A Miniature Universe That Tries to Count Itself
Let us imagine a very small universe.
Not one with galaxies and quarks, but a tiny thought-experiment universe consisting of a finite number of identical atoms. Say: 100 of them. No more.
These atoms can move, interact with one another, and take on simple states. And—this is crucial—they are required to do just two things:
- to count
- to be counted
Nothing more is demanded.
The Simple Task
The universe is supposed to determine how many atoms it contains.
So it must do something like:
“There are N atoms.”
This sounds harmless. But now things become interesting.
In order to hold on to “N,” the universe needs a representation of that number.
A trace. A mark. A state.
No matter how primitive:
- a single atom taking on a particular state
- a group of atoms forming a pattern
- a configuration that “stands for a number”
In short:
to represent a number requires physical resources.
The First Paradox
Assume the universe contains exactly 100 atoms.
To represent “100,” some of those atoms must take on a state that encodes this number.
But those atoms are then no longer fully available to be counted themselves.
That is:
A portion of the atoms is required to represent the number that is meant to count all atoms.
Suddenly it becomes clear:
The universe cannot fully count itself
without changing itself in the process.
Counting Consumes Substance
Suppose it takes 5 atoms to encode a number in a stable way.
Then the following happens:
- Initially, there are 100 atoms.
- 5 are “used up” to store the number.
- 95 countable atoms remain.
But now the number is no longer correct.
So one would have to say:
“There are 95 atoms.”
But that statement again requires representation.
Which again requires atoms.
Which again reduces the count.
A strange effect emerges:
The more precisely the universe tries to count itself, the less completely it can be counted.
The Surprising Insight
The problem is not logical arithmetic.
It is physical self-reference.
The number being counted and the number doing the counting share the same material basis.
Or put differently:
The roots of the number and the object being counted grow in the same soil.
This creates a kind of feedback loop:
- counting alters what is being counted
- representation consumes substance
- representation is never free
Cross-Layer Relativity in the Miniature Universe
Now the real point comes into view.
The “number of atoms” is not only relative to its representation,
but also relative to the other numbers that are represented at the same time.
If the miniature universe also wants to store the number “2” or “7,” it requires additional physical structure.
These additional representations, in turn, influence the conditions under which all other numbers exist.
Thus:
The value of a number is relative to the physical instantiation of other numbers.
Not only:
- nnn is relative to xxx
but also:
- x1x_1x1 is relative to x2x_2x2
and therefore:
- n1n_1n1 is indirectly relative to n2n_2n2
through their shared physical basis.
An Unexpected Result
In this miniature universe, mathematics is not an external observer.
It is part of the system.
Counting is a physical act.
Representation consumes resources.
Representations compete.
And for this reason, there is no such thing as a completely “neutral” number.
Not because mathematics is wrong—
but because it must always take place somewhere.
What This Shows
This thought experiment makes something very sober visible:
- numbers require carriers
- carriers are limited
- carriers influence one another
- therefore numbers influence one another—indirectly
This is not magic. It is physics.
And this is precisely where the idea of cross-layer relativity emerges:
Not only are numbers relative to one another,
not only are numbers relative to their physical form,
but numbers are relative to the physical forms of other numbers.
Here is the paper on the topic in a more formal and official version: