Education tethers meanings to symbols so deeply that we can react with shock or anger to them. It empowers us to co-ordinate our efforts for common objectives, but it also traps our train of thought along invisible tracks. This paper seeks to articulate a radical idea in which symbols, tethers and meanings are processed in accordance with a new principle, but being a paper intended to be read by educated people it has to achieve this using symbols, tethers and meanings that its readers have internalised. The following notes are offered as a guide to help the reader to construe the paper as intended.

The idea that the concept of balance is flawed came explosively in 1982 during a class on bookkeeping at the counting house of Deloitte Haskins and Sells in London. It brought a short career in accountancy and business to an abrupt end and led to embarkation on a new one in innovation and patent law. An education in accountancy teaches you to use a method of counting based on the concepts of balance, equations and nothing. An education in patent law teaches you how to construe the words used by inventors in their patent specifications to describe and define their inventions. The intended meanings are often different from their usual meanings, because the subject matter is an invention new to man. However, neither of these disciplines admits of the possibility that meanings and symbols can exchange their tethers. It comes across as a radical idea, because it appears to threaten to deconstruct everything that people believe in.

Development of the radical idea was frozen in response to its hostile reception in 1982, but was resumed in 2009, and taken to a point when it was believed to be ready for submission to Nature. The rejection was humbling, but the idea is too compelling to abandon, so began a quest to find a way to describe and define it clearly and concisely in a way that would enable the reader to use it, like any other invention being communicated to the public through a patent specification. That has proved to be incredibly hard!!!

“*When we a start thinking about a symmetric pair of opposites and then start to think about a mid-point between them (zero), there is another pair of opposites waiting ready to come into mind*.”

Expressed in different words, when we start thinking about the two opposite sides of a relationship defining a distinct object: what that object is and what it is not, and then start to think about a mid-point between them, two opposite sides of another relationship defining another distinct object come into mind. The difference between two things, whether they are the same or different, is always two things, not one. For example, the difference between black and black or white and white is black and white, and the difference between black and white or white and black is black and white. However, there are hidden logical rules of great simplicity that determine these outputs, and from application of these simple rules emerges a control system. These rules are not symmetric, where the order of the comparison does not matter, but antisymmetric, where reversing the order inverts the output.

Chiralkine counting works on the principle of order, not balance. It is dual. An object is a relationship between what it is and what it is not. When an object is counted, that relationship is replaced by another one.

“*The train of human thought is best described by the switching of two antisymmetric waves having the same phase between a symmetric and two alternative antisymmetric couplings.*“

For a very long time it troubled me that there are two, not three possible antisymmetric couplings, because my intuition was telling me that there was a deep connection with a spin system consisting of two spin 1/2 particles and that can exist in one singlet and three triplet forms. Now I understand that the intuition was correct, but that the symmetric and two antisymmetric couplings are all triplet forms. Below is a diagram reproduced from the page on my website devoted to spinors.

The top triplet state is symmetric, because each pair of symbols denoting what an object is and is not are aligned. The other two triplet states are antisymmetric, because the members of each pair of symbols denoting what an object is and is not are exchanged. The singlet state is simply a scalar: triplet states of each kind can stack up one on another of their same kind.

There are six ways of ordering four different objects each defined by a relationship between what it is and what it is not. In one pair the symbols represented by B and W in the drawings are tethered consistently to same, not different and not same, different. In the other two pairs the meanings are mixed up.

My website explains how application of simple logical rules generates the triplet states. These rules are grounded in two mutually exclusive ways for drawing and erasing distinctions: exclusive OR (XOR) and not exclusive OR (XNOR). Each of these mutually exclusive ways is defined by a truth table having four columns dealing with the four different ways in which “same” and “different” can be compared to generate an output of “same” or “different”. The two ways interpenetrate, such that there are six, not eight states: six being the number of ways of ordering four different objects.

The system is chiral, because there are two ways of ordering four different objects in 3D space: projecting towards the corners of a tetrahedron. The diagrams in the paper are based on Fischer projections used by organic chemists to model the 3D structure of a chiral tetrahedral molecule as it is rotated in 3D space. The cube contains both chiral forms embedded in it, so each face of the cube contains two opposed objects belonging to one of the chiral tetrahedrons and two belonging to the other. This is illustrated in the diagram below where the four members of each enantiomer of a chiral tetrahedron are all black or all white.

“*Accordingly, two such non-interfering wave functions 180ยบ out of phase could describe all possible motions of the relative position of an antagonistic muscle pair*.”

Each member of a pair of opposed faces of the cube is related to the other by rotation through 180 degrees, with exchange of black and white.

“*HIDDEN COUPLING BETWEEN ANTISYMMETRIC WAVES IN THE BRAIN COULD BE LIMITING THE RELIABILITY OF THE SCIENTIFIC METHOD*“

Mathematics and science are constructed based on counting in accordance with the principle of a balance and equations. When meanings are tethered to symbols in this way, information about relationships is hidden and lost. Mathematics and science are extremely powerful approaches to modelling nature quantitatively in terms of laws, but the concepts of the balance of nature and nothing (zero) need to be complimented with their dual partners to see the whole picture.

“*To understand better how the human mind works, you have to think of the markings on two Taijitu symbols coupled in motion*.”

The form of the symbol chosen to represent chiralkine shares features of the Taijitu symbol, because that is how it works.

“*the symmetry errors that are presently being made in language and the systems that are used to structure how people co-operate*“.

Inequality keeps rising, financial systems keep failing, voting fails to allocate power and resources in accordance with voter’s wishes, conflicts keep breaking out all over the world, and mankind’s relationship with the planet is disintegrating. The root cause of all this is maths! We are doing it wrong!