When Mathematics Became a Proposition: III. When the Unknown Became the Identified

Suppose we begin with:

x + 2 = 5

We know there is an unknown quantity in the equation.

But x is not yet the whole of the Identified.

The identifying relation is between:

x + 2 and 5

The expression x + 2 is what is being identified with 5.

To find the value of x, we transform the equation:

x + 2 = 5

x = 5 − 2

x = 3

At first glance, this looks like a sequence of calculations.

But something more interesting is happening.

Each step produces another clause.

And the clauses progressively reorganise the relation being construed.

In the first equation, the quantity containing the unknown is:

x + 2

In the second, the unknown has been isolated:

x

And in the final equation:

x = 3

x has become the Identified.

3 has become its Identifier—the value that identifies what x is.

The same movement can be seen across the metafunctions.

At the beginning, x + 2 is the Theme of the clause and the Identified/Token of the identifying relation.

By the end:

x = 3

we have:

Theme ^ Rheme

x ^ = 3

and:

Identified/Token ^ Identifier/Value

x ^ 3

The unknown has become the thing being talked about, the thing being identified, and the Subject of the proposition.

Its value has become what is said about it, what identifies it, and the Complement of the clause.

The solution is therefore not simply a number.

It is a new clausal configuration in which the unknown has been brought into the position where its value can be stated.

This also explains why solving an equation is not simply a matter of manipulating symbols until a number appears.

Each transformation has to preserve the mathematical relation being construed.

The new clause must remain valid in relation to the preceding one.

That is why:

x + 2 = 5

can become:

x = 5 − 2

and then:

x = 3

but not just any arbitrary expression.

The transformations are constrained.

Something has to remain true as the equation changes.

And now we can see something that was hidden in the apparently simple notation.

Solving an equation is a process in which one clause is progressively elaborated into another until the unknown can be identified.

The final equation does more than give us an answer.

It construes the answer as the value of the unknown.

The unknown has become the Identified.

And that raises a new question.

If every step in solving is another clause, what kind of relation holds between all those clauses?

They are not separate statements.

They form one continuing mathematical argument.

And that is where the interpersonal dimension begins to come into view.