Well, ask the question, should the bigger brain receive a million dollar, or do you not care?
I've always maintained that in order to solve this issue we must first solve the question of, what does it even mean to say that a physical system is implementing a particular algorithm? Does it make sense to say that an algorithm is only approximately implemented? What if the algorithm is something very chaotic such as prime-checking, where approximation is not possible?
An algorithm should be a box that you can feed any input into, but in the real, causal world, there is no such choice, any impression that you "could" input anything into your pocket calculator is due to the counterfactuals your brain can consider purely because it has some uncertainty about the world (an omniscient being could not make any choice at all! -- assuming complete omniscience is possible, which I don't think it is, but let us imagine the universe as an omniscient being or something).
This leads me to believe that "anthropic binding" cannot be some kind of metaphysical primitive, since for it to be well-defined it needs to be considered by an embedded agent! Indeed, I claimed that recognizing algorithms "in the wild" requires the use of counterfactuals, and omniscient beings (such as "the universe") cannot use counterfactuals. Therefore I do not see how there could be a "correct" answer to the problem of anthropic binding.
Fantastic work!
How do we express the way that the world might be carved up into different agent-environment frames while still remaining "the same world"? The dual functor certainly works, but how about other ways to carve up the world? Suppose I notice a subagent of the environment, can I switch perspective to it?
Also, I am guessing is that an "embedded" cartesian frame might be one where i.e. where the world is just the agent along with the environment. Or something. Then, since we can iterate the choice function, it ould represent time steps. Though we might in fact need sequences of agents and environments. Anyway, I can't wait to see what you came up with.
There are two theorems. You're correct that the first theorem (that there is an unprovable truth) is generally proved by constructing a sort of liar's paradox, and then the second is proved by repeating the proof of the first internally.
However I chose to take the reverse route for a more epistemological flavour.
But we can totally prove it to be consistent, though, from the outside. Its sanity isn't necessarily suspect, only its own claim of sanity.
If someone tells you something, you don't take it at face value, you first verify that the thought process used to generate it was reliable.
You are correct. Maybe I should have made that clearer.
My interpretation of the impossibility is that the formal system is self-aware enough to recognize that no one would believe it anyway (it can make a model of itself, and recognizes that it wouldn't even believe it if it claimed to be consistent).
It's essentially my jumping off point, though I'm more interested in the human-specific parts than he is.
The relevance that I'm seeing is that of self-fulfilling prophecies.
My understanding of FEP/predictive processing is that you're looking at brains/agency as a sort of thermodynamic machine that reaches equilibrium when its predictions match its perceptions. The idea is that both ways are available to minimize prediction error: you can update your beliefs, or you can change the world to fit your beliefs. That means that there might not be much difference at all between belief, decision and action. If you want to do something, you just, by some act of will, believe really hard that it should happen, and let thermodynamics run its course.
More simply put, changing your mind changes the state of the world by changing your brain, so it really is some kind of action. In the case of predict-o-matic, its predictions literally influence the world, since people are following its prophecies, and yet it still has to make accurate predictions; so in order to have accurate beliefs it actually has to choose one of many possible prediction-outcome fixed points.
Now, FEP says that, for living systems, all choices are like this. The only choice we have is which fixed point to believe in.
I find the basic ideas of FEP pretty compelling, especially because there are lots of similar theories in other fields (e.g. good regulators in cybernetics, internal models in control systems, and in my opinion Löb's theorem as a degenerate case). I haven't looked into the formalism yet. I would definitely not be surprised to see errors in the math, given that it's very applied math-flavored and yet very theoretical.
Excellent post, it echoes much of my current thoughts.
I just wanted to point out that this is very reminiscent of Karl Friston's free energy principle.
The reward-based agent’s goal was to kill a monster inside the game, but the free-energy-driven agent only had to minimize surprise. [...] After a while it became clear that, even in the toy environment of the game, the reward-maximizing agent was “demonstrably less robust”; the free energy agent had learned its environment better.
Yes, I am arguing against the ontological realism of anthropic binding. Beyond that, I feel like there ought to be some way of comparing physical systems and having a (subjective) measure of how similar they are, though I don't know how to formalize it.
It is for example clear that I can relate to a dolphin, even though I am not a penguin. Meaning that the penguin and I probably share some similar subsystems, and therefore if I care about the anthropic measure of my subsystems then I should care about penguins, too.