We might translate Gaudiya Vedanta’s acintya bhedabheda as “trans-rational monistic dualism.” It is a form of substance dualism, but one in which the natural world (maya-sakti) is also one with consciousness (jiva-sakti), despite their larger difference from one another. They are one with one another in that they are both saktis of Bhagavan. They are different from one another in that one is experiential and the other is non-experiential.
In Gaudiya ontology the world is really there but we only experience an idea of it. It could also be called idealistic dualism perhaps. But this is Hane Htut Maung’s term for his position, and his form of it acknowledges only correlation between the natural and the spiritual (objective process and subjective experiences). He understands subjective experience to supplement the physical world as an extra fact, preserving causal closure while remaining ontologically distinct from the physical world. But here we are getting away from Gaudiya Vedanta it would seem. And causal closure is far from a scientific fact.
While Advaita Vedanta does not posit the idea that consciousness is an agent of action, but rather a witness, an awareness alone, Gaudiya Vedanta does. The jivatma is an agent (doer), experiencer of qualia, and an apprehender, among other things. Somehow it moves matter. The Bhagavata describes it’s doing so as more or less magical. It gives the example of a magnet moving iron filings by is magical power (sakti), without itself undergoing transformation. Such a folk science explanation of how consciousness is a causal agent in the world is probably not much more convincing to a naturalist that the centuries long argument of Christianity that attributes God’s intervention in the world to his miraculous power.
But as I mentioned in my earlier comment, I believe it is more reasonable to be a mysterian with regard to exactly how consciousness moves matter rather than to embrace the performative contradiction that does not acknowledge the causal efficacy of consciousness staring us in the face. All of us think, feel, and will and live our lives as if consciousness has such efficacy, but exactly how it does remains a mystery that there is no need to solve.
What the objective world is, is another related question. Is there any “thing” out there? And the quantum perspective really brings our understanding into question. The teaching of the Bhagavatam is that it is impossible to fully grasp. So how does the jivatma move “what” might be the question? Chomsky reasons that while we thought we took the ghost out of the machine, in fact the machine has been taken out. The mechanistic world is in question. As Russell writes in his study of matter, all we really “know” is our own consciousness. And I might add, we don’t know it very well. But I am not suggesting pure idealism.
That said, when I say there is no need to solve the mystery, I mean that there are better things to do. Solving it proves the existence of the immaterial, the supernatural. So that would be good, if possible to do so objectively. I now doubt it is possible. But it has no bearing on those who have ruci.
The naturalist project is doomed to failure if it involves demonstrating that consciousness is reducible to matter.
While there is a rationally unknowable aspect of Gaudiya Vedanta, does it include knowing how the jiva moves matter? I do not think it explicitly says this. It does not seem particularly interested in this detail. God wills and “glances” at the world and it begins to move. Then within the world the jiva also moves matter. Today’s more rational world wants a more detailed explanation because it seems that movement requires a point of contact and their appears to be universal physical closure to some (I don’t agree). So the Bhagavata gives an answer but it causes the modern world to ask further. And its answer is also interpreted differently by different systems of Vedanta, sometimes very differently.
I think that the naturalist method could lead objective persons to conclude that consciousness is not part of or does not emerge from the brain. But I would think that they would have to go much further to turn to mysticism. Still I think that will happen. Modern science as a whole was born Christian. In its adolescence it became agnostic. Now in its adult life it is heavily influenced by atheism. But it if it so live into the wisdom of old age, I argue that it must become a mystic.