Repeated use of equations to manage systems causes quantities to accumulate asymmetrically on the two sides of a relationship.
Hidden in plain sight, mathematics is governed not by equations, but by the relational symmetry of Order 4 acting on four pairs of inputs 10, 10, 00 and 11 to generate four oppositional pairs. Mathematics has been constructed absolutely in equations on the principle of balance. Equations treat the two sides of a relationship asymmetrically, cancelling out relational symmetry, which manifests in the asymmetric accumulation of quantities on the two sides of relationships. Relational symmetry must be maintained to control a system, and needs to be restored in a system that has become unbalanced.
Working on the principle of a balance, the outputs of inputs 10, 01, 00 and 11 are 1, 1, 0 and 0: 1 means distinct, countable and 0 means the same, not countable. This is consistent with the truth table in Boolean logic for exclusive OR (XOR) wherein 1 means true and 0 means false. For example when 1 = 1, 1 – 1 = 0.
The truth table for not exclusive OR (XNOR) is the mirror image of that for XOR. It exchanges the meanings of 1 and 0. By convention the symbols 1 and 0 are still given the absolute meanings true and false, but working in XNOR is equivalent to working in XOR where 1 means false and 0 means true, and on the principle of balance where 1 means same, not countable and 0 means distinct countable. For example, when 0 = 0, 0 – 0 = 1.
What is hidden in plain sight is that the meanings for 1 and 0 (distinct, countable and same, not countable or true and false) require the relational symmetry of Order 4 to enable their construction. 1 and 0 are mirror symmetric, like the two sides of a coin, one side of which can be seen from the front and the other side by reflection in a mirror. When inputs 1 and 1 generate 0 from the front, them inputs 0 and 0 generate 1 by reflection in a mirror.
The relational symmetry of Order 4 will not allow the outputs of inputs 10, 01, 00 and 11 to be generated independently in the front and mirror views. There are only six, not eight ways of ordering them in rings. The truth tables for XOR and XNOR interpenetrate. The meanings of 1 and 0 are thus defined relationally: they are relational objects. They operate in mirrored cycles. You cannot cancel 1 and 1 down to 0 in a cycle unless you can also cancel 0 and 0 down to 1 in the mirror. For example, in the exchange of goods and services, you cannot cancel down (from -1 to 0) created debt unless in the mirror you also cancel down (+1 to 0) the credit created with it. If you do, you lose relational symmetry, because you are leaving quantities of relational objects incompletely counted – like leaving the ends of chains free instead of connected to form rings.
