Related materials: axiomatics, proof, reification, form as a condition of description, theses.
Introduction
In ordinary language, “subjective” almost immediately means personal opinion, taste, or a lack of objectivity. For the status of an axiom, a different word is needed here: subject-relative. It means that a proposition has this status only through its relation to a subject.
An axiom is accepted without proof within a particular system. The unchangeability of this acceptance exists in relation to the system, not as a property of reality itself. This document establishes this relation and the transition in which the foundation of a system is mistaken for the foundation of reality.
The structure of a set of initial propositions and rules of inference is examined in axiomatics. How a proof remains within its own foundations is examined in proof. The question of how a conclusion that is necessary within a system relates to reality is also left open there. Here, the status of the axiom itself is established.
Definition
An axiom is a proposition accepted without proof within a particular system and used as a foundation for further inferences.
The system within which this is possible is not given together with reality. The subject draws its boundaries: chooses the language, admissible objects, operations, and rules of inference, and decides which propositions within those boundaries are to be regarded as initial.
The unchangeability of an axiom exists relative to that system. Reality contains no propositions that are in themselves exempt from further distinction.
Every axiom is subject-relative because it is an axiom only relative to a system of distinctions drawn by a subject.
There are no axioms in reality. An axiom arises only within a system formed by a subject.
Both formulas preserve one status from two sides. Within a system an axiom exists: it is accepted without proof and serves as the foundation for further inferences. In the structure of reality it does not: it arises only together with the system formed by the subject. In the discussion of proof, this was already stated as follows:
An axiom can condition a system, but it cannot condition the reality in which the axiom itself was created.
Subject-Relativity and Rigor
The subject-relative nature of an axiom is compatible with the rigor of a system. A system may be internally consistent, practically powerful, and shared by many. Inferences within it may be necessary.
An arbitrary foundation would be one indifferent to the structure of its own system. Rigor operates within the system and leaves the foundation of that system as the foundation of that system. Rigor does not turn it into a property of reality independent of the subject.
Three Statuses
A proposition may pass through three statuses.
Assertion. A subject asserts something.
Axiom. A subject fixes the assertion as not subject to proof within a system and adopts it as a foundation for further inferences.
Reification. The status the proposition held within the system is transferred to reality itself.
Fixing an axiom still preserves the foundation as the foundation of the system. The substitution begins with the following step:
we accepted this as the foundation of our description
→ this is the foundation of the world
The system is local. The absoluteness of an axiom holds only within this locality. When a local foundation is taken as the foundation of reality, the condition for constructing the system is collapsed into the structure of reality.
This transition is reification: the method of description acquires the status of the intrinsic structure of what is described.
Localization and Thesis
The rigidity of an axiom arises through localization. A system’s boundary is drawn before a proposition receives its status as an initial proposition. Within that boundary, an axiom means:
No questions are asked below this point within this system.
The breadth of an assertion does not itself confer this rigidity. A narrow proposition may be more rigid than a broad one. A thesis may encompass a more general relation and remain a point below which questions can still be asked. Broad coverage does not take it beyond the boundaries within which it was expressed: this is a local suspension of thinking, not an immovable general foundation. An axiom narrows the domain and fixes the status. A thesis allows broad coverage and further movement of distinction.
An axiom is narrower but more rigid. A thesis is broader but more mobile.
