FORMAL SYSTEMSMETA-MATHEMATICSOne conception of meta-mathematics is the 'traditional' one, which says that meta-mathematics is simply critical theory about mathematics. In my view, that usage belongs either to philosophy of mathematics, or to mathematics itself.In my own usage, meta-mathematics is particularly the formal extension of mathematics through mathematical OR philosophical theories, derived from the word 'meta-' meaning after.Formalism has always had the reputation of being metaphysical, and it is no less the case in meta-mathematics (under my definition).Here are four areas to study, to pique your interest in my definition of Meta-Mathematics.1. Category Theory[+], an extension of set theory. This is related to modules, which are effectively relativistic versions of set groupings. A key feature of category theory in my view is a system I invented called coherent Categorical Deduction. This form of analogous to / replaces the plus symbol from mathematics.2. Paradoxical Formalism [ - ]. Paradox theory is a way of relating the ultimate bounds of knowledge in a formalized way. Paradox is analogous to the original, unconscious meaning of levels of analysis. Level 1 of paradox is analogous to / replaces the minus symbol, and typically is represented with potential dualities. At the level of duality, these are solvable with categorical deduction. Otherwise, they are solved with paroxysm, which creates level 2. In level 2 [=], the Paroxysmic Method is introduced, solving the paradox. Level 2 in my theory of Paradoxical Formalism is analogous to / replaces the equals sign symbol or absolute conditioning, by creating commutative levels of equivalency. Level 3 if there is one [represented by three horizontal lines] might deal with Synergies or Synergasms.3. Judgment Theory [ X ]. This set of theories has to do with formal iteration and consolidation of sets. For most purposes it is already covered by Category Theory and Paradoxical Formalism. However, it is an important teaching tool having to do with relative absoluteness, the completeness of sets, and the incidents in which systems emerge (typically in neutral, optimal, conditioned states). In my theory, Judgment theory is analogous to / replaces the multiplication sign, and can also be interchangeable with theories of exponential growth, which relate with theories of completeness and modularization.4. Metalogical Theory [ / ]. Metalogical Theory is the extension of category theory for non-closed sets, specifically infinite sets that exist within a boundary. Proportional methods are used to yield relations between parts of the internal set, or between the internal and the external. As you might predict, in my theory, Metalogical Theory is analogous to / replaces the division sign in mathematics.5. Entity Theory [ { } --> || ]. In addition, there is what might be called entity theory, similar to judgment theory. Entity theory deals with the real or theoretical status of symbols or other concepts as entities within the system. This relates with such concepts as evaluation and parsing processes, each of which serves the role of proving a function for an entity. Not only does proving in-terms-of-entity seem necessary, but this type of approach is also preferable because of its capacity to define systems elements on-the-fly. Thus, it is analogous to / replaces proof theory from mathematics.  BACK TO SYSTEMS