7. Visibility and Conviction

We have seen how recursion tables are constructed.
We have seen how they define functions.
And we have seen how they can be used to reason about those functions.

What remains is to look at what kind of understanding they produce.

In each example, the same situation appeared.

All possible cases were visible.
The structure of the data was reflected in the definition.
The process of computation could be followed step by step.

Because of this, certain questions no longer arise.

We do not need to ask whether a case has been forgotten.
We do not need to ask whether the function will eventually return a result.
We do not need to rely on an external argument to justify the definition.

Instead, we can look at the table and see how it works.

This is a particular kind of convincing force.

It does not come from authority, or from derivation, or from appeal to general principles.
It comes from the possibility of inspection.

When all cases are present and the process is visible, conviction forms easily. Not because it is imposed, but because there is nothing left unclear in how the definition proceeds.

This does not mean that recursion tables replace formal reasoning. They cannot. There is an upper bound to the complexity they can handle.

But recursion tables can make some mathematical objects of lower complexity intuitively comprehensible to a human mind, much like truth tables do for propositional logic. They allow us to see, before formal verification, what it means for a function to be total and terminating.

They are a pedagogical tool. And they are also an instance of a more general pattern.

Conviction forms through different mechanisms: perception, repetition, formal systems, social agreement, and others. In some cases, conviction is stabilized by direct visibility.

Recursion tables belong to this class.

They make a definition convincing by exposing its structure completely. Nothing is hidden, and nothing is left implicit. The table can be inspected, and the behavior of the function can be followed without interruption.

This is why they are effective for teaching, but also why they are interesting in their own right.

They show, in a simple and concrete setting, how conviction can arise from making a process visible.