The Key Idea – Resolution of Zero

Key thoughts

Information about economic relationships between people is coded and processed using accounts.

Accounts are constructed from two symbols, 1 and 0.  The 1s are tallied and the 0s are not.

1 and 0 are defined by the two sides of a distinction, like the opposed sides of a coin, which can be seen exclusively as heads or tails (is up, not down).

Accounting works on the principle of a balance, like the mixing of blue and yellow to give green:

arithmetic in chiralkine numbers

Each person’s sale is another’s purchase? So why is it harder to sell things (and so pay off debt) than it is to buy things (and so get into debt)? Is one side of a relationship between two people engaged in exchange privileged over the other? Does this have something to do with the way in which 1, 1 cancels down to 0, 0, so that there is only one zero number?

When two opposites like +1 and -1 interact, they should change one another in complementary ways to produce two new opposites, not combine into zero.

The Idea: Resolution of Zero:

Split zero into two opposites: change from the principle of a balance (-1 + 1 = 0 = +1 -1) to the principle of order:

The equation (-1 +1 = 0) splits, so that each of +1, -1 and 0 has a dual:

The operations of addition and cancellation are now replaced with new operations where the “numbers” act on one another in complementary ways, like self-aware beings. This ensures that whenever they act on one another, they produce two new numbers, one for each being, not combine into zero.

One side of a relationship is no longer privileged over the other.

On one side, 0 turns 1 into 1 and 1 turns 0 into 1. On the other side, 0 turns 1 into 0 and 1 turns 0 into 0.

There are now two kinds of +1, two kinds of -1 and two kinds of zero.

In order to be able to make use of this insight, each of the six differently coloured numbers needs to be defined by symbols that can be used in an accounting system. Ordered pairs of 1s and 0s can no longer be used, because each number now has a dual. The solution to this problem lies in using four ordered symbols instead of two. The particular order of the four symbols does not matter. What does matter is that the two sides are mirror images of one another.

The numbers have both a colour and a sign.

In the statements on the left side 0 is privileged over 1, because it alone turns itself into itself. In the statements on the right side 1 is privileged over 0, because it alone turns itself into itself. Overall, neither of 1 and 0 is privileged over the other.

On the left side, the 0s are symbols that are tallied and the 1s are symbols that are not. On the right side, the 1s are symbols that are tallied and the 0s are symbols that are not.

The six coloured numbers expressed as four ordered symbols are called chiralkine numbers. Instead of using the symbols 1 and 0; a symbol and a vacant space; ↑ and ↓; or + and – can be used. The use of + and – is preferred over 1 and 0, because 1 is generally understood to be a symbol that is tallied and 0 a symbol that is not. Whichever two symbols are used, they represent the two sides of a distinction. Whatever meanings they signify on one side, they signify the opposite on the other.

As the system has been developed, it has turned out that the left and right sides in the table above correspond with the inputs and output for an XOR (exclusive or) and an XNOR (exclusive nor) logic gate. In a NOR gate, 0 signifies same and 1 signifies different. In an XNOR gate, the two symbols have the opposite significations. In the table below, + has been equated with 1 and – with 0, but the opposite significations could equally well have been chosen. The meanings are defined relative to one another.

It has also turned out that chiralkine numbers behave as unitary quaternions and can be related directly to the symmetry properties of chiral tetrahedral molecules.

Chiral tetrahedral molecules are composed of four different atoms or groups that project towards the corners of a tetrahedron. There are two different ways in which the groups can be arranged in 3D space, corresponding with the difference between the left and right hands. These are called enantiomers. The four different groups can be identified by a colour and their arrangement in 3D space by a + or – sign.

The + and – enantiomers of a chiral tetrahedral molecule are mutually exclusive (XOR/XNOR). The eight objects of the two enantiomers if taken together (interpenetrating) would point to the corners of a cube. Each corner and face of the cube corresponds with a quaternion.

The quaternions defining the faces of the cube form the truth tables for XOR and XNOR logic gates. They are chiralkine numbers.

On one side, it is the – signs that are tallied. On the other side it is the + signs.

The pattern of alternation of signs in the columns of the truth tables for XOR and XNOR sets up a cycle, whereby it is possible to transition from one side to the other.

The ordered symbols form into triplets going down their columns. In one column these triplets alternate – + – + – +. In the other three columns they alternate + + + – – -. These columns of alternating signs provide a hidden tunnel between the two opposite, mutually exclusive worlds of XOR and XNOR.

Each step through a cycle corresponds with flipping two of the four signs on both sides, as if the two sides have exchanged signs. The exchange interaction is like money changing hands, but where two opposite kinds of money are swapped over.

It is as if the two opposite kinds of money change colour when they are being created, exchanged and redeemed.

Each exchange interaction can be related to perceptual switching as experienced through the Necker cube effect, where an observer breaks the symmetry of a hexagon to perceive a cube projecting towards or away from the observer.

This can be related back to the symmetry properties of chiral tetrahedral molecules, to the quaternions and to the exclusive logic gates XOR and XNOR

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The cycle can also be visualised as rotation through a Möbius strip. A Möbius strip has two sides locally and one side globally. Imagine the yellow, blue and green discs arranged along a strip which is given a half twist then joined to form a Möbius strip. One side of each disc is labelled + and the other -. As you go around the strip (two full 360 degree rotations), each colour is visited twice; like the head and tail of a coin.

It does not matter which direction of rotation around the cycle is selected (clockwise or anticlockwise) as long as all processing is consistent (yellow to blue to green or blue to yellow to green). If the distinction between the two different orders is not maintained then the system reverts to operating on the principle of a balance, where -0 and +0 are the same 0.

Chemists use the term “resolution” to describe the separation of the two enantiomers of a chiral molecule from a 1:1 mixture of the two, known as a racemic mixture. So the resolution of a 1:1 mixture of the two enantiomers of an amino acid DL affords the D enantiomer separate from the L enantiomer. The switching in coding of relationships from the principle of a balance to the principle of order is a resolution of zero. It breaks the symmetry of the equation +1-1 = 0 such that there are now four distinguishable states: (1 0), (0 1), (0 0) and (1 1). “Resolution of Zero” is also the title of a novel I wrote several years ago to try to get across this very concept in a non-mathematical way.

Early in the development of chiralkine systems, the letters A (1 0), C (0 1), D (1 1) and L (0 0) were chosen to signify the kinds of chiralkine number. D and L signify dark and light (polarities). They also have a special meaning to a chemist: separately they signify the dextro (D) and levo (L) rotatory forms of a chiral molecule that in a 1:1 racemic mixture are denoted DL. A and C signify anticlockwise and clockwise (movement). Hence the name chiralkine came from a combination of chiral, as in handed and kine, as in kinetic or movement: movement of the hands. Much of the early development of chiralkine systems grew out of exploring links between the four different states A, C, D and L and the four different groups that define chiral tetrahedral molecules. These links turned out to be profound.

The system has therefore been re-engineered from one based on the principle of a balance to one based on the principle of order.

Chiralkine numbers, each composed of strings of four ordered distinctions, are useful for coding and processing information about relationships without privileging one side of a relationship over another. They can link the interests of people together. They can be used to control the exchange of goods and services, taxation, voting, decision making and dispute resolution. They may be useful in the construction of a social robot: one that can perceive both sides of a relationship. They can also be used in a game, which may prove useful in the treatment of brain injury and stroke.