Nobody really understands spinors. They are said to be related in some way to the “square root” of geometry.

A thought is offered: a challenge for anyone reading this who is interested in spinors: could the properties of spinors be related fundamentally to self-reference, such that two spin states are each defined by itself relative to its counterpart? When a number is squared (or negated), it is compared with itself.

The Pauli matrices are constructed out of 0, +1, -1, +i and -i. The square roots of 1 are ±1. The square roots of -1 are ±i. 0 means the difference between two objects that are the same. The conventional multiplication rules are privileged in the construction of the numbers used in the matrices. In other words, the matrices are constructed out of numbers, to each of which has been assigned a meaning. At the base of this assignment of meanings are XOR and XNOR. Spinor particles in the physical world have no meaning: only physical properties through which they can interact.

So could new mathematics defined in four ordered polarities, each of which has no meaning, be used to construct representations of spinors that make geometric sense (accord with the rotational properties of space)?