The way we count today works on the principle of balance and equations. If two objects are the same, then they balance. The difference between them is nothing. If two objects are not the same, then they are distinct. We assign the meaning same to 0 and the meaning distinct to 1. Hence 1 = 1 and 1 – 1 = 0.
Whenever an object is counted it is erased (/) from consideration and a new object recording the count is brought into consideration.
The erasure of a counted object (/) works on the principle of a balance and an equation: 1 – 1 = 0. The order of the mutually negating numbers on the left side of the equation does not matter (1 – 1 = -1 +1 = 0).
Another way to visualise this is to think in colour. In the picture below, an object is shown as a yellow circle set against a green background, representing zero and a negative object is shown as a blue circle set against a green background. According to the principle of an equation, zero is defined as the difference between two objects that are the same, which can be obtained by subtracting an object from itself or adding an object to its negative (anti) self. The colours yellow and blue have been selected because they can combine to give green.
Counting in an accordance of the principle of an equation ignores the crossings that separate objects being counted. It conflates crossings with nothing (zero).
Objects are perceived to have fixed locations, but crossings are not, because the order in which objects are counted does not matter. Crossings are delocalized. They can follow any path between the objects that they connect.