Proof

Do not confuse with verification: proof establishes the necessity of a conclusion from foundations, while verification means that an assertion passes through a particular testing procedure.

Initial Construction

Proof never begins from zero.

Every proof already has:

• initial propositions;
• rules of inference;
• a language;
• criteria of identity;
• criteria for permissible transitions;
• a conception of contradiction;
• a conception of what counts as proven at all;
• a criterion for a proof’s completeness.

Even the simplest form:

If A, and if B follows from A, then B,

already accepts the rules of logical inference.

These rules cannot simultaneously be used to prove their own unconditional truth without creating a circle. To prove the rules of the first system, a metasystem is required. But the metasystem, too, has its own rules and foundations.

Consequently, proof does not exist by itself.

There is only:

proof within a particular system of foundations;

or:

proof on condition that certain axioms, rules, and distinctions are accepted.

That is why the phrase “this has been proven” almost always contains a hidden remainder:

this is proven for someone who already accepts the particular method of proof.

Proof and Axiom

An axiom precedes proof and determines its possible space. It establishes:

• what may appear as an object;
• what will count as a distinction;
• which questions can be asked at all;
• which answers can be regarded as admissible;
• which operations are permitted;
• which conclusions will appear necessary;
• which criteria of truth operate within the system.

An axiom does not merely participate in proof. It creates the space within which proof becomes possible.

Therefore, proof does not move beyond an axiom toward neutral truth. It unfolds the consequences of a previously accepted foundation.

Proof does not say:

Here is truth without a position.

It says:

If these foundations and rules are accepted, this conclusion becomes necessary.

An axiom does not distort a neutral field of proof. There is no such field before the axiom. Without an axiom, there is no space of proof at all.

At the same time, an axiom does not create reality. It creates a particular contour of reasoning within reality. Its binding force operates inside the construction it establishes and does not extend backward to the reality within which the axiom itself was introduced.

An axiom may condition a system, but it cannot condition the reality within which the axiom itself was created.

Neutral Proof Does Not Exist

Neutral proof does not exist in the sense of proof that depends on no previously accepted foundations.

If one removes the initial propositions, rules of inference, language, criteria of identity, contradiction, permissible transition, and completeness, neutral proof does not appear. Proof disappears.

Thus neutral proof is not a rare or unattainable variety of proof, but an internally contradictory expression:

Proof without prior foundations is proof without the conditions that make proof possible.

Every proof is already on the side of its axioms and rules of inference. It does not pass through axioms to neutral reality. It unfolds what becomes necessary after a specific system of distinctions is accepted.

Proof does not eliminate an axiom. It makes the consequences of the axiom binding within the space created by it.

Regress of Justification

To prove a proposition, a criterion of proof is required. But an attempt to prove the unconditional truth of the criterion itself requires another criterion. Proving the second requires a third.

Three possible outcomes arise:

1. an infinite regress of criteria;
2. a circle in which the system justifies its own rules using those very rules;
3. a halt at a foundation accepted without proof within the given construction.

Moving to a metasystem does not eliminate the problem of foundations. It transfers it to the next level, since the metasystem also has its own rules and initial propositions.

It does not follow that proof is useless or arbitrary. It follows only that its necessity is relative to the conditions that make proof possible and must not be presented as free of presuppositions.

Proof Through the Incompatibility of an Object’s Conditions

The discussion of neutrality contains a separate form of proof: proof through the incompatibility of the conditions for the very existence of the object.

Its structure is as follows:

1. Neutrality is defined as judgment without a prior position.
2. Every judgment necessarily contains a prior distinction.
3. A prior distinction already constitutes a position.
4. Therefore, judgment without a position is impossible.
5. Therefore, neutrality as judgment does not exist.

This is not an inference from lack of evidence to the denial of existence. The assertion is not “neutrality has not been found,” but “the conditions under which it would have to exist destroy the very possibility of judgment.”

Here it is important to distinguish:

We were unable to prove that an object exists; therefore it does not exist.

from:

The conditions for the existence of the supposed object are mutually incompatible; therefore such an object is impossible within the given definitions.

The second is a proof of impossibility, but it too depends on accepted definitions, rules of inference, and criteria of contradiction.

Central Formulas

Every proof is an inference within a system of axioms, rules, language, and criteria of being proven. It may be necessary relative to the system, but it is not free of presuppositions.

Eliminating foundations does not cleanse proof of its position; it destroys proof as such.

Proof does not create unconditional truth. It establishes the binding force of the transition from accepted foundations to a conclusion within a particular space of proof.

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Analyses That Have Not Yet Been Conducted

The questions listed below are touched upon by the current construction but have not yet been examined. They are not established propositions, but a program for further investigation.

1. Definition of Proof

The conditions of proof have been enumerated, but what precisely turns a sequence of statements into a proof has not yet been established. Proof must be distinguished from:

• inference;
• argument;
• explanation;
• demonstration;
• testimony;
• confirmation;
• persuasion and rhetorical influence.

2. The Bearer of Provenness

It has not been established to what the property “being proven” belongs:

• to the thesis itself;
• to the chain of transitions;
• to the thesis’s relationship with a system;
• to the relationship between proof and a consciousness capable of verifying it;
• to the practice of a community recognizing certain rules.

Without this, the expression “the thesis has been proven” remains ontologically unexamined.

3. Proof and Consciousness

It has not been investigated whether proof exists without consciousness that distinguishes premises, transitions, and conclusions. At least three versions are possible:

• proof exists as an objective structure of relations;
• proof arises only in an act of understanding;
• a structure exists independently, but becomes proof only for consciousness within the corresponding system of distinctions.

4. Acceptance of Foundations

The phrase “if the foundations are accepted” has not yet been analyzed. It is unclear:

• who or what accepts them;
• whether acceptance is psychological agreement, practical action, or logical fixation;
• whether a conclusion can be binding for someone who does not recognize the initial system;
• whether using a foundation differs from believing it is true.

5. Truth, Necessity, and Provenness

Three properties have not yet been separated:

• a thesis is true;
• a thesis is necessary within a system;
• a thesis is proven using the means of a system.

It must be established whether what is true may be unprovable, what is proven may be untrue outside the system, and what is necessary may depend on the method of formalization.

6. Formal and Empirical Proof

The current construction primarily describes formal inference. It has not examined:

• mathematical proof;
• empirical confirmation;
• causal explanation;
• historical establishment of an event;
• judicial proof;
• proof in everyday reasoning.

It is necessary to check whether these are varieties of one phenomenon, or whether the word “proof” designates different operations here.

7. Completeness and Closure

A criterion of completeness was mentioned, but it has not been established why a proof is regarded as finished. It is necessary to examine:

• where a sufficient chain of foundations ends;
• what permits one not to prove every intermediate transition;
• how implicit knowledge and obviousness affect completion;
• whether completeness belongs to the text, the system, or the act of checking.

8. Error and False Proof

If proof always exists within a system, it is still necessary to determine what counts as error inside that system. One must distinguish:

• an invalid transition according to accepted rules;
• a hidden or substituted premise;
• a contradictory system of foundations;
• a valid inference from false premises;
• a change in the meaning of terms during the reasoning;
• a proof valid in one system and invalid in another.

9. Translation Between Systems

It has not been examined how to compare proofs constructed on different axioms and rules. Criteria are needed for:

• translatability of concepts;
• preservation of inference during translation;
• compatibility and incompatibility of systems;
• choice of a metasystem for comparison;
• identifying when a dispute over a conclusion is actually a dispute over foundations.

10. The Strength of Proof

It is unclear whether all proofs possess the same status. It is necessary to investigate what determines their strength:

• the minimality of premises;
• the transparency of rules;
• the reproducibility of verification;
• stability under changes in wording;
• the breadth of systems in which a conclusion remains valid;
• the capacity to reveal one’s own limits of applicability.

11. Proof and Reality

It has been said that an axiom creates a space of proof but not reality. It remains unexamined how proof can speak about reality if its binding force is created within a system. Two extremes must be avoided:

• identifying systemic necessity with the structure of reality;
• concluding that dependence on foundations renders every proof arbitrary and incapable of saying anything about reality.

12. Self-Application of the Thesis About Proof

The central construction itself must be tested:

“Every proof depends on foundations.”

If it is proposed as proven, its foundations and rules of inference must be stated explicitly. If it has another status, that status must be defined: description, principle, result of self-observation of thinking, or ontological conclusion. Without such an examination, the theory of proof risks concealing its own system of foundations precisely where it demands that they be revealed in other proofs.