The λ-Calculus

The λ-calculus is a scientific model for Nature’s rules of computation. Laws that govern every species of dynamic life and every act of programmed software. The mathematician and scholar Jacob Bronowski, who published The Ascent of Man, also published poems. The Abacus and the Rose explain the intimate relationship between mathematics and the humanities as the underlying force of nature.

The force that makes the winter grow

Its feathered hexagons of snow

And drives the bee to match at home,

Their calculated honeycomb,

Is abacus and rose combined.

An icy sweetness fills my mind, 

A sense that under thing and wing,

Lies, taut yet living, coiled, the spring.

The λ-calculus is the spring that powers the machinery of life. It was codified by the American mathematician Alonzo Church in the 1930s, who at that same time was Alan Turing’s doctoral tutor at Princeton University and a professorial colleague of John von Neumann. Unsurprisingly, Church guided his student’s discovery of the so-called ‘Turing’s α-machine.’ Turing’s atomic computer exactly matches the atomic form Church defined as a λ-calculus Application.
What a shame that von Neumann, after discussions with Turing and without consulting Church, proposed his overstretched architecture. It was due to von Neumann’s premature publication, all too quickly adopted by General-Purpose Computer Science. The von Neumann architecture unscientifically assumes blind trust in the software quality, exposing the General-Purpose Computer to undetected cybercrime and remote hacking. Using a Church-Turing Machine adds the check and balance of the λ-calculus and by adding a directional DNA hierarchy to copy Nature’s Universal Model of Computation.
The λ-calculus is very simple. It defines clear binding rules for substituting variables as abstract mathematical functions. By starting from the same atomic form, the theorems and axioms that rule the λ-calculus also apply to Turing’s α-machine as simple substitution rules for endless, eternal computational binding. Rules that level the playing field of cyberspace. These mechanical rules of Capability Limited Addressing apply the Church-Turing Thesis to computer science, fairly and in full, preventing both malware and hacking.  In the simplest form, terms are built using the three rules applying access rights with Capability-Keys to Variables, Abstractions, and Applications within a λ-calculus Namespace.

Syntax	Computation	Description
x	Variable {Access.ID}	An immutable token as a symbol in expression; a value, a parameter, or mathematical abstraction.
(λx.M)	Abstraction {Class.Function}	A function definition (M is a lambda term). The variable x becomes bound in the expression.
(M N)	Application {Namespace.Thread}	Applying a function to an argument. M and N are lambda terms.