Does Gödel’s theorem imply that there are questions about the universe, or about mathematics, whose answers we can never learn? Are these questions at least comprehensible to us?
Note that this seems like a category error: mathematics is not physics, and building models of the universe is independent of mathematical theorems about soundness and consistency. Physicists (and humans in general) constantly use a self-contradictory patchwork of abstractions to make sense of the world. Yes, really.
If you want to investigate whether “there are questions about the universe… whose answers we can never learn”, then you need to focus on learning. Specifically, a model of the universe an embedded agent has is a lossy compression of the “reality”, so the question to ask would be “what kind of possible worlds compatible with observations would limit the extent of useful lossy compression?” Here is an example: most lossy image/sound and data compression algorithms are based on Digital Cosine Transform. At some point as the required compression degree gets too high, say, because we want so many details to be compressed in the number of bits that fit into the human brain, the compression fidelity might become unacceptably low, hitting the limits of our understanding, even in principle.
You sort of hint at this approach with your mention of empiricism, there is no need to worry about the Loeb obstacle at all.
I agree, and one could think of this in terms of markets: a market cannot capture all information about the world, because it is part of the world.
But I disagree that this is fundamentally unrelated—here too the issue is that it would need to represent states of the world corresponding to what belief it expresses. Ultimately mathematics is supposed to represent the real world.
Ultimately mathematics is supposed to represent the real world.
Well, I think a better way to put it is that mathematics is sometimes a part of some models of the world. The relationship is world → inputs → models <-> math. Whether the part of mathematics that deals with self-reference and soundness and completeness of formal systems corresponds to an accurate and useful model of the world is not at all obvious. So, yeah, some parts of mathematics lossily represent some parts of the world. But it is a pretty weak statement.
Half-joking: Find me a mathematical statement that purports to not describe the real world and I’ll show you a mathematical statement that describes patterns in constructions of symbols made of the real world. Territory comes first; math can show us perfect patterns in the territory, but by the same token, it can only ever be map. The map, being part of the territory, can then make insightful statements about itself, through many layers of patterns of patterns of patterns of patterns, but you’ve never seen a mathematical statement that wasn’t in the territory.
Of course math is in the territory, everything is. Embedded agency and all. But it does not mean that the particular part of the territory that is a specific part of math is also a lossy compression of a larger part of the territory.
You overstate your case. The universe contains a finite amount of incompressible information, which is strictly less than the information contained in ZF+ωCK. That self-reference applies to the universe is obvious, because the universe contains computer programs.
The point is the universe is certainly a computer program, and that incompleteness applies to all computer programs (to all things with only finite incompressible information). In any case, I explained Godel with an explicitly empirical example, so I’m not sure what your point is.
That’s about as much of an argument as saying that the universe is contained in the decimal expansion of Pi, therefore Pi has all the information one needs.
Note that this seems like a category error: mathematics is not physics, and building models of the universe is independent of mathematical theorems about soundness and consistency. Physicists (and humans in general) constantly use a self-contradictory patchwork of abstractions to make sense of the world. Yes, really.
If you want to investigate whether “there are questions about the universe… whose answers we can never learn”, then you need to focus on learning. Specifically, a model of the universe an embedded agent has is a lossy compression of the “reality”, so the question to ask would be “what kind of possible worlds compatible with observations would limit the extent of useful lossy compression?” Here is an example: most lossy image/sound and data compression algorithms are based on Digital Cosine Transform. At some point as the required compression degree gets too high, say, because we want so many details to be compressed in the number of bits that fit into the human brain, the compression fidelity might become unacceptably low, hitting the limits of our understanding, even in principle.
You sort of hint at this approach with your mention of empiricism, there is no need to worry about the Loeb obstacle at all.
I agree, and one could think of this in terms of markets: a market cannot capture all information about the world, because it is part of the world.
But I disagree that this is fundamentally unrelated—here too the issue is that it would need to represent states of the world corresponding to what belief it expresses. Ultimately mathematics is supposed to represent the real world.
Well, I think a better way to put it is that mathematics is sometimes a part of some models of the world. The relationship is world → inputs → models <-> math. Whether the part of mathematics that deals with self-reference and soundness and completeness of formal systems corresponds to an accurate and useful model of the world is not at all obvious. So, yeah, some parts of mathematics lossily represent some parts of the world. But it is a pretty weak statement.
Half-joking: Find me a mathematical statement that purports to not describe the real world and I’ll show you a mathematical statement that describes patterns in constructions of symbols made of the real world. Territory comes first; math can show us perfect patterns in the territory, but by the same token, it can only ever be map. The map, being part of the territory, can then make insightful statements about itself, through many layers of patterns of patterns of patterns of patterns, but you’ve never seen a mathematical statement that wasn’t in the territory.
Of course math is in the territory, everything is. Embedded agency and all. But it does not mean that the particular part of the territory that is a specific part of math is also a lossy compression of a larger part of the territory.
That’s syntax, not semantics.
HUH. iiiiinteresting...
You overstate your case. The universe contains a finite amount of incompressible information, which is strictly less than the information contained inZF+ωCK. That self-reference applies to the universe is obvious, because the universe contains computer programs.The point is the universe is certainly a computer program, and that incompleteness applies to all computer programs (to all things with only finite incompressible information). In any case, I explained Godel with an explicitly empirical example, so I’m not sure what your point is.
That’s about as much of an argument as saying that the universe is contained in the decimal expansion of Pi, therefore Pi has all the information one needs.
It’s really not, that’s the point I made about semantics.Eh that’s kind-of right, my original comment there was dumb.