The function of the Four Worlds in the evolution of matter is identical to the function of the the four levels of the human soul. They are the delineation of the evolution of matter from the pure, amorphous spiritual Substance, or light, that was the first emanation of the Creator, into the solid tangible matter that makes up the form of physically perceivable universe in which we live.
Atziluth—The Archetypical World
The first and highest of the Four Worlds, corresponding to Fire of the elements, Chiah of the soul, and Yod of the Tetragrammaton, is called Atzilut. The term Atzilut is usually translated as "Emanation", but literally means "closeness". This is the "World", or primordial Substance which is the first emanation out of God's unique and pure Essence, and is therefore the "World" closest to Divinity. This Substance corresponds to Philosophic Fire, which - as opposed to literal fire - is best described as "living light". The Substance of "living light" is intangible and has no definite, specific Form, but its dynamic qualities of vitality and illumination are distinguishable. Life and light are the two qualities that must be postulated as being prevalent in any Divinely creative act. Since these qualities can be discerned as having existence distinct from Essential Divinity, the emanation of the primal "living light" (Substance) is the preliminary phase of Divine Creation. This phase is what is called Atzilut. As this is the primal spiritual Substance from which all other matter evolves, it corresponds to that which is called Chiah in the evolution of souls.
Briah—The Creative World
The next phase, corresponding to Water of the elements, Neshamah of the soul, and the first Heh of the Tetragrammaton, is called the World of Briah, which translates as "Creation". Like Neshamah, the function of Briah is to define specific Form and function in the amorphous energy of Atzilut. This is the evolutionary stage where matter begins to condense, "solidify", and to acquire specific, distinguishable qualities of it's own. This is because the Divine Unity is indivisible. Specification, which is division from that Unity, cannot be classified as Divine because the condensation of primal Substance into specific Form begins to occur at this stage. Thus, Briah corresponds to Neshamah, the stage where the human soul acquires individual personality and identity.
Yetzirah—The Formative World
It must be understood that Briah is a purely embryonic stage where specification and differentiation are applied only in an abstract sense, identifying the potential function of specific Forms and forces. The actual separation and division of these different qualities occurs in the next phase of evolution, called Yetzirah, which literally translates as "Formation". The world of Yetzirah corresponds to Air of the elements, Ruach of the soul, and Vau of the Tetragrammaton. As Air is generally associated with the intellect, this is the stage where differentiation of qualities (analysis) is applied to matter, and where these qualities are combined (synthesis) and Formed into archetypal conceptual compositions with a view towards various specific applications. This is the function of the human intellect with which an individual theoretically experiments with various courses of action with the intellect and comes to a decision that the individual thinks will best serve the primal motivation or desire. Yetzirah corresponds to Ruach in that this is the stage where a specific compound is identified and given personal existence in the same way that an individual human personality is given identity and necessitates a physical vehicle.
Assiah—The Material World
Here we come naturally to the fourth and final stage of development, called Assiah, which translates as "Action". The World of Assiah corresponds to Earth of the elements, Nefesh of the soul, the human body, and the final Heh of the Tetragrammaton. Assiah is the actual physical universe in which all things live and carry out their functions. This is the final proving ground where all the preceding Worlds and Spheres are actualized. The results are judged according to how well they perform their intended purpose, which is conceived in Briah and gestates and takes Form in Yetzirah. In correspondence, the human body is the proving ground of the soul, where the soul is evaluated according to how well it fulfills it's intended purpose (Thelemically, the Higher, or Pure, Will).
