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On Evidence and Proof: A Recursive Model

The Successor Study Case

Consider the arithmetical claim:

For every natural number $A$, there exists a successor $B$ such that $B = A + 1$.

How do we prove this? We look for $B$. We find it. We repeat. The sequence never fails—so we call it true.

But notice:

• We never look for a failure mode—a case where $A$ has no successor.

• We treat the absence of failure, repeated indefinitely, as proof.

• Yet an infinite repetition can never be completed—only performed.

If you ask: "Show me all successors without failing," the request is impossible to fulfill. The proof is not given—it is performed, and the performance is never finished.

This reveals a structural confirmation bias:

We search for confirming instances (successors) and ignore disconfirming possibilities (failure modes). We treat the absence of failure as evidence, and we call the accumulated performance proof.

Definitions

Let:

• $C =$ confirmation bias—a function that favors a prior frame

• $P₀ =$ a prior proof—an earlier threshold, frame, or conclusion

• $E =$ evidence—the output of $C$ acting on $P₀$

Then:

$$E = C(P₀)$$

Evidence is not discovered—it is generated by a bias acting on a prior proof.

Now let:

• $E₁$ = a specific piece of evidence

• $\sum{E}$ = the sum of all prior evidence

Then:

$$P = E₁ \text{ such that } \sum{E} < E₁$$

Proof is a threshold—a single piece of evidence that exceeds the cumulative total of prior evidence, thereby interrupting the performance of confirmation bias.

The Successor Case in These Terms

In the successor case, the prior proof is the axiom every number has a successor. The confirmation bias looks for successors and finds them. Each found successor is a piece of evidence. Proof is the decision to stop looking for a failure mode—the moment we declare the rule universally true.

But that declaration is not a logical necessity—it is a threshold we set. We choose to stop. We call that stop proof.

The Recursive Loop

Proof itself is also constructed from evidence:

$$P₀ = f(E₀)$$

Where $f$ is the process by which evidence is assembled into a proof.

Thus:

$$E = C(P₀) = C(f(E₀))$$

And:

$$P = E₁ \text{ such that } ΣE < E₁$$

And:

$$E₁ = C(P₁)$$

So the full system is:

$$P₀ → E → P → E₁ → P₁ → ...$$

There is no beginning—only recursive dependence.

Implications for the Successor Case

The successor case illustrates that what we call mathematical proof is not a logical terminus—it is a practical stop. The axiom is a prior proof, instituted rather than discovered. The search for successors is a performed confirmation bias. Each found successor is evidence. The declaration of universality is a threshold—a decision to stop searching for a failure mode. Proof, in this view, is not a logical necessity but a pragmatically set stop.

What This Does Not Claim

This model does not claim that reasoning is arbitrary—it is constrained by context, prior frames, and practice. It does not claim that proof is impossible—only that it is instituted, not discovered. It does not reject evidence—it locates it as produced, not given. It does not deny the utility of arithmetic—it describes its epistemic structure as performative.

Conclusion

We have formalized evidence and proof as mutually recursive, performance-based, threshold-dependent, and non-foundational. The successor case demonstrates that even in arithmetic, proof is not a logical terminus—it is a practical stop: the moment we choose to interrupt the infinite performance of confirmation bias and call the result true.

This model does not solve the regress—it represents it—and suggests that practice, not foundation, is the ground we actually operate from.