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In what follows the term transcendence refers to an entity or agent beyond all physical laws. (It is not used in the Kantian sense.) In contrast, immanence refers to all operational, intrinsic physical means available to embedded observers [500, 538] from within some universe.

Suppose that transcendent agents, interact with a(n) (in)deterministic universe via suitable interfaces. In what follows we shall refer to the transcendental universe as the beyond.

1 Gaming Metaphor

For the sake of metaphorical models, take Eccles’ mind-brain model [191], or consider a virtual reality, and, more particular, a computer game. In such a gaming universe, various human players are represented by avatars. There, the universe is identified with the game world created by an algorithm (essentially, some computer program), and the transcendental agent is identified with the human gamer. The interface consists of any kind of device and method connecting a human gamer with the avatar. Like the god Janus in the Roman mythology, an interface possesses two faces or handles: one into the universe, and a second one into the beyond.

Human players constantly input or inject choices through the interface, and vice versa. In this hierarchical, dualistic scenario, such choices need not solely (or even entirely) be determined by any conditions of the game world: human players are transcendental with respect to the context of the game world, and are subject to their own universe they live in (including the interface). Nevertheless the game world itself is totally deterministic in a very specific way: it allows the player’s input from beyond; but other than that it is created by a computation. One may think of a player as a specific sort of indeterministic (with respect to intrinsic means) oracle, or subprogram, or functional library.

Another algorithmic metaphor is an operating system, or a real-time computer system, serving as context. (This is different from a classical Turing machine, whose emphasis is not on interaction with some user-agent.) The user is identified with the agent. Any user not embedded within the context is thus transcendent with respect to this computation context. In all these cases the real-time computer system acts deterministically on any input received from the agent. It observes and obeys commands of the agent handed over to it via some interface. An interface could be anything allowing communication between the real-time computer system and the (human) agent; say a touch screen, a typewriter(/display), or any brain-computer interface.

2 How to Acknowledge Intentionality?

The mere existence of gaps in the causal fabric cannot be interpreted as sufficient evidence for the existence of providence or free will, because these gaps may be completely supplied by creatio continua.

As has already been observed by Frank [219, Kapitel III, Sects. 14 and 15], in order for any miracle or free will to manifest itself through any such gap in the natural laws, it needs to be systematic, according to a plan and intentional (German planmäßig). If there were no possibilities to inject information or other matter or content into the universe from beyond through such gaps, there would be no possibility to manipulate the universe, and therefore no substantial choice.

Alas, intentionality may turn out to be difficult or even impossible to prove. How can one intrinsically decide between chance on the one hand, and providence, or some agent executing free will through the gap interface, on the other hand? The interface must in both cases employ gaps in the intrinsic laws of the universe, thereby allowing steering and communicating with it in a feasible, consistent manner. That excludes any kind of immanent predictability of the signals emanating from it. (Otherwise, the behaviour across the interface would be predictable and deterministic.) Hence, for an embedded observer [538] employing intrinsic means which are operationally available in his universe [500], no definite criterion can exist to either prove or falsify claims regarding mere chance (by creatio continua) versus the free choice of an agent. Both cases – free will of some agent as well as complete chance – express themselves by irreducible intrinsic indeterminism.

Suppose an agent or gamer is immersed in such dualistic environment and experiences “both of its sides” through the interface but has no knowledge thereof. (C.f the metaphors “we are the dead on vacation” by Godard [241], or of the “brain in the vat” employed by Descartes [166, Second meditation, 26–29] and Putnam [422, Chap. 1], among others.) Then the agent’s knowledge of the beyond amounts to ineffability [294]. However, ineffability is neither necessary nor sufficient for dualism; and could also be a mere subjective illusion, constructed by the agent in a deparate attempt to make sense and create meaning from his sensory perceptions, very much like brain hallucinations [419]. And yet, ineffability might present some hint on metaphysics.

For the sake of an example, suppose for a moment that we would possess a sort of ‘Ark of the Covenant,’ an oracle which communicates to us the will of the beyond, and, in particular, of divinity. How could we be sure of that? (Sarfatti, in order to investigate the paranormal, attempted to build what he called an Eccles telegraph by connecting a radioactive source to a typewriter.) This situation is not dissimilar to problems in recognizing hypercomputation, that is, computational capacities beyond universal computation [334]; in particular also to zero knowledge proofs [65, 425].