Formal Systems

A formal system can specify objects, stabilize symbols, restrict permitted operations, and preserve intermediate steps.

Another person can inspect the same structure, and errors can sometimes be localized.

A result can be reconstructed rather than merely asserted.

Counting provides a simple example.

A pile of stones lies on a table.

You count twelve.

I count thirteen.

We can count again, perhaps moving each stone as it is counted or arranging them in a row. We might pair each stone with a numbered marker.

If we disagree again, we may be able to locate the skipped or duplicated stone.

The procedure does more than produce a number.

It organizes the disagreement.

Measurement adds further constraints.

Two people disagree about the length of a table.

A measuring tape can reorganize the problem, but it does not give us a quantity independently of the conditions of measurement.

The measurement depends on standardized units, a suitable instrument, correct placement, an adequate reading, and sometimes calibration and uncertainty estimates.

Much of the stability associated with measurement is engineered.

Conditions have been constructed under which certain disagreements become easier to investigate.

Diagrams do something different.

A diagram can make several relations visible at once. They can be pointed at, compared, rearranged, and criticized. A geometrical construction, a circuit diagram, a state machine, and a dependency graph differ greatly in purpose, but all can make relations available for visual inspection.

Logic makes constraint more explicit still.

Once a formal language and rules are operative, a derivation can be constructed step by step.

Each transformation can be checked.

If two people disagree, they can move backward until they find the first point at which their judgments diverge.

Was a premise introduced?

Or a rule applied incorrectly?

Was a variable substituted where it should not have been?

Formalization does not remove the need for judgment.

Someone still has to decide what to formalize.

The notation has to be understood. Premises have to be supplied. Rules have to be interpreted and applied.

The formal system itself has to be appropriate to the task.

But within the system, some kinds of disagreement can become sharply constrained.

Across these examples, parts of inquiry move outside immediate memory and become easier to repeat or inspect. Intermediate states persist, some moves can be restricted, and errors can sometimes be localized. Other people can reconstruct what happened.

Disagreement can move from: "Your conclusion is wrong" to: "This step does not follow."

That is a substantial change in the conditions under which disagreement can be examined and convictions reconsidered.

The constraint is nevertheless local.

Formal systems can make parts of inquiry explicit, repeatable, and inspectable, but they do not replace the wider orientation in which those parts operate. And their constraints carry force only if the system itself is convincing.

A derivation still begins from premises. A measurement still depends on instruments, procedures, and interpretation. A formal model still has to be connected to some problem or domain.

Nor does all conviction formation proceed through reconstructible steps. Perception, practice, narrative, and affect can alter orientation in different ways.

Formal systems are not a general model of conviction formation. They are an especially explicit way of constraining parts of inquiry.