In order to understanding Jonathan Edwards' views on personal identity, one must understand his occasionalism, according to which God is the only true cause of all occurrences. Since each occurrence in spacetime is a singular event which accords by bare decree, it follows for Edwards that the human person is merely an aggregate of thoughts, perceptions, sensations, etc. throughout time, whose apparent continuity and solidarity is an illusion upheld by God's bare decree:
Edwards' mental phenomenalism is a natural extension of his occasionalism and views on substance. If God is the only real cause of spatio-temporal phenomena, he is the only real cause of “thoughts” or “perceptions.” If a substance is what “subsists by itself,” “stands underneath,” and “keeps up” a set of properties, then a mental substance can only be what subsists by itself, stands underneath, and keeps up mental properties. It follows that the concept of mental substance either denotes nothing or refers to God's causal activity. “What we call spirit,” then, “is nothing but a composition and series of perceptions [mental events]…connected by…laws” (“Notes on Knowledge and Existence”; Edwards 1957–, vol. 6, 398).
Mental and physical substance are thus identical with God's causal production of the mental events constituting minds and the sensible ideas or “sensations” which constitute bodies “according to…methods and laws ” which he has freely established (“The Mind,” no. 13; Edwards 1957–, vol. 6, 344). God is thus the only true substance as well as the only true cause(Wainwright, 2012).
For Edwards, God's decree is the only true cause and He is likewise the only true "substance." Rather than constituting distinct substances, therefore, the human subject is an ontologically derivative epiphenomena sustained by God's bare decree. There is a very interesting sense, therefore, in which Edwards' philosophical and theological anthropology is radically anti-essentialist. Gordon Clark, also an occasionalist, takes what seems to be a somewhat similar position:
…a man is a congeries, a system, sometimes an agglomeration of miscellany, but at any rate a collection of thoughts. A man is what he thinks: and no two men are precisely the same combination.
This is true of the Trinity also, for although each of the three Persons is omniscient, one thinks “I or my collection of thoughts is the Father,” and the second thinks, “I or my thoughts will assume or have assumed a human nature.” The Father does not think this second thought, nor does the Son think the first. (The Trinity, Individuation)
The propositions which constitute God's thoughts do not proceed by way of temporal succession, and so there is perfect unity and stasis. But not so with humans. Our thoughts are in a state of continual flux. Clark therefore seems to give a comparably anti-essentialist account of the human person.
Wainwright, William, "Jonathan Edwards", The Stanford Encyclopedia of Philosophy (Winter 2012 Edition), Edward N. Zalta (ed.), URL = <plato.stanford.edu/archives/win2012/entries/edwards/>.