If the “boring view” of reality is correct, then you can never predict anything irreducible because you are reducible. You can never get Bayesian confirmation for a hypothesis of irreducibility, because any prediction you can make is, therefore, something that could also be predicted by a reducible thing, namely your brain.
Some boxes you really can’t think outside. If our universe really is Turing computable, we will never be able to concretely envision anything that isn’t Turing-computable—no matter how many levels of halting oracle hierarchy our mathematicians can talk about, we won’t be able to predict what a halting oracle would actually say, in such fashion as to experimentally discriminate it from merely computable reasoning.
I don’t quite understand this one. How does “you are reducible” imply “you cannot conceive anything nonreducible”? Human beings with their merely Turing-complete brains can understand the concept of a non-Turing-computable problems. If our universe turns out to be more than Turing computable, and aliens give us a box that can map an integer to an integer by a non-computable function together with a verbal description of the function (say, “N → busy-beaver(N)”), we will be able to use it, and understand what it does and why it is useful. Even though we will not be able to predict the exact outputs without a similar box, we could conceive what would the output look like (“like an integer bigger than X and smaller than Y”). Correspondingly, I see no impossibility in that a reducible brain can imagine what a non-reducible universe would look like.
Say, suppose there is a universe made of three types of things: ghosts, transistors and billiard balls. Transistors and billiard balls can form structures that compute functions up to primitive recursive. Billiard balls can interact with ghosts and transistors, acting as an interface between two. Ghosts can directly interact only with billiard balls. Every ghost observes the state of billiard balls around itself every five seconds and outputs one of actions: haunt, spook or wail, that affect the billiard balls in some way. The computation performed by a ghost is Turing-complete, but not primitive recursive. Thus, ghosts can never be reduced to transistors and billiard balls. Creatures made of transistors can observe billiard balls and infer the existence of ghosts. They will obviously not be able to form a complete model of a ghost, but they could make statistical observations about them. They could form primitive recursive statements, such as “a ghost spooks 50% of the time regardless of billiard balls around, except if it was surrounded by four balls in pyramidal pattern 5 seconds ago, in which case it always haunts”. These statements will not describe the entire behavior of a ghost, but they will be conceivable, imaginable and detectable by transistor-creatures. And it, I suppose, is a probable thought that can occur to a transistor-creature—“what if ghosts are not computable?” (in their definition of computability that is merely primitive recursive).
In the same way, I see no trouble in visualizing a world which is just like ours, but contains a non-reducible-to-quarks, non-computable (by my definition of computability that is merely Turing computable) ghost that reads the state of quarks and produces a behavior that is outside of the box I’m thinking in. It will be my problem, not Universe’s.
There’s a difference between an existence proof and a constructive proof. We can talk about existence proofs for, “Here’s what happens when we hook a magical Halting Oracle to a Turing Machine and run certain programs.” We do not have any constructive proof of how a Halting Oracle would behave.
Just because you can say, “Imagine we had a thing with these properties” doesn’t mean you know how to build such a thing.
I don’t quite understand this one. How does “you are reducible” imply “you cannot conceive anything nonreducible”? Human beings with their merely Turing-complete brains can understand the concept of a non-Turing-computable problems. If our universe turns out to be more than Turing computable, and aliens give us a box that can map an integer to an integer by a non-computable function together with a verbal description of the function (say, “N → busy-beaver(N)”), we will be able to use it, and understand what it does and why it is useful. Even though we will not be able to predict the exact outputs without a similar box, we could conceive what would the output look like (“like an integer bigger than X and smaller than Y”). Correspondingly, I see no impossibility in that a reducible brain can imagine what a non-reducible universe would look like.
Say, suppose there is a universe made of three types of things: ghosts, transistors and billiard balls. Transistors and billiard balls can form structures that compute functions up to primitive recursive. Billiard balls can interact with ghosts and transistors, acting as an interface between two. Ghosts can directly interact only with billiard balls. Every ghost observes the state of billiard balls around itself every five seconds and outputs one of actions: haunt, spook or wail, that affect the billiard balls in some way. The computation performed by a ghost is Turing-complete, but not primitive recursive. Thus, ghosts can never be reduced to transistors and billiard balls. Creatures made of transistors can observe billiard balls and infer the existence of ghosts. They will obviously not be able to form a complete model of a ghost, but they could make statistical observations about them. They could form primitive recursive statements, such as “a ghost spooks 50% of the time regardless of billiard balls around, except if it was surrounded by four balls in pyramidal pattern 5 seconds ago, in which case it always haunts”. These statements will not describe the entire behavior of a ghost, but they will be conceivable, imaginable and detectable by transistor-creatures. And it, I suppose, is a probable thought that can occur to a transistor-creature—“what if ghosts are not computable?” (in their definition of computability that is merely primitive recursive).
In the same way, I see no trouble in visualizing a world which is just like ours, but contains a non-reducible-to-quarks, non-computable (by my definition of computability that is merely Turing computable) ghost that reads the state of quarks and produces a behavior that is outside of the box I’m thinking in. It will be my problem, not Universe’s.
There’s a difference between an existence proof and a constructive proof. We can talk about existence proofs for, “Here’s what happens when we hook a magical Halting Oracle to a Turing Machine and run certain programs.” We do not have any constructive proof of how a Halting Oracle would behave.
Just because you can say, “Imagine we had a thing with these properties” doesn’t mean you know how to build such a thing.