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Metatheoretical Considerations

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This section of the Wiki is devoted to some metatheoretical considerations. Essentially, it contains some half-baked ideas which need the "attention of an expert", as they say in Wikipedia, as I do not have the necessary mathematical knowledge to develop them further, or to know whether they are wrong, trivial, or already done.

Theories as Models Isomorphic to Reality

 stb  The aim of a theory in any field is essentially to build a model that is isomorphic to reality, so that its behavior will mimic exactly the behavior of the physical or other system that it tries to describe.

Let us see how this works for physical systems. We have the space of all possible states of the system we study. This is usually called phase space, but it would be more exact to call it states space. Each point of this space represents a state of the system. Physical laws determine the movement of the system along these points. When the system is at a specific state-point, in the next time step it could conceivably pass to any other state-point, or at least to any other "neighbouring" state-point. Physical laws determine to which of all the possible state-points it will pass. (So if we hold an apple and release it, the apple will move in a specific direction with a specific speed.)

Here there is an interesting parallel with information. When we do not know the state of a system, information restricts the possible alternative states the system could be in. In the evolution of a system, physical laws restrict the possible alternative states that the system can pass to.

Theories as Vector Spaces of Valid Statements

 stb  Consider the set of all symbols used in a theory. Take the subset of well-formed combinations of these symbols, or theorems. Take a space whose points represent these theorems, a theorem space. Usually there is a category of symbols such as =, `=>` and `<=>` that act as "vectors", in the sense that they allow as to move from one point of the theorem space to another. Axioms act as starting points or points of entry to the theorem space. From these, we contruct more complex vectors (for instance, `(a+b)^2=a^2+b^2+2ab`, and so on). Using these, we expand the area of "true" theorems within the theorem space, that is, the theorems to which we can arrive at starting from the axiom entry points and using the vectors that the rules of the theory allow us.

Theory Development as a Decision Tree

 stb  A similar formulation to the previous one. The movement within the theorem space can be seen as a decision tree, in the sense that at each point there is a number of alternative ways to continue the course of theory development, and the theorists decide which to follow. Their choices are influenced both by theoretical and by sociological factors.