Groups are the New Integers

Simple Units that combine to express complexity as order

The integers 1,2,3… represent a powerful abstraction for many real things, and they evolved into arithmetic, and algebra to become the tools of engineering and technology. Similar role is emerging for a more elevated abstraction: groups. A group is a collection of things where symmetry is present. A system is symmetrical if it is left unchanged under some action. And as such it is an abstraction of order itself: order is an underlying constant in overlying dynamics. In recent times groups have become building blocks in modern physics, chemistry, biology, economics, and finance, as well as in cryptography and such.

Groups are simply a collection of things where any two of those things can be ‘reduced’ to a single thing of same collection. That implies that any large number of things may be reduced to a single thing, by applying the ‘two reduced to one’ again and again. The ‘things’ are organized as an item X and its symmetrical X’ such that when the two are reduced to one, they produce a NULL element: X*X’ = Null. Where the ‘Null’ serves as the axis of symmetry: X * Null = Null * X = X. That is it. A collection of n things may be arranged in an astronomical number of groups (n^(n^2)), which like integers combine in various ways to express order in the complexity of reality.

The pursuit of knowledge amounts to first integer-modeling of reality, then to group-modeling. Reality though is rich with randomness. The new definition of randomness is absence of symmetry. Much that appears as random is eventually carved with integers, and groups, and randomness recedes.

Randomness as Absence of Symmetry