One 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

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
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.