Supernumbers

In Bijectivism, I introduced a philosophical stance that treats reversibility as non-negotiable, exposing the limits of defining inverses as single-valued. The Supernumbers project is my attempt to instanciate that philosophy into actual numbers that we can manipulate with standard mathematics.

When Does Information Really Get Lost?

We are used to a simple idea:

when a transformation is not invertible, information has been lost.

Take a familiar example. The function

\[ f(x) = x^2 \]

maps both \(2\) and \(-2\) to \(4\).

From the value \(4\), we cannot recover the original input. The conclusion seems unavoidable: information has been destroyed.

But this conclusion depends on a quiet assumption.

We decided that the result of a transformation should be represented as a single value.

What if that choice is the problem?

The hidden structure of a value

When we compute:

\[ f(2) = 4 \]

we typically retain only the output: \(4\).

But the transformation itself does not forget that both \(2\) and \(-2\) lead to the same result. This structure still exists—it is just not carried forward.

More precisely, for a given output \(y\), there is a set of inputs:

\[ S_y = \{ x \mid f(x) = y \} \]

For \(y = 4\), we have:

\[ S_4 = \{2, -2\} \]

This set is usually treated as secondary, or even discarded.

But nothing forces us to do that.

A different representation

Instead of representing a value as a single number, we can represent it together with its admissible origins.

We write:

\[ (4, \{2, -2\}) \]

This object contains:

• the observed value

• the structure of how it can arise

We call such an object a supernumber.

In this representation, the ambiguity of \(4\) is not a problem—it is made explicit.

Invertibility revisited

Now consider the original issue: invertibility.

The classical inverse of \(f(x) = x^2\) fails because it attempts to recover a single input.

But if we allow the result to be:

\[ f^{-1}(4) = \{2, -2\} \]

then the inverse always exists.

More generally, for any transformation \(f\), we can define:

\[ f^{-1}(y) = \{ x \mid f(x) = y \} \]

Every transformation becomes invertible—once we accept that outputs carry their full generative structure.

What changed?

Nothing about the transformation itself.

What changed is the representation.

We moved from:

value = output

to:

value = output + admissible origins

Under the first representation, information appears to be lost. Under the second, it was never destroyed—it was only omitted.

Composition without collapse

This difference becomes more visible under composition.

In standard computation, composing functions tends to flatten structure. Intermediate distinctions disappear as soon as we move to the next step.

In the supernumber view, this does not happen.

Each step carries its own structure forward. Instead of collapsing intermediate information, composition builds nested layers of origins.

The result is not a single value, but a structured object that records how it came to be.

What this reveals

This perspective shifts a familiar conclusion.

Non-invertibility is not simply a property of transformations.

It is also a consequence of what we choose to remember.

By representing values as isolated outputs, we collapse distinctions that were present all along. By restoring those distinctions, invertibility reappears—not as a special case, but as a general property.

Closing

We often say that information is lost when we cannot recover the past from the present.

But this assumes that the present is just a point.

If instead we allow it to carry its structure, then the past was never lost—it was only hidden by our representation.