God’s complexity cost is not just relatively big like any intelligent mind (such as the witch) would be, but literally infinite if we say that God is omniscient: If God is a “halting oracle”,
Is that an argument against the Mathematical Universe Hypothesis? Wouldn’t the ultimate ensemble have to include a halting oracle?
Well, the relationship between infinite extent and infinite complexity is tricky. Everyone in the rationalsphere knows that pi has an infinite decimal expansion, and also that the digits can be generated by a finite program. “Every mathematical entity exists, and only mathematical entities exist” is likewise a brief compression of the MUH.
You can’t prove a halting oracle exists inductively. How could you? Solomonoff induction is doing induction perfectly, and a halting oracle is not even in the hypothesis space, because the space contains only computable functions, and the halting problem is not decidable. And even if it were, what use would that hypothesis be to you? You can’t get any predictions from running a program on a halting oracle machine when you don’t even have one.
Tegmark claims that the hypothesis has no free parameters and is not observationally ruled out. Thus, he reasons, it is preferred over other theories-of-everything by Occam’s Razor. Tegmark also considers augmenting the MUH with a second assumption, the computable universe hypothesis (CUH), which says that the mathematical structure that is our external physical reality is defined by computable functions.
So this is a point that Tegmark himself considers fair. The CUH would not have a halting oracle.
I find the MUH philosophically dubious. I also disagree with Wikipedia’s characterization of the CUH as adding an additional hypothesis on top of MUH (I’m not sure if that’s how Tegmark sees it, or if that was just an interpolation by the editor). Instead, the CUH is throwing out the dubious axiom that allows things like uncomputable sets to exist, which means by Occam’s razor, I think the CUH is the simpler hypothesis. I don’t exactly buy the CUH either, but I don’t have a better idea.
How is that relevant? It is perfectly possible for a mathematical universe to be a form of Platonic realism.
I disagree with your interpretation of “perfectly possible”, but even if I hypothetically grant you that a halting oracle exists, how can an agent ever be rationally justified in believing that it does? It’s something that takes an infinite amount of evidence to prove. The method clearly can’t be induction.
I think you are missing some things that are quite basic: essentially no one believes in things like the Mathematical Universe on the basis of empiricism or induction. Instead, Occams razor is the major factor.
Note that by things like MUH include MWI. It is straightforwardly impossible to prove MWI or any other interpretation on the basis of evidence, because they make the same predictions. So the argument given for MWI is in terms of simplicity and consilience.
Not many people here reject all reasoning of that type. Many reject it selectively.
The simplicity criterion means MUH is preferable to CUH, since CUH has an additional constraint.
Is that an argument against the Mathematical Universe Hypothesis? Wouldn’t the ultimate ensemble have to include a halting oracle?
Well, the relationship between infinite extent and infinite complexity is tricky. Everyone in the rationalsphere knows that pi has an infinite decimal expansion, and also that the digits can be generated by a finite program. “Every mathematical entity exists, and only mathematical entities exist” is likewise a brief compression of the MUH.
You can’t prove a halting oracle exists inductively. How could you? Solomonoff induction is doing induction perfectly, and a halting oracle is not even in the hypothesis space, because the space contains only computable functions, and the halting problem is not decidable. And even if it were, what use would that hypothesis be to you? You can’t get any predictions from running a program on a halting oracle machine when you don’t even have one.
From the Wikipedia article:
So this is a point that Tegmark himself considers fair. The CUH would not have a halting oracle.
How is that relevant? It is perfectly possible for a mathematical universe to be a form of Platonic realism.
Which implies that the MUH might.
I find the MUH philosophically dubious. I also disagree with Wikipedia’s characterization of the CUH as adding an additional hypothesis on top of MUH (I’m not sure if that’s how Tegmark sees it, or if that was just an interpolation by the editor). Instead, the CUH is throwing out the dubious axiom that allows things like uncomputable sets to exist, which means by Occam’s razor, I think the CUH is the simpler hypothesis. I don’t exactly buy the CUH either, but I don’t have a better idea.
I disagree with your interpretation of “perfectly possible”, but even if I hypothetically grant you that a halting oracle exists, how can an agent ever be rationally justified in believing that it does? It’s something that takes an infinite amount of evidence to prove. The method clearly can’t be induction.
I think you are missing some things that are quite basic: essentially no one believes in things like the Mathematical Universe on the basis of empiricism or induction. Instead, Occams razor is the major factor.
Note that by things like MUH include MWI. It is straightforwardly impossible to prove MWI or any other interpretation on the basis of evidence, because they make the same predictions. So the argument given for MWI is in terms of simplicity and consilience.
Not many people here reject all reasoning of that type. Many reject it selectively.
The simplicity criterion means MUH is preferable to CUH, since CUH has an additional constraint.