perhaps unfathomably rich in detail, while the conceptual systems they embody are digital, propositional, sparsely-defined intentional abstractions.
A cognitive model is a conceptual system, which can be formalized if a conscious agent can stand in for the intentionality of the system and exhaustively express its elements. If a system is thus potentially formalizable in this sense, let us call it an informal system. Then, not only cognitive models but all intentional creations are informal systems. Human culture as a whole and all its particular manifestations are informal systems. What is not such a system is the world-in-itself, the transcendent analog domain that can never be completely captured in any formal expression. All that exists can thus be divided in two categories: the domain of intentional creations, which can be formalized, and the transcendent causal world, which cannot.
Cultural institutions are informal systems, embodied in their physical "interpretations". The human mind has gone to great lengths to remove itself from Nature by imposing an intentional world of its own design-- culture-- upon the analog or natural world. The humanly invented world is the one in which the mind seems master-- but at the peril of what this mastery may fail to encompass. Culture, as informal system, is a kind of theory of reality, like the airplane is a theory of flight. The degree to which culture is out of harmony with Nature (that is, reality) is the degree to which the theory does not fly.
Human culture mediates our relationship with the natural world, in much the same way that experience is mediated by cognitive models. This is true also of particular institutions. A business or a corporation is an informal system. So is a science or a school, a religion or a church, a state or a government, an art or an opus, a language or a library or a book. Cultural creations are externalized cognitive models, with a dual existence as self-contained games and as metaphors that point to reality.