"In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory consists of an axiomatic system and all its derived theorems."
https://en.wikipedia.org/wiki/Axiomatic_system
1. All maths are dependent upon an inherent set of axioms, this in itself is an axiom and necessitates the quantitative nature of mathematics as having a form of alternation with qualitative logic.
2. These maths, dependent upon further axioms, are in turn justified by the theories they produce.
3. These theories show the inherent relations of the axioms, and how they exist through each other.
4. One set of axioms results in a theory with the theory resulting in a further axiom.
5. The axiom produces the theorem, in turn the theorem justifies if not "produces" the axiom. A circularity occurs, leading to a fallacy of circularity in one respect while simultaneously necessitating mathematics replicating itself into further mathematics as one set of axioms results in another.
(A ⇆ (T=(A ⇆ A))) ↔ A
This cyclical nature to the maths as both axiom and theory, is in itself an axiom and theory simultaneously where the axiom as a constant and the theory as an approximate. Hence the axiom leading to the theory as multiple axioms necessitates a tri-fold structure within the axiom itself, through this dichotomy of "axiom" and "theory", where mathematics is a product of perpetual synthesis.
a. The axiom diverges into another axiom and itself: A → (A,A)
b. These axioms converge into a theory: (A⇆A) → T
c. This theory, as an axiom in itself cycles in the process: A(T) → (A,A)
6. This perpetual synthesis, inherent within the foundations of mathematics, necessitates mathematics as going through a constant state of perpetual progression where not only is it never complete, except as a process of self-referencing where it contains all of its own answers, but fundamentally is an adaptation to chaos, through the potential nature of time relative to the observers in one respect while simultaneously inverting a point 0 into quantities.
7. The dichotomy between the axiom and theory as axiom, necessitates mathematics as fundamentally derived from a point 0 within the subjective nature of the observer where each "measurement" as an extension of the observer, as well as the agreement of the observers (as subject to time) lending itself to "objectivity", comes from "nothing".
This point can be derived from the premise that all axioms, setting a foundation for the theory, occurs first in the progressive nature of time and is founded in the observer.
8. All axioms in themselves are quantifiable as 1. As 1 axiom requires a relation to another axiom, all axioms are premised in arithmetic as one axiom is effectively added to another resulting in an inherent set of axioms existing as 1 in themselves. Hence all mathematics, derived from a set of axioms is premised in a base form of arithmetic, however this base form of arithmetic in itself is a system of axioms subject to its own nature where the addition of one axiom to another results in a theory, as well as there subtraction or seperation.