When defining an object, the relationship between what it is and what it is not is crucial, yet equations privilege one side over the other. The difference between two objects that are “equal” – the two sides of an equation – is not nothing (as equations assume), but rather two distinct aspects of a relationship. To maintain true symmetry, objects must be counted twice, in a cycle or ring, ensuring both sides of the defining relationship are treated equally. Equations inherently cause inequality because they treat the two sides of a relationship defining a countable object, is and not, asymmetrically.
Equation-based mathematics is based logically on drawing a distinction between 1 and 0 using XOR, such that 0 identifies with itself as itself, but 1 does not, treating 1 as distinct and countable and 0 as not. This generates three rotational presentations of the triangle defining 1 in relation to 0:
It takes just two steps to leave and return to balance (closure), corresponding with the creation of -1 and +1 from 0 and their mutual negation back to 0.
Consider, for example, the process for counting one.
The logic actually requires that one (+1) is recorded by turning 0 (balance) into +1 using -1!
Traditional mathematics gets around this by deeming that there are two interchangeable forms of 0 (balance). However, there is a problem with this approach. There is no logical (commutative) mechanism whereby these two forms can be interconverted.
Equation-based mathematics could alternatively be based logically on drawing a distinction between 1 and 0 using XNOR, the mirror opposite of XOR, such that 1 identifies with itself as itself, but 0 does not, treating 0 as distinct and countable and 1 as not.
However, when this logic is exploited to construct two mirror opposite equation-based systems, the two operate independently. Both of these equation-based systems are embedded in chiralkine counting, but chiralkine also exploits hidden logical couplings between the two to create a system that treats the two sides of relationships defining countable objects symmetrically.
These hidden couplings arise, because conservation of mirror symmetry requires that there be an equal number of commutative and non-commutative interactions between 1 and 0. All of the non-commutative interactions are hidden in an equation-base system. Putting the two together results in two inputs always giving two mirror opposite outputs (e.g. 1(1) turns 1(2) into 1 and 1(2) turns 1(1) into 0).
Summary Drawing
The drawing below summarises why equations are inherently asymmetric and how switching from counting based on the principle of balance, in which symbols are defined absolutely, to counting based on the principle of symmetry, in which symbols are defined relationally, reveals hidden non-commutative interactions.
Three steps are required to rotate from the original position to the position where all “is” and “not” pairs have been aligned with their counterpart.
Money, Debt, and the Inequality of Chains
In the modern economy, banks create money (+1) and debt (-1) out of nothing (zero) through a simple accounting step designed to solve the double coincidence of wants problem inherent in barter. When a person offers something that another wants, the seller receives money (+1), and the buyer takes on debt (-1). While this seems symmetrical, it isn’t in practice: society views money (+1) as an asset to be spent, but the debt (-1) must be repaid.
Critically, there is no obligation for the seller to spend the money so that it circulates back to the debtor, creating an imbalance. Over time, more people buy goods and services from fewer suppliers, concentrating wealth and expanding inequality. Governments attempt to address this inequality by borrowing more money from banks (who create it from nothing and assign the debt to governments) to redistribute it, but this only amplifies the problem, as public debt grows uncontrollably.
This instability is a direct consequence of equations and the chains they create. Chains of money and debt lack the symmetry of rings, where offers and wants are paired and balanced. They leave a terminal is and not unpaired. Without symmetrical counting, economic systems are destined to collapse under the weight of inequality.
Elections and the Divergence from Democracy
Another consequence of equation-based systems is the growing divergence between election results and the actual wishes of voters. In authoritarian systems, this issue is stark due to controlled candidate selection. However, it is increasingly evident in mature democracies with two dominant parties, such as the UK (Labour and Conservative) and the USA (Democrat and Republican). Voters lose faith in democracy when the system no longer reflects their will.
A cast vote is defined by the relationship between “for” and “against.” Yet, electoral systems fail to treat these sides symmetrically. A vote for one candidate is not inherently a vote against another, but many systems implicitly treat it as such. This asymmetry leads to scenarios where a candidate with the most votes can win even if the majority of voters oppose them. Furthermore, tactical voting emerges when voters choose a candidate not because they support them, but to block another.
Even abstentions, whether active (protesting the election itself) or passive (not participating), are not distinguished in traditional systems, reducing nuanced voter behavior to a single, ignored zero. Equations erase the complexity of these relationships, undermining democracy itself.
A Symmetrical Alternative: Chiralkine Counting
Chiralkine counting offers a potential solution by treating the two sides of relationships symmetrically. In this framework, exchanges – whether of goods, services, or votes – must form closed rings, where every offer and want is paired and balanced. Unlike chains, which create inequality, rings preserve symmetry and prevent the accumulation of imbalances.
In elections, chiralkine counting could ensure that every “for” and “against” is treated as a distinct yet equal component of the voting relationship. This approach could eliminate tactical voting, provide clear distinctions between active and passive abstention, and better align results with the will of the voters.
By embracing symmetrical counting, chiralkine provides an alternative to the inequality embedded in equation-based systems. Whether applied to economics, democracy, or other social systems, it represents a paradigm shift toward fairness and balance.
Acknowledgement: revised with the assistance of Chat GPT.