Axioms are Non-Foundational; Pre-Foundational Emergence of Philosophy, Religion, Sciences, Maths/Logics as Self-Scaling Events;
Math and all intellectual endeavors ranging to the logics, sciences, philosophies, theology etc., which will be called "X", are grounded in axioms.
This is obvious. Nothing deep. Shallow. Necessary within standard given human constructs.
Because it is obvious it should not be assumed or given little attention to.
The axiom: "Self-Evident Truth".
If the axiom is observed as applied to itself, which must meta-logically occur for the simple reason that an ""axiom as foundational" is an axiom", then what remains is:
1. Self-Evident Truth = Axiom
2. Self = Axiom
3. Evident = Axiom
4. Truth = Axiom
And the reason for this is that all that constitutes "Axiom" must effectively be an axiom if the definition of an axiom is a self-evident truth....if the definition of an axiom is not a self-evident truth than distinction is prior to axioms as the definition is purely distinction.
So to go further (* * marks the repetition of what equates to what is within the equated):
1. Self = *Self*-Evident Truth
2. Evident = Self-*Evident* Truth
3. Truth = Self-Evident *Truth*
4. Self-Evident Truth = *Self-Evident Truth*
So from this angle an axiom, by design, repeats itself within itself, at relative scales of distinctions. This can be seen in the list presented where "* *" observes each distinction within the definition of axiom repeating in scale as the definition of the axiom.
This scaling can be further derived from the prior meta-argument:
1. Self-Evident Truth = Axiom
2. Self-Evident Truth = (((Axiom)Axiom)Axiom)
3. Axiom = (((Axiom)Axiom)Axiom)
4. Axiom(Axiom -> Axiom)
In these respects the assertion of an axiom is the assertion of further axioms derived from it thus relegating epistemological foundations to assertions as boundaries.
By degree the distinction of both axiom and the definition of the axiom is holographic when applied to itself. This results in the axiom, by its own nature, holographic.
This holographic nature is recursive. It is recursive precisely because a fixed point repeating across scale. The axiom is a scale of axioms and these axioms are in themselves axioms thus relegating the axiom containing itself as a self-scaling epistemological event at minimum.
However the axiom as a self-scaling event, by its own nature, requires
1. The distinction of the axiom from its definition (other axioms).
2. The distinction of the definition (other axioms) from the axiom.
3. The distinction of axiom from axiom.
4. Axiom as distinction it is derived from.
In these respects the application and derivation of an axiom is the application of a boundary by which further boundaries scale. X effectively becomes of holographic construct of the axioms it is grounded upon thus effectively resulting in all X being recursive derivations by default.
However the nature of distinction, the axiom is subject to reveals there is a meta-layer beyond axioms themselves, the event of distinction.
Now re-reading the axiom derivation argument above, what we can see when observing the axiom(s) as events of distinctions is effectively a self-embedding self-contrast as the event of distinction itself.
This must be emphasized: a self contained, self contrast as distinction itself.
To view this intuitively we can refer to base level topology, one line segment: ●●
There is only ● when observing ●●.
● self contrasts as ●● given ●●.
● is self contained as its own context as ●●.
Both happen simultaneously.
So going back to the "axiom of axiom" derivative argument, the axiom is self scaling as self-contained self-contrast. This necessitates the axiom as a distinction, subject to distinction.
Thus distinction is pre/post/mediatory/trans axiomatic.
And a proto logical/mathematical foundation is revealed.
One that side-steps the inherent belief or assumption required for X and instead reveals a pre subject/object event that allows for each and yet is transcendental (not necessarily in a purely mystical sense), while allowing for percievable grounds.
No assumption is required as an assumption is subject to distinction.
No doubt is required as a doubt is subject to distinction.
No subject is required as subject is subject to distinction.
No object is required as object is subject to distinction.
Ethical systems have grounding, outside of authoritative or belief driven assertions, as distinction is:
Recursive: you reap as you sow.
Reflexive: treat others as you would be treated.
Boundaries: "Do/Do not Y"
Transcendental: Self-Lessness
While dually the constitutive paradigm of subjectivity and objectivity that is under broad philosophical debate:
Subjective: the focal point of attention is that by which they arise and dissolve
Objective: alignment of distinctions result in unity across attention spans
Religiosity and mystical experiences can align with rational and analytic experience as
Religious: the event of distinction itself is omnipresent and absolute
Mystical: the event of distinction itself is transcendental.
Rational: the relation of distinctions is a ratio, reasoning is application of relations of distinctions
Analytical: distinction effectively is the cut from which categories arise.
What remains is distinction:
1. X is a distinction
2. "Ethical systems" is a distinction.
3. "Religious, Mystical, Rational, Analytical Experience" are distinctions
4. "Indistinction", "transcendence of distinction" is a distinction.
Thus distinction is self-embedding and what remains of X, and its various scalings, is a self scaling event via axioms.