Axiomatic hierarchy and layering/ stratification

We need an allomathematics (allo- means different) = logical foundations like mathematics which is the quantum foundations, but based on a program which yields quantum mechanics, and the program shouldn't be a list of axioms but internally flexible to be a quantum computer.

We should mimic the foundations of mathematics and differentiate to build quantum mechanics.

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Axiomatic hierarchy and layering/ causopermeable stratification

I mainly created this page for ontoaxiomatics = the axiomatics of substantiality = physical systems which don't have the ability to have intricate but unjustified complexity, and instead they have better causal coherence, and still few elements which have the only justification: these givens work well for creating a physical system.

(problem: causal gaps on the foundations of mathematics; solution: we can introduce self-interactions and biased approaches which create infinite different physical foundations and universes [which aren't the sole possible physical axiomatic system])

But the **axiomatic hierarchy/ stratification** is a wider issue which includes also nonphysical calculatory/ calculational and proof axiomatic systems like the infinite possible geometries and the infinite possible [allo]mathematics.

In open axiomatics/ open axiomaticity systems we have to build new axioms (for example in further mathematics open axiomaticity becomes exposed), usually by putting in hierarchical order the existing axioms and by constructing new ones.

## **Why common mathematics (and common geometry) aren't physics/ aren't physically substantial?**

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**Because their axioms aren't causally linked either by an algorithm, or by an evolving/ branching algorithmoid (the algorithmoid is an algorithm-like program which either has internal switches which affect the program itself or which [the program] can internally evolve and/ or multifurcate.**

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## physical axiomatic prerequisites/ prerequisites for physical axiomatics

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axiomatic prerequisites for physical foundations

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1. Any physical axiomatic system (ontoaxiomatics) is necessarily based on mathematical (common mathematics) or allomathematical (mathematics with different axiomatics) logic. The supernatural is false-reported empiricism without any specific logical foundations, thus it has no identity; it cannot ever be.

2a. Nothing (here I don't speak about nihilogony) is self-evident (see also: nihilogony versus logicogony); the ontoaxiomatics = physical axiomatics should describe everything within it per ontoaxiomatic system (infinite different are possible). An ontological foundations isn't supposed to leave open gaps to be interpreted; except if there is a particular algorithm for that especially when open axiomatics is used a a technique to manage the problems of the axiomatic systems (problems of axiomaticity).

2b. Physical axiomatics (ontoaxiomatics) should be in most cases an algorithm (axiomatic systems can be: lists, algorithms or hybrid). An algorithm provides more coherence to the physical axiomatics than separate axioms which might look logical, but not necessarily causally connected.

In mathematics we can describe geometry, but the axiomatics doesn't include the exact instructions for making the shapes. Ontoaxiomatics = physical axiomatics necessary should include everything, even making the shapes for example; nothing is self-evident and exosystemic.

3. Physical axiomatics (ontoaxiomatics) should manage the problems of the axiomatic systems (problems of axiomaticity: axiomatic incompleteness, axiomatic inconsistency and axiomatic incalculability). It is impossible to avoid in tandem these problems, but there are axiomatic workarounds, like Heisenberg uncertainty principle (it isn't fundamental in limited systems, but the universe is open-ended; see: the infinite series of Wigner's friends/ infinite Wigner's friends). Informational, physical entropy and ontological entropy (the expansion of the universe itself) are ontoaxiomatic workarounds = axiom building algorithms, because universes are ontological Turing machines (a universe making and being subcategory of the Turing machine); their own axiomatics, hardware and software all expressed in the same manner. David Deutsch and Chiara Marletto disagree and claim that infinite or a vast number of different Deutschian constructors can exist in the same universe; but causopermeability = causal permeability per different order of magnitude and phenomenon remains.

4. Physical axiomatics (ontoaxiomatics) which necessarily are infinite allomathematics per different universe, don't have to be very friendly for everyday calculations by humans; it's not a prerequisite. These allomathematics might be even hard or impossible to generalize and describe common mathematical problems other than their original allophysics = physics of different universes.

5. Infinite physioallomathematics = allomathematics which are also physical axiomatics (ontoaxiomatics) can exist. There is no omniontoaxiomatics neither omniaxiomatics; there are no general rules for all axiomatic systems (δεν υπάρχει η καθολική [παντο]αξιωματικότητα); only general logical guides. Thus there cannot be proven that some specific universe is the absolute way to make universes. Any possible universe-like cosmos, necessarily exists. It's not a superior cosmos. Usually cosmoi/ worlds of medium complexity might harbor intelligent life; because universes with simple physical foundations usually have bacteria-like life and universes with very complicated physical foundations are unstable for intelligent life; but these are general guides and not absolute proofs. A negligible percentage of universes is intelligent-life-yielding. No thinker–observer causes any universe; physical existence and thought are both procedures; no procedure can be proven as mathematically superior (except if we use weak axioms; but weak axioms aren't fundamental and usually are a trick used by mathematicians to solve further mathematics problems when proper axioms are missing; the computational-calculational-proof systems humans and other thinkers use, aren't necessarily physical foundations; they don't usually meet the prerequisites for physical axiomatics = physical foundations).

6. Physical axiomatics should be self-executable. It requires to start operating itself without exoaxiomatical intervention. The formulas and the axioms should meet the self-executability criterion. It is possible with coherent algorithmic foundations which don't have to be tautological to the axiomatic list of common mathematics, but due to logic the physical algorithm is transcribable to common mathematics. Common mathematics isn't self-executable because it includes too many things and it has logical gaps on its foundations. Any effort to bridge the holes of the mathematical foundations isn't a correct approach of general mathematics as a measuring-calculational-proof-science axiomatic system but (these erroneous efforts to bridge the gaps of the mathematical foundations) these efforts only create sub-mathematic and not the unbiased general mathematics, which is not physics = it isn't a substantiality = common mathematics doesn't exist as a physical system; it's simply a scientific tool. If any allomathematics has well justified foundations, it requires self-interactions and it is one out of the many physics, but not the sole physics. A well justified foundations via axiomatic self-interactions which include variations like the multiverse, is necessary for the **self-executability of the physical foundations**. **The physical self-executability is tautological/ identical to the physical self-causation due to logic itself (logicogony and not nihilogony; infinite physical allomathematics are possible)**.

## comment 1: Self-evidence and self-causation of any physical and calculational proof system foundations

The self-evidence and/or self-causation of the physical foundations; necessarily self-evidently/ self-causally produces infinite different physical worlds (universes) and calculational proof systems (all the infinite different geometries and all the infinite different [allo]mathematics with any possible axiomatic foundations).

What is the difference between physical universe-like [substantialities] systems and calculational and measuring proof systems (like all [allo]geometries and all [allo]mathematics [allo-: different])?

A wide physical system requires a more coherent axiomatics/ axiomaticity. Usually an algorithmic axiomatics and not a mere list of axioms (hybrid [algorithm + list] axiomatics is an alternative).

Axioms should be stratified in wide universe-like [substantiality] physical systems. The first layer of the physical foundations is in tandem arbitrary and logical (logic here means mathematical and allomathematical logic; allomathematics means mathematics with different axiomatics or axiomatic algorithm, still rigorous). Arbitrary and logical = self-evident and self-caused. The universe we live in is anthropically arbitrary (thinkers dwell in the puny percentage of intelligent-life-yielding cosmoi/ universes). It is arbitrary because infinite other universes are logically possible. Mathematical and allomathematical logic = physicalism/ antisupernatural foundations are necessary because the supernatural isn't only unreachable and unteachable, but it isn't also scientifically hypothesizeable, because it could not have any specific logical foundations, thus it cannot have exact characteristics, thus the supernatural is impossible; it cannot exist. Supernaturalists only empirically or via imaginary empiricism describe their experienced realities, but empiricism without scientific method is prone to mistakes and biases (hallucination, intoxication, misunderstanding, etc.), but also the supernaturalists never elaborate on the physical foundations (deep causality of specific procedures without causal violations and uncausal fragmentation) at an axiomatic level. The supernaturalists complete their theoretical gaps within their own brains and range of biases and don't produce or even care to produce a complete systemic foundations/ axiomatics.

Most universes or cosmoi (not necessarily exactly like our universe) have very simple foundations; but still simple foundations can produce complexity.

Universes with very simple and very complicated foundations probably aren't suitable for intelligent life. The best option for life is universes with medium complexity foundations. Medium complex for stability and not simple foundations for [allo]biochemical complexity.

Many physicists and mathematicians base their axiomatics/ foundations on nihilogony (nilogony: cosmogony from nothing/ a universe from nothing; mathematical foundations from the null set, etc.). They are wrong.

Any axiomatic foundations is based on mathematical and allomathematical logic itself. If you say: nothing[ness] organizes itself and performs tasks; the most important thing isn't the Nothing (which doesn't and cannot exist and shouldn't be confused with the void); instead it is the **logical procedurality** (procedurality: the state of procedural function/ usually complicated procedures).

Thus any axiomatic foundations is based on logic (not common sense logic because it's not axiomatic, based on biases, and the biases differ per thinker and aren't coherently organised in a manner that produces an axiomatic system).

**logicogony versus nihilogony**: Logic and not nothing[ness] (the Nothing) is the foundations of any axiomatic system, physical or calculational proof system.

## comment 2: physical axiomatic prerequisites/ prerequisites for physical axiomatics

I work on physical axiomatics (scientophysicalism/ scientific physicalism, see: David Deutsch, David Tong, Max Tegmark); the hypernym of quantum foundations for any possible universe-like cosmos/ world.

(I work on other topics also.)

For the physical foundations of our specific universe, David Hilbert initiated the axiomatization of physics, but he didn't have modern particle accelerator data, and David Tong (University of Cambridge) is way more correct (David Hilbert suggested many things proven wrong like the supposed simplicity of the deeper physical foundations; a modern approach is topological quantum field theory, but not everyone tries to describe basic quantum field theory as a high-order complexity of the more fundamental topological quantum field theory).

Max Tegmark for example tries to calculationally describe the physical foundations,

David Hilbert tried to axiomatically describe the physical foundations.

If we had the final theory of our particular universe (it's an open system with open axiomatics, thus by definition there is no final theory; only better approximations)... thus if we had the semblance of a final theory of our particular local universe (which isn't the Everything which is not a single system and also is undefinable)... its calculational and its axiomatic foundations would be equivalent, simply with different notation.

Humanity is very afar from reaching these goals = quantum foundations and hypernymously/ superordinately: foundations of substantiality = foundations of any other wide universe-like physical system as a self-sufficient ontological Turing machine (a subcategory of the Turing machine, which is in tandem its own axiomatics, hardware and software and IS ITSELF [identity and not mere description] a particular universe).

If you don't care about these ideas, deliver them to someone who does.

I am not an important person.

I never said I am correct.

We humans should work on these ideas.

I am not biased to promote a specific idea.

We should understand the foundations of substantiality/ wide universe-like physical systems and their foundational (= self-causation due to allomathematical logic of any possible physical/ universe-like foundations) prerequisites.

* scientophysicalist = scientific physicalist: physicalist focusing on the physical foundations in general; not only of our universe

## Quantum computations [calculations] are more substantial than classical calculations

This means that quantum mathematics, when formulated as an allomathematics with allomathematical foundations, is more coherent than classical mathematics, but narrower. Common mathematics is wider and can express way more logical procedures but it doesn't meet the criteria of physics/ substantiality = (which means that) mathematics can describe many different physics but it isn't itself physics on its foundations.

Quantum computation is foundationally more coherent, but also it is less generic.

see: quantum computing

## links

* search on Bing AI chat (or other platform): Which are the axiomatic prerequisites for physical foundations?
* Four Significant Axiomatic Systems and Some of the Issues Associated with Them - Stefan Mykytiuk and Abe Shenitzer - JSTOR

## videos

* The story of mathematical proof – with John Stillwell – The Royal Institution (it is related, because proof is based on axioms; but questions about the axioms are exposed for example in "physical axiomatic systems" [ontoaxiomatics] and when comparing alternative axiomatic systems)
* The Search for a Theory of Everything – with Yang-Hui He – The Royal Institution – YouTube
* without a single theory of one single coherent everything: On the origin of time - with Thomas Hertog – The Royal Institution – YouTube

## index

* universal axiomatic system = universal axiomaticity (Greek: καθολική αξιωματικότητα/ παντοαξιωματικότητα) = omniaxiomatics = omniaxiomaticity (it doesn't exist/ it is impossible)