Participation refers to the ontodynamic relationship between two causally related (say Emanator and Emanatee) entities. In this, the inferior (created, formed, dependent) entity always participates in the superior (source, foundational, emanating, causal) entity or entities
The concept or theory of Participation (Greek metechein, to partake of or share in, hence methekton, the participable (the causal reality) and metechon, the particant (the dependent reality) goes back to Plato but was only fully developed by the Neoplatonists and specifically Proclus, who explained it in great detail. A comprehensive description of Proclus' metaphysics regarding this can be found in Lucas Siorvanes' invaluable book Proclus: Neoplatonic Philosophy and Science pp.71-86
What Participation means is that every lesser (the Greeks would say sense-reality, but i'd say, in this example - surface consciousness) reality partakes of or shares in the characteristics of its superior (say spiritual or archetypal) reality.
I hope to at some point provide a write-up of Proclus' position on this, but until then the reader is referred to Siorvanes book or to Laurence J. Rosan's sadly out of print The Philosophy of Proclus. The Final Phase of Ancient Thought (New York: Cosmos, 1949).
I will give a few examples here of how I understand Participation
In religious belief or faith, the individual worshipper or devotee is caught up in and uplifted by a larger reality, which he or she calls God (or usually, the God of his or her religion, this is especially the case in literalist or fundamentalist religions. In Christianity this experience of a greater, uplifting, personality or presence is called the Holy Spirit. In some forms of Sufism, both genuine but also pop- and watered down Sufism, this is called baraka. In Hinduism it is bhakti (path of devotion) or ananda (bliss), and in the pop-guru Siddha Yoga (and maybe other pop guru sects) it is called shaktipat
To use (or express in) Aurobindonian concepts, this experience comes about through an inner opening or awakening that allows the surface consciousness or outer being to participate in the nature of the divine irradiation, either a pure experience in (and transformation through) the Divine Soul or (more often) a mixed experience in the Intermediate Zone
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