Randomness, Mystery, Symmetry, Religions and a Mathematical Construct called “group”

Our fears and our rationality are inseparable

Fear of the Unknown skips on no one, randomness is unsettling, to order we yearn, a cornerstone we desire. Before we created an omnipotent picture of divinity, we perceived God as the constant, the unchanging axis of life and reality. The story of science is a search for order, pattern, stability, predictability. Newton is celebrated because he explained the complexity of motions, collisions, movement, flights and falls, using three immutable, “forever” reliable principles and formulas. Thermodynamics tells us that as much as we build, organize and systematize our neighborhood of reality — by and large everything devolves into greater and greater randomness, disorder, unpredictability.

In this battle royal between order and disorder, between randomness and symmetry we use mathematics to guide us. First principle: the measure of order and constancy is symmetry. Symmetry is the property of constancy under action. A system that has a feature that remains the same when parts of it change is a symmetric system. Randomness is simply a measure of lack of symmetry. And symmetry itself is measured through a simple mathematical construct called a ‘group’.

A group is a collection of different things over which we define an action. The action condenses two things in the group into any of the things of the same group. That means the action can condense any large number of things into a single thing. A basic condensation is (X,Y) → Z. But then (W,X,Y) → U because (W,X,Y) → (W,(X,Y)) → (W,Z) → U. Condensation is a drive to the essence, the kernel, the ‘fixed part’. It is easy to see that such condensation defines sequences of things that are organized in a “fixed” cycle. And the cycles can start anywhere. In addition any thing in the group has a symmetric thing such that one condenses a thing and its symmetrical counterpart, the result is a NULL thing, which is part of the group. When the NULL is condensed with any other thing, X, the result is X. And that’s it — that is a ‘group’.

Much as integers are the basis of science and order on the first level, so groups are the basis of tracking order in physics, economics, biology, history and much more. Symmetries are numerous, and they combine to additional symmetries, expressing order not readily visible.

Much as science is a chase of hidden symmetries, so are mysticism, cults, and religions. They are all in a chase for more and more symmetries. The chase for the constant, the reliable, the forever; the axis of rotation, of exchange, of mirroring; the immutable center. And in that respect there are no atheists among us, we are all deeply religious, searching for the ever-standing rock, around which all rotates, all moves — and all stays the same: God.

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