We count objects that are distinct, not the same.

Drawing a distinction separates what an object is from what it is not. An object is in fact a relationship between these two opposites.

In the sections above, we explored how a sheep (a) can be counted by making a tally mark, then the tally mark can be counted by sliding a bead across a wire in an abacus (c).

Each “what is an object” is separated by a “what is not an object” and each “what is not an object is separated by a “what is an object”. The two are mutually exclusive. The same number is counted whether we count “what is an object”s or the “what is not an object”s.

For example, consider the example of four circles as we have discussed earlier.

Here a distinction is drawn between a circle and a crossing. The circles and crossings are mutually exclusive. A circle is not a crossing and a crossing is not a circle. Each kind separates the other kinds.

We have been taught to think that the crossing is simply nothing, zero: that it is not countable. We think that what we are counting are distinct objects, circles, not pairs of circles and crossings defining a relationship. Hence when we count a circle, record the count and cancel out the circle and a crossing, the result of that cancellation is conflated with “not a circle” – a crossing.

We pretend that the crossings, which separate the circles one from another, are not there.

We do this, because we work on the principle of a balance. The difference between two things that are the same is nothing (zero). However, there is a problem with doing this, which we ignore.

According to the principle of a balance, the difference between a circle and a circle is nothing. The difference between a crossing and a crossing is also nothing. If a crossing is nothing, then the meaning of nothing is self-referential. However, it follows that the meaning of a circle cannot be self-referential.