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I have had many non-discursive moments with God. A non-discursive moment is an experience that cannot be adequately captured in words (discourse). It is an experience that is real, but one cannot talk about it. Words don't do it justice; indeed, words will be misleading.
Many of these moments come during extended times of praying. Others come while just going through the day - in my home, walking, driving, with my Jesus-community, doing dishes while looking out the window at my bird feeders, whatever and wherever.
The more I take extended times of praying, the more I am susceptible to non-discursive encounters with God. I don't seek them. They find me, or, rather, God finds me, in the depths of my heart where words are not needed. This is a knowing that refuses to be imprisoned by ideation.
Thomas Merton referred to this as "contemplation." He writes:
"Contemplation is the highest expression of man’s intellectual and spiritual life. It is that life itself, fully awake, fully active, fully aware that it is alive. It is spiritual wonder. It is spontaneous awe at the sacredness of life, of being. It is gratitude for life, for awareness and for being. It is a vivid realization of the fact that life and being in us proceed from an invisible, transcendent and infinitely abundant Source. Contemplation is, above all, awareness of the reality of that Source. It knows the Source, obscurely, inexplicably, but with a certitude that goes both beyond reason and beyond simple faith. For contemplation is a kind of spiritual vision to which both reason and faith aspire, by their very nature, because without it they must always remain incomplete. Yet contemplation is not vision because it sees “without seeing” and knows “without knowing.” It is a more profound depth of faith, a knowledge too deep to be grasped in images, in words or even in clear concepts. It can be suggested by words, by symbols, but in the very moment of trying to indicate what it knows the contemplative mind takes back what it has said, and denies what it has affirmed. For in contemplation we know by “unknowing.” Or, better, we know beyond all knowing or “unknowing.”" (Merton, New Seeds of Contemplation, pp. 1-2)
This is similar to what some have called "mindfulness"; viz., a hyper-awareness of self in the present moment.
For me, as a follower of Jesus, the source of non-discursive experiences is God. This is, I believe, the kind of thing Greg Boyd is talking about in his book Present Perfect. Presently, God is with me, and I am neither remembering past events nor anticipating something to come in the future. This is "Emmanuel" taken to its completion. This is "fulfillment."
Seek Him. The more you seek Him, the more you will be found by Him.