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Theories as Vector Spaces of Valid Statements

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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, the 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 to starting from the axiom entry points and using the vectors that the rules of the theory allow us.