Not extra-natural; extra-material. You're unnecessarily conflating matter and nature.
Aristotle's arguments for the existence of the soul are the best positive proof for its existence. The soul (or form) provides a coherent explanation for many difficult philosophical subjects, such as universals, the problem of change, and our ability to know things with certainty.
And they can keep on claiming that no matter how much evidence keeps piling up showing that the mind is indeed the work-product of the physical brain. But without any positive evidence for such a supernatural entity - "soul", "hylomorphic form", or whatever - that claim becomes vacuous.
The existence of the soul can also be argued negatively, by disproving materialist accounts of the mind.
The materialist may scoff at this approach, but as Lewis relished in pointing out, the materialist has his own problems: The materialist who debunks everyone elses ideas as the subrational products of their brain chemistry or environment cannot avoid being debunked himself. If he is honest, says Lewis, the materialist will have to admit that his own ideas are merely the "epiphe-nomenon which accompanies chemical or electrical events in a cortex which is itself the by-product of a blind evolutionary process." If all thoughts are merely the products of non-rational causes, this includes the materialists own thoughts. In other words, there is no reason according to materialism for materialism itself to be regarded as true.Materialist accounts of the mind are self-refuting, necessitating an ultimately non-materialistic account of the mind.
But if a rock falls to the ground & breaks up into 3 pieces, we now have four entities: Three smaller rocks and one triangle. Is the triangle material? If not, then where did it come from? It wouldn't exist if not for the 3 rocks sitting in a plane.
[Note that the notion of "triangle" is an abstraction, since we can know "triangle" (without seeing a physical representation of one) as a three sided, closed geometric figure.]
This is the problem of change. How can one substance (a rock) become another substance (a triangle composed of three rocks)? Aristotle argues that the original rock possessed "triangle" in potency. When the rock broke apart, the triangle came into existence, or into "act." So the change can be described as beginning with a rock in act (possessing triangle in potency and three pieces in potency), to a triangle in act composed of three rocks in act.
This explanation may sound arcane, but it is the only coherent explanation for the problem of change that I know. The history of the problem and Aristotle's solution are very interesting. (See the link above)
The triangle is a higher-order system than the rocks themselves, but there's nothing supernatural about it.
But there's nothing material about it either. "Triangle" is a principle.
It's perfectly material, in that it's made up of the three material rocks & nothing else.
Not so. A triangle is a closed, three-sided geometric figure. I can know "triangle" without experiencing one through my senses, just as I can know a "sixteen-sided, closed geometric figure" without experiencing one through my senses.
And yet it has exactly zero mass.
Because it's a non-material principle which is apprehended by an (ultimately) non-material mind.
You're arguing against a rather naive materialism. Or more accurately, you're arguing in favor of the fallacy of composition. Remember: When things can come together in specific ways, they become components of the newly emerged complex entity (for example: an organism). An organism can be explained in terms of its component parts, but that doesn't mean it is nothing more than its parts. And yet the organism didn't "come from" anywhere else but the parts. You don't need to conjure up a supernatural intelligence, person, or whatever to explain the rise of an emergent property.
The existence of the soul can also be argued negatively, by disproving materialist accounts of the mind.The materialist may scoff at this approach, but as Lewis relished in pointing out, the materialist has his own problems: The materialist who debunks everyone elses ideas as the subrational products of their brain chemistry or environment cannot avoid being debunked himself. ...Materialist accounts of the mind are self-refuting, necessitating an ultimately non-materialistic account of the mind.